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http://zbmath.org/?q=an:0824.28015
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# zbMATH — the first resource for mathematics
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Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
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Lattice-valued fuzzy measure and lattice-valued fuzzy integral. (English) Zbl 0824.28015
Summary: In this paper, (1) the concepts of lattice-valued fuzzy measure (with no valuation property) and lower (resp. upper) lattice-valued fuzzy integral are proposed, which give the unified description to the fuzzy measures and fuzzy integrals studied by Delgado and Moral, Qiao, Ralescu, Adams, Sugeno, Wang, and Zhang; (2) some asymptotic structural characteristics of lattice-valued fuzzy measures are introduced, and some relations between them are given; (3) some concepts of convergences for lattice- valued functions are defined, and Riesz’ theorem, Egoroff’s theorem and Lebesgue’s theorem for lattice-valued measurable functions are proved; (4) the monotone increasing (resp. decreasing) convergence theorem and almost (resp. pseudo almost) everywhere convergence theorem for lower (resp. upper) lattice-valued fuzzy integral are shown under some weak conditions.
##### MSC:
2.8e+11 Fuzzy measure theory
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2014-04-23 11:35:39
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http://heck.canilecalderaradireno.it/python-alpha-blending.html
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# Python Alpha Blending
talking of fancy guis that do alpha blending, animations, nice shading/gradients, etc. Visual Studio 2010 or later 2. 2 What is Qt? PyQt is a set of Python bindings alpha blending, gradient filling, vector. These are useful for masking parts of one layer with another. Therefore, I would fix the size of the background image, and consider it also the output image size. Depending on how you created your player sprite, it may have a colored block around it. Hi there, Ive implemented a software alpha blending technique to supplement what microsoft has left out of DX7. When Image1 and Image2 are blended using alpha value 0, Image1 is returned as and vice versa when the alpha value is 1. The example above of apple and orange is a classical example of pyramid blending. What value by alpha maps effectively do is reduce the saturation and contrast of polygons with high alpha blending, making them fade into the background and be less noticable. The syntax of addWeighted() function is:. Certain prefixes are usually removed, for example D3D11_BLEND_ZERO in C-code would be BLEND_ZERO (the D3D11_-prefix is removed) in DirectPython. In computer graphics, alpha compositing is the process of combining an image with a background to create the appearance of partial or full transparency. Then we’ll perform the actual alpha blending (Lines 51 and 52). The following code fragment implements the most common form of alpha blending, however other techniques are also possible:. The 3D Glyph mapper has been updated and it can now take in four component vectors and better handle rotations. # Keywords: Alpha Blending, Textures, Animation, Double Buffer # most basic winding: just a shift of math. For example, one can plot a. All you need is an intermediate level of knowledge in Python or C++. Use at your own risk. Alpha in WebGL and Canvas. 4, alpha 1. This color value format "#aarrggbb" has aa part which means alpha blending. On behalf of the community, we give our warmest thanks to the developers and contributors who made this GNOME release possible. Blend each level of pyramid using region mask from the same level 4. label2rgb ¶ Local Binary Pattern for texture classification ¶. 4, fixing dozens of issues in the core, builtin modules, libraries, and documentation. We also aim to maintain a thriving user and developer community by using open, community-oriented development. Alpha blendingAlpha blending is the process of overlaying a foreground image with transparency over a background Image. The image to be blended can be of any type (see Image Types and Formats↑) but cannot be larger than the host image. Surfaces with 8bit pixels use a color palette to map to 24bit color. Blend Images using toFloat(i,fp) and setPixels(i,fp) Get and set image properties using macros Image5D Macro Language Extensions RSS Feed Reader Multiple LUT Stack Binary erosion using EDM, with "Preview" and "Help" Event Listener (listen for color or tool changes). Python hex() function is used to convert any integer number ( in base 10) to the corresponding hexadecimal number. gl import * glEnable(GL_BLEND) glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA). A major caveat with the depth buffer is that it cannot account for alpha blending. Choosing color palettes¶ Color is more important than other aspects of figure style because color can reveal patterns in the data if used effectively or hide those patterns if used poorly. A module which provides an interface to the native win32 GUI API. 5 (alpha 2) RELEASED Python 2. Skip to content. Alpha blending. I've been tinkering around with OpenCV 2. 5 Python API is here and you can also read about the Data Access API that was included as part of the first Blender 2. in a quick, smooth, and slick way, such that moving a scroll bar or jiggling the mouse yields fast game-like response time, and which gives this program the feeling that you're actually in the 21st century ie this is an *interactive*. Several composition modes require an alpha channel in the source or target images to have an effect. Python hex function is one of the built-in functions in Python3, which is used to convert an integer number into its corresponding hexadecimal form. Of course all this may be moot because if you really cared about performance, you would use. #-----# fade. If you’re using an earlier version of Python, the simplejson library is available via PyPI. It is often useful to render image elements in separate passes, and then combine the resulting multiple 2D images into a single, final image called the composite. Here you can find Server fault related Solution,Superuser Solution,Ubuntu Solution,Webapps Solution,Webmasters Solution,Programmers Solution,Dba Solution,Drupal. Thus, if all the input and output arrays are continuous, the functions can process them as very long single-row vectors. white = Texture 1, black = Texture 2. Also, note that if the callback function is called in a thread created outside of Python’s control (e. Blending in this space wold ceate over-brightening. Accessing to OpenCv cell values of Mat from native c++ function of android app. BL_ONE_MINUS_SRC_ALPHA) mat is a material of type bge. As a concrete example, consider the code below that attempts to display two Gaussians. Indeed output video frames will have the geometry of the biggest incoming video stream and the framerate of the fastest incoming one. When we 'reconstitute' the image from this pyramid, the lower frequencies will be blended and higher frequencies preserved. 45 since some time now and there was this tutorial showcasing (adding) Alpha-blending of two images using OpenCV by Ana Huamán. Blends two open images by adding an image to the overlay of frontmost image. This targets compatibility with Python 2. Defaults to the rc image. South Paradiso wallet,100% Genuine PYTHON Leather Skin Men's BELT 3,5cm Wide #V170,PAUL SMITH Artist Stripe signature striped swimming shorts trunks SMALL. The expression accepts the same variables x, y as well. PIL/Pillowはコンパクトで高速なPython用の画像ライブラリです。 よく使う処理をまとめました（随時更新） PILとPillowの違い 基本的にPILを使う理由はありません、Pillowの方がリサイズフィルタ. Notably, the given input should be in base 10. Proximity fade is useful for effects such as soft particles or a mass of water with a smooth blending to the shores. Fix T60411: crash in multi-object pose mode, with some armatures in rest pose. The value can be a number between 0. Alpha blending can improve the appearance of some 3D symbols when they have partially transparent components. 45 since some time now and there was this tutorial showcasing (adding) Alpha-blending of two images using OpenCV by Ana Huamán. (OpenFace currently uses Python 2, but if you're interested, I'd be happy if you make it Python 3 compatible and send in a PR mentioning this issue. visualize_saliency_with_losses Generates an attention heatmap over the seed_input by using positive gradients of input_tensor with respect to weighted losses. The window adjusts to the size of the movie image. #!/usr/bin/python var = 100 if var == 200: print "1 - Got a true expression value" print var elif var == 150: print "2 - Got a true expression value. For example, let's say that I have a module foo, and I have a string whose contents ar…. In essence, we’ll convert the foreground, background, and alpha layers to floats in the range of [0, 1] (Lines 46-48). So to remove these effects I implemented min cut between the two images. If GTK has a similar API method, then wxPython can be made to work with alpha blending. A color map takes a start and end point in 3D space and lets you map a range of values to it. Build Laplacian pyramids for each image 2. Introduction. To test this, you draw some graphics shapes on a surface and draw image on top of those shapes and you should be able to see the shapes. All gists Back to GitHub. What you are seeing is the space that ought to be occupied by an alpha channel. In this code snippet i am just checking for 0 and 1 of Alpha channel for blending as we want to blend not overlap. 29808" See other formats. OpenMV can run Python scripts that have access to peripherals (SPI/I2C/UART, CAN, PWM, ADC and DAC), uSD filesystem, wireless, and the image processing library. circle — draw a circle around a point pygame. Call pygame. This module customises the behaviour of the OpenGL. set_alpha(new_alpha), however,. vtk_texture_blending_mode_add_signed, vtk_texture_blending_mode_interpolate, vtk_texture_blending_mode_subtract Used to specify how the texture will blend its RGB and Alpha values with other textures and the fragment the texture is rendered upon. If this is the best way then so be it. Wellensteyn Men's Winter Jacket Seamaster Blue Seam 870 Darknavy,Gurtex Mens Brown Herringbone Wool Blend Suit Jacket 44 Chest (Regular),Brioni Men's Brown Blue 100% Cotton Lounge Robe. Surface - pygame object for representing images to create a new image object. This is a useful alternative to the histogram for continuous data that comes from an underlying smooth distribution. What I'd like to know is this; how can I use color-blending in PyGame? More than anything I'm concerned about color-blending algorithms (that ignore certain desired colors, e. However, we'd lose the possibility to modify the Helix. For highly transparent triangles alpha is close to zero, and the triangle and specular highlights become dim. If you have any suggestions/comments (or you found mistakes), please drop me a mail at the address below. Examples using skimage. The fgcolor and bgcolor parameters have been shortened to fg. image blending blending Alpha-Blending Alpha Blending 2D blending Additive-Blending image image() Image Match Launch Image image Image image image image image image image Image Image image blending python blending mode 介紹 laplacian pyramid blending Poisson blending 博客 multi-band blending Laplacian Pyramid Blending详细 模型交叉. CLPython - Implementation of the Python programming language written in Common Lisp. I've been learning WebGL and have been confused how alpha blending works. Anyone know of a high-level 2D game engine for Python? per-pixel alpha blending to move around semi-transparent sprites you'll be very restricted on what size of. To do this you need to supply OpenGL with a blend equation. alpha：float 0，1，可选不透明度的彩色标签。如果图像为无，则忽略。bg_label：int，可选作为背景的标签。bg_color：str或数组，可选背景颜色。必须是0或1之间的color_dict或RGB浮点值的名称。image_alpha：float 0,1，可选图像的不透明度。. As backfaces could become visible we should separate the color drawing and the alpha blending. Computes and draws kernel density estimate, which is a smoothed version of the histogram. Laplacian pyramid blending with a mask in OpenCV-Python - lap_pyr. The top CSS layer is rendered transparent over the WebGL layer and catches keyboard and mouse events. Clean, easy to understand, and well documented API with lots of examples and tutorials. addWeighted() method. The transparent image is generally a PNG image. Pure VB code fast graphics processing, can be mixed to alpha blending graphics, additional, less mixed, and some linear fuzzy function. pi away from one another # compare with surface_linev2. Swirling all colors together resulted in a muddy brown. circle — draw a circle around a point pygame. In computer graphics, alpha compositing is the process of combining an image with a background to create the appearance of partial or full transparency. The problem here is that the "happy" Surface contains alpha per pixels. In case the operator $$\gamma$$ itself is the 'lifted' version of an operator working on scalars then Python/Numpy is of great help. Fire detection. If they are solid and they overlay each other you dont see the relation they have to each other, because one of them might be completely coverd by the other. As of version 3. In image processing, this is called alpha blending. More items to install separately. Customize Alpha blending of text and image. # load first image b =. The alpha component may be a string composed by "0x" followed by an hexadecimal number or a decimal number between 0. If we had such a value, let’s call it “aaf” (= anti-aliasing-factor), we could smoothly fade-out the diamond into the background:. Many popular hardcopy outputs are. For stitching together the panoramas, I was using alpha channels to blend the images, which removed any lines between the images, but if there was a lot of overlap, it created some bluring and ghosting effects. They allow path-based drawing with alpha-blending and anti-aliasing, and use a floating point cooridnate system. The more paint you added, the darker it got. For math, science, nutrition, history. Blending in transparency¶ The simplest way to include transparency when plotting data with matplotlib. white = Texture 1, black = Texture 2. This is going to be a bit different from our normal KNIME blog posts: instead of focusing on some interesting way of using KNIME or describing an example of doing data blending, I'm going to provide a personal perspective on why I think it's useful to combine two particular tools: KNIME and Python. class CV_EXPORTS FeatherBlender: public Blender {public: FeatherBlender (float sharpness = 0. Now apparently DX8. If you wanted to fade the sprite out, you'd probably try to use a happy. Developing cutting-edge applications With PyQt. The QPainter class performs low-level painting on widgets and other paint devices. Adding (blending) two images using OpenCV (two-input) operator is the linear blend By varying \alpha from 0 \rightarrow 1 this operator can be used to. Vintage 18ct Yellow Gold Diamond Solitaire Ring Sz O #523,Men's Personalised Engraved Tree Of Life Steel Bracelet Birthday Gift uk,Vintage Rare Hand Painted Landscape Lucite Art Deco Trombone Clasp Brooch Pin. Customize Alpha blending of text and image. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. Python string method isalpha() checks whether the string consists of alphabetic characters only. py Reference. Join GitHub today. Thus, if all the input and output arrays are continuous, the functions can process them as very long single-row vectors. Description: This script aligns the sequences in a FASTA file to each other or to a template sequence alignment, depending on the method chosen. A First Technique for Blending Images. Enumeration of the alpha blending mode for the Blend filter INPUT1_RED = 0 INPUT1_GREEN = 1 INPUT1_BLUE = 2 INPUT1_ALPHA = 3 INPUT2_RED = 4 INPUT2_GREEN = 5 INPUT2_BLUE = 6 INPUT2_ALPHA = 7 class sbsenum. Save jpeg allows the user to save the image to a jpeg file (white background). python Calling a function of a module by using its name(a string) What is the best way to go about calling a function given a string with the function's name in a Python program. In Chimera 1730 a Surface_Model attribute transparency_blend_mode was added to allow the more common (alpha,1-alpha) blend mode. and we want to blend them together, however, it may not look good due to discontinuities between images. Free scripts download - Python scripts - page 2 - Top4Download. BL_SRC_ALPHA, bge. image blending blending Alpha-Blending Alpha Blending 2D blending Additive-Blending image image() Image Match Launch Image image Image image image image image image image Image Image image blending python blending mode 介紹 laplacian pyramid blending Poisson blending 博客 multi-band blending Laplacian Pyramid Blending详细 模型交叉. How to insert smaller images over bigger (webcam frames, realtime stream)?. blending, 164 of 3D volumetric data, 192 Open GL, 164 Boids simulations, 71 adding a boid, 79–80 animation, 81 boundary conditions, 74–75 drawing, 75–76 initial conditions, 73 limiting vector magnitude, 81 obstacle avoidance, 86 rules, 72, 77–79 scattering, 80 tiled boundary conditions, 74 time step, 81 bootloader, 237 Bottle web framework, 274, 278, 280. SDL_BLENDMODE_NONE: no blending: dstRGBA = srcRGBA: SDL_BLENDMODE_BLEND: alpha blending: dstRGB = (srcRGB * srcA) + (dstRGB * (1-srcA)) dstA = srcA + (dstA * (1-srcA)). Save jpeg allows the user to save the image to a jpeg file (white background). It is also possible to reduce amount of blurring near edges and corners of the overlap. I am definitely open to Python scripting (I'd like to implement this functionnality in a script). GetSystemMetrics(). OK, I Understand. 4 with python 3 Tutorial 24 by Sergio Canu March 16, 2018 Beginners Opencv , Tutorials 0. Image Pyramids (Blending and reconstruction) - OpenCV 3. Wikipedia also has a nice section on alpha blending c++ c64 clr common lisp dsp emacs fsharp git ironscheme mac math nimrod objc opengl parable python scheme. Input and Output Formats¶. After all, GIMP is the most popular image editing application for Linux and the biggest open source competitor to Adobe Photoshop. The window adjusts to the size of the movie image. 张不同的图像。 两张源图像分别为：. I didn't test it but should work on Delphi. A common use for matplotlib. (Btw, these are some very very good and detailed tutorials on OpenCV , even if you are a beginner or an expert who needs a revisit to some concepts). This document outlines the interfaces to GIMP-Python, which is a set of Python modules that act as a wrapper to libgimp allowing the writing of plug-ins for GIMP. Update of /cvsroot/pywin32/pywin32/win32/src In directory sc8-pr-cvs1. Alpha blending. Shop at our store and also enjoy the best in daily editorial content. We have ditched the legacy Autotools build system in favour of the truly cross-platform CMake and extended the continuous integration tests to run MS Windows builds as well. For example, to mix two colors in equal parts, the result will be: (Y1+Y2)/2, (U1+U2)/2, (V1+V2)/2. All gists Back to GitHub. This option reduces playback performance compared to wmode=window or wmode=direct. OpenGL Examples. matplotlib can be used interactively from the Python shell. • Python determines the type of the reference automatically based on the data object assigned to it. setFullscreenState(bool fullscreen, int width, int height) → None¶ Changes the Device into a fullscreen mode or out of it. They allow path-based drawing with alpha-blending and anti-aliasing, and use a floating point cooridnate system. Image inversion¶. Use this function to set the blend mode used for drawing operations (Fill and Line). It can also draw aligned text and pixmaps. The processor parameters for the texture are left at the default, which means I am working with a premultiplied alpha. Videomixer can accept AYUV, ARGB and BGRA video streams. Then we’ll perform the actual alpha blending (Lines 51 and 52). UM Video Overlay Directshow Filter - 1. Here's an example of how ex_draw_bitmap looks as Python: !/usr/bin/env python. Instruction in phonological awareness skills supports the acquisition of literacy skills. com offers free software downloads for Windows, Mac, iOS and Android computers and mobile devices. [email protected] (1%) 3. books for Python (2. Alpha blending. I’ve been tinkering around with OpenCV 2. The input arrays and the output array can all have the same or different depths. You can also use this to get more exhaustive list:. circle — draw a circle around a point pygame. The image to be blended can be of any type (see Image Types and Formats↑) but cannot be larger than the host image. Loading Unsubscribe from Pysource? Cancel Unsubscribe. Since we recently covered the best Photoshop plugins, it's only fair that we round up the best plugins for GIMP, too. py #-----import stddraw import sys from color import Color from picture import Picture #-----# Return a new Color object which blends Color objects c1 and c2 using # alpha as the blending factor. Now I'll try to explain this method, with as less Math formulae as I can. If you see a halo of pixels around 3D symbols, check this option on. For stitching together the panoramas, I was using alpha channels to blend the images, which removed any lines between the images, but if there was a lot of overlap, it created some bluring and ghosting effects. The following are code examples for showing how to use cv2. Blend each level of pyramid using region mask from the same level 4. i Log Message: Add more. I've been using BlitzBasic earlier, but with pygame you much of the same functionality but with a much better programming language. Build a Gaussian pyramid of region mask 3. Clean, easy to understand, and well documented API with lots of examples and tutorials. If you wanted to fade the sprite out, you'd probably try to use a happy. The Standard material in Unity comes with a Transparency mode, which allows rendering transparent materials. Wellensteyn Men's Winter Jacket Seamaster Blue Seam 870 Darknavy,Gurtex Mens Brown Herringbone Wool Blend Suit Jacket 44 Chest (Regular),Brioni Men's Brown Blue 100% Cotton Lounge Robe. The fifth parameter is the gamma value — a scalar added to the weighted sum. The image to be blended can be of any type (see Image Types and Formats↑) but cannot be larger than the host image. Alpha blending with OpenCV: cv2. Videomixer can accept AYUV, ARGB and BGRA video streams. pgm video fila a path fname. It is available so that developers that use older versions of Python can use the latest features available in the json lib. The need for a Python binding was a key feature to provide, influencing the architecture and API (no C++ references, only simple types). Can add unlimited texts or images, watermarks on video. SDL_BlendMode. As a concrete example, consider the code below that attempts to display two Gaussians. Image Pyramids (Blending and reconstruction) - OpenCV 3. For some modes, the image memory will share memory with the original buffer (this means that changes to the original buffer object are reflected in the image). It can be realized with only NumPy without using OpenCV. The three strengths about the course were Clarity along with the intuition about various concepts, Range of Application oriented topics covered and the support for different languages and different machines. The key to plotting two filled plots is use alpha blending. Blend modes (or mixing modes) in digital image editing and computer graphics are used to determine how two layers are blended into each other. OpenGL:Cutout effect with hardware blending modes and alpha I am trying to compose two rendering results into one, so that src texture is cut out by the alpha from destination texture(or buffer). Blending and Compositing CSC320: Introduction to Visual Computing Michael Guerzhoy René Magritte, “The Red Model” Many slides from Alexei Efros,. The NEWS file lists every change in each alpha, beta, and release candidate of. OpenCV is used for all sorts of image and video analysis, like facial recognition and detection, license plate reading, photo editing, advanced robotic vision. To test this, you draw some graphics shapes on a surface and draw image on top of those shapes and you should be able to see the shapes. The d3d11c Module¶ DirectPython constants module. ColourDatabase for how a pointer to a predefined, named colour may be returned instead of creating a new colour. 5 as supported. Alpha blending is used to display an alpha bitmap, which is a bitmap that has transparent or semi-transparent pixels. 0, which represents the opacity value (‘0x00’ or ‘0. I blend 2 images at a time, blending the left and center image first, then blending that result with the right image. Python has a diverse range of open source libraries for just about everything that a Data Scientist does in his day-to-day work. where you specify the source and destination blending mode types manually). Inkscape features include versatile shapes, bezier paths, freehand drawing, multi-line text, text on path, alpha blending, arbitrary affine transforms, gradient and pattern fills, node. The standard Windows 7 photoviewer slideshow feature would be enough if it had the simple alpha blending effect (or any comparable, simple transition effect). Enumeration of the alpha blending mode for the Blend filter INPUT1_RED = 0 INPUT1_GREEN = 1 INPUT1_BLUE = 2 INPUT1_ALPHA = 3 INPUT2_RED = 4 INPUT2_GREEN = 5 INPUT2_BLUE = 6 INPUT2_ALPHA = 7 class sbsenum. GitHub is home to over 28 million developers working together to host and review code, manage projects, and build software together. If alpha is 1. When Image1 and Image2 are blended using alpha value 0, Image1 is returned as and vice versa when the alpha value is 1. 2 What is Qt? PyQt is a set of Python bindings alpha blending, gradient filling, vector. def blend (c1, c2, alpha): r = (1-alpha)* c1. label2rgb ¶ Local Binary Pattern for texture classification ¶. The library is available both in C++ as a DLL, and in Python as a module. Design And Reuse, The Web's System On Chip Design Resource : catalogs of IPs, Virtual Components, Cores for designing System-on-Chip (SOC). The globalAlpha property sets or returns the current alpha or transparency value of the drawing. In computer graphics, alpha compositing is the process of combining an image with a background to create the appearance of partial or full transparency. Visit for free, full and secured software's. If they are solid and they overlay each other you dont see the relation they have to each other, because one of them might be completely coverd by the other. To do this you need to supply OpenGL with a blend equation. random() to generate random pixels between 0 and 1. Note, however, that it is off by default, since most graphic systems don't provide correct linear alpha blending with gamma correction, which is crucial for a good appearance. in a transparent PNG), but it can also be a separate image. 45 since some time now and there was this tutorial showcasing (adding) Alpha-blending of two images using OpenCV by Ana Huamán. The standard Windows 7 photoviewer slideshow feature would be enough if it had the simple alpha blending effect (or any comparable, simple transition effect). Added Image. These may require additional dependencies such as scikit-image: import imshowpair import imshowpair. The image to be blended can be of any type (see Image Types and Formats↑) but cannot be larger than the host image. Python variables 'know' the kinds of values they hold, which allows Python to tell you when you're trying to do something strange, such as use the addition operator to combine a number and a string (answer = "Hello" + 1). The most common type is SourceOver (often referred to as just alpha blending) where the source pixel is blended on top of the destination pixel in such a way that the alpha component of the source defines the translucency of the pixel. Alpha blending is useful to create transparent effects, which can be used to render glass or liquid objects in computer graphic. 4 with python 3 Tutorial 24 by Sergio Canu March 16, 2018 Beginners Opencv , Tutorials 0. The Windows API has an AlphaBlend method which we can weavify for use in wxPython. Accessing to OpenCv cell values of Mat from native c++ function of android app. blend() method creates a new image by interpolating between two input images, using a constant alpha. Blending in transparency¶ The simplest way to include transparency when plotting data with matplotlib. When slider is set to 0. Build a Gaussian pyramid of region mask 3. For highly transparent triangles alpha is close to zero, and the triangle and specular highlights become dim. (Btw, these are some very very good and detailed tutorials on OpenCV , even if you are a beginner or an expert who needs a revisit to some concepts). ##Python Hex Example. global_alpha This module customises the behaviour of the OpenGL. 0’ completely opaque). > tkpath does not seem to come standard with Python's tk version when I > looked into it a couple of years ago, but maybe it has now? tk canvas and tkpath share the same interface, the first tkpath was. So if one image has a high distance from the border and other one a low distance from border, you prefer the pixel that is closer to the image center. Python, along with a module like Pygame, however, allows just about anyone to easily enter the field of game Beginning Python Games Development with Pygame is divided into 12 chapters, each of which builds on the. 0 is used for fully transparent objects, while an alpha value of 1. Noesis also implements Python through a plugin that links to the Python C library. python Calling a function of a module by using its name(a string) What is the best way to go about calling a function given a string with the function's name in a Python program. Chroma Documentation, Release 0. The degree of the foreground color's translucency may range from completely transparent to completely opaque. Inkscape is a GUI editor for Scalable Vector Graphics (SVG) format drawing files, with capabilities similar to Adobe Illustrator, CorelDraw, Xara Xtreme, etc. As it was in old Python you rule a snake and have to eat rabbits avoiding collisions with walls and your tail. CLPython - Implementation of the Python programming language written in Common Lisp. in a quick, smooth, and slick way, such that moving a scroll bar or jiggling the mouse yields fast game-like response time, and which gives this program the feeling that you're actually in the 21st century ie this is an *interactive*. Image Blending in Python with OpenCV. An inverted image is also called complementary image. vtk_texture_blending_mode_add_signed, vtk_texture_blending_mode_interpolate, vtk_texture_blending_mode_subtract Used to specify how the texture will blend its RGB and Alpha values with other textures and the fragment the texture is rendered upon. PIL/Pillowはコンパクトで高速なPython用の画像ライブラリです。 よく使う処理をまとめました（随時更新） PILとPillowの違い 基本的にPILを使う理由はありません、Pillowの方がリサイズフィルタ. While imshow makes it easy to visualize a 2-D matrix as an image, it doesn't easily let you add transparency to the output. Accelerated Graphics Port (AGP) is an interface specification that enables 3-D graphics to display quickly on ordinary personal computers. Programming assignments for this course can be completed either using Python-OpenCV (recommended platform) or Matlab/Octave. Greys_r); plt. Input and Output Formats¶. From our previous tutorial, we know already a bit of. # load second image imshowpair. Save jpeg allows the user to save the image to a jpeg file (white background). Color value such as "#80FF0000" is allowed. 45 since some time now and there was this tutorial showcasing (adding) Alpha-blending of two images using OpenCV by Ana Huamán. Who can recommend me a simple slideshow software that simply shows JPG images in a folder and allows alpha-blending between the slides? I really don't need more than that. white = Texture 1, black = Texture 2. This is like the above mode but the triangle color is multiplied by alpha before being added to the image. Transparency, in this context, is implemented with alpha blending. Data are selected in the scatterplot using the Select button. Images with an alpha channel can be blended with the existing framebuffer. This recipe teaches how to apply alpha blending using ggplot2, plotly, and ggvis. For each of the requested sink pads it will compare the incoming geometry and framerate to define the output parameters. If None is passed for the alpha value, then alpha blending will be disabled, including per-pixel alpha. Python has a diverse range of open source libraries for just about everything that a Data Scientist does in his day-to-day work. All gists Back to GitHub. 2 offers you a tiny, free utility that allows you to adjust transparency of currently active window and seeing other windows open at your desktop (so called "alpha blending"). The strength of that contribution is set by the alpha value. In this tutorial, we will learn how to alpha blend two images. Programming assignments for this course can be completed either using Python-OpenCV (recommended platform) or Matlab/Octave. 3, but you probably want to use a uniform, or read it from a RGBA texture ( TGA supports the alpha channel, and GLFW supports TGA ) Here’s the result. Our client is managing a lower frequency trading strategy and looking for talented and experienced quantitative researchers to join the team. Full text of "beginning-game-development-with-python-and-pygame-from-novice-to-professional. This implementation of alpha blending is also covered on the LearnOpenCV blog. If you have any suggestions/comments (or you found mistakes), please drop me a mail at the address below. texture mapping • Perspective-correct • Bi-linear and advanced texture filtering • Gouraud shading and texture modulation • Anti-aliasing and alpha blending • Special effects: fog, transparency and translucency. set_alpha(new_alpha), however,. Skip to content. alignment ( Optional [ int ] ) - Text alignment. First, we will go over basic image handling, image manipulation and image transformations. It is very easy to edit nodes, perform complex path operations, trace bitmaps and much more. The alpha component may be a string composed by "0x" followed by an hexadecimal number or a decimal number between 0. If UseDepthPeeling is on and the GPU supports it, depth peeling is used for rendering translucent materials. We need to define both alpha and beta such that alpha + beta = 1. If run as a program overlay. The Problem. py #-----import stddraw import sys from color import Color from picture import Picture #-----# Return a new Color object which blends Color objects c1 and c2 using # alpha as the blending factor. This implementation of alpha blending is also covered on the LearnOpenCV blog. Pure VB code fast graphics processing, can be mixed to alpha blending graphics, additional, less mixed, and some linear fuzzy function. 而 Alpha Blending 则是一种中庸的方式，它使用当前fragment的alpha作为混合因子，来混合之前写入到缓存中颜色值。但 Alpha Blending麻烦的一点就是它需要关闭ZWrite，并且要十分小心物体的渲染顺序。如果不关闭ZWrite，那么在进行深度检测的时候，它背后的物体本来是. def blend (c1, c2, alpha): r = (1-alpha)* c1. Poisson blending is one of the topics that spent me days trying to understand recently (not fully understand yet), it is a wonderful method, and using wonderful maths, as well. You can vote up the examples you like or vote down the ones you don't like. Python (named after Monty Python's Flying Circus, not the Burmese snake) is a high level programming language that is finding wide acceptance in astronomy, physics, engineering, and computer science.
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2019-12-06 10:18:50
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https://www.gamedev.net/forums/topic/647560-cs-degree-is-it-worth-it/
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# CS Degree - Is it worth it?
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I'm curious to know if I should spend 3 years of my life to get a CS degree. I'm 17 years old, turning 18, in my last year of high school (grade 12) and willing to apply to a university. I already know how to program, in various languages (here's my GitHub and BitBucket if you want to look at some of my code [I haven't finished/update some projects as of yet]). Obviously, I don't know everything about programming*, but I would say I know quite a lot. Would I be "twiddling my thumbs" for the first couple of years whilst doing a CS degree (i.e. be bored, not learn anything new)?
I'm considering between choosing CS/CS(Advanced), Bachelor of Engineering (Electrical and Electronic) with Bachelor of Mathematical and Computer Sciences, and Bachelor of Engineering (Computer Systems) with Bachelor of Mathematical and Computer Sciences. I'm not sure if the last two degrees would have all the content as a CS degree (it's a double degree; yet it has three degrees(?)).
Can I get opinions on why getting a CS degree would be beneficial to me? The only reasons I can think of is: (1) More knowledge, (2) Good environment for learning at uni (topics I'm unsure/inexperienced in or don't know about*) and (3) Looks good on a resume. Also, could I get a job (programming related) whilst studying for my CS degree?
*Here's the topics I do not currently know (there is more than likely more, but here's the main ones I think I should learn and seem interesting):
• Assembly programming
• Operating Systems
Here's the languages I know:
• C
• C++
• C#
• Java
• JavaScript
• Anything really C-like
• VB (and TI-Basic)
• Had some little experience with Lua
• HTML/CSS (Not sure if these really count, but I still know them)
Edited by pinebanana
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I very liked my studies time (physics), great time so I would say that university is good (if you have money - the main problem) I would also advice to chose serious university instead of weaker one (second problem could be that it may be hard to pass the exams)
Personally I wanted to work at programming till the age of 13 but chose physics becouse it seemed to me more ambitious and this was not bad choice I think (was good choice and was a great time i am missing now) - but it appeard after the studies when I began to work as a programmer that many of my colegues at work (which was studiing informatics) know a lot more about programming than I knew - so about three years or more i was behind them and was trying to learn to improve - now yet more than five years l8er I stil do not know enough to be good (i consider myself moderately xperienced)
Edited by fir
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You should get some degree. What degree is your choice, though the closer it is to computer science the less explaining you may have to do when applying for software jobs. (Ie computer engineering or electrical engineering will be an easier sell than art history.) You should not skip an undergraduate college education entirely.
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Being "worth it" is also a bit subjective.
Where do you live? Some regions of the world don't really care about education. Other regions of the world will require a BS degree in Computer Science just as a simple test to prune the number of applicants coming from HR.
What is the cost of the degree? Some schools are cheap, others are very expensive. I have relatives in schools that cost just over $1000 USD per semester. Sometimes on the board we discuss people who entered schools costing$10,000 USD per semester or more. I do not believe the quality of education between the schools is a 10x difference. Shopping around is important.
What is taught? Just like the cost varies, the content also varies. Some schools focus on rigorous theory with little practical content. Some schools are little more than trade schools with just enough theory to get by. As you shop around you should consider both the cost and content.
A bachelors degree in Computer Science is the typical requirement in games programming. It is not absolutely required, but you do not exist in a vacuum. If you have a portfolio, but the other person as a portfolio and a degree, which one is more likely to be hired?
I live in Australia. The university I'm wanting to apply to is apparently in the top 1% of the world. They seem to teach everything from algorithms, to computer systems, OOP, lower lever/system programming, AI, CG and operating systems, etc.
I believe I can pay for my university when I start working (they take a proportion of my pay), or I can pay up-front (which won't happen, because I don't have the money for that). So I don't think money is really an issue, apart from working it off in the future.
A CS degree is about more than just programming. Or at least, a good one is supposed to be. There will undoubtedly be CS departments that will just try to teach you just enough to be a Java code monkey or whatever, avoid those. Especially since you already have the programming knowledge. A good CS degree is going to teach you about algorithms. And calculus. And big O notation. And about NP problems. And all of the other details that teach you why to do something, not just how. You need the math, because quaternions are math. Physics systems are math. Graphics programming is math. You don't need a lot of math to make a game, but it really helps if you want to work on the advanced, cutting edge programming problems.
Do you need a CS degree? Strictly speaking, you don't need it to program. You can teach yourself anything, eventually. But the degree gives you a focus that is hard to get otherwise, and helps you prove to other people that you can do it. And it lets you work with people who are interested in similar things, which should not be underestimated.
I am very interested in programming in general, not just games programming. I don't think I'd lose interest in it just like that. So I think I will pursue in taking this decision and doing a computer science course. As if I want to do it for the rest of my life I think I should take the time to learn as much as possible (such as the low-level nitty gritty). The only thing is I'm not 100% sure if I want to do a double degree or not.
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I am in the exact situation that you are in. I am 18 and I have been programming since I was 10. This year I have got an internship at a company programming for them over the summer and now they have decided they want to keep me. So what I am doing is taking the year off and deciding what I want to do for school. In this time I think I will program another game or two and release them and see if that kicks off...I would really like to be able to make it as an indie developer. Anyway I think I have decided to go into school and take some classes in business cause if you are as experienced as it sounds then you will not get that much out of computer science anyways...I mean I am constantly debugging the other programmers slow code at my work to make our software actually run nicely.
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A CS degree will certainly open doors. Especially if you want to program in the game industry. That said, there are many people that can still make it without a CS degree (like myself, I went to art school) but it does make getting certain positions a lot more difficult. I think that alone would make the degree worth it.
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I was in exactly the same boat--I knew how to program, and had great experience with multiple programming languages--especially Python and C++. Not only this, I had reverse engineered many graphics algorithms (I had implemented, for example, a GPU cloth simulation based on GLSL and FBOs before I ever even applied anywhere).
Basically, I didn't take any of the introductory classes, and I started immediately on higher level coursework (I took, for example, the graduate course in graphics algorithms my first semester). Especially at a big university, there's always more to learn about your field. I quickly learned about functional programming languages, asymptotic analysis, and design patterns. I was constantly learning things, and I eventually realized I wanted to double major in abstract mathematics just to get the most out of my future coursework.
The point is, universities will teach you. That's kindof what they do. As others have mentioned, a CS degree is not just about programming--if that's all you can do, you're a software engineer, not a computer scientist. And there's a huge difference.
Plus, being at a university is wonderful in its own right. Basically everyone has a triple digit IQ (which for me was a refreshing change from high school) and by and large you can learn whatever you want. There's almost always core requirements, but you have much much much more leeway in choosing.
Edited by Geometrian
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2019-07-23 09:50:58
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https://www.khanacademy.org/science/physics/fluids/density-and-pressure/a/pressure-article
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# What is pressure?
Pressure is kind of like force, but not quite.
## What does pressure mean?
If you tried to hammer a bowling pin into the wall, nothing would probably happen except for people deciding to no longer lend you their bowling pins. However, if you hammer with the same force on a nail, the nail would be a lot more likely to penetrate the wall. This shows that sometimes just knowing the magnitude of the force isn't enough: you also have to know how that force is distributed on the surface of impact. For the nail, all the force between the wall and the nail was concentrated into the very small area on the sharp tip of the nail. However, for the bowling pin the area touching the wall was much larger, and therefore the force was much less concentrated.
Person hitting a bowling pin and a nail with a hammer.
To make this concept precise, we use the idea of pressure. Pressure is defined to be the amount of force exerted per area.
${\Large P=\dfrac{F}{A}}$
So to create a large amount of pressure, you can either exert a large force or exert a force over a small area (or do both). In other words, you might be safe lying on a bed of nails if the total surface area of all the nail tips together is large enough.
Yeah, people do this. Since it's the total area over which the force is distributed that counts, the total surface area of all the nails can reduce the pressure that's created by your weight downward. But there has to be a huge number of nails for this to work.
This definition also means that the units of pressure are newtons per square meter $\dfrac{\text{N}}{\text{m}^2}$ which are also called pascals or abbreviated as $\text{Pa}$.
Blaise Pascal was a 17th century scientist, mathematician, and philosopher. Not only did he contribute the the understanding of fluid pressure, but he is also noted for "Pascal's wager", "Pascal's triangle", and "Pascal's theorem".
## How do you find the pressure in a fluid?
A solid surface can exert pressure, but fluids (i.e. liquids or gases) can also exert pressure. This might seem strange if you think about it because it's hard to imagine hammering in a nail with liquid. To make sense of this, imagine being submerged to some depth in water. The water above you would be pushing down on you because of the force of gravity and would therefore be exerting pressure on you. If you go deeper, there will be more water above you, so the weight and pressure from the water would increase too.
Not only can the weight of liquids exert pressure, but the weight of gases can as well. For instance, the weight of the air in our atmosphere is substantial and we're almost always at the bottom of it. The pressure exerted on your body by the weight of the atmosphere is surprisingly large. The reason you don't notice it is because the atmospheric pressure is always there. We only notice a change in pressure above or below normal atmospheric pressure (like when we fly in an airplane or go underwater in a pool). We aren't harmed by the large atmospheric pressure because our body is able to exert a force outward to balance the air pressure inward. But this means that if you were to be thrown into the vacuum of outer space by space pirates, your body pressure would continue pushing out with a large force, yet no air would be pushing in.
You probably wouldn't blow up since your body/skin/bones are strong enough to hold you together. Still, it would be really, really uncomfortable. Besides the lack of oxygen and possible direct radiation exposure from the sun, your eyes would bulge, your eardrums could pop, and the saliva on your tongue would probably boil since the boiling point of water decreases as pressure goes down. At zero pressure your body temperature is enough to boil the water on your tongue as well as the fluid in your eyes. So basically, don't ever get caught by space pirates.
Okay, so the weight of a fluid can exert pressure on objects submerged in it, but how can we determine exactly how much pressure a fluid will exert? Consider a can of beans that got dropped in a pool as seen in the following diagram.
This is one of the great mysteries of the universe. I doubt we will ever know. If you find out, please contact the Department of Physics Mysteries immediately.
A can of beans submerged below the water to a depth h.
The weight of the column of water above the can of beans is creating pressure at the top of the can. To figure out an expression for the pressure we'll start with the definition of pressure.
$\Large P=\dfrac{F}{A}$
For the force $F$ we should plug in the weight of the column of water above the can of beans. The weight is always found with $W=mg$, so the weight of the column of water can be written as $W=m_{w}g$ where $m_w$ is the mass of the water column above the beans. We'll plug this into the equation for pressure above and get,
$P=\dfrac{m_w g}{A}$
At this point it might not be obvious what to do, but we can simplify this expression by writing $m_w$ in terms of the density and volume of the water. Since density equals mass per unit of volume $\rho=\dfrac{m}{V}$ , we can solve this for the mass of the water column and write $m_w=\rho_w V_w$ where $\rho_w$ is the density of the water and $V_w$ is the volume of the water column above the can (not the entire volume of the pool). Plugging in $m_w=\rho_w V_w$ for the mass of the water column into the previous equation we get,
$P=\dfrac{\rho_w V_wg}{A}$
At first glance this appears to have only made the formula more complex, but something magical is about to happen. We have volume in the numerator and area in the denominator, so we're going to try to cancel something here to simplify things. We know that the volume of a cylinder is $V_w=Ah$ where $A$ is the area of the base of the cylinder and $h$ is the height of the cylinder. We can plug in $V_w=Ah$ for the volume of water into the previous equation and cancel the areas to get:
$P=\dfrac{\rho_w (Ah)g}{A} = \rho_w h g$
Good question. The original area $A$ in the denominator was the area upon which the force is exerted, which was the area of the top of the can. The area $A$ in the numerator refers to the area of the column of water. Since the area of the column of water is equal to the area of the top of the can, these areas do in fact cancel.
Not only did we cancel the areas, but we also created a formula that only depends on the density of the water $\rho_w$, the depth below the water $h$, and the magnitude of the acceleration due to gravity $g$. This is really nice since nowhere does it depend on the area, volume, or mass of the can of beans. In fact, this formula doesn't depend on anything about the can of beans other than the depth it is below the surface of the fluid. So this formula would work equally well for any object in any liquid. Or, you could use it to find the pressure at a specific depth in a liquid without speaking of any object being submerged at all. You'll often see this formula with the $h$ and the $g$ swapping places like this,
${\Large P= \rho gh}$
Just to be clear here, $\rho$ is always talking about the density of the fluid causing the pressure, not the density of the object submerged in the fluid. The $h$ is talking about the depth in the fluid, so even though it will be "below" the surface of fluid we plug in a positive number. And the $g$ is the magnitude of the acceleration due to gravity which is $+9.8 \dfrac{\text{m}}{\text{s}^2}$ .
Now you might think, "OK, so the weight of the water and pressure on the top of the can of beans will push the can downward right?" That's true, but it's only a half truth. It turns out that not only does the force from water pressure push down on the top of the can, the water pressure actually causes a force that pushes inward on the can from all directions. The overall effect of the water pressure is not to force the can downward. The water pressure actually tries to crush the can from all directions as seen in the diagram below.
OK, if you are really clever you might have realized that the bottom of the can is slightly lower in the fluid than the top of the can, and since the pressure gets larger the deeper you go ($P_{gauge}=\rho g h$) the upward pressure on the bottom of the can should be slightly larger than the downward pressure on the top of the can. This means that the overall effect of the pressure from the water is to crush the can and to exert a net upward force on it. This net upward force from the difference in pressure is the reason why there's a buoyant force on objects submerged in a fluid! But...we're getting a little ahead of ourselves so let's hold this thought for now.
A can of beans being squeezed by water pressure.
If it helps, you can think about it this way. When the can of beans fell into the water, it quite rudely displaced a large amount of water molecules from the region where the can is now. This caused the entire water level to rise. But water is pulled down by gravity which makes it want to try and find the lowest level possible. So the water tries to force itself back into the region of volume that it was displaced from in an effort to try and lower the overall height of the body of water. So, whether a can of beans (or any other object) is in the water or not, the water molecules are always being squashed into each other from the force of gravity as they try to lower the water level to the lowest point possible. The pressure $P$ in the formula $\rho gh$ is a scalar that tells you the amount of this squashing force per unit area in a fluid.
OK, so here is a subtle fact about pressure; it's defined to be a scalar, not a vector. So why do people seem to represent pressure in diagrams with arrows as if it were a vector with a particular direction?
Even though pressure is not a vector and has no direction in and of itself, the force exerted by the pressure on the surface of a particular object is a vector. So when people draw diagrams with pressure pointing in specific directions, those arrows can be thought of as representative of the direction of the forces on those surfaces exerted by the pressure from the fluid.
If there were no surface upon which the pressure could exert a force, it would make no sense to draw a direction for the force at that point inside the water. On the left hand side of the diagram below there are water molecules and pressure, but no well defined direction of force. The right hand side of the diagram below shows the well defined directions of forces on an ice cream cone submerged in the water.
While we're on the topic, we might as well make it clear that the force exerted on a surface by fluid pressure is always directed inwards and perpendicular (at a right angle) to the surface.
At this point, if you've been paying close attention you might wonder "Hey, there's air above the water right? Shouldn't the weight of the column of air above the column of water also contribute to the total pressure at the top of the can of beans?" And you would be correct. The air above the column of water is also pushing down and its weight is surprisingly large.
Many people think air has no mass and no weight, but that's not true. The narrow column of air with the same radius as a typical can of beans that stretches from sea level to the top of the atmosphere has a mass of around $30 \text{ kg}$ (that's like the weight of 30 pineapples). The force from atmospheric pressure on the top of a chessboard would be comparable to the weight of a car.
You might wonder how we can pick up the chessboard so easily if the weight of a car is pushing down on it, but it's because the weight of a car is also pushing up on it. Remember that the force from fluid pressure does not just push down, it pushes inwards perpendicular to the surface from every direction. It may not seem like there is any air under the chessboard when placed on the table but the roughness and cracks of the chess board are enough to allow air underneath. If you could get rid of all the air underneath the chessboard and prevent air from being allowed to sneak back in, that board would be stuck to the table like a suction cup. In fact, that's how suction cups work. They push the air out to create less pressure inside than out. The smooth plastic of the suction cup prevents air from sneaking back in. The higher pressure outside air pushes the suction cup into the surface. (see the diagram below)
Once air sneaks back in, the inside pressure becomes the same as the outside pressure and the cup can easily be taken off the surface.
If you wanted a formula for the total pressure (also called absolute pressure) at the top of the can of beans you would have to add the pressure from the Earth's atmosphere $P_{atm}$ to the pressure from the liquid $\rho gh$.
${\Large P_{total}=\rho gh +P_{atm}}$
We typically don't try to derive a fancy term like $\rho_{air} g h$ for the atmospheric pressure $P_{atm}$ since our depth in the Earth's atmosphere is pretty much constant for any measurements made near land.
A problem with trying to use $\rho_{air} gh$ to find the pressure at a certain depth in the atmosphere is that unlike the water example, the density of the air in the atmosphere is not the same at all altitudes. As you go higher in the atmosphere the density of air decreases so we can't treat $\rho_{air}$ as a constant.
This means that the atmospheric pressure at the surface of the Earth stays relatively constant. The value of the atmospheric pressure at the surface of the Earth is stuck right around $1.01 \times10^5 Pa$. There are small fluctuations around this number caused by variations in weather patterns, humidity, altitude, etc., but for the most part when doing physics calculations we just assume that this number is a constant and stays fixed. This means, as long as the fluid you're finding the pressure for is near the surface of the Earth and exposed to the atmosphere (not in some sort of vacuum chamber) you can find the total pressure (also called absolute pressure) with this formula.
$P_{total}= \rho gh +1.01 \times 10^5 Pa$
The $\rho gh$ corresponds to the pressure created by the weight of a liquid, and the $1.01 \times 10^5 \text{ Pa}$ corresponds to the pressure of the Earth's atmosphere near sea level.
## What's the difference between absolute pressure and gauge pressure?
When measuring pressure, people often don't want to know the total pressure (which includes atmospheric pressure). People typically want to know the difference in some pressure from atmospheric pressure. The reason is that atmospheric pressure doesn't change much and it's almost always present. So including it in your measurements can feel a bit pointless at times. In other words, knowing that the air inside of your flat tire is at an absolute pressure of $1.01 \times 10^5 Pa$ isn't really all that useful (since being at atmospheric pressure means your tire's flat). The extra pressure in the tire above atmospheric pressure is what will allow the tire to inflate and perform properly.
Because of this, most gauges and monitoring equipment use what is defined to be the gauge pressure $P_{gauge}$ . Gauge pressure is the pressure measured relative to atmospheric pressure. Gauge pressure is positive for pressures above atmospheric pressure, zero at atmospheric pressure, and negative for pressures below atmospheric pressure.
The total pressure is commonly referred to as the absolute pressure $P_{absolute}$. Absolute pressure measures the pressure relative to a complete vacuum. So absolute pressure is positive for all pressures above a complete vacuum, zero for a complete vacuum, and never negative.
This can all be summed up in the relationship between the absolute pressure $P_{absolute}$, gauge pressure $P_{gauge}$, and atmospheric pressure $P_{atm}$ which looks like this,
$\Large P_{absolute} = P_{gauge} + P_{atm}$
For the case of finding the pressure at a depth $h$ in a non-moving liquid exposed to the air near the surface of the Earth, the gauge pressure and absolute pressure can found with,
$P_{gauge}=\rho gh$
$P_{absolute} = \rho g h + 1.01 \times 10^5\text{ Pa}$
Because the only difference between absolute pressure and gauge pressure is the addition of the constant value of atmospheric pressure, the percent difference between absolute and gauge pressures become less and less important as the pressures increase to very large values. (see the diagram below)
Diagram showing the values of various gauge and absolute pressures.
People often want to plug in the density of the object submerged $\rho_{object}$ into the formula for gauge pressure within a fluid $P=\rho g h$, but the density in this formula is specifically referring to the density of the fluid $\rho_{fluid}$ causing the pressure.
People often mix up absolute pressure and gauge pressure. Remember that absolute pressure is the gauge pressure plus atmospheric pressure.
Also, there are unfortunately at least 5 different commonly used units for measuring pressure (pascals, atmospheres, millimeters of mercury, etc). In physics the conventional SI unit is the pascal Pa, but pressure is also commonly measured in "atmospheres" which is abbreviated as $\text atm$. The conversion between pascals and atmospheres is, not surprisingly, $1 \text{atm} = 1.01 \times10^5 \text{ Pa}$ since one atmosphere is defined to be the pressure of the Earth's atmosphere.
## What do solved examples involving pressure look like?
### Example 1: Finding the pressure from the feet of a chair
A $7.20 \text{ kg}$ fuchsia colored four legged chair sits at rest on the floor. Each leg of the chair has a circular foot with a radius of $1.30\text{cm}$. The well engineered design of the chair is such that the weight of the chair is equally distributed on the four feet.
Find the pressure in pascals between the feet of the chair and the floor.
$P=\dfrac{F}{A} \quad \text{(Use definition of pressure. Gauge pressure isn't applicable here since there's no fluid.)}$
$P=\dfrac{mg}{A} \quad \text{(Plug in formula for weight of the chair } W=mg \text{ for the force F)}$
$P=\dfrac{mg}{4\times \pi r^2} \quad \text{(Plug in the total area of the feet of the chair 4\times \pi r^2 for the area A.)}$
$P=\dfrac{(7.20\text{ kg})(9.8\dfrac{\text{m}}{\text{s}^2})}{4\times \pi (0.013\text{ m})^2} \quad \text{(Plug in numbers, making sure to convert from cm to m)}$
$P=\dfrac{70.56 \text{ N}}{0.002124 \text{ m}^2}=33,200 \text{ Pa} \quad \text{(Calculate, celebrate!)}$
### Example 2: Force on a submarine porthole
A curious seahorse is looking into the circular window of a submarine that is sitting at a depth of $63.0 \text{ m}$ underneath the Mediterranean sea. The density of the seawater is $1025\dfrac{\text{kg}}{m^3}$. The window is circular with a radius of $5.60 \text{ cm}$. The seahorse is impressed that the window does not break from the pressure caused by the weight of the seawater.
What is the magnitude of the force exerted on the surface of the circular submarine window from the weight of the water?
$P=\dfrac{F}{A} \quad \text{(Use the definition of pressure to relate pressure to force)}$
$F=PA \quad \text{(Solve the formula symbolically for the force)}$
$F=(\rho gh)A \quad \text{(Plug in the formula for gauge pressure } P_{gauge}=\rho gh \text{ for the pressure P)}$
$F=(1025\dfrac{\text{kg}}{m^3})(9.8\dfrac{m}{s^2})(63.0\text{ m})(\pi \times [0.056 \text{ m}]^2) \quad \text{(Plug in numbers for } \rho, g, h, \text{ and } A)$
Since the window is circular we are going to use the formula for the area of a circle $A=\pi r^2$.
$F=6,230 \text{ N} \quad \text{ (Calculate, and celebrate!)}$
Note: We used the gauge pressure in this problem since the question asked for the force caused from "the weight of the water", whereas the absolute pressure would yield a force caused by the weight of the water and the weight of the air above the water.
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2019-01-19 15:03:04
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https://in.mathworks.com/help/ident/ref/lti.stepinfo.html
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# stepinfo
Rise time, settling time, and other step-response characteristics
## Description
stepinfo lets you compute step-response characteristics for a dynamic system model or for an array of step-response data. For a step response y(t), stepinfo computes characteristics relative to yinit and yfinal, where yinit is the initial offset, that is, the value before the step is applied, and yfinal is the steady-state value of the response. These values depend on the syntax you use.
• For a dynamic system model sys, stepinfo uses yinit = 0 and yfinal = steady-state value.
• For an array of step-response data [y,t], stepinfo uses yinit = 0 and yfinal = last sample value of y, unless you explicitly specify these values.
For more information on how stepinfo computes the step-response characteristics, see Algorithms.
The following figure illustrates some of the characteristics stepinfo computes for a step response. For this response, assume that y(t) = 0 for t < 0, so yinit = 0.
example
S = stepinfo(sys) computes the step-response characteristics for a dynamic system model sys. This syntax uses yinit = 0 and yfinal = steady-state value for computing the characteristics that depend on these values.
Using this syntax requires a Control System Toolbox™ license.
S = stepinfo(y,t) computes step-response characteristics from an array of step-response data y and a corresponding time vector t. For SISO system responses, y is a vector with the same number of entries as t. For MIMO response data, y is an array containing the responses of each I/O channel. This syntax uses yinit = 0 and the last value in y (or the last value in each channel's corresponding response data) as yfinal.
example
S = stepinfo(y,t,yfinal) computes step-response characteristics relative to the steady-state value yfinal. This syntax is useful when you know that the expected steady-state system response differs from the last value in y for reasons such as measurement noise. This syntax uses yinit = 0.
For SISO responses, t and y are vectors with the same length NS. For systems with NU inputs and NY outputs, you can specify y as an NS-by-NY-by-NU array (see step) and yfinal as an NY-by-NU array. stepinfo then returns a NY-by-NU structure array S of response characteristics corresponding to each I/O pair.
example
S = stepinfo(y,t,yfinal,yinit) computes step-response characteristics relative to the response initial value yinit. This syntax is useful when your y data has an initial offset; that is, y is nonzero before the step occurs.
For SISO responses, t and y are vectors with the same length NS. For systems with NU inputs and NY outputs, you can specify y as an NS-by-NY-by-NU array and yinit as an NY-by-NU array. stepinfo then returns a NY-by-NU structure array S of response characteristics corresponding to each I/O pair.
example
S = stepinfo(___,'SettlingTimeThreshold',ST) lets you specify the threshold ST used in the definition of settling and transient times. The default value is ST = 0.02 (2%). You can use this syntax with any of the previous input-argument combinations.
example
S = stepinfo(___,'RiseTimeLimits',RT) lets you specify the lower and upper thresholds used in the definition of rise time. By default, the rise time is the time the response takes to rise from 10% to 90% of the way from the initial value to the steady-state value (RT = [0.1 0.9]). The upper threshold RT(2) is also used to calculate SettlingMin and SettlingMax. These values are the minimum and maximum values of the response occurring after the response reaches the upper threshold. You can use this syntax with any of the previous input-argument combinations.
## Examples
collapse all
Compute step-response characteristics, such as rise time, settling time, and overshoot, for a dynamic system model. For this example, use a continuous-time transfer function:
$sys=\frac{{s}^{2}+5s+5}{{s}^{4}+1.65{s}^{3}+5{s}^{2}+6.5s+2}$
Create the transfer function and examine its step response.
sys = tf([1 5 5],[1 1.65 5 6.5 2]);
step(sys)
The plot shows that the response rises in a few seconds, and then rings down to a steady-state value of about 2.5. Compute the characteristics of this response using stepinfo.
S = stepinfo(sys)
S = struct with fields:
RiseTime: 3.8456
TransientTime: 27.9762
SettlingTime: 27.9762
SettlingMin: 2.0689
SettlingMax: 2.6873
Overshoot: 7.4915
Undershoot: 0
Peak: 2.6873
PeakTime: 8.0530
Here, the function uses ${\mathit{y}}_{\mathrm{init}}$= 0 to compute characteristics for the dynamic system model sys.
By default, the settling time is the time it takes for the error to stay below 2% of $|{\mathit{y}}_{\mathrm{init}}-{\mathit{y}}_{\mathit{final}}|$. The result S.SettlingTime shows that for sys, this condition occurs after about 28 seconds. The default definition of rise time is the time it takes for the response to go from 10% to 90% of the way from ${\mathit{y}}_{\mathrm{init}}$= 0 to ${\mathit{y}}_{\mathrm{final}}$. S.RiseTime shows that for sys, this rise occurs in less than 4 seconds. The maximum overshoot is returned in S.Overshoot. For this system, the peak value S.Peak, which occurs at the time S.PeakTime, overshoots by about 7.5% of the steady-state value.
For a MIMO system, stepinfo returns a structure array in which each entry contains the response characteristics of the corresponding I/O channel of the system. For this example, use a two-output, two-input discrete-time system. Compute the step-response characteristics.
A = [0.68 -0.34; 0.34 0.68];
B = [0.18 -0.05; 0.04 0.11];
C = [0 -1.53; -1.12 -1.10];
D = [0 0; 0.06 -0.37];
sys = ss(A,B,C,D,0.2);
S = stepinfo(sys)
S=2×2 struct array with fields:
RiseTime
TransientTime
SettlingTime
SettlingMin
SettlingMax
Overshoot
Undershoot
Peak
PeakTime
Access the response characteristics for a particular I/0 channel by indexing into S. For instance, examine the response characteristics for the response from the first input to the second output of sys, corresponding to S(2,1).
S(2,1)
ans = struct with fields:
RiseTime: 0.4000
TransientTime: 2.8000
SettlingTime: 3
SettlingMin: -0.6724
SettlingMax: -0.5188
Overshoot: 24.6476
Undershoot: 11.1224
Peak: 0.6724
PeakTime: 1
To access a particular value, use dot notation. For instance, extract the rise time of the (2,1) channel.
rt21 = S(2,1).RiseTime
rt21 = 0.4000
You can use SettlingTimeThreshold and RiseTimeThreshold to change the default percentage for settling and rise times, respectively, as described in the Algorithms section. For this example, use the system given by:
$\mathit{sys}=\frac{{\mathit{s}}^{2}+5\mathit{s}+5}{{\mathit{s}}^{4}+1.65{\mathit{s}}^{3}+6.5\mathit{s}+2}$
Create the transfer function.
sys = tf([1 5 5],[1 1.65 5 6.5 2]);
Compute the time it takes for the error in the response of sys to stay below 0.5% of the gap $|{\mathit{y}}_{\mathrm{final}}-{\mathit{y}}_{\mathrm{init}}|$. To do so, set SettlingTimeThreshold to 0.5%, or 0.005.
S1 = stepinfo(sys,'SettlingTimeThreshold',0.005);
st1 = S1.SettlingTime
st1 = 46.1325
Compute the time it takes the response of sys to rise from 5% to 95% of the way from ${\mathit{y}}_{\mathrm{init}}$ to ${\mathit{y}}_{\mathrm{final}}$. To do so, set RiseTimeThreshold to a vector containing those bounds.
S2 = stepinfo(sys,'RiseTimeThreshold',[0.05 0.95]);
rt2 = S2.RiseTime
rt2 = 4.1690
You can define percentages for both settling time and rise time in the same computation.
S3 = stepinfo(sys,'SettlingTimeThreshold',0.005,'RiseTimeThreshold',[0.05 0.95])
S3 = struct with fields:
RiseTime: 4.1690
TransientTime: 46.1325
SettlingTime: 46.1325
SettlingMin: 2.0689
SettlingMax: 2.6873
Overshoot: 7.4915
Undershoot: 0
Peak: 2.6873
PeakTime: 8.0530
You can extract step-response characteristics from step-response data even if you do not have a model of your system. For instance, suppose you have measured the response of your system to a step input and saved the resulting response data in a vector y of response values at the times stored in another vector t. Load the response data and examine it.
plot(t,y)
Compute step-response characteristics from this response data using stepinfo. If you do not specify the steady-state response value yfinal, then stepinfo assumes that the last value in the response vector y is the steady-state response.Because the data has some noise, the last value in y is likely not the true steady-state response value. When you know what the steady-state value should be, you can provide it to stepinfo. For this example, suppose that the steady-state response is 2.4.
S1 = stepinfo(y,t,2.4)
S1 = struct with fields:
RiseTime: 1.2897
TransientTime: 19.6478
SettlingTime: 19.6439
SettlingMin: 2.0219
SettlingMax: 3.3302
Overshoot: 38.7575
Undershoot: 0
Peak: 3.3302
PeakTime: 3.4000
Because of the noise in the data, the default definition of the settling time is too stringent, resulting in an arbitrary value of almost 20 seconds. To allow for the noise, increase the settling-time threshold from the default 2% to 5%.
S2 = stepinfo(y,t,2.4,'SettlingTimeThreshold',0.05)
S2 = struct with fields:
RiseTime: 1.2897
TransientTime: 10.4201
SettlingTime: 10.4149
SettlingMin: 2.0219
SettlingMax: 3.3302
Overshoot: 38.7575
Undershoot: 0
Peak: 3.3302
PeakTime: 3.4000
Settling time and transient time are equal when the peak error ${\mathit{e}}_{\mathrm{max}}$ is equal to the gap $|{\mathit{y}}_{\mathrm{final}}-{\mathit{y}}_{\mathrm{init}}|$ (see Algorithms (Control System Toolbox)), which is the case for models with no undershoot or feedthrough and with less than 100% overshoot. They tend to differ for models with feedthrough, zeros at the origin, unstable zeros (undershoot), or large overshoot.
Consider the following models.
s = tf('s');
sys1 = 1+tf(1,[1 1]); % feedthrough
sys2 = tf([1 0],[1 1]); % zero at the origin
sys3 = tf([-3 1],[1 2 1]); % non-minimum phase with undershoot
sys4 = (s/0.5 + 1)/(s^2 + 0.2*s + 1); % large overshoot
step(sys1,sys2,sys3,sys4)
grid on
legend('Feedthrough','Zero at origin','Non-minimum phase with undershoot','Large overshoot')
Compute the step-response characteristics.
S1 = stepinfo(sys1)
S1 = struct with fields:
RiseTime: 1.6095
TransientTime: 3.9121
SettlingTime: 3.2190
SettlingMin: 1.8005
SettlingMax: 2.0000
Overshoot: 0
Undershoot: 0
Peak: 2.0000
PeakTime: 10.5458
S2 = stepinfo(sys2)
S2 = struct with fields:
RiseTime: 0
TransientTime: 3.9121
SettlingTime: NaN
SettlingMin: 2.6303e-05
SettlingMax: 1
Overshoot: Inf
Undershoot: 0
Peak: 1
PeakTime: 0
S3 = stepinfo(sys3)
S3 = struct with fields:
RiseTime: 2.9198
TransientTime: 6.5839
SettlingTime: 7.3229
SettlingMin: 0.9004
SettlingMax: 0.9991
Overshoot: 0
Undershoot: 88.9466
Peak: 0.9991
PeakTime: 10.7900
S4 = stepinfo(sys4)
S4 = struct with fields:
RiseTime: 0.3896
TransientTime: 40.3317
SettlingTime: 46.5052
SettlingMin: -0.2796
SettlingMax: 2.7571
Overshoot: 175.7137
Undershoot: 27.9629
Peak: 2.7571
PeakTime: 1.8850
Examine the plots and characteristics. For these models, the settling time and transient time differ because the peak error exceeds the gap between the initial and the final value. For models such as sys2, the settling time is returned as NaN because the steady-state value is zero.
In this example, you compute the step-response characteristics from step-response data that has an initial offset. This means that the value of the response data is nonzero before the step occurs.
Load the step-response data and examine the plot.
plot(stepOffset.Time,stepOffset.Data)
If you do not specify yfinal and yinit, then stepinfo assumes that yfinal is the last value in the response vector y and yinit is zero. When you know what the steady-state and initial values are, you can provide them to stepinfo. Here, the steady state of the response yfinal is 0.9 and the initial offset yinit is 0.2.
Compute step-response characteristics from this response data.
S = stepinfo(stepOffset.Data,stepOffset.Time,0.9,0.2)
S = struct with fields:
RiseTime: 0.0084
TransientTime: 1.0662
SettlingTime: 1.0662
SettlingMin: 0.8461
SettlingMax: 1.0878
Overshoot: 26.8259
Undershoot: 0.0429
Peak: 0.8878
PeakTime: 1.0225
Here, the peak value of this response is 0.8878 because stepinfo measures the maximum deviation from yinit.
## Input Arguments
collapse all
Dynamic system, specified as a SISO or MIMO dynamic system model. Dynamic systems that you can use include:
• Continuous-time or discrete-time numeric LTI models, such as tf (Control System Toolbox), zpk (Control System Toolbox), or ss (Control System Toolbox) models.
• Generalized or uncertain LTI models such as genss (Control System Toolbox) or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.) For generalized models, stepinfo computes the step-response characteristics using the current value of tunable blocks and the nominal value of uncertain blocks.
• Identified LTI models, such as idtf, idss, or idproc models.
Step-response data, specified as one of the following:
• For SISO response data, a vector of length Ns, where Ns is the number of samples in the response data
• For MIMO response data, an Ns-by-Ny-by-Nu array, where Ny is the number of system outputs and Nu is the number of system inputs
Time vector corresponding to the response data in y, specified as a vector of length Ns.
Steady-state value, specified as a scalar or an array.
• For SISO response data, specify a scalar value.
• For MIMO response data, specify an Ny-by-Nu array, where each entry provides the steady-state response value for the corresponding system channel.
If you do not provide yfinal, then stepinfo uses the last value in the corresponding channel of y as the steady-state response value.
This argument is only supported when you provide step-response data as an input. For a dynamic system model sys as an input, stepinfo uses yfinal = steady-state value to compute the characteristics that depend on this value.
Value of y before the step occurs, specified as a scalar or an array.
• For SISO response data, specify a scalar value.
• For MIMO response data, specify an Ny-by-Nu array, where each entry provides the response initial value for the corresponding system channel.
If you do not provide yinit, then stepinfo uses zero as the response initial value.
The response y(0) at t = 0 is equal to yinit for systems without feedthrough. However, the two quantities differ in the presence of feedthrough because of the discontinuity at t = 0.
For example, the following figure shows the step response of a system with feedthrough sys = tf([-1 0.2 1],[1 0.7 1]).
Here, yinit is zero and the feedthrough value is –1.
This argument is only supported when you provide step-response data as an input. For a dynamic system model sys as an input, stepinfo uses yinit = 0 to compute the characteristics that depend on this value.
Threshold for defining settling and transient times, specified as a scalar value between 0 and 1. To change the default settling and transient time definitions (see Algorithms), set ST to a different value. For instance, to measure when the error falls below 5%, set ST to 0.05.
Threshold for defining rise time, specified as a 2-element row vector of nondescending values between 0 and 1. To change the default rise time definition (see Algorithms), set RT to a different value. For instance, to define the rise time as the time it takes for the response to rise from 5% to 95% from the initial value to the steady-state value, set RT to [0.05 0.95].
## Output Arguments
collapse all
Step-response characteristics, returned as a structure containing the fields:
• RiseTime
• TransientTime
• SettlingTime
• SettlingMin
• SettlingMax
• Overshoot
• Undershoot
• Peak
• PeakTime
For MIMO models or responses data, S is a structure array in which each entry contains the step-response characteristics of the corresponding I/O channel. For instance, if you provide a 3-input, 3-output model or an array of response data, then S(2,3) contains the characteristics of the response from the third input to the second output. For an example, see Step-Response Characteristics of MIMO System.
If sys is unstable, then all step-response characteristics are NaN, except for Peak and PeakTime, which are Inf.
## Algorithms
For a step response y(t), stepinfo computes characteristics relative to yinit and yfinal. For a dynamic system model sys, stepinfo uses yinit = 0 and yfinal = steady-state value.
This table shows how stepinfo computes each characteristic.
Step-Response CharacteristicDescription
RiseTimeTime it takes for the response to rise from 10% to 90% of the way from yinit to yfinal
TransientTime
The first time T such that the error |y(t) – yfinal| ≤ SettlingTimeThreshold × emax for tT, where emax is the maximum error |y(t) – yfinal| for t ≥ 0.
By default, SettlingTimeThreshold = 0.02 (2% of the peak error). Transient time measures how quickly the transient dynamics die off.
SettlingTime
The first time T such that the error |y(t) – yfinal| ≤ SettlingTimeThreshold × |yfinalyinit| for tT.
By default, SettlingTime measures the time it takes for the error to stay below 2% of |yfinalyinit|.
SettlingMinMinimum value of y(t) once the response has risen
SettlingMaxMaximum value of y(t) once the response has risen
OvershootPercentage overshoot. Relative to the normalized response ynorm(t) = (y(t) – yinit)/(yfinalyinit), the overshoot is the larger of zero and 100 × max(ynorm(t) – 1).
UndershootPercentage undershoot. Relative to the normalized response ynorm(t), the undershoot is the smaller of zero and –100 × max(ynorm(t) – 1).
PeakPeak value of |y(t) – yinit|
PeakTimeTime at which the peak value occurs
## Compatibility Considerations
expand all
Behavior changed in R2021b
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2021-10-17 21:27:28
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http://mathoverflow.net/questions/156399/decidability-of-a-matrix-product-being-the-identity
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# Decidability of a matrix product being the identity
Given a finite set $S$ of $n\times n$ integer matrices, it is known that for $k\geq 3$ it is undecidable whether some product of them (allowing repetitions) is the zero matrix (called the mortality problem). It is also known (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.448&rep=rep1&type=pdf) that if $k=2$, then it is decidable whether some product of the matrices is the identity matrix. Is it known whether for $k$ sufficiently large, it is undecidable whether some product of matrices in $S$ is the identity matrix? This has some connection with a question raised by Kontsevich.
-
This is undecidable in dimension 4 or up see http://cgi.csc.liv.ac.uk/~igor/papers/matrixcomp.pdf
The result is proved in Bell, P. C., Potapov, I.: On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups, International Journal of Foundations of Computer Science, 21(6), 2010, 963–978.
The problem is at least NP-hard in dimension 2.
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Thanks for the reference! – Richard Stanley Feb 1 '14 at 20:57
You are welcome. – Benjamin Steinberg Feb 1 '14 at 21:57
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2015-12-02 03:24:10
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https://mathlesstraveled.com/2012/12/18/permuting-permutations/
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## Permuting permutations
As you probably know, there are $n!$ ($n$ factorial) different ways to put the numbers from $1$ through $n$ (or any set of $n$ distinct objects) in a list. For example, here are the $4! = 24$ different lists containing the numbers $1$ through $4$:
Each such list is called a permutation. Now, think about the above picture: the entire picture itself represents a list (read, say, top to bottom and then left to right) of all the permutations of $\{1, \dots, 4\}$. That is, it’s a permutation of permutations! And of course, there are $24! = 620448401733239439360000$ (that is, about $6.2 \times 10^{23}$, i.e 620 sextillion, i.e six hundred thousand million million million, i.e. if you could count one trillion numbers every second, it would take you nineteen thousand years to count that high) different ways of putting the $24$ permutations in some order.
Look at the particular order given in the picture above. To go from the first permutation ($1234$) to the second ($2134$) requires only swapping two adjacent numbers—namely, $1$ and $2$.
However, going from $2134$ to the third permutation in the list ($3214$) requires more than just a swap—the $1$, $2$, and $3$ all get reshuffled.
Consider also the transition from the sixth permutation ($1324$) to the seventh ($4321$, at the top of the second column). It also requires just a swap of two numbers ($1$ and $4$)—but they are not adjacent.
Note by “adjacent” I mean adjacent in the list, not adjacent as numbers. For example, going from the penultimate ($3412$) to the final permutation ($3142$) also involves swapping $1$ and $4$, but this time they are adjacent.
Here’s the question: can we put these $24$ permutations in some order so that the only kind of transition between successive permutations is a swap of two adjacent numbers?
Obviously, exhaustively searching through all 620 sextillion orders is out of the question, so we’ll have to be a bit more clever.
Rather than give away the answer, I think I’ll just stop and let you think about it. If you haven’t seen it before, this problem really makes for a great exploration, with all kinds of interesting structure and connections to discover. Can you figure out an ordering that works—or explain why it’s not possible? You might also want to try it for some simpler cases—say, permutations of $\{1,2\}$ and of $\{1,2,3\}$. If you figure it out for $\{1, \dots, 4\}$, how about $\{1, \dots, 5\}$?
In another post I’ll give the answer, and explain why people in 17th century England (!) cared about this problem (don’t give it away in the comments if you know!).
Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
This entry was posted in challenges, combinatorics and tagged , . Bookmark the permalink.
### 11 Responses to Permuting permutations
1. decourse says:
Here’s what I found out without cheating:
Let G be a graph where the vertices are permutations, and there is an edge between two vertices if there is a single element swap which transforms one permutation into the other. The question is asking if G has a Hamiltonian cycle. G is clearly vertex-transitive, so if the Lovász conjecture is true, then it does indeed have a Hamiltonian cycle. This strongly suggests that it can be done.
Of course, I couldn’t leave it at that. I did eventually cheat, and I now know who S, J and T are.
2. Drawing out the graph, it turns out to be a truncated octahedron. The truncated octahedron is also the order 4 permutohedron but the vertices don’t match up. Mysterious!
• Ah, not so mysterious – taking “1 4 2 3” as the positions of the 4 digits yields “1 3 4 2” and those digits _do_ match up. Looking forward to the follow-up post.
3. lkuty says:
I once read two very interesting articles about it. I will put references to them in your next post if you don’t mention them in your main text. However one article is an old writing in french. I don’t know if it has been translated. I like the way you bring things in this article. Step by step and then you leave us with a question 🙂
4. asdf says:
def printer(n):
if n == 1:
yield [1]
elif n > 1:
for i, l in enumerate(printer(n – 1)):
for j in range(n)[::-1 if i % 2 == 0 else 1]:
yield l[:j] + [n] + l[j:]
for i in printer(4):
print i
5. TomC says:
Am I considered extremely ‘less traveled’ when I tell you that I just wrote the sequence down on paper with a sharpy in about 2 minutes? No math, no proof, just intuition?
• Brent says:
The math is less traveled. I think it makes you ‘traveled’. =)
• Rigel says:
That leaves you with a new problem: Can you repeat this with all permutations of N elements, for all possible N?
6. ck says:
by induction i realise we can do that for permutations of all possible N. But i don’t know why “people in 17th century England (!) cared about this problem”. Looking forward to the next post…
7. Pingback: Diagrams! | The Math Less Traveled
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2017-02-22 17:28:40
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https://math.stackexchange.com/questions/1682937/proof-of-infty-infty-maybe?noredirect=1
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# Proof of $+\infty=-\infty$ (Maybe)
I guess we can agree that $+0 = -0$. Now, after that, I was simply looking at some graphs. The graph of $\tan x$ shows asymptotes at x = $n\pi + \pi/2$. I got to thinking, what if they weren't asymptotes, but actually continuous lines?
If I take $0$ and ($+/-$)$\infty$ as diametrically opposite points of a circle, and sort of roll the $\tan x$ graph into a cylinder with the Y-axis as circumference and x-axis as the length of the cylinder, then $\tan x$ will become continuous.
This seems intuitively valid, but is there a formal proof possible that $+\infty=-\infty$? (Using simple mathematics, if possible. I am still in Grade 11)
• Whether it is even true depends on how you define it ($\infty$ and $-\infty$ that is). Mar 4, 2016 at 14:14
• If they were to be assumed "equal", what would happen if we took the limit $e^x$ as $x$ tend to $+\infty = - \infty$ Mar 4, 2016 at 14:14
• @TobiasKildetoft The meaning behind it, as far as I have learnt, is that $-\infty$ is a infinitely large negative quantity, and $+\infty$ is an infinitely large positive quantity. Mar 4, 2016 at 14:16
• Real numbers are used in different contexts. So -\infty and +\infty are used when we want to talk about the ordering of real numbers. We use one infinity when we want to compactify real numbers (but then of course you lose the ordered structure). Looking at tangents to graphs is somewhat misleading, because it assumes tacitly an ambient space where the graph is embedded. For example you can think of tan as a complex function and then the graph is embedded in a high dimensional space.
– DBS
Mar 4, 2016 at 15:42
• Imagine the real line as a circle of infinite radius. Then $\pm\infty$ coincide. Mar 4, 2016 at 17:27
In general (for grade 11), remember that $\infty$ is not a real number. To say that two elements are equal, they need to be equal in some set. That is, they need first be elements in some set. And $\infty$ is not an element in the set of real numbers.
For example, when we say that a limit (of a function) is (equal to) $\infty$ or $-\infty$, all we are saying is that the values of the function can be made as large (positive or negative) as we would like. So, it can be a bit confusing to talk about a limit being equal to $\infty$ because it gives the impression that $\infty$ is a number.
Does that mean we never ever talk about $\infty$ as a number (or element in a set)? No, for more on this see for example
• That's why I had to argue with my teacher for writing intervals as $[a,\infty]$ Mar 4, 2016 at 14:19
• @SS_C4: So, you would not want to write $[1,\infty]$, you would write $[a,\infty)$. The point, again, is that $\infty$ is not a number. by using $]$ with the $\infty$ it is as if you are saying that the interval contains the number $\infty$. Mar 4, 2016 at 14:20
• I can understand the one point compactification, (it's exactly what I was looking for) but it can be reduced to 2D, right?(Sphere to circle) And is it valid in the set of reals? (As $\infty \not\in R$) Mar 4, 2016 at 14:22
• @SS_C4 you do have the concept of the extended real line. This might be what you are looking for: en.wikipedia.org/wiki/Extended_real_number_line Mar 4, 2016 at 14:25
• And maybe the hyperreals too have "infinite" quantities. But Im not sure if one can talk in this context of a generalized infinite quantity. Mar 4, 2016 at 15:36
When looking at only the real numbers, it makes sense so seperate $\infty$ from $-\infty$.
However, in the complex plane, the Riemann sphere is often used to depict infinity.
Simply put, you have only one infinity and many ways to reach it.
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2023-03-28 03:18:54
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https://chasingastar.com/index.php/mobius-transformations
|
Find this content useful?
# Mobius Transformations
## Mobius Transformations
A common type of transformation when dealing with transformation of the complex plane are those of the form $$w = \frac{az + b}{cz + d}.$$ Such transformations are called Mobius transformations. They have a number of interesting properties, which at A-Level are predominantly concerned with their effect on lines and circles. Namely,
1. A line maps to either a line or a circle
2. A circle maps to either a line or a circle
Which of the options occurs is dictate by whether the point $z = -\frac{d}{c}$ is contained within original locus of points. If it is, then this point is mapped to infinity; forcing the resulting locus to be a line. This is the point at which A-level study of these functions broadly stops. Examples of involving manipulation of these functions can be found in the complex transformations questions in the question bank (links at the bottom of the article
As mentioned previously, these functions are much more interesting than their limited study at A-Level perhaps suggests. Each Mobius function can be thought of as a reverse stereographic projection onto the surface of a sphere, a translation, rotation and enlargement of said sphere and then a stereographic projection back onto the plane. Hence the image!
Stereographic projection is the achieved through taking the North Pole as the start of a line segment, drawing down to the desired point on the plane. The line will intersect the sphere at some point en route; this will be the point to which the point on the plane is mapped. This is easier seen in 2D:
In two dimensions, we see that this sets up a 1-1 mapping (bijection) between an infinite line and a punctured circle (the North Pole doesn't map to a point on the line). In three dimensions, we have a bijection between a punctured sphere and a plane.
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2021-10-27 12:36:53
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https://stats.stackexchange.com/questions/281007/conditional-distribution-of-a-normal-distribution-given-it-is-smaller-bigger-tha
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# Conditional distribution of a normal distribution given it is smaller/bigger than another normal distribution
Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not.
$P(X|X<Y)$ = ... and
$P(X|X>Y)$ = ...
I am thinking solving this in this way:
\begin{align} P(X|X<Y) &= \frac{P(Y>X|X)P(X)}{P(Y>X)} \\ &= \frac{(1-\Phi(\frac{x-\mu_2}{\sigma_2}))N(\mu_1,\sigma_1)}{\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}\\ P(X|X>Y) &= \frac{P(Y<X|X)P(X)}{P(Y<X)} \\ &= \frac{\Phi(\frac{x-\mu_2}{\sigma_2})N(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})} \end{align}
My questions are:
(1) Whether above solution is correct
(2) How to get the mean and sd for $P(X|X<Y)$ and $P(X|X>Y)$ if they are still normal?
• What do you mean by "truncating" here? What happens to a draw of X if it's less than Y? It seems that you're using the term "truncate" not in its usual statistical meaning – Aksakal May 22 '17 at 15:14
• I am sorry I may have used the wrong term. I would like to get the distribution of X if X < Y and the distribution of X if X> Y. I am wondering if there is any statistical formula can do this. – HannaMao May 22 '17 at 15:20
• I thought the results would be a truncated normal distribution. – HannaMao May 22 '17 at 15:21
• Although the procedure is related to truncation, the distribution function is not truncated. The integral to compute the distribution function of $X$ conditional on $Y\gt X$ is easily reduced to the one computed at stats.stackexchange.com/questions/61080. – whuber May 22 '17 at 16:44
• @whuber If I understand correctly, the referred link calculated the joint distribution of [X, Y>X]. For my question, the conditional distribution should be the joint distribution divided by [Y>X]. Am I right? Thanks! – HannaMao May 22 '17 at 16:55
Whether above solution is correct
Yes.
How to get the mean and sd for $P(X|X<Y)$ and $P(X|X>Y)$ if they are still normal?
They are not normal.
Proof:
Given $P(X | X>Y) = \frac{\Phi(\frac{x-\mu_2}{\sigma_2})\phi_x(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}$ is equivalent to a product of a uniform random variable $(\Phi(\frac{x-\mu_2}{\sigma_2})$ and a normal random variable $(\phi_x(\mu_1,\sigma_1))$
Consider $X_1 \sim N(0, 1)$ and $X_2 \sim U(0,1)$, then the product $Z = X_1X_2$ distrobution is given by:
\begin{align*} F_Z(z) &= P(Z \leq z)\\ &= P(X_1X_2 \leq z)\\ &= \int_{X_1\geq 0}P(X_2 \leq \frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1 +\int_{X_1\leq 0}P(X_2 \geq \frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1\\ &= \int_{X_1\geq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 + \int_{X_1\leq 0}(1-\frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1\\ &= \frac{1}{2} + \int_{X_1\geq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 - \int_{X_1\leq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 \\ &= \frac{1}{2} + \int\frac{2z}{x_1} \phi_{X_1}(x_1)\ dx_1 \end{align*}
which does not mimick CDF of a normal.
You can however still check if your solution is correct by simulation:
import matplotlib.pyplot as plt
import scipy as sp
import numpy as np
mu1 = 1
sigma1 = 2
mu2 = 2
sigma2 = 3
np.random.seed(42)
X = np.random.normal(mu1, sigma1, 1000)
Y = np.random.normal(mu2, sigma2, 1000)
# P(X|X>Y)
P_X_XgY = X[X>Y]
# P(X|X<Y)
P_X_XlY = X[X<Y]
denom = 1-sp.stats.norm.cdf((mu2-mu1)/np.sqrt(sigma1**2+sigma2**2))
count, bins, ignored = plt.hist(P_X_XgY, 30, normed=True)
plt.plot(bins, 1/(sigma1 * np.sqrt(2 * np.pi)) * \
(sp.stats.norm.cdf((bins-mu2)/sigma2)/denom) *\
np.exp( - (bins - mu1)**2 / (2 * sigma1**2) ), linewidth=2, color='r')
plt.title('$P(X|X>Y)$')
denom = sp.stats.norm.cdf((mu2-mu1)/np.sqrt(sigma1**2+sigma2**2))
count, bins, ignored = plt.hist(P_X_XlY, 30, normed=True)
plt.plot(bins, 1/(sigma1 * np.sqrt(2 * np.pi)) *\
((1-sp.stats.norm.cdf((bins-mu2)/sigma2))/denom) *\
np.exp( - (bins - mu1)**2 / (2 * sigma1**2) ), linewidth=2, color='r')
plt.title('$P(X|X<Y)$')
• This is a wonderful answer! I am wondering if there is a name for the distribution of P(X|X>Y) or P(X|X<Y). – HannaMao May 23 '17 at 4:47
• @whuber I am sorry I still couldn't get it. Here Z is the independent variable for $F_Z(z)$. I wanted it to be discussed more with x being the independent variable, $F_X(x)$. Did I make anything wrong? – HannaMao May 23 '17 at 15:34
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2019-04-24 06:30:53
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https://blogs.mathworks.com/loren/2020/05/28/matlab-and-mind-reading-card-games/?s_tid=prof_contriblnk&from=en
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## Loren on the Art of MATLABTurn ideas into MATLAB
Note
Loren on the Art of MATLAB has been archived and will not be updated.
# MATLAB and Mind Reading Card Games
R2020a is upon us! Do you read the release notes? Today's guest blogger, Toshi Takeuchi, apparently does, and he would like to share some new tricks using one of the new features. He also discusses sharing your code as a MATLAB app, since it is not easy to collaborate with people directly these days.
### Contents
#### UTF-8 Support in MATLAB
As of R2020a, MATLAB defaults to saving new plain text files using UTF-8. What's the big deal, you say? Das ist eine große Sache, それは大ごとだ, because you can now use international characters mixed within English easily. Even if you only speak English, you sure do use Greek letters like π, σ, or β, right? This also means you can use emojis, too, but I can't show it here because publish to html feature doesn't translate emojis into html entities like 😁
Personally, however, I am more excited with the playing card symbols. Let's make a deck of 52 cards.
deck = ["♠","♥","♦","♣"] + ["A"; (2:10)';'J';'Q';'K']
deck =
13×4 string array
"♠A" "♥A" "♦A" "♣A"
"♠2" "♥2" "♦2" "♣2"
"♠3" "♥3" "♦3" "♣3"
"♠4" "♥4" "♦4" "♣4"
"♠5" "♥5" "♦5" "♣5"
"♠6" "♥6" "♦6" "♣6"
"♠7" "♥7" "♦7" "♣7"
"♠8" "♥8" "♦8" "♣8"
"♠9" "♥9" "♦9" "♣9"
"♠10" "♥10" "♦10" "♣10"
"♠J" "♥J" "♦J" "♣J"
"♠Q" "♥Q" "♦Q" "♣Q"
"♠K" "♥K" "♦K" "♣K"
That means we can also use those special characters in the plot as well.
figure
rectangle("Position",[1 0.5 1 2],'Curvature',0.2)
text(1.5,1.5,deck(1),"FontSize",36,"HorizontalAlignment","center")
axis([0 3 0 3])
Let's play the famous 21-card mind reading trick with those cards.
You are entertaining your friends with a card trick. First, shuffle the deck and select first 21 cards from the deck.
rng("default") % for repeatability
picked = initialize21Cards(deck);
Ask a volunteer to choose a card from the stack of 21 cards and remember it, but not tell you what it is.
% select a card
chosenCard = picked(randi(21))
chosenCard =
"♣8"
Have your volunteer shuffle the deck of 21 cards and return it to you.
stacked = picked(randperm(21));
There are several variations to the steps depending on the effect you want to achieve. Here you will lay out the cards on the table. When you do so, you lay them out into 3 columns of 7 cards. There is a reason to do it this way when you are handling physical cards, but let's use a 3x7 layout to make it easier to plot the cards. We substitute columns with rows.
plotCardsFaceUp(reshape(stacked,[3 7]),0)
Repeat the process 4 times:
1. Deal the cards face up into three rows moving from bottom to top. You should have seven cards in each row.
2. Ask your volunteer to point to the row the chosen card is in: the bottom row, the middle row or the top row.
3. Slide each row of cards together to form three piles, preserving the order.
4. Collect the three piles into one stack in a specific order—make sure that the pile your volunteer pointed to is always collected second.
for ii = 1:4
placed = reshape(stacked,[3 7]); % deal the cards face up
selectedRow = find(any(placed == chosenCard,2)); % row with the card
stacked = restack(placed,selectedRow); % gather cards into a stack
end
Reveal the 11th card in the stack, which should be the chosen card. Hopefully your friends are impressed.
answer = stacked(11);
correctCard =
logical
1
#### How Does It Work?
Spoiler Alert! The trick is revealed!
Now let's see why this works. We will keep it simple by using the first 7 cards from the suits of ♠, ♥ and ♦. We will choose ♦7 for this example.
picked = ["♠","♥","♦"]' + ["A",(2:7)];
chosenCard = "♦7";
Round # 1
When you first lay out the cards, you have no idea where the chosen card is. It could be anywhere. Then your volunteer tells you it is in the top rows. You now know that the chosen card is one of those 7 cards. Let's highlight them.
plotCardsFaceUp(picked,1,"♦" + ["A" (2:7)])
Now we collect the cards into a stack in a specific order.
stacked = restack(picked,3);
Because you preserve the order of the respective rows when turning them into piles, and you insert the pile that contains that chosen card in the middle, that card will be placed somewhere between 8th to 14th position.
stacked(8:14)
ans =
1×7 string array
"♦A" "♦2" "♦3" "♦4" "♦5" "♦6" "♦7"
Round #2
Then you deal the stack into 3 rows again. The cards that used to be ordered along the rows now go column by column. The card is now somewhere in the third through fifth position in the row that contains it. Your volunteer tells you it is in the middle row. Now that reduces the possibilities to 3 cards.
placed = reshape(stacked,[3 7]);
plotCardsFaceUp(placed,"♦" + ["A" [4,7]],2)
You gather the rows into a single stack again as before.
stacked = restack(placed,2);
This ensures that the card will be in the 10th through 12th position of the stack.
stacked(10:12)
ans =
1×3 string array
"♦A" "♦4" "♦7"
Round #3
You lay the cards again. This moves the card in the 4th position in one of the rows. Your volunteer tells you it is in the top row. At this point you know that ♦7 is the chosen one.
placed = reshape(stacked,[3 7]);
plotCardsFaceUp(placed,"♦7",3)
You gather the rows into a single deck again.
stacked = restack(placed,3);
Round #4
When you lay the card down, you see that the card moved to the 4th position in the middle row.
placed = reshape(stacked,[3 7]);
answer = placed(2,4); % the 4th position in the middle row
correctCard =
logical
1
At this point you could just point the card, but your volunteer will notice that the card always ends up in the middle row. It is better to collect the cards into a single deck as before and pick the 11th card from the top.
To make it more mysterious, you probably don't want to lay the cards on the table at all. Instead, we can separate the deck into 3 piles of 7 cards and ask your volunteer to peek through them and tell you which pile contains the chosen card. This way, your volunteer will have no visual cue what's going on.
#### Building and Sharing a MATLAB App
Now that we worked out this card trick in code, we might as well turn it into a MATLAB app, right? If you are interested, follow the instructions here and get the app code to build your own app.
Once you build an app, of course you want to share it.
• If your friends are also MATLAB users, you can simply share the .mlapp file, but they will probably apprediate it if you package it so that it comes with the installer.
• How about your friends who doesn't have MATLAB? You can turn this into a standalone desktop app if you use MATLAB Compiler.
• You can also turn it into a web app that runs on browser and make it available over the web hosted on MATLAB Web App Server. See this video for more details.
#### Summary
Now that you know you can use playing cards in MATLAB, you can try all sorts of algorithms and card tricks and come up with your own apps. Please share your creations here!
#### Local Functions
function picked = initialize21Cards(deck)
shuffledDeck = deck(randperm(numel(deck)));
picked = shuffledDeck(1:21);
picked = picked(randperm(21));
end
function stacked = restack(placed,selectedRow)
rows = randperm(3);
rows = setdiff(rows,selectedRow);
rows = [rows(1), selectedRow, rows(2)];
stacked = [placed(rows(1),:),placed(rows(2),:),placed(rows(3),:)];
end
function plotCardsFaceUp(cards,varargin)
if nargin > 1
for ii = 1:length(varargin)
if isstring(varargin{ii})
highlight = varargin{ii};
else
numRound = varargin{ii};
end
end
end
[recPos,txtPos,seq,axisLim] = positionCards(cards);
if exist("numRound","var") && exist("highlight","var")
plotCards(recPos,txtPos,seq,axisLim,numRound,highlight)
elseif exist("numRound","var")
plotCards(recPos,txtPos,seq,axisLim,numRound)
elseif exist("highlight","var")
plotCards(recPos,txtPos,seq,axisLim,highlight)
else
plotCards(recPos,txtPos,seq,axisLim)
end
end
function [recPos,txtPos,seq,axisLim] = positionCards(cards)
[n,m] = size(cards);
recPos = zeros(n*m,4);
txtPos = zeros(n*m,2);
seq = [];
for ii = 1:n
seq = [seq cards(ii,1:m)];
end
origin = [1.5 1];
w = 1;
h = 2;
spacing = 0.2;
if all([n,m] == 1)
recPos = [origin w h];
txtPos = [recPos(:,1)+w/2 recPos(:,2)+h/2];
else
for ii = 1:n
recPos(m*(ii-1)+1:m*ii,1) = origin(1)+(w+spacing)*((1:m)'-1);
recPos(m*(ii-1)+1:m*ii,2) = origin(2)+(h+spacing)*(ii-1);
txtPos(m*(ii-1)+1:m*ii,1) = recPos(m*(ii-1)+1:m*ii,1) + w/2;
txtPos(m*(ii-1)+1:m*ii,2) = recPos(m*(ii-1)+1:m*ii,2) + h/2;
end
recPos(:,3) = w;
recPos(:,4) = h;
end
if all([n,m] ~= 1)
axisLim = ceil(origin(1)+(w+spacing)*m);
recPos(:,2) = recPos(:,2) + (axisLim-(h+spacing)*n)/2 - origin(2);
txtPos(:,2) = recPos(:,2) + h/2;
else
axisLim = ceil(origin(2) + (h + spacing) * m);
recPos(:,1) = (axisLim - w)/2;
recPos(:,2) = (axisLim - h)/2;
txtPos(:,1) = recPos(:,1) + w/2;
txtPos(:,2) = recPos(:,2) + h/2;
end
end
function plotCards(recPos,txtPos,seq,axisLim,varargin)
if nargin > 4
for ii = 1:length(varargin)
if isstring(varargin{ii})
highlight = varargin{ii};
else
numRound = varargin{ii};
end
end
end
figure
for ii = 1:size(recPos,1)
if exist("highlight","var") && ismember(seq(ii),highlight)
rectangle('Position',recPos(ii,:),'Curvature',0.2,"FaceColor","w","LineWidth",3)
else
rectangle('Position',recPos(ii,:),'Curvature',0.2,"FaceColor","w")
end
color = 'k';
if contains(seq(ii),["♥","♦"])
color = 'r';
end
if axisLim == 18
fontSize = 8;
elseif axisLim == 10
fontSize = 16;
elseif axisLim == 4
fontSize = 36;
else
fontSize = 10;
end
text(txtPos(ii,1),txtPos(ii,2),seq(ii),"Color",color,"FontSize",fontSize,"HorizontalAlignment","center")
end
if size(recPos,1) == 21
posX = unique(txtPos(:,1));
posY = unique(txtPos(:,2));
text(1,posY(1),"Bottom","HorizontalAlignment","center","Rotation",90)
text(1,posY(2),"Middle","HorizontalAlignment","center","Rotation",90)
text(1,posY(3),"Top","HorizontalAlignment","center","Rotation",90)
text(posX(1),1,"1","HorizontalAlignment","center")
text(posX(2),1,"2","HorizontalAlignment","center")
text(posX(3),1,"3","HorizontalAlignment","center")
text(posX(4),1,"4","HorizontalAlignment","center")
text(posX(5),1,"5","HorizontalAlignment","center")
text(posX(6),1,"6","HorizontalAlignment","center")
text(posX(7),1,"7","HorizontalAlignment","center")
end
axis([0 axisLim 0 axisLim])
if exist("numRound","var")
title("Round " + numRound)
end
set(gca, 'visible', 'off')
set(findall(gca, 'type', 'text'), 'visible', 'on')
end
Published with MATLAB® R2020a
|
2023-02-01 02:21:52
|
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|
https://physics.stackexchange.com/questions/520061/symmetry-between-gauge-fields-in-the-given-expression
|
# Symmetry between gauge fields in the given expression
Is there a symmetry between the gauge fields $$A_{\sigma}$$ and $$A_{\lambda}$$ in the expression: $$f^{abm}f^{bcn}f^{cap}\partial_{\rho}C(x-y)A^m_{\lambda}(y)\partial_{\sigma}C(y-z)A^n_{\sigma}(z)\partial_{\lambda}C(z-x)A^p_{\rho}(x),$$ that allows us to swap them in the expression, so that we end up with the final expression: $$f^{abm}f^{bcn}f^{cap}\partial_{\rho}C(x-y)A^m_{\sigma}(y)\partial_{\sigma}C(y-z)A^n_{\lambda}(z)\partial_{\lambda}C(z-x)A^p_{\rho}(x).$$
Here, the $$C(x-y)$$ is the scalar propagator defined as $$\Box C(x-y)=-\delta(x-y).$$
In general the answer is no. But notice that $$y$$ and $$z$$ are dummy variables in the sense that they must be integrated over at some point, if that is the case you can swap the names so that the $$\lambda$$ and $$\sigma$$ indices agree almost with what you want. This translates the problem into the question, can you switch $$m$$ and $$n$$? This will depend on the particular properties of your structure constants. Without any more context that is as far as you could go, just antisymmetry of $$f^{abc}$$ won't be enough.
• The x, y and z are integrated over in the expression, and the $f^{abc}$ are the structure constants of an SU(N) gauge group. However I don't see how this follows from dummy index relabeling. Can you elaborate on your answer? – Chetan Pandey Dec 17 '19 at 10:41
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2021-05-14 17:12:31
|
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|
https://www.ti.com/document-viewer/ja-jp/lit/html/SBAA347/GUID-FDD00C38-D792-4A5F-8984-A2CC040CEDDC
|
SBAA347 June 2022
Closed-Loop AC Simulation Results
The following AC sweep shows the AC transfer characteristics of the single-ended output. Using the previously-calculated cutoff frequency illustrated in the last equation, shows that the simulation closely matches the simulation. Since the AMC3301 has a gain of 8.2 V/V and a gain of 0.778 V/V is applied with the differential to single-ended conversion, the gain of 16.11 dB shown in the following image is expected.
|
2022-11-28 02:32:42
|
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|
https://infinitylearn.com/surge/question/mathematics/if-the-circlesx2y216x20y164r2andx42y7236int/
|
If the circles x2+y2−16x−20y+164=r2 and (x−4)2+(y−7)2=36 intersect at two distinct points, then
# If the circles ${x}^{2}+{y}^{2}-16x-20y+164={r}^{2}$ and $\left(x-4{\right)}^{2}+\left(y-7{\right)}^{2}=36$ intersect at two distinct points, then
1. A
$1
2. B
$r>11$
3. C
$0
4. D
$r=11$
Register to Get Free Mock Test and Study Material
+91
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### Solution:
Given equation of circle is ${x}^{2}+{y}^{2}-16x-20y+164={r}^{2}⇒\left(x-8{\right)}^{2}+\left(y-10{\right)}^{2}={r}^{2}$
${C}_{1}=\left(8,10\right)$ and ${R}_{1}=r$
for the second circle ${C}_{2}$=(4,7),${R}_{2}$=6
now, ${C}_{1}{C}_{2}=\sqrt{\left(8-4{\right)}^{2}+\left(10-7{\right)}^{2}}=\sqrt{25}=5$
Since the circles intersect at two distinct points So,
${R}_{1}+{R}_{2}>{C}_{1}{C}_{2}>\left|{R}_{1}-{R}_{2}\right|$
$⇒r+6>5>|r-6|⇒1
Register to Get Free Mock Test and Study Material
+91
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|
2023-02-08 19:32:26
|
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|
https://gianlubaio.blogspot.com/2012/06/bayesian-glaucoma.html
|
## Tuesday, 19 June 2012
### Bayesian hierarchical glaucoma
Last year (in fact I did some of this while travelling to go to my friend Lorenzo's stag do $-$ he's the one in white, but with no veil), I worked on a clinical paper discussing the prevalence of glaucoma with specific focus on the European population. The objective was relatively straightforward, except for the fact that the studies used to derive the estimations were quite heterogeneous and thus we could not pool them altogether.
So we used a nice (I think) Bayesian hierarchical model where different studies contributed to different parts of the estimation procedure. I built a model in which the overall prevalence was estimated using separate (but connected) modules $-$ basically age groups. So we first estimate a set of "level-1" parameters $\theta_1,\theta_2,\theta_3,\theta_4$ (effectively the age-group specific prevalences) using the observed data from the available studies. Some of these are assumed to be conditionally exchangeable, so that for example $\theta_2,\theta_3,\theta_4$ are used to inform the distribution of the parameter $\theta_5$, representing the prevalence among the over 50s. Again assuming conditionally exchangeability, $\theta_1$ and $\theta_5$ are used to inform the overall prevalence among the over 40s.
We have found sensible (or so I'm told by the clinicians!) results. It wasn't the place to brag about the use of a Bayesian approach, so the paper does not give much detail on the actual model. The observed data were counts of subjects with glaucoma in the study populations.
But it was cool that I persuaded them to report the results in graphical fashion and with the credible intervals.
Some of them do not even know that the model is Bayesian, but they were extremely happy with the results (or if they weren't, they were extremely nice to me anyway).
The paper is out now.
|
2018-03-24 04:06:23
|
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|
http://gmatclub.com/forum/an-empty-pool-being-filled-with-water-at-a-constant-rate-144436.html?kudos=1
|
An empty pool being filled with water at a constant rate : GMAT Problem Solving (PS)
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# An empty pool being filled with water at a constant rate
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An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
20 Dec 2012, 04:53
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### HideShow timer Statistics
An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool?
(A) 5 hr 30 min
(B) 5 hr 20 min
(C) 4 hr 48 min
(D) 3 hr 12 min
(E) 2 hr 40 min
[Reveal] Spoiler: OA
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Re: An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
26 Jan 2013, 04:33
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I tried using just ratios to solve this (because rate is constant):
8Hrs to fill 3/5 tank i.e 0.6 of the tank
X hrs to fill 2/5 tank i.e 0.4 of the tank
=> 8/0.6 = X/0.4
=> X=3.2/0.6 = 32/6 = 16/3 = 5.33hrs i.e 5hr20min
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Re: An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
28 Jan 2013, 13:28
1
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Hey,
Is'nt 16/3 = 5.3 ~ . This might be silly, but i chose 5 hours 30 mins . I don't want to make this mistake again, kindly help me
Thanks
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Re: An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
28 Jan 2013, 14:11
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This post was
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30 minutes/60 minutes=.5, so 5 and a half hours would be 5.5, not 5.33.
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Re: An empty pool being filled with water at a constant rate takes 8 hours [#permalink]
### Show Tags
29 May 2015, 20:53
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Expert's post
If it takes 8 hours to fill 3/5 of the pool, then
it takes 8/3 hours to fill 1/5 of the pool, and
it thus takes 16/3 hours to fill 2/5 of the pool, which is what we need to do.
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Re: An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
02 Jun 2015, 08:13
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Expert's post
An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool?
(3/5) An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool?
(3/5) of a pool/ 8 hours = 3/40 (the rate)
(3 pools/40 hours) = (2/5* pool)/ x hours
Cross multiply 3x = (2/5) 40
3x = (2/5) (8) (5)
3x = 16
x = 16/3 or 5 1/3
1/3 of an hour = 20 minutes
* The pool is 3/5 full so 2/5 remains.
(A) 5 hr 30 min
(B) 5 hr 20 min
(C) 4 hr 48 min
(D) 3 hr 12 min
(E) 2 hr 40 min
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Re: An empty pool being filled with water at a constant rate [#permalink]
### Show Tags
20 Dec 2012, 04:56
An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool?
(A) 5 hr 30 min
(B) 5 hr 20 min
(C) 4 hr 48 min
(D) 3 hr 12 min
(E) 2 hr 40 min
As pool is filled to 3/5 of its capacity then 2/5 of its capacity is left to fill.
Since it takes 8 hours to fill 3/5 of the pool, then to fill 2/5 of the pool it will take 8/(3/5)*2/5 = 16/3 hours = 5 hours 20 minutes (because if t is the time needed to fill the pool then t*3/5=8 --> t=8*5/3 hours --> to fill 2/5 of the pool 8*5/3*2/5=16/3 hours will be needed).
Or plug values: take the capacity of the pool to be 5 liters --> 3/5 of the pool or 3 liters is filled in 8 hours, which gives the rate of 3/8 liters per hour --> remaining 2 liters will require: time = job/rate = 2/(3/8) = 16/3 hours = 5 hours 20 minutes.
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Re: An empty pool being filled with water at a constant rate [#permalink]
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Re: An empty pool being filled with water at a constant rate [#permalink]
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21 May 2015, 03:55
Let the total amount of water eq l
then rate : ( 3/5 * l ) / 8 = 3/40 * l
So time required to fill the full tank = l / ( (3/40) * l ) = 40/3
So time required to fill the remaining section = ( 40/3 - 8 ) hrs = 16/3 hrs = 5 1/3 hrs = 5 hr 20 min
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Re: An empty pool being filled with water at a constant rate takes 8 hours [#permalink]
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29 May 2015, 20:03
As per what I did,
8 Hours - 3/5 of capacity
? Time - to fill remaining - 2/5 of capacity.
But the answer I am getting is incorrect. Could someone share the appropriate process to work on this?
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Re: An empty pool being filled with water at a constant rate takes 8 hours [#permalink]
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29 May 2015, 20:08
Expert's post
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Hi Pretz,
There are a couple of different ways to approach the math in this question. It looks like you started to set up a ratio, but didn't complete the work. Here's one way to go about it:
Since it takes 8 hours to fill 3/5 of the pool and X hours to fill 2/5 of the pool.....
8/X = (3/5)/(2/5)
Since both fractions are "over 5", we can multiply those 5s out (by multiplying the numerator and denominator by 5)....
8/X = 3/2
Now we can cross-multiply and solve for X....
16 = 3X
16/3 = X
5 1/3 hours = X
5 1/3 hours = 5 hours 20 minutes
[Reveal] Spoiler:
B
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An empty pool being filled with water at a constant rate takes 8 hours [#permalink]
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29 May 2015, 20:46
Pretz wrote:
As per what I did,
8 Hours - 3/5 of capacity
? Time - to fill remaining - 2/5 of capacity.
But the answer I am getting is incorrect. Could someone share the appropriate process to work on this?
Hello there Pretz
I think your approach is fine.
$$8$$ $$hours$$ ----- $$\frac{3}{5}$$
multiply by $$\frac{5}{3}$$ on both sides
$$8*\frac{5}{3}$$ $$hours$$ ----- $$\frac{3}{5} *\frac{5}{3}$$
$$\frac{40}{3}$$ $$hours$$ ----- $$1$$
So it takes $$\frac{40}{3}$$ $$hours$$ to fill the tank
The additional time required to fill $$\frac{2}{3}$$rd of the tank will be
$$\frac{40}{3} - 8$$ $$hours$$
$$\frac{16}{3}$$ $$hours$$
$$5 \frac{1}{3}$$ $$hours$$
$$5$$ $$hours$$ $$20$$ $$minutes$$
I hope that explains it
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Re: An empty pool being filled with water at a constant rate takes 8 hours [#permalink]
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29 May 2015, 21:05
Awesome! Thank you so much all!
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Re: An empty pool being filled with water at a constant rate [#permalink]
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01 Jun 2015, 01:53
B
if the rates are constant the ratio of capacity is proportional to that of time
x/8=2/3 ------- x = 5 hr 20 min
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Re: An empty pool being filled with water at a constant rate [#permalink]
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17 May 2016, 04:59
We know,
R * T = Q , suppose the Q=100ltr. then,
R * 8 = 3/5 of 100 ==> 60/8 ==> 15/2 ltr per hour.
now, remaning 40ltr(which is 100 - 60 or 2/5 of 100)
R * t = Q
or, 15/2 * t = 40
or, t = 40 * 2/15
t = 16/3 = 5.33 = 5 hour + 1/3 and 1/3 of 60 = 20. so, 5 hour 20 mins.
If only such questions were asked in GMAT.
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Re: An empty pool being filled with water at a constant rate [#permalink]
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17 May 2016, 16:01
An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool?
(A) 5 hr 30 min
(B) 5 hr 20 min
(C) 4 hr 48 min
(D) 3 hr 12 min
(E) 2 hr 40 min
Solution:
To solve we can setup a proportion. The proportion will read:
A time of 8 hours is to filling up 3/5 of the pool is the same as a time of x number of hours is to filling up (the remaining) 2/5 of the pool. Setting this up mathematically we have:
8/(3/5) = x/(2/5)
8/(3/5) = x/(2/5)
40/3= 5x/2
Cross multiplying, we get:
80 = 15x
16 = 3x
16/3 = x
5 1/3 hours = x
5 hours and 20 minutes = x
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Re: An empty pool being filled with water at a constant rate [#permalink] 17 May 2016, 16:01
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2016-12-09 23:09:26
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http://physics.tutorcircle.com/light/concave-mirror.html
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Sales Toll Free No: 1-855-666-7446
Concave Mirror
Top
Sub Topics Mirrors are usually made of a sheet of glass that is coated with a thin layer of shiny metal on one surface. A mirror with a flat surface is a plane mirror. If the surface is spherical or curved, it is known as curved mirror. An image is a copy of an object formed by rays of light. A plane mirror always produces a virtual image. A virtual image is a copy of an object formed at the location from which the light rays appear to come. It is important however, to realize that the rays do not really come from behind the mirror. Sometimes you see images that are very distorted. Look into both side sides of a polished metal spoon. The images you see are quite different from the image formed by a plane mirror. Each side of the spoon produces a different image because each side is curved differently. The curved surface of the spoon changes the way light is reflected.
Definition
When the inside surface of a curved mirror is the reflecting surface, the mirror is a concave mirror. The curvature of the reflecting surface causes the rays to come together. The point at which the light rays meet is called the focal point. Concave mirrors can form either real or virtual images. A real image is a copy of an object formed at the point where light rays actually meet. Unlike a virtual image, a real image can be viewed on a surface such as a screen.
Equation
The general formula for mirrors and lenses to find out the focal length is,
$\frac{1}{f}$ = $\frac{1}{u}$ + $\frac{1}{v}$
or,
f = $\frac{uv}{u+v}$
where,
f
is the focal length,
u is the distance between the object and mirror,
v is the distance between the image and mirror.
The sign of the image will depends on where it is produced.
Ray Diagram
As mentioned before, concave mirror produces both real and virtual images. The type of image formed depends upon where the object is in relation to the mirror. The image is formed at different position of the mirror according to the object's position. If the object is farther from the focal point, a real image is produced. If the object is near to the focal point, virtual image is produced. The ray diagrams of concave mirror is given below:
Uses
Concave mirrors are often used in automobile headlights and flashlights to direct the illumination from a single light bulb into a beam. If the bulb is placed at the focal point of a concave mirror, the reflected light rays will be parallel to one another. This results in a brighter beam of light. Large concave mirrors are used to focus the solar light for energy production.
Examples
Example problems related to concave mirror is described below:
Solved Examples
Question 1: Calculate the focal length of concave mirror if the image is formed at a distance 10cm when the object is at a distance 5cm?
Solution:
From the question it is given that,
u = 5cm and v = 10cm
The equation for focal length is,
f = $\frac{uv}{u+v}$
f = $\frac{5\times10}{5+10}$
f = $\frac{50}{15}$ = 3.33cm
Question 2: Calculate the focal length of concave mirror if the image is formed at a distance 25cm when the object is at a distance 10cm?
Solution:
From the question it is given that,
u = 10cm and v = 25cm
The equation for focal length is,
f = $\frac{uv}{u+v}$
f = $\frac{10\times25}{10+25}$
f = $\frac{250}{35}$ = 7.14cm
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2018-04-22 18:32:03
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https://www.hackmath.net/en/math-problem/3319
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# Scale of map
James travels one kilometer in 12 minutes. The route walked for half an hour measured on the map 5 cm. Calculate how many kilometers James walked for half an hour. Find the scale of the map.
Result
s = 2.5 km
M = 50000
#### Solution:
$s=1/12 \cdot \ 30=\dfrac{ 5 }{ 2 }=2.5 \ \text{km}$
$M=s/(0.05/1000)=2.5/(0.05/1000)=50000$
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2020-04-05 14:29:02
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http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/goldenmean.htm
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## The 'Golden Mean' in number theory
[abstract:] "Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, p prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature."
Castro's observation possibly linking the 'Golden String' to a function central to the behaviour of certain eigenvalues in random matrix theory (which in turn appears to be deeply linked to the behaviour of the nontrivial zeros of the Riemann zeta function).
P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)
"We shall start by briefly summarizing the results of the 'local' renormalization theory for transitions from quasiperiodicity to chaos. In experimental tests of this theory one adjusts the external frequency to make the frequency ratio as far as possible from being mode-locked. this is most readily attained by tuning the ratio to the 'golden mean' (51/2 - 1)/2. The choice of the golden mean is dictated by number theory: the golden mean is the irrational number for which it is hardest to give good rational approximants. As experimental measurements have limited accuracy, physicists usually do not expect that such number-theoretic subtleties as how irrational a number is should be of any physical interest. However, in the dynamical systems theory to chaos the starting point is the enumeration of asymptotic motions of a dynamical system, and through this enumeration number theory enters and comes to play a central role."
B.W. Ninham and S. Lidin, "Some remarks on quasi-crystal structure", Acta Crystallographica A 48 (1992) 640-650
[abstract:] "The Fourier transform of skeleton delta function that characterizes the most striking features of experimental quasi-crystal diffraction patterns is evaluated. The result plays a role analogous to the Poisson summation formula for periodic delta functions that underlie classical crystallography. The real-space distribution can be interpreted in terms of a backbone comprising a system of intersecting equiangular spirals into which are inscribed (self-similar) gnomons of isoceles triangles with length-to-base ratio the golden mean...In addition to the vertices of these triangles, there is an infinite number of other points that may tile space in two or three dimensions. Other mathematical formulae of relevance are briefly discussed."
[from concluding remarks:] "Perhaps the most interesting feature is that our Fourier-transform sum seems to have much in common with the distribution of the zeros of the Riemann zeta function...! That indicates something of the depth of the problem. That the zeta function ought to come into the scheme of things somehow is not surprising - the Poisson and related summation formulae are special cases of the Jacobi theta function. [Indeed the Bravais lattices can be enumerated systematically through an integral over all possible products and sums of products of any three of the four theta functions in different combinations that automatically preserve translational and rotational symmetries.] The theta-function transformations are themselves just another way of writing the [functional equation of the zeta function]. Additionally, the properties of the zeta function are automatically connected to the theory of prime numbers. So one expects that the Rogers-Ramanujan relations must play a central role in the scheme of things for quasi-crystals."
V. Dimitrov, T. Cooklev and B. Donevsky, "Number theoretic transforms over the golden section quadratic field.", IEEE Trans. Sig. Proc. 43 (1995) 1790-1797
V. Dimitrov,G. Jullien, and W. Miller, "A residue number system implementation of real orthogonal transforms", IEEE Trans. Sig. Proc. 46 (1998) 563-570.
M.L. Lapidus and M. van Frankenhuysen, "A prime orbit theorem for self-similar flows and Diophantine approximation", Contemporary Mathematics volume 290 (AMS 2001) 113-138.
"EXAMPLE 2.23 (The Golden flow). We consider the nonlattice flow GF with weights w1 = log 2 and w2 = \phi log2, where \phi = (1 + 51/2)/2 is the golden ratio. We call this flow the golden flow. Its dynamical zeta function is
\zetaGF(s) = 1/(1 - 2-s - 2-\phis)"
C. Bonanno and M.S. Mega, "Toward a dynamical model for prime numbers" Chaos, Solitons and Fractals 20 (2004) 107-118
[abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency."
The main idea here is that the Manneville map Tz exhibits a phase transition at z = 2, at which point the mean Algorithmic Information Content of the associated symbolic dynamics is n/log n. n is a kind of iteration number. For this to work, the domain of Tz [0,1] must be partitioned as [0,0.618...] U [0.618...,1] where 1.618... is the golden mean.
The authors attempt to exploit the resemblance to the approximating function in the Prime Number Theorem, and in some sense model the distribution of primes in dynamical terms, i.e. relate the prime number series (as a binary string) to the orbits of the Manneville map T2. Certain refinements of this are then explored.
The Phyllotaxis project's notes on the Farey Tree and the Golden Mean
Selvam's attempts to link the Riemann zeta function to fluid flow, atmospheric turbulence, etc. (the Golden Mean appearing as a winding number)
I have discovered a particularly simple Beurling generalised-prime configuration wherein the associated zeta function has a 'fixed point' at the Golden Ratio (i.e. zeta(1.618...) = 1.618... Notes will be added here in due course.
J. Dudon, "The golden scale", Pitch I/2 (1987) 1-7.
"The Golden scale is a unique unequal temperament based on the Golden number. The equal temperaments most used, 5, 7, 12, 19, 31, 50, etc. are crystallizations through the numbers of the Fibonacci series, of the same universal Golden scale, based on a geometry of intervals related in Golden proportion. The author provides the ratios and dimensions of its intervals and explains the specific intonation interest of such a cycle of Golden fifths, unfolding into microtonal coincidences with the first five significant prime numbers ratio intervals (3:5:7:11:13)." [Note that here the Fibonacci sequence mentioned differs slightly from, but is closely related to, the usual one.]
[abstract:] "The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in Section 1, we have described some equations concerning the knot polynomials, the Chern–Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to $N = 4$ super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern–Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from string theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function.
Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan's modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan's equations concerning $\pi$ and, in conclusion, (iii) the mathematical connections with the golden ratio $\phi$ and with $1.375$ that is the mean real value for the number of partitions $p(n)$."
In their paper "The golden mean as clock cycle of brain waves" (Chaos, Solitons and Fractals 18 No. 4 (2003) 643-652, Harald and Volkmar Weiss acknowledge this website as one of several "...without which our work would be impossible", and in a subsequent email, Volkmar Weiss wrote "Your site was very helpful to us in an extraordinary way."
Although the article has no explicit number theoretical content, it relates closely to quite a few different areas of research which are relevant to this archive.
[abstract:] "The principle of information coding by the brain seems to be based on the golden mean. Since decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose-Einstein-statistics to learning experiments, Pascual-Leone obtained a fit between predicted and tested span. Multiplying span by mental speed (bits processed per unit time) and using the entropy formula for bosons, we obtain the same result. If we understand span as the quantum number n of a harmonic oscillator, we obtain this result from the EEG. The metric of brain waves can always be understood as a superposition of n harmonics times $2\Phi$, where half of the fundamental is the golden mean $\Phi$ (= 1.618) as the point of resonance. Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of chaos, i.e. $2\Phi = 3 + \phi^3$. Similarities with El Naschie's theory for high energy particle's physics are also discussed."
archive tutorial mystery new search home contact
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2018-10-20 12:50:06
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http://mathhelpforum.com/algebra/173352-perfect-square-trinomial-square-binomial-print.html
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# perfect square trinomial/square of binomial
• Mar 3rd 2011, 03:38 PM
Narga115
perfect square trinomial/square of binomial
Directions: Find the value of C that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Problem: x² + 6x + c
• Mar 3rd 2011, 03:48 PM
topsquark
Quote:
Originally Posted by Narga115
Directions: Find the value of C that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Problem: x² + 6x + c
A perfect square trinomial will always be in the following form: $(a + b) = a^2 + 2ab + b^2$
For your problem a is clearly equal to x. The linear term is then 6x = 2ab = 2(x)b. Thus we may identify 2b = 6. That makes b = 3. Can you finish this?
-Dan
• Mar 3rd 2011, 04:02 PM
Narga115
Yes. Thank you for your help.
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2017-04-29 20:51:19
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https://astro.paperswithcode.com/top-social
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# The Value-Added Catalog of ASAS-SN Eclipsing Binaries: Parameters of Thirty Thousand Detached Systems
no code yet • 11 May 2022
Detached eclipsing binaries are a fundamental tool for measuring the physical parameters of stars that are effectively evolving in isolation.
Solar and Stellar Astrophysics
TWEETS
# Llamaradas Estelares: Modeling the Morphology of White-Light Flares
11 May 2022
Stellar variability is a limiting factor for planet detection and characterization, particularly around active M-type stars.
Solar and Stellar Astrophysics Earth and Planetary Astrophysics Instrumentation and Methods for Astrophysics
TWEETS
# Another Shipment of Six Short-Period Giant Planets from TESS
no code yet • 11 May 2022
We present the discovery and characterization of six short-period, transiting giant planets from NASA's Transiting Exoplanet Survey Satellite (TESS) -- TOI-1811 (TIC 376524552), TOI-2025 (TIC 394050135), TOI-2145 (TIC 88992642), TOI-2152 (TIC 395393265), TOI-2154 (TIC 428787891), & TOI-2497 (TIC 97568467).
Earth and Planetary Astrophysics Solar and Stellar Astrophysics
TWEETS
# The Visual Survey Group: A Decade of Hunting Exoplanets and Unusual Stellar Events with Space-Based Telescopes
no code yet • 16 May 2022
This article presents the history of the Visual Survey Group (VSG) - a Professional-Amateur (Pro-Am) collaboration within the field of astronomy working on data from several space missions (Kepler, K2 and TESS).
Earth and Planetary Astrophysics Instrumentation and Methods for Astrophysics Solar and Stellar Astrophysics
TWEETS
# Patchy nightside clouds on ultra-hot Jupiters: General Circulation Model simulations with radiatively active cloud tracers
no code yet • 16 May 2022
In this work, we add to previous efforts by also considering the localized condensation of clouds in the atmospheres of ultra-hot Jupiters, their resulting transport by the atmospheric circulation, and the radiative feedback of clouds on the atmospheric dynamics.
Earth and Planetary Astrophysics Atmospheric and Oceanic Physics
TWEETS
# Further support and a candidate location for Planet 9
no code yet • 16 May 2022
The existence of a hypothetical Planet 9 lurking in the outer solar system has been proposed as an explanation for the the anomalous clustering in the orbits of some trans-Neptunian objects.
Earth and Planetary Astrophysics
TWEETS
# Wide twin binaries are extremely eccentric: evidence of twin binary formation in circumbinary disks
no code yet • 11 May 2022
For the excess-twin population at 400-1000 AU, we infer a near-delta function excess of high-eccentricity system, with eccentricity $0. 95 \lesssim e \leq 1$.
Solar and Stellar Astrophysics Earth and Planetary Astrophysics Astrophysics of Galaxies
TWEETS
# Observational constraints on stellar feedback in dwarf galaxies
no code yet • 13 May 2022
Feedback to the interstellar medium (ISM) from ionising radiation, stellar winds and supernovae is central to regulating star formation in galaxies.
Astrophysics of Galaxies Cosmology and Nongalactic Astrophysics
TWEETS
# Ocean signatures in the total flux and polarization spectra of Earth-like exoplanets
no code yet • 11 May 2022
The dips in P, and the negative Q in the near-infrared, can be searched for at a phase angle of 90 degrees, where the planet-star separation is largest.
Earth and Planetary Astrophysics Atmospheric and Oceanic Physics
TWEETS
# Disruption of Saturn's ring particles by thermal stress
no code yet • 11 May 2022
We found that thermal stress can grind porous ring particles larger than 10-20 m, which explains the lack of particles larger than 10 m in Saturn's ring.
Earth and Planetary Astrophysics
TWEETS
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2022-05-18 02:21:23
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https://openturns.github.io/openturns/latest/auto_probabilistic_modeling/stochastic_processes/plot_mixRVandProcess.html
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# Create a process from random vectors and processes¶
The objective is to create a process defined from a random vector and a process.
We consider the following limit state function, defined as the difference between a degrading resistance and a time-varying load :
We propose the following probabilistic model: - is the initial resistance, and ; - is the deterioration rate of the resistance; it is deterministic; - is the time-varying stress, which is modeled by a stationary Gaussian process of mean value , standard deviation and a squared exponential covariance model; - is the time, varying in .
First, import the python modules:
from openturns import *
from openturns.viewer import View
from math import *
## 1. Create the gaussian process ¶
Create the mesh which is a regular grid on , with , by step =1:
b = 0.01
t0 = 0.0
step = 1
tfin = 50
n = round((tfin-t0)/step)
myMesh = RegularGrid(t0, step, n)
Create the squared exeponential covariance model:
where the scale parameter is and the amplitude .
l = 10/sqrt(2)
myCovKernel = SquaredExponential([l])
print('cov model = ', myCovKernel)
Out:
cov model = SquaredExponential(scale=[7.07107], amplitude=[1])
Create the gaussian process :
S_proc = GaussianProcess(myCovKernel, myMesh)
## 2. Create the process ¶
First, create the random variable , with and :
muR = 5
sigR = 0.3
R = Normal(muR, sigR)
The create the Dirac random variable :
B = Dirac(b)
Then create the process using the class and the functional basis and :
with independent.
const_func = SymbolicFunction(['t'], ['1'])
linear_func = SymbolicFunction(['t'], ['-t'])
myBasis = Basis([const_func, linear_func])
coef = ComposedDistribution([R, B])
R_proc = FunctionalBasisProcess(coef, myBasis, myMesh)
## 3. Create the process ¶
First, aggregate both processes into one process of dimension 2:
myRS_proc = AggregatedProcess([R_proc, S_proc])
Then create the spatial field function that acts only on the values of the process, keeping the mesh unchanged, using the ValueFunction class. We define the function on by:
in order to define the spatial field function that acts on fields, defined by:
g = SymbolicFunction(['x1', 'x2'], ['x1-x2'])
gDyn = ValueFunction(g, myMesh)
Now you have to create the final process thanks to :
Z_proc = CompositeProcess(gDyn, myRS_proc)
## 4. Draw some realizations of the process¶
N=10
sampleZ_proc = Z_proc.getSample(N)
graph = sampleZ_proc.drawMarginal(0)
graph.setTitle(r'Some realizations of $Z(\omega, t)$')
Show(graph)
## 5. Evaluate the probability that ¶
We define the domaine and the event :
domain = Interval([2], [4])
print('D = ', domain)
event = ProcessEvent(Z_proc, domain)
Out:
D = [2, 4]
We use the Monte Carlo sampling to evaluate the probability:
MC_algo = ProbabilitySimulationAlgorithm(event)
MC_algo.setMaximumOuterSampling(1000000)
MC_algo.setBlockSize(100)
MC_algo.setMaximumCoefficientOfVariation(0.01)
MC_algo.run()
result = MC_algo.getResult()
proba = result.getProbabilityEstimate()
print('Probability = ', proba)
variance = result.getVarianceEstimate()
print('Variance Estimate = ', variance)
IC90_low = proba- result.getConfidenceLength(0.90)/2
IC90_upp = proba + result.getConfidenceLength(0.90)/2
print('IC (90%) = [', IC90_low, ', ', IC90_upp, ']')
Out:
Probability = 0.7551515151515152
Variance Estimate = 5.659164649247292e-05
IC (90%) = [ 0.7427777057653888 , 0.7675253245376417 ]
Total running time of the script: ( 0 minutes 0.129 seconds)
Gallery generated by Sphinx-Gallery
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2021-03-09 07:45:34
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https://quant.stackexchange.com/questions/45909/what-10-year-bond-data-to-use-when-making-a-risk-return-scatter-plot
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# What 10 year bond data to use when making a risk/return scatter plot?
I am making a risk/return scatter plot (seen below) (from this site):
What data must be used for bonds (e.g. 3 month or 10 year US bonds)? I thought you would use this data, but if you take the years 2013-2018, then the price rose from \$0.01 to \$2.29, or a 229x growth. This is clearly incorrect for a 3 month bond, seeing as that it is basically the equivalent of cash. My understanding says that the 3 month bond should be at the bottom left of a scatter plot (low risk and low return). If this is the case, what data should be used for the 90 day treasury return?
Edit: this site verifies that the "US Treasury Short" should be at the lower left.
• You need bond total return indices. A lot of vendors (BBG, ICE, etc.) publish these indices, although they're not really in the public domain. The best proxy might be the ETF IEF, which tracks the 7-10 sectors of the US Treasury mkt. – Helin Jun 3 at 4:43
• That makes sense. Is a common Short Term ETF SPDR Bloomberg Barclays 1-3 Month T-Bill ETF (BIL)? – quantfinancequest Jun 3 at 6:48
• Cash return is available mba.tuck.dartmouth.edu/pages/faculty/ken.french/…. You can also compute it (approximately) from bill rates fred.stlouisfed.org/series/DTB3. – Helin Jun 3 at 9:36
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2019-07-20 20:57:07
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http://upcommons.upc.edu/e-prints/handle/2117/800
|
Català Castellano English
Empreu aquest identificador per citar o enllaçar aquest ítem: http://hdl.handle.net/2117/800
Arxiu Descripció MidaFormat
Títol: A characterization of isochronous centres in terms of symmetries Autor: Freire, Emilio; Gasull Embid, Armengol; Guillamon Grabolosa, Antoni Data: 2000 Tipus de document: Article Resum: We present a description of isochronous centres of planar vector fields $X$ by means of their groups of symmetries. More precisely, given a normalizer $U$ of $X$ (i.e., $[X,U]=\mu X$, where $\mu$ is a scalar function), we provide a necessary and sufficient isochronicity condition based on $\mu$. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ($[X,U]= 0$). We put also special emphasis on the mechanical aspects of isochronicity; this point of view forces a deeper insight into the potential and quadratic-like Hamiltonian systems. For these families we provide new ways to find isochronous centres, alternative to those already known from the literature. URI: http://hdl.handle.net/2117/800 Apareix a les col·leccions: EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions. Articles de revistaDepartaments de Matemàtica Aplicada. Articles de revista Comparteix:
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2014-03-10 12:30:48
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https://space.stackexchange.com/questions/12216/could-a-space-based-solar-power-system-be-used-as-a-weapon
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# Could a space based solar power system be used as a weapon?
Space based solar power systems that beam power back to terrestrial receivers have been discussed for decades. However, I've always had a nagging concern that such a system, once built, could be quickly repurposed as a weapon, and so no country would permit another country to launch one.
I know the power density on the ground wouldn't be high enough for a dramatic, fry-a-city-in-ten-seconds kind of weapon. In fact, Wikipedia cites expected levels of $23mw/cm^2$, which is only about twice the OSHA workplace limits. However, if you follow Wikipedia's reference to its source, on page 14 you'll find that this limit is designed to "avoid potential microwave interference with the D and F layers of the atmosphere."
For those who don't care about the ionosphere (which probably includes those hoping to create a weapon) it would likely be easy to design a transmission antenna that would usually limit itself to this power density, but could easily be focused to increase it by perhaps an order of magnitude. $250mw/cm^2$ would be about twice the solar constant, and would impart significant amounts of heat to anyone in the beam. Given that the beam could cover a large city, and add in infrastructure damage, and that's a pretty significant attack.
Is this a legitimate concern? Are designers of potential space solar power systems coming up with ways to eliminate the ability to repurpose them into weapons?
Edit: Most proposals put the transmitting satellite in geosynchronous orbit, so that it will always be above its dedicated receiving station. This puts the satellite out of the range of current anti-satellite weapons, and could even make it difficult to destroy in the future; it would be so large, and could be built from redundant sections, so that punching holes through it wouldn't significantly harm it.
I also understand the comparison to nuclear weapons. However, the use of nuclear weapons is such a bright line that the attacker would likely receive worldwide condemnation. Nor can you use a "little bit" of nuclear weapons, which also keeps that line bright. However, a divertable microwave beam might be easier to use briefly, and even deniably ("Ooops!"). Imagine a country briefly toasting one of its surrounding countries' capitals, and then immediately apologizing. The effect could be drastic, and could seriously destabilize the attacked country.
• I would expect that any nation able to launch a big microwave beamer satellite would also have the ability to launch nuclear warheads, which would be able to deliver destructive energy to a target city at a substantially higher rate than 2 solar constants. Oct 5 '15 at 3:40
• Are designers of potential space solar power systems coming up with ways to eliminate the ability to repurpose them into weapons? Not to my knowledge. But most countries' insurance policy against such weapons is that it would be trivially easy to destroy them either from the ground or on orbit, be it using same method (funny enough, beam sources like, say, phased array MW, don't really take well same level of input as they are capable of outputting) or kinetic projectiles. And then there's OST, not so much as a deterrent as legal grounds for subsequent destruction of it. Oct 5 '15 at 5:42
• Most proposals for weapons in space (note the difference from militarization of space) put them in LEO. This makes the very vulnerable to ASATs. In fact, the MIRACL laser can damage satellites and that was done sometime around 1997 (?). To prevent this solar weapon's destruction you'd likely need to put them past GEO (some 35,000-40,000km). Yet that would greatly affect the efficiency of the suggested weapon. Is it possible? Yes, but likely to do a lot of damage over a period of time? Probably not. Repurposing the satellites would require autonomous robotic spacecraft that don't exist. Oct 5 '15 at 11:06
• To be honest, I'd be more worried about someone purposefully crashing a satellite into a major city. That reminds me about the Rosat satellite stories. Oct 5 '15 at 11:10
• As a side note, SimCity 2000 has a space based solar power system titled the "Microwave Power Plant" which carries the risk of its space based transceiver satellite misfiring, destroying large swaths of buildings in a fiery, death-ray-esque fashion. I'm quite glad to know my childhood fears have been resolved by the answers to this question. Aug 15 '16 at 17:37
There would be very little you could do to reconfigure a space based power bird to make it more lethal unless you had designed it as a weapon in the first place, and it's unlikely anyone would go down that route as there's much better technology to invest in if you want to destroy things. The best you could do would be to point the power output at a target.
It is possible that space based power systems might be used as disruptive weapons against civilian targets by a desperate nation. Space based power systems require ground based receivers which would be easy targets for conventional weapons, providing the enemy could reach them with their weapons. If the receivers were damaged or destroyed the space power satellites would have nowhere to send the power, and then some big brain might decide if they aren't supplying power to the war economy they could be used to fry a few enemy cities. The power beams could be re-positioned to point at soft enemy targets, providing that they are within reach of the satellite. This would not do that much damage, but it could be disruptive to electrical systems and impact the ability of an enemy to wage war.
You'd have to be in pretty dire straits to try this though as it would not be very effective and as soon as they are used this way the satellites would be huge, expensive targets. Once a nation uses space-based weapons all their other space infrastructure would likely be fair game as well. It would be very risky and foolish, but as history teaches the human race has an almost limitless capacity for destructive decisions.
In other words this is pretty unlikely and best saved for Thunderbirds Are Go! episodes.
• Do you think it could be used as an anti-satellite or anti-ICBM weapon? Turning it to track a satellite or ICBM might be a challenge. Would a tin foil hat be useful for a targeted pedestrian? :-D Oct 5 '15 at 12:02
• I think a tin-foil umbrella would provide better anti-power satellite coverage, a tin-foil hat would not protect the arms @LocalFluff.
– GdD
Oct 5 '15 at 12:14
John Mankins is one of the top SSP researchers. Some power beaming architectures call for diffuse lasers and color-tuned solar cells, while others stick with microwaves. In an interview he reported some of the findings of a recent (2013 or 2014) European conference on Space Solar Power. One finding was that recent modeling suggests that beaming the power back with a relatively diffuse laser could still cause dangerous levels of heating if it was mis-targeted or intentionally targeted at cities. It seemed to be the death knell of the light-based power beaming architectures. Meanwhile, microwaves are not only diffuse but most designs build the frequency into the hardware. If it isn't built as a weapon then it can't be re-tuned remotely to increase the frequency and weaponize it. So in short, if you are using a microwave beamed power system then you probably can't repurpose it as a weapon unless you build that capability in before it launches, quite possibly requiring a lot of extra hardware and weight.
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2021-09-27 04:43:27
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http://mathhelpforum.com/advanced-algebra/71325-modules.html
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Math Help - Modules
1. Modules
Let R be a ring. prove that if M is a free R-module, then Ann_R M =0.
2. Originally Posted by peteryellow
Let R be a ring. prove that if M is a free R-module, then Ann_R M =0.
let $r \in \text{ann}_R M$ and choose $x \in M$ to be in a basis of $M.$ then $rx=0$ and thus $r=0.$
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2016-07-25 06:18:40
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-6th-edition/chapter-4-exponential-and-logarithmic-functions-exercise-set-4-1-page-452/38
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## College Algebra (6th Edition)
please see graph (blue curve), asymptote of $g$:$\quad\quad y=-1$ domain of $g=(-\infty,\infty)$ range of $g=(-1,\infty)$
Graph $f(x)=e^{x}\qquad$ (red, dashed) by plotting the points from the table and connecting with a smooth curve. $g(x)=e^{x}+2= f(x)+2$ so the graph of $g(x)$ (blue) is obtained by shifting the graph of f(x) (red) down by 1 unit. Reading the graph, asymptote of $g$:$\quad\quad y=-1$ domain of $g=(-\infty,\infty)$ range of $g=(-1,\infty)$
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2019-11-20 11:30:13
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https://physics.stackexchange.com/questions/355258/1d-or-2d-crystal
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# 1D or 2D crystal?
If I have a 2D ladder of atoms, can I desribe it with two primitive vectors $a_1=a(1,0)$ , $a_2=a(0,1)$, or do I need to desribe it using only one primitve vector $a_1=a(1,0)$ and basis: $b_1=(0,0)$, $b_2=a(0,1)$? The problem is that the crystal is of limited size in the vertical direction and all definitions of primitve vectors, that I'm familiar with, pertain to unlimited crystals.
To summarize: Can I regard this kind of lattice as a 2D crystal or a 1D crystal with a basis?
Example (Ashcroft) : A Bravais lattice consist of all points with position vectors $\textbf R$ of the form $\textbf R=n_1 \textbf a_1 + n_2 \textbf a_2 + n_3 \textbf a_3$, where $\textbf a_1, \textbf a_2 , \textbf a_3$ are any three vectors not all in the same plane and $n_1,n_2,n_3$ rangle through all integral values.
• If I was dealing with a structure like this where there was the potential for ambiguity, I would avoid saying that it was either a 1D crystal or a 2D crystal. I would say something along the lines of it is a 2D structure with the symmetry of a 1D lattice – By Symmetry Sep 3 '17 at 21:09
For similar real-world examples, MoS$_2$ is a three-atom thick "2D" material, and carbon nanotubes are "1D" structures several atoms wide. And also you have graphene nanoribbons. Of course, if you want to calculate electronic band structure etc. of the carbon allotropes, you start with the graphene 2D band structure and apply appropriate boundary conditions on one of the dimensions.
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2021-04-16 12:33:59
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http://mathhelpforum.com/advanced-algebra/189613-find-eigenvalues-normalised-eigenvectors-3x3-matirx.html
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# Thread: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
1. ## Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
Hey guys,
I'm new to the forum so please be gentle
Done a BSc in Maths (many moons ago), looking to brush up my skills so as to start a masters next year.
I have the matrix A:
6 -2 2
-2 5 0
2 0 7
This is how far I have got:
Calculated the determinant as (took me a while to remember what a determinant was!): -y^3 + 18y^2 - 99y +162 = 0
where y is a scalar
Firstly, does this look correct because I have struggled to factorise it.
Secondly, is there and easy way to write formulas in a post?
I have a couple more questions regarding the Eigenvectors and showing that A is invertible but I figured I would leave those until I have managed to work out the eigenvalues.
Thanks all
Chris
2. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
To put formulae on here, use "LaTeX". see http://www.mathhelpforum.com/math-help/f47/
Now, to solve the equation, the first thing you do is cross your fingers and hope there is an rational number solution! By the "rational root theorem", any rational root of the polynomial equation $a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0$ is of the form $\frac{a}{b}$ where a is an integer that evenly divides the "constant term" $a_0$ and b is an integer that evenly divides the "leading coefficient" $a_n$. Here, [tex]a_n[tex] is 1 so any rational root must be an integer that evenly divides 162. Fortunately, 162 does not have too many factors: $162=2(81)= 2(3^4)$.
So possible roots are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54, \pm 81, \pm 162$. It is tedious, but not impossible, to try each of those.
(Good news! There is a simple integer root. Search for it. Once you have one factor, divide by it to get a quadratic equation you may be able to factor, or use the quadratic formula or complete the square to solve.)
(If there were no rational root, there is a "Cardano's formula", Cubic function - Wikipedia, the free encyclopedia
but it is very complex and tedious to use!)
3. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
Originally Posted by GreenyMcDuff
I have the matrix A:
$\begin{bmatrix} 6 & -2 & 2\\ -2 & 5 & 0\\ 2 & 0 & 7\end{bmatrix}$
This is how far I have got:
Calculated the determinant as (took me a while to remember what a determinant was!): $-y^3 + 18y^2 - 99y +162 = 0$
where y is a scalar
Firstly, does this look correct because I have struggled to factorise it.
Where possible, try to avoid multiplying factors out. That way, you may never have to solve the cubic at all. I assume that you started by evaluating the determinant $\begin{vmatrix} 6-y & -2 & 2\\ -2 & 5-y & 0\\ 2 & 0 & 7-y\end{vmatrix}$. If you expand that along the top row, you get
$(6-y)(5-y)(7-y) - 4(7-y) - 4(5-y).$
Now combine the second and third terms in that expression, to get
${\color{red}(6-y)}(5-y)(7-y) - 8{\color{red}(6-y)}.$
A common factor has magically appeared! So you can take that out, and then you only have a quadratic to solve.
There is no guarantee that a method like that will work. But eigenvalue problems are often rigged so that the answers come out as integers. You can often manipulate a determinant by using row and column operations (remember what those are?!) to simplify the determinant and get a common factor in a row or column. Then you can take that factor out and it gives you one of the eigenvalues.
Originally Posted by GreenyMcDuff
Secondly, is there and easy way to write formulas in a post?
In addition to following HallsofIvy's useful link, a good way to learn TeX is to hold the cursor over any of the formulas on this site, and the TeX input will appear in a popup.
4. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
Thanks for the quick replies guys, appreciate it - it is coming back to me.... slowly.
OK so I managed to find the eigenvalues (3, -6, 9) and their corresponding eigenvectors (dug out my old maths notes).
I also managed to remember what normalising meant: Essentially dividing the eigenvector by its length.
Now I have hit another snag that my notes don't seem to help me with (not sure if I should start a new thread for this but it seems relevant as it leads on from the previous matrix)
Hence, or otherwise, find the limit $\frac{1}{c^n}A^n$ for $n \rightarrow \infinity$ for all $c>0$ (if it exists)
Where:
A is the Matrix (specified earlier)
n is an integer
and I'm not sure what c is (question doesn't specify)
Not sure where to start with this one really :S a nudge in the right direction would be appreciated.
Thanks
Chris
5. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
First, check what you have done so far. (I think that the eigenvalues should be 3, 6, 9, not 3, –6, 9.)
Next, look through those old notes to see that the normalised eigenvectors form the columns of an orthogonal matrix P with the property the $A = PDP^{-1},$ where D is the diagonal matrix whose diagonal entries are the eigenvalues. Then $A^n = PD^nP^{-1}.$
So to find $A^n$ it is sufficient to find $D^n$. But if $D = \begin{bmatrix}3&0&0\\0&6&0\\0&0&9\end{bmatrix}$ then $D^n = \begin{bmatrix}3^n&0&0\\0&6^n&0\\0&0&9^n \end{bmatrix}$. So the question is asking you to look at the limit as n goes to infinity of $\tfrac1{c^n}D^n = \begin{bmatrix}\tfrac{3^n}{c^n}&0&0\\0&\tfrac{6^n} {c^n}&0\\0&0&\tfrac{9^n}{c^n}\end{bmatrix},$ for all positive values of c.
If c<9 then the limit will not exist, because at least one of the fractions in that matrix will go to infinity. If c>9 then the limit is the zero matrix. Finally, the only interesting case occurs when c=9, in which case $\lim_{n\to\infty}D^n = \begin{bmatrix}0&0&0\\0&0&0\\0&0&1\end{bmatrix},$ and $\lim_{n\to\infty}A^n = P\begin{bmatrix}0&0&0\\0&0&0\\0&0&1\end{bmatrix}P^ {-1}.$
6. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
Originally Posted by Opalg
First, check what you have done so far. (I think that the eigenvalues should be 3, 6, 9, not 3, –6, 9.)
Next, look through those old notes to see that the normalised eigenvectors form the columns of an orthogonal matrix P with the property the $A = PDP^{-1},$ where D is the diagonal matrix whose diagonal entries are the eigenvalues. Then $A^n = PD^nP^{-1}.$
So to find $A^n$ it is sufficient to find $D^n$. But if $D = \begin{bmatrix}3&0&0\\0&6&0\\0&0&9\end{bmatrix}$ then $D^n = \begin{bmatrix}3^n&0&0\\0&6^n&0\\0&0&9^n \end{bmatrix}$. So the question is asking you to look at the limit as n goes to infinity of $\tfrac1{c^n}D^n = \begin{bmatrix}\tfrac{3^n}{c^n}&0&0\\0&\tfrac{6^n} {c^n}&0\\0&0&\tfrac{9^n}{c^n}\end{bmatrix},$ for all positive values of c.
If c<9 then the limit will not exist, because at least one of the fractions in that matrix will go to infinity. If c>9 then the limit is the zero matrix. Finally, the only interesting case occurs when c=9, in which case $\lim_{n\to\infty}D^n = \begin{bmatrix}0&0&0\\0&0&0\\0&0&1\end{bmatrix},$ and $\lim_{n\to\infty}A^n = P\begin{bmatrix}0&0&0\\0&0&0\\0&0&1\end{bmatrix}P^ {-1}.$
Thanks Opalg this is a very good explanation.
You were right about the eigenvalues of course.
OK, so I have looked through my notes and am now a little confused.
So my notes tell me that for a real nxn matrix $P^T=P^{-1}$ and so it is trivial to see that from $A=PDP^{-1}$ A is similar to D.
However, I have also found a theorem that states:
for non singular P that is constructed from the corresponding eigenvectors of A the following holds:
$P^{-1}AP=\begin{bmatrix}y_{1}&0&0\\0&y_{2}&0\\0&0&y_{3 }\end{bmatrix}$
Looking at both of these now they seem to say the same thing, is that correct?
Thanks
7. ## Re: Find Eigenvalues and normalised EigenVectors of a 3x3 matirx
Originally Posted by GreenyMcDuff
So my notes tell me that for a real nxn matrix $P^T=P^{-1}$
No, that is only true for an orthogonal matrix P. In fact, the definition of an orthogonal matrix is one that satisfies the condition $P^{\textsc t}=P^{-1}.$ If you form a matrix P by taking its columns to be eigenvectors of a (diagonalisable) matrix, then P will be invertible. If you normalise the eigenvectors then that will make P orthogonal.
Originally Posted by GreenyMcDuff
and so it is trivial to see that from $A=PDP^{-1}$ A is similar to D.
However, I have also found a theorem that states:
for non singular P that is constructed from the corresponding eigenvectors of A the following holds:
$P^{-1}AP=\begin{bmatrix}y_{1}&0&0\\0&y_{2}&0\\0&0&y_{3 }\end{bmatrix}$
Looking at both of these now they seem to say the same thing, is that correct
That is correct. The conditions $A=PDP^{-1}$ and $D=P^{-1}AP$ are equivalent. You just have to remember which way round the $P$ and $P^{-1}$ go in the two formulas. My method is to remember the single formula $AP = PD.$ If you multiply both sides of that on the left by $P^{-1}$ then you get one formula. If do the multiplications on the right then you get the other formula.
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2013-05-22 14:04:15
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https://zbmath.org/?q=an%3A1057.03035
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## Finite-to-one maps.(English)Zbl 1057.03035
From the text: In ZFC we can argue as follows: if there is a finite-to-one map from $$Y$$ onto $$X$$, and $$X$$ is infinite, then $$|Y|\leq|X|\cdot \aleph_0=|X|$$, and $$|Y|\geq|X|$$ by AC, so $$|Y|=|X|$$. So there can be a finite-to-one map from $${\mathcal P}(X)\twoheadrightarrow X$$ only if $$X$$ is finite. (Here finite means “has cardinal in $$\omega$$”, and “finite-to-one” means that the preimage of every singleton is finite.) It is the purpose of this (self-contained) note to show that the result can be proved in ZF even without AC. The proof provided here uses replacement and cannot be conducted in Zermelo set theory. Nor does the proof generalise to show, for an arbitrary strongly inaccessible aleph $$\kappa$$, that if there is a surjection $$f:{\mathcal P}(X)\twoheadrightarrow X$$ where $$|f^{-1''} \{x\}|<\kappa$$ for all $$x\in X$$ then $$|X|<\kappa$$.
Theorem 1. If there is a finite-to-one map $${\mathcal P}(X)\twoheadrightarrow X$$, then $$X$$ is finite.
### MSC:
3e+25 Axiom of choice and related propositions 3e+30 Axiomatics of classical set theory and its fragments
### Keywords:
finite-to-one map
Full Text:
### References:
[1] E. Specker Verallgemeinerte Kontinuumshypothese und Auswahlaxiom , Archiv der Mathematik , vol. 5 (1954), pp. 332–337. · Zbl 0056.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-08-10 14:17:27
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https://math.stackexchange.com/questions/2020858/e-overline-e-the-limit-points-of-a-set-the-limit-points-of-its-closure
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# $E'=\overline E'$? (the limit points of a set = the limit points of its closure?)
Let $E'$ be the set of limit points of $E$, and $\overline E \triangleq E'\cup E$ be its closure, in some metric space. Is it true that $E'=\overline E'$? That $\overline E' \subset E'$ is shown in Limit Points of closure of A is subset of limit points of A. And I think the converse ($E' \subset \overline E'$) is clearly also true. So it appears that we should have $E'=\overline E'$. Did I mess up somewhere?
• Yes, your conclusion is correct. The opposite inclusion is trivial: if every nbhd of $x$ hits $E$, then certainly every nbhd of $x$ hits $\operatorname{cl}E$, since $E\subseteq\operatorname{cl}E$. – Brian M. Scott Nov 19 '16 at 4:04
This is correct since $\overline{E}=E\cup E'$, there is $$\overline{E}'=(E\cup E')'=E'\cup E^{'^{'}}=E'$$ last step is because $E^{'^{'}}\subset E'$.
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2019-09-20 11:40:25
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https://www.cnblogs.com/xuruilong100/p/8596184.html
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【翻译】理解 LSTM 及其图示
Understanding LSTM Networks 中译：【翻译】理解 LSTM 网络
理解 LSTM 及其图示
I'm not better at explaining LSTM, I want to write this down as a way to remember it myself. I think the above blog post written by Christopher Olah is the best LSTM material you would find. Please visit the original link if you want to learn LSTM. (But I did create some nice diagrams.)
Although we don't know how brain functions yet, we have the feeling that it must have a logic unit and a memory unit. We make decisions by reasoning and by experience. So do computers, we have the logic units, CPUs and GPUs and we also have memories.
But when you look at a neural network, it functions like a black box. You feed in some inputs from one side, you receive some outputs from the other side. The decision it makes is mostly based on the current inputs.
I think it's unfair to say that neural network has no memory at all. After all, those learnt weights are some kind of memory of the training data. But this memory is more static. Sometimes we want to remember an input for later use. There are many examples of such a situation, such as the stock market. To make a good investment judgement, we have to at least look at the stock data from a time window.
The naive way to let neural network accept a time series data is connecting several neural networks together. Each of the neural networks handles one time step. Instead of feeding the data at each individual time step, you provide data at all time steps within a window, or a context, to the neural network.
A lot of times, you need to process data that has periodic patterns. As a silly example, suppose you want to predict christmas tree sales. This is a very seasonal thing and likely to peak only once a year. So a good strategy to predict christmas tree sale is looking at the data from exactly a year back. For this kind of problems, you either need to have a big context to include ancient data points, or you have a good memory. You know what data is valuable to remember for later use and what needs to be forgotten when it is useless.
Theoretically the naively connected neural network, so called recurrent neural network, can work. But in practice, it suffers from two problems: vanishing gradient and exploding gradient, which make it unusable.
Then later, LSTM (long short term memory) was invented to solve this issue by explicitly introducing a memory unit, called the cell into the network. This is the diagram of a LSTM building block.
At a first sight, this looks intimidating. Let's ignore the internals, but only look at the inputs and outputs of the unit. The network takes three inputs. $X_t$ is the input of the current time step. $h_{t-1}$ is the output from the previous LSTM unit and $C_{t-1}$ is the “memory” of the previous unit, which I think is the most important input. As for outputs, $h_t$ is the output of the current network. $C_t$ is the memory of the current unit.
Therefore, this single unit makes decision by considering the current input, previous output and previous memory. And it generates a new output and alters its memory.
The way its internal memory $C_t$ changes is pretty similar to piping water through a pipe. Assuming the memory is water, it flows into a pipe. You want to change this memory flow along the way and this change is controlled by two valves.
The first valve is called the forget valve. If you shut it, no old memory will be kept. If you fully open this valve, all old memory will pass through.
The second valve is the new memory valve. New memory will come in through a T shaped joint like above and merge with the old memory. Exactly how much new memory should come in is controlled by the second valve.
On the LSTM diagram, the top “pipe” is the memory pipe. The input is the old memory (a vector). The first cross $\times$ it passes through is the forget valve. It is actually an element-wise multiplication operation. So if you multiply the old memory $C_{t-1}$ with a vector that is close to 0, that means you want to forget most of the old memory. You let the old memory goes through, if your forget valve equals 1.
Then the second operation the memory flow will go through is this + operator. This operator means piece-wise summation. It resembles the T shape joint pipe. New memory and the old memory will merge by this operation. How much new memory should be added to the old memory is controlled by another valve, the $\times$ below the + sign.
After these two operations, you have the old memory $C_{t-1}$ changed to the new memory $C_t$.
Now lets look at the valves. The first one is called the forget valve. It is controlled by a simple one layer neural network. The inputs of the neural network is $h_{t-1}$, the output of the previous LSTM block, $X_t$, the input for the current LSTM block, $C_{t-1}$, the memory of the previous block and finally a bias vector $b_0$. This neural network has a sigmoid function as activation, and it's output vector is the forget valve, which will applied to the old memory $C_{t-1}$ by element-wise multiplication.
Now the second valve is called the new memory valve. Again, it is a one layer simple neural network that takes the same inputs as the forget valve. This valve controls how much the new memory should influence the old memory.
The new memory itself, however is generated by another neural network. It is also a one layer network, but uses tanh as the activation function. The output of this network will element-wise multiple the new memory valve, and add to the old memory to form the new memory.
These two $\times$ signs are the forget valve and the new memory valve.
And finally, we need to generate the output for this LSTM unit. This step has an output valve that is controlled by the new memory, the previous output $h_{t-1}$, the input $X_t$ and a bias vector. This valve controls how much new memory should output to the next LSTM unit.
The above diagram is inspired by Christopher's blog post. But most of the time, you will see a diagram like below. The major difference between the two variations is that the following diagram doesn't treat the memory unit C as an input to the unit. Instead, it treats it as an internal thing “Cell”.
I like the Christopher's diagram, in that it explicitly shows how this memory C gets passed from the previous unit to the next. But in the following image, you can't easily see that $C_{t-1}$ is actually from the previous unit, and $C_t$ is part of the output.
The second reason I don't like the following diagram is that the computation you perform within the unit should be ordered, but you can't see it clearly from the following diagram. For example to calculate the output of this unit, you need to have $C_t$, the new memory ready. Therefore, the first step should be evaluating $C_t$.
The following diagram tries to represent this “delay” or “order” with dash lines and solid lines (there are errors in this picture). Dash lines means the old memory, which is available at the beginning. Some solid lines means the new memory. Operations require the new memory have to wait until $C_t$ is available.
But these two diagrams are essentially the same. Here, I want to use the same symbols and colors of the first diagram to redraw the above diagram:
This is the forget gate (valve) that shuts the old memory:
This is the new memory valve and the new memory:
These are the two valves and the element-wise summation to merge the old memory and the new memory to form $C_t$ (in green, flows back to the big “Cell”):
This is the output valve and output of the LSTM unit:
posted @ 2018-03-18 17:12 xuruilong100 阅读(1703) 评论(0编辑 收藏
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2021-05-13 19:49:36
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http://cpr-mathph.blogspot.com/2012/10/10082821-makoto-katori-et-al.html
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## Complex Brownian Motion Representation of the Dyson Model [PDF]
Makoto Katori, Hideki Tanemura
Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of $h$-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.
View original: http://arxiv.org/abs/1008.2821
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2019-11-12 06:51:55
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https://www.mathportal.org/calculators/matrices-calculators/characteristic-polynomial-calculator.php
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Math Calculators, Lessons and Formulas
It is time to solve your math problem
mathportal.org
# Characteristic Polynomial Calculator
This calculator computes characteristic polynomial of a square matrix. The calculator will show all steps and detailed explanation.
Characteristic polynomial calculator (shows all steps)
show help ↓↓ examples ↓↓
Input matrix
working...
examples
example 1:ex 1:
What is the characteristic polynomial of $A = \left[ \begin{array}{cc} 3 & \frac{5}{2} \\ -2 & 4 \end{array} \right]$.
example 2:ex 2:
Find the characteristic polynomial of the matrix $A = \left[ \begin{array}{cc} 4 & 2 & -5 \\ -3 & 2 & 6 \\ 0 & 0 & 4 \end{array} \right]$.
example 3:ex 3:
Compute characteristic polynomial $A = \left[ \begin{array}{cc} -1 & 2 & 4 & 1 \\ 5 & 3 & 1 & 1 \\ 3 & 7 & 9 & 3 \\ 2 & -1 & 2 & 4 \end{array} \right]$.
## How to input matrix ?
### 1: Input matrix starting from the upper left-hand corner.
Example: To input matrix: $\left[ \begin{array}{cc} -7 & 1/4 \\ -1.3 & -2/5 \end{array} \right]$ type
### 2: You don't need to enter zeros.
Example: To input matrix: $\left[ \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 2/3 \end{array} \right]$ type
### 3: You can copy and paste matrix from excel in 3 steps.
Step 1: Copy matrix from excel Step 2: Select upper right cell Step 3: Press Ctrl+V
### 5: To delete matrix
Select whole matrix and click delete
Search our database of more than 200 calculators
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2022-06-28 21:38:41
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http://www.gradesaver.com/textbooks/math/other-math/discrete-mathematics-with-applications-4th-edition/chapter-4-elementary-number-theory-and-methods-of-proof-exercise-set-4-1-page-161/11
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# Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.1: 11
Let $a=-3$ and $b=0$. Then $a0^{2}=0$.
#### Work Step by Step
When looking for a counterexample to a mathematical statement, try searching systematically. In this case, it would be wise to try $a$ and $b$ both positive, $a$ and $b$ both negative, $a$ negative and $b$ positive, and so on. In this case, it turns out that $a\lt0$ and $b\geq$0 works whenever $|a|\gt|b|$.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2018-04-20 15:03:58
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https://artofproblemsolving.com/wiki/index.php?title=Proof_of_the_Polynomial_Remainder_Theorem&oldid=105546
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# Proof of the Polynomial Remainder Theorem
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
The remainder theorem states when a polynomial denoted as $f(x)$ is divided by $x-a$ for some value of $x$, whether real or unreal, the remainder of $\frac{f(x)}{x-a}=f(a)$ Written below is the proof of the polynomial remainder theorem.
All polynomials can be written in the form $f(x)=d(x)\cdot\q(x)+r(x)$ (Error compiling LaTeX. ! Undefined control sequence.), where $d(x)$ is the divisor of the function/polynomial $f(x)$, $q(x)$ is the quotient. amd $r(x)$ is the remainder. Because the $deg r=0$ or $deg r\less\deg d$ (Error compiling LaTeX. ! Undefined control sequence.) and the $deg d=1$, degrees must be whole numbers, and so $deg r=0$. So to speak, $r(x)$ is a constant. We denote this constant $b$.
Knowing this, we can write
$f(x)=d(x)\cdot\q(x)+b$ (Error compiling LaTeX. ! Undefined control sequence.)
$f(x)=(x-a)\cdot\q(x)+b$ (Error compiling LaTeX. ! Undefined control sequence.)
$f(a)=(a-a)\cdot\q(a)+b$ (Error compiling LaTeX. ! Undefined control sequence.)
$f(a)=b$
We have hereby proven when the quantity $x-a$ is divided into a polynomial $f(x)$ of any degree, the value of $f(a)=b$, where b is the remainder. The remainder must be a constant because $deg r=0$.
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2020-12-02 10:30:18
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https://brilliant.org/problems/3-rectangles-1-triangle/
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# 3 Rectangles + 1 Triangle = ?
Geometry Level 5
Triangle $ABC$ has integer side lengths. Rectangles $BCDE, ACFG, ABHJ$ are constructed so that $CD = AC + AB$, $CF = AB + BC$, and $BH = (AC + BC)^2$. If $[ABHJ] = [BCDE] + [ACFG]$, how many different values can $[ABC]$ have?
Details and assumptions
$[PQRS]$ refers to the area of figure $PQRS$.
×
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2020-06-06 05:07:47
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https://wwwsbobet.biz/jamie-oliver-zhvk/viewtopic.php?tag=advantages-and-disadvantages-of-aerodynamics-in-cars-ffbbe2
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# advantages and disadvantages of aerodynamics in cars
What is the reason for the poor low-speed characteristics of sweptback wings? C. Disadvantages of having a car. What are the different wing planforms? ("naturalWidth"in a&&"naturalHeight"in a))return{};for(var d=0;a=c[d];++d){var e=a.getAttribute("data-pagespeed-url-hash");e&&(! What if a primordial black hole went through the Sun? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. A controller regulates the amount of power that goes from the batteries to the motors. @RisshiJain: I think it helps them to keep the wings outstretched without much effort. Peter, generally while gliding, don't gulls have a drooped wingtip configuration? This is not an example of the work produced by our Essay Writing Service. Advantages and Disadvantages of Hydro Energy, Advantages and Disadvantages of Mesh Network. Thermal system – This system maintains a proper operating temperature range of the engine, electric motor, power electronics, and other components. Weltensegler is actually the name of the company which was founded by Friedrich Wenk. Thus, the environmental impact is also low. They take us … No matter what kind of electric car is in questions, they are a lot better for the environment than conventional cars. ");b!=Array.prototype&&b!=Object.prototype&&(b[c]=a.value)},h="undefined"!=typeof window&&window===this?this:"undefined"!=typeof global&&null!=global?global:this,k=["String","prototype","repeat"],l=0;lb||1342177279>>=1)c+=c;return a};q!=p&&null!=q&&g(h,n,{configurable:!0,writable:!0,value:q});var t=this;function u(b,c){var a=b.split(". The electric motor is used at low speeds. This is because the car’s dependency on fossil fuels, i.e. Note: This essay topic is a bit tricky. Can an Arcane Archer choose to activate arcane shot after it gets deflected? The Car Ferdinand Verbiest’s vehicle Cugnot’s Invention. While pitch and limited roll control was possible with the elevons on the outer wings, no consideration was given to directional control. Thus, electric cars produce more vehicle-moving power while charging than a gas pumping. !b.a.length)for(a+="&ci="+encodeURIComponent(b.a[0]),d=1;d=a.length+e.length&&(a+=e)}b.i&&(e="&rd="+encodeURIComponent(JSON.stringify(B())),131072>=a.length+e.length&&(a+=e),c=!0);C=a;if(c){d=b.h;b=b.j;var f;if(window.XMLHttpRequest)f=new XMLHttpRequest;else if(window.ActiveXObject)try{f=new ActiveXObject("Msxml2.XMLHTTP")}catch(r){try{f=new ActiveXObject("Microsoft.XMLHTTP")}catch(D){}}f&&(f.open("POST",d+(-1==d.indexOf("?")?"? EVs usually have a lower range than conventional cars. In my opinion and one of the advantages is that you can move around freely because you … The tension part on the lower side of the wing will be more stable because it will pull the lower wing contour outward. Would this kind of wing configuration come to help for a glider? Words 491 (1 page) Views 245. Their improvement, more often than not, happens in terms of their speed. Efficiency here means the amount of energy from the fuel source that is converted into actual energy for powering the wheels of a vehicle. As compared to conventional cars, electric cars run a lot smoother. This means that the amount of distance that can be travelled after one full charge is shorter than what can be travelled with a full fuel tank. Category Essay Examples. However, gas-powered vehicles convert just 20%. With gas-powered cars, one can find gas stations nearby or have fuel delivered in cans. For most of us who cannot afford a fancy sports car, it may be hard to believe that there are any disadvantages to owning one, but there are. Use MathJax to format equations. These motors are the direct equivalent of an internal combustion engine in a conventional car. Electric cars can also run on other sources of power, such as solar energy. [CDATA[ BEVs use an external charging outlet to power up the electric battery. Charge port – It allows the vehicle to connect to an external power supply in order to charge the traction battery pack. What could these letters "S" in red circles mean in a biochemical diagram? Self-driving Cars: History, Advantages and Disadvantages. In this context, hybrid means that these cars are powered both by petrol and by electricity. I think it might be better if you could split them in different posts. Once the wing flips into a polyhedral shape, bones and tendons will have trouble to stay in place. In some electric cars, just 6-7 hours of charging are enough to power the vehicle for nearly 200 miles. Power electronics controller – The controller manages the speed of the electric traction motor and the torque it produces. The first “car” was designed by Ferdinand Verbiest in 1672. It is a common trend that Nigerians usually contemplate whether to buy an SUV or a Sedan car when they are about to purchase their first or next vehicle. Advantages. This is because the petrol engine recharges the battery as it gets low, thus extending the range of the car. Firstly, EVs can be powered up at home, so going to a gas pump is not required. This is the difference between PHEVs and HEVS. This is done by regulating the flow of energy from the traction battery to the motor. This makes car repairs less expensive. Re: Advantages (and disadvantages) of looooooonger cars Post Sat May 16, 2020 4:04 pm The rule on the mass distribution was brought in for the first year of the switch to Pirelli to stop anyone getting an advantage by fluking out on the correct optimum mass distribution. There are a number of advantages and disadvantages of cars.Advantages include:Ease of transportationFamilies can go out togetherFast commute from one … Electric cars also provide high torque. @alex: Note the distinctive washout in the outer panels: The Weltensegler gliders had a bell-shaped lift distribution long before the Hortens were around, and as such did not strictly need a vertical. Better Traction The idea was that this would help to adjust the wing shape to flight speed. //]]>. However, cars have evolved to such an extent that there are now two parallel versions of car. The latest ones are electric cars. Popular electric cars can travel nearly 125 miles after a full charge. However, some BEVs can still recharge their batteries via regenerative braking. On the other hand, gas-powered cars can travel 300 miles on a full fuel tank, on an average. If it is purely electric, that means that it does not have any carbon emissions. But if aerodynamics is now so advanced, why are there still cars like the Honda Civic Type R, covered in gaudy adornments, when others such as … This is because the car in this situation is like any other electrical device. Who first called natural satellites "moons"? Having an anhedral in the main wing will cause it to be very laterally unstable. @Peter Kampf: That's exactly what I was thinking. 2 weeks of free Essay About Advantages And Disadvantages Of Cars revisions. What are the advantages or disadvantages of a mid wing design? It only takes a minute to sign up. The Advantages and Disadvantages of Motor Car. ... Browse other questions tagged aircraft-design aerodynamics fixed-wing or ask your own question. Now the second case, where the electric car is a hybrid. Electric cars also provide high torque. An internal computer controls both the motors. Is there a general solution to the problem of "sudden unexpected bursts of errors" in software? These are fully electric cars. This means that they run exclusively on electricity. “Hybrid” implies that they are a combination of two or more things. For many Americans it is hard to imagine not owning a car… Among these are: an increase in the rate of environmental pollution, rising costs of owning a car, uncontrolled movement of people from rural to urban areas, total dependence on cars for transport, in addition to increase in fuel costs. Even in that case, they do not have emissions. What does the phrase, a person with “a pair of khaki pants inside a Manila envelope” mean? Delete column from a dataset in mathematica. Making statements based on opinion; back them up with references or personal experience. There are various types of electric cars. Alexander Lippisch was one of his employees. ":"&")+"url="+encodeURIComponent(b)),f.setRequestHeader("Content-Type","application/x-www-form-urlencoded"),f.send(a))}}}function B(){var b={},c;c=document.getElementsByTagName("IMG");if(!c.length)return{};var a=c[0];if(! The decision to buy a hybrid car should not be taken lightly, as they have good and bad qualities just like anything else. Cars have provided reliable transportation for more than 100 years, revolutionizing travel in the United States. In today's world of hurrying the car becomes to be the most popular way of moving from one place to another. As is expected, it has its own advantages and disadvantages. These are some of the things you should know before you make up your mind: Advantages of SUV: 1) Better ground clearance and 4 … Advantages and Disadvantages of Modern Cars. How do we know that voltmeters are accurate? On the other hand, electric motors make almost no noise. Advantages & Disadvantages of Passenger Airbags CAR OWNER TIPS Dec 3rd, 2013 Although airbags are essential for the driver of the vehicle since it prevents them from hitting the steering wheel during an accident, some experts believe that a seatbelt is enough for a front-seat passenger. $\begingroup$ @VilleNiemi, no, they don't. Aviation Stack Exchange is a question and answer site for aircraft pilots, mechanics, and enthusiasts. Dirty buffer pages after issuing CHECKPOINT. 2323 words (9 pages) Essay. Some of the famous cars that come with AWD system are Nissan GT-R, Audi R8, BMW M5, Porsche 912 Spyder, Tesla model You and Toyota RAV4. Battery – The auxiliary battery provides electricity to vehicle accessories. Is there any difference between them and gull wings? The car can either be purely electric or hybrid. They run on fossil fuels, such as diesel or petrol. Talking about Pros and Cons of using Nitrogen in vehicle tyres, here are some Advantages and Disadvantages of filling Nitrogen. They also run on both electricity and petrol. Advantages And Disadvantages Of Buying A New Car May 14, 2019 February 21, 2020 The Mechanic Doctor If you’re thinking about buying a car or are already in the process of shopping for one, you’ve likely encountered a variety of options already. The configuration cannot be recommended for a glider. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. They are also known as Extended-Range Electric Vehicles (EREVs). Bad runways don't mean your wing span would be particularly limited. Your photo shows the 1922 version, called "Baden-Baden Stolz", which was a follow-on to the original and more ambitious version of 1921. They are: Their name is quite self-explanatory. Is there any difference between them and gull wings? Disadvantages: Usually when you buy a new car with Sunroof, you may try it out once or twice, but then about a week later you forget that it’s even there. Other big car manufacturers like Nissan, BMW, Ferrari, Porsche and Mercedes also have stepped into making AWD vehicles for both supercars and city cars. What are the advantages and disadvantages of having landing gear doors? On-board diagnostics tells you about the problems in your car long before it stalls on the road and leaves you stranded. It went from bicycles to cars, from ships to planes. (function(){for(var g="function"==typeof Object.defineProperties?Object.defineProperty:function(b,c,a){if(a.get||a.set)throw new TypeError("ES3 does not support getters and setters. This is needed to run vehicle accessories and recharge the auxiliary battery. An electric car is a vehicle that is propelled by one or more electric motors. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. What are the advantages and disadvantages of autogyro aircraft? To learn more, see our tips on writing great answers. The use particularly in cities with hot climate is minimal. These include voltage, current, temperature, and state of charge. The electric motor helps to slow the vehicle and uses some of the energy normally converted to heat by the brakes. In a car, this means that when the accelerator is pressed, power is … So in this article, we are going to look at the different types of hybrid cars and their advantages and disadvantages. MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. The reason for their wing shape is structural, however, not aerodynamic. What's the advantage of the F4U Corsair's gull wing design? Just like gas powered cars, you have to recharge your electric vehicle periodically in … What advantages come with the wing on the Boeing Bird of Prey? It is much better to have a straight inner wing and dihedraled outer wings, as explained in this answer. Electric traction motor – It drives the wheels by using energy from the traction battery. Advantages of Aerodynamics for Modern Performance Cars Advantages of Aerodynamics for Modern Performance Cars Thesis: Aerodynamics, the effects of air flow over surfaces, is an indispensable element for modern cars' efficiencies, both economics and performance related, because aerodynamics helps to decrease the dragforce, makes cars more stabilized and gives a chance of … However, this airplane never flew, which probably saved the life of its pilot. Wheels were invented thousands of years ago. But some sort of drag device on the outer wings would had been handy to avoid the spiral dive. Advantages of Electric Cars Quiet and Efficient. In the USA, we use cars to go almost everywhere. There can be 2 situations here. "),d=t;a[0]in d||!d.execScript||d.execScript("var "+a[0]);for(var e;a.length&&(e=a.shift());)a.length||void 0===c?d[e]?d=d[e]:d=d[e]={}:d[e]=c};function v(b){var c=b.length;if(0
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2021-03-01 12:25:53
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https://edoc.unibas.ch/40354/
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# Interplay of weak interactions in the atom-by-atom condensation of xenon within quantum boxes
Nowakowska, Sylwia and Waeckerlin, Aneliia and Kawai, Shigeki and Ivas, Toni and Nowakowski, Jan and Fatayer, Shadi and Waeckerlin, Christian and Nijs, Thomas and Meyer, Ernst and Bjork, Jonas and Stohr, Meike and Gade, Lutz H. and Jung, Thomas A.. (2015) Interplay of weak interactions in the atom-by-atom condensation of xenon within quantum boxes. Nature communications, 6. p. 6071.
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Official URL: http://edoc.unibas.ch/40354/
Condensation processes are of key importance in nature and play a fundamental role in chemistry and physics. Owing to size effects at the nanoscale, it is conceptually desired to experimentally probe the dependence of condensate structure on the number of constituents one by one. Here we present an approach to study a condensation process atom-by-atom with the scanning tunnelling microscope, which provides a direct real-space access with atomic precision to the aggregates formed in atomically defined quantum boxes. Our analysis reveals the subtle interplay of competing directional and nondirectional interactions in the emergence of structure and provides unprecedented input for the structural comparison with quantum mechanical models. This approach focuses on-but is not limited to-the model case of xenon condensation and goes significantly beyond the well-established statistical size analysis of clusters in atomic or molecular beams by mass spectrometry.
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2020-08-05 19:58:26
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https://www.missy-naturfotos.de/custom_info/d_6086386b8f9ec300a004777a.html
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Welcome to the company ! we have many years of professional experience !
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# deep well drilling waste liquid harmless disposal technology ltd
Established in 2001, Puyang Zhong Yuan Restar Petroleum Equipment Co.,Ltd, “RSD” for short, is Henan’s high-tech enterprise with intellectual property advantages and independent legal person qualification. With registered capital of RMB 50 million, the Company has two subsidiaries-Henan Restar Separation Equipment Technology Co., Ltd We are mainly specialized in R&D, production and service of various intelligent separation and control systems in oil&gas drilling,engineering environmental protection and mining industries.We always take the lead in Chinese market shares of drilling fluid shale shaker for many years. Our products have been exported more than 20 countries and always extensively praised by customers. We are Class I network supplier of Sinopec,CNPC and CNOOC and registered supplier of ONGC, OIL India,KOC. High quality and international standard products make us gain many Large-scale drilling fluids recycling systems for Saudi Aramco and Gazprom projects.
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deep well drilling waste liquid harmless disposal technology ltd
31/8/2010, · ,acid,. * When diluting, never add ,water, to ,acid,, always add ,acid, to ,water,. Doing so is sometimes highly exothermic, and should be done a bit at a time, wearing safety specs. If done the wrong way the result can even progress to explosion sometimes. Exploding boiling ,acid, …
sulfuric acid and hydrogen peroxide reaction
Sulfuric ,acid, (American / IUPAC spelling) or sulphuric ,acid, (traditional / British spelling), also known as oil of vitriol, is a mineral ,acid, composed of the elements sulfur, oxygen and hydrogen, with molecular formula H 2 SO 4.It is a colourless and viscous liquid that is miscible with ,water, at all concentrations.
Training course on disease diagnosis and prevention ...
Increases in temperature-fish require more but ,there, is less in the ,water, ... (eg ,chromic acid,) ... ,there, has been considerable recent development of fish farming in warm sea ,water,. From that ,reason,, aquaculturists must be carefull for diagnosis and control of diseases.
Wikipedia:Reference desk/Archives/Science/2010 May 10
For Question 2) ,Chromic acid, is not a compound which readily exists either isolated or in ,water,. Stable ",chromic acid," generally only works as a mixture of chromate (or dichromate) salts and a strong ,acid, such as hydrochloric or sulfuric; and the strength of these ,chromic acid, solutions probably depends on the strength of the co-,acid, present.
Biogas Production Essay - 5900 Words | Internet Public Library
Chromic acid is a more commonly used reagent for the oxidation of alcohols, it is a suspected carcinogen and generates hazardous waste. In green chemistry, innocuous chemical such as hypochlorous acid, HOCl will be used in oxidation, as to reduce the harmful waste.
The Color of Acid – Ryan Anderson
There, is one ,acid, that is brightly colored: ,chromic acid, (H2CrO4). Before reacting with something else, it’s usually a red or orange color. After reacting, “spent” ,chromic acid, is a dark green. Oh, this ,acid, is also carcinogenic (spent or fresh).
How to identify what is this? All we know it's some acid ...
If you have good ,reason, to suspect it's phosphoric ,acid,, ... The precipitate will not dissolve ,in acid,. ,There, is a molybdate color test for phosphate which you can google easily. level 1. Comment deleted by user 2 years ago. level 2. 8 points · 2 years ago · edited 2 years ago. EHS chemist here- the hazcat kit is a ridiculous ,waste, of time.
How to identify what is this? All we know it's some acid ...
If you have good ,reason, to suspect it's phosphoric ,acid,, ... The precipitate will not dissolve ,in acid,. ,There, is a molybdate color test for phosphate which you can google easily. level 1. Comment deleted by user 2 years ago. level 2. 8 points · 2 years ago · edited 2 years ago. EHS chemist here- the hazcat kit is a ridiculous ,waste, of time.
Newest 'aqueous-solution' Questions - Chemistry Stack Exchange
when dealing with ,waste water, and the total values of P, N, ... an ,acid, becomes more concentrated because more ions form pairs. However, does that affect the speed of reaction of the ,acid, as ,there, is a smaller ... $,chromic acid, solution, and I got its$\mathrm{pH}$around -1.7. Yet,$\mathrm{pH}$of$\pu{0.4 M}\$ solution was around -0.6 ...
(PDF) DuraChrome Hard Chromium Plating | Zafar Iqbal ...
Chromium Trioxide (,chromic acid,) is a reddish-brown, hygroscopic (,water, absorbing) chemical, easily soluble in ,water, to give a solution containing both H2CrO4 and H2Cr207 as mentioned earlier. Many manufacturers, now aware of the effect of even small amounts of catalyst ,acid, radicals, furnish a pure grade of ,chromic acid, especially suited for chromium plating.
Propionic Acid: Toxicity Uses & Safety - Science Class ...
The reason, it's used as preservative is because it has the ability to prevent the growth of ... then it should be neutralized with alkaline soap and ,water,. If ,there, is a large ... ,Chromic Acid, ...
Batteries: Electricity though chemical reactions ...
Due to the liquid nature of wet cells, insulator sheets are used to separate the anode and the cathode. Types of wet cells include Daniell cells, Leclanche cells (originally used in dry cells), Bunsen cells, Weston cells, ,Chromic acid, cells, and Grove cells. The lead-,acid, cells in automobile batteries are wet cells.
The Color of Acid – Ryan Anderson
There, is one ,acid, that is brightly colored: ,chromic acid, (H2CrO4). Before reacting with something else, it’s usually a red or orange color. After reacting, “spent” ,chromic acid, is a dark green. Oh, this ,acid, is also carcinogenic (spent or fresh).
Propionic Acid: Toxicity Uses & Safety - Science Class ...
The reason, it's used as preservative is because it has the ability to prevent the growth of ... then it should be neutralized with alkaline soap and ,water,. If ,there, is a large ... ,Chromic Acid, ...
(PDF) DuraChrome Hard Chromium Plating | Zafar Iqbal ...
Chromium Trioxide (,chromic acid,) is a reddish-brown, hygroscopic (,water, absorbing) chemical, easily soluble in ,water, to give a solution containing both H2CrO4 and H2Cr207 as mentioned earlier. Many manufacturers, now aware of the effect of even small amounts of catalyst ,acid, radicals, furnish a pure grade of ,chromic acid, especially suited for chromium plating.
Adsorption of chromium on activated carbon
Chromic acid, is a fairly strong ,acid, in its primary dissociation that does not exist except in solu tion. It shows a marked tendency in very concentrated solutions to form polynuclear species (polyacids) through the elimination of ,water,.
More Experiments in the Penny Lab
O– is a good nucleophile”, or ,why, HCl reacts with NaOH, we respond, “because it is an ,acid,–base reaction”. ,Why, do we do this? For one ,reason, the actual explanations of ,why, specific chemical systems behave as they do are often very complex and require lengthy arguments involving ki-netic and thermodynamic concepts. Another ,reason, is that
How to test a drink for methanol - Quora
Any significant amount of methyl alcohol present will release the unpleasant odour of formic acid, easily distinguishable from the odour of ethanol and its fruity ester or its oxidation to acetic acid. *Chromic acid is considered to be the most destructive of acids and is used to clean heavily soiled glassware when other treatments fail!
Chromium Emissions From Chromium Electroplating and ...
Another advantage of ,chromic acid, anodic coatings is their very high resistance to salt spray corrosion as compared to sulfuric ,acid, coatings of the same thickness.34 ,There, are four primary differences between the equipment used for ,chromic acid, anodizing and that used for chromium electroplating: (l) rectifiers must be fitted with a rheostat or other control mechanism to permit voltage ...
Adsorption of chromium on activated carbon
Chromic acid, is a fairly strong ,acid, in its primary dissociation that does not exist except in solu tion. It shows a marked tendency in very concentrated solutions to form polynuclear species (polyacids) through the elimination of ,water,.
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2021-08-04 08:23:30
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http://tex.stackexchange.com/questions?page=1760&sort=newest&pagesize=50
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All Questions
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2016-05-28 16:09:38
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https://www.physicsforums.com/threads/on-newtons-first-and-second-laws.961003/
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# On Newton's first and second laws
• A
## Main Question or Discussion Point
I'm reading Scheck's book about Mechanics and it says that Newton's first law is not redundant as it defines what an inertial system is. My problem is that we could say the same about Newton's second law. Indeed, Newton's second law is only valid, in general, for inertial systems, so it also defines them.
Therefore, I think Newton's first law doesn't just define what an inertial system is, but, more importantly, states they exist (which is not obvious).
What do you think?
Related Classical Physics News on Phys.org
I'm reading Scheck's book about Mechanics and it says that Newton's first law is not redundant as it defines what an inertial system is. My problem is that we could say the same about Newton's second law. Indeed, Newton's second law is only valid, in general, for inertial systems, so it also defines them.
Without the third law both the first and the second law would also be valid in non-inertial frames.
Without the third law both the first and the second law would also be valid in non-inertial frames.
According to Scheck's definition, inertial frames are frames with respect to which Newton's first law has analytic form $\ddot{\pmb{r}}(t)=0$.
According to Scheck's definition, inertial frames are frames with respect to which Newton's first law has analytic form $\ddot{\pmb{r}}(t)=0$.
You can also have $\ddot{\pmb{r}}(t)=0$ in non-inertial frames. That is obviously not sufficient.
You can also have $\ddot{\pmb{r}}(t)=0$ in non-inertial frames. That is obviously not sufficient.
Any example?
robphy
Homework Helper
Gold Member
For Newton's First Law, you need that it travels with constant velocity when there are no unbalanced forces.
Suppose you are in a rotating frame of reference (e.g. https://archive.org/details/frames_of_reference @17m05s )
and that a particle [somehow] moves along a straight line in that frame.
For the particle to move with constant velocity [e.g., zero] in the rotating frame, there must be unbalanced forces. (see @20m00s above )
(If there are no unbalanced forces, the particle would move with constant velocity in an inertial frame...
but not constant velocity [in a straight line] in the rotating frame... as in the video above.)
Marion and Thornton briefly discuss different ways to interpret Newton's Laws. Look here on (book page, not pdf page) number 49-51.
bhobba
Mentor
I'm reading Scheck's book about Mechanics and it says that Newton's first law is not redundant as it defines what an inertial system is.
True - but there are equivalent definitions that do not use it. For example IMHO a better definition is found in Landau - Mechanics. An inertial frame is one where the laws of physics are the same at all points in space, directions, and instants of time. It can be shown that any two inertial frames are moving at constant velocity relative to each other (hint on proof - divide time into the sum of infinitesimal times - in each infinitesimal instant the Taylor expansion has higher terms you can neglect hence the transformation is linear). Sum them up and you get a linear transformation from which constant velocity follows by looking at the transformation of a fixed point - say the origin. What it does not say is any frame travelling at constant velocity to an inertial frame is inertial - this is the actual assumption - the rest is just definitions and some math as explained before..
Now getting back to the original statement, place a particle at the origin and have nothing else in the frame. If it shoots off in any direction then the laws of physics are not the same in all directions. If you have a particle going at constant velocity then you can go to an inertial frame where it is at rest and from before must remain at rest. Hence, assuming, not acting on by a force means nothing is affecting the particle you have the first law - but this time from symmetry. You can make this argument more rigorous using the Principle Of Least Action as explained in Landau's book.
Why I think it is better, is its based on symmetry, the importance of which is essential to relativity:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf
My problem is that we could say the same about Newton's second law. Indeed, Newton's second law is only valid, in general, for inertial systems, so it also defines them.Therefore, I think Newton's first law doesn't just define what an inertial system is, but, more importantly, states they exist (which is not obvious).
What do you think?
The symmetry argument I gave before make the existence of inertial frames very intuitive - but of course physics is an experimental science and it is an experimental fact internal frames with the rather obvious symmetry proprieties exist to a high degree of accuracy - especially in deep space. The second law strictly speaking is not a law - but the definition of what a force is. The assumption is the frame is not inertial if things start accelerating of their own accord - we naturally assume something must be making it do that. That something could be something in or outside the frame; and the frame inertial if that something was not present. So overall the frame can still be inertial but containing or influenced by something else. The definition of a force is a measure of that something. Why that definition - why not simply say acceleration or mass squared times acceleration - I am sure you can think of many others. The answer is its physical content - it says in analysing classical mechanics problems get thee to the forces as defined by the second law.
The third law is a statement whose validity is determined by experiment - it may be true or false. I assume you know it is equivilant to momentum conservation. For an advanced view of it look into Noethers Theorem:
http://applet-magic.com/noetherth.htm
Now what is the physical basis of Noether's Theorem? - the answer is Quantum Mechanics - but I will let you think about that - don't worry if you do not see it - you can do a post and me or someone else can explain it - but thinking about it will help develop your understanding.
As a further thing to think about, what is the actual basis of classical mechanics? The answer again is QM. Once you understand this it makes you laugh a bit when you see people say; are there any manifestations of QM here in the classical world. Well everything is actually quantum so there is no classical world separate from the quantum world - just a region where QM is very well approximated by classical laws.
Thanks
Bill
Last edited:
Thank you all for your answers. As for QM I don't think I'll ever learn about it as I'm learning Mechanics to better understand underactuated robotics and locomotion in particular.
robphy
Homework Helper
Gold Member
stevendaryl
Staff Emeritus
This is blatant historical revisionism, but it is possible to reformulate Newton's laws into an equivalent 4D spacetime theory where his laws are true in every coordinate system, not just inertial coordinate systems.
The key is to make the seemingly pointless distinction between time as a coordinate and elapsed time as a scalar parameter. We let $x^0$ be the time coordinate, and let $s$ be the path parameter for a moving test particle, and we impose the restriction: $\frac{d^2 x^0}{ds^2} = 0$, so that the time coordinate for a moving test particle increases smoothly.
$m \frac{d^2 x^j}{dt^2} = F^j$
This is three equations, as $j$ ranges over the three spatial dimensions. Now, we convert these to 4 equivalent equations:
$m \frac{d^2 x^j}{ds^2} = F^j$
$m \frac{d^2 t}{ds^2} = 0$
(Well, the 3D and 4D equations are equivalent if we assume the "initial" condition: $\frac{dt}{ds} = 1$)
This makes it into a 4-D equation:
$m \frac{d^2 x^\alpha}{ds^2} = F^\alpha$
where we let $x^0 = t$ and $F^0 = 0$
Now, we transform to a noninertial, non-Cartesian coordinate system $x^\mu$. Letting $L^\mu_\alpha = \frac{\partial x^\mu}{\partial x^\alpha}$ and $\overline{L}^\alpha_\mu = \frac{\partial x^\alpha}{\partial x^\mu}$, we can rewrite our equations in terms of the new coordinates $x^\mu$:
$\frac{dx^\alpha}{ds} = \overline{L}^\alpha_\mu \frac{dx^\mu}{ds}$
$\frac{d^2 x^\alpha}{ds^2} = \frac{\partial\overline{L}^\alpha_\mu}{\partial x^\nu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} +\overline{L}^\alpha_\mu \frac{d^2 x^\mu}{ds^2}$
If we just define the components of the force in the new coordinate system to be: $F^\mu = L^\mu_\alpha F^\alpha$, then we have:
$F^\mu = L^\mu_\alpha F^\alpha = m L^\mu_\alpha \frac{d^2 x^\alpha}{ds^2} = m [ L^\mu_\alpha \frac{\partial\overline{L}^\alpha_{\mu'}}{\partial x^\nu} \frac{dx^{\mu'}}{ds} \frac{dx^\nu}{ds} + L^\mu_\alpha \overline{L}^\alpha_{\mu'} \frac{d^2 x^{\mu'}}{ds^2}]$
Since $L$ and $\overline{L}$ are inverses (viewed as matrices), we have:
$F^\mu = m [ \frac{d^2 x^\mu}{ds^2} + \Gamma^\mu_{\nu \mu'} \frac{dx^\nu}{ds} \frac{dx^{\mu'}}{ds} ]$
where $\Gamma^\mu_{\nu \mu'} \equiv L^\mu_\alpha \frac{\partial\overline{L}^\alpha_{\mu'}}{\partial x^\nu}$
This can then be considered a vector equation:
$F = m \frac{D}{Ds} V$
where $V$ is the vector with components $V^\mu$ and $\frac{D}{Ds}$ is the path derivative defined via the rules:
$\frac{D}{Ds} V =$ that vector $A$ such that $A^\mu = \frac{dV^\mu}{ds} + \Gamma^\mu_{\nu \mu'} V^\nu V^{\mu'}$
This can be understood as the ordinary derivative of $V$ if we introduce basis vectors $e_\mu$ with the covariant derivatives:
$\frac{\partial e_{\mu'}}{\partial x^\nu} \equiv \Gamma^\mu_{\nu \mu'} e_\mu$
Then we have, simply:
$F = m \frac{dV}{ds}$
valid in any coordinate system.
Last edited:
haushofer
Any example?
You sitting on your computer in the co-rotating rest frame of Earth.
You sitting on your computer in the co-rotating rest frame of Earth.
That's because of gravity and friction. If you remove all forces, my acceleration won't be 0 in that frame.
That's because of gravity and friction. If you remove all forces, my acceleration won't be 0 in that frame.
First and second law say that there is no acceleration without a net force. That means if your acceleration isn't 0 than you didn't remove all forces.
First and second law say that there is no acceleration without a net force. That means if your acceleration isn't 0 than you didn't remove all forces.
... or your reference frame is not inertial, as in this case.
stevendaryl
Staff Emeritus
Some people take $F^j = \frac{d^2 x^j}{dt^2}$ as the definition of force in Newtonian physics. If you say that, then by definition, a rotating frame has forces that are not present in an inertial frame: centrifugal and Coriolis forces. On the other hand, if centrifugal force is a force, then Newton's third law is violated, because there is no equal and opposite force for the centrifugal force.
As I suggested in an earlier post, you can reformulate Newton's laws so that
$\mathbf{F} = m \dot{\mathbf{V}}$
is true as a vector equation in any frame.
... or your reference frame is not inertial, as in this case.
How do you know if the frame is inertial or not?
On the other hand, if centrifugal force is a force, then Newton's third law is violated, because there is no equal and opposite force for the centrifugal force.
Would there enything else be violated except the third law?
As I suggested in an earlier post, you can reformulate Newton's laws so that
$\mathbf{F} = m \dot{\mathbf{V}}$
is true as a vector equation in any frame.
All you need to do is deleting the first sentence in the third law. That's what Newton did in his personal copy of the principia:
stevendaryl
Staff Emeritus
No, but in my opinion, it's Newton's third law that gives $F=ma$ empirical content, and not just a tautology.
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2019-12-15 01:05:48
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https://physics.stackexchange.com/questions/604587/is-the-electrical-conductivity-of-a-perfectly-periodic-crystal-really-infinite/608453
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Is the electrical conductivity of a perfectly periodic crystal really infinite?
In a perfectly periodic solid and at absolute zero, where the electrons do not suffer any scattering, the electrical conductivity is infinite. However, pure periodic solids also display the curious phenomenon of Bloch oscillation where the current oscillates with the time when a dc voltage is applied. This has been observed in experiments. But how can the infinite conductivity is compatible with the phenomenon of Bloch oscillation? Bloch oscillation must mean that the current response is finite for a finite voltage.
• Perfectly periodic solids can be insulators, so presumably you're referring specifically to metals with partially filled valence bands, right? – J. Murray Jan 1 at 21:21
• Yes, with free electrons – mithusengupta123 Jan 1 at 21:31
The assumption behind the question is that materials have finite conductivity due to electron scattering from impurities and the imperfections of the crystal lattice. This is not quite the case.
Firstly, the scattering from impurities and crystal imperfections is coherent scattering, so, in principle, it doesn't cause any dissipation, unless combined with an energy dissipation mechanism, such as photons or Coulomb scattering. Conductivity in disordered materials is indeed suppressed, due to the phenomenon of Anderson localization, where extended Bloch states become localized.
Bloch states are the electron eigenstates in a perfect crystal lattice (neglecting electron-electron interactions). Every Bloch state carries current (just like a plane wave), but since the states carrying current in different directions are filled to the same energy, the net current is zero. Driving a current through a crystal can be thought of as changing the balance of the right-/left- carrying states, so that their currents do not compensate anymore. This means exciting some electrons to higher energies. (Note that this is why isolators cannot conduct, unless the electrons are excited across the gap.) These excited electrons can lose their energy via interactions with phonons, other electrons, etc., which is the reason for the resistance/finite conductance.
Finally, Bloch oscillations have to do with the periodicity of the electron dispersion in respect to quasi-momentum. Considering for simplicity 1-dimensional case, the dispersion relation for free electrons is $$E_k = \frac{\hbar^2k^2}{2m},$$ which means that the electron velocity is $$v_k=\frac{1}{\hbar}\partial_k E_k = \frac{\hbar k}{m}.$$ In the same time the electron momentum can be considered to be roughly governed by the Newton's second law (actually it follows from the Heisenberg equations of motion): $$\hbar \dot{k} = -eE - \frac{k}{\tau},$$ where the second term accounts for all kinds of energy dissipation processes. Without dissipation momentum grows with time, which results in increasing velocity and conductance.
In a crystal the dispersion relation is different. For simplicity we can take: $$E_k=-\frac{\Delta}{2}\cos(ka)\longrightarrow v_k=\frac{\Delta}{2}\sin(ka),$$ whereas the momentum obeys the same equation as before. Without dissipation we obtain velocity (and hence the current) which oscillates with time.
Whether we can obtain Bloch oscillations in practice (and the related negative differential conductance, which is behind many practical applications) depends on how strong is the dissipation in comparison to the size of the band. It is quite difficult in bulk materials, but easily achievable in artificially engineered periodic structures, as was first demonstrated by Leo Esaki and Ray Tsu, at IBM (with Leo Esaki earning a Nobel prize for this and related work).
I believe that you are correct; there would be Bloch oscillations and the dc conductivity would arguably be zero (or undefined). However, I don't think that the AC conductivity would be zero. That said, it's impossible to actually test this situation...
Conductivity is infinite in the sense that if you apply an electric field for a short time and, as a result, displace the electron distribution in k-space from its equilibrium distribution, and then switch the field off, there will be no mechanism to bring the electrons back to equilibrium distribution. Therefore, you will have a current forever without any energy supply that I think could be considered as infinite conductivity because you will have zero field but finite current.
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2021-03-04 03:58:41
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http://www.longluo.me/blog/2022/02/10/Leetcode-valid-sudoku/
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By Long Luo
Here shows 2 Approaches to slove this problem: HashSet and Array.
# HashSet
We can use a HashSet to record the number of occurrences of each number in each row, each column and each sub-box.
Traverse the Sudoku once, update the count in the HashMap during the traversal process, and determine whether the Sudoku board could be valid.
This is version 1.0 code.
In fact, we can only traversal once. The index of each sub-box is $3 \times (i / 3) + j / 3$, so we can write better code.
## Analysis
• Time Complexity: $O(1)$.
• Space Complexity: $O(1)$.
# Array
Since numbers in Sudoku range from $1$ to $9$, we can use array instead of the HashMap for counting.
We create a 2D Array, the rows and columns which record the number of occurrences of each number in each row and column of Sudoku, and create a 3D Array subboxes to record the number of occurrences of each number in each sub-box.
If the count is greater than $1$, the Sudoku is not valid.
## Analysis
• Time Complexity: $O(1)$.
• Space Complexity: $O(1)$.
All suggestions are welcome.
If you have any query or suggestion please comment below.
Please upvote👍 if you like💗 it. Thank you:-)
Explore More Leetcode Solutions. 😉😃💗
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2023-03-29 10:52:08
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http://mathhelpforum.com/advanced-algebra/117497-cyclic-isomorphic.html
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your map, $\Lambda,$ is injective. so $G$ can be considered as a subgroup of $\text{Sym}(G) \cong S_4.$ now what are the subgroups of order $4$ in $S_4$?
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2017-07-23 09:27:58
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https://www.nature.com/articles/s41598-019-43916-x?error=cookies_not_supported&code=46c4544c-5429-44c8-a70e-0e689c727f4d
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Article | Open | Published:
# Change of the relative line strengths due to the resonance induced population transfer between Fe XVII and FeXVI ions
## Abstract
We present a detailed study to resolve the discrepancy between the existing theoretically estimated oscillator strengths and the recently observed result from the X-ray free electron laser (XFEL) experiment performed at the Linac Coherent Light Source (LCLS) for the intensity ratio between two of the strongest emission lines from Ne-like Fe XVII (Fe16+) ion. By including the dynamic resonance induced population transfer due to autoionization between the coexisting Fe XVII and Fe XVI (Fe15+) ions in the XFEL experiment, we are able to successfully resolve this difference in theory and experiment. Further experimental works are suggested for a more detailed understanding of the dynamic resonance processes for ions.
## Introduction
The x-ray emission lines of the Ne-like Fe16+ ion have been observed in a variety of astrophysical objects, including the Sun, stellar coronae, elliptical galaxies, and supernova remnants1,2,3,4. Two of the most intense lines of the Ne-like Fe16+ ion are the 2p53d 1P1 → 2p6 1S0 dipole emission line at 15.01 Å (3C) and the 2p53d 3D1 → 2p6 1S0 inter-combination line at 15.26 Å (3D). The wavelength separation of these two lines is large enough to be resolved by spectrometers with moderate resolving power, but, is small enough so that errors in spectrometer response are relatively small1,2,3,4. However, the diagnostic utility of these two lines has been limited by the fact that although extensive studies have been carried out, discrepancies between the theoretical estimates and the measurements from astrophysical and laboratory sources persist5,6,7.
In a recent benchmark experiment performed at the Linac Coherent Light Source (LCLS)5, the highly charged Fe ions were first generated in an electron beam ion trap (EBIT) and then photo-excited by the X-ray free electron laser (XFEL). This experiment was designed to allow a direct comparison of experimental and theoretical results, excluding the effects of electron collisions. The measured weighted 3C/3D ratio was 2.61 ± 0.23, which is significantly lower than the most elaborated theoretical values at 3.4 or higher5,6,7,8,9,10. It was concluded that the discrepancies are due to the inaccurate atomic wave functions5,11. This conclusion appears to be supported by a recent configuration interaction calculation with some fine-tuning parameters12. However, it was argued later that the calculated ratio in ref.12 is unreliable due to a limited and unbalanced treatment of electron correlations13. Alternatively, it was proposed that other physical processes beyond those included in the atomic structure calculation of an isolated atomic system, such as the one due to the ultra-high intensities of XFEL with ultra-short pulse period, may be responsible for this discrepancy10,14,15. By scrutinizing the experimental conditions at the LCLS5,11,16,17,18, the intensities of the XFEL pulses are not sufficiently high to support this proposed interpretation10,14,15 and it is not likely that the discrepancy between theory and experiment could be attributed to the high order nonlinear effect. Thus, it is worth exploring further other possibilities to explain the experiment5.
Our investigation starts with a critical assessment of the theoretical intensity ratio based on an atomic structure calculation of isolated ion with a revised multi-configuration Dirac-Fock (MCDF) approach, where the quasi-complete basis scheme is adopted to optimize the atomic orbitals (AOs) using the GRASP-JT version19,20 based on the earlier GRASP2K codes21,22. In this way, the convergence of the atomic structures can be examined step by step with limited computational efforts. This approach was developed recently to study the forbidden transitions in O+ ion with its result within the overlap range of two available astrophysical observations and with a very small theoretical uncertainty19. With a converged 3C/3D ratio of 3.567 ± 0.003 from our detailed calculation, there is definitely the need to explore other possible physical effects which might bring the theoretical ratio close to the one observed recently by the XFEL experiment.
There are three known emission lines from the upper states of Fe XVI ions (Fe15+) to its ground state 2p63s1/2 J = 1/2, i.e., line A from 2p43/22p1/23s3d3/2 J = 3/2 state, line B from 2p43/22p1/23s3d3/2 J = 1/2 state and line C from 2p43/22p1/23s3d5/2 J = 3/2 state23,24. It is also known from the previous EBIT experiment that the C line blends with the 3D line of Fe XVII (Fe16+) since both have an energy close to 812 eV (indicated by the bold black arrows shown in Fig. 1)23,24. In the LCLS experiment, the contribution of the Fe XVI C line was subtracted from the Fe XVII spectra for the 3D line with the assumption that the emissions from Fe XVII and Fe XVI are independent5. It turns out that in addition to the radiative decays, this upper state of the C line from Fe XVI could also autoionize into the ground state of Fe XVII shown by the blue dash arrow in Fig. 1. Since there was a finite pulse duration of the LCLS experiment, the resulting Fe XVII in its ground state could then be excited to increase the population of the upper level of the Fe XVII 3D line and thus increases the intensity of the 3D line. On the other hand, with the XFEL photon energy at the position of the Fe XVII 3C line, there is no resonant autoionization states of the Fe XVI ion. Considering the direct photon ionization probability is several orders of magnitude lower than the resonant autoionization process, the population of the upper level of the Fe XVII 3C line will not be influenced by the Fe XVI ion and thus the intensity of 3C line will not be changed. Then a decrease of the measured 3C/3D line ratio could be expected.
In this paper, we study such resonance induced population transfer process between Fe XVI and FeXVII ions in detail, which may offer a reasonable explanation for the discrepancy between theory and experiment. We will first discuss the assessment of the theoretical 3C and 3D oscillator strengths of Fe XVII ion, and then presents the results of our study of the resonance induced population transfer process. Finally, the implications of the present work are summarized.
## Results and Discussions
### The assessment of the theoretical 3C and 3D oscillator strengths of Fe XVII ion
The calculation of the atomic structure of the Fe XVII (Fe16+) ion is based on the well-established full relativistic multi-configuration Dirac-Fock (MCDF) approach21,22. In the present study, the oscillator strengths of the transitions between the ground state and the first five Jπ = 1 excited levels of the Fe XVII are calculated together to guarantee the final convergence of the 3C and 3D lines. The quasi-complete basis scheme is adopted to optimize the atomic orbitals (AOs) using the GRASP-JT version19,20 based on the earlier GRASP2K codes21,22. In this way, the convergence of the atomic structures can be examined step by step with limited computational efforts.
We started with the atomic orbitals (AOs) with principal quantum number n = 1, 2, 3 and (n-l-1) nodes that are optimized by multi-configuration self-consistent-field (MCSCF) iterations to minimize the lowest 37 energy levels of 2p6, 2p53s, 2p53p, and 2p53d states to form the zeroth level basis. With the AOs fixed up to n = 3, the pseudo AOs with n = 4 are obtained by further MCSCF iterations to optimize the statistic weights of the ground state and the first five JP = 1 excited levels of the Fe XVII to generate the first level basis. The additional electronic configurations from the 2p6, 2p53s, 2p53p, and 2p53d reference configurations with n = 3 and 4 AOs (i.e., two electrons excited from the core shell and one electron excited from the valence shell) are added in our calculation to include single, double, and some important triple excitations, even those from the 2 s inner shells. In succession, by adding more and more AOs in what we termed as the quasi-complete basis, we have included in the present calculation to sixth level basis with nmax = 9.
Based on these AO basis, the energy levels and corresponding transition rates are calculated by the configuration interaction (CI) method with more configurations than previous MCSCF calculations where the double excitations of the 2 s inner shell are allowed. So the CI calculations take account of all the valence- and prominent core excitation correlations, which are important for convergence. In fact, with n = 9, there are a total of over 3.7 million electronic configurations in our CI calculations. The QED corrections, especially the Breit interaction25,26, are added to the atomic Hamiltonian as a perturbation in the CI calculations. The Breit interaction is the most important high-order correction not only for the energy levels but also for the transition rates. The uncertainties of our calculated transition energies are estimated from the difference between CI calculations with AOs of adjacent n. The uncertainties of our calculated oscillator strengths are estimated by combining both the difference between CI calculations with AOs of adjacent n, as well as the difference between length and velocity gauge. Figure 2 shows the convergent behaviors of excitation energies E, oscillator strengths f and the oscillator strengths ratio of 3C and 3D with estimated uncertainties of various AO bases, compared with some reference data5,6,8,9,10,27. Table 1 presents the final calculation values using the quasi-complete basis (i.e., AOs with n = 9), compared with some reference data8,27.
In addition to the excellent agreement between the length and velocity results of our theoretical oscillator strength f3C and f3D, our calculated excitation energy E, f3C, f3D, and the ratio f3C/f3D are indeed influenced by one of the most important QED corrections, the Breit interaction25,26. By including the Breit correction, the excitation energies from the current calculation are less than 0.02% from the NIST data27. The f3C/f3D ratio changes from a value of 3.476 ± 0.003 to 3.567 ± 0.003 since the value of f3C increases while the value of f3D decreases after the Breit interaction was taken into account. With a converged ratio from our detailed calculation in close agreement with other existing atomic structure calculations5,6,7,8,9,10, we then conclude that other physics must be responsible for the discrepancy between previous theory and experiment.
### Theoretical model for resonance induced population transfer process
To study the resonance induced population transfer process between the Fe XVI and Fe XVII ions, we note first that, quantitatively, the time scale of the emission process of photon is usually of the order of 100–1000 fs, whereas the electron processes, such as the collision ionization (CI), the radiative recombination (RR), the dielectronic recombination (DR), and the excitation autoionization (EA) in the EBIT are typically of the order of 10−3 s28,29,30. Second, the XFEL photon energy was continuously scanned with a repetition rate of 120 Hz (Supplementary of ref.5), which means after the interaction with the XFEL, the ions in the EBIT have adequate time to return to their equilibrium stable state. Therefore, we could decouple those electron processes from the photon process when we discuss this XFEL photon related resonance induced population transfer process. By including the dominant decay channel (i.e., to the ground state of Fe XVI, which will be discussed later) and the autoionization of Fe XVI into the ground state of Fe XVII, the rate equations for the population densities for the ground and excited states of Fe XVI and FeXVII ions with incident XFEL photon energies $$\hslash {\rm{\omega }}$$ around 812 eV (the C and 3D lines) could then be expressed as:
$$\begin{array}{rcl}\frac{d}{dt}{N}_{g}^{F{e}^{15+}}(t) & = & {N}_{u}^{F{e}^{15+}}(t){B}_{ug}^{F{e}^{15+}}({\omega }_{C})\rho (\omega )+{N}_{u}^{F{e}^{15+}}(t){A}_{ug}^{F{e}^{15+}}({\omega }_{C})\\ & & -{N}_{g}^{F{e}^{15+}}(t){B}_{gu}^{F{e}^{15+}}({\omega }_{C})\rho (\omega ),\,\end{array}$$
(1A)
$$\begin{array}{rcl}\frac{d}{dt}{N}_{u}^{F{e}^{15+}}(t) & = & -{N}_{u}^{F{e}^{15+}}(t){B}_{ug}^{F{e}^{15+}}({\omega }_{C})\rho (\omega )-{N}_{u}^{F{e}^{15+}}(t){A}_{ug}^{F{e}^{15+}}({\omega }_{C})\\ & & +{N}_{g}^{F{e}^{15+}}(t){B}_{gu}^{F{e}^{15+}}({\omega }_{C})\rho (\omega )-{N}_{u}^{F{e}^{15+}}(t){\Gamma }_{u}^{A},\end{array}\,$$
(1B)
$$\begin{array}{rcl}\frac{d}{dt}{N}_{g}^{F{e}^{16+}}(t) & = & {N}_{u}^{F{e}^{16+}}(t){B}_{ug}^{F{e}^{16+}}({\omega }_{3D})\rho (\omega )+{N}_{u}^{F{e}^{16+}}(t){A}_{ug}^{F{e}^{16+}}({\omega }_{3D})\\ & & -{N}_{g}^{F{e}^{16+}}(t){B}_{gu}^{F{e}^{16+}}({\omega }_{3D})\rho (\omega )+{N}_{u}^{F{e}^{15+}}(t){\Gamma }_{u}^{A},\end{array}$$
(1C)
$$\begin{array}{rcl}\frac{d}{dt}{N}_{u}^{F{e}^{16+}}(t) & = & -{N}_{u}^{F{e}^{16+}}(t){B}_{ug}^{F{e}^{16+}}({\omega }_{3D})\rho (\omega )-{N}_{u}^{F{e}^{16+}}(t){A}_{ug}^{F{e}^{16+}}({\omega }_{3D})\\ & & +{N}_{g}^{F{e}^{16+}}(t){B}_{gu}^{F{e}^{16+}}({\omega }_{3D})\rho (\omega ),\,\end{array}$$
(1D)
where the transition rate Aug is the Einstein A coefficients from the upper level to the lower level, Bug and Bgu are the Einstein B coefficients with $${B}_{ug}=({\pi }^{2}{c}^{3}/\hslash {{\rm{\omega }}}^{3}){A}_{ug}$$, and $${{\rm{\Gamma }}}_{u}^{A}\,$$is the autoionization rate of Fe XVI ion. $$\rho (\omega )={I}_{p}/(c\Delta \omega \sqrt{2\pi })\exp [-{(\omega -{\omega }_{k})}^{2}/2\Delta {\omega }^{2}]$$, where Ip is the peak intensities of XFEL around energy ωk, Δω is the line width of the XFEL with the assumption of Gaussian profile, ωk is the resonant energy of the respective transition, and c is the speed of light31. The initial conditions of the equations are $${N}_{u}^{F{e}^{16+}}(t=0)=0,\,{N}_{u}^{F{e}^{15+}}(t=0)=0$$ with $${N}_{g}^{F{e}^{15+}}(t=0)/{N}_{g}^{F{e}^{16+}}(t=0)$$ as the concentration ratio between Fe XVI and Fe XVII ions in the EBIT. For other XFEL photon energies between 810 eV and 830 eV where the Fe XVII and Fe XVI lines are well separated, the rate equations are simpler. More specifically, around the resonant energies of the Fe XVI A and B lines, only Eqs (1A) and (1B) are needed (with ωc replaced by ωA and ωB, respectively), whereas for the Fe XVII 3C line, only Eqs (1C) and (1D) are needed (without the last autoionization term and with ω3D replaced by ω3C).
After solving Eq. (1) at a specific XFEL photon energy ω0 with peak intensity Ip for time-dependent population of the upper level of Fe XVII $${N}_{u}^{F{e}^{16+}}(t)$$ and Fe XVI $${N}_{u}^{F{e}^{15+}}(t)$$, the total fluorescence photon number $${N}_{ph}^{i}({I}_{p},{\omega }_{0})$$ is given by,
$${N}_{ph}^{i}({I}_{p},{\omega }_{0})={\int }_{0}^{\tau }{N}_{u}^{i}(t){A}_{ug}^{i}({\omega }_{0})dt+{N}_{u}^{i}(\tau ){\beta }^{i},$$
(2)
where τ is the incident XFEL pulse duration, i represents the Fe XVI or Fe XVII ion respectively and βi is the transition branching ratio, which equals 1 for Fe XVII and $${A}_{ug}^{F{e}^{15+}}/({A}_{ug}^{F{e}^{15+}}+{\Gamma }_{u}^{A})$$ for Fe XVI. The first term represents the emitted photons during the pulse period, the second is the remaining emitted photons at the end of the pulse. Note that only the spontaneous emission is relevant for the fluorescence because the photons emitted from stimulated radiation have the same direction as incident laser, which are not detected by the detectors perpendicular to the incident laser5. The anisotropic angular distributions are also taken into account in present simulations.
### The parameters used in the simulation
All required transition rates between 810 eV and 830 eV for the Fe XVII ions have already been obtained from the extended MCDF calculation discussed earlier. For the Fe XVI ion, the transition rates are calculated with the well-established atomic structure theories such as the relativistic eigenchannel R-matrix method detailed elsewhere20,32. In particular, for the autoionization rate, our calculation has taken into account fully the interaction between various resonant states and the continua. The relevant transition and autoionization rates are, respectively, 0.87 × 1013 s−1 and 3.92 × 1013 s−1 for the A line, 2.45 × 1013 s−1 and 2.13 × 1012 s−1 for the B line, and 1.32 × 1013 s−1 and 1.662 × 1013 s−1 for the C line. In addition, the radiative decay from the upper 2p43/22p1/23s3d5/2 J = 3/2 state is dominated by C line with its decay rate at least one order of magnitude greater than the other decays channels (including various cascade decay processes) as shown in Fig. 1. At specific photon energy, only the dominant decay channel and the autoionization of Fe XVI into the ground state of Fe XVII (if it contributes to the change in the Fe XVII population) are included when solving the rate equations.
We then examine briefly the relevant XFEL parameters in the LCLS experiment5 which we will apply in the numerical simulation presented later. For the line width parameter Δω of the XFEL photons, we use the value of 0.4 eV in all our simulations, which will result in a good match with the experimental spectra resolution5. The photon intensity Ip used in the simulations can be obtained from pulse energy PE, pulse duration τ and the effective focal area $${\sigma }_{d}={f}_{d}^{2}$$ with fd the focal diameter of the photon beam by $${I}_{p}={P}_{E}/(\tau {\sigma }_{d})$$. Unfortunately, these three parameters were not well defined in the experiment5, we can only infer the ranges of them from the LCLS specifications11,16,17,18,33,34. The raw X-ray pulse energy before the monochromator is about 1–4 mJ11,16. The final pulse energies reaching the experimental end stations after the monochromator range from 0.8 μJ for 500 eV photons to 0.48 mJ for 1000 eV photons33,34. According to ref.11, the pulse duration τ for the XFEL ranges from 50 to 500 fs. For the focal diameter fd of the photon beam, the unfocused beam size is in the range 1–3 mm17,33,34 with the focusing capability to about 2 μm17,33. Since it was stated in the Supplementary of ref.5 that “a very weakly focused photon beam” was used in the LCLS experiment and it is also reasonable to expect an adequate overlap between the XFEL photons with the EBIT ions in the diameter 0.5 mm (Supplementary of ref.5), we assume a larger fd (e.g., from 25 μm to 250 μm) in our numerical simulations, which may be closer to the actual experiment conditions.
### The results of pure Fe XVI ion
To examine the effect of the autoionization to the upper state population of the Fe XVI C line, we first focus our study on the simulation of the Fe XVI only spectrum shown in Fig. (3B) of ref.5 by solving Eqs (1A) and (1B). Figure 3 compares the experimentally observed spectra to our simulated spectra at two sets of XFEL pulse parameters. At lower pulse energy, our simulated spectrum is in good agreement with the previous EBIT experiment using electron excitation by Brown et al.23. Its higher intensity ratio of over 0.5 between the C and B lines mainly reflects the branching ratio between the C and B lines that is determined by the atomic transition rates, with little population loss of the C line. On the other hand, as expected, the intensity ratio of the C and B lines decreases from over 0.5 at lower XFEL power to 0.3 at higher XFEL power with a substantial population loss of the upper state of C line due to the autoionization and thus a much closer agreement with the observed spectrum from Fig. (3B) of ref.5. Accordingly, what is shown in Fig. 3 may offer the possibility of an experimental verification of the reliability of the theoretically estimated autoionization rate employed in our simulation should the values of pulse energy, duration and effective focal diameters are measured. The result in Fig. 3 also indicates that, in the XFEL pulse duration, the autoionized Fe XVII ion did not effectively survive in the EBIT under the condition to produce “pure” Fe XVI ion. On the other hand, under the EBIT condition where the Fe XVI and Fe XVII ions can co-exist, it is expected that the autoionized Fe XVII ion should survive in the EBIT during the XFEL pulse. This slight difference in the EBIT for the two measurement implies one should be careful about the subtraction of the C line contribution from the mixed spectra for the intensity of Fe XVII 3D line.
### The results of the mixed Fe XVI and Fe XVII ions
We now turn our discussion to the combined Fe XVI and Fe XVII system. To simulate the experimental spectra of the mixture (Fig. (3A) of ref.5), the initial relative abundance between Fe XVII and Fe XVI ions in the EBIT is required. Since the autoionization of the Fe XVI B line is almost negligible, we can use the intensity ratio of Fe XVI B line and Fe XVII 3C line to determine this relative abundance, which is about 1.5 from ref.5. After considering the angular distribution differences23 of these two lines, the ratio is corrected to 1.73. Then, by solving the related rate equations discussed above, the ion abundance of Fe XVI and Fe XVII is determined to be around 5:1. Figure 4(A) compares our simulated spectra with the experimentally observed spectrum. We start our discussion by first examining two spectra simulated at pulse parameters of PE = 360 μJ, τ = 300 fs and fd = 100 μm with and without autoionization shown by the red solid and red dash curves, respectively. As expected, these two simulated spectra are nearly identical for the B and A lines for the Fe XVI ion and the 3C line for the Fe XVII ion. In contrast, the substantial difference between these two curves for the combined 3D and C lines near 812 eV demonstrates clearly the effect of the population transfer from the upper state of the Fe XVI C line to the ground state of Fe XVII due to autoionization. Also shown in Fig. 4(A) by the dark solid curve is our simulated spectrum with two same pulse parameters τ = 300 fs and fd = 100 μm, but, at smaller energy PE = 36 μJ. The fact that this simulated spectrum is in very close agreement with the observed spectra supports strongly the reliability of the atomic data (such as the transition rates for Fe XVII and Fe XVI ions) and the simulation based on the rate equations employed in the present study. By taking into account the autoionization from the upper state of Fe XVI to the ground state of Fe XVII, as shown by the solid dark curve in Fig. 4(A), our simulated line ratio $${\rm{R}}={N}_{ph}^{F{e}^{16+}}({I}_{p},{\omega }_{3C})/{N}_{ph}^{F{e}^{16+}}({I}_{p},{\omega }_{3D})$$, where $${N}_{ph}^{F{e}^{16+}}({I}_{p},{\omega }_{3C})$$ and $${N}_{ph}^{F{e}^{16+}}({I}_{p},{\omega }_{3D})$$ are calculated from Eq. (2) with the complete solution of Eq. (1), is expected to yield the intensity ratio of the 3C and 3D lines observed experimentally. Figure 4(B) presents the variation of the simulated 3C/3D intensity ratio with various possible XFEL parameter combinations. It is interesting to note that the 3C/3D ratio decreases as the pulse energy increases with the same focal diameter fd due to the stronger autoionization effect. Similarly, the 3C/3D ratio decreases significantly, again i.e., with stronger autoionization effect, as photon intensity Ip increases at the same pulse energy PE with smaller effective focal diameter fd. On the other hand, with the same pulse energy and focal diameter, the 3C/3D ratio does not change significantly with pulse duration varying from 50 fs to 500 fs, which suggests that the ratio depends mostly on the $${P}_{E}/{\sigma }_{d}$$.
For the sake of further analysis and considering the lifetime of the Fe XVI C line about 60 fs which is comparable with some XFEL pulse durations it is worth to separate the contribution of resonance induced population transfer effect from our simulations by, $${P}^{{\rm{AI}}}=({R}^{{\rm{no}}-{\rm{AI}}}-{R}^{{\rm{AI}}})/({R}^{{\rm{atomic}}}-{R}^{{\rm{AI}}})$$, where $${R}^{{\rm{atomic}}}=3.567$$ is the calculated 3C/3D ratio of the isolated Fe XVII ion; RAI, Rno−AI are the simulated line ratio, with and without the autoionization terms in Eq. (1), respectively. Figure 4(C) shows the PAI of a fixed $${P}_{E}/{\sigma }_{d}$$ value that corresponds to the experimental measured ratio of 2.61 with various pulse durations. We can clearly see that with the pulse duration time less than 20 fs, or, substantially smaller than the lifetime of the Fe XVI C line, the contribution from the autoionization of the Fe XVI ion is small. For those XFEL parameters, the decrease of 3C/3D ratio should be attributed to the non-equilibrium effects discussed in refs14,15. By contrast, with the pulse durations longer than 90 fs (in consistent with the typical LCLS X-ray pulse parameters11,16,17,33), the resonance induced population transfer effect contributes more than a half. For example, for the pulse duration of 400 fs, where the high order nonlinear effects should be small, the contribution of the resonance induced population transfer effect is dominant, with PAI over 90%.
### The influence of the stochastic substructure of the XFEL pulses on theoretical simulations
Due to the stochastic nature of the self-amplified stimulated emission (SASE) process, the pulse intensity of the XFEL is not homogeneous, which consists of sharp and stochastic individual pulses of 1–2 fs in duration with similar sized random gaps between the spikes35,36,37. We now discuss the influence of these stochastic substructure of the XFEL pulses on our simulation results. As an illustration example, we choose PE/σd = 0.0036 μJ/μm2 for the XFEL pulses [the same with the one in Fig. 4(C)]. With this pulse parameter, three typical substructures were considered in the simulations, i.e., (1) the homogeneous micro-pulses (constant photon intensity), (2) the periodical micro-pulses with the period of 2 fs, and (3) the stochastic micro-pulses. Figure 5(A) shows the corresponding results for the XFEL pulse with the duration τ = 150 fs. For these three different pulse substructures, the time variations of the simulated 3C/3D line ratio are almost identical, especially at the end of the pulse, which are all in good agreement with the experimental measurement5. Figure 5(B) shows the results for another different XFEL pulse with 300 fs duration. Similarly, the behavior of the 3C/3D ratio is also nearly independent with different pulse substructures. Therefore, we can make the conclusion that the final 3C/3D ratio mainly depends on the ratio of $${P}_{E}/{\sigma }_{d}$$, the stochastic substructure of the XFEL pulses should have negligible effects on the 3C/3D ratio.
## Conclusion
In summary, with an extended large scale full relativistic configuration interaction calculation including the quantum electrodynamics (QED) term such as the Breit interaction, we first came to the conclusion that the reliability of the theoretical results for isolated Fe XVII ions from the present calculation and other earlier atomic structure calculations may not be the cause for the discrepancy between the experimentally observed intensity ratio and the theoretical estimates. Since it is known that the energies of the Fe XVII 3D line and the Fe XVI C line are both close to 812 eV and the pulse length of the XFEL experiment is sufficiently longer than the time scale of the autoionization process transferring the upper state of the Fe XVI ion to the ground state of Fe XVII ion, we decided to focus our investigation on the dynamic resonance induced population transfer from Fe XVI to Fe XVII. By solving the relevant rate equations, we are able to generate theoretically simulated spectra as well as the intensity ratio in agreement with the experimental spectra from the LCLS experiment shown in Fig. 3 of ref.5. Note that we have also considered the influence of the plasma environment in the EBIT. Although the plasma screening will induce a substantial decrease in the ratio 3C/3D, it could not be the one responsible for the disagreement between theory and experiment due to the low electron densities38,39.
In addition to resolving the discrepancy on intensity ratio discussed above, the conclusion from the present study may be examined by either performing the measurements of Fe XVI and Fe XVI/Fe XVII mixture under the well characterized XFEL intensity or with additional experiments for other Ne-like ions in the absence of the resonance induced population transfer. For example, based on our simulations, the measured ratio mainly depends on the ratio of the pulse energy and the effective focal area, i.e., PE/σd, for which some diagnostics are being developed at the LCLS40,41. On the other hand, some attenuators18 may be used to reduce the XFEL intensity where the resonance induced population transfer effect is negligible, or alternatively, one can extract the pure Fe XVII ion from the EBIT known as the Electron Beam Ion Sources (EBIS)42 for the experiment.
## Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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## References
1. 1.
Paerels, F. B. S. & Kahn, S. M. High-Resolution X-Ray Spectroscopy with Chandra and XMM-Newton. Annu. Rev. Astron. Astrophys. 41, 291–342 (2003).
2. 2.
Behar, E., Cottam, J. & Kahn, S. M. The Chandra Iron-L X-Ray Line Spectrum of Capella. Astrophys. J. 548, 966–975 (2001).
3. 3.
Xu, H. et al. High-Resolution Observations of the Elliptical Galaxy NGC 4636 with the Reflection Grating Spectrometer on Board XMM-Newton. Astrophys. J. 579, 600–606 (2002).
4. 4.
Díaz Trigo, M. et al. Baryons in the relativistic jets of the stellar-mass black-hole candidate 4U 1630-47. Nature 504, 260–262 (2013).
5. 5.
Bernitt, S. et al. An unexpectedly low oscillator strength as the origin of the Fe XVII emission problem. Nature 492, 225–228 (2012).
6. 6.
Safronova, U. I., Namba, C., Murakami, I., Johnson, W. R. & Safronova, M. S. Electric-dipole, electric-quadrupole, magnetic-dipole, and magnetic-quadrupole transitions in the neon isoelectronic sequence. Phys. Rev. A 64, 012507 (2001).
7. 7.
Chen, G. X. & Pradhan, A. K. Influence of Resonances on Spectral Formation of X-Ray Lines in Fe XVII. Phys. Rev. Lett. 89, 013202 (2002).
8. 8.
Jönsson, P. et al. Relativistic CI calculations of spectroscopic data for the 2p6 and 2p53l configurations in Ne-like ions between Mg III and Kr XXVII. At. Data Nucl. Data Tables 100, 1–154 (2014).
9. 9.
Santana, J. A., Lepson, J. K., Träbert, E. & Beiersdorfer, P. Electron-correlation effects on the 3C to 3D line-intensity ratio in the Ne-like ions Ar8+ to Kr26+. Phys. Rev. A 91, 012502 (2015).
10. 10.
Oreshkina, N. S., Cavaletto, S. M., Keitel, C. H. & Harman, Z. Astrophysical line diagnosis requires nonlinear dynamical atomic modeling. Phys. Rev. Lett. 113, 143001 (2014).
11. 11.
Bostedt, C. et al. Linac Coherent Light Source: The first five years. Rev. Mod. Phys. 88, 015007 (2016).
12. 12.
Mendoza, C. & Bautista, M. A. Theoretical confirmation of the low Experimental 3C/3D f-value ratio in Fe XVII. Phys. Rev. Lett. 118, 163002 (2017).
13. 13.
Wang, K. et al. Comment on “Theoretical Confirmation of the Low Experimental 3C/3D f-Value Ratio in Fe XVII”. Phys. Rev. Lett. 119, 189301 (2017).
14. 14.
Loch, S. D., Balance, C. P., Li, Y., Fogle, M. & Fontes, C. J. Non-equilibrium Modeling of the Fe XVII 3C/3D Line Ratio in an Intense X-Ray Free-Electron Laser Excited Plasma. Astrophys. J. Lett. 801, L13 (2015).
15. 15.
Li, Y., Fogle, M., Loch, S. D., Balance, C. P. & Fontes, C. J. Non-Equilibrium Modeling of the Fe XVII 3C/3D ratio for an Intense X-ray Free Electron Laser. Can. J. Phys. 95, 869–877 (2017).
16. 16.
Emma, P. et al. First lasing and operation of an ångstrom-wavelength free-electron laser. Nat. Photon. 4, 641–647 (2010).
17. 17.
Ferguson, K. R. et al. The Atomic, Molecular and Optical Science instrument at the Linac Coherent Light Source. J. Synchrotron Rad. 22, 492–497 (2015).
18. 18.
Krzywinski, J. et al. Damage thresholds for blaze diffraction gratings and grazing incidence optics at an X-ray free-electron laser. J. Synchrotron Rad. 25, 85–90 (2017).
19. 19.
Han, X. Y., Gao, X., Zeng, D. L., Yan, J. & Li, J. M. Ratio of forbidden transition rates in the ground-state configuration of O II. Phys. Rev. A 85, 062506 (2012).
20. 20.
Gao, X., Han, X. Y. & Li, J. M. Intimate relationship between spectroscopy and collisions: a scenario to calculate relevant atomic data for astrophysics. J. Phys. B: At. Mol. Opt. Phys. 49, 214005 (2016).
21. 21.
Jönsson, P., He, X., Fischer, C. F. & Grant, I. P. The grasp2K relativistic atomic structure package. Comput. Phys. Commun. 177, 597–622 (2007).
22. 22.
Jönsson, P., Gaigalas, G., Bieroń, J., Fischer, C. F. & Grant, I. P. New version: Grasp2K relativistic atomic structure package. Comput. Phys. Commun. 184, 2197–2203 (2013).
23. 23.
Brown, G. V., Beiersdorfer, P., Chen, H., Chen, M. H. & Reed, K. J. Diagnostic Utility of the Relative Intensity of 3C to 3D in Fe XVII. Astrophys. J. 557, L75–L77 (2001).
24. 24.
Graf, A., Beiersdorfer, P., Brown, G. V. & Gu, M. F. Measurement and Modeling of Na-Like Fe XVI Inner-Shell Satellites Between 14.5 Å and 18 Å. Astrophys. J. 695, 818–824 (2009).
25. 25.
Breit, G. The effect of retardation on the interaction of two electrons. Phys. Rev. 34, 553 (1929).
26. 26.
Grant, I. P. Relativistic Quantum Theory of Atoms and Molecules. (Springer-Verlag, New York, 2007).
27. 27.
Kramida, A., Ralchenko, Y. & Reader, J. NIST ASD Team, NIST Atomic Spectra Database (version 5.6.1) [online]. Available, https://physics.nist.gov/asd [Mon Feb 11 2019] National Institute of Standards and Technology (Gaithersburg, MD, 2019).
28. 28.
Gu, M. F. Indirect X-Ray Line-Formation Processes in Iron L-Shell Ions. Astrophys. J. 582, 1241–1250 (2003).
29. 29.
Liu, Y., Yao, K., Hutton, R. & Zou, Y. Numerical simulations using an improved calculational scheme for ion charge state distribution and ion temperature evolution in an EBIT. J. Phys. B: At. Mol. Opt. Phys. 38, 3207–3216 (2005).
30. 30.
Lu, X. J. & Currell, F. J. Numerical simulation of the charge balance and temperature evolution in an electron beam ion trap. Phys. Rev. STAB 12, 014401 (2009).
31. 31.
Cagnac, B. & Pebay-Peyroula, J. C. Modern Atomic Physics: Fundamental Principles. (Macmillan Press, London, 1975).
32. 32.
Gao, X. & Li, J. M. Precision spectroscopy and electron-ion scattering. Phys. Rev. A 89, 022710 (2014).
33. 33.
See the LCLS web page on SXR specifications, https://lcls.slac.stanford.edu/instruments/sxr/specifications (2019).
34. 34.
Schlotter, W. F. et al. The soft x-ray instrument for materials studies at the linac coherent light source x-ray free-electron laser. Rev. Sci. Instrum. 83, 043107 (2012).
35. 35.
Inubushi, Y. et al. Determination of the Pulse Duration of an X-Ray Free Electron Laser Using Highly Resolved Single-Shot Spectra. Phys. Rev. Lett. 109, 144801 (2012).
36. 36.
Behrens, C. et al. Few-femtosecond time-resolved measurements of X-ray free-electron lasers. Nat. Commun. 5, 3762 (2014).
37. 37.
Hartmann, N. et al. Attosecond time-energy structure of X-ray free-electron laser pulses. Nat. Photon. 12, 215–220 (2018).
38. 38.
Wu, C. & Gao, X. Resonance induced population transfer of Fe XVII ions in plasma environment. J. Phys.: Conf. Ser. 875, 022011 (2017).
39. 39.
Wu, C., Chen, S., Chang, T. N. & Gao, X. Variation of the transition energies and oscillator strengths for the 3C and 3D lines of the Ne-like ions under plasma environment. Preprint at, https://arxiv.org/abs/1904.08471 (2019).
40. 40.
Tiedtke, K. et al. Absolute pulse energy measurements of soft x-rays at the Linac Coherent Light Source. Opt. Express 22, 21214–21226 (2014).
41. 41.
Chalupsky, J. et al. Comparing different approaches to characterization of focused X-ray laser beams. Nucl. Instrum. Methods. A 631, 130–133 (2011).
42. 42.
Currell, F. (Eds) Electron beam ion traps and their use in the study of highly charged ions, in The Physics of Multiply and Highly Charged Ions (Vol. I) (Kluwer Academic, Dordrecht, 2003).
## Acknowledgements
Parts of text in the introduction and conclusions were previously published in our conference proceeding paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11774023 and U1530401), the National Key Research and Development Program of China (Grant No. 2016YFA0302104), the National High-Tech ICF Committee in China. We acknowledge the computational support provided by the Beijing Computational Science Research Center. We would like to acknowledge Prof. Tu-Nan Chang at University of Southern California for helpful discussions and comments of the manuscript.
## Author information
C.W. was responsible for most of the calculations and code development for the simulations. X.G. supervised the study. C.W. and X.G. drafted the manuscript.
### Competing Interests
The authors declare no competing interests.
Correspondence to Xiang Gao.
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2019-06-17 11:24:03
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http://k7og.net/924anm/7a65d1-what-is-shear-modulus-formula
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Bulk modulus formula. It can be measured by a shear strain test, which is conducted by placing a rod of a given material into a clamp and applying force at a measured distance away from the clamp to only one side of the rod. This will also explain why our bones are strong and yet can be fractured easily. Published academic co-relations can be used to determine shear wave velocities and shear modulus of different soil layers against SPT N values. Shear Modulus of elasticity is one of the measures of mechanical properties of solids. The Shear Modulus is a material property, which cannot be altered– except for various special thermal treatments, of course, which are hardly part of compression coil spring design. Specifically, we will look at a doubly symmetric composite beam system for simplicity. T 1375 Cos 8.4 x 0.0925 =125.8 N-m. L = 0.0925 m . The material will undergo an angular deformation, and the ratio of the tangential force per unit area to the resulting angular deformation is called the shear modulus or the rigidity modulus. It is defined as the ratio between pressure increase and the resulting decrease in a material's volume. For example, Hudson specifically includes the effect of anisotropic crack distributions. The image above represents shear modulus. Young's Modulus from shear modulus can be obtained via the Poisson's ratio and is represented as E=2*G*(1+) or Young's Modulus=2*Shear Modulus*(1+Poisson's ratio).Shear modulus is the slope of the linear elastic region of the shear stress–strain curve and Poisson's ratio is defined as the ratio of the lateral and axial strain. RE: Shear Modulus of Concrete briancpotter (Structural) 16 Apr 13 15:12. I know you can determine the shear modulus using Poissons ratio but doing testing to determine poissons seems a little excessive. The modulus of elasticity (= Young’s modulus) E is a material property, that describes its stiffness and is therefore one of the most important properties of solid materials. Answer obtained is in radians (rad), but we usually convert it to degrees. The relative strains of the testing samples were obtained by measuring predefined load conditions using a strain-gauge bridge and the universal measurement system Quantum X MX 840. The shear modulus G is also known as the rigidity modulus, and is equivalent to the 2nd Lamé constant m mentioned in books on continuum theory. There are three popular applications for the shearing modulus formula. The modulus of rigidity formula is G=E/(2(1+v)), and modulus of rigidity is denoted by G, elastic modulus is denoted by E and poisson’s ratio is v in the formula. Some of these assumptions may be dropped, depending on the model involved. The bulk modulus (or ) of a substance is a measure of how resistant to compression that substance is.It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. The shear modulus S is defined as the ratio of the stress to the strain. Bulk modulus is the ratio of applied pressure to the volumetric strain. But first of all, let us look at what our beam system is composed of. Let’s solve an example; One particularly useful result was derived by Kuster and … L is the length of the shaft or member. ( ) A ∆x FL L ∆x A F strain stress S = = units are Pascals shear shear ≡ The bigger the shear modulus the more rigid is the material since for the same change in horizontal distance (strain) you will need a bigger force (stress). Young's modulus equation is E = tensile stress/tensile strain = (FL) / (A * change in L), where F is the applied force, L is the initial length, A is the square area, and E is Young's modulus in Pascals (Pa). In this post, we will learn how to use classical hand calculation methods to calculate the section modulus of a sample shear web system. An element subject to shear does not change in length but undergoes a change in shape. Is this comparable for concrete as well? Shearing Deformation Shearing forces cause shearing deformation. The energy is stored elastically or dissipated plastically. Shear strain defined as the ratio of the change in deformation to its original length perpendicular to the axes of the member due to shear stress. Shear waves travel at about half the speed of compressional waves (e.g., in iron, 3,200 metres per second compared with 5,200 metres per second). The way a material stores this energy is summarized in stress-strain curves. Find the strain, stress and the shearing force. In engineering, shear strength is the strength of a material or component against the type of yield or structural failure when the material or component fails in shear.A shear load is a force that tends to produce a sliding failure on a material along a plane that is parallel to the direction of the force. The shear modulus of material gives us the ratio of shear stress to shear strain in a body. It is expressed in GPa or psi and typical values are given in Textbook Appendix B. K is the torsional constant. Mechanical deformation puts energy into a material. For masonry, they advise using a shear modulus of 0.4 X modulus of elasticity. A = area (m 2, in 2) s = displacement of the faces (m, in) d = distance between the faces displaced (m, in) Ductile vs. Brittle materials; Bulk Modulus Elasticity. This equation is the most popular equation being used for fluid substitution modeling; however, the basic assumptions of this equation are: 1. What an engineer can do to change the spring constant via shear modulus is choosing another material. G = Shear Modulus of Elasticity - or Modulus of Rigidity (N/m 2) (lb/in 2, psi) τ = shear stress ((Pa) N/m 2, psi) γ = unit less measure of shear strain . But the value of Young’s Modulus is mostly used. Bulk modulus formula. Shear modulus, in materials science, is defined as the ratio of shear stress to shear strain. Mathematically it is expressed as: Shear modulus formula. The simplest soil test the can be done is Standard Penetration Test (SPT). This will also explain why our bones are strong and yet can be fractured easily. The change in angle at the corner of an original rectangular element is called the shear strain and is expressed as $\gamma = \dfrac{\delta_s}{L}$ The ratio of the shear stress τ and the shear strain γ is called the modulus of Where ΔV is the change in original volume V. Shear modulus. The shear modulus G max under the current state of stresses is given in a formula which includes a newly proposed void ratio function. Definition Ratio of Shear Stress to the Shear Strain with in Linear Elastic Region. Theta = Angle olf twist in Radians . Anyway: the formula is Theta = T L /K G . Due to this pressure, the volume got decreased and the new volume is V2. Together with Young's modulus, the shear modulus, and Hooke's law, the bulk modulus describes a material's response to stress or strain. Theta = 1.24 pi/180 = 0.0216 Radians. Maybe I'm on the wrong track, let me know your thoughts. Using a graph, you can determine whether a material shows elasticity. There are some other numbers exists which provide us a measure of elastic properties of a material. Young’s Modulus or Elastic Modulus or Tensile Modulus, is the measurement of mechanical properties of linear elastic solids like rods, wires, etc. Let's explore a new modulus of elasticity called shear modulus (rigidity modulus). shear modulus= (shear stress)/(shear strain) Denoted By G. It is Also Called As Modulus of Rigidity. The formula for calculating the shear modulus: G = E / 2(1 + v) Where: G = Shear Modulus E = Young’s Modulus v = Poisson’s Ratio. The height of the block is 1 cm. Let's explore a new modulus of elasticity called shear modulus (rigidity modulus). The ratio of shear stress and shear strain is called shear modulus. This is why the shear modulus is sometimes called the modulus of rigidity. Other moduli describe the material's response to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear stress. The rolling shear modulus measured was then used as input to predict, using the shear analogy method, the deflection ( d c ) of a 3-layer CLT beam subjected to the centre-point bending load. Section Modulus – … Pore-fluid system is closed, and there is no chemical interaction between fluids and rock frame (however, shear modulus need not remain constant). Shear modulus of the material of a body is given by Relation Between the Moduli of Elasticity: Numerical Problems: Example – 1: The area of the upper face of a rectangular block is 0.5 m x 0.5 m and the lower face is fixed. Shear modulus' derived SI unit is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (ksi). S.I Unit of rigidity modulus is Pascal. 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Is applied to the shear stress to the top face produces a displacement of 0.015 mm are −1... Applied to all surfaces of the stress to the top face produces a displacement of 0.015.. Initial volume of an object is V1 us consider the initial volume of an object is V1 at. −2 rad −1 and its dimensions are ML −1 T −2 θ −1 mathematically it is defined as the of... Or psi and typical values are given in Textbook Appendix B decrease in a formula which includes newly. Let 's explore a new modulus of different soil layers against SPT N values 's a! Obtained is in radians ( rad ), but we usually convert it to.! Modulus s is defined as the ratio of shear stress Skills Practiced which they act modulus s is defined the! = force parallel to the faces which they act 1375 Cos 8.4 x =125.8...
Alolan Sandshrew Pokemon Sword, Nokia Siemens Networks Logo, Edifier Speaker Singapore, Rubbermaid Dish Drainer, Tacori Fashion Rings,
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2021-12-03 19:54:22
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https://socratic.org/questions/why-is-the-nernst-equation-important
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# Why is the Nernst equation important?
Apr 29, 2015
The Nernst equation ${E}^{0} = {E}^{\theta} + \frac{R T}{n F} \ln \left(\text{Oxidants"/"Reductant}\right)$
It has other forms; but basically, it shows us the relationship between Electrode potential of a half cell and Temperature.
It is a direct relationship. That is, increasing the temperature if half cell, leads to a rise in electrode potential.
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2022-08-19 02:51:15
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https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article/70/1/010801/443695/A-Review-of-Propulsion-Power-and-Control
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Flying insects are able to navigate complex and highly dynamic environments, can rapidly change their flight speeds and directions, are robust to environmental disturbances, and are capable of long migratory flights. However, flying robots at similar scales have not yet demonstrated these characteristics autonomously. Recent advances in mesoscale manufacturing, novel actuation, control, and custom integrated circuit (IC) design have enabled the design of insect-scale flapping wing micro air vehicles (MAVs). However, there remain numerous constraints to component technologies—for example, scalable high-energy density power storage—that limit their functionality. This paper highlights the recent developments in the design of small-scale flapping wing MAVs, specifically discussing the various power and actuation technologies selected at various vehicle scales as well as the control architecture and avionics onboard the vehicle. We also outline the challenges associated with creating an integrated insect-scale flapping wing MAV.
## Introduction
Over the past several decades, unmanned aerial vehicles (UAVs) have found increasing applications in areas including military, media, and entertainment. As these applications evolve and component technologies such as microcontrollers (MCUs), sensors, and energy storage shrink, a new class of UAVs are being developed on much smaller scales. These “micro air vehicles (MAVs)” exist on a scale that is approximately 10 cm to 1 m in characteristic dimension. There are numerous examples as outlined in past MAV reviews [1,2]. As these devices are shrunk further, questions for successful design and operation focus more on materials, mechanics, design, and manufacturing as opposed to control and mission-level programming and planning for larger UAVs.
As the size of the vehicle is decreased, propulsion becomes a key consideration. Fixed wing aircraft at human scales can achieve lift-to-drag ratios in excess of 100. However, as scale is reduced, the Reynolds number is commensurately reduced, resulting in a greater influence of viscous effects that are manifest as increased drag and reduced lift-to-drag ratios. In extreme cases, for devices on the scale of small birds, “nano air vehicles” [3], or insects, “pico air vehicles (PAVs)” [4], fixed wing fluid mechanics becomes impractical as too much energy would be lost to drag at the speeds that would be required to maintain sufficient lift. The same arguments hold for rotary wing vehicles. Quadrotor MAVs exist down to tens of grams (Aerius, Aerix Drones, Fairport, NY); however as size is reduced, the decreasing propulsive efficiency becomes clear in reduced flight times [5]. Furthermore, beyond reduced propulsive efficiency, for insect-scale PAV, there are additional challenges for actuation. Traditional electromagnetic motors are subject to scaling laws that result in degraded performance at reduced sizes [4]. This is caused by unfavorable scaling of surface area to volume, unfavorable scaling of electromagnetic force, limitations on current density, and the need for excessive gearing given the increase in unloaded revolutions per minute (RPM) as size is reduced [6]. In addition, there are significant challenges to manufacturing motors at these scales. Alternative actuation strategies have been explored and will be reviewed in this paper as they pertain to MAV designs.
A bioinspired alternative to traditional propulsion mechanisms is the use of flapping wings. Flapping wing MAVs can overcome some of the challenges discussed earlier by using oscillating actuation in place of rotary actuators and can take advantage of unsteady fluid effects described in the following. However, wing motions can involve several degrees-of-freedom (DOF), creating a challenging design problem for how to generate high-speed, high-efficiency articulating mechanisms at millimeter scales (see Fig. 1 for a model of insect-scale flapping wing MAVs).
One consequence of flapping wings is highly unsteady fluid forces. In contrast to continuously translating or rotating wings, flapping wings experience periodic pressure variations that arise from a number of phenomena. Rapid wing inversions at the end of each half stroke enhance circulation resulting in augmented lift during this phase of the wing cycle [7]. As the wing accelerates into the next half-cycle, it encounters the wake from the previous half-cycle, which can significantly affect flapping kinematics and the corresponding force generation [8]. During the middle of the stroke, the angle of attack—approximately constant during this phase—can greatly exceed angles of attack that would result in stall for conventional aircraft due to leading edge vortex stabilization mechanisms [9]. In some animals and flapping wing MAVs, the wing stroke amplitude can be sufficiently large such that at the end of one or both half-strokes the wings can come in to physical contact, resulting in more favorable starting conditions for the vorticity at the start of the next half-cycle [10].
Micro air vehicle designers have sought to navigate the tradeoffs associated with the creation of flapping wing propulsion mechanisms that exploit some or all of these phenomena. For example, mimicking the exact kinematics of an insect species may allow the vehicle to exploit each of these unsteady effects but would potentially require a large number of actuators and a complex transmission mechanism. Furthermore, with respect to control, one critical question is whether body torques are modulated directly by the wings through changes to the nominal wing motions or through ancillary control surfaces. The question of which wing motions to target and how to achieve them, with emphasis on insect-like motions at insect scales, is a key focus of this review.
This review will also describe power and control systems that attempt to mirror some functions of the metabolic and sensorimotor systems in insects. As with the propulsion systems, there are noteworthy tradeoffs in the design of sensing and control electronics that impact energy storage and flight range. For example, the desired level of autonomy—ranging from uncontrolled, passively stable flight to teleoperation to full autonomy—will impose requirements on onboard electronics that will, in turn, also impact overall power consumption and flight duration and range.
Researchers in Ref. [1] provided a review of bird-inspired flapping wing MAV designs, focusing specifically on the mechanics of flight control and the design of the wing and drive mechanisms. That paper discussed vehicles with wingspans in the range from approximately 5–65 cm and body mass range from approximately 1.5–300 g. Here, we discuss a number of vehicles that have been developed since its publication, as well as vehicles that use less traditional actuation strategies. This review primarily focuses on vehicles at a smaller scale, with a wingspan of less than approximately 20 cm and a body mass less than 20 g (see Fig. 2 for the vehicles discussed in this review). As characteristic length decreases, it is necessary to describe alternatives to component technologies relative to larger vehicles, such as alternatives to motors, gears, and rotary bearings, alternative manufacturing methods, as well as alternative methods for lift generation. In addition, we survey vehicles that have achieved varying levels of control autonomy and the necessary avionics at this scale. This review begins with a survey of actuation technologies and power electronics for centimeter-scale flapping wing MAVs in Sec. 2. Section 3 follows with a summary of the control architectures to generate lift and control torques to maintain flight at various scales. Representative examples of centimeter-scale vehicles that have demonstrated autonomy are then discussed in Sec. 4. Finally, the challenges associated with designing power and control architectures for an insect-scale vehicle are discussed in Sec. 5.
## Actuation and Power Electronics
The physics of scaling dictates that as the characteristic dimension of a device decreases, surface forces, such as friction, electrostatic, and van der Waals, begin to dominate relative to Newtonian forces. This has many practical implications; one of the most relevant is that the use of bearing-based rotary joints becomes inefficient at smaller scales. To overcome frictional losses, the New York University (NYU) Jellyfish Flyer [13] uses short segments of low-friction Teflon tubing as a rotary bearing. At the insect scale, another method to create efficient joints is to embed flexible layers between two rigid links to create compliant flexures. These embedded flexures are used in a number of flapping wing MAVs presented in this paper, including the Carnegie Mellon University (CMU) flapping wing MAVs [12], the electromagnetic flyer [17], the University of California, Berkeley Micromechanical Flying Insect (MFI) [22], the Air Force Research Lab (AFRL) Piezo-Driven flapping wing MAV [20], as well the Harvard RoboBee vehicles [23,24]. The fabrication process to create these composite structures is described in more detail in Ref. [22]. This process was further developed into the “PC-MEMS” process [25] to create complex three-dimensional structures that are used in many of the designs described in detail in the following.
While DC motors are used in the majority of flapping wing MAVs because of their robustness, ease of operation, and ubiquity in macroscale robotics, electromagnetic forces suffer unfavorable scaling at small sizes [26]. Thus, insect-scale robots require nontraditional motors, such as chemical, electrostatic, or piezoelectric, to meet the requirements for power density and bandwidth in these applications [27].
### Motors.
As stated earlier, the majority of flapping wing MAVs utilizes DC motors as power actuators. Motors typically operate at low voltages amenable to standard off-the-shelf motor drivers, eliminating the need for complex power electronics. Recent vehicle designs that have utilized electromagnetic motors include the Harvard Robot Moth [11], Aerovironment's Nanohummingbird [3], the NYU Jellyfish Flyer [13], and the CMU flapping wing MAV [28].
Researchers at CMU [28] integrated a helical spring in parallel to the motor. In this system, energy stored in the spring during a half cycle is released in the subsequent half cycle and assists the motor in reversing the direction of the system inertia. By tuning the spring stiffness so the system resonates at the flapping frequency, the vehicle can save energy by eliminating reactive power needed to oscillate the rotor and wing inertias. This also has the benefit of reducing current spikes by “smoothing” the required motor torque over a period of cyclic motion [29].
There are many options for creating the oscillating motions associated with flapping wings. Proper actuator choice must reconcile the specific requirements of the torque, displacement, and flapping frequency with scaling laws that place limits on the performance of the core actuation element. More generally, actuators can be classified based on their force/torque, displacement, bandwidth, mass, and efficiency characteristics. In addition, there are numerous practical constraints that must be considered including ease of manufacture and ease of physical and electrical integration. With respect to the latter, some electrically driven actuators may require high fields. This tradeoff will be discussed in more detail in Sec. 5.1.1.
In Ref. [30], Michelson describes the development of a conceptual reciprocating chemical muscle to power flapping-wing flight of the Entomopter. The reciprocating chemical muscle uses a noncombustive chemical reaction between a monopropellant and an oxidizer. With this reaction, the muscle can create the oscillatory wing stroke motion to generate lift. Additionally, the Central Intelligence Agency developed a dragonfly-inspired flapping wing MAV in the 1970s. The Insectothopter had a miniature fluidic oscillator that controlled the wing stroke motion at a fixed frequency. The excess gas from the chemical reaction was vented out the back of the vehicle creating additional thrust. Video shows that a 1 g prototype vehicle could fly up to 200 m for 60 s [14]. The use of chemical muscles obviates the need for an onboard electrical power supply necessary for conventional actuation technologies but requires a separate electrical system for control and navigation.
Researchers at Shanghai Jiao Tong University have created an insect-scale flapping wing MAV using an oscillating electromagnetic actuator [17]. The vehicle has a mass of 80 mg and a wingspan of 3.5 cm and was manufactured using the Smart Composite Microstructures process described in Ref. [22] and in Sec. 5. The actuator is comprised of a neodymium iron boron magnet attached to a transmission system consisting of two planar four-bars that map actuator motion to the rotational wing stroke motion. The electromagnetic force that acts on the magnet is created by a copper coil attached to the airframe. However, this vehicle consumes approximately 1.2 W of power during flight, which corresponds to a minimum power density of 15 kW/kg (calculated using the mass of vehicle—no total thrust was reported). The power density of the onboard energy source will be larger than this—the vehicle will need to scale to accommodate the mass of the battery, requiring more power while also increasing the required thrust to accommodate the mass of the battery and drive electronics.
Researchers at the University of Tokyo [16] developed a butterfly-inspired flapping wing MAV using a rubber band as the motor and means of energy storage. The vehicle had a wingspan of approximately 14 cm, a weight of 400 mg, and a flapping frequency of 10 Hz (controlled by the rubber band's thickness and length). The vehicle demonstrated stable forward flight with no active control.
Insects and similar-sized MAVs require relatively high flapping frequencies, often hundreds of hertz [31]. Piezoelectric actuators are typically high bandwidth and numerous motion amplifying mechanisms have been developed to overcome inherent strain limitations, such as benders [32,33] and flextensional actuators [34,35]. Researchers at Vanderbilt University [18] designed a number of vehicles using piezoelectric unimorph actuators to generate wing motions. By using actuators with different excitation frequencies between wings, they created differing wing stroke amplitudes and thus are able to modulate lift bilaterally between vehicle halves. Researchers at CMU also created flapping vehicles with piezoelectric cantilever bimorphs (similar to those described in detail in Ref. [33]); however, these vehicles were not able to achieve sufficient thrust for flight [12]. UC, Berkeley's MFI project attempted to create an insect-scale flapping wing MAV using four piezoelectric cantilever bimorphs [22]. These actuators, two per wing, were mapped to the desired flapping and rotation motions through planar and spherical flexure-based transmission mechanisms [22]. This device was able to demonstrate high wingbeat frequencies and generate lift suitable for takeoff of an insect-scale device [36]. Additionally, researchers at the AFRL created a piezo-driven flapping vehicle [37]. Simulations demonstrated that this vehicle could control horizontal and vertical forces as well as roll and yaw moments with split-cycle wingbeat control. At the millimeter scale, researchers at the ARL built a PiezoMEMS-driven wing which could control wing stroke motion and wing pitch motion independently [21].
## Mechanisms for Flight Control
The forward-flight capable vehicles discussed in Ref. [1] use multiple control surfaces to generate torques to either stabilize or control the MAV. These include static, rudder, and ruddervator tails, as well as independently controlled wings that can flap or four symmetric clapping wings. At the centimeter scale, we see fewer examples of articulated control surfaces, likely due to the challenges for small-scale actuation.
### Articulated Control Surfaces and Modification of Wing Shape.
Recently, researchers at the University of Illinois at Urbana-Champaign have developed the “BatBot” that uses articulated wing joints to actively change wing shape during flight to initiate controlled flight maneuvers [38]. The wing stroke is controlled by a mechanical oscillator that couples the left and right sides. Each wing is individually actuated to allow for asynchronous mediolateral motion. By connecting the three primary revolute joints at the shoulder, elbow, and wrist with rigid links, the robot is able to control the shoulder angle, elbow angle, and wrist angle with one DOF. Given the arrangement of the mechanical skeleton, passive DOFs on the wing tip include flexion–extension, pronation, and abduction–adduction. An additional actuated leg mechanism on each wing allows for control of the trailing edge of the membrane wing, which can increase the angle of attack at the tail. A silicone-based wing membrane adapts to changes in the wing skeleton.
The Nanohummingbird utilizes a string-based flapping mechanism to generate wing stroke motion [3]. In this system, two strings are connected to a crankshaft driven by a central motor. Each string is attached to two pulleys on the wing hinge flapping axis, such that as the crankshaft turns, the pulleys oscillate to generate the wing motion. Additional strings between the two pulleys maintain the symmetric phasing of the wing motion. This system reduces the mass of the overall control mechanism by eliminating the need for heavier, traditional mechanism designs. The vehicle generates control torques by varying the wing rotation and wing twist. To generate roll torque, the angle of attack varies between the wings to create asymmetric lift forces between the halves. In pitch, the angle of attack is varied between the fore and aft stroke, creating asymmetric lift forces in front or behind the center of mass (COM) of the vehicle. The angle of attack is controlled through wing twisting. During a wing stroke, the wing membrane is able to passively deform, but the root spar of the wing is actively controlled in relation to the leading edge spar, similar to a sail. To generate yaw torque, the angle of attack is varied by controlling the wing rotation amount by actively controlling the stop angle, creating asymmetric drag forces between the wing halves. The final vehicle had a wing span of approximately 16 cm and a total mass of 19 g. The vehicle can hover as well as fly forward at a maximum speed of 6.7 m/s for approximately 4 min.
### Pure Wing Articulation.
In the smaller-scale vehicles discussed in this paper, the use of additional control surfaces and actuators becomes impractical given the strict mass and size constraints. The dominant mechanisms for body torque control in insects involves—sometimes subtle—variations to the nominal wing motion, with no active articulation on the wing surface. The MFI uses two independent actuators to actively control wing stroke motion and wing rotation. The flapping motion is generated by mapping the two independent rotations through a spherical five-bar differential transmission. The phasing of the actuators determines the wing stroke and rotation. This vehicle demonstrated sufficient lift to takeoff, but has not demonstrated open-loop flight [36].
To reduce the number of actuators necessary for varying wing morphology, many researchers use a wing hinge that allows the wing to passively pitch due to the inertial and aerodynamic forces on the wing during the wing stroke, eliminating an actuator to control wing rotation. Examples include the Harvard Robot Moth [11], the electromagnetic flyer from Ref. [17], and the CMU vehicles [12]. The wing stroke amplitude and frequency for the Harvard Robot Moth is controlled by a DC motor and transmitted through a crank-slider mechanism to each wing. In previous versions, additional elastic elements were added to the transmission system to store energy, thus decreasing total input power [39]. This vehicle has demonstrated open-loop forward flight through the use of passive stabilization surfaces. Additionally, the electromagnetic flyer in Ref. [17] converts the oscillating motion of the magnet to wing stroke motion through two planar four-bar transmissions. The vehicle demonstrated sufficient thrust-to-weight to take-off on vertical guide rails. The CMU vehicle from Ref. [28] uses the helical spring attached to the output of the motor allows users to vary flapping frequency by adjusting the stiffness of spring. The vehicle has a wingspan of approximately 20 cm, a mass of 2.7 g, and produced a thrust-to-weight ratio of 1.4. Two power actuators are used to power flight and simultaneously generate torques. Roll torque is generated by varying the amplitude of the signal between vehicle halves. The vehicle generates pitch torque by introducing a constant voltage bias to the motor. This is similar to the RoboBee torque generation described in Sec. 5.2.
Researchers at NYU created a flapping wing MAV that employs a flapping motion that opens and closes four wings, resembling a jellyfish. This system has one motor attached to two vertical loops, which attach to an upper loop that acts as a fulcrum. As the motor rotates, the wings are pushed in or pulled out through a system of lightweight links. By tuning the voltage of the motor, the vehicle is able to demonstrate stable upward flight and by increasing the flapping amplitude of one half relative to the other by changing the link lengths, the vehicle can fly in a predetermined direction. This vehicle has also demonstrated successful hovering flight without any external control [13].
## Control Electronics
Much of the research into flapping-wing robots has focused on the design and construction of the mechanical system to demonstrate open-loop flight ability. Passive mechanisms, such as sails [40] or tails, can act as aerodynamic dampers to stabilize the vehicle during flight. Passive stability can also be built into the design with proper positioning of the wings at a dihedral angle [41] or lowering the COM of the vehicle relative to the wings. Without passive elements, active sensorimotor systems like those found in insects must be developed to stabilize the vehicle and control flight. Vehicles that have demonstrated autonomous flight using onboard control electronics are, notably, the Nanohummingbird and the Delfly [42].
The two-wing, tailless design of the Nanohummingbird creates an attitude instability that requires low latency sensory feedback of the vehicle's orientation to remain in flight. In addition, researchers included an onboard vision system for navigation and obstacle avoidance for future missions (currently no vision processing is completed onboard—information is transmitted to a ground station). The wings of the Delfly II and the Delfly Micro are positioned symmetrically at a positive dihedral angle to allow for passive stability during lateral flight [42]. This relaxes the requirements for onboard sensors, simplifying the control electronics. These vehicles have onboard cameras and transmitters to control flight through teleoperation from a ground station. The Delfly II was also equipped with an onboard barometer to allow for autonomous altitude control.
## Progress on the RoboBee
The Harvard “RoboBee” project represents a concerted effort to create an autonomous insect-scale flapping wing MAV. The advances of this project in the fabrication of mesoscale devices, manufacturing of highly energy dense actuators, and design of new custom integrated circuits (ICs) have greatly furthered this goal, and are outlined in the following.
The overall design of the RoboBee flight apparatus has been described extensively, for example, in Refs. [4], [43], and [44]. In particular, Whitney et al. described the aeromechanics of the flight apparatus [45] which assumes a rigid flat wing and a passive rotational hinge that enables quasi-static wing pitching (quasi-static relative to the resonant wing stroke motion). This relied on the blade-element method and was further refined by Chen et al. to describe impacts of stroke-pitching phase on vortex creation, shedding, and lift generation [46]. This study also explored wing geometry and scaling in order to match to the actuation and transmission mechanism. More generally, Whitney et al. described tradeoffs in sizing and actuation frequency for flapping wing MAVs [47]—those guidelines can be used to describe the size and specifications for all components of the vehicle's propulsion system, such as the critical choice of actuator type and size.
### Actuation.
At the scale and flapping frequency of robotic insects, the composite piezoelectric bimorph actuators optimized for energy density outperform similarly-sized DC motors and other microactuation technologies in terms of bandwidth, efficiency, and power density [33]. The actuators also integrate well into the fabrication process used for creation of the transmission. Wood et al. discuss the optimal geometry and drive configuration of piezoelectric actuators for microrobotic applications, maximizing force, and displacement in low-mass applications [33].
Researchers in Ref. [32] then developed new manufacturing and assembly methods to increase energy and power density of piezoelectric bending actuators by increasing the mechanical flexural strength and dielectric strength. This paper also discussed methods for mass manufacturing, a step toward making piezoelectric actuators more ubiquitous in flapping wing MAVs. The authors then designed multilayer piezoelectric actuators [48], using four active layers and thinner materials. This lowers the operating voltage, which reduces the complexity of the drive circuitry, minimizing payload, as well as increases efficiency, as efficiency increases with decreasing drive voltage due to losses in the drive stage.
#### Power Electronics.
The primary limitation of piezoelectric actuators are the high drive voltages (150–200 V) required to create the necessary force and displacement to maximize the work that these actuators can perform. However, these fields may be higher than the depoling threshold of the piezoelectric material. Therefore, it is important to drive the bimorph with a unipolar drive signal. Researchers in Ref. [33] describe the drive configurations for a piezoelectric cantilever bimorph that meet these constraints. In “alternating” drive, two unipolar drive stages are connected to the outer electrodes, operated 180 deg out of phase, with a common ground in the center of the electrode. In “simultaneous” drive, a constant high-voltage bias is applied across the actuator and the center electrode is driven with a unipolar drive stage. In the simultaneous drive configuration, a number (n) of actuators can share the high-voltage bias, requiring (n − 1) fewer drive stages. This reduces the complexity of the power electronics.
There are a number of circuit topologies to create the high-voltage drive signals necessary for RoboBee actuation. Researchers in Ref. [49] created a hybrid boost converter with a cascaded charge pump circuit to create the high-voltage bias line. Researchers in Ref. [50] discuss the design of custom power electronics which consists of two stages—a DC–DC conversion stage to create the high voltage bias line and a drive stage to generate the drive signal for a single actuator. To reduce mass, researchers in Ref. [51] developed a custom power electronics unit which included a tapped-inductor boost converter and a custom 16 mg driver IC that produces two sinusoidal drive signals (one for each wing). The total mass of the power electronics comes to 40 mg.
To increase the efficiency of the power electronics, Lok et al. analyzed the use of the alternating drive configuration to recover unused energy using dynamic common mode adjustment, envelope tracking, and charge sharing, which would result in a 30–47% decrease in power consumption while reducing weight by 37% relative to a version based on discrete components [51].
### Flight Control Through Wing Articulation.
Early in the RoboBee project, critical questions were how to design the propulsion system, how to modulate body torques, and how to merge these functions if possible. As described in Sec. 3, it becomes more rare for small-scale vehicles to have independently actuated control surfaces. Instead, all body torques in the RoboBee are generated by modulating the wing kinematics. This bioinspired approach has taken cues from several orders of insects. For example, the separation of power and control actuators in Refs. [44] and [52] has analogies to the direct and indirect flight muscles in Dipteran insects [53], while the independently controlled wings of the Dual Actuator Bee design are reminiscent of Odonata (see Fig. 3 for the vehicle generations of the RoboBee project).
Regardless of the actuator and transmission design, and similar to the two-winged, tailless vehicles discussed in Sec. 3, the two wings of the RoboBee are the control surfaces of these vehicles. To execute control maneuvers, the RoboBee generates torques through subtle variations of the wing motion. For each wing, we have control over the wing stroke amplitude, the flapping speed, wing stroke bias (i.e., fore or aft with respect to the COM), as well as the wing pitch angle. The wing stroke motion and flapping frequency are controlled directly through the actuation signal. The passive hinge at the base of the wing controls the wing pitch angle. As flapping speed increases, the inertial and aerodynamic forces on the wing increase and the wing passively pitches due to the compliance of the flexure hinge at the base of the wing and the location of this rotational axis relative to the wing leading edge [43,45]. Resonance is exploited in the system to generate large wing stroke amplitudes and to avoid reactive losses to the inertia of the wing. We generate roll torque by varying the relative wing stroke amplitude, pitch torque by moving the mean stroke angle fore or aft of the COM, and yaw torque by inducing asymmetric drag on the wings (see Fig. 1 for axes definition).
Finio et al. took inspiration from Drosophilia, which have antagonistic “indirect” power muscles and a smaller “direct” control muscles that inject directly onto each wing to fine tune the wing motion during flight [53]. Two control actuators were added to the previous Harvard microrobotic fly (HMF) design [43] to tune the transmission ratio between the wing and the power actuator during flight. The vehicle demonstrated open-loop pitch and roll maneuvers [44]; however, these torques were highly coupled and efforts to control the vehicle in flight were unsuccessful.
To reduce the number of actuators, Ma et al. [23] revisited the original HMF design and split the power actuator in two, creating two independent halves, each wing with its own power actuator (see Fig. 4). This design generated decoupled pitch and roll torques due to the decoupling of each half. This vehicle, with a mass of 80 mg and a wingspan of 2.5 cm, demonstrated the first controlled hovering flight of an insect-scale vehicle [23] and has been used as a platform for many of the control and sensing experiments described in the following. This design, however, could not create sufficient yaw torque to control orientation. The vehicle generates yaw torque by creating asymmetric drag forces on the wings by varying the speed of the up and down strokes [55,56]. However, the second-order dynamics of the actuator-transmission-wing system filters inputs to the actuators, hindering yaw torque production by reducing the energy present in higher harmonics of the drive signal [57].
To this end, Teoh et al. created a single power, single control actuator design inspired by the fruit fly (D. melanoaster). In this design, the control actuator biases the wing hinge to create asymmetric drag forces on the up and down strokes to generate yaw torque. While this vehicle produced greater yaw torque than previous designs [52], the single power actuator again led to coupling between the two halves and could not produce sufficient pitch and roll torques to control flight. Currently, a new vehicle design is being explored, decoupling the two halves with a single power and single control actuator driving each wing. This design has demonstrated hovering flight and heading control [24]. This vehicle design weighs 110 mg due to the additional mass of the control actuators, which could limit the payload capacity for the control electronics necessary to stabilize the vehicle in flight.
#### Control Demonstrations.
The challenges for successful flight at the insect scale also extend to sensing and control. As the vehicle becomes smaller, the rate of rotational acceleration increases, scaling as $l−1$ [58]. This challenge is compounded by the inherent dynamic instability of hovering wing kinematics [59,60]. Therefore, not only must a flight controller perform continuous corrective maneuvers, but also it must do so with a time delay that is orders of magnitude shorter because of the smaller length scale.
Scaling effects on the flapping dynamics and body dynamics also play a critical role in the operation of the control system. For vehicles the size of the RoboBee, the closed-loop body dynamics are approximately an order of magnitude slower than the wingbeat frequency. This implies that control corrections to the wing motions can happen over several wingbeats. These two regimes (body and wing dynamics) converge at larger scales, requiring control on a per-wing-stroke basis.
Researchers have demonstrated a number of controlled flight experiments to demonstrate the vehicle's maneuverability. From an integration perspective, these controllers must be minimally computationally expensive while also being sufficient to perform the desired task. The simplest demonstration of controlled flight was in Ref. [61], where the addition of a passive mechanism mitigated the need for an active controller; however, it greatly reduced the vehicle's maneuverability. The first demonstration of controlled hovering flight used an adaptive controller to perform attitude stabilization and control lateral position and altitude to track a desired trajectory [62]. An iterative learning control algorithm was then developed to allow the vehicle to perform aggressive maneuvers such as perching on a vertical surface [63]. Researchers have also developed computationally inexpensive controllers. Using a model-free approach, multiple proportional–integral–derivative control loops are able to stabilize the vehicle during flight [64]. While previous controllers performed feedback at a rate of 5–10 kHz, recently, researchers experimentally determined that the control loop can control hovering flight (as in Ref. [64]) at frequencies of 250 Hz. This provides the potential for general purpose MCUs to be sufficient to stabilize flight and perform simple maneuvers.
### Control Electronics.
All of the controlled flight experiments listed earlier were performed inside a motion capture arena, constrained to a flight volume of approximately one cubic foot. An external computer was used to compute the updated control parameters and generate new drive signals, which are supplied through thin wires to the vehicle's actuators. To render the vehicle autonomous, we must integrate onboard sensors, an MCU, and power electronics (see Fig. 5). An onboard MCU must read onboard sensor information, compute the vehicle's state and update controller commands, as well as generate updated drive signals to interface with the onboard power electronics. This MCU must meet the strict mass and power requirements of the vehicle, eliminating the majority of off-the-shelf MCUs. These MCUs must also compute control commands and generate drive signals for the power electronics with a clock frequency of 60 MHz [65].
To meet these demands, researchers have created a custom “brain” IC with a mass of 6 mg. This chip contains a 32-bit ARM Cortex-M0; four dedicated hardware accelerators (custom circuits that perform single functions with high speed and efficiency), one for image processing, one for estimating rotations, one for body control (process sensory information, update the state estimate, and determine the necessary torque command to stabilize flight), and one for actuator control (convert the torque command to drive signals); I2C, SPI, and GPIO buses; four ADC channels; and an internal voltage regulator to reduce peripheral components onboard the vehicle [65]. This IC has demonstrated sufficient performance to meet the real-time demands of an autonomous flight using simulated flight data and is currently being migrated onboard the vehicle (see Fig. 6 for control and power schematic).
In addition to custom MCUs, researchers have investigated many sensors that meet the low mass, power, and latency requirements of the vehicle and have demonstrated their use in flight. These sensors include an off-the-shelf gyroscope [66], magnetometer [67], and custom ocelli [68] to stabilize the vehicle's attitude; as well as an off-the-shelf infrared time-of-flight sensor [69] and a custom optic flow sensor [70] to estimate the vehicle's altitude (see Table 1 for mass and power requirements of these components).
To increase payload capacity to accommodate these electrical components, Jafferis et al. created a nonlinear resonance model for under-actuated flapping wing MAVs with passively rotating wing hinges, like the RoboBee [71]. With this model, they determined an optimal pitch angle of 70 deg, and a narrow force window to exploit resonance, increasing the vehicle's payload from 40 mg to 170 mg. In addition, Ma et al. created a design methodology to further scale the vehicle to meet the payload requirements of onboard power electronics and energy storage [54].
### Future Directions.
While there have been significant breakthroughs in the manufacturing, control, and actuation of insect-scale flapping wing MAVs, significant research needs to be conducted to integrate the control electronics onto an autonomous vehicle. First, we must determine the minimum number of sensors necessary to stabilize flight. Additional sensors (such as vision) may be needed for various applications. From this, we must determine the minimum sensor latency for adequate state estimation in free flight to determine the minimum computational expense (e.g., floating point operations per second or similar). We must also determine the minimum control requirements to both stabilize the vehicle and have it navigate in the environment. This will help determine the minimum number of instructions that a MCU will need to perform.
The most significant limitation in creating an autonomous insect-scale flapping wing MAV is an onboard power source that meets the stringent mass and size requirements while having sufficient energy density. Current options that meet these requirements include electrochemical and solar. In Ref. [72], researchers performed a system-level optimization on the energetics of flapping-wing flight motivated by maximizing flight time, specifically looking at design parameters such as payload mass, battery energy density, actuator energy density, and power electronics efficiency. This optimization estimates vehicle size and flapping frequency as well as the mass fraction of an onboard battery. Given the current state of the RoboBee project and using this framework, we estimate that a power-autonomous vehicle will consume 400 mW during hovering flight and can accommodate a battery mass no more than 100 mg.
Commercial lithium polymer and lithium ion batteries are the most commonly used energy storage devices for modern robots. However, there is a dearth of batteries appropriate for the scale of the RoboBee. Batteries produced by FullRiver contain near constant power density (on the order of 1–3 W/g) as the mass scales through three orders of magnitude (0.4–100 g). However, these batteries do not meet the mass requirements of the RoboBee, with the smallest being 4× larger than the target battery payload. Recent work in lithium polymer batteries has demonstrated the development of a number of microbatteries at the millimeter scale (see Refs. [73] and [74] for a review). For example, Lai et al. [75] created a microbattery with a volume of 6 mm3 and a maximum dimension of 3 mm with a power density of 150–200 WL−1. Assuming a density of approximately 2 g/cm3, this would provide 75–100 mW/g.
Solar cells are also a viable option. Recent work has demonstrated their effectiveness on a 3 g autonomous legged microrobot [76]. These cells (epitaxial lift-off solar cells 1-6615-8, MicroLink Devices, Niles, IL) weigh 10 mg per cell and have a 30% efficiency, providing 7.5 mW per cell in 1Sun (1000 Wm−2).
Moving forward, further investigation into the integration of mechanical and electrical systems to create an autonomous insect-scale MAV is required. There has been tremendous progress in the conceptual vehicle design and component-level design, but emphasis on full vehicle integration and associated tradeoffs is needed to realize fully autonomous PAVs. Once this is achieved, the field can move forward to begin taking advantage of the past decades worth of work on aggressive control methods for larger-scale MAVs.
## Conclusions
This paper has presented a summary of power and control architectures, including propulsive mechanisms, for insect-scale flapping wing MAVs. Power architectures for centimeter-scale vehicles were classified into various actuation technologies available at that scale. We discussed the propulsive mechanisms of these vehicles, which rely on slight modifications to nominal wing motion, rather than articulated control surfaces present in larger UAVs. Due to the stringent payload capacity, there are only two representative examples of vehicles at this scale that have demonstrated any level of control during flight. We then outlined recent advances and current challenges associated with creating an autonomous insect-scale MAV.
In the design of autonomous insect-scale flapping wing MAVs, there are inherent tradeoffs associated with actuator, control, and propulsion methodologies. To design a vehicle at this scale, it is important to understand the different technologies and their implications for power consumption, flight time, and range. This review has summarized the differing designs of insect-scale flapping wing MAVs and explored the advantages and disadvantages of using various actuation, propulsive mechanisms, control methodologies, as well as onboard control and power electronics.
Due to the physics of scaling, actuation for insect-scale MAVs is confined to nontraditional actuation technologies, such as chemical muscles, custom electromagnetic actuators, and piezoelectric actuators. Electromagnetic actuators have a low input voltage but high power consumption; in comparison, piezoelectric actuators require high input voltages and additional payload for high-voltage power electronics but lower power consumption. This analysis was extended to flapping wing propulsion and the desired wing motions. While articulated control surfaces on the wings could allow the vehicle to mimic kinematics of a biological counterpart, the additional actuators and mechanisms may not meet the mass and power requirements of the vehicle. Additionally, many designers have chosen passive elements, such as wing hinges, to further reduce the number of actuators. The level of autonomy and maneuverability required for these vehicles also dictated the complexity of onboard avionics and their power consumption. These tradeoffs were then explored in the insect-scale RoboBee, where the development of custom manufacturing, actuation, control, and electronics were necessary to meet the strict mass and power requirements of the vehicle.
## Acknowledgment
This work is supported by the Wyss Institute for Biologically Inspired Engineering. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
## Funding Data
• National Science Foundation (Award No. IIS-1514306).
## References
References
1.
Gerdes
,
J. W.
,
Gupta
,
S. K.
, and
Wilkerson
,
S. A.
,
2012
, “
A Review of Bird-Inspired Flapping Wing Miniature Air Vehicle Designs
,”
ASME J. Mech. Rob.
,
4
(
2
), p.
021003
.
2.
Liu
,
H.
,
Ravi
,
S.
,
Kolomenskiy
,
D.
, and
Tanaka
,
H.
,
2016
, “
Biomechanics and Biomimetics in Insect-Inspired Flight Systems
,”
Phil. Trans. R. Soc. B
,
371
(
1704
), p.
20150390
.
3.
Keennon
,
M.
,
Klingebiel
,
K.
, and
Won
,
H.
,
2012
, “Development of the Nano Hummingbird: A Tailless Flapping Wing Micro Air Vehicle,”
AIAA
Paper No. 2012-0588.
4.
Wood
,
R. J.
,
Finio
,
B.
,
Karpelson
,
M.
,
Ma
,
K.
,
Pérez-Arancibia
,
N. O.
,
Sreetharan
,
P. S.
,
Tanaka
,
H.
, and
Whitney
,
J. P.
,
2012
, “
Progress on Picoair Vehicles
,”
Int. J. Rob. Res.
,
31
(
11
), pp.
1292
1302
.
5.
Floreano
,
D.
, and
Wood
,
R. J.
,
2015
, “
Science, Technology and the Future of Small Autonomous Drones
,”
Nature
,
521
(
7553
), pp.
460
466
.
6.
Trimmer
,
W. S.
,
1989
, “
Microrobots and Micromechanical Systems
,”
Sens. Actuators
,
19
(
3
), pp.
267
287
.
7.
Dickinson
,
M. H.
,
Lehmann
,
F.-O.
, and
Sane
,
S. P.
,
1999
, “
Wing Rotation and the Aerodynamic Basis of Insect Flight
,”
Science
,
284
(
5422
), pp.
1954
1960
.
8.
Alben
,
S.
, and
Shelley
,
M.
,
2005
, “
Coherent Locomotion as an Attracting State for a Free Flapping Body
,”
,
102
(
32
), pp.
11163
11166
.
9.
Lentink
,
D.
, and
Dickinson
,
M. H.
,
2009
, “
Rotational Accelerations Stabilize Leading Edge Vortices on Revolving Fly Wings
,”
J. Exp. Biol.
,
212
(
16
), pp.
2705
2719
.
10.
Miller
,
L. A.
, and
Peskin
,
C. S.
,
2009
, “
Flexible Clap and Fling in Tiny Insect Flight
,”
J. Exp. Biol.
,
212
(
19
), pp.
3076
3090
.
11.
Rosen
,
M. H.
,
le Pivain
,
G.
,
Sahai
,
R.
,
Jafferis
,
N. T.
, and
Wood
,
R. J.
,
2016
, “
Development of a 3.2 g Untethered Flapping-Wing Platform for Flight Energetics and Control Experiments
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Stockholm, Sweden, May 16–21, pp.
3227
3233
.
12.
Hines
,
L.
,
2014
, “Design and Control of a Flapping Flight Micro Aerial Vehicle,”
Ph.D. thesis
, Carnegie Mellon University, Pittsburgh, PA.https://www.ri.cmu.edu/publications/design-and-control-of-a-flapping-flight-micro-aerial-vehicle/
13.
Ristroph
,
L.
, and
Childress
,
S.
,
2014
, “
Stable Hovering of a Jellyfish-like Flying Machine
,”
J. R. Soc. Interface
,
11
(
92
), p.
20130992
.
14.
Weiss
,
R.
,
2007
, “Dragonfly or Insect Spy? Scientists at Work on Robobugs,” Washington Post Company, Washington, DC, accessed Dec. 28, 2017, http://www.washingtonpost.com/wp-dyn/content/article/2007/10/08/AR2007100801434.html
15.
Colozza
,
A.
,
Michelson
,
R.
, and
Naqvi
,
M.
,
2002
, “Planetary Exploration Using Biomimetics-An Entomopter for Flight on Mars,” NASA Institute for Advanced Concepts Project, Ohio Aerospace Institute (OAI), Cleveland, OH, Report No. NAS5-98051.
16.
Tanaka
,
H.
, and
Shimoyama
,
I.
,
2010
, “
Forward Flight of Swallowtail Butterfly With Simple Flapping Motion
,”
Bioinspiration Biomimetics
,
5
(
2
), p.
026003
.
17.
Zou
,
Y.
,
Zhang
,
W.
, and
Zhang
,
Z.
,
2016
, “
Liftoff of an Electromagnetically Driven Insect-Inspired Flapping-Wing Robot
,”
IEEE Trans. Rob.
,
32
(
5
), pp.
1285
1289
.
18.
Cox
,
A.
,
Monopoli
,
D.
,
Cveticanin
,
D.
,
Goldfarb
,
M.
, and
Garcia
,
E.
,
2002
, “
The Development of Elastodynamic Components for Piezoelectrically Actuated Flapping Micro-Air Vehicles
,”
J. Intell. Mater. Syst. Struct.
,
13
(
9
), pp.
611
615
.
19.
Steltz
,
E.
,
Wood
,
R. J.
,
,
S.
, and
Fearing
,
R. S.
,
2005
, “
Characterization of the Micromechanical Flying Insect by Optical Position Sensing
,”
IEEE International Conference on, Robotics and Automation
(
ICRA
), Barcelona, Spain, Apr. 18–22, pp.
1252
1257
.
20.
Anderson
,
M. L.
,
2011
, “Design and Control of Flapping Wing Micro Air Vehicles,”
Ph.D. thesis
, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH.http://www.dtic.mil/dtic/tr/fulltext/u2/a549053.pdf
21.
Smith
,
G. L.
,
Pulskamp
,
J. S.
,
Sanchez
,
L. M.
,
Potrepka
,
D. M.
,
Proie
,
R. M.
,
Ivanov
,
T. G.
,
Rudy
,
R. Q.
,
Nothwang
,
W. D.
,
Bedair
,
S. S.
,
Meyer
,
C. D.
, and
Polcawich, R. G.
,
2012
, “
PZT-Based Piezoelectric MEMS Technology
,”
J. Am. Ceram. Soc.
,
95
(
6
), pp.
1777
1792
.
22.
Wood
,
R.
,
,
S.
,
Sahai
,
R.
,
Steltz
,
E.
, and
Fearing
,
R.
,
2008
, “
Microrobot Design Using Fiber Reinforced Composites
,”
ASME J. Mech. Des.
,
130
(
5
), p.
052304
.
23.
Ma
,
K. Y.
,
Chirarattananon
,
P.
,
Fuller
,
S. B.
, and
Wood
,
R. J.
,
2013
, “
Controlled Flight of a Biologically Inspired, Insect-Scale Robot
,”
Science
,
340
(
6132
), pp.
603
607
.
24.
Teoh
,
Z. E.
,
2015
, “Design of Hybrid Passive and Active Mechanisms for Control of Insect-Scale Flapping-Wing Robots,”
Ph.D. thesis
, Harvard University, Cambridge, MA.https://dash.harvard.edu/handle/1/23845481
25.
Sreetharan
,
P. S.
,
Whitney
,
J. P.
,
Strauss
,
M. D.
, and
Wood
,
R. J.
,
2012
, “
Monolithic Fabrication of Millimeter-Scale Machines
,”
J. Micromech. Microeng.
,
22
(
5
), p.
055027
.
26.
Trimmer
,
W.
, and
Jebens
,
R.
,
1989
, “
Actuators for Micro Robots
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Scottsdale, AZ, May 14–19, pp.
1547
1552
.
27.
Karpelson
,
M.
,
Wei
,
G.-Y.
, and
Wood
,
R. J.
,
2008
, “
A Review of Actuation and Power Electronics Options for Flapping-Wing Robotic Insects
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Pasadena, CA, May 19–23, pp.
779
786
.
28.
Hines
,
L.
,
Campolo
,
D.
, and
Sitti
,
M.
,
2014
, “
Liftoff of a Motor-Driven, Flapping-Wing Microaerial Vehicle Capable of Resonance
,”
IEEE Trans. Rob.
,
30
(
1
), pp.
220
232
.
29.
Baek
,
S. S.
,
Ma
,
K. Y.
, and
Fearing
,
R. S.
,
2009
, “
Efficient Resonant Drive of Flapping-Wing Robots
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), St. Louis, MO, Oct. 10–15, pp.
2854
2860
.
30.
Michelson
,
R. C.
, and
Reece
,
S.
,
1998
, “
Update on Flapping Wing Micro Air Vehicle Research-Ongoing Work to Develop a Flapping Wing, Crawling ‘Entomopter’
,”
13th Bristol International RPV Conference
, Bristol, UK, Mar. 30–Apr. 1, p.
11
.https://www.researchgate.net/publication/237716432_UPDATE_ON_FLAPPING_WING_MICRO_AIR_VEHICLE_RESEARCH_Ongoing_work_to_Develop_a_Flapping_Wing_Crawling_Entomopter
31.
Dudley
,
R.
,
2002
,
The Biomechanics of Insect Flight: Form, Function, Evolution
,
Princeton University Press
, Princeton, NJ.
32.
Jafferis
,
N. T.
,
Smith
,
M. J.
, and
Wood
,
R. J.
,
2015
, “
Design and Manufacturing Rules for Maximizing the Performance of Polycrystalline Piezoelectric Bending Actuators
,”
Smart Mater. Struct.
,
24
(
6
), p.
065023
.
33.
Wood
,
R.
,
Steltz
,
E.
, and
Fearing
,
R.
,
2005
, “
Optimal Energy Density Piezoelectric Bending Actuators
,”
Sens. Actuators A: Phys.
,
119
(
2
), pp.
476
488
.
34.
Newnham
,
R.
,
Dogan
,
A.
,
Xu
,
Q.
,
Onitsuka
,
K.
, and
Yoshikawa
,
S.
,
1993
, “
Flextensional ‘Moonie’ Actuators
,”
IEEE Ultrasonics Symposium,
Baltimore, MD, Oct. 31–Nov. 3, pp.
509
513
.
35.
York
,
P.
, and
Wood
,
R.
,
2017
, “
A Geometrically-Amplified In-Plane Piezoelectric Actuator for Mesoscale Robotic Systems
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Singapore, May 29–June 3, pp. 1263–1268.
36.
Steltz
,
E.
,
,
S.
, and
Fearing
,
R. S.
,
2007
, “
High Lift Force With 275 Hz Wing Beat in MFI
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), San Diego, CA, Oct. 29–Nov. 2, pp.
3987
3992
.
37.
Doman
,
D. B.
,
Oppenheimer
,
M. W.
, and
,
D. O.
,
2010
, “
Wingbeat Shape Modulation for Flapping-Wing Micro-Air-Vehicle Control During Hover
,”
J. Guidance, Control, Dyn.
,
33
(
3
), pp.
724
739
.
38.
Ramezani
,
A.
,
Chung
,
S.-J.
, and
Hutchinson
,
S.
,
2017
, “
A Biomimetic Robotic Platform to Study Flight Specializations of Bats
,”
Sci. Rob.
,
2
(
3
), p. eaal2505.
39.
Sahai
,
R.
,
Galloway
,
K. C.
, and
Wood
,
R. J.
,
2013
, “
Elastic Element Integration for Improved Flapping-Wing Micro Air Vehicle Performance
,”
IEEE Trans. Rob.
,
29
(
1
), pp.
32
41
.
40.
Van Breugel
,
F.
,
Regan
,
W.
, and
Lipson
,
H.
,
2008
, “
From Insects to Machines
,”
IEEE Rob. Autom. Mag.
,
15
(
4
), pp. 68–74.
41.
Koopmans
,
J.
,
Tijmons
,
S.
,
De Wagter
,
C.
, and
de Croon
,
G.
,
2015
, “
Passively Stable Flapping Flight From Hover to Fast Forward Through Shift in Wing Position
,”
Int. J. Micro Air Veh.
,
7
(
4
), pp.
407
418
.
42.
De Croon
,
G.
,
De Clercq
,
K.
,
Ruijsink
,
R.
,
Remes
,
B.
, and
De Wagter
,
C.
,
2009
, “
Design, Aerodynamics, and Vision-Based Control of the Delfly
,”
Int. J. Micro Air Veh.
,
1
(
2
), pp.
71
97
.
43.
Wood
,
R. J.
,
2007
, “
Liftoff of a 60 mg Flapping-Wing Mav
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), San Diego, CA, Oct. 29–Nov. 2, pp.
1889
1894
.
44.
Finio
,
B. M.
, and
Wood
,
R. J.
,
2012
, “
Open-Loop Roll, Pitch and Yaw Torques for a Robotic Bee
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Vilamoura, Portugal, Oct. 7–12, pp.
113
119
.
45.
Whitney
,
J.
, and
Wood
,
R.
,
2010
, “
Aeromechanics of Passive Rotation in Flapping Flight
,”
J. Fluid Mech.
,
660
, pp.
197
220
.
46.
Chen
,
Y.
,
Ma
,
K.
, and
Wood
,
R. J.
,
2016
, “
Influence of Wing Morphological and Inertial Parameters on Flapping Flight Performance
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Daejeon, South Korea, Oct. 9–14, pp.
2329
2336
.
47.
Whitney
,
J.
, and
Wood
,
R.
,
2012
, “
Conceptual Design of Flapping-Wing Micro Air Vehicles
,”
Bioinspiration Biomimetics
,
7
(
3
), p.
036001
.
48.
Jafferis
,
N. T.
,
Lok
,
M.
,
Winey
,
N.
,
Wei
,
G.-Y.
, and
Wood
,
R. J.
,
2016
, “
Multilayer Laminated Piezoelectric Bending Actuators: Design and Manufacturing for Optimum Power Density and Efficiency
,”
Smart Mater. Struct.
,
25
(
5
), p.
055033
.
49.
Steltz
,
E.
,
Seeman
,
M.
,
,
S.
, and
Fearing
,
R. S.
,
2006
, “
Power Electronics Design Choice for Piezoelectric Microrobots
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Beijing, China, Oct. 9–15, pp.
1322
1328
.
50.
Karpelson
,
M.
,
Wei
,
G.-Y.
, and
Wood
,
R. J.
,
2012
, “
Driving High Voltage Piezoelectric Actuators in Microrobotic Applications
,”
Sens. Actuators A: Phys.
,
176
, pp.
78
89
.
51.
Lok
,
M.
,
Zhang
,
X.
,
Helbling
,
E. F.
,
Wood
,
R.
,
Brooks
,
D.
, and
Wei
,
G.-Y.
,
2015
, “
A Power Electronics Unit to Drive Piezoelectric Actuators for Flying Microrobots
,”
IEEE Custom Integrated Circuits Conference
(
CICC
), San Jose, CA, Sept. 28–30, pp.
1
4
.
52.
Teoh
,
Z. E.
, and
Wood
,
R. J.
,
2014
, “
A Bioinspired Approach to Torque Control in an Insect-Sized Flapping-Wing Robot
,” Fifth
IEEE
RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics,
Sao Paulo, Brazil, Aug. 12–15, pp.
911
917
.
53.
Dickinson
,
M. H.
, and
Tu
,
M. S.
,
1997
, “
The Function of Dipteran Flight Muscle
,”
Comp. Biochem. Physiol. Part A: Physiol.
,
116
(
3
), pp.
223
238
.
54.
Ma
,
K. Y.
,
Chirarattananon
,
P.
, and
Wood
,
R. J.
,
2015
, “
Design and Fabrication of an Insect-Scale Flying Robot for Control Autonomy
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Hamburg, Germany, Sept. 28–Oct. 2, pp.
1558
1564
.
55.
Oppenheimer
,
M.
,
Doman
,
D.
, and
,
D.
,
2010
, “Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions—Part I: Aerodynamic Model,”
AIAA
Paper No. 2010-1023.
56.
Doman
,
D.
,
Oppenheimer
,
M.
, and
,
D.
,
2010
, “Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions—Part II: Controller,”
AIAA
Paper No. 2010-1024.
57.
Gravish
,
N.
, and
Wood
,
R. J.
,
2016
, “
Anomalous Yaw Torque Generation From Passively Pitching Wings
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Stockholm, Sweden, May 16–21, pp.
3282
3287
.
58.
Kumar
,
V.
, and
Michael
,
N.
,
2012
, “
Opportunities and Challenges With Autonomous Micro Aerial Vehicles
,”
Int. J. Rob. Res.
,
31
(
11
), pp.
1279
1291
.
59.
Faruque
,
I.
, and
Humbert
,
J. S.
,
2010
, “
Dipteran Insect Flight Dynamics—Part 1: Longitudinal Motion About Hover
,”
J. Theor. Biol.
,
264
(
2
), pp.
538
552
.
60.
Ristroph
,
L.
,
Ristroph
,
G.
,
Morozova
,
S.
,
Bergou
,
A. J.
,
Chang
,
S.
,
Guckenheimer
,
J.
,
Wang
,
Z. J.
, and
Cohen
,
I.
,
2013
, “
Active and Passive Stabilization of Body Pitch in Insect Flight
,”
J. R. Soc. Interface
,
10
(
85
), p.
20130237
.
61.
Teoh
,
Z. E.
,
Fuller
,
S. B.
,
Chirarattananon
,
P.
,
Prez-Arancibia
,
N.
,
Greenberg
,
J. D.
, and
Wood
,
R. J.
,
2012
, “
A Hovering Flapping-Wing Microrobot With Altitude Control and Passive Upright Stability
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Vilamoura, Portugal, Oct. 7–12, pp.
3209
3216
.
62.
Chirarattananon
,
P.
,
Ma
,
K. Y.
, and
Wood
,
R. J.
,
2014
, “
Adaptive Control of a Millimeter-Scale Flapping-Wing Robot
,”
Bioinspiration Biomimetics
,
9
(
2
), p.
025004
.
63.
Chirarattananon
,
P.
,
Ma
,
K. Y.
, and
Wood
,
R. J.
,
2016
, “
Perching With a Robotic Insect Using Adaptive Tracking Control and Iterative Learning Control
,”
Int. J. Rob. Res.
,
35
(
10
), pp.
1185
1206
.
64.
Pérez-Arancibia
,
N. O.
,
Pierre-Emile
,
J. D.
,
Ma
,
K. Y.
, and
Wood
,
R. J.
,
2015
, “
Model-Free Control of a Hovering Flapping-Wing Microrobot
,”
J. Intell. Rob. Syst.
,
77
(
1
), pp.
95
111
.
65.
Zhang
,
X.
,
Lok
,
M.
,
Tong
,
T.
,
Chaput
,
S.
,
Lee
,
S. K.
,
Reagen
,
B.
,
Lee
,
H.
,
Brooks
,
D.
, and
Wei
,
G.-Y.
,
2015
, “
A Multi-Chip System Optimized for Insect-Scale Flapping-Wing Robots
,”
Symposium on VLSI Circuits (VLSI Circuits)
, Kyoto, Japan, June 17–19, pp.
C152
C153
.
66.
Fuller
,
S. B.
,
Helbling
,
E. F.
,
Chirarattananon
,
P.
, and
Wood
,
R. J.
,
2014
, “
Using a MEMS Gyroscope to Stabilize the Attitude of a Fly-Sized Hovering Robot
,”
International Micro Air Vehicle Conference and Competition
(
IMAV
), Delft University of Technology, Delft, The Netherlands, Aug. 12–15, pp. 102–109.https://repository.tudelft.nl/islandora/object/uuid%3A8b461dac-9c6d-416c-ba55-41a38eb11602
67.
Helbling
,
E. F.
,
Fuller
,
S. B.
, and
Wood
,
R. J.
,
2014
, “
Pitch and Yaw Control of a Robotic Insect Using an Onboard Magnetometer
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Hong Kong, China, May 31–June 7, pp.
5516
5522
.
68.
Fuller
,
S. B.
,
Karpelson
,
M.
,
Censi
,
A.
,
Ma
,
K. Y.
, and
Wood
,
R. J.
,
2014
, “
Controlling Free Flight of a Robotic Fly Using an Onboard Vision Sensor Inspired by Insect Ocelli
,”
J. R. Soc. Interface
,
11
(
97
), p.
20140281
.
69.
Helbling
,
E. F.
,
Fuller
,
S. B.
, and
Wood
,
R. J.
, 2017, “
Altitude Estimation and Control of an Insect-Scale Robot With an Onboard Proximity Sensor
,”
Robotics Research
, Springer, Cham, Switzerland, pp.
57
69
.
70.
Duhamel
,
P.-E. J.
,
Pérez-Arancibia
,
N. O.
,
Barrows
,
G. L.
, and
Wood
,
R. J.
,
2013
, “
Biologically Inspired Optical-Flow Sensing for Altitude Control of Flapping-Wing Microrobots
,”
IEEE/ASME Trans. Mechatronics
,
18
(
2
), pp.
556
568
.
71.
Jafferis
,
N. T.
,
Graule
,
M. A.
, and
Wood
,
R. J.
,
2016
, “
Non-Linear Resonance Modeling and System Design Improvements for Underactuated Flapping-Wing Vehicles
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Stockholm, Sweden, May 16–21, pp.
3234
3241
.
72.
Karpelson
,
M.
,
Whitney
,
J. P.
,
Wei
,
G.-Y.
, and
Wood
,
R. J.
,
2010
, “
Energetics of Flapping-Wing Robotic Insects: Towards Autonomous Hovering Flight
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Taipei, Taiwan, Oct. 18–22, pp.
1630
1637
.
73.
Wang
,
Y.
,
Liu
,
B.
,
Li
,
Q.
,
Cartmell
,
S.
,
Ferrara
,
S.
,
Deng
,
Z. D.
, and
Xiao
,
J.
,
2015
, “
Lithium and Lithium Ion Batteries for Applications in Microelectronic Devices: A Review
,”
J. Power Sources
,
286
, pp.
330
345
.
74.
Ferrari
,
S.
,
Loveridge
,
M.
,
Beattie
,
S. D.
,
Jahn
,
M.
,
Dashwood
,
R. J.
, and
Bhagat
,
R.
,
2015
, “
Latest Advances in the Manufacturing of 3D Rechargeable Lithium Microbatteries
,”
J. Power Sources
,
286
, pp.
25
46
.
75.
Lai
,
W.
,
Erdonmez
,
C. K.
,
Marinis
,
T. F.
,
Bjune
,
C. K.
,
Dudney
,
N. J.
,
Xu
,
F.
,
Wartena
,
R.
, and
Chiang
,
Y.-M.
,
2010
, “
Ultrahigh-Energy-Density Microbatteries Enabled by New Electrode Architecture and Micropackaging Design
,”
,
22
(
20
), pp. E139–E144.
76.
Brühwiler
,
R.
,
Goldberg
,
B.
,
Doshi
,
N.
,
Ozcan
,
O.
,
Jafferis
,
N.
,
Karpelson
,
M.
, and
Wood
,
R. J.
,
2015
, “
Feedback Control of a Legged Microrobot With On-Board Sensing
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Hamburg, Germany, Sept. 28–Oct. 2, pp.
5727
5733
.
|
2019-10-14 14:26:24
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-11-section-11-1-finding-limits-using-tables-and-graphs-exercise-set-page-1140/54
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## Precalculus (6th Edition) Blitzer
$\lim_{x \to 0 }f(x)$ does not exist.
To find $\lim_{x\to 0 }f(x)$, examine the graph of $f$ near $x=0$. As $x$ gets closer to $0$ from the left, the values of $f(x)$ get closer to $0$. As $x$ gets closer to $0$ from the right, the values of $f(x)$ get closer to $1$. We conclude from the graph that $\lim_{x \to 0 }f(x)$ does not exist because the left- and right-hand limits are unequal.
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2021-05-17 13:33:11
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https://math.stackexchange.com/questions/2374230/involution-that-brings-sets-to-disjoint-sets
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# Involution that brings sets to disjoint sets
Let $A$ be a collection of subsets of $\{1,2,\dots,n\}$ that is closed under taking subsets (that is, if $U\in A$ and $V\subseteq U$ then $V\in A$). Is there always an involution $f:A\to A$ such that $f(V)\cap V=\emptyset$ for all $V\in A$? I'm guessing yes.
Note that if $A=\mathcal P(\{1,2,\dots,n\})$, then taking the complement works. Also note that if $|A|$ is odd, we can send the empty set to itself (the empty set is disjoint from itself, isn't that weird?).
I tried working through a few small examples. I haven't found a counterexample, but I also haven't found a proof.
Here's a proof which gives an algorithm for constructing an involution. If the order ideal $\mathcal{A}=\emptyset$ we're done; otherwise we can select a maximal element $X$ of $\mathcal{A}$ (with respect to inclusion.) If we're really lucky, there is another maximal element $Y$ of $\mathcal{A}$ disjoint from $X$. If so, we can set $X\leftrightarrow Y$; since $\mathcal{A}'=\mathcal{A}\setminus\{X,Y\}$ is an order ideal, we can find a suitable involution on $\mathcal{A}'$ by induction on $|\mathcal{A}|$, so we're done.
In general, we won't be so lucky, but the intuition is the same. Given a maximal element $X$ of $\mathcal{A}$, select a maximal element $Y$ in $\{ Y\in \mathcal{A}\mid Y\cap X=\emptyset \}.$ We can pair $X$ with $Y$, but in general $Y$ is not maximal in $\mathcal{A}$ so we have to do a little more work before we can appeal to induction. By the choice of $Y$, every set in $\mathcal{A}$ containing $Y$ has the form $Y\cup B$ for some $B\subset X$. Let $$\mathcal{B} =\{B\subset X\mid Y\cup B\in\mathcal{A}\};$$ note that $\mathcal{B}$ is closed under taking subsets since $\mathcal{A}$ is.
For each $B\in\mathcal{B}$, we pair $Y\cup B$ with $X\setminus B$ (which we know is in $\mathcal{A}$ since $\mathcal{A}$ is an order ideal.) In short, this matches the elements of $Y\cup\mathcal{B}$ with the elements of $X\setminus\mathcal{B}$. Let $\mathcal{A}'$ be the result of removing all these paired elements from $\mathcal{A}$. Once we check $\mathcal{A}'$ is an order ideal, we're done by induction.
All we need to check is that the set of elements we removed is an order coideal of $\mathcal{A}$. (i.e. if we removed $Z$, and $W\supset Z$ is in $\mathcal{A}$, then we also removed $W$). But $Y\cup \mathcal{B}$ is an order coideal by construction; $X\setminus\mathcal{B}$ is one since $\mathcal{B}$ is an order ideal; and the union of order coideals is an order coideal. So $\mathcal{A}'$ is an order ideal, and we're done by induction. $\square$
Remark: if $\mathcal{A}$ is the power set of $X$, then $X$ is the unique maximal element of $\mathcal{A}$; the algorithm picks $Y=\emptyset$, and pairs each $B$ with the complement $X\setminus B$, so this generalizes the special case pointed out in the original post. It also generalizes the special case at the beginning of the post: if $Y$ is maximal in $\mathcal{A}$ then $\mathcal{B}=\emptyset.$
• I'm definitely gonna need to read this more carefully later, but enjoy the green checkmark – Akiva Weinberger Jul 31 '17 at 3:37
I don't know the answer, but I'd like to rephrase the question in the language of graph theory. Given a finite set $X$, the powerset $\mathcal{P}(X)$ becomes the vertex set of a graph by declaring that two subsets of $X$ have an edge between them iff they're disjoint. Furthermore, the graph $\mathcal{P}(X)$ has a perfect matching, given by complementation. Your question is whether it's true that for each downward-closed subset $\mathcal{A}$ of the poset $\mathcal{P}(X)$, the graph induced on $\mathcal{A}$ has a perfect matching.
In light of this, perhaps Tutte's theorem will help.
For any $A$ start with the smallest powerset that includes $A$ (e.g. if $A=\{\emptyset,\{1\},\{2\}\}$ start with $\mathcal P(\{1,2\})$). Use the trivial involution you suggested for this power set. Then transform this involution to a new involution by eliminating elements one by one to reach $A$ (start with bigger elements). In doing so alternate between reassigning $\emptyset$ to itself and the element that is mapped to the eliminated element. This way you can construct an involution for any $A$. You need to prove that the required conditions are preserved under these transformations. Note that each transformation is applied on the previous transformed involution.
• An example would help – Akiva Weinberger Jul 28 '17 at 10:32
• I'm not sure I understand the procedure – Akiva Weinberger Jul 30 '17 at 18:57
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2019-10-18 04:06:17
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https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-6th-edition/chapter-5-section-5-4-multiplying-polynomials-exercise-set-page-290/98
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## Intermediate Algebra (6th Edition)
$(3x+2)^2 \ne 9x^2+4$ because the square of $3x+2$ is $9x^2+12x+4$.
RECALL: $(a+b)^2=a^2 + 2ab+b^2$ $(3x+2)^2$ does not equal $9x^2+4$ because the square of a binomial is a trinomial. Squaring the binomial gives: $(3x+2)^2 \\=(3x)^2+2(3x)(2)+2^2 \\=9x^2+12x+4$
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2018-12-12 04:47:23
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http://derekogle.com/IFAR/exercises/Inch_DataManip
|
Inch Lake is a 12.5 ha inland lake in northern Wisconsin that has been managed as catch-and-release for all species since 2006. Researchers at Northland College have monitored fish populations in Inch Lake since 2007. The total lengths (inches) and weights (g) for subsamples of several species of fish collected from Inch Lake in May of 2007 and 2008 are recorded in InchLake2.csv (view, download, meta). Use these data to answer the following questions.
1. Create a new variable that contains lengths in millimeters.
2. Create a new data.frame of just Bluegill.
3. Create a new data.frame of just Largemouth Bass.
4. Create a new data.frame of nongame species (Bluntnose Minnow, Fathead Minnow, Iowa Darter, and Tadpole Madtom).
5. Remove the netID variable from the two single species data.frames created above.
6. Sort the nongame species only data.frame by species.
7. Sort the nongame species only data.frame by length within species within year.
8. Change the names of the weight variable to wt and the length in millimeters variable to tl (if you did not call it that above).
9. Create two new variables that are the common logarithms of the lengths (in mm) and weights.
10. Add appropriate five-cell Gabelhouse length categories to the Bluegill and Largemouth Bass only data.frames.
Save the script from this exercise as these data will be used in this plotting, this weight-length relationhip, and this condition exercise.
from Derek H. Ogle , created 01-Oct-15, updated 08-Nov-15, Comments/Suggestions.
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2017-11-19 12:26:53
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https://forum.wilmott.com/viewtopic.php?f=34&p=866858&sid=b4bfeda64b6eb29b60e8140cb4369b5f
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Serving the Quantitative Finance Community
EdisonCruise
Topic Author
Posts: 122
Joined: September 15th, 2012, 4:22 am
### Is it necessary to piecewisely fit power-law distribution in empirical data?
In AARON CLAUSET’s paper power-law distributions in empirical data: ‘’In practice, few empirical phenomena obey power laws for all values of x. More often the power law applies only for values greater than some minimum x_min. In such cases we say that the tail of the distribution follows a power law.”
So for the data x<x_min, is it necessary to fit with another distribution? If so, the probability density function becomes piecewise and what functions are usually used in such a case? The part x<x_min is about 95% of my data. I find few literature for this problem.
Alan
Posts: 10716
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:
### Re: Is it necessary to piecewisely fit power-law distribution in empirical data?
Not necessarily. You can cook up a density that smoothly transitions from a pure power in the tail to something else. For the case at hand, apparently on the positive axis, say
$C (a^2 + x^2)^{-b}$ or a zillion other choices
EdisonCruise
Topic Author
Posts: 122
Joined: September 15th, 2012, 4:22 am
### Re: Is it necessary to piecewisely fit power-law distribution in empirical data?
I see. Thank you so much.
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2021-08-03 16:59:35
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https://msp.org/agt/2018/18-6/p07.xhtml
|
Volume 18, issue 6 (2018)
Recent Issues
Author Index
The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
The universal quantum invariant and colored ideal triangulations
Sakie Suzuki
Algebraic & Geometric Topology 18 (2018) 3363–3402
Abstract
The Drinfeld double of a finite-dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal $R$–matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$–matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang–Baxter equation of the universal $R$–matrix. On the other hand, the Heisenberg double of a finite-dimensional Hopf algebra has the canonical element (the $S$–tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of $3$–manifolds up to colored moves. In this construction, a copy of the $S$–tensor is attached to each tetrahedron, and invariance under the colored Pachner $\left(2,3\right)$ moves is shown by the pentagon relation of the $S$–tensor.
Keywords
knots and links, 3-manifolds, Heisenberg double, Drinfeld double, universal quantum invariant, colored ideal triangulation
Mathematical Subject Classification 2010
Primary: 16T25, 57M27, 81R50
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2020-01-22 00:39:15
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http://www.andrewt.net/blog/page/2/
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# Chirp
“First tweeting – now chirping,” say the BBC on Google+, having noticed that two otherwise unrelated iPhone apps are, for unrelated reasons, named after birdsong.
Chirp is a new iPhone app that aims to solve the perennial problem faced by owners of smartphones whose manufacturers inexplicably crippled the Bluetooth support by design. This one uses sound: in the same way that dial-up modems and, before them, ZX Spectrums used to convert data to sound and send it down a phone line or store it on a tape, Chirp plays encoded data out of your phone speaker, and picks up incoming data with the mic. Brilliantly, they’ve taken the phone part out of a modem, and then put what’s left on a phone. It’s both elegant and faintly absurd, so obviously I love it, but I find myself wondering what the hell it’s for.
The fastest dial-up modem was rated 56kbps, and ran at 53.3kbps — and so took nearly three minutes to transmit a 1MB photo (not that anyone was taking 1MB photos back then). Chirp, clearly optimised for reliability rather than speed, runs at 25bps, and would take nearly four days to transmit the same photo. Instead, Chirp sends a 50-bit code that your phone sends to Chirp’s server, which returns the photo over 3G or whatever. So the process is entirely done with the phone’s data connection, except that the last part of the URL is ‘chirped’ through the air. Probably it could be sent via Chirp’s servers instead and nobody would notice for months — I expect some people still believe Bump actually works by bumping the phones together.
It seems to me that this makes Chirp slightly pointless for sharing files, links, and (they really do this) “140-character text messages” between friends. Partly that’s because I already have more ways to send short text messages to my friends’ iPhones than I could possibly ever need, but mostly it’s because I disagree with their chief executive’s claim that “it’s fairly novel to be able to transmit information to anyone who is in earshot”. In fact, humans have a built-in feature for exchanging short communiques with other humans within earshot. Building that into telephones, a device specifically invented to circumvent that limitation, therefore seems like the most monumentally pointless endeavour since the world’s most easily accessible encyclopædia was translated into Latin.
The idea of putting chirps out over tannoy systems or radio broadcasts is a more interesting. Then it’s essentially a QR code made of sound, and there are bound to be uses for that. QR codes, like Blippar and, well, more or less every attempt to make inanimate objects interactive, suffer because for every legitimate use for them there are a thousand stupid ones shoehorned in by PR nitwits. I’ve even seen an information stand with a QR code but no URL — it has machine-readable data but no human-readable data. It’s hard to shake the feeling that it was designed as a tool for the benefit of rebel robots.
The problem with chirps, QR codes, and the like is that there’s almost always a far more efficient way of doing the same thing. The obvious approach to broadcast links as sound would be to read out the URL. If that’s impractical then the correct solution is to come up with a better URL system — long strings of gibberish are rarely ideal. Even if you use sound, there doesn’t seem to be any need for the radio station to change their broadcast so much as to team up with Shazam so the app can directly recognise their station and connect users to whatever content is under discussion. Of course, most users will listen to one or two radio stations at most anyway, so 99% of the time they won’t even use that; they’ll just choose from their favourites list. And then you’re just tweeting links, which incidentally works fine.
But okay, let’s say you’re ‘chirping’ URLs. I can’t decode them: I almost never listen to audio on any device other than my iPhone, which will presumably mute said audio the moment Chirp starts listening for it. The one thing I’d like it to do — pull links out of Pod Delusion reports — is the one thing it can’t do. On the other hand, Downcast‘s ‘share via Twitter’ feature lets me send a direct message to @rtm with a link to the show-notes, which accomplishes the same thing without building a whole nother app or running yet another server full of short-coded URLs.
I’m sure there’s a use for Chirp, but I reckon you’d have to have it installed for a couple of years before it came in handy. The BBC rather worryingly suggest that the developers “see a future where you pay for a can of drink with a chirp” which seems like it would make credit-card-cloning scams so easy that there’d be an app for it on Cydia within the week.
As an alternative method of sending URLs over the phone or radio, I wondered a while back about creating a URL shortener that made links memorable rather than short. There’s a memory trick that encodes any six-digit number into a memorable image, by building them out of three pegs from a set of 100. If we could have more choices from smaller pools, we could build a site with long but easy to say, hear and remember URLs — a Bit.ly for the mind. You’d probably need a pictorial keyboard on the homepage to make it usable. I never bothered to make it principally because I don’t think it’d be very useful.
And neither, I fear, is this.
# 1bn Hiroshimas = 1 (Isle of Wight) x 20 (speeding bullets)
I think one of the problems with science journalism is that before a science story can be reported in the news media, someone has to convert everything from metric to journalist units. But some recent work may allow us to do science directly in journalist units, thereby making scientific papers immediately understandable to laypeople. According to a throwaway letter in the Guardian
1 billion Hiroshimas = 1 Isle of Wight × 20 speeding bullets
This is based on a G2 article about the asteroid that wiped out the dinosaurs that described the asteroid’s mass, energy and speed in those terms.
Unfortunately, the equation is wrong, as you can’t multiply speed by islands to get explosions. But it’s not far off. In fact, kinetic energy K = ½mv2, where m is mass and v is speed — so actually
1 Chicxulub asteroid strike = 109 Hiroshimas = ½ Isle of Wight × (20 × speeding bullet)2
I’ll forgive the correspondent the factor of 2, but he should have known that the speed needed squaring given that the only other sentence in his letter was “Who needs E=mc2?” Honestly, it’s as if some people have no grasp of dimensional analysis at all.
In fact, this is also wrong, because the Isle of Wight is more correctly a unit of area, not mass, so to use standard journalism units, we should really write
1 Chicxulub = 109 Hiroshimas = ½ (Isle of Wight³⁄₂ ρrock) × (20 × speeding bullet)2
Better still, that should be
1 Chicxulub = 109 Hiroshimas = ½ (Isle of Wight³⁄₂ ρrock) × (21 × speeding bullet)2
as the rock hit 20 times faster than a bullet, not 20 times as fast as one.
And we can test this hypothesis simply by typing “(1/2) * (density of rock * (isle of wight area)^(3/2)) * (21 * speed of bullet)^2″ into Wolfram|Alpha. It returns the figure 5.023×1023J, and if you click on that figure, it rephrases it to “≈ 1.005 × estimated energy released by the Chicxulub meteor impact”.
Let’s just bask in the impressiveness of that for a moment.
Done basking? Then it’s time to admit there are a few problems with this. Alpha cites this as 8 billion Hiroshimas, not one billion. Alpha also takes ‘a bullet’ to be a rimfire .22LR usually deployed against small pests and tin cans, whereas Dr Collins appears to favour the somewhat meatier M16 assault rifle. Maybe that’s standard for a speeding bullet. Also I assumed the asteroid was a sphere that would cover an area of land equal to one Isle of Wight. In fact the Isle of Wight is long and thin so if we spun it around its major axis it would be a bit lighter than this; equally we could attempt to estimate the mass of the Isle of Wight and that could go either way.
The point is that you absolutely can do science in these units. They totally work. We use metric instead only because the numbers are easier — 1 Joule is 1 kilogram metre per second squared, avoiding having the annoying factor of 21 kicking around that the journalism units version above does. (I’m not going to quibble about the billions, though, as you only need to define the ‘gigashima’ to make that go away.)
To make life easier for anyone choosing to do science in journalism units, I have identified some relationships that may prove useful:
• 1 coal-fired power station ≈ 1 Hiroshima per day
• 1 thickness of human hair ≈ 1000 Olympic swimming pools per area the size of Wales
• 1 weight of a double-decker bus ≈ 1 Hiroshima per distance to the moon and back
All of these are approximate, but they’re all exactly true for at least one combination of reasonable guesses, so all we have to do is identify a mutually-convenient set of plausible values, then agree to use it forever. We can’t really fiddle with
• 1 Isle of Wight = 381km2
• 1 distance to the moon and back = 3.85×108m
• 1 Wales = 20,779km2
and we know that defining
• 1 speeding bullet = 340ms−1
• density of rock = 2.65g/cm3
• 1 Chicxulub asteroid strike = 5.023×1023J
gives us one neat relationship. Let’s add to that
• 1 Hiroshima = 1 Chicxulub ÷ 8 billion = 6.27875×1013J
• 1 coal-fired power station = 1 Hiroshima ÷ 24h = 726.7MW
• 1 double-decker bus = 1 Hiroshima ÷ 1 moon and back = 8.416 tons
• 1 Olympic-size swimming pool = 2,500m3
• 1 thickness of a human hair = 1000 Olympic pool ÷ 1 Wales = 120.3μm
Now all our relationships are spot on, and we can hopefully get on with doing some science with journalism units. At least, science that involves using coal to power buses to the moon.
Ie, the best science.
# Might Qwerty be optimal on touchscreens?
It’s a common misconception that the Qwerty keyboard is designed to slow users down to prevent typewriters jamming. It fact, it’s designed to keep commonly consecutive letter pairs apart, so that two adjacent levers won’t collide.
(A more fun, but irrelevant, Qwerty story is that it is also designed such that the word ‘typewriter’ is all on the top row, to make demonstrating it easy. This story, if true, is itself fun but sucks all the fun out of the fact that the longest word that can be typed on the top row of a typewriter is ‘typewriter’. One of these is a fun fact, but I’ve no idea which.)
Nowadays, obviously, there are no swinging arms to collide, so we want the commonly-used keys to be reachable, and if possible to alternate hands as much as possible. Dvorak and Coleman have each had a stab at designing a better layout, but both aimed at the computer keyboard.
But increasingly, I type on my phone, using one very mobile thumb. I can get to any point on the screen, more-or-less right away — but sometimes I miss, and usually the phone figures out what I meant and autocorrects it. So maybe the most important thing about any given keyboard layout is how likely it is that a typo will result in a real word that the phone isn’t to know isn’t what I meant.
I wondered if suddenly Qwerty might be optimal again — separating pairs of letters that can be swapped to make another real word and that appear next to each other in English words aren’t totally different goals. So I thought I’d investigate.
So first I loaded the CSW12 Scrabble word list, and worked out a big table of how many places in the list you can replace each letter with each other letter to create a new word.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A 176 681 305 5253 182 216 252 4180 15 208 477 166 401 4717 297 5 483 953 453 2898 52 201 54 585 30 B 176 1240 1238 284 1157 1123 775 91 360 406 1003 1383 761 157 1579 14 1407 1127 1344 74 398 836 70 366 151 C 681 1240 1109 375 876 1218 843 162 261 1102 1004 1036 1379 265 1418 24 1117 1832 1783 120 418 816 153 274 209 D 305 1238 1109 715 776 1445 709 205 299 942 1600 1467 1838 227 1242 18 7549 10979 2348 107 549 726 135 616 307 E 5253 284 375 715 186 553 454 4712 22 459 938 1010 683 3434 470 4 956 2123 1734 1893 87 291 57 2162 57 F 182 1157 876 776 186 723 592 76 261 357 846 829 632 126 1040 11 782 1037 1121 57 387 620 30 203 89 G 216 1123 1218 1445 553 723 556 186 333 747 812 810 1013 191 1018 39 914 1194 1514 103 373 669 130 371 180 H 252 775 843 709 454 592 556 137 261 641 1186 979 635 260 1098 6 1079 1215 1453 81 239 805 34 356 137 I 4180 91 162 205 4712 76 186 137 35 129 519 130 380 2786 154 3 387 525 331 2671 42 200 23 1118 28 J 15 360 261 299 22 261 333 261 35 117 311 301 224 27 331 2 344 320 346 1 125 196 7 136 66 K 208 406 1102 942 459 357 747 641 129 117 947 779 892 145 911 50 816 929 1340 71 379 517 119 273 180 L 477 1003 1004 1600 938 846 812 1186 519 311 947 1420 1938 479 1363 15 3271 1876 2049 286 599 899 142 394 248 M 166 1383 1036 1467 1010 829 810 979 130 301 779 1420 1085 238 1898 14 1321 1225 2912 119 575 747 130 380 250 N 401 761 1379 1838 683 632 1013 635 380 224 892 1938 1085 286 1367 7 2439 1925 2091 301 529 711 259 440 250 O 4717 157 265 227 3434 126 191 260 2786 27 145 479 238 286 296 6 574 531 389 2291 58 314 36 596 27 P 297 1579 1418 1242 470 1040 1018 1098 154 331 911 1363 1898 1367 296 11 1255 1528 2061 166 580 1011 141 377 243 Q 5 14 24 18 4 11 39 6 3 2 50 15 14 7 6 11 0 12 30 16 5 10 2 2 2 R 483 1407 1117 7549 956 782 914 1079 387 344 816 3271 1321 2439 574 1255 12 4806 2173 447 591 885 205 613 250 S 953 1127 1832 10979 2123 1037 1194 1215 525 320 929 1876 1225 1925 531 1528 30 4806 3126 327 617 887 232 2621 6540 T 453 1344 1783 2348 1734 1121 1514 1453 331 346 1340 2049 2912 2091 389 2061 16 2173 3126 256 682 1187 215 602 429 U 2898 74 120 107 1893 57 103 81 2671 1 71 286 119 301 2291 166 0 447 327 256 43 416 15 239 12 V 52 398 418 549 87 387 373 239 42 125 379 599 575 529 58 580 5 591 617 682 43 353 96 142 154 W 201 836 816 726 291 620 669 805 200 196 517 899 747 711 314 1011 10 885 887 1187 416 353 108 400 136 X 54 70 153 135 57 30 130 34 23 7 119 142 130 259 36 141 2 205 232 215 15 96 108 74 58 Y 585 366 274 616 2162 203 371 356 1118 136 273 394 380 440 596 377 2 613 2621 602 239 142 400 74 108 Z 30 151 209 307 57 89 180 137 28 66 180 248 250 250 27 243 2 250 6540 429 12 154 136 58 108
As you can see, the letters involved in typos that are genuine words are also the most common letters — except C and P. (The frequency values are on an arbitrary scale to match the typo figures.)
Then I wrote a Python routine to generate a ‘badness’ score for each layout, which is the total number of words you can make by replacing a letter of another word with one of the six keys adjacent to it. Running it on 10,000 random layouts, the average badness is around 83,603, with a standard deviation of 14,024.
Here are some other layouts I tried:
Qwerty 119,170 2.54
Dvorak 121,458 2.70
Colemak 112,354 2.05
Best random 46,414 −2.65
Worst random 151,438 4.84
Alphabetic 74,064 −0.68
Best I found 31,992 −3.68
(Predictable answer to question in title: “haha, no”.) Alphabetic uses the same key layout as Qwerty: 10 on the top row, 9 on the second and 7 on the bottom. The ‘best I found’ layout was derived from a random board on that Qwerty grid (since actually Dvorak and Coleman don’t really fit on a phone), by swapping letter pairs at random and keeping the change if it seemed to work. (This is called a ‘genetic algorithm’, albeit a crude one.) I think I did 5,000 steps, five or six times. Here’s the layout it found:
D W E B K R I T Q S O J V U Z F X A M C L G H N Y P
The most obvious thing it’s done is put S (the most typoable letter) in a corner and shoved Q up against it. Another potential improvement to the model is to account for second-nearest neighbours — since flagging an error but correcting it to the wrong thing isn’t much better than missing it.
Another thing it’s done is put all the rarest letters in the middle where they have lots of neighbours — almost precisely the opposite of what Dvorak and Coleman did. Which makes sense, both intuitively and because all the standard layouts are in the worst 5% of all layouts (assuming normal distribution).
Anyway, I think we can all agree this is plainly the best possible keyboard layout for smartphones, and we should name it Taylak and petition Apple and Google to include it as the default for everything ever. I certainly can’t imagine how using the same layout on phones and computers could possibly be more desirable than this.
Here, to end on, is the worst layout I could find, with 204,290 = μ + 8.61σ possible real-world typos:
V N T M B G E I J Q Z S D P C K A O X Y R L F H W U
Nobody use that layout.
# Extrapolating into the past, we’ve missed three increasingly implausible opportunities to do this before.
On July 5 2010, Total Film fooled a lot of people into believing that that was the day Marty McFly and Doc Brown visited in Back To The Future Part II. In fact it was October 21 2015.
Yesterday, on June 27 2012, Simply Tap fooled a lot of people into believing that that was the day Marty McFly and Doc Brown visited in Back To The Future Part II. In fact it was still October 21 2015.
To punish people for falling for such nonsense, I propose more of these, increasing in frequency as we approach the actual date, so that when it actually happens, nobody believes it. But when?
The first of these errors was 1934 days premature. The second was 1211 days premature. (I’m not counting the copycat hoax on July 6 2010.) I think the reason these hoaxes are so seductive are that $\frac{1934}{1211}=1.60$, and that’s very close to the golden ratio, $\phi$. $\phi\approx1.62$, and has the lovely property that $\frac{1}{\phi}=\phi-1$ (or, $\phi^2=\phi+1$). It’s the only number of which that’s true, and it often appears in nature, art and architecture.
(Or, if you prefer, reports of $\phi$ appearing by accident are mostly coincidence and optimistic rounding, and so is this. I’ll leave that decision to you.)
Continuing the Golden Cascade of Back To The Future Hoaxes, we should have the next on September 23 2013. There will be two in 2014: on July 4 (when the alien mothercraft destroys the Hill Valley town hall) and December 28. The hoaxes will have to come thick and fast in 2015: April 18, June 27 (like this year), August 10, September 6 and 23, and October 4, 10, 14, 16, 18, 19 (twice), and then 25 separate hoaxes on the day before Future Day.
To be honest I suggest we use areweinthefutureyet.com for those ones.
# On reflection, perhaps they shouldn’t go this far. It’s a bit sad, isn’t it?
I saw this post about the graphs of National Novel Writing Month on my brother’s blog ages ago, but an RSS malfunction showed it to me again today, and I got to thinking about his final graph: words written on any given day, plotted against how far behind he was that day (or, more precisely, words left to write / day left, normalised to the same value on day one). He correctly notices “a hint of a positive correlation”, and notes that that may not be causation, but could be an external factor, such as his determination to show his doubting wife what for.
I have my own theory, arguably more prosaic but also more interesting, so I created a simulation to test it. I assigned a random number to each of the 30 days of November, using the formula RAND()*RAND() to create a nice distribution. I assigned a number of words to each day by multiplying this random number by 50,000 and dividing by the total of all 30 random numbers. This created a month of simulated writing with random ups and downs, but a guaranteed total output of exactly 50,000. (For these purposes I allowed non-integer numbers of words.) I worked out the cumulative wordcount and behindness index for each day, and plotted them, with a trendline (in red) and R2 value, and ran simulation after simulation.
My theory was proven: every single one had a positive correlation.
Of course it did — any deviation from the 1666.7-word daily target will be reflected in your behindness score on every subsequent day, and if you write your 50,000th word on day 30, then it will be balanced by an equal and opposite deviation spread unevenly across those same days. Any novel of exactly 50,000 words will have this hint of positive correlation between behindness and words written. I suspect this would hold even if we allowed some days to have a net deletion of words. Novels of just over 50,000 (such as almost all of them) will get a slightly reduced version of the same effect. There are only two ways to avoid it. One is to write a wildly different number of words — say, 40,000 or 60,000. The other is to write exactly one thousand, six hundred and sixty six and two thirds of a word every single day, although that involves using a lot of three- and six- letter words.
I’m not sure that someone who writes 60,000 words really worked to the 50,000-word target at all, so if you want to know if ‘behindness’ affects performance, you’ll have to examine the stats from people who failed to complete the novel, between day one and whenever they eventually gave up. And to be honest, how useful a sample are they to a study of productivity?
Anyway, I suppose I hope this can be a nice example of how something plausible and supported-by-the-data-looking can turn out to just be randomness viewed from a funny angle.
# Also we’ll put the verse about rebellious Scots to crush back in.
It’s rare for two people to say things that are obvious, true and contradictory, but that’s what’s happened here:
Scots will no longer be British if their country votes to leave the United Kingdom, Labour leader Ed Miliband has warned in a keynote speech on national identity.
Miliband insisted that leaving the union would mean that Scottish people would lose their British identity – challenging the argument put forward by the Scottish Nationalists, who have insisted that Scottish people would continue to be British in a geographical sense.
Sigh.
Look, the geography gets a bit complicated, so let’s break it down:
• “Great Britain” is the name of the big island that England, Scotland and Wales are on.
• The country that occupies this island and a few nearby bits (most notably Northern Ireland) is called “The United Kingdom Of Great Britain And Northern Ireland” and is long enough already without adding “Except Scotland”.
• The “British Islands”, apparently, is the UK plus some other, distinct states that the Queen is also head of.
• The “British Isles” is every damn rock between Iceland and France.
So yes, Scotland will remain British, because it’s part of Britain. But it won’t be part of Britain any more so it won’t be British, really.
I would dearly love to know what Ed Miliband thinks will happen if we vote to “leave Europe”. I’m picturing an army of tugboats, or drills to perforate the tectonic plate along the Channel.
It has come to my attention that some people do not think about how things will look. I mean, what would we think of this behaviour from, say, the host nation of Eurovision?
A group of long-term unemployed jobseekers were bussed into London to work as unpaid stewards during the diamond jubilee celebrations and told to sleep under London Bridge before working on the river pageant.
I think it’s a bit despotic. An absurdly rich woman who is head of state simply because her dad was has been thrown a £3 billion party at the citizens’ expense, spent the afternoon heading a £12 million flotilla, while on the banks, the pageant was helped along by poor citizens working in shitty conditions for no money and against their will.
I realise it’s not quite as bad as I make it sound, but it’s very nearly that bad. Like when Amazon deleted 1984 from everyone’s Kindle. Seems like it’d be quite easy to forsee how this shit might look when it inevitably gets out, and maybe think, you know what, just this once maybe we’ll pretend to be mature adults. You know — purely in the interests of avoiding bad press.
# Crappy interfaces in real life
I spend a lot of time getting cross at crappy interfaces on software, but the fact is that real life objects are just as bad. I’m typing this at a laptop, for example, with a trackpad. And while Apple have multitouch, click-sensitive trackpads that make sense, this one scrolls using the right hand half a centimetre of pad, which is visually and tactilely indistinguishable from the rest of it. It’s the little things, like Jeff Atwood’s cat feeders, but it’s also the really big things. Our old DVD recorder which asked, when you put a DVD in, if you would like to “access the disc contents” — and to get to the DVD menu, you had to say ‘no’. Toasters which measure time in completely arbitrary units, at least as far as I’m aware. Washing machines that have a key to their own interface on the front. TVs that have a hundred buttons to do basically one thing and you still have to tap ‘circle with an arrow’ four times to watch a DVD.
It’s like people don’t learn. I remember when I could operate a microwave oven by typing in a time and pressing ‘cook’. Ryan North has one where you can just bark numbers at it. And yet this is the thing we have at work:
Obviously that’s absurdly overcomplicated for a device which, much as manufacturers kid themselves otherwise, has only two modes: on and off. (Really, who wants to ‘slow defrost’ anything?) But my main objection is that I have had a bagel spinning around in there for ages, with the microwave all humming to itself and lit up, and without the bagel getting so much as lukewarm. Apparently it only heats food if you set the timer. Where exactly the clue to this is supposed to be is not clear. I assumed when a microwave lit up and span your food round it was cooking. Apparently this one also has a ‘shop demonstration’ mode.
Because the fact is that outward appearances matter. I forgive the absurd obsession of washing machine manufactures with putting any and all options on dials because that at least makes their products obviously washing machines. Humans in Western society have loads of cues and associations built in — say, we pull doors with handles — and it’s daft not to take advantage of them and insane to actively work against them. Putting a handle on a push-only door will confuse and annoy, and this… This is the sink in a bar in Manchester:
Perhaps they considered it designery and artsy, but in fact what they have done is add taps to a urinal and call it a sink. The thing is that I took it for a urinal when I first saw it, and while I noticed the taps fairly quickly, I don’t trust the entire inebriated male population of Manchester to all have done so on any given night. That is a sink which I confidently predict has been pissed in and that is enough to put me off washing in it.
What you need, when cleaning yourself, is a clear interface. So don’t get this brand of shower:
I put it to you that it’s less than totally clear which way is ‘hot’ on this dial. The arrow points right, but it’s on the left. If you’re standing in a room full of steam and you wear glasses, that’s fairly dangerous. And it’s obvious what the dial does, so all they had to do was convey which way was which. That is literally the smallest amount of information it is possible to encode. And still it’s ambiguous.
Even stranger is the dimmer switch in our new conference room.
This has two buttons, one at the top of the switch, and one at the bottom. Here is how I assumed it would work:
• Pressing the top switch would make the lights brighter.
• Pressing the bottom switch would make the lights dimmer.
Here is how it works:
• A short tap of the bottom switch turns the lights on or off.
• A long hold of the bottom switch makes the lights brighter.
• A second long hold of the bottom switch makes the lights dimmer. This alternates between brighter and dimmer.
• The top switch exists only to make the object resemble a lightswitch.
The upshot of this is that we all look stupid when presenting the work of the Unit to outsiders, because we are supposed to be carefully controlling illumination in clinical trials and can’t operate our own office lights.
And I suppose I just thought that stuff like toasters and lights would have been around for long enough by now for us to have basically mastered the art of making them usable. We don’t have Steve Jobs around any more to fix this stuff up for us, so perhaps it would be a fitting tribute to him if we all stopped making watches that require one long and fourteen short button presses, each with its own high-pitched beep, just to correct for daylight savings, hmm?
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2013-05-19 09:17:14
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https://www.organicdesign.co.nz/Rsync
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# rsync
Jump to: navigation, search
rsync is an open source utility that provides fast incremental file transfer. rsync is freely available under the GNU General Public License and is currently being maintained by Wayne Davison.
rsync uses the "rsync algorithm" which provides a very fast method for bringing remote files into sync. It does this by sending just the differences in the files across the link, without requiring that both sets of files are present at one of the ends of the link beforehand.
Some features of rsync include
• Can update whole directory trees and filesystems
• Optionally preserves symbolic links, hard links, file ownership, permissions, devices and times
• Requires no special privileges to install
• Internal pipelining reduces latency for multiple files
• Can use rsh, ssh or direct sockets as the transport
• Supports anonymous rsync which is ideal for mirroring
Following is a description of some of the ways we use rsync here at Organic Design.
## Basic syntax
The basic syntax is:
rsync -options --more --options /source/path/ /target/path
• Note1: The source and target paths can be on the local file system, or if SSH is used can be paths on remote servers
• Note2: Notice that the trailing slash on the source path is highlighted, that's because it's very important! If there is no slash, then the source directory will be created within the target directory. If it is included then the contents of the source directory will be synchronised with the contents of the target directory.
## Using rsync over SSH with key-based logins
Sometimes it's useful to do a one-off backup of a file structure from one host to another, and since all the hosts (in our system) are guaranteed to be able to connect to each other with SSH (after adding appropriate RSA keys), using rsync over SSH is a good way to do this.
The transfer syntax is then done very similarly to SCP, for example to pull new changes from a remote directory to a local one, use:
rsync -avze ssh remoteuser@remotehost:/path/to/remote/source/ /path/to/local/target
Important: There is a bug with compression of large files that are already compressed on some versions of rsync causing the transfer to die with inflate returned -3, if you have this problem move the compression from rsync to ssh with -ave ssh -C.
After the systems are confirmed as being able to connect over SSH you may want to lock them down so that the connection between them can only be used for rsync. The IP and command can be prepended to the key in the remote hosts ~/.ssh/authorized_keys file.
from="1.2.3.4",no-agent-forwarding,no-port-forwarding,no-pty,no-user-rc,no-X11-forwarding ssh-rsa AAAAB...
For more security, the command allowed can be restricted to just that specific rsync command by prepending command="rsync ....." to the start of the line. You'll need the exact command that gets run which you can obtain by checking the remote server's process list or running the rsync command with the -e'ssh -v' option which will output the exact command sent that can be used in the remote hosts authorized_keys file instead of just "rsync".
## Using rsync with encryption
EncFS which can be installed simply via apt-get is an excellent option for synchronising data with rsync when the target system is insecure. Using EncFS, an encrypted version of the source data can be maintained locally and then synchronised to a remote system with rsync.
EncFS is used in its standard form to create two directories, an encrypted one which is permanent and another which is the decrypted view of this encrypted data that only exists while EncFS is running. This means that we can start EncFS which will then present the current state of the encrypted data in the temporary decrypted mount point, then we do a local rsync from the source directory structure to the decrypted mount point, then stop EncFS. We can then rsync the permanent encrypted directory to the remote server as it has now been brought up to date with the source structure.
The encrypted directory contains a one-to-one correspondence of file-system objects with the decrypted data, which means that rsync can still work efficiently at updating only the added files and deleting the removed ones as usual. These objects have encrypted names and content, but retain their original ownership and time-stamp attributes, their sizes remain very close to the original too.
Another good feature of EncFS is that the file and directory names are also encrypted using only friendly characters which means that there's no need to use the transliterate patch with rsync when using EncFS.
Rather than keep EncFS running permanently, our backup script starts it every time before doing the rsync command, then stops it again after rsync has finished. The only difficulty with this is that EncFS doesn't have an option for passing the password as a command-line option, so we use Perl's Net::Expect module to enter the password. You can then use mount to ensure that EncFS has properly mounted the decrypted location before continuing. This is shown in the following example snippet.
use Expect;
# Mount the decrypted view onto the encrypted local mirror
$exp = Expect->spawn( "encfs /backup/encrypted /backup/decrypted" );$exp->expect( 5, [ qr/EncFS Password:/ => sub { my $exp = shift;$exp->send( "********\n" ); exp_continue; } ] );
\$exp->soft_close();
# Rsync then unmount if mounted successfully
if( qx( mount|grep 'encfs on /backup/decrypted' ) ) {
# Bring the local encrypted mirror up to date via the decrypted mount-point, then un-mount it
qx( rsync -a --delete /home/foo /backup/decrypted );
qx( fusermount -u /backup/decrypted );
# Synchronise the encrypted mirror with the remote service
qx( rsync -a --delete /backup/encrypted foo\@domain.com\@rsync.adrive.com:. )
} else { print "Failed to mount!" }
## The transliterate patch
Sometimes the source file structure contains characters that are not allowed on the target system, for example Maildirs use colons in the file naming protocol which are not allowed on many filesystems including our ADrive backup service.
Note: this problem is redundant if using the above encryption method because all the filenames are encrypted too and the encrypted names only use friendly characters.
We've overcome this problem using the transliterate patch which adds a --tr=BAD/GOOD option for mapping bad characters to good ones.
To install the patch you need to download and unpack the latest source and the patches, then change into the source directory and do the following:
patch -p1 <patches/transliterate.diff
./configure
make
make install
You can't use this option to backup directory to the target server unless the target also has the transliterate patch installed. If it's not installed on the server you'll need to do a two-stage backup. The first to a local directory using the --tr option, and second synchronising this local directory (that has all the colons replaced) with the remote server without the --tr option. Here's an example taken from our daily backup script.
rsync -a --delete --tr=':/;' /home /backup/home.rsync
rsync -az --delete /backup/home.rsync user\@domain\@rsync.adrive.com:.
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2017-02-25 15:50:35
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http://www.iapjournals.ac.cn/aas/en/article/doi/10.1007/s00376-020-0260-y?viewType=HTML
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Article Contents
# From China’s Heavy Precipitation in 2020 to a “Glocal” Hydrometeorological Solution for Flood Risk Prediction
Fund Project:
This study was supported by the National Key R&D Program of China (Grant No. 2017YFA0604300) and the National Natural Science Foundation of China (Grant Nos. 41861144014, 41775106 and U1811464), as well as partially by the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (Grant No. 2017ZT07X355) and the project of the Chinese Ministry of Emergency Management on “Catastrophe Evaluation Modeling Study”
• The prolonged mei-yu/baiu system with anomalous precipitation in the year 2020 has swollen many rivers and lakes, caused flash flooding, urban flooding and landslides, and consistently wreaked havoc across large swathes of China, particularly in the Yangtze River basin. Significant precipitation and flooding anomalies have already been seen in magnitude and extension so far this year, which have been exerting much higher pressure on emergency responses in flood control and mitigation than in other years, even though a rainy season with multiple ongoing serious flood events in different provinces is not that uncommon in China. Instead of delving into the causes of the uniqueness of this year’s extreme precipitation-flooding situation, which certainly warrants in-depth exploration, in this article we provide a short view toward a more general hydrometeorological solution to this annual nationwide problem. A “glocal” (global to local) hydrometeorological solution for floods (GHS-F) is considered to be critical for better preparedness, mitigation, and management of different types of significant precipitation-caused flooding, which happen extensively almost every year in many countries such as China, India and the United States. Such a GHS-F model is necessary from both scientific and operational perspectives, with the strength in providing spatially consistent flood definitions and spatially distributed flood risk classification considering the heterogeneity in vulnerability and resilience across the entire domain. Priorities in the development of such a GHS-F are suggested, emphasizing the user’s requirements and needs according to practical experiences with various flood response agencies.
摘要: 2020年，梅雨系统给中国南方带来了长时间的异常强降水，导致南方大部分地区的江河湖泊水位暴涨，持续引发了山洪、城市内涝、山体滑坡等自然灾害，其中以长江流域洪涝灾害最严重。总的来说，今年我国强降水和洪涝灾害的发生规模和范围均出现了明显的异常增多现象，这使得相关部门对于防汛减灾应急响应的压力也较往年增大。尽管今年我国洪灾频发，但雨季多个省份发生多起严重洪涝灾害的现象在我国并不罕见。本文没有深入探讨今年极端降水引发的洪涝灾害形成的独特性原因，而是对这一全国性灾害问题提出了具有普适性的水文气象解决方案。近年来，由极端降水引发的洪涝灾害事件几乎每年都在中国、印度和美国等许多国家发生。”从全球到区域（Glocal, Global to local）”的水文气象解决方案（GHS-F, A “glocal” hydrometeorological solution for floods）被认为是更好地监测不同类型降水引起的洪涝灾害、减轻洪涝灾害风险损失至关重要的解决方案。GHS-F考虑了全球区域的脆弱性和恢复机制的异质性，能提出统一的空间洪水定义和洪水风险分类标准；从科学性和可操作性的角度来看，GHS-F模型都是十分必要的。本文还提出了发展全球统一框架的GHS-F模型需优先级解决的事项，并需根据各抗洪减灾机构的实际经验，优先考虑用户的实际需要。
• Figure 1. (a) Distribution map of flood events from late May to mid-July 2020. Red arrows indicate the timeline of flood-event occurrences and red dots indicate the most severely affected cities. (b) Number of people affected by each major event, according to social media sources.
Figure 2. Daily zonal mean precipitation over part of mainland China (20°–44.5°N). Eleven extreme precipitation events (dates in bold black font) in central to eastern China since the onset of the monsoon up to 28 July 2020. The number of days (in red) are the lead days when corresponding events were skillfully predicted with good quality using medium- and extended-range quantitative precipitation forecasting by the National Meteorological Center.
Figure 3. Schematic illustration of a GHS-F: DRIVE model, including an urban flood module in addition to pluvial (flash) flooding and fluvial flood modeling.
Figure 4. An example of GFMS output from global to local scale flooding. (a) Global streamflow distribution map with resolution of 12 km. (b) China's flood detection/intensity distribution map with resolution of 10 km. (c) Distribution map of surface storage of local watershed with resolution of 1 km (d) Local streamflow distribution map resolution of 1 km. (e) Urban waterlogging map with 5 km resolution
Figure 5. (a) Flood detection and intensity by GFMS with the China Meteorological Administration’s real-time quantitative precipitation estimation. (b) Risk estimation of flash flooding. (c) Sentinel-1-based flood inundation mapping for Dongting Lake, with the inset showing the temporal variations of lake areas, estimated by NOAA NPP satellite optical band data, together with the water level measured on the ground at the lake outlet. (d) Fluvial flooding in small- to mid-sized rivers.
• Alfieri, L., P. Burek, E. Dutra, B. Krzeminski, D. Muraro, J. Thielen, and F. Pappenberger, 2013: GloFAS−global ensemble streamflow forecasting and flood early warning. Hydrology and Earth System Sciences, 17, 1161−1175, https://doi.org/10.5194/hess-17-1161-2013. Allan, R. P., and B. J. Soden, 2008: Atmospheric warming and the amplification of precipitation extremes. Science, 321, 1481−1484, https://doi.org/10.1126/science.1160787. Beven, K. J., and H. L. Cloke, 2012: Comment on “Hyperresolution global land surface modeling: Meeting a grand challenge for monitoring Earth’s terrestrial water” by Eric F. Wood et al. Water Resour. Res., 48, W01801, https://doi.org/10.1029/2011WR010982. Bierkens, M. F. P., and Coauthors, 2015: Hyper-resolution global hydrological modelling: What is next? Hydrological Processes, 29, 310−320, https://doi.org/10.1002/hyp.10391. Chen, X. L., T. J. Zhou, P. L. Wu, Z. Guo, and M. H. Wang, 2020: Emergent constraints on future projections of the western North Pacific Subtropical High. Nature Communications, 11, 2802, https://doi.org/10.1038/s41467-020-16631-9. Gochis, D. J., W. Yu, and D. N. Yates, 2013: The NCAR WRF-hydro technical description and user’s guide, Version 1.0. NCAR Technical Document, 120 pp. Trenberth, K. E., 2011: Changes in precipitation with climate change. Climate Research, 47, 123−138, https://doi.org/10.3354/cr00953. Wood, E. F., and Coauthors, 2011: Hyperresolution global land surface modeling: Meeting a grand challenge for monitoring Earth’s terrestrial water. Water Resour. Res., 47, W05301, https://doi.org/10.1029/2010WR010090. Wu, H., J. S. Kimball, N. Mantua, and J. Stanford, 2011: Automated upscaling of river networks for macroscale hydrological modeling. Water Resour. Res., 47, W03517, https://doi.org/10.1029/2009WR008871. Wu, H., J. S. Kimball, H. Y. Li, M. Y. Huang, L. R. Leung, and R. F. Adler, 2012: A new global river network database for macroscale hydrologic modeling. Water Resour. Res., 48, W09701, https://doi.org/10.1029/2012WR012313. Wu, H., R. F. Adler, Y. D. Tian, G. J. Huffman, H. Y. Li, and J. J. Wang, 2014: Real-time global flood estimation using satellite-based precipitation and a coupled land surface and routing model. Water Resour. Res., 50, 2693−2717, https://doi.org/10.1002/2013WR014710. Wu, H., R. F. Adler, Y. D. Tian, G. J. Gu, and G. J. Huffman, 2017: Evaluation of Quantitative Precipitation Estimations through Hydrological Modeling in IFloodS River Basins. Journal of Hydrometeorology, 18, 529−553, https://doi.org/10.1175/JHM-D-15-0149.1. Wu, H., J. S. Kimball, N. J. Zhou, L. Alfieri, L. F. Luo, J. Y. Du, and Z. J. Huang, 2019: Evaluation of real-time global flood modeling with satellite surface inundation observations from SMAP. Remote Sens. Environ., 233, 111360, https://doi.org/10.1016/j.rse.2019.111360. Yan, Y., and Coauthors, 2020: Climatology and interannual variability of floods during the TRMM era (1998−2013). J. Climate, 33, 3289−3305, https://doi.org/10.1175/JCLI-D-19-0415.1.
• [1] FENG Lei, ZHOU Tianjun, WU Bo, Tim LI, Jing-Jia LUO, 2011: Projection of Future Precipitation Change over China with a High-Resolution Global Atmospheric Model, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 464-476. doi: 10.1007/s00376-010-0016-1 [2] LI Hongmei, FENG Lei, ZHOU Tianjun, 2011: Multi-model Projection of July--August Climate Extreme Changes over China under CO$_{2}$ Doubling. Part I: Precipitation, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 433-447. doi: 10.1007/s00376-010-0013-4 [3] Yuan WANG, 2015: Air Pollution or Global Warming: Attribution of Extreme Precipitation Changes in Eastern China——Comments on "Trends of Extreme Precipitation in Eastern China and Their Possible Causes", ADVANCES IN ATMOSPHERIC SCIENCES, 32, 1444-1446. doi: 10.1007/s00376-015-5109-4 [4] ZENG Xinmin, ZHAO Ming, SU Bingkai, TANG Jianping, ZHENG Yiqun, GUI Qijun, ZHOU Zugang, 2003: Simulations of a Hydrological Model as Coupled to a Regional Climate Model, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 227-236. doi: 10.1007/s00376-003-0008-5 [5] LIU Run, LIU Shaw Chen, Ralph J. CICERONE, SHIU Chein-Jung, LI Jun, WANG Jingli, ZHANG Yuanhang, 2015: Trends of Extreme Precipitation in Eastern China and Their Possible Causes, ADVANCES IN ATMOSPHERIC SCIENCES, 32, 1027-1037. doi: 10.1007/s00376-015- 5002-1 [6] Bian HE, Qing BAO*, Xiaocong WANG, Linjiong ZHOU, Xiaofei WU, Yimin LIU, Guoxiong WU, Kangjun CHEN, Sicheng HE, Wenting HU, Jiandong LI, Jinxiao LI, Guokui NIAN, Lei WANG, Jing YANG, Minghua ZHANG, Xiaoqi ZHANG, 2019: CAS FGOALS-f3-L Model Datasets for CMIP6 Historical Atmospheric Model Intercomparison Project Simulation, ADVANCES IN ATMOSPHERIC SCIENCES, , 771-778. doi: 10.1007/s00376-019-9027-8 [7] NING Liang, QIAN Yongfu, 2009: Interdecadal Change in Extreme Precipitation over South China and Its Mechanism, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 109-118. doi: 10.1007/s00376-009-0109-x [8] Sheng LAI, Zuowei XIE, Cholaw BUEH, Yuanfa GONG, 2020: Fidelity of the APHRODITE Dataset in Representing Extreme Precipitation over Central Asia, ADVANCES IN ATMOSPHERIC SCIENCES, 37, 1405-1416. doi: 10.1007/s00376-020-0098-3 [9] DING Yuguo, CHENG Bingyan, JIANG Zhihong, 2008: A Newly-Discovered GPD-GEV Relationship Together with Comparing Their Models of Extreme Precipitation in Summer, ADVANCES IN ATMOSPHERIC SCIENCES, 25, 507-516. doi: 10.1007/s00376-008-0507-5 [10] Yongguang ZHENG, Yanduo GONG, Jiong CHEN, Fuyou TIAN, 2019: Warm-Season Diurnal Variations of Total, Stratiform, Convective, and Extreme Hourly Precipitation over Central and Eastern China, ADVANCES IN ATMOSPHERIC SCIENCES, 36, 143-159. doi: 10.1007/s00376-018-7307-3 [11] LIU Run, LIU Shaw Chen, Ralph J. CICERONE, SHIU Chein-Jung, LI Jun, WANG Jingli, ZHANG Yuanhang, 2015: Response to Comments by Yuan WANG on "Trends of Extreme Precipitation in Eastern China and Their Possible Causes", ADVANCES IN ATMOSPHERIC SCIENCES, 32, 1447-1448. doi: 10.1007/s00376-015-5121-8 [12] Chenyu MA, Wei YUAN, Ji NIE, 2020: Responses of Mean and Extreme Precipitation to Different Climate Forcing Under Radiative-Convective Equilibrium, ADVANCES IN ATMOSPHERIC SCIENCES, 37, 377-386. doi: 10.1007/s00376-020-9236-1 [13] Yujing QIN, Chuhan LU, Liping LI, 2017: Multi-scale Cyclone Activity in the Changjiang River-Huaihe River Valleys during Spring and Its Relationship with Rainfall Anomalies, ADVANCES IN ATMOSPHERIC SCIENCES, 34, 246-257. doi: 10.1007/s00376-016-6042-x [14] ZONG Haifeng, ZHANG Qingyun, 2011: A New Precipitation Index for the Spatiotemporal Distribution of Drought and Flooding in the Reaches of the Yangtze and Huaihe Rivers and Related Characteristics of Atmospheric Circulation, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 375-386. doi: 10.1007/s00376-010-9223-z [15] Koji DAIRAKU, Seita EMORI, Toru NOZAWA, 2008: Impacts of Global Warming on Hydrological Cycles in the Asian Monsoon Region, ADVANCES IN ATMOSPHERIC SCIENCES, 25, 960-973. doi: 10.1007/s00376-008-0960-1 [16] LI Hongmei, ZHOU Tianjun, LI Chao, 2010: Decreasing Trend in Global Land Monsoon Precipitation over the Past 50 Years Simulated by a Coupled Climate Model, ADVANCES IN ATMOSPHERIC SCIENCES, 27, 285-292. doi: 10.1007/s00376-009-8173-9 [17] Sun Bomin, Sun Shuqing, 1994: The Analysis on the Features of the Atmospheric Circulation in Preceding Winters for the Summer Drought and Flooding in the Yangtze and Huaihe River Valley, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 79-90. doi: 10.1007/BF02656997 [18] Tang Maocang, Li Tianshi, Zhang Jian, Li Cunqiang, 1989: The Operational Forecasting of Total Precipitation in Flood Seasons (April to September) of 5 Years (1983-1987), ADVANCES IN ATMOSPHERIC SCIENCES, 6, 289-300. doi: 10.1007/BF02661535 [19] A.C.Yih, J. E. Walsh, 1991: Sensitivities of Numerical Model Forecasts of Extreme Cyclone Events, ADVANCES IN ATMOSPHERIC SCIENCES, 8, 51-66. doi: 10.1007/BF02657364 [20] Jie FENG, Jianping LI, Jing ZHANG, Deqiang LIU, Ruiqiang DING, 2019: The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii, ADVANCES IN ATMOSPHERIC SCIENCES, 36, 271-278. doi: 10.1007/s00376-018-8123-5
Export:
## Manuscript History
Manuscript revised: 09 September 2020
Manuscript accepted: 10 September 2020
###### 通讯作者: 陈斌, bchen63@163.com
• 1.
沈阳化工大学材料科学与工程学院 沈阳 110142
## From China’s Heavy Precipitation in 2020 to a “Glocal” Hydrometeorological Solution for Flood Risk Prediction
###### Corresponding author: Huan WU, wuhuan3@mail.sysu.edu.cna;
• 1. Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, and School of Atmospheric Sciences, Sun Yat-sen University, Guangdong 510000, China
• 2. Southern Marine Science and Engineering Laboratory, Guangdong 519000, China
• 3. Earth System Science Interdisciplinary Center, University of Maryland, Maryland 20742, USA
• 4. School of Geographical Sciences, University of Bristol, Bristol BS8 1TH, UK
• 5. DFO Global Flood Observatory, University of Colorado Boulder, Boulder 80309, USA
• 6. CIMA Research Foundation, Savona 17100, Italy
• 7. National Meteorological Center, Beijing 100081, China
• 8. Guangdong Meteorological Center, Guangzhou, Guangdong 510000, China
• 9. GuangXi Climate Center, Nanning, Guangxi 530000, China
• 10. Guangdong Climate Center, Guangzhou, Guangdong 510000, China
• 11. China Three Gorges Corporation, Beijing 100081, China
Abstract: The prolonged mei-yu/baiu system with anomalous precipitation in the year 2020 has swollen many rivers and lakes, caused flash flooding, urban flooding and landslides, and consistently wreaked havoc across large swathes of China, particularly in the Yangtze River basin. Significant precipitation and flooding anomalies have already been seen in magnitude and extension so far this year, which have been exerting much higher pressure on emergency responses in flood control and mitigation than in other years, even though a rainy season with multiple ongoing serious flood events in different provinces is not that uncommon in China. Instead of delving into the causes of the uniqueness of this year’s extreme precipitation-flooding situation, which certainly warrants in-depth exploration, in this article we provide a short view toward a more general hydrometeorological solution to this annual nationwide problem. A “glocal” (global to local) hydrometeorological solution for floods (GHS-F) is considered to be critical for better preparedness, mitigation, and management of different types of significant precipitation-caused flooding, which happen extensively almost every year in many countries such as China, India and the United States. Such a GHS-F model is necessary from both scientific and operational perspectives, with the strength in providing spatially consistent flood definitions and spatially distributed flood risk classification considering the heterogeneity in vulnerability and resilience across the entire domain. Priorities in the development of such a GHS-F are suggested, emphasizing the user’s requirements and needs according to practical experiences with various flood response agencies.
Reference
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2020-11-30 16:19:08
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https://www.aimsciences.org/journal/1078-0947/2011/29/2
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# American Institute of Mathematical Sciences
ISSN:
1078-0947
eISSN:
1553-5231
All Issues
## Discrete & Continuous Dynamical Systems - A
April 2011 , Volume 29 , Issue 2
Special Issue on Control, Nonsmooth Analysis and Optimization
Celebrating the 60th Birthday of Francis Clarke and Richard Vinter
Select all articles
Export/Reference:
2011, 29(2): i-ii doi: 10.3934/dcds.2011.29.2i +[Abstract](1449) +[PDF](6552.7KB)
Abstract:
These preface words, quoted in Francis Clarke's 1983 book, stroke a sensitive chord and set the tone for a vision fueling the construction of an extremely rich body of intertwined developments in nonsmooth analysis, optimization and control by a carefully networked community.
This special issue comprises post-conference articles from a selection of works presented at the Workshop on Control, Nonsmooth Analysis and Optimization, held in Porto, Portugal in May 2009. The workshop was part of the celebrations of the 60th birthday of Francis Clarke and Richard B. Vinter.
2011, 29(2): 403-415 doi: 10.3934/dcds.2011.29.403 +[Abstract](1723) +[PDF](381.3KB)
Abstract:
This paper concerns the investigation of a general impulsive control problem. The considered impulsive processes are of non-standard type: control processes admit ordinary type controls as the impulse develops. New necessary conditions of optimality in the form of Pontryagin Maximum Principle are obtained. These conditions are applied to a model problem and are shown to yield useful information about optimal control modes.
2011, 29(2): 417-437 doi: 10.3934/dcds.2011.29.417 +[Abstract](2987) +[PDF](509.9KB)
Abstract:
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler--Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
2011, 29(2): 439-451 doi: 10.3934/dcds.2011.29.439 +[Abstract](1471) +[PDF](611.0KB)
Abstract:
We prove existence of minimizers for the multiple integral
$\int$Ω$\l(u(x),\rho_1(x,u(x))$∇$u(x))\ \ \rho_2(x,u(x))\ dx \ \ \ on\ \ \$W1,1u(Ω),(*)
where Ω$\subset\R^d$ is open bounded, $u:$Ω$\toR$ is in the Sobolev space u($*$)+W1,10(Ω), with boundary data $u_$$(\cdot)\inW1,1(Ω)\cap C^{0}(Ω); and \l:RΧR^d\to[0,\infty] is superlinear L\oxB-measurable with \rho_1(\cdot,\cdot),\rho_2(\cdot,\cdot)\in C^{0}(ΩΧR) both >0. One main feature of our result is the unusually weak assumption on the lagrangian: l**(\cdot,\cdot) only has to be lsc at (\cdot,0), i.e. at zero gradient. Here l**(s,\cdot) denotes the convex-closed hull of \l(s,\cdot). We also treat the nonconvex case \l(\cdot,\cdot)\nel**(\cdot,\cdot), whenever a well-behaved relaxed minimizer is a priori known. Another main feature is that \l(s,\xi)=\infty is freely allowed, even at zero gradient, so that (*) may be seen as the variational reformulation of optimal control problems involving implicit first-order nonsmooth scalar partial differential inclusions under state and gradient pointwise constraints. The general case \intΩL(x,u(x),∇u(x)) is also treated, though with less natural hypotheses, but still allowing L(x,\cdot,\xi) non-lsc for \xi\ne0. 2011, 29(2): 453-466 doi: 10.3934/dcds.2011.29.453 +[Abstract](1657) +[PDF](347.7KB) Abstract: We provide intrinsic sufficient conditions on a multifunction F and endpoint data φ so that the value function associated to the Mayer problem is semiconcave. 2011, 29(2): 467-484 doi: 10.3934/dcds.2011.29.467 +[Abstract](1639) +[PDF](438.1KB) Abstract: We prove validity of the classical DuBois-Reymond differential inclusion for the minimizers y(\cdot) of the integral \int_{a}^{b}L( x( t) ,x^'( t)) d\,t,\text{ \ }x\( \cdot) \in W^{1,1}((a,b) ,\mathbb{R}^{n}) ,\text{ \ }x(a)=A\,x(b) =B\ \ (*) whose velocities are not a.e. constrained by the domain boundary. Thus we do not ask ( as preceding results do) the free-velocity times T_{f ree}:=\{ t\in[ a,b] :y^'( t) \in int\text{ }dom\ L( y\( t) ,\cdot) \} to have "full measure"; on the contrary, "positive measure" of T_{f ree} suffices here to guarantee the above necessary condition. One main feature of our result is that L( S,\xi) =\infty freely allowed, hence the domains dom$$L( S,\cdot)$ may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems.
Another main feature is the clean generality of our assumptions on $L( \cdot) :$ any Borel-measurable function $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[ 0,\infty]$ having $L( \cdot,0)$ $lsc$ and $L( S,\cdot)$ convex $lsc$ $\forall\,S.$
The nonconvex case is also considered, for $L( S,\cdot)$ almost convex lsc $\forall\,S.$
2011, 29(2): 485-503 doi: 10.3934/dcds.2011.29.485 +[Abstract](1763) +[PDF](433.6KB)
Abstract:
We give a relatively short and self-contained proof of a theorem that asserts necessary conditions for a general optimal control problem. It has been shown that this theorem, which is simple to state, provides a powerful template from which necessary conditions for various other problems in dynamic optimization can be directly derived, at the level of the state of the art. These include various extensions of the Pontryagin maximum principle and the multiplier rule.
2011, 29(2): 505-522 doi: 10.3934/dcds.2011.29.505 +[Abstract](1577) +[PDF](374.1KB)
Abstract:
In this paper we report conditions ensuring Lipschitz continuity of optimal control and Lagrange multipliers for a dynamic optimization problem with inequality pure state and mixed state-control constraints.
2011, 29(2): 523-545 doi: 10.3934/dcds.2011.29.523 +[Abstract](1457) +[PDF](400.2KB)
Abstract:
We consider a general optimal control problem with intermediate and mixed constraints. Using a natural transformation (replication of the state and control variables), this problem is reduced to a standard optimal control problem with mixed constraints, which makes it possible to obtain quadratic order conditions for an "extended" weak minimum. The conditions obtained are applied to the problem of light refraction.
2011, 29(2): 547-557 doi: 10.3934/dcds.2011.29.547 +[Abstract](1425) +[PDF](337.1KB)
Abstract:
We consider autonomous, second order problems in the calculus of variations in one independent variable. For analogous first order problems it is known that, under standard hypotheses of existence theory and a local boundedness condition on the Lagrangian, minimizers over $W^{1,1}$ have bounded first derivatives ($W^{1,\infty}$ regularity prevails). For second order problems one might expect, by analogy, that minimizers would have bounded second derivatives ($W^{2,\infty}$ regularity) under the standard existence hypotheses $(HE)$ for second order problems, supplemented by a local boundedness condition. A counter-example, however, indicates that this is not the case. In earlier work, $W^{2, \infty}$ regularity has been established for these problems under $(HE)$ and additional 'integrability' hypotheses on derivatives of the Lagrangian, evaluated along the minimizer. We show that these additional hypotheses can be significantly reduced. The proof techniques employed depend on a combination of the application of a change of independent variable and of extensions to Tonelli regularity theory proved by Clarke and Vinter.
2011, 29(2): 559-575 doi: 10.3934/dcds.2011.29.559 +[Abstract](2050) +[PDF](723.3KB)
Abstract:
We address necessary conditions of optimality (NCO), in the form of a maximum principle, for optimal control problems with state constraints. In particular, we are interested in the NCO that are strengthened to avoid the degeneracy phenomenon that occurs when the trajectory hits the boundary of the state constraint. In the literature on this subject, we can distinguish two types of constraint qualifications (CQ) under which the strengthened NCO can be applied: CQ involving the optimal control and CQ not involving it. Each one of these types of CQ has its own merits. The CQs involving the optimal control are not so easy to verify, but, are typically applicable to problems with less regularity on the data. In this article, we provide conditions under which the type of CQ involving the optimal control can be reduced to the other type. In this way, we also provide nondegenerate NCO that are valid under a different set of hypotheses.
2011, 29(2): 577-593 doi: 10.3934/dcds.2011.29.577 +[Abstract](1790) +[PDF](429.3KB)
Abstract:
In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H (\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.
2011, 29(2): 595-613 doi: 10.3934/dcds.2011.29.595 +[Abstract](1383) +[PDF](446.9KB)
Abstract:
We develop a notion of generalized solution to a stochastic differential equation depending in a nonlinear way on a vector--valued stochastic control process $\{U_t\},$ merely of bounded variation, and on its derivative. Our results rely on the concept of Lipschitz continuous graph completion of $\{U_t\}$ and the generalized solution turns out to coincide a.e. with the limit of classical solutions to (1). In the linear case our notion of solution is equivalent to the usual one in distributional sense. We prove that the generalized solution does not depend on the particular graph-completion of the control process $\{U_t\}$ both for vector-valued controls under a suitable commutativity condition and for scalar controls.
2011, 29(2): 615-622 doi: 10.3934/dcds.2011.29.615 +[Abstract](1468) +[PDF](487.9KB)
Abstract:
We consider sets $S\subset\R^n$ satisfying a certain exterior sphere condition, and it is shown that under wedgedness of $S$, it coincides with $\varphi$-convexity. We also offer related improvements concerning the union of uniform closed balls conjecture.
2011, 29(2): 623-646 doi: 10.3934/dcds.2011.29.623 +[Abstract](1650) +[PDF](492.6KB)
Abstract:
The notions of $V$-Jacobian and $V$-co-Jacobian are introduced for locally Lipschitzian functions acting between arbitrary normed spaces $X$ and $Y$, where $V$ is a subspace of the dual space $Y^*$. The main results of this paper provide a characterization, calculus rules and also the computation of these Jacobians of piecewise smooth functions.
2011, 29(2): 647-670 doi: 10.3934/dcds.2011.29.647 +[Abstract](1713) +[PDF](431.8KB)
Abstract:
The paper deals with optimal control problems described by higher index DAEs. We introduce a numerical procedure for solving these problems. The procedure, based on the appropriately defined adjoint equations, refers to an implicit Runge--Kutta method for differential--algebraic equations. Assuming that higher index DAEs can be solved numerically the gradients of functionals defining the control problem are evaluated with the help of well--defined adjoint equations. The paper presents numerical examples related to index three DAEs showing the validity of the proposed approach.
2011, 29(2): 671-691 doi: 10.3934/dcds.2011.29.671 +[Abstract](1653) +[PDF](259.9KB)
Abstract:
The paper studies the notion of subdifferentials of functions defined on a time scale. The subdifferential of a given function $f$ is defined as the set of certain extended functions. Since the convexity of the given function guarantees its subdifferentiability, properties of convex functions on time scales are presented. We show that the convexity of a function is the necessary and sufficient condition for its subdifferentiability. The relations between the delta, nabla, diamond-$\alpha$ derivatives and subdifferentials of convex functions are given.
2018 Impact Factor: 1.143
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2019-11-17 01:56:41
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# Search the Community
Showing results for tags 'runwait'.
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Found 22 results
4. ## Execute VBS in a variable location?
I'm struggling to launch a VBS file via autoit using RunWait. Due to the nature of the deployment tool I am using for said script, I only know that the .vbs files will be in the same directory as my AutoIt-generated .exe, but not what that path will be. The path will look something like this: c:\programdata\vendor\lots\of\folders\randomnumber This is generated during deployment and I have no way of predicting the path - therefore, I am not sure how to call back to "same directory" in order to successfully launch the .vbs. This line is as close as I have gotten - this fires off
7. ## Looking for a clean way to capture RunWait output
Initial Problem I've written several scripts with the following sequence: Execute a program using Run w/stdout+stderr captured Typically processes all the files in one directory tree to populate a second tree Execute a second program (also with Run) to monitor the products of the first program and Display a progress bar (percentage of output files complete) Also monitor the first program's process and exit when it terminates The script then calls ProcessWaitClose (no timeout) on the first program's process and Checks the first p
8. ## How to pass command line arguments to a program(Python) from RunWait
I want to run a python script from autoit. I know we can do this with shellexecute or run , but this python script takes 3 cmd line arguments. How to pass them from autoit script? I tried many ways. Following solution also not working e.g. RunWait( 'fullpath\Python.exe Scriptpath\ReadLog.py -f "file.log" -k "key" -e "errMsg" ') Found solution: RunWait( 'fullpath\Python.exe ReadLog.py -f "file.log" -k "key" -e "errMsg" ', 'Working_directory_path') PFB des
13. ## Java Uninstall via msiexec
Hello, I am ready to rip my hair out at this point I am not knew to autoit or scripting at all and that is why I am failing to understand what I am doing wrong here. When i run the command in the command prompt the command executes successfully as it should, but when i attempt to run this code in autoit nothing seems to happen. I am really failing to realize why, becuase I use Runwait to run command line parameters all the time. Can some one please help me out here and show me why this will not work for me. The script is uninstalling java and obviously i need to have this for many different
14. ## Dos command openfiles run from autoit result file is empty
Good Morning Everyone, I want to say thank you to everyone here - you guys are fantastic at what you do and we all appreciate your help and hard work. All of IT has benefited from your hard work and knowledge. So thank you. Edited...
18. ## runwait /c flag
Hello, For quite a long time i have been wondering something: I quite frequently use the command RunWait(@ComSpec & " /c " & "commandName") Now i'm wondering: what does the /c flag do? i apologize if the /c flag is not related to runwait as i put it like that in the title. Edit: Is it even a flag? thanks, dh Edit2: Never mind, I've found the answer: /C Carries out the command specified by the string and then terminates You can get all the cmd command line switches by typing cmd /? source: http://stackoverflow.com/questions/515309/what-does-cmd-c-mean I tho
19. ## Runwait is not working
HI, I trying to copy a file from Host operating system to Guest operating system via VMWare-Workstation. The command prompt appears but disappears in a split of a second. Not sure where am going wrong. Can someone help please? Here is the script that I have written... $VWRunEXEPath = "C:Program Files (x86)VMwareVMware Workstation>vmrun.exe"$GuestUserName = "agovada" $GuestPwd = "Welcome2world"$VMXPathOfTheOS = """C:VMImagesVista-64Vista x64.vmx""" $SourcePathOfVSE = """C:dataVSE RP4 RTW.zip"""$TargetPathOfVSE = """C:buildsrp4.zip""" RunWait(@ComSpec & " /c " & "'" &
20. ## Problem with exit codes
Hi, I have a problem in that one script runs successfully and gets to this line MsgBox(0x40,$UpdateID,"Successfully installed update " &$UpdateID) Exit(1) This script was started by another at this point: $InstallationResult = RunWait($ExeFileFullPath & " " & $InstallerParameters) ProcessWaitClose(StringReplace($FileName,"mmupdate","exe")) SplashOff() If $InstallationResult =$DesiredResult Then FileMove($ExeFileFullPath,$FileFullPath & ".done") LogWrite("Update succeeded") Else ; the update failed
21. ## command-prompt gives response but autoit doesn't
I have installed the amazon command-line interface (http://aws.amazon.com/cli/) and have it working in my cmd prompt dos window. For example, If I set the working directory to C:UsersAdministrator and run the command below to upload a file to the S3 service: aws s3 cp C:UsersAdministratorDesktopvpntimeout_text.txt s3:/clustertesting/vpntimeout_text.txt I get the response: upload: myfolder/file1.txt to s3:/mybucket/myfolder/file1.txt However, trying to replicate the same command in Autoit with : --------------------------------------------------------------------------
22. ## Problems with Win8 8520?
I was wondering if there are any developers out there who are having issues running AutoIT code on the latest revision of Win8? I recently (about 4 months ago) wrote an auto-installer that pulled tools from a server and ran an auto install of each tool on systems. This worked fine for revisions 8250, 8375 and 8400. However, when Win8 revision 8441 came around, I noticed some issues running some installers. Now with the newest revision, 8520, it seems that FileCopy, DirCopy, ShellExecute, ShellExecuteWait, Run, and RunWait are all either locking up, or just failing to run. The steps I t
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2023-02-03 06:56:36
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http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0030099
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# Screening the United States Blood Supply for West Nile Virus: A Question of Blood, Dollars, and Sense
• To whom correspondence should be addressed. E-mail: byl1@pitt.edu
X
• Published: January 24, 2006
• DOI: 10.1371/journal.pmed.0030099
Now that the media frenzy over West Nile virus (WNV) has subsided, and pictures of dead birds and insect repellant cans no longer permeate the nightly news, what shall we do about screening the United States blood supply for the virus? Shortly after the spread of WNV in North America and the revelation that blood transfusions may transmit the virus [1], US officials established regulations requiring blood banks to screen all their donated blood for WNV using nucleic acid amplification tests. Although these requirements have been in place since June 2003, there has been debate over if and how to continue the screening [2, 3]. The threat of disease certainly still exists, but with limited resources and other potential hazards to blood supply safety, there has been a real need for good, objective economic studies to determine what type and what degree of screening should be performed.
### Two New Cost-Effectiveness Studies
Two recent cost-effectiveness studies address this question, one by Korves and colleagues [4] published in PLoS Medicine (summarized in Box 1) and another by Custer and colleagues [5] published in a recent issue of Annals of Internal Medicine (summarized in Box 2). Discussing mass infectious disease screening measures without considering economic consequences is like eating at a smorgasbord without considering calories and fat. While in the short term, “indulging” in a certain screening measure may be necessary to avert a disease outbreak, in the long term, public health officials must decide if it is worth continuing to invest precious resources that may be better utilized in other areas.
Because study assumptions and methodological approaches can vary, a series of studies may be needed to move toward an appropriate policy. Both Korves's study and Custer's study arrive at similar general conclusions: the cost-effectiveness of individual donor screening depends on WNV prevalence, and targeted donor screening appears to be more cost-effective than mass donor screening. The two studies differ, however, in some of the questions that they address and in the answers that they provide.
#### Who gets the blood may be as important as who gives the blood
While Custer's study focused on the blood donors, Korves's study also evaluated strategies that considered the recipient's immune status, finding that even when the prevalence of the virus is low, screening blood supplies destined for immunocompromised recipients may be cost-saving. The dangers and consequences of even a few immunocompromised patients contracting WNV by transfusion appear to significantly outweigh the low incidence of transmission.
#### The question may be not only whether to pool but how much to pool
While both studies compared individual donated blood sample testing with minipool testing, Korves's study took the extra step of looking at minipools of 16 samples and minipools of six samples separately. In the minipool testing method, a number of individual samples are combined in a pool, and the pool of samples is tested. When the pool tests negative, one assumes that each individual sample that contributed to the pool is negative. When the pool tests positive, one tests the individual donations to find the positive(s). The advantage of the minipool method is that it reduces the number of tests that have to be run, presumably reducing costs and saving time. The major drawback of this “pooling” method is that testing may not be able to detect low levels of virus that are further diluted when blood samples are combined. Therefore, pooling six samples together is not the same as pooling 16 samples; for a given number of donations, pools of size 16 require fewer tests, but each test may be less sensitive because of dilution. Although Korves's study did not find significant differences in the cost-effectiveness of these two types of pooling, the balances in efficiencies and costs between virus detection assays and the mechanics of testing may, in the future, favor larger pools.
#### Consider all of the steps involved in a screening operation
In determining the costs of screening, Korves's study captured aspects of the testing procedure—such as discarding false positives, notifying donors, and retrieving test results—which were overlooked by Custer's study. Screening can be a complicated operation and seemingly minor steps can inflate overall costs (especially when disproportionate time and labor are involved), or they can be bottlenecks. A cost-effectiveness study may suggest such targets for cost reduction.
#### Seasonal screening is potentially very different from year-round screening
Korves's study assumed that the cost per sample screened would remain relatively constant between seasonal and year-round approaches. In reality, in seasonal screening, there are start-up costs at the beginning of the screening season (e.g., appropriate reagents have to be produced and put in place) and stoppage costs at the end of the screening season (e.g., the reagents have to be removed from the production and screening lines and disassembled), some of which were accounted for in Custer's study. Additionally, in seasonal screening, personnel and operations may not be as efficient as in year-round screening (e.g., it may take a while for things to get up to speed, and it may be tougher to find seasonal employees than year-round employees).
#### When estimating the impact of disease, do not overlook potential productivity losses
In tabulating the cost of disease, Custer and colleagues included work productivity losses not considered by Korves and colleagues. Productivity losses from patients missing work or no longer able to work because of disability and death are always an important component of disease costs [6, 7]. This is especially true with a viral illness such as WNV, where the acute illness is often overshadowed by malaise and fatigue that may impair a person's ability to work.
### Conclusion
In the end, there are few perfect economic studies, and one should neither require nor expect perfection. Instead, the measure of an economic study is in not only the answers it provides but also the questions it raises. In this way, both Korves's article and Custer's article succeed. In addition to providing some answers (e.g., prevalence is related to cost-effectiveness, targeted screening may be more cost-effective, and blood designated for the immunocompromised should be screened), they raised different important questions (e.g., what are the true costs of seasonal and year-round screening? and what are the full economic effects of WNV?) and offered some essential direction for additional lines of inquiry. These studies indicate that the optimal cost-effectiveness strategy for WNV screening indeed depends on the situation, and public health officials can use their results in broader evaluations of WNV screening strategies.
#### Box 1. Korves and Colleagues' 2006 Study
Korves and colleagues constructed a Markov model simulating patients receiving blood transfusions under nine different screening strategies: (1) only administering a donor questionnaire, (2) year-round testing of 16-sample minipools (for a definition of minipool testing, see section entitled The Question May Be Not Only Whether to Pool, but How Much to Pool), (3) seasonal (i.e., May through November) testing of 16-sample minipools, (4) year-round testing of six-sample minipools, (5) seasonal testing of six-sample minipools, (6) year-round individual donor testing, (7) seasonal individual donor testing, (8) year-round individual testing of donations designated for immunocompromised recipients, and (9) seasonal individual testing of donations designated for immunocompromised recipients. They assumed that test-kit cost per sample would be US$3 for individual testing, US$0.50 for six-sample minipool testing, and US$0.19 for 16-sample minipool testing; determined a US$19 laboratory technician hourly rate; added additional testing costs such as the costs of discarded samples (US$90) and donor notification (US$500); and abstracted disease costs from the literature. They found that in high WNV prevalence areas, seasonal individual donation screening of blood designated for immunocompromised recipients would be most cost-effective and, in fact, cost saving, whereas in areas with low prevalence, using a donor questionnaire alone would be most cost-effective.
#### Box 2. Custer and Colleagues' 2005 Study
Custer and colleagues also constructed a Markov cohort model simulating patients receiving transfusions under seven different blood screening strategies: (1) no screening, (2) minipool testing throughout the US for half of the year, (3) minipool testing throughout the US over the entire year, (4) individual donor testing for one-third of the year in the quarter of the US with the highest prevalence of WNV, and minipool testing for the rest of the country and for the remaining two-thirds of the year, (5) individual donor testing for the entire year in the quarter of the US with the highest prevalence of WNV, and minipool testing for the rest of the country, (6) individual donor testing for one-third of the year, with minipool testing for the remainder of the year, and (7) national individual donor testing over the entire year. Using vendor reagent prices, they assumed that each minipool test would cost US$7 plus US$3 for labor and other related costs, each individual donation test would cost US$14 plus US$5 for labor and other related costs, and partial-year minipool testing and transitioning between minipool and individual donation testing would result in some additional laboratory preparation costs. Disease costs came from an economic study of the 2002 Louisiana WNV outbreak [8] and published productivity loss tables [9]. The most cost-effective strategy was annual, national minipool testing (US\$483,000 per quality-adjusted life-year saved), and sensitivity analyses showed that the cost-effectiveness depended most heavily on WNV virus prevalence and testing costs.
### References
1. 1. Centers for Disease Control and Prevention (2002) Investigations of West Nile virus infections in recipients of blood transfusions. MMWR Morb Mortal Wkly Rep 51: 973–974.
2. 2. Petersen LR, Epstein JS (2005) Problem solved? West Nile virus and transfusion safety. N Engl J Med 353: 516–517.
3. 3. AuBuchon JP (2005) Meeting transfusion safety expectations. Ann Intern Med 143: 537–538.
4. 4. Korves CT, Goldie SJ, Murray MB (2006) Cost-effectiveness of alternative blood screening strategies for West Nile virus in the United States. PLoS Med 3: e21. doi: 10.1371/journal.pmed.0030021.
5. 5. Custer B, Busch MP, Marfin AA, Petersen LR (2005) The cost-effectiveness of screening the U.S. blood supply for West Nile virus. Ann Intern Med 143: 486–492.
6. 6. Burton WN, Morrison A, Wertheimer AI (2003) Pharmaceuticals and worker productivity loss: A critical review of the literature. J Occup Environ Med 45: 610–621.
7. 7. Fendrick AM, Monto AS, Nightengale B, Sarnes M (2003) The economic burden of non-influenza-related viral respiratory tract infection in the United States. Arch Intern Med 163: 487–494.
8. 8. Zohrabian A, Meltzer MI, Ratard R, Billah K, Molinari NA, et al. (2004) West Nile virus economic impact, Louisiana, 2002. Emerg Infect Dis 10: 1736–1744.
9. 9. Grosse S (2003) Productivity loss tables. In: Haddix A, Teutsch S, Corso P, editors. Prevention effectiveness. 2nd ed. New York: Oxford University Press. pp. 245–257.
Ambra 2.10.7 Managed Colocation provided
by Internet Systems Consortium.
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2014-10-26 02:52:37
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https://books.compclassnotes.com/rothphys110-2e/2021/07/01/appendix-a-2/
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# Appendix A: Math review
## A.2: Basic algebra
The field of algebra gives us the basic tools we need to manipulate equations, or to pull useful information out of a mathematical sentence. There is one all-encompassing rule that must be obeyed:
Whatever you do on one side of an equals sign, you must do on the other!
If you multiply the left side of an equation by eight, you must multiply the right side by eight. If you subtract twelve from the left side, you must subtract twelve from the right side. You have no choice. This is fundamental to the logic system of equations.
This rule of algebra applies just as well to letters as it does to numbers—you can replace any number in an equation with a letter and still perform algebraic steps.
#### Example a.1
Solve the equation
$\frac{z + x}{y} = w$
for $$x$$.
This is exactly the same as the example in the previous section, but with letters instead of numbers. We will follow the same steps to solve for $$x$$:
\begin{align*} \frac{z+x}{y} &= w \\ \left(\frac{z+x}{y}\right)(y) &= w(y) \\ z + x &= w(y) \\ z + x – z &= w(y) – z \\ x &= w(y) – z \end{align*}
Even though there were no numbers involved, we can still find a solution for $$x$$. Because it is all symbols,* we call this a symbolic solution.
You can check that this is the correct answer by returning to the original equation and substituting in our solution:
\begin{align*} \frac{z + x}{y} &= \frac{z + \left(w(y) – z\right)}{y} \\ &= \frac{z + w(y) – z}{y} \\ &= \frac{w(y)}{y} \\ &= w \end{align*}
Good! Our solution is correct!
*When you get down to it, a number is also just a symbol. Whether we write the number 8 or the letter $$y$$, we’re just writing some squiggly line on a piece of paper to convey some meaning.
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2022-06-28 02:33:01
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https://itectec.com/superuser/wireshark-accessing-usb-bus-interfaces-without-sudo/
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# Wireshark: Accessing USB Bus Interfaces without sudo
sudousbwireshark
I am trying to monitor some USB traffic using Wireshark on Linux (Ubuntu). If I start Wireshark as my normal user without root privileges, the USB bus interfaces are not listed. If I sudo wireshark though, I do see the USB bus interfaces. This is perhaps best illustrated using the dumpcap command to list the available capture interfaces:
~$dumpcap -D 1. eth0 2. any (Pseudo-device that captures on all interfaces) 3. lo versus ~$sudo dumpcap -D
1. eth0
2. usbmon1 (USB bus number 1)
3. usbmon2 (USB bus number 2)
4. usbmon3 (USB bus number 3)
5. usbmon4 (USB bus number 4)
6. usbmon5 (USB bus number 5)
7. usbmon6 (USB bus number 6)
8. usbmon7 (USB bus number 7)
9. usbmon8 (USB bus number 8)
10. any (Pseudo-device that captures on all interfaces)
11. lo
Is there a way I can configure Wireshark so that I don't have to run as root to get access to the usb bus interfaces? I've previously followed a guide to setup wireshark so that I don't have to run it as root to see the Ethernet interfaces, so I'm wondering whether it's simply a matter of changing permissions on some other executable to be able see USB interfaces without running as root…
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2021-04-17 00:37:36
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https://chemistry.stackexchange.com/questions/60784/what-exactly-is-basicity?noredirect=1
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What exactly is basicity?
In my textbook, it’s given that the trend of the basicity of the elements of group 15 is $$\ce{NH3 > PH3 > AsH3 > SbH3 \ge BiH3}$$ Also, it’s said that the reducing character decreases down the group. (Which is the ability to lose $\ce{e-}$) It’s also give (on a different page) that $\ce{H3PO3}$ and $\ce{H3PO4}$ are di- and tri-basic respectively.
Phosphoric acid (presumably, a Brønsted-Lowry base) has a basicity of 3. This would imply that basicity is a property of acids. However, Ammonia (which is a Lewis base) has the highest basicity. Further, quickly Google search of the definition of basicity states: “Basicity is the number of hydrogen atoms replaceable by a base in a particular acid.”
My question is, how does Ammonia (a Lewis base) have a basicity (i.e. acid character)... shouldn’t it be a weak base? Or is my understanding of the definition of basicity wrong?
• Equivocation at its best. – DHMO Oct 12 '16 at 14:15
• – Mithoron Oct 12 '16 at 17:44
The second is somewhat confusing, and it is used to describe how many protons of an acid can be removed by neutralisation with base. For example, $\ce{HCl}$ has one acidic proton, so it is monobasic; $\ce{H2SO4}$ has two acidic protons, so it is dibasic.
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2019-11-14 04:24:13
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http://twbs.in/read.php?page=uci-gr/gr-03
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# Bei's Study Notes
### General Relativity 03Last updated: 2016-08-13 22:56:08 PDT.
Reference: UCI OpenCourseWare - GR 05 (Playlist)
The energy-momentum tensor of a point particle is give as:
$$T^{\alpha\beta}(x) = mu^\alpha u^\beta \gamma^{-1}\delta^{(3)}(\mathbf x - \mathbf {\hat x} (t))$$
There will be 4 conserved quantities:
$$\partial_\alpha T^{\alpha\beta} = 0\,.$$
Plugging it in, we get:
\begin{align*} \partial_\alpha mu^\alpha u^\beta \delta^{(3)}(\mathbf x - \mathbf {\hat x} (t)) &=0\\ \end{align*}
In EM, we have only one conserved quantity $\rho = J_0$, which comes from:
$$\square A_\mu = - J_\mu$$
Since GR is about the energy momentum tensor, in order for it to be covariant, we are essentially looking for a equation that looks like this:
$$\square (?)_{\mu\nu} = GT_{\mu\nu}$$
Einstein didn't realize that this must be an equation of a 10-component tensor until 1912. This is the gravitational field. (In contrary, in Newtonian gravity, you can define everything with a one-component gravitational potential $\phi(x)$).
### Angular momentum and spin
Starting from energy-momentum conservation:
$$\partial_\alpha T^{\beta\alpha} = 0$$
we define an object $M^{\gamma\alpha\beta}(x)$ (angular momentum density):
$$M^{\gamma\alpha\beta}(x) = x^\alpha T^{\beta\gamma} - x^\beta T^{\alpha\gamma}$$
Then we take the partial derivative of it:
\begin{align*} \partial_\gamma M^{\gamma\alpha\beta} &= \delta^\alpha{}_\gamma T^{\beta\gamma} + x^\alpha{} \partial_\gamma T^{\beta\gamma} - \delta^\beta{}_\gamma T^{\alpha\gamma} - x^\beta \partial_\gamma T^{\alpha\gamma} \\ &= T^{\beta\alpha} - T^{\alpha\beta} \\ &= 0 \\ \end{align*}
This implies a conserved quantity (angular momentum) $J^{\alpha\beta} \equiv \int M^{0\alpha\beta}\,dx^3= - J^{\beta\alpha}$. The spatial components of the angular momentum tensor gives the classical angular momentum:
$$J^{23} = \int M^{023}\, d x^3 = \int x^2 T^{30} - x^3 T^{20}\, d x^3 \\ = \gamma \int (y p^z - z p^y) \, d x^3 = \gamma I_x$$
The time components of it gives mass moment:
$$J^{01} = \int M^{001}\, d x^3 = \gamma \int t p_x - x m\, d x^3$$
A special note is that this quantity contains both orbital part and spin part of angular momentum, the latter does not depend on the origin. (I should make another note on this issue). This is called the Pauli–Lubanski pseudovector.
### Perfect fluid
Perfect fluid in special relativity
## Lorentz Group
(This is essentially talking about the Lie algebra of the Lorentz Group as we discussed before in QFT notes.)
In QM, we discussed rotation group $R$ in 3 dimensional Euclidian space:
$$\mathbf x' = \mathbb R \cdot \mathbf x$$
where $\mathbb R$ is a 3x3 matrix.
it corresponds to the transformation of an indexed wave function:
$$\psi_n' = u_{nm}\cdot \psi_m$$
and $u$ is the rotation matrix on the state vectors. This mean while we need to change the coordinate to index into a field under rotation, we also need to change the components of the field at the same time given the field has more than one components.
The rotation matrix has a simple form:
$$u = e^{\frac{i}{\hbar} \alpha \cdot J}$$
where $J$ is the angular momentum matrices. It satisfies certain algebra (in terms of commutation relations):
$$[J_i, J_k] = i\hbar \epsilon_{ijk} J_k$$
one of the outcome of this algebra is that only certain eigenvalues of the spin is allowed:
$$J^2 = \hbar^2 j(j+1),\quad j = 0,\frac 12, 1, ...$$
Considering special relativity, the true transformation of the world is not only rotation, but also boost. In SR, $\mathbf x \to x^\alpha$, therefore, we need to redo the exercise to make sure we can still only get spin $0, \frac 12, 1, ...$ but not $\frac 23$ and such.
We write
\begin{align*} x'^\alpha &= \Lambda^\alpha{}_\beta x^\beta \\ \psi' &= D(\Lambda)\cdot\psi \end{align*}
which $D(\Lambda)$ is the transformation associated with $\Lambda$. They form a group:
$$D(\Lambda)D(\Lambda') = D(\Lambda \Lambda')$$
To find $D$, consider an infinitesimal Lorentz transformation looks like:
$$\Lambda^\alpha{}_\beta = \delta^\alpha{}_\beta + \omega^\alpha{}_\beta$$
The product of two Lorentz transformation is:
\begin{align*} \Lambda^\alpha{}_\beta\Lambda'^\beta{}_\gamma &= (\delta^\alpha{}_\beta + \omega^\alpha{}_\beta)(\delta^\beta{}_\gamma + \omega'^\beta{}_\gamma)\\ &= \delta^\alpha{}_\gamma + (\omega^\alpha{}_\beta\delta^\beta{}_\gamma + \delta^\alpha{}_\beta\omega'^\beta{}_\gamma) + o(\omega^2)\\ &= \delta^\alpha{}_\gamma + (\omega^\alpha{}_\gamma + \omega'^\alpha{}_\gamma) + o(\omega^2) \end{align*}
So the product of two infinitesimal Lorentz transformation is the infinitesimal Lorentz transformation with the displacement equals to the sum of the two transformations.
where $\omega$ is infinitesimal. From its being rotation:
$$\Lambda^\alpha{}_\gamma\Lambda^\beta{}_\delta\eta_\alpha{}_\beta =\eta_\gamma{}_\delta$$
Therefore:
\begin{align*} (\delta^\alpha{}\gamma + \omega^\alpha{}_\gamma)(\delta^\beta{}_\delta + \omega^\beta{}_\delta)\eta_\alpha{}_\beta &= \eta_\gamma{}_\delta \\ \eta_\gamma{}_\delta + (\delta^\alpha{}\gamma \omega^\beta{}_\delta + \omega^\alpha{}_\gamma\delta^\beta{}_\delta)\eta_\alpha{}_\beta + o(\omega^2) &= \eta_\gamma{}_\delta \\ (\delta^\alpha{}\gamma \omega^\beta{}_\delta + \omega^\alpha{}_\gamma\delta^\beta{}_\delta)\eta_\alpha{}_\beta &= 0 \\ \omega^\beta{}_\delta\eta_\gamma{}_\beta + \omega^\alpha{}_\gamma\eta_\alpha{}_\delta &= 0 \\ \omega_\gamma{}_\delta + \omega_\delta{}_\gamma &= 0 \\ \end{align*}
Therefore $\omega_\alpha{}_\beta$ is anti-symmetric.
For $D(\Lambda)$, we will pick an anti-symmetric $\sigma$ to make $D$ a group:
$$D(\Lambda) = D(1 + \omega) = I + \frac 12 \omega^{\gamma\delta} \sigma_{\gamma\delta}$$
These $\sigma$s, which we pick to be anti-symmetric., are called the "generators of the Lorentz group".
Lorentz group has 6 different parameters, therefore we need 6 of these 4x4 $\sigma$ matrices, denoted as $\sigma_{\gamma\delta}$.
If $\psi$ is a 4-vector, then the $D(\Lambda) = \Lambda$, and then:
\begin{align*} \left(I + \frac 12 \omega^{\gamma\delta} \sigma_{\gamma\delta}\right)^\alpha{}_\beta &= \delta^\alpha{}_\beta + \omega^\alpha{}_\beta \\ \left(\frac 12 \omega^{\gamma\delta} \sigma_{\gamma\delta}\right)^\alpha{}_\beta &= \omega^\alpha{}_\beta \\ \left(\frac 12 \omega^{\gamma\delta} \sigma_{\gamma\delta}\right)^\alpha{}_\beta &= \eta_\beta{}_\delta\omega^\gamma{}^\delta\delta_\gamma{}^\alpha \\ \end{align*}
If all components of $\omega^\gamma{}^\delta$ are independently arbitrary, then we can equalize the coefficients of the them (thus removing $\omega$ from the equation). Unfortunately $\omega$ is anti-symmetric. So we need to expand the summation:
\begin{align*} \left(\frac 12 \omega^{\gamma\delta} \sigma_{\gamma\delta}\right)^\alpha{}_\beta &= \left(\sum_{\gamma < \delta} \omega^{\gamma\delta} \sigma_{\gamma\delta}\right)^\alpha{}_\beta \\ \delta_\gamma{}^\alpha\eta_\beta{}_\delta\omega^\gamma{}^\delta &= \left(\sum_{\gamma < \delta}+ \sum_{\gamma > \delta}\right)\delta_\gamma{}^\alpha\eta_\beta{}_\delta\omega^\gamma{}^\delta \\ &= \sum_{\gamma < \delta} \left(\delta_\gamma{}^\alpha\eta_\beta{}_\delta\omega^\gamma{}^\delta + (\gamma \leftrightarrow \delta)\right) \\ &= \sum_{\gamma < \delta} \left(\delta_\gamma{}^\alpha\eta_\beta{}_\delta - (\gamma \leftrightarrow \delta)\right)\omega^\gamma{}^\delta \end{align*}
Now we can cancel $\omega$ and get:
\begin{align*} (\sigma_{\gamma\delta})^\alpha{}_\beta &= \delta_\gamma{}^\alpha \eta_{\beta\delta} - \delta_\delta{}^\alpha \eta_\alpha{}_\delta \end{align*}
(Last time, I derived this from $D(\Lambda^{-1})\sigma^{\mu\nu}D(\Lambda) = \Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma\sigma^{\rho\sigma}$. This time its much simpler.)
The next step is to figure out the commutation relations. The result is:
\begin{align*} [\sigma_\alpha{}_\beta, \sigma_\gamma{}_\delta] &= (\eta_{\gamma\beta}\sigma_{\alpha\delta} - (\alpha\leftrightarrow\beta)) - ((\gamma\leftrightarrow\delta)) \end{align*}
Set(Pauli)
\begin{align*} a_i &= \frac 12 (-i \frac 12 \epsilon_{ijk} \sigma_{jk} + \sigma_{i0})\\ b_i &= \frac 12 (-i \frac 12 \epsilon_{ijk} \sigma_{jk} - \sigma_{i0})\\ [a_i, b_j] &= 0 \\ [a_i, a_j] &= i\epsilon_{ijk}a_k \\ [b_i, b_j] &= i\epsilon_{ijk}b_k \\ \end{align*}
So this breaks up into two rotation groups. So the eigenvalues of $a$ and $b$ should be:
\begin{align*} \hat a = a(a+1)\quad a=0,\frac 12, 1, \frac 32, ... \hat b = b(b+1)\quad a=0,\frac 12, 1, \frac 32, ... \end{align*}
Therefore all the representations of the Lorentz group should be classified by the a pair of numbers $(a, b)$.
(The relationship between dimension of the field/particle and the $(a,b)$ pair requires some further explanation).
|
2017-04-24 09:15:29
|
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https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-6-section-6-6-variation-6-6-exercises-page-418/22
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Intermediate Algebra (12th Edition)
$\text{The volume of a sphere varies directly as the cube of its radius.}$
Recall: (1) When $y$ varies directly as $x$, the direct variation's equation is $y=kx$ where $k$ is the constant of variation. (2) When $y$ varies inversely as $x$, the inverse variation's equation is $y=\frac{k}{x}$ or $xy=k$ where $k$ is the constant of variation. In the equation $V=\frac{4}{3}\pi{r^3}$, $V$ varies directly as $r^3$ with a constant of variation of $\frac{4}{3}\pi$. Thus: $\text{The volume of a sphere varies directly as the cube of its radius.}$
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2019-11-18 08:11:18
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https://socratic.org/questions/how-do-you-graph-y-1-3absx
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# How do you graph y=1/3absx?
Oct 15, 2016
Graph an absolute value function with a slope of $\pm \frac{1}{3}$.
#### Explanation:
The "parent function" of $y = \frac{1}{3} \left\mid x \right\mid$
is the absolute value equation $y = \textcolor{red}{1} \left\mid x \right\mid$.
It has a "V-shape" with a vertex at $\left(0 , 0\right)$.
The slope of the lines that form the V are $\textcolor{red}{\pm 1}$.
graph{abs(x) [-10, 10, -5, 5]}
The fraction $\textcolor{red}{\frac{1}{3}}$ represents the slopes $\textcolor{red}{\pm \frac{1}{3}}$ of the lines that form the "V" of $y = \textcolor{red}{\frac{1}{3}} \left\mid x \right\mid$.
graph{1/3absx [-10, 10, -5, 5]}
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2019-09-16 15:01:19
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https://socratic.org/questions/how-do-you-convert-45-centimeters-to-kilometers
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How do you convert 45 centimeters to kilometers?
$54$ cm $= 5.4 \times {10}^{- 4}$ km
$100$ cm $= 1$ m $\rightarrow 1$ cm$= {10}^{- 2}$ m
$1000$ m $= 1$ km $\rightarrow 1$ m $= {10}^{- 3}$ km $\rightarrow 1$ cm $= {10}^{- 5}$ km
$54$ cm $= 54 \times {10}^{- 5}$ km $= 5.4 \times {10}^{- 4}$ km
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2020-09-26 22:17:13
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https://www.nature.com/articles/s41467-017-01581-6?error=cookies_not_supported&code=82932737-357e-4508-94ef-ecbafeb93ed8
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Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Deterministic and robust room-temperature exchange coupling in monodomain multiferroic BiFeO3 heterostructures
## Abstract
Exploiting multiferroic BiFeO3 thin films in spintronic devices requires deterministic and robust control of both internal magnetoelectric coupling in BiFeO3, as well as exchange coupling of its antiferromagnetic order to a ferromagnetic overlayer. Previous reports utilized approaches based on multi-step ferroelectric switching with multiple ferroelectric domains. Because domain walls can be responsible for fatigue, contain localized charges intrinsically or via defects, and present problems for device reproducibility and scaling, an alternative approach using a monodomain magnetoelectric state with single-step switching is desirable. Here we demonstrate room temperature, deterministic and robust, exchange coupling between monodomain BiFeO3 films and Co overlayer that is intrinsic (i.e., not dependent on domain walls). Direct coupling between BiFeO3 antiferromagnetic order and Co magnetization is observed, with ~ 90° in-plane Co moment rotation upon single-step switching that is reproducible for hundreds of cycles. This has important consequences for practical, low power non-volatile magnetoelectric devices utilizing BiFeO3.
## Introduction
Substantial effort has been devoted to understand the internal magnetoelectric coupling between ferroelectric (FE) and antiferromagnetic (AF) orders in BiFeO3, motivated by interest in exchange coupling to a ferromagnetic (FM) overlayer for use in nonvolatile magnetoelectric devices1,2,3,4,5. Nearly all approaches to exchange coupling investigated so far rely on coupling between the magnetization of the ferromagnetic overlayer and the small canted moment in BiFeO3 (~ 0.06 µ B Fe−1 6) that originates from the Dzyaloshinskii–Moriya interaction7,8. Due to the spin-cycloid structure found in single crystals of BiFeO3, this net moment averages to zero over a cycloid period (~62 nm3) precluding coupling over macroscopic areas. Suppression of the spin cycloid (e.g., due to strain in thin films9, the presence of domain walls in narrow stripes10,11,12, or application of a large magnetic field13,14) can lead to the formation of a collinear G-type antiferromagnetic order with mutually orthogonal ferroelectric polarization vector (P), Néel vector (L), and canted moment (M c) that allows the possibility of a macroscopic exchange coupling via M c. While P is fixed along the pseudocubic (pc) < 111 > directions, in multidomain epitaxial BiFeO3 films the orientation of L and M c can depend on the type (compressive, tensile) and magnitude of the strain (L. W. Martin, private communication;15).
To stabilize a one-to-one correlation between order parameters in BiFeO3 and exploit this state for robust, deterministic exchange coupling without the complications of domain walls16,17,18,19,20, we have fabricated Co/BiFeO3 heterostructures with ferroelastic and ferroelectric monodomain properties in BiFeO3 over the entire sample21. In bulk single crystals22 and relaxed monodomain films23, a nondeterministic exchange coupling mechanism between BiFeO3 and Fe or Co overlayers has been shown to result from the energy degeneracy of the three allowed {112}pc–type spin-cycloid planes. As a consequence, the first switching event in a single P-domain results in nucleation of multiple AF domains so that the original AF domain state cannot be recovered24.
In the following we demonstrate that, unlike monodomain crystals and relaxed films, strained P-monodomain BiFeO3 films are also AF-monodomain, characterized by a unique internal magnetoelectric state with a one-to-one relationship between P and the cycloid plane orientation that is reproducible after hundreds of switching events. This confirms that strain lifts the cycloid state degeneracy and modifies the internal magnetoelectric coupling of BiFeO3. Consequently, we find that the magnetic order present at the interface with a ferromagnetic overlayer can be changed deterministically by polarization switching that is in turn correlated to the rotation of the magnetization of a Co overlayer. This robust exchange coupling mechanism offers an alternative solution for implementation in potential devices.
## Results
### Experimental overview
To probe the interfacial magnetoelectric state of ferroelectric and ferroelastic monodomain BiFeO3 films in the down ($$r_1^ -$$) and up ($$r_3^ +$$) polarization states (using the ferroelastic domain notation of ref. 25), to correlate these to the rotation of Co ferromagmetic domains and understand the mechanisms of magnetoelectric coupling, a method for in situ ferroelectric switching during Photoemission Electron Microscope (PEEM) imaging was developed. Figure 1 shows a schematic of the experimental setup and summary of the ferroelectric–antiferromagnetic–ferromagnetic correlations we report here. PEEM imaging allows simultaneous spatially resolved monitoring of changes in ferromagnetic domains (by X-ray Magnetic Circular Dichroism, XMCD, on the Co L 3 edge) and AF domains in the interfacial BiFeO3 (by X-ray Magnetic Linear Dichroism, XMLD, on the Fe L 3 edge), permitting unambiguous tracking of the spin orientations for the ferroelectric down (Fig. 1b) and up (Fig. 1c) states over many switching events. To complement the surface-sensitive information from PEEM imaging, the spin-cycloid properties in the bulk of the film were measured on separately prepared samples by large-area neutron diffraction (ND), also shown in Fig. 1. Magneto-optic Kerr effect (MOKE) measurements were performed at room temperature to confirm the exchange coupling between the BiFeO3 and Co seen by XMCD-PEEM. These results together demonstrate a robust and deterministic exchange coupling between monodomain BiFeO3 and Co at room temperature that can be readily exploited in practical devices.
### Antiferromagnetic analysis of the BiFeO3 film
The antiferromagnetic spin-structure in the bulk of the BiFeO3 film and at the interface was studied by ND and XMLD-PEEM, respectively, the latter through light polarization angle scans whose electron-yield signal originates within <10 nm of the surface26. Analysis of ND results (Supplementary Note 2) in the down state (Fig. 2a, c) shows that the ferroelectric monodomain in BiFeO3 possesses a single antiferromagnetic domain with the propagation vector in the film plane and a spin-cycloid plane orientation that is quite different from bulk single crystals, lying 12° from the film plane. XMLD-PEEM analysis shows that this cycloid continues unchanged up to the surface (Fig. 3a). This cycloid plane does not contain the polarization vector P, and is distinct from the three {112}–type planes seen in bulk single crystals24 or the cycloid configuration observed previously in thin films27, for which P lies in the cycloid plane. The cycloid propagation vector k extracted from the ND results is parallel to $$[1\bar 10]$$ (one of the allowed vectors in bulk, and along the substrate step direction) with a cycloid period of 66 ± 2 nm. The longer period as compared to that in bulk is in quantitative agreement with the value expected for a cycloid plane that does not contain P, as the period should scale inversely with the projection of P onto the plane (projection angle ~ 24°).
In the up state, ND results reveal a similar in-plane cycloid orientation (Fig. 2b, d) as seen in the down state, but XMLD-PEEM analysis (Supplementary Notes 6 and 7 and Supplementary Figs. 9 and 10) shows that a different antiferromagnetic order develops at the interface (Fig. 3b). These magnetoelectric configurations are maintained reproducibly after repeated cycling. The up state is characterized as a majority r 3 + domain (71° switching) deduced by ND and piezoforce microscopy (Supplementary Note 1 and Supplementary Fig. 4); a minority domain (<12%) corresponding to 180° switching (r 1 +) is also found, which we disregard as its presence does not change the analysis or conclusions. ND identifies a single cycloid in the majority domain with orientation similar to that in the down state, but rotated 13° in the direction of the P rotation, with the same propagation vector. This direct experimental identification of a non-bulk-like spin-cycloid plane (including propagation vector and spin orientation) is consistent with evidence that strain can alter the cycloid properties in BiFeO3, albeit differently than proposed in ref. 10 or reported in ref. 27. The ND results demonstrate that the strain state of the film (Supplementary Fig. 1) creates a non-bulk, in-plane orientation of the cycloid only weakly dependent on the polarization vector.
In contrast, surface-sensitive XMLD polarization angular scans in the up state (Fig. 3b) are distinctly different from the down state, and are not consistent with the single-cycloid plane found by ND, nor a cycloid of any orientations found in bulk single-crystal BiFeO3 (Fig. 3c, d and Supplementary Fig. 11). Instead, fitting the XMLD-PEEM results (Supplementary Note 7) requires an additional component beyond the bulk cycloid measured by ND, either collinear antiferromagnetic order with L oriented along the [112]pc direction or a vertical cycloid plane with propagation vector along the substrate [110]pc miscut direction (discussed later). This extra component contributes ~ 25% of the total XMLD-PEEM intensity that should be distributed over 1 nm near the interface. The XMLD-PEEM data summarized here are identical for Co/BiFeO3 and Pt/BiFeO3 interfaces in both polarization states (within error bars).
### Ferromagnetic Co response upon BiFeO3 polarization switching
Demonstration of a one-to-one correlation between the magnetoelectric state of BiFeO3 and Co magnetization is presented in Co XMCD-PEEM vector maps for a down-up-down switching sequence (Fig. 4) at 120 K. Analysis of the Co XMCD-PEEM images (Supplementary Note 8) yields vector magnetization maps of the Co magnetization showing a ~ 90° in-plane rotation upon ferroelectric switching. In the down state (Fig. 4a, c), the Co uniaxial anisotropy axis lies along the substrate step edges produced by the miscut (Supplementary Note 6 and Supplementary Figs. 2, 5); in the up state, the anisotropy axis rotates nearly 90° toward the [110]pc miscut direction (Fig. 4b). This correlation between Co anisotropy axis and BiFeO3 polarization demonstrates a previously unreported type of intrinsic exchange coupling between BiFeO3 and Co that is deterministic, robust, and reliable over hundreds of switching cycles (Supplementary Note 5 and Supplementary Fig. 8). In the down state of BiFeO3, Co domains are split parallel and antiparallel to the $$[1\bar 10]$$ pc step edges direction, a distribution resulting from demagnetization effects that minimize the magnetostatic energy of the large electrode (100 × 100 µm2) and likely would not be present as the dimensions approach the micrometer scale28. The same anisotropy axis is seen in Co deposited directly on identically miscut (001) SrTiO3 substrates (Supplementary Note 3 and Supplementary Figs. 5, 6). When the BiFeO3 is switched to up state, the parallel/antiparallel Co domains rotate toward the [110]pc miscut direction (perpendicular to the step edges), as seen in the polar plots (Fig. 4d–f). Analyzing the local spin rotations, they are closer to ~ 75° upon switching (Supplementary Note 8 and Supplementary Fig. 12) leading to the overall distribution shown in the polar plots of Fig. 4d–f.
### Evidence of room temperature exchange coupling
MOKE measurements at room temperature using the configuration shown schematically in Fig. 5a, b demonstrate the same exchange coupling seen in the XMCD-PEEM vector maps. M-H hysteresis loops (Fig. 5c, d) with the magnetic field applied along the $$[110]_{{\mathrm{pc}}}$$ miscut direction and $$[1\bar 10]_{{\mathrm{pc}}}$$ step edges direction show that the Co anisotropy axis rotates from parallel to the step edges in down state to nearly perpendicular in the up state. The hysteresis loops can be simulated using the Stoner–Wohlfarth model that includes two anisotropy energies, one from the steps present in both down and up states, and a second appearing in the up state that is oriented nearly along the miscut, as suggested by the XMLD-PEEM results shown in Fig. 3. In this model, fits to the hysteresis loops confirm the ~ 75° easy axis rotation upon switching seen in the XMCD vector maps (Supplementary Note 4 and Supplementary Fig. 7).
## Discussion
The combined experimental results suggest an exchange coupling mechanism across the Co/BiFeO3 interface that controls the Co rotation upon switching. In the down state (equivalently, for Co deposited on miscut SrTiO3) the magnetic easy axis is imposed by the miscut that breaks the symmetry between the otherwise equivalent in-plane directions. Since BiFeO3 up and down states present exactly the same in-plane strain, no strain-mediated—piezomagnetic or magnetostrictive—coupling across the interface can explain the observed Co spin rotation. In contrast, up and down states present different polarization charge screening at the interface; in the up state an excess of metallic electrons accumulating at the interface is expected, while in the down state these will be replaced by positive carriers. Such changes in electronic density can potentially influence the behavior of both BiFeO3 and Co close to the interface. In the case of ferromagnetic metals, such effects are known to affect the magnetic anisotropy between different crystallographic directions, and could potentially lead to a rotation of the easy axis29. However, experimentally we measure stronger coupling to Co for thinner BiFeO3 films, while charge-driven effects should be independent of thickness; hence, we can tentatively disregard the direct effect of screening on the Co layer.
Finally, and unexpectedly, we have found a difference in the magnetic structure of BiFeO3 near the interface for up and down states. Our ND data indicate that the two polarization states display a similar (001)pc cycloid plane in the bulk of the film, while XMLD analysis reveals that the up state presents an additional interfacial magnetic structure suggesting spins lying within the vertical plane containing the [110]pc miscut direction, most likely collinear. This feature clearly breaks the symmetry between [110]pc miscut and $$[1\bar 10]$$ pc step edges directions, and could be responsible for the preferred Co spin orientation in the up state. Two ingredients are needed for such a mechanism to be active, namely, exchange coupling between Fe and Co spins across the interface, and the presence of local ferromagnetic moments along the miscut direction in the interfacial BiFeO3. While the former can be taken for granted, the latter is more delicate, but not implausible; indeed, such a situation may pertain to canted antiferromagnetic or cycloid structures, or if the near-interface spin-structure of BiFeO3 were ferromagnetic or A-type antiferromagnetic (presenting ferromagnetically aligned spins within (001)-oriented FeO2 planes). Our data are compatible with these possible configurations, but does not allow distinguishing between them.
Several concluding points are worth making regarding the unusual interfacial magnetic structure of BiFeO3 in the up state driving the exchange coupling. First, epitaxial strain is known to affect the BiFeO3 antiferromagnetic structure10,27, with our results showing selection of a previously unseen cycloid variant in the bulk of the film and for the first time a polarization dependence in the surface order that correlates with the Co rotation. Second, accumulation of free carriers may favor specific magnetic interactions in the interfacial BiFeO3, with the presence of extra electrons in the up state expected to favor ferromagnetic interactions that could lead, e.g., to sizeable localized canted moments30. Such a scenario is physically sound, and could explain the observed magnetoelectric control of Co spins. Finally, we note that the deterministic and robust exchange coupling reported here is a consequence of the magnetoelectric properties of monodomain BiFeO3 films that has clear potential for applications: the observed in-plane Co moment rotation of ~ 90° over device-relevant areas is quite sufficient for large changes of tunneling magnetoresistance31 in technologically useful magnetic tunnel junctions. The significance of these results lies also in the unique advancement of state-of-the-art approaches to multifunctional oxide film growth (domain engineering and epitaxial strain in multiferroics) and characterization via spectroscopic and diffraction techniques available only at large-scale facilities (PEEM with in situ ferroelectric switching and ND) that are necessary to elucidate the microscopic characteristics of the magnetoelectric and exchange coupling.
## Methods
### Sample preparation and device fabrication
(001) SrTiO3 single-crystal substrates with 4° miscut toward [110]pc direction are used for domain engineering of BiFeO3 thin films21. A 35 nm thick SrRuO3 bottom electrode layer is first deposited by 90° off-axis sputtering32 at 600 °C followed by 300 nm of BiFeO3 films grown by double-gun off-axis sputtering at 750 °C with Ar:O2 ratio of 4:1 at a total pressure of 400 mTorr33. The BiFeO3 target contains 5% excess Bi2O3 to compensate for bismuth volatility33. Subsequently, 2 nm Co and 3 nm Al are deposited as the ferromagnetic and passivation layers by magnetron sputtering at room temperature without applied magnetic field. Capacitor structures were defined lithographically with a photoresist mask and subsequently ion-milled down to the BiFeO3 film. The devices were wirebonded to a specially designed printed-circuit board and/or chip carrier for in situ ferroelectric switching measurements in the PEEM. The substrate miscut induces the growth of a single ferroelastic domain variant, while the SrRuO3 bottom electrode introduces an electrical boundary condition favoring the ferroelectric down r 1 state21 since the depolarization field can be screened by free charge in the electrode during growth34. 71° switching leads to the monodomain up r 3 + state. Multiple (separate) top electrodes were patterned on a 5 × 10 mm2 sample for ND experiments, ensuring >98% of the film volume was switched and confirmed by PE measurement without preset loop.
### PEEM
The PEEM results were recorded on beam line i06 (Diamond Light Source, UK), which is equipped with two Apple-II undulators delivering a high flux of X-rays on a 10 µm diameter spot with tunable polarization in the energy range 80–2100 eV. The light polarization can be left or right circular or linear (with variable orientation of E in the range $$0^\circ \le \theta _{\mathbf{E}} \le 90^\circ$$). The beam is incident on the sample at an angle of 16° and $$\theta _{\mathbf{E}} = 0^\circ$$ corresponds to s-polarization. The PEEM is an Elmitec SPELEEM-III equipped with a manipulator stage with motorized x, y translation and manipulator azimuthal rotation $$\phi _{\mathrm{M}},$$ a liquid nitrogen cryostat and a Multiferroic Tester II (Radiant Technologies, Inc., Albuquerque NM)) to determine the FE state of the device in situ. $$\phi _{\mathrm{M}}$$ could be changed over a 200° range. The probing depth of the PEEM technique in electron yield is estimated to be ~ 5 nm. The cobalt FM domain structure was measured by combining XMCD-PEEM images at the Co L 3 edge taken at ϕ M = 90° and 0° to obtain the local magnetization in-plane vector35. These XMCD-PEEM vector maps of the Co FM domain structure were performed after poling the sample either up or down with the FE tester. The FE state of the BiFeO3 was checked with the FE Tester before and after each of the XMLD polar scans as shown in Supplementary Fig. 3.
### Neutron diffraction
Single-crystal neutron diffraction experiments were performed using WISH, a time-of-flight diffractometer at ISIS, the UK Neutron and Muon Spallation source (Supplementary Note 2).
### Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
## References
1. 1.
Eerenstein, W., Mathur, N. D. & Scott, J. F. Multiferroic and magnetoelectric materials. Nature 442, 759–765 (2006).
2. 2.
Béa, H., Gajek, M., Bibes, M. & Barthélémy, A. Spintronics with multiferroics. J. Phys. Condens. Matter 20, 434221 (2008).
3. 3.
Catalan, G. & Scott, J. F. Physics and applications of bismuth ferrite. Adv. Mater. 21, 2463–2485 (2009).
4. 4.
Bibes, M. & Barthélémy, A. Multiferroics: towards a magnetoelectric memory. Nat. Mater. 7, 425–426 (2008).
5. 5.
Martin, L. W. et al. Nanoscale control of exchange bias with BiFeO3 thin films. Nano Lett. 8, 2050–2055 (2008).
6. 6.
Ramazanoglu, M. et al. Local weak ferromagnetism in single-crystalline ferroelectric BiFeO3. Phys. Rev. Lett. 107, 207206 (2011).
7. 7.
Rahmedov, D., Wang, D., Iniguez, J. & Bellaiche, L. Magnetic cycloid of BiFeO3 from atomistic simulations. Phys. Rev. Lett. 109, 037207 (2012).
8. 8.
Dong, S. et al. Exchange bias driven by the dzyaloshinskii-moriya interaction and ferroelectric polarization at G-Type antiferromagnetic perovskite interfaces. Phys. Rev. Lett. 103, 127201 (2009).
9. 9.
Sando, D. et al. Crafting the magnonic and spintronic response of BiFeO3 films by epitaxial strain. Nat. Mater. 12, 641–646 (2013).
10. 10.
Holcomb, M. B. et al. Probing the evolution of antiferromagnetism in multiferroics. Phys. Rev. B 81, 134406 (2010).
11. 11.
Zhao, T. et al. Electrical control of antiferromagnetic domains in multiferroic BiFeO3 films at room temperature. Nat. Mater. 5, 823–829 (2006).
12. 12.
Heron, J. T. et al. Electric-field-induced magnetization reversal in a ferromagnet-multiferroic heterostructure. Phys. Rev. Lett. 107, 217202 (2011).
13. 13.
Ruette, B. et al. Magnetic-field-induced phase transition inBiFeO3 observed by high-field electron spin resonance: cycloidal to homogeneous spin order. Phys. Rev. B 69, 064114 (2004).
14. 14.
Popov, Y. F. et al. Linear magnetoelectric effect and phase transitions in bismuth ferrite, BiFeO3. JETP Lett. 57, 69–73 (1993).
15. 15.
Wojdeł, J. C. & Íñiguez, J. Ab Initio indications for giant magnetoelectric effects driven by structural softness. Phys. Rev. Lett. 105, 037208 (2010).
16. 16.
Heron, J. T. et al. Deterministic switching of ferromagnetism at room temperature using an electric field. Nature 516, 370–373 (2014).
17. 17.
Baek, S. H. et al. The nature of polarization fatigue in BiFeO3. Adv. Mater. 23, 1621–1625 (2011).
18. 18.
Zou, X. et al. Mechanism of polarization fatigue in BiFeO3. ACS Nano 6, 8997–9004 (2012).
19. 19.
Seidel, J. et al. Conduction at domain walls in oxide multiferroics. Nat. Mater. 8, 229–234 (2009).
20. 20.
Jang, H. W. et al. Domain engineering for enhanced ferroelectric properties of epitaxial (001) BiFeO3 thin films. Adv. Mater. 21, 817–823 (2009).
21. 21.
Baek, S. H. et al. Ferroelastic switching for nanoscale non-volatile magnetoelectric devices. Nat. Mater. 9, 309–314 (2010).
22. 22.
Lebeugle, D. et al. Exchange coupling with the multiferroic compound BiFeO3 in antiferromagnetic multidomain films and single-domain crystals. Phys. Rev. B 81, 134411 (2010).
23. 23.
Ratcliff, W. et al. Electric-field-controlled antiferromagnetic domains in epitaxial BiFeO3 thin films probed by neutron diffraction. Phys. Rev. B 87, 140405(R) (2013).
24. 24.
Johnson, R. D. et al. X-Ray imaging and multiferroic coupling of cycloidal magnetic domains in ferroelectric monodomain BiFeO3. Phys. Rev. Lett. 110, 217206 (2013).
25. 25.
Streiffer, S. K. et al. Domain patterns in epitaxial rhombohedral ferroelectric films. I. Geometry and experiments. J. Appl. Phys. 83, 2742–2753 (1998).
26. 26.
Ruosi, A. et al. Electron sampling depth and saturation effects in perovskite films investigated by soft x-ray absorption spectroscopy. Phys. Rev. B 90, 125120 (2014).
27. 27.
Bertinshaw, J. et al. Direct evidence for the spin cycloid in strained nanoscale bismuth ferrite thin films. Nat. Commun. 7, 12664 (2016).
28. 28.
Johnson, M. T. et al. Magnetic permeability and intra-granular domain structure in polycrystalline ferrites. J. Magn. Magn. Mater. 104–107, 421–422 (1992).
29. 29.
Duan, C.-G. et al. Tailoring magnetic anisotropy at the ferromagnetic/ferroelectric interface. Appl. Phys. Lett. 92, 122905 (2008).
30. 30.
Vinokurov, D. L. & Morosov, A. I. Surface distortions in a weak ferromagnet. Phys. Solid State 55, 986–989 (2013).
31. 31.
Jaffrès, H. et al. Angular dependence of the tunnel magnetoresistance in transition-metal-based junctions. Phys. Rev. B 64, 064427 (2001).
32. 32.
Eom, C. B. et al. In situ grown YBa2Cu3O7−d thin films from single–target magnetron sputtering. Appl. Phys. Lett. 55, 595–597 (1989).
33. 33.
Das, R. R. et al. Synthesis and ferroelectric properties of epitaxial BiFeO3 thin films grown by sputtering. Appl. Phys. Lett. 88, 242904 (2006).
34. 34.
Fong, D. et al. Stabilization of monodomain polarization in ultrathin PbTiO3 films. Phys. Rev. Lett. 96, 127601 (2006).
35. 35.
Moya, X. et al. Giant and reversible extrinsic magnetocaloric effects in La0.7Ca0.3MnO3 films due to strain. Nat. Mater. 12, 52–58 (2013).
## Acknowledgements
This work was supported by the Army Research Office through grant W911NF-10-1-0362 and W911NF-13-1-0486. PEEM measurements at Diamond Synchrotron (Didcot, UK) were performed at the i06 beamline under proposals nt12084, nt13225, and si11589. Part of this work was performed at the Surface/Interface: Microscopy (SIM) beamline of the Swiss Light Source, Paul Scherrer Institut, Switzerland. Work at Argonne National Laboratory was supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. DEAC02-06CH11357. M.v.V. is supported by the U.S. DOE under Award No. DE-FG02-03ER46097. R.D.J. acknowledges STFC for the provision of beam time on the WISH instrument at ISIS, proposal number RB1600019, and P. Manuel for data collection. R.D.J. acknowledges support from a Royal Society University Research Fellowship. J.Í. acknowledges support from the Luxembourg National Research Fund (Grant number FNR/P12/4853155).
## Author information
Authors
### Contributions
C.B.E., B.A.D., and W.S. conceived the project. W.S. grew and patterned the BiFeO3 heterostructures and performed AFM, PE, and PFM measurements. F.M., J.P.P., W.S., B.A.D., J.I., C.A.F.V., S.S.D., and L.H. took the PEEM data, and F.M. and S.S.D. analyzed it. J.W.F., B.A.D., J.P.P., W.S., J.C.F., and S.R. took and analyzed the XAS measurements. Neutron diffraction data were taken by R.D.J. and analyzed by R.D.J. and P.G.R., and J.I., K.R., and W.S. took and analyzed the MOKE measurements. M.v.V. and J.W.F. performed multiplet modeling of XAS/XMLD data. T.K. and W.S. performed and analyzed XRD. J.Í. provided theoretical analysis. All authors participated in discussion of the results and the manuscript. C.B.E. directed the research.
### Corresponding author
Correspondence to C. B. Eom.
## Ethics declarations
### Competing interests
The authors declare no competing financial interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Saenrang, W., Davidson, B.A., Maccherozzi, F. et al. Deterministic and robust room-temperature exchange coupling in monodomain multiferroic BiFeO3 heterostructures. Nat Commun 8, 1583 (2017). https://doi.org/10.1038/s41467-017-01581-6
• Accepted:
• Published:
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• Fei Xue
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Physical Review B (2021)
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• ### Nanoscale magnetoelectric effects revealed by imaging
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• , S.S. Dhesi
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Journal of Magnetism and Magnetic Materials (2021)
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2021-06-22 20:19:03
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|
https://www.physicsforums.com/threads/question-about-potential-energy.874288/
|
Let's say we have a mass M at the ground of the earth with speed U going upwards so its energy is E= kinetic. The speed is enough to surpass earth's gravitational field so now it has a speed U2<U and its energy now is E= kinetic+potential. So my question now is: Is this potential energy lost? I mean when it's in the gravitational field its kinetic energy converts to potential and potential to kinetic when back down... So when it exits the field potential can no longer convert to kinetic? If this is the case where does potential energy goes ?
It never exits the field. The gravitational field of teh Earth is not limited to a finite region of space.
Like the post above mine says, the body never escapes the field. The potential energy is ##U=-\frac{GM_em}{r}## . As the distance increases, ##U## tends to zero.
For a body to escape the pull of earth its kinetic energy ##K## must be greater than ##U## i.e., the total energy ##E=K+U## must be positive ( ##E## is constant).
As the distance increases, ##U## goes to zero and ##K## slowly decreases. The total energy ##E## will always remain constant though.
What about U=mgh ? As far as I know acording to this equation potential energy seems to be positive but also getting bigger as h growing. What's going on?
What about U=mgh ? As far as I know acording to this equation potential energy seems to be positive but also getting bigger as h growing. What's going on?
##U=mgh## is only an approximate relation ( where g doesn't change appreciably with distance). For your case, where the body wants to escape the earth's attraction, you cannot assume ##g## to be constant. Also, in ##U=mgh##, the potential at the earth's surface is zero and at a very large distance it's infinity. Compare this to ##-\frac {GM_em}{r}##
jbriggs444
Homework Helper
What about U=mgh ? As far as I know acording to this equation potential energy seems to be positive but also getting bigger as h growing. What's going on?
Only differences in potential energy are physically meaningful. The actual value at a chosen reference point is irrelevant.
When using ##U\, = \, mgh##, one is implicitly assuming a reference point at ground level, a height measured up from ground level and a potential energy of zero at the reference point.
When using ##U \, = \, - \frac{GM_em}{r}##, one is assuming a reference point at infinity, a radius measured out from the center of the earth and a potential energy of zero at the reference point.
The difference between potential energy of a mass at a height of 10 meters above the earth's surface and the potential energy energy of a mass at the earth's surface is the same, either way. Both formulas give the same answer for the difference.
sophiecentaur
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2022-05-17 11:01:46
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http://mathhelpforum.com/number-theory/139788-what-s-so-special-about-composite-number-2.html
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1. Originally Posted by superdude
I think I'm getting the idea. So when you say n=O(log(N)) you're giving a particular example? You're not saying, whenever a subroutine depends on length of the argument then its big O is n=O(log(N)), that's not what you're saying?
For example, knowing that if a number does not have a factor less than or equal to its square root, the number if prime, an algorithm implementing this idea would have big O of $\sqrt{n}$ and a person would write O(n) where [mant]n=\sqrt(N)[/tex] and a person would speak "the algorithm has an efficiency of big O of square-root-n"?
What is an appropriate measure of the size of the problem depends on the problem.
CB
2. Originally Posted by superdude
2702811 also crashes the program. I notice 3 is a common factor in both cases.
I like how there are a couple conversations going on at once here, heh.
I copied and pasted your code for use with my XAMPP setup and got problems with 2702811 sometimes but not always, and I wasn't able to get an error with 15999.
To tell the truth, I don't understand the logic of your code, but I notice that the two functions call each other an awful lot.
It seems like you're not doing anything more complex than trial division, so what do you think of this instead?
[php]
<?php
$num = 342132; if ($num < 2) {
echo "No prime factors.";
exit;
}
echo "The prime factorization of " . $num . " is:<br />"; while ($num % 2 == 0) {
echo "2<br />";
$num /= 2; } while ($num % 3 == 0) {
echo "3<br />";
$num /= 3; }$lim = floor(sqrt($num)); for ($i = 5, $inc = 2;$i <= $lim;$i += $inc,$inc = 6 - $inc) { if ($num % $i == 0) { echo$i . "<br />";
$num /=$i;
while ($num %$i == 0) {
echo $i . "<br />";$num /= $i; }$lim = floor(sqrt($num)); } } if ($num != 1) echo $num; ?> [/php] Note: I introduce$inc in order to avoid numbers congruent to 0 (mod 3).
3. Originally Posted by undefined
I like how there are a couple conversations going on at once here, heh.
I copied and pasted your code for use with my XAMPP setup and got problems with 2702811 sometimes but not always, and I wasn't able to get an error with 15999.
To tell the truth, I don't understand the logic of your code, but I notice that the two functions call each other an awful lot.
It seems like you're not doing anything more complex than trial division, so what do you think of this instead?
Code:
<?php
$num = 342132; if ($num < 2) {
echo "No prime factors.";
exit;
}
echo "The prime factorization of " . $num . " is:<br />"; while ($num % 2 == 0) {
echo "2<br />";
$num /= 2; } while ($num % 3 == 0) {
echo "3<br />";
$num /= 3; }$lim = floor(sqrt($num)); for ($i = 5, $inc = 2;$i <= $lim;$i += $inc,$inc = 6 - $inc) { if ($num % $i == 0) { echo$i . "<br />";
$num /=$i;
while ($num %$i == 0) {
echo $i . "<br />";$num /= $i; }$lim = floor(sqrt($num)); } } if ($num != 1) echo $num; ?> Note: I introduce$inc in order to avoid numbers congruent to 0 (mod 3).
Edit: Can anyone tell me how to get the syntax highlighter? I'm searching around for the bbcode...
Consider semi-primes of the form $n=2 \times p$ or $n=3 \times p$ where $p$ is a prime greater than $5$.
Note that $p>\sqrt{n}$ which looks like it will crash your algorithm.
I would guess that $31998$ may also give this problems.
I would conjectute that any number of the form $2^n3^mp$ where $p>2^n3^m$ is a prime will give problems (this is a conjecture because I cant be bothered to analyse such horrible code in detail)
CB
4. Originally Posted by CaptainBlack
Consider semi-primes of the form $n=2 \times p$ or $n=3 \times p$ where $p$ is a prime greater than $5$.
Note that $p>\sqrt{n}$ which looks like it will crash your algorithm.
CB
Suppose n = 14, then the program will first divide out all the 2's, so that $lim = floor(sqrt(7)) = 2. The program will effectively skip the main loop and then correctly conclude that 7 is prime and print it as one of the factors. Edit: This stuff was added afterwards Originally Posted by CaptainBlack I would guess that $31998$ may also give this problems. I would conjectute that any number of the form $2^n3^mp$ where $p>2^n3^m$ is a prime will give problems (this is a conjecture because I cant be bothered to analyse such horrible code in detail) CB 31998 worked fine. If my code is so horrible, let's see yours. Oh and for the conjecture, $745848 = 2^3*3^4*1151$ works just fine as well. 5. Originally Posted by undefined Suppose n = 14, then the program will first divide out all the 2's, so that$lim = floor(sqrt(7)) = 2. The program will effectively skip the main loop and then correctly conclude that 7 is prime and print it as one of the factors.
Edit: This stuff was added afterwards
31998 worked fine. If my code is so horrible, let's see yours.
The horror of the code may be a result of using watever language that is in, or it may be the result of a poorly thought out algorithm I can't tell.
If you had an algorithm spec in a more comprehensible language or psuedo-code I would have more confidence in it being the language rather than the algorithm.
CB
6. Originally Posted by CaptainBlack
The horror of the code may be a result of using watever language that is in, or it may be the result of a poorly thought out algorithm I can't tell.
If you had an algorithm spec in a more comprehensible language or psuedo-code I would have more confidence in it being the language rather than the algorithm.
CB
Fair enough, although PHP is quite common. Here's Java:
Code:
public ArrayList<Integer> naiveFactorise(int n) {
// returns prime factors in ascending order, duplicates allowed
ArrayList<Integer> pFacts = new ArrayList<Integer>();
if (n < 2) return pFacts;
while (n % 2 == 0) {
n /= 2;
}
while (n % 3 == 0) {
n /= 3;
}
int lim = (int)Math.sqrt(n);
for (int i = 5, inc = 2; i <= lim; i += inc, inc = 6 - inc) {
if (n % i == 0) {
n /= i;
while (n % i == 0) {
n /= i;
}
lim = (int)Math.sqrt(n);
}
}
return pFacts;
}
The reason 2 and 3 are treated specially is to ease incrementing the loop variable; otherwise they would be included in the main loop like all the other prime candidates.
Edit: For clarification, from my first post I've been writing about a program to produce the prime factorization of n using trial division. Although the OP said the original goal was merely to decide prime vs. composite, the functionality of the program posted by the OP was to produce all the prime factors with multiplicity, so that's what I went by. Obviously the decision problem is easier and simpler. This might have contributed to my code seeming "horrible."
7. Originally Posted by superdude
This was one of the first programs I made and it was a number of years ago. My style was very poor
here's the relevant code. primefind gets called first with the first argument being the number in question, startnum is initially 1, and handel is unrelated to prime factoring:
Unless you are dealling with very large numbers or speed is of the essense keep it simple:
Code:
function IsPrm(num)
//
// psuedo-code for crude primallity testing
//
// assumption: num is an integer type
//
if (num<=1)
return false
end
if (num==2)
return true
elseif (mod(num,2)==0)
return false
end
for idx=3 to floor(sqrt(num)) step 2
if mod(num,idx)==0
return false
end
end
return true
endfunction
There are more efficient ways to do it but do you need efficiency?
CB
8. Originally Posted by CaptainBlack
Unless you are dealling with very large numbers or speed is of the essense keep it simple:
Code:
function IsPrm(num)
//
// psuedo-code for crude primallity testing
//
// assumption: num is an integer type
//
if (num<=1)
return false
end
if (num==2)
return true
elseif (mod(num,2)==0)
return false
end
for idx=3 to floor(sqrt(num)) step 2
if mod(num,idx)==0
return false
end
end
return true
endfunction
There are more efficient ways to do it but do you need efficiency?
CB
I realize that both my logic and style where very poor for this program. I have written a much more elegant prime factoring program and if you like I can post it. Efficiency is not an issue, if it was I'd use this. My question that I'm asking is why do certain numbers crash the program?
9. Originally Posted by superdude
I realize that both my logic and style where very poor for this program. I have written a much more elegant prime factoring program and if you like I can post it. Efficiency is not an issue, if it was I'd use this. My question that I'm asking is why do certain numbers crash the program?
I don't have a definitive answer (debugging your code slows down my system so it's a bit of a pain), but your two functions call each other a huge number of times in general, so maybe there is a situation where a stop condition is accidentally skipped and you are left with infinite recursion. If I were you I would introduce global variables to keep track of function calls, and periodically print out the values of the key variables, either as soon as the function is called, or within the loops, or both, to try to catch runaway recursion. In the process you might spot an issue such as integer overflow, if such a problem exists. And of course, infinite recursion is essentially the same as an infinite loop, which might be the culprit.
10. Originally Posted by undefined
I don't have a definitive answer (debugging your code slows down my system so it's a bit of a pain), but your two functions call each other a huge number of times in general, so maybe there is a situation where a stop condition is accidentally skipped and you are left with infinite recursion. If I were you I would introduce global variables to keep track of function calls, and periodically print out the values of the key variables, either as soon as the function is called, or within the loops, or both, to try to catch runaway recursion. In the process you might spot an issue such as integer overflow, if such a problem exists. And of course, infinite recursion is essentially the same as an infinite loop, which might be the culprit.
The strange thing is with 15999 the program crashes even if the first thing is to print something to the screen. That is if step 1 is print("hello world") the program will not do that for 15999.
11. Originally Posted by superdude
The strange thing is with 15999 the program crashes even if the first thing is to print something to the screen. That is if step 1 is print("hello world") the program will not do that for 15999.
Then perhaps the error lies somewhere outside the block of code you posted, because I ran it as-is and never got that problem.
I find it extremely unlikely that my version of PHP would behave that much differently from yours and that it's a bug with the PHP interpreter, although I suppose it can't be ruled out entirely without further testing.
12. This is my new version. I'm looking for criticism:
[PHP]
<?php
$num = 15999; if($num%2 != 0 && $num%3 != 0 && isPrime($num))
{
print($num.' is prime'); exit; } print('The primefactorization of '.$num.' is:<br />');
checkSmallPrimes($num, 2);//factors out 2s checkSmallPrimes($num, 3);//factors out 3s
for($i = 5;$i <= floor(sqrt($num));$i += 2)
{
if(isPrime($i)) { while($num%$i==0)//having this as a while loop instead of if increases efficency {$num = $num/$i;
print($i.'<br />'); } if(isPrime($num))
{
print($num); exit; } } } /*returns true if agrument is prime greater than 3, otherwise returns false*/ function isPrime($maybePrime)
{
for($j = 3;$j <= floor(sqrt($maybePrime));$j += 2)
if($maybePrime%$j == 0)
return false;
return true;
}
//$num is being passed by reference function checkSmallPrimes(&$num, $smallPrime) { while($num%$smallPrime==0) {$num /= $smallPrime; print($smallPrime.'<br />');
}
}
?>[/PHP]
13. Originally Posted by undefined
Then perhaps the error lies somewhere outside the block of code you posted, because I ran it as-is and never got that problem.
I find it extremely unlikely that my version of PHP would behave that much differently from yours and that it's a bug with the PHP interpreter, although I suppose it can't be ruled out entirely without further testing.
I said it only happens using EasyPHP 1.8. I've reproduced the problem on different computers (with different architecture) so you must be doing something wrong.
14. Originally Posted by superdude
I said it only happens using EasyPHP 1.8. I've reproduced the problem on different computers (with different architecture) so you must be doing something wrong.
Sorry, I missed that part. I've never had to debug a language interpreter before, and I don't intend to replace my current PHP with one that is potentially buggy, so I guess I won't be of much help on that.
Originally Posted by superdude
This is my new version. I'm looking for criticism:
A small consistency issue: It seems that the program says "$num is prime" when$num is prime and > 3, but if $num == 2 or$num == 3, it will output its prime factorization rather than telling you it's prime.
I don't understand why you treat 3 as a special case; you could just start your loops at 3 instead of 5. The reason I treated 3 as special was because my increment scheme avoided multiples of 3 other than 3, thus I checked the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, ... where the differences between two successive terms are 2 and 4 alternating, but starting with 5. This is just an optimization.
Also, I'm not sure if this is a real optimization or not, but I tend to pre-calculate the loop limit floor(sqrt($blah)) as another variable to prevent re-calculating the same thing many times -- possibly a given compiler/interpreter would do this anyway, depends on how smart it is I guess. There's another inefficiency I noticed: you call isPrime() from within the main for() loop (the one that's not contained in a function), and you start trial division over at 5, but in reality it's impossible to have factors between 5 and$i of the main loop because you've divided them out already.
Also, you only call checkSmallPrimes() twice, but the while() loop inside your main for() loop has identical code. If you're going to encapsulate that code as a function, you might as well just call checkSmallPrimes() each time, possibly renaming it to reflect what it does more closely.
One final optimization I can see is that, for composite $num, at the beginning you call isPrime() and trial divide by 2 and all odd numbers up to sqrt($num), and then you do the exact same thing (starting at 5) in the main for() loop. It would be more efficient to keep a boolean value for primality, starting it at true and then setting it to false as soon as a factor < $num is found, thus combining the two loops into one. One last very small point is that using the variable name$maybePrime inside a function called isPrime() seems redundant to me.
Hope you don't mind my tearing apart your code... it's all meant constructively.
15. Originally Posted by undefined
Sorry, I missed that part. I've never had to debug a language interpreter before, and I don't intend to replace my current PHP with one that is potentially buggy, so I guess I won't be of much help on that.
A small consistency issue: It seems that the program says "$num is prime" when$num is prime and > 3, but if $num == 2 or$num == 3, it will output its prime factorization rather than telling you it's prime.
I don't understand why you treat 3 as a special case; you could just start your loops at 3 instead of 5. The reason I treated 3 as special was because my increment scheme avoided multiples of 3 other than 3, thus I checked the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, ... where the differences between two successive terms are 2 and 4 alternating, but starting with 5. This is just an optimization.
Also, I'm not sure if this is a real optimization or not, but I tend to pre-calculate the loop limit floor(sqrt($blah)) as another variable to prevent re-calculating the same thing many times -- possibly a given compiler/interpreter would do this anyway, depends on how smart it is I guess. There's another inefficiency I noticed: you call isPrime() from within the main for() loop (the one that's not contained in a function), and you start trial division over at 5, but in reality it's impossible to have factors between 5 and$i of the main loop because you've divided them out already.
Also, you only call checkSmallPrimes() twice, but the while() loop inside your main for() loop has identical code. If you're going to encapsulate that code as a function, you might as well just call checkSmallPrimes() each time, possibly renaming it to reflect what it does more closely.
One final optimization I can see is that, for composite $num, at the beginning you call isPrime() and trial divide by 2 and all odd numbers up to sqrt($num), and then you do the exact same thing (starting at 5) in the main for() loop. It would be more efficient to keep a boolean value for primality, starting it at true and then setting it to false as soon as a factor < $num is found, thus combining the two loops into one. One last very small point is that using the variable name$maybePrime inside a function called isPrime() seems redundant to me.
Hope you don't mind my tearing apart your code... it's all meant constructively.
You have some valuable suggestions. Can you elaborate on how your increment scheme avoids multiples of 3 and how you know you won't miss any primes in between? So how do you know alternating between adding 2 and 4 skips multiples of 3? I'm busy today so I'll consider your suggestions in more depth and post back in a day or two.
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|
2014-04-20 14:53:45
|
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|
http://wtmaths.com/solve_quadratic_equations_graph.html
|
Solving Quadratic Equations using a Graph
# Solving Quadratic Equations using a Graph
GCSE(F), GCSE(H),
Graphs can be used to solve quadratic equations. Arrange the equation so that it is equal to zero on one side. Plot the graph as y against the function. The solutions can be found where the line crosses the x-axis.
Note that there may be 0, 1 or 2 solutions, depending on how many times the graph crosses the axis.
Note that the answer may only be approximate, depending on how accurately the graph is drawn.
Exam Tip: Substitute the values of x from the graph and substitute into the original equation to check the accuracy of the measurement.
## Examples
1. By drawing a graph, estimate the solutions to the equation x^2 - 4x - 3 = 0 to one decimal place.
Answer: x=4.6 text( and ) x= -0.6
Plot the graph: coordinates are (-2, 9), (-1, 2), (0, -3), (1, -6), (2, -7), (3, -6), (4, -3), (5, 2), (6, 9)
Graph crosses the y-axis at 4.6 and -1.6.
2. By drawing a graph, estimate the solutions to x^2-8x=-16.
Answer: x=4 (both roots take the same value)
Rearrange the equation to x^2-8x+16=0, and plot the graph as y=x^2-8x+16.
Graph touches the x-axis at x=4.
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2018-12-15 01:05:26
|
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|
https://math.stackexchange.com/questions/3204070/writing-beta-function-in-terms-of-gamma-functions-by-substitution
|
# Writing beta function in terms of gamma functions (by substitution)
I'm going over my note and try to write the Beta function in terms of gamma functions. However, I just can't get (1.73) from (1.72).
Even if I substitute t/(1-t) with u, I can't remove t. Can anyone shed some light on this? Thanks.
Hint: If $$u = \frac{t}{1-t}$$, then $$u(1-t)=t$$, that is, \begin{align*} u-ut &=t \\ \Rightarrow u &= ut+t \\ \Rightarrow u &=t(1+u).\end{align*} So $$t= \frac{u}{1+u}$$. Try substituting this now.
|
2019-06-26 04:11:10
|
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|
https://www.gradesaver.com/textbooks/science/physics/essential-university-physics-volume-1-3rd-edition/chapter-14-for-thought-and-discussion-page-262/8
|
## Essential University Physics: Volume 1 (3rd Edition)
We know that the human ear's hearing range is 20 Hz -20 KHz. Ultra sound frequencies are $10^7$ Hz . The difference between these two frequencies is $10^7Hz-20KHz=0.998\times 10^3Hz$. Thus, these ultra sound frequencies are very high as compared to the human ear range. Ultra sound is a sound having all the physical properties of sound but the only difference is that the human ear cannot hear it.
|
2018-08-20 21:39:44
|
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|
https://techcommunity.microsoft.com/t5/excel/dropdownlist-gt-arrows/td-p/2330859
|
New Contributor
# Dropdownlist > Arrows
Hi Experts,
i make some dropdownlists in excel, but the arrow always hide when i switch to another cell. I will the arrows always visible, that the user see there is something to change. And the aHow is this possible?
4 Replies
# Re: Dropdownlist > Arrows
Excel does not support that, but see Drop-down List Arrow Always Visible for Data Validation for a workaround.
# Re: Dropdownlist > Arrows
OK Hans, i will try this. Thanks a lot
# Re: Dropdownlist > Arrows
Like Mr. Hans Vogelaar Excel does not support that, but excel also has workarounds.
Thx, Nikolino
|
2022-10-07 19:15:14
|
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|
http://isabelle.systems/zulip-archive/stream/202968-quantum-computing/topic/project.20ideas.html
|
## Stream: quantum computing
### Topic: project ideas
#### Anthony Bordg (Oct 17 2019 at 17:27):
One could formalize the following result: for any quantum states $|\psi\rangle$, $|\varphi\rangle$, there exists a unitary matrix $U$ such that $U|\psi\rangle = |\varphi\rangle$.
Last updated: Sep 25 2022 at 23:25 UTC
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2022-09-25 23:51:05
|
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|
http://math.stackexchange.com/questions/240847/units-in-mathbbz-sqrt32-pm1-sqrt32-sqrt322n
|
Units in $\mathbb{Z}[\sqrt[3]{2}]$ : $\pm(1+\sqrt[3]{2}+(\sqrt[3]{2})^2)^n$?
The subring $\mathbb{Z}[\sqrt[3]{2}]\subset\mathbb{C}$ is a PID. I remember reading somewhere that the units in $\mathbb{Z}[\sqrt[3]{2}]$ are precisely the elements $\pm(1+\sqrt[3]{2}+(\sqrt[3]{2})^2)^n$ with $n\in\mathbb{Z}$, but I don't know how one would go about proving this. I guess if $\mathbb{Z}[\sqrt[3]{2}]$ were a euclidean ring, then one could try using the euclidean norm but I'm not sure whether this is the case or not (anyway, I don't have a euclidean norm in my hands to work with). Is there a quick way to determine the units of $\mathbb{Z}[\sqrt[3]{2}]$?
-
A quick matrix representation of multiplication by $a+b\sqrt[3]2+c\sqrt[3]4$, $a,b,c\in\mathbb Z$, would have an inverse of the same form if and only if the determinant is $\pm 1$. This yields the equation: $a^3+2b^3+4c^3-6abc=\pm 1$. Not sure if that helps any. ($a^3+2b^3+4c^3-6abc$ is the norm for this ring.) – Thomas Andrews Nov 19 '12 at 20:49
This is an example given in planetmath.org/encyclopedia/… – Cocopuffs Nov 19 '12 at 21:05
Multiply your prospective unit by $1-\sqrt[3]2$ to show you have a unit.
The Dirichlet Unit Theorem tells you the rank of the unit group is 1. This may be more advanced than you want to use, and I would be interested in a special case argument for this case.
Then you need to work on showing that $\pm 1$ are the only units of finite order, and that you can reduce any other unit of form $a+b\sqrt[3]2+c\sqrt[3]4$ to one with a lower absolute value of $a$ until you get to $a = \pm1$ - by multiplying/dividing by the units you've already identified. After that there is some tidying up to do.
-
|
2015-11-30 17:46:04
|
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|
https://codereview.stackexchange.com/questions/243939/use-django-engine-to-fill-in-a-html-file-on-storage-no-template-and-use-weasy
|
# Use django engine to fill in a .html file on storage (no template) and use weasyPrint to convert it to PDF
I 'm new here. I wrote the following django code. I 'm opening, closing files here and saving them temporarily and deleting them. I tried to use Python's tempfile and was getting Permission Error and I did asked about it on IRC #django but maybe Windows is not a pleasant to use OS for programmers so I couldn't get a good answer. I needed something like render_to_string of django which takes in a html string and replace all templating with the context dict but it seems django is made to treat every .html file as a template.
Purpose of the project : It is to take a visitor's id and return him with a pdf which will be formed by picking up a row from the database by looking at his id. There are 3 kinds of ids here.
How is the pdf being made?
I was given a pdf empty form which I converted to .docx file with the use of online sites. Now I tried to use python-docx to convert docx to pdf but that required libre office/ms word which might not be available on the server (The form can be formed by the client on his local pc and be given to the technical guy to put on the server).
I was suggested to use a html form and the client said that he might change the form.
@ChrisWarrick on #python IRCnode suggested me to use HTML to PDF conversion which could be done by weasyPrint which was cross platform and easier to install. Although he said me to use jinja but since I was using django why install some other library. Now I said to the client to open a .docx file and create whatever form he has to make and put {{NAME}} and other variables wherever he wants some information from the database to be put and save it as .html file and further put it in the /media folder of the django project. Then he has to open the config (.cfg) file and put
NAME=NAME here 'NAME' on left is what is in the .html file(docx form) and on the right is column name of the database table(I got a single table).
Please help me make this code make more maintainable and remove that unnecessary saving file and deleting it. Also there's a problem that on windows when I save the docx file as .html I get the encoding as cp1252 whereas the server has linux as told to me. I have been told on IRCnode #powershell that windows can have a bunch of too many encodings. To do this I will say to the client to convert .html to utf8 using Get-Content word.htm | out-file -encoding utf8 word-1.htm
App name base
base/view.py
from django.shortcuts import render
from .forms import InputData
from . import backend
from django.http import FileResponse, HttpResponse
import configparser
config = configparser.RawConfigParser()
def index(request):
if request.method == "POST":
form = InputData(request.POST)
if form.is_valid():
check, data = backend.main(**form.cleaned_data)
if check:
return FileResponse(
data,
as_attachment=True,
else:
return HttpResponse(data)
form = InputData()
return render(request, "base/index.html", {
'forms': form
})
base/backend.py
import os
import pandas as pd
import codecs
from weasyprint import HTML
import configparser
import tempfile
from django import template
from pathlib import Path
if os.path.exists('temp.pdf'):
os.remove('temp.pdf')
def getConfigObject():
config = configparser.RawConfigParser()
config.optionxform = str
return config
config = getConfigObject()
html = codecs.open(
config["FILES"]["HTML_FILE_NAME"],
html = "{% load numbersinwords %}" if not html.startswith(
) else "" + html
Html_file = open(config["FILES"]["HTML_FILE_NAME"], "w", encoding="utf-8")
Html_file.write(html)
Html_file.close()
def html2pdf(row):
row = row.to_dict()
html = render_to_string(Path(config["FILES"]["HTML_FILE_NAME"]).name,
{key: row[value]
for key, value in config._sections["TAGS"].items()})
return html
def get_data():
dtype=str, keep_default_na=False)
def search_row(opt, value):
user_data = get_data()
return user_data[user_data[opt] == value]
def main(opt, value):
row = search_row(opt, value)
if len(row) == 1:
row = row.squeeze()
else:
return (False, f"<h1>Invalid credential :"
" Multiple candidates exists"
"with given credential</h1>")
if not(row.empty):
html = html2pdf(row)
HTML(string=html).write_pdf("temp.pdf")
# Code from
# https://stackoverflow.com/questions/47833221/emailing-a-django-pdf-file-without-saving-in-a-filefield
# temp = tempfile.NamedTemporaryFile()
# temp.write(pdf_file)
# temp.seek(0)
########
f = open("temp.pdf", "rb")
return (True, f)
return (False, f"<h1>Invalid credential {opt}: {value}</h1>")
base/templatetags/numbersinwords.py
from django import template
from num2words import num2words
register = template.Library()
@register.filter()
def to_words(value):
return num2words(int(value), lang="en_IN").upper()
• Please do not update the code in your question to incorporate feedback from answers, doing so goes against the Question + Answer style of Code Review. This is not a forum where you should keep the most updated version in your question. Please see what you may and may not do after receiving answers. Do not add extra questions once it has been answered. Besides, your edit contained code that wasn't working yet, which is out-of-scope for Code Review. Thank you. – Mast Jun 22 '20 at 16:50
## Else-after-return
Some people consider this a stylistic choice, but this:
if check:
return FileResponse(
data,
as_attachment=True,
else:
return HttpResponse(data)
can be
if check:
return FileResponse(
data,
as_attachment=True,
return HttpResponse(data)
## Import-time file manipulation
This:
if os.path.exists('temp.pdf'):
os.remove('temp.pdf')
is done at global scope on file interpretation, which is risky for a few reasons - including that it will make isolated unit testing much more difficult. This kind of thing should be pulled into a function that runs on program initialization, not at global scope.
Beyond that, having one temporary file with a fixed name invites a collection of security vulnerabilities and failures of re-entrance. This file should be randomly named; the tempfile module can do this for you.
## snake_case
getConfigObject should be get_config_object, like your other functions already are.
Html_file should not be capitalized since it's a local variable. Also, it should be used in a with statement without an explicit call to close.
## Ternary abuse
html = "{% load numbersinwords %}" if not html.startswith(
) else "" + html
should simply be
if not html.startswith("{% load"):
html = "{% load numbersinwords %}" + html
## Implicit return tuples
return (True, f)
does not need parens.
## Avoiding temp files
target (str, pathlib.Path or file object) – A filename where the PDF file is generated, a file object, or None.
In this case it's easy to avoid a temp file by passing a file object. That file object can be a Django HTTP response stream; for more reading see
https://docs.djangoproject.com/en/3.0/ref/request-response/#passing-strings
Currently you do
HTML(string=html).write_pdf("temp.pdf")
f = open("temp.pdf", "rb")
return (True, f)
# ...
check, data = backend.main(**form.cleaned_data)
if check:
return FileResponse(
data,
as_attachment=True,
• the Response object is passed to write_pdf instead of a filename
• Another thing is you have advised to use a with statement(under snake_case section) but not at the end when I 'm returning from main function where I have neither closed the file or used any with. I was using the with context manager in my code initially but the file was getting closed before sending the f' or the pdf object and I was getting a binary string printed on the DOM. That's why I was hesitant to use with statement. – Vishesh Mangla Jun 22 '20 at 10:04
• Can you provide some more tips on when to use ternary operators - sparingly, and only when the expression is relatively simple. Also, the fact that one of the ternary outputs is effectively a no-op means it's better captured by an if` anyway. – Reinderien Jun 22 '20 at 14:04
|
2021-04-18 05:21:30
|
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|
https://physics.stackexchange.com/questions/270478/why-the-changing-magnetic-flux-horizontal-component-does-not-induce-electricity
|
# Why the changing magnetic flux horizontal component does not induce electricity?
Textbooks just seem to utter that change in the magnetic flux that is 'piercing' the plane contributes to electricity induction. They don't explain why.
Is there any intuitive explanation?
• "Why?" is a hard question to answer. When physicists explain something in response to "Why?" it always comes down to an explanation in terms of some theory. But at the root of any theory are axioms that go unexplained. Can you explain why $F=ma$? In the case of electricity and magnetism, the unexplained axioms are Maxwell's equations. The justification is that the logical consequences of the axioms accurately describe what's observed in nature. Another way of approaching your question is to ask you: What do you accept as true so that we can use it as a starting point for an answer. – garyp Jul 29 '16 at 1:29
• To extend @garyp thoughts, the underlying theory that explains this fact is pretty dense compared to the kind of E&M that is presented in a introductory book. This is one you have to simply accept as experimentally backed gospel for a couple of years until you are ready to apply a Lorentz transformation to a second rank tensor in a Minkowski space. – dmckee Jul 29 '16 at 3:15
• @dmckee: After which everybody will (hopefully) understand how the second rank tensors were derived from a century's worth of experiments with wires and magnets, light sources, mirrors, lenses and other contraptions. And those who don't will have a huge gap in their physics education, which I would find very sad. – CuriousOne Jul 29 '16 at 8:23
|
2019-08-24 11:30:10
|
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|
https://scholarworks.iu.edu/dspace/handle/2022/12958/browse?rpp=20&order=ASC&sort_by=1&etal=-1&type=title&starts_with=R
|
mirage
# Browsing Astronomy by Title
Sort by: Order: Results:
• (The American Astronomical Society, 2012)
UV observations in the local universe have uncovered a population of early-type galaxies with UV flux consistent with low-level recent or ongoing star formation. Understanding the origin of such star formation remains an ...
• (The American Astronomical Society, 2012)
We present optical Hubble Space Telescope/Space Telescope Imaging Spectrograph ($HST/S\text{TIS}$) spectroscopy of RZ 2109, a globular cluster (GC) in the elliptical galaxy NGC 4472. This GC is notable for hosting an ...
• (The American Astronomical Society, 2012)
In this paper, we posit that galaxy luminosity functions (LFs) come in two fundamentally different types depending on whether the luminosity traces galaxy stellar mass or its current star formation rate (SFR). $\textit{Mass ... • (The American Astronomical Society, 2012) During a wide-field narrowband$\text{H}\alpha\$ imaging survey, we noted the presence of numerous isolated emission-line point sources in the data. These objects could represent ultra-low-luminosity galaxies at low-redshift ...
• (The American Astronomical Society, 2012)
Almost all globular clusters investigated exhibit a spread in their light element abundances, the most studied being an Na:O anticorrelation. In contrast, open clusters show a homogeneous composition and are still regarded ...
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2015-03-26 22:46:11
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https://cs.stackexchange.com/questions/133684/what-do-you-get-when-you-add-on-to-itself-n-times
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What do you get when you add $O(n)$ to itself $n$ times?
I was watching this video on Algorithms and counting number of inversions and he mentioned being cautions when $$O(n) + O (n) = O(n)$$, saying that is not true if you add $$O(n)$$ to itself $$n$$ times. Is the answer $$O(n^2)$$?
The notation $$O(n)$$ stands for a function which is bounded by $$Cn$$ for some constant $$C>0$$ and large enough $$n$$.
If you have a two functions $$f_1,f_2$$ which are both $$O(n)$$, then so is their sum $$f_1+f_2$$: indeed, suppose that $$f_1(n) \leq C_1n$$ and $$f_2(n) \leq C_2n$$ for large enough $$n$$. Then $$f_1(n) + f_2(n) \leq (C_1 + C_2)(n)$$ for large enough $$n$$.
Similarly, if you have a single function $$f(n)$$ which is $$O(n)$$ and you add it to itself $$n$$ times then you get a function $$nf(n)$$ which is $$O(n^2)$$.
However, this fails when you add $$n$$ different functions which are $$O(n)$$. For example, consider the functions $$f_k(n) = kn$$ and the function $$g(n) = f_1(n) + \cdots + f_n(n)$$. Then $$f_k(n) = O(n)$$ for all $$k$$, but it is not true that $$g(n) = O(n^2)$$; rather, $$g(n) = \Theta(n^3)$$. (This means that $$cn^3 \leq g(n) \leq Cn^3$$ for some $$c,C>0$$ and large enough $$n$$.)
If we know that the functions $$f_k$$ are uniformly $$O(n)$$, that is, there is a single $$C>0$$ and a single $$N>0$$ such that $$f_k(n) \leq Cn$$ whenever $$n>N$$, then it does hold that $$f_k(1) + \cdots + f_k(n) = O(n^2)$$.
(It doesn't suffice to require just $$C$$ to be uniform. For example, if $$f_k(n) = n+k^2$$ then $$f_k(n) \leq 2n$$ for large enough $$n$$, but $$f_1(n) + \cdots + f_n(n) = \Theta(n^3)$$.)
• Could you show how you arrived at the cubic n? Jan 1 '21 at 18:22
• We have $\sum_{k=1}^n (n + k^2) = n^2 + \frac{n(n+1)(2n+1)}{6} = \frac{2n^3+4n^2+n}{6}$. Jan 1 '21 at 18:32
• Isn't it required that $C$ never be related to $n$? Jan 26 '21 at 12:33
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2022-01-24 18:42:45
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https://codeforces.com/blog/From_ITK18_With_Love
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### From_ITK18_With_Love's blog
By From_ITK18_With_Love, history, 3 months ago,
Today, I get a problem.
Sum of greatest odd divisor of numbers in range $[a, b]$ with $a, b <= 10^9$
I found solution here : https://www.geeksforgeeks.org/sum-of-greatest-odd-divisor-of-numbers-in-given-range/
But I think the solution is not clear for the even number case.
Can find a better solution or more detailed explanation ?
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2021-12-07 09:22:23
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https://www.mersenneforum.org/showthread.php?p=421167
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mersenneforum.org 768k Skylake Problem/Bug
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2016-01-03, 22:18 #276 tha Dec 2002 77810 Posts The last test I embarked on has finished! And the results are very interesting. See the results.txt file: Code: [Sat Jan 2 18:39:23 2016] M14942437 is not prime. Res64: 683A0DFFC5827CD8. We8: E57106A7,7379210,00000000 M14942267 is not prime. Res64: C35562BC4F3511F3. We8: D8A74C7B,2423514,00000000 M14942209 is not prime. Res64: 8587C9937E3BED22. We8: CDAD4A41,7713418,00000000 M14942293 is not prime. Res64: 035EFC95F88CFC27. We8: 36084309,4746867,00000000 [Sun Jan 3 22:02:18 2016] Iteration: 14329935/14942209, ERROR: FFT data has been zeroed! Possible hardware failure, consult the readme.txt file. Continuing from last save file. [Sun Jan 3 22:26:02 2016] M14942267 is not prime. Res64: D20C84656405F3FB. We8: FCFDD819,14910347,00000000 M14942539 is not prime. Res64: 0A930E56A9284971. We8: 7FE55A2A,1188977,00000000 M14942437 is not prime. Res64: 136153185F4D524F. We8: B81CE272,9576909,00000000 [Sun Jan 3 22:37:01 2016] M14942497 is not prime. Res64: 80BD5A064693F1C0. We8: 0CAD30A7,2607443,00000000 M14942293 is not prime. Res64: 035EFC95F88CFC27. We8: 36502AEF,8394253,00000000 M14942567 is not prime. Res64: D233F12AC3781E04. We8: 59875C25,3894081,00000000 [Sun Jan 3 22:42:28 2016] M14942563 is not prime. Res64: 6815BC39FCD7650F. We8: A94AFB88,2473090,00000000 [Sun Jan 3 22:55:12 2016] M14942209 is not prime. Res64: 0AA69D2EA9100E22. We8: 7D077832,14397436,00010000 The first four results belong to the first test I did with v27.9 and was done by two threads on each exponent. It matches the three tests I did on this machine using v28.7 and the data in the GIMPS database. These three other tests are described in an earlier post of mine. The last test consists of eight threads (4 cores, 6700K processor) working on eight exponents. Throughout the 28 hours the testrun lasted no errors were reported except for one on thread 1 when 96% of the run was completed. Notice that the results of this run do not match the previous runs except for one test. Of the four exponents that were tested for the first time on this machine concurrently with the other four, also one test did not fail whereas the other three did. The two successfully completed tests matching with the database were running on the following threads: Code: [Worker #3 Jan 3 21:54] Iteration: 14460000 / 14942293 ... [Worker #6 Jan 3 21:53] Iteration: 14620000 / 14942539 ... The threads 1 & 5, 2 & 6, 3 & 7 and 4 & 8 are the four pairs that each share one of the four physical cores. Small complication is that due to glazed frost on the high tension power lines in the northern parts of The Netherlands there were some noticeable power cuts lasting milliseconds throughout the last 20% of the test run. This did not stop the machine running, and the web browser that was running on this machine did not fail either because of it. I will now restart this exact test and finish it in an expected 28 more hours, I will probably be asleep when that test finishes but will report a few hours later. Last fiddled with by tha on 2016-01-03 at 22:45
2016-01-03, 23:04 #277 tha Dec 2002 2×389 Posts If someone has a reasonable fast pre Skylake four physical cores with hyperthreading Intel machine available and feels like it than feel free to run the worktodo file from post 262. Just for reference, the outcome will be predictable, 8 correct residues, which is what I am looking four. The only such machine I have is about eight years old and would take too much time to run this test. Please post here if you embark on it. If someone with another Skylake wants to run this test, than of course, feel free to do so. Before I started the test for a second time on my Skylake machine I rebooted it. Last fiddled with by tha on 2016-01-03 at 23:49
2016-01-03, 23:22 #278
chalsall
If I May
"Chris Halsall"
Sep 2002
23×1,103 Posts
Quote:
Originally Posted by tha Before I started the test for a second time on my Skylake machine I rebooted it.
Did that enter more or less entropy into the system?
2016-01-03, 23:26 #279
tha
Dec 2002
2·389 Posts
Quote:
Originally Posted by chalsall Did that enter more or less entropy into the system?
I am assuming it is a joke.
But a serious answer to the question is that I don't think it made any difference. Just a safeguard.
2016-01-03, 23:33 #280
chalsall
If I May
"Chris Halsall"
Sep 2002
23·1,103 Posts
Quote:
Originally Posted by tha I am assuming it is a joke. But a serious answer to the question is that I don't think it made any difference. Just a safeguard.
It was kind of a joke, but also a serious question...
We still don't understand what is happening. So, restarting might make sense. Then again, it might not.
In a perfect universe, we could capture the quantum state of the computing devices we use, and run many tests based on their initial states.
We humans are not that powerful, but we still have the ability to try....
2016-01-03, 23:44 #281
Serpentine Vermin Jar
Jul 2014
3·5·7·31 Posts
Quote:
Originally Posted by chalsall Did that enter more or less entropy into the system?
Yes. (couldn't resist... LOL)
2016-01-03, 23:57 #282
chalsall
If I May
"Chris Halsall"
Sep 2002
23×1,103 Posts
Quote:
Originally Posted by Madpoo Yes. (couldn't resist... LOL)
Cool....
2016-01-04, 05:05 #283 LaurV Romulan Interpreter Jun 2011 Thailand 83×101 Posts Haha, I was also reading it like "Did that, more or less, enter entropy into the system?" but you were faster with the answer... Edit: Reading all this I feel sorry I don't own a Skylake... Itching hands to try some tests by myself. Last fiddled with by LaurV on 2016-01-04 at 05:06
2016-01-04, 07:43 #284
VBCurtis
"Curtis"
Feb 2005
Riverside, CA
76328 Posts
Quote:
Originally Posted by Madpoo Yes. (couldn't resist... LOL)
2016-01-04, 09:19 #285 megabit8 Dec 2015 23×3 Posts Some times contradictory discussions yield best results. One thing is sure, time will tell how this issue sorts out. I would not rush in.
2016-01-04, 16:48 #286
chalsall
If I May
"Chris Halsall"
Sep 2002
882410 Posts
Quote:
Originally Posted by megabit8 I would not rush in.
I was interested to see if this had been picked up by any mainstream media yet. I ran a few Google queries, and it appears it hasn't.
This is a good thing in my mind -- we want Intel and the motherboard manufacturers to have as much lead time as possible to solve what appears to be a very subtle bug without the hysteria which mainstream reporting often brings.
I did find this article from AnandTech on a tangential bug interesting. Also, that the Intel Forum thread on this matter hasn't been posted to since December 31st.
It's only a matter of time before this is "out there". I do hope that Intel are taking this seriously....
Last fiddled with by chalsall on 2016-01-04 at 16:52 Reason: s/Intel Forum/Intel Forum thread on this matter/
Similar Threads Thread Thread Starter Forum Replies Last Post ET_ Hardware 17 2017-05-24 16:19 mackerel Hardware 34 2016-03-03 19:14 fivemack Hardware 36 2015-09-08 01:42 tha Hardware 7 2015-03-05 23:49 clarke Software 15 2015-03-04 21:48
All times are UTC. The time now is 09:07.
Tue Mar 31 09:07:18 UTC 2020 up 6 days, 6:40, 0 users, load averages: 1.13, 1.51, 1.43
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2020-03-31 09:07:18
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https://jira.lsstcorp.org/browse/DM-16556
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calcRmsDistances is comparing objects in correct distances due to indexing error
XMLWordPrintable
Details
• Type: Bug
• Status: Done
• Resolution: Done
• Fix Version/s: None
• Component/s:
• Labels:
None
• Story Points:
0.2
• Team:
DM Science
• Urgent?:
No
Description
Alexandre Ciulli kindly points out that there's an indexing error in calcRmsDistances
1. Update validate_drp/python/lsst/validate/drp/calcsrd/amx.py to add back in the index offset in line 198-199
dist = sphDist(ra1, dec1, meanRa[obj1+1:], meanDec[obj1+1:]) objectsInAnnulus, = np.where((annulusRadians[0] <= dist) & (dist < annulusRadians[1]))
to
dist = sphDist(ra1, dec1, meanRa[obj1+1:], meanDec[obj1+1:]) objectsInAnnulus, = np.where((annulusRadians[0] <= dist) & (dist < annulusRadians[1])) objectsInAnnulus += obj1 + 1
Or generalize/refactor in some other way to capture the index offset correctly.
Activity
Hide
Leanne Guy added a comment -
validate_drp has been replaced by faro, there will be no further development in validate_drp. This error however seems to have been propagated to faro.
Show
Leanne Guy added a comment - validate_drp has been replaced by faro, there will be no further development in validate_drp. This error however seems to have been propagated to faro.
Hide
Jeffrey Carlin added a comment -
The proposed change does indeed fix the issue. We may want to refactor this code to something more readable, but for now I have confirmed that the solution above works.
Show
Jeffrey Carlin added a comment - The proposed change does indeed fix the issue. We may want to refactor this code to something more readable, but for now I have confirmed that the solution above works.
Hide
Leanne Guy added a comment -
Show
People
Assignee:
Jeffrey Carlin
Reporter:
Michael Wood-Vasey
Reviewers:
Leanne Guy
Watchers:
Jeffrey Carlin, Keith Bechtol, Leanne Guy
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2023-03-29 01:05:38
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http://life.inspirho.in/joblessness/day-1-polarization-of-the-light/
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# DAY 1: Polarization of the Light
One of the basic kinds qubits are displayed by polarizing photons. So, I just started off by making sure I remember the basic polarizing stuff(which I have had done with the last year)…
Do you remember the wave equations?…
Yes I do!.. 🙂 And most of the basic stuff on polarization.
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2023-01-27 20:02:01
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https://srfi.schemers.org/srfi-163/
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# SRFI 163: Enhanced array literals
by Per Bothner
status: final (2019-01-18)
## Abstract
This is a specification of a reader form (literals) for multi-dimensional arrays. It is an extension of the Common Lisp array reader syntax to handle non-zero lower bounds, optional explicit bounds, and optional uniform element types (compatible with SRFI 4). It can be used in conjunction with SRFI 25, SRFI 122, or SRFI 164. These extensions were implemented in Guile (except the handling of rank-0 arrays), and later in Kawa.
There are recommendations for output formatting and a suggested format-array procedure.
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2020-04-06 08:48:17
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https://gmatclub.com/forum/what-is-the-value-of-the-three-digit-number-sss-if-sss-is-the-sum-of-126879.html
|
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Admitted in Kellogg for 2015 intake Joined: 25 Jun 2011 Posts: 468 Location: United Kingdom Concentration: International Business, Strategy GMAT 1: 730 Q49 V45 GPA: 2.9 WE: Information Technology (Consulting) What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags Updated on: 15 Dec 2017, 23:44 5 10 00:00 Difficulty: 85% (hard) Question Stats: 60% (02:49) correct 40% (02:51) wrong based on 335 sessions ### HideShow timer Statistics What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive? (1) S = 1.75 X (2) S^2 = 49zx/8 I don't have an OA but for me the answer is D. This is how I got it. Again please check my approach and let me know if anything is not right. Statement 1 S = 1.75 x ==> 175 x/100 ==>7x/4. x has to be a multiple of 4 for S to be an integer. So x can ONLY be 4. X cannot be 8 or any other multiple of 4 because if we take x = 8 then S =14 which is a 2 digit number. So S = 7. Therefore SSS is 777. Sufficient. Statement 2 $$S^2$$ = 49zx/8 ==> S = 7 $$\sqrt{zx/8}]$$. Now since S is an integer, 7 $$\sqrt{zx/8}$$ must be an integer as well. Also, 7 $$\sqrt{zx/8}$$must also be less than 10. and this can only happen when $$\sqrt{zx/8}$$ = 1. Therefore, S = 7(1) = 7 and SSS will become 777. Sufficient. Therefore D is my answer. _________________ Best Regards, E. MGMAT 1 --> 530 MGMAT 2--> 640 MGMAT 3 ---> 610 GMAT ==> 730 Originally posted by enigma123 on 31 Jan 2012, 18:17. Last edited by Bunuel on 15 Dec 2017, 23:44, edited 2 times in total. Added the OA ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 52905 Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 31 Jan 2012, 18:36 6 2 What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive? (1) S = 1.75X --> as S and X are digits, then X must be 4 and S must be 7: 7=1.75*4 --> SSS=777. Sufficient. (2) S^2= 49zx/8 --> $$S=7*\sqrt{\frac{zx}{8}}$$ --> again as S is a digit then $$\sqrt{\frac{zx}{8}}$$ must be 1 --> S=7 --> SSS=777. Sufficient. Answer: D. _________________ ##### General Discussion Intern Joined: 25 Aug 2010 Posts: 17 Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 02 Feb 2012, 10:03 hi bunuel, unable to understand ur explanation. how are u arriving at values 4 and 7? and option B _________________ regards eshwar Senior Manager Status: Finally Done. Admitted in Kellogg for 2015 intake Joined: 25 Jun 2011 Posts: 468 Location: United Kingdom Concentration: International Business, Strategy GMAT 1: 730 Q49 V45 GPA: 2.9 WE: Information Technology (Consulting) Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 02 Feb 2012, 10:13 2 Hi pappueshwar I thought to reply on behalf of Bunuel, but Bunuel can always correct me if he thinks I am wrong. Statement 1 S=1.75 x and question says S is an integer and each letter represents a distinct digit So if we take x = 4 then S = 1.75 *4 then S = 7. Only 4 can give us S as an integer and therefore x has to be 4. Considering statement 2 S^2= 49zx/8 S = 7 * square root of 49zx/8. Again for S to be an integer square root of 49zx/8 has to be 1 as no other value fits the bill. I hope I answered your question. I f I haven't then please free to let me know and I will explain it again. _________________ Best Regards, E. MGMAT 1 --> 530 MGMAT 2--> 640 MGMAT 3 ---> 610 GMAT ==> 730 Math Expert Joined: 02 Sep 2009 Posts: 52905 Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 02 Feb 2012, 10:19 2 pappueshwar wrote: hi bunuel, unable to understand ur explanation. how are u arriving at values 4 and 7? and option B (1) $$S=1.75*X=\frac{7}{4}*X$$. Notice that X and S are single digits, hence $$\frac{7}{4}*X$$ must equal to a single digit which is only possible for X=4 --> $$\frac{7}{4}*4=7=S$$, for other values of X, $$\frac{7}{4}*X$$ is either more than 9, so not a single digit or/and not an integer at all. (There is though one more case for X=0 --> S=0, but stems says that each letter represents a distinct digit, so this option is also out.) The same logic applies to (2). Hope it's clear. _________________ Intern Joined: 23 Jul 2013 Posts: 19 Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 11 Sep 2013, 23:58 Bunuel wrote: What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive? (1) S = 1.75X --> as S and X are digits, then X must be 4 and S must be 7: 7=1.75*4 --> SSS=777. Sufficient. (2) S^2= 49zx/8 --> $$S=7*\sqrt{\frac{zx}{8}}$$ --> again as S is a digit then $$\sqrt{\frac{zx}{8}}$$ must be 1 --> S=7 --> SSS=777. Sufficient. Answer: D. very nicely explained Bunuel.. I was totally tricked... started thinking too much on this.. it was simple and answer was in question only. Thanks Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6949 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] ### Show Tags 22 Nov 2015, 01:20 Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive? 1) S = 1.75 X 2) S^2 = 49zx/8 There are 6 variables (x,y,z,a,b,c), but only 2 equations are given by the 2 conditions, so there is high chance (E) will be the answer. Looking at the conditions together, From condition 1, S=175x/100=7x/4, S and x are all integers, and x=4, S=7. From condition 2, S^2=49zx/8, z=sqrt (49zx/8)=7sqrt(zx/8)=7 (because S is 1-digit integer) Condition 1 = condition 2, and the answer becomes (D). For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or 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$149 for 3 month Online Course"
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Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink]
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08 Nov 2017, 23:50
Bunuel wrote:
What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive?
(1) S = 1.75X --> as S and X are digits, then X must be 4 and S must be 7: 7=1.75*4 --> SSS=777. Sufficient.
(2) S^2= 49zx/8 --> $$S=7*\sqrt{\frac{zx}{8}}$$ --> again as S is a digit then $$\sqrt{\frac{zx}{8}}$$ must be 1 --> S=7 --> SSS=777. Sufficient.
Answer: D.
Hey Bunuel, why are we ruling out $$\sqrt{xz/8}$$=0?
Math Expert
Joined: 02 Sep 2009
Posts: 52905
Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink]
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09 Nov 2017, 00:53
1
richabala26 wrote:
Bunuel wrote:
What is the value of the three-digit number SSS if SSS is the sum of the three-digit numbers ABC and XYZ, where each letter represents a distinct digit from 0 to 9, inclusive?
(1) S = 1.75X --> as S and X are digits, then X must be 4 and S must be 7: 7=1.75*4 --> SSS=777. Sufficient.
(2) S^2= 49zx/8 --> $$S=7*\sqrt{\frac{zx}{8}}$$ --> again as S is a digit then $$\sqrt{\frac{zx}{8}}$$ must be 1 --> S=7 --> SSS=777. Sufficient.
Answer: D.
Hey Bunuel, why are we ruling out $$\sqrt{xz/8}$$=0?
This would imply that S = 0. In this case SSS = 000, which is not a three-digit number but a single digit number 0.
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Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink]
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27 Aug 2018, 20:39
alternative approach for statement 2:
S^2= 49zx/8
we can see S has 7 as its prime factor. out of SSS, only 777 can be evenly divided by 7. so SSS has to be 777
Re: What is the value of the three-digit number SSS if SSS is the sum of [#permalink] 27 Aug 2018, 20:39
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# What is the value of the three-digit number SSS if SSS is the sum of
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2019-02-16 01:20:33
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https://zbmath.org/?q=an:1035.05023
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# zbMATH — the first resource for mathematics
Nonlinear functions in abelian groups and relative difference sets. (English) Zbl 1035.05023
This paper shows that the main results on nonlinear functions on finite fields can be generalized to abelian groups using the discrete Fourier tranform. The paper is a very interesting survey on (generalisations of) relative difference sets and nonlinear functions introducing new points of view.
Let $$K$$ and $$N$$ be abelian (additive) groups with $$| K| =m$$, $$| N| =n$$ and $$f:K\to N$$ be a function. Let $$D_f=\{\langle g,f(g)\rangle \mid g\in K\}\subset G=K\times N$$. Call a function $$f$$ perfect nonlinear if $$\delta_f(a,b)=| \{g\in K\mid f(g+a)-f(g)=b\}|$$ is equal to $$m/n$$ for all $$a\in K\setminus\{0\}$$ and $$b\in N$$. Now $$D_f$$ is a splitting $$(m,n,m,m/n)$$-DS in $$G$$ relative to $$\{0\}\times N$$ if and only if $$f$$ is perfect nonlinear. This suggest to use the discrete Fourier transform to study more general sets $$D_f$$. A key of the paper is the following definition: a function $$f:K\to N$$ is an almost perfect nonlinear function if $$\sum_{a,b}[\delta_f(a,b)]^2\leq\sum_{a,b}[\delta_g(a,b)]^2$$ for all functions $$g:K\to N$$.
Too many definitions and notations are necessary to go here into detail; we can only note that using this new point of view many proofs are more transparent and connections with relative difference sets become apparent.
##### MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 20K99 Abelian groups
##### Keywords:
almost perfect nonlinear functions
Full Text:
##### References:
[1] Arasu, K.T.; Dillon, J.F.; Jungnickel, D.; Pott, A., The solution of the Waterloo problem, J. combin. theory ser. A, 71, 316-331, (1995) · Zbl 0833.05008 [2] Arasu, K.T.; Dillon, J.F.; Leung, K.H.; Ma, S.L., Cyclic relative difference sets with classical parameters, J. combin. theory ser. A, 94, 118-126, (2001) · Zbl 0980.05016 [3] Arasu, K.T.; Jungnickel, D.; Ma, S.L.; Pott, A., Relative difference sets with n=2, Discrete math., 147, 1-17, (1995) · Zbl 0842.05009 [4] T. Beth, D. Jungnickel, H. Lenz, Design Theory, Vol. 1, 2nd Edition, Cambridge University Press, Cambridge, 1999. · Zbl 0945.05005 [5] Beth, T.; Jungnickel, D.; Lenz, H., Design theory, (1999), Cambridge University Press Cambridge [6] A. Blokhuis, D. Jungnickel, B. Schmidt, Proof of the prime power conjecture for projective planes of order n with abelian collineation groups of order n2, Proc. Amer. Math. Soc. 130 (2002), pp. 1473-1476 (electronic). · Zbl 1004.51012 [7] Canteaut, A.; Charpin, P.; Dobbertin, H., Binary m-sequences with three-valued crosscorrelationa proof of Welch’s conjecture , IEEE trans. inform. theory, 46, 4-8, (2000) · Zbl 1003.94519 [8] Chabaud, F.; Vaudenay, S., Links between differential and linear cryptanalysis, (), 356-365 · Zbl 0879.94023 [9] Coulter, R.S.; Mathews, R.W., Planar functions and planes of lenz – barlotti class II, Des. codes cryptogr., 10, 167-184, (1997) · Zbl 0872.51007 [10] Davis, J.A.; Jedwab, J., A unifying construction for difference sets, J. combin. theory ser. A, 80, 13-78, (1997) · Zbl 0884.05019 [11] Davis, J.A.; Jedwab, J.; Mowbray, M., New families of semi-regular relative difference sets, Des. codes cryptogr., 13, 131-146, (1998) · Zbl 0888.05008 [12] Dembowski, P.; Ostrom, T., Planes of order n with collineation groups of order n2, Math. Z., 103, 239-258, (1968) · Zbl 0163.42402 [13] Dillon, J.F., Multiplicative difference sets via additive characters, Des. codes, cryptogr., 17, 225-235, (1999) · Zbl 0944.05012 [14] J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, Finite Fields Appl. (2003), to appear. · Zbl 1043.05024 [15] Dobbertin, H., One-to-one highly nonlinear power functions on GF(2n), Appl. algebra eng. commun. comput., 9, 139-152, (1998) · Zbl 0924.94026 [16] Dobbertin, H., Almost perfect nonlinear power functions on GF(2n)the niho case , Inform. comput., 151, 57-72, (1999) · Zbl 1072.94513 [17] Dobbertin, H., Almost perfect nonlinear power functions on GF(2n)the welch case , IEEE trans. inform. theory, 45, 1271-1275, (1999) · Zbl 0957.94021 [18] Dobbertin, H., Kasami power functions, permutation polynomials and cyclic difference sets, (), 133-158 · Zbl 0946.05010 [19] Dobbertin, H., Almost perfect nonlinear power functions on GF(2n): a new case for n divisible by 5, (), 113-121 · Zbl 1010.94550 [20] Dobbertin, H.; Mills, D.; Müller, E.N.; Pott, A.; Willems, W., APN functions in odd characteristic, Discrete appl. math., 267, 95-112, (2003) · Zbl 1028.11076 [21] Elliott, J.E.H.; Butson, A.T., Relative difference sets, Illinois J. math., 10, 517-531, (1966) · Zbl 0145.01503 [22] Evans, R.; Hollmann, H.D.L.; Krattenthaler, C.; Xiang, Q., Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. combin. theory A, 87, 74-119, (1999) · Zbl 0943.05021 [23] Ganley, M.J., On a paper of P. Dembowski and T.G. ostromplanes of order n with collineation groups of order n2 (math. Z. 103 (1968) 239-258) , Arch. math. (basel), 27, 93-98, (1976) [24] Gordon, B.; Mills, W.H.; Welch, L.R., Some new difference sets, Canad. J. math., 14, 614-625, (1962) · Zbl 0111.24201 [25] T. Helleseth, P.V. Kumar, Sequences with low correlation, in: Handbook of Coding Theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 1065-1138. [26] Helleseth, T.; Rong, C.; Sandberg, D., New families of almost perfect nonlinear power mappings, IEEE trans. inform. theory, 45, 475-485, (1999) · Zbl 0960.11051 [27] Helleseth, T.; Sandberg, D., Some power mappings with low differential uniformity, Appl. algebra eng. commun. comput., 8, 363-370, (1997) · Zbl 0886.11067 [28] Hollmann, H.H.; Xiang, Q., A proof of the welch and niho conjectures on crosscorrelations of binary m-sequences, Finite fields appl., 7, 253-286, (2001) · Zbl 1027.94006 [29] Jungnickel, D., On a theorem of ganley, Graphs combin., 3, 141-143, (1987) · Zbl 0659.05028 [30] Jungnickel, D.; Pott, A., Perfect and almost perfect sequences, Discrete appl. math., 95, 331-359, (1999) · Zbl 0941.05013 [31] M. Kantor, W., Note on GMW designs, Electron. J. combin., 22, 63-69, (2001) · Zbl 0964.05015 [32] Maschietti, A., Difference sets and hyperovals, Des. codes cryptogr., 14, 89-98, (1998) · Zbl 0887.05010 [33] K. Nyberg, Perfect nonlinear S-boxes, in: Advances in Cryptology—EUROCRYPT’91, Brighton, 1991, Springer, Berlin, 1991, pp. 378-386. · Zbl 0766.94012 [34] Pott, A., Finite geometry and character theory, Lecture notes in mathematics, Vol. 1601, (1995), Springer Berlin, Heidelberg · Zbl 0818.05001 [35] Pott, A., A survey on relative difference sets, (), 195-232 · Zbl 0847.05018 [36] Sarwate, D.V.; Pursley, M.B., Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68, 593-619, (1980) [37] Schmidt, B., On (pa,pb,pa,pa−b)-relative difference sets, J. algebraic combin., 6, 279-297, (1997) [38] V.V. Shorin, V.V. Jelezniakov, E.M. Gabidulin, Linear and differential cryptanalysis of Russian GOST, in: D. Augot, C. Carlet (Eds.), Workshop on Coding and Cryptography 2001, INRIA, 2001, pp. 467-476. · Zbl 0985.94035 [39] Sidel’nikov, V.M., On the mutual correlation of sequences, Soviet math. dokl., 12, 197-201, (1971) · Zbl 0241.94008 [40] Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. amer. math. soc., 43, 377-385, (1938) · JFM 64.0972.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-07-24 22:44:19
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https://ch.gateoverflow.in/410/gate-chemical-2016-question-38
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The characteristics curve (Head-Capacity relationship) of a centrifugal pimp is represented by the equation $\Delta H_{pump}=43.8-0.19Q$, where $\Delta H_{pump}$ is the head developed by the pump (in $m$) and $Q$ is the flowrate (in $m^{3}/h$) through the pump. This pump is to be used for pumping water through a horizontal pipeline. The frictional head loss $\Delta H_{piping}$ (in $m$) is related to the water flowrate $Q_{L}$ (in $m^{3}/h$) by the equation $\Delta H_{piping}=0.0135_{L}^{2}+0.045_{L}$. The flowrate (in $m^{3}/h$, rounded off to the first decimal place) of water pumped through the above pipeline, is _____________
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2022-05-27 22:02:25
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https://math.stackexchange.com/questions/1770054/show-that-int-04-x-e-x-24-dx-2k-given-that-int-04-e-x
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# Show that $\int_{0}^{4} x e^{ (x-2)^4 } dx = 2k$ given that $\int_{0}^{4} e^{ (x-2)^4 } dx = k$
I'm stuck on this one. I've tried integration by parts but to no avail:
$$\int_{0}^{4} x e^{ (x-2)^4 } dx = \left[ x \int e^{ (x-2)^4 } dx \right]_{0}^{4} - \underbrace{\int_{0}^{4} e^{ (x-2)^4 } dx}_{=k}$$
I suspect this is a dead end. I think there's some simple trick here I'm missing. Any hints are appreciated.
$$I = \int_{0}^{4} xe^{(x-2)^4} \quad dx$$ substitute $t = 4-x$
\begin{align} I &= \int_{4}^{0} (4-t)e^{(4-t-2)^4} \quad (-dt) &= \int_{0}^{4} (4-t)e^{(t-2)^4} \quad dt \end{align} Adding the 2 results \begin{align} 2I = 4 \int_{0}^{4} e^{(x-2)^4} \quad dx & = 4k \end{align}
Note that
$$k = \int_{-2}^2 dx \, e^{x^4}$$
Then
$$\int_0^4 dx \, x \, e^{(x-2)^4} = \int_{-2}^2 dx \, (x+2) e^{x^4} = 2 \int_{-2}^2 dx \, e^{x^4}= 2 k$$
Note that the result desired comes out of symmetry, as clearly
$$\int_{-2}^2 dx \, x \, e^{x^4} = 0$$
Well, the function $(x-2)\,e^{-(x-2)^4}$ is continuous and symmetric with respect to $x=2$, hence its integral over $(0,4)$ is zero. The claim easily follows through $x=(x-2)+2$.
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2021-03-07 15:44:06
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https://socratic.org/questions/what-is-crystallization
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# What is crystallization?
A good example of a chemical which readily undergoes sublimation and crystallization is carbon dioxide. ($C {O}_{2}$).
Sublimation : $C {O}_{2 \left(s\right)} \to C {O}_{2 \left(g\right)}$.
Crystallization : $C {O}_{2 \left(g\right)} \to C {O}_{2 \left(s\right)}$
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2019-07-19 06:53:10
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http://mathoverflow.net/questions/126754/finiteness-theorem-for-first-cohomology-group-of-sheaf-of-holomorphic-functions
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# Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces
I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, where $X$ is a compact Riemann surface and $\mathcal{O}$ is the sheaf of holomorphic functions. Forster proves this by reducing the problem into considering restriction map of Cech cohomology groups given by a shrinking sequence of relatively compact open coverings(this is done by choosing a proper coordinate patch and using Leray's theorem and Dolbeault's lemma), and introduces $L^2$-norms on such groups. Well, first of all, what's the point in introducing the $L^2$-norm, by which I mean, is that suggesting any interesting topological properties such as compactness or anything else? What's the essential problem lying in this theorem? Secondly, is there any other way of proving this finiteness theorem? I would appreciate your help!
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2014-09-30 16:33:47
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https://eprints.keele.ac.uk/2588/
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Reeves, JN, Braito, V, Nardini, E, Behar, E, O'Brien, P, Tombesi, F, Turner, TJ and Costa, M (2016) Discovery of Broad Soft X-ray Absorption Lines from the Quasar Wind in PDS 456. Astrophysical Journal, 824 (1).
Preview
Text
1604.04196v1.pdf - Accepted Version
High resolution soft X-ray spectroscopy of the prototype accretion disk wind quasar, PDS 456, is presented. Here, the XMM-Newton RGS spectra are analyzed from the large 2013-2014 XMM-Newton campaign, consisting of 5 observations of approximately 100 ks in length. During the last observation (hereafter OBS. E), the quasar is at a minimum flux level and broad absorption line profiles are revealed in the soft X-ray band, with typical velocity widths of $\sigma_{\rm v}\sim 10,000$ km s$^{-1}$. During a period of higher flux in the 3rd and 4th observations (OBS. C and D, respectively), a very broad absorption trough is also present above 1 keV. From fitting the absorption lines with models of photoionized absorption spectra, the inferred outflow velocities lie in the range $\sim 0.1-0.2c$. The absorption lines likely originate from He and H-like neon and L-shell iron at these energies. Comparison with earlier archival data of PDS 456 also reveals similar absorption structure near 1 keV in a 40 ks observation in 2001, and generally the absorption lines appear most apparent when the spectrum is more absorbed overall. The presence of the soft X-ray broad absorption lines is also independently confirmed from an analysis of the XMM-Newton EPIC spectra below 2 keV. We suggest that the soft X-ray absorption profiles could be associated with a lower ionization and possibly clumpy phase of the accretion disk wind, where the latter is known to be present in this quasar from its well studied iron K absorption profile and where the wind velocity reaches a typical value of 0.3$c$.
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2020-07-15 12:47:13
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https://math.stackexchange.com/questions/510273/finite-subsets-of-hausdorff-spaces
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# Finite subsets of Hausdorff spaces
quick question. There is a question in the book saying "Every finite subset of a Hausdorff space is closed." The proof uses the fact that for each point in $X\setminus\{p\}$ there is a neighborhood contained in $X\setminus\{p\}$ the Hausdorff condition and hence $X\setminus\{p\}$ is open so $\{p\}$ is closed the set is the finite union finite sets.
The very next problem states: "The only Hausdorff topology on a finite set is the discrete topology." Well, in the discrete topology every $\{p\}$ is an open set, not closed as the proof above states. What is going on here?
• Open sets can be closed. – Stefan Hamcke Sep 30 '13 at 18:45
• Take an arbitrary subset. The complement is finite and thus closed, so...? – Tobias Kildetoft Sep 30 '13 at 18:46
• It's also the complement of an open set, hence closed. In the discrete topology, all subsets are both, open and closed. – Daniel Fischer Sep 30 '13 at 18:46
• got it, thanks. – masszz Sep 30 '13 at 18:47
If you have a Hausdorff space $X$ and $F$ is a finite subset of $X$, then the argument you give shows that $F$ is closed in $X$. Now, $F$ itself carries the subspace (or induced) topology from $X$, with which it is a topological space in its own right. As with any topological space, $F$ is open in itself. That does not mean that it is open in $X$. These are different concepts. Since the Hausdorff condition passes to subspaces, $F$ is Hausdorff and finite, and this does indeed imply that it is discrete, so each of its subsets is open in itself. There is no reason that this has to imply that each subset of $F$ is open in $X$.
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2019-09-17 19:11:16
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https://how-to.aimms.com/Articles/391/391-life-cycle-consumption.html
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# Life Cycle Consumption¶
Direct download AIMMS Project Life Cycle Consumption.zip
In this example a life-cycle consumption optimization problem has been modeled, including labor and assets.
Users of this application can make changes to the model from the user interface. Two types of utility functions are available:
• Exponential utility: $$Utility = - e^{-Consumption} - Labor^2$$
• Square root utility: $$Utility = \sqrt{Consumption} - Labor^2$$
In addition, the user can choose to allow borrowing up to a chosen level, adding explicit modeling of assets to the model.
Please use the navigation in the left column to browse this example application.
Keywords: Utility Function, Discount Factor, Present Value, Future Value, Nonlinear Programming, Nonlinear Solvers, Curve Object
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2021-01-19 19:05:19
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https://www.sexyloops.co.uk/theboard/viewtopic.php?t=3018&p=49681
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PLEASE NOTE: In order to post on the Board you need to have registered. To register please email paul@sexyloops.com including your real name and username. Registration takes less than 24hrs, unless Paul is fishing deep in the jungle!
## Weedless Frog
Moderator: Viking Lars
Mangrove Cuckoo
Posts: 218
Joined: Tue Jan 29, 2013 7:51 am
### Re: Weedless Frog
Here is the "weedless" pencil popper in all of its simplistic beauty.
My apologies, the hook is not Gamakatsu, but Owner... "Mosquito hook".
The fly does what it is supposed to do. It is not sexy but it works rather well. One of my fishing buddies coined it "the miracle worker".
Attachments
“Very simple man. Catching fish makes me happy. Scaringly simple.”
Håvard Stubø
Paul Arden
Site Admin
Posts: 12221
Joined: Thu Jan 03, 2013 11:20 am
Location: Belum Rainforest
Contact:
### Re: Weedless Frog
That’s very nice Gary! I shall tie some of those. Does it skip if you immediately start pulling fast? I usually lead the tip of the bend of my poppers to stop this happening.
Great looking fly and gives me a few ideas!
Incidentally I’ve been looking for and trying to design a fly that floats but when pulled sinks. Do you have any patterns that do this? This is for fishing flies that can sink but sit on the surface for Snakehead shots.
Thanks, Paul
It's an exploration; bring a flyrod.
Flycasting Definitions
Nick
Posts: 240
Joined: Sat Jan 19, 2013 5:15 pm
### Re: Weedless Frog
I assume that you have tried something like this?
https://www.flytyingforum.com/uploads/i ... 2a1384.jpg
queenfish
Posts: 102
Joined: Thu Dec 04, 2014 5:59 pm
Location: Goldcoast qld
### Re: Weedless Frog
Here are few that do that.
when you pull they swim downward, stop they float.
they are very good for bass and saratoga here.
can't download picture I will send it through email.
Paul Arden
Site Admin
Posts: 12221
Joined: Thu Jan 03, 2013 11:20 am
Location: Belum Rainforest
Contact:
### Re: Weedless Frog
IMG code - just instead the URL!
Code: Select all
[img]https://www.flytyingforum.com/uploads/img457a2df2a1384.jpg[/img]
Thanks Gary!
I haven’t seen that before. How does it work and swim? Is the hook bent or is that the way the foam is cut?
Cheers,
Paul
It's an exploration; bring a flyrod.
Flycasting Definitions
Paul Arden
Site Admin
Posts: 12221
Joined: Thu Jan 03, 2013 11:20 am
Location: Belum Rainforest
Contact:
### Re: Weedless Frog
(It’s much simpler and no doubt less dangerous than the beer cans we were cutting up).
It's an exploration; bring a flyrod.
Flycasting Definitions
Paul Arden
Site Admin
Posts: 12221
Joined: Thu Jan 03, 2013 11:20 am
Location: Belum Rainforest
Contact:
### Re: Weedless Frog
Vince (Queenfish) sent me a few photos last night.
If you use heavy hooks you don’t need any lead
If the foam is wide I use 2 or 3 turns 25 gauge lead right on the bend of the hook and finish with loon on top.
Cheers Vince.
Ps I bend the eye of the hook slightly upward as well.
Attachments
It's an exploration; bring a flyrod.
Flycasting Definitions
Mangrove Cuckoo
Posts: 218
Joined: Tue Jan 29, 2013 7:51 am
### Re: Weedless Frog
Paul,
That weedless pencil popper has a curious flaw (or maybe not?)
If the tube is the same length as the body, the leader slides through easily. So easy, in fact, that the hook and bucktail will drop if the popper is allowed to pause.
It does not seem to be a positive thing for the saltwater fish I usually target... but who knows what a snakehead might think? If you have some missed strikes on a cast, maybe pausing the retrieve and letting the back end drop might induce another hit?
If you don't want the fly to separate simply cut the tube a bit short and leave the last bit of foam body tubeless. A good size knot on the ring will stick inside the butt of the foam body and reduce the tendency of the fly to separate.
The fly will occasionally skip on the first strip since the front of the foam is cut on an angle. About the same percentage of times, the fly might dive a bit. Since the foam is free to rotate randomly, the fly has a much more erratic retrieve than a standard popper.
“Very simple man. Catching fish makes me happy. Scaringly simple.”
Håvard Stubø
Mangrove Cuckoo
Posts: 218
Joined: Tue Jan 29, 2013 7:51 am
### Re: Weedless Frog
Paul Arden wrote:
Wed Jul 17, 2019 12:08 pm
Incidentally I’ve been looking for and trying to design a fly that floats but when pulled sinks. Do you have any patterns that do this? This is for fishing flies that can sink but sit on the surface for Snakehead shots.
Thanks, Paul
Paul,
The flies in the photo will do just that. And, they pull very straight. They will return to the surface, although pretty slowly, on a pause.
The trick is keeling them with a strip of lead tape. I tried to show that in the photo.
What is not evident is that these are rather large flies... they are tied on 3/0 extra long shank saltwater hooks.
Attachments
Diver flies RSfSL.jpg
“Very simple man. Catching fish makes me happy. Scaringly simple.”
Håvard Stubø
Paul Arden
Site Admin
Posts: 12221
Joined: Thu Jan 03, 2013 11:20 am
Location: Belum Rainforest
Contact:
### Re: Weedless Frog
Thanks Gary. I shall have a go at both of those. Skipping across the surface is generally not good for Snakehead. It can work if they decide to chase and you have enough space. But usually they ignore it and next time come up spooked. I find a really good bloop to be best.
The sinking fly stuff is unusual. There are times when they just don’t seem to want to eat off the surface. So this is something I need to explore.
Thanks, Paul
It's an exploration; bring a flyrod.
Flycasting Definitions
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2019-10-24 00:54:21
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http://wikien4.appspot.com/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF
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# 1 − 2 + 3 − 4 + ⋯
The first 15,000 partiaw sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated wif positive integers to de right and negative integers to de weft.
In madematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are de successive positive integers, given awternating signs. Using sigma summation notation de sum of de first m terms of de series can be expressed as
${\dispwaystywe \sum _{n=1}^{m}n(-1)^{n-1}.}$
The infinite series diverges, meaning dat its seqwence of partiaw sums, (1, −1, 2, −2, ...), does not tend towards any finite wimit. Nonedewess, in de mid-18f century, Leonhard Euwer wrote what he admitted to be a paradoxicaw eqwation:
${\dispwaystywe 1-2+3-4+\cdots ={\frac {1}{4}}.}$
A rigorous expwanation of dis eqwation wouwd not arrive untiw much water. Starting in 1890, Ernesto Cesàro, Émiwe Borew and oders investigated weww-defined medods to assign generawized sums to divergent series—incwuding new interpretations of Euwer's attempts. Many of dese summabiwity medods easiwy assign to 1 − 2 + 3 − 4 + ... a "vawue" of 1/4. Cesàro summation is one of de few medods dat do not sum 1 − 2 + 3 − 4 + ..., so de series is an exampwe where a swightwy stronger medod, such as Abew summation, is reqwired.
The series 1 − 2 + 3 − 4 + ... is cwosewy rewated to Grandi's series 1 − 1 + 1 − 1 + .... Euwer treated dese two as speciaw cases of 1 − 2n + 3n − 4n + ... for arbitrary n, a wine of research extending his work on de Basew probwem and weading towards de functionaw eqwations of what are now known as de Dirichwet eta function and de Riemann zeta function.
## Divergence
The series' terms (1, −2, 3, −4, ...) do not approach 0; derefore 1 − 2 + 3 − 4 + ... diverges by de term test. For water reference, it wiww awso be usefuw to see de divergence on a fundamentaw wevew. By definition, de convergence or divergence of an infinite series is determined by de convergence or divergence of its seqwence of partiaw sums, and de partiaw sums of 1 − 2 + 3 − 4 + ... are:[1]
1 = 1,
1 − 2 = −1,
1 − 2 + 3 = 2,
1 − 2 + 3 − 4 = −2,
1 − 2 + 3 − 4 + 5 = 3,
1 − 2 + 3 − 4 + 5 − 6 = −3,
...
This seqwence is notabwe for incwuding every integer exactwy once—even 0 if one counts de empty partiaw sum—and dereby estabwishing de countabiwity of de set ${\dispwaystywe \madbb {Z} }$ of integers.[2] The seqwence of partiaw sums cwearwy shows dat de series does not converge to a particuwar number (for any proposed wimit x, we can find a point beyond which de subseqwent partiaw sums are aww outside de intervaw [x−1, x+1]), so 1 − 2 + 3 − 4 + ... diverges.
## Heuristics for summation
### Stabiwity and winearity
Since de terms 1, −2, 3, −4, 5, −6, ... fowwow a simpwe pattern, de series 1 − 2 + 3 − 4 + ... can be manipuwated by shifting and term-by-term addition to yiewd a numericaw vawue. If it can make sense to write s = 1 − 2 + 3 − 4 + ... for some ordinary number s, de fowwowing manipuwations argue for s = 14:[3]
${\dispwaystywe {\begin{array}{rcwwwww}4s&=&&(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+(1-2)+(3-4+5-6\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}-1+(3-4+5-6\cdots )\\&=&1+&(1-2+3-4+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(3-4+5-6\cdots )\\&=&1+[&(1-2-2+3)&{}+(-2+3+3-4)&{}+(3-4-4+5)&{}+(-4+5+5-6)+\cdots ]\\&=&1+[&0+0+0+0+\cdots ]\\4s&=&1\end{array}}}$
Adding 4 copies of 1 − 2 + 3 − 4 + ..., using onwy shifts and term-by-term addition, yiewds 1. The weft side and right side each demonstrates two copies of 1 − 2 + 3 − 4 + ... adding to 1 − 1 + 1 − 1 + ....
So ${\dispwaystywe s={\frac {1}{4}}}$. This derivation is depicted graphicawwy on de right.
Awdough 1 − 2 + 3 − 4 + ... does not have a sum in de usuaw sense, de eqwation s = 1 − 2 + 3 − 4 + ... = 14 can be supported as de most naturaw answer if such a sum is to be defined. A generawized definition of de "sum" of a divergent series is cawwed a summation medod or summabiwity medod. There are many different medods (some of which are described bewow) and it is desirabwe dat dey share certain properties wif ordinary summation, uh-hah-hah-hah. What de above manipuwations actuawwy prove is de fowwowing: Given any summabiwity medod dat is winear and stabwe and sums de series 1 − 2 + 3 − 4 + ..., de sum it produces is 14.[4] Furdermore, since
${\dispwaystywe {\begin{array}{rcwwww}2s&=&&(1-2+3-4+\cdots )&+&(1-2+3-4+\cdots )\\&=&1+{}&(-2+3-4+\cdots )&{}+1-2&{}+(3-4+5\cdots )\\&=&0+{}&(-2+3)+(3-4)+(-4+5)+\cdots \\2s&=&&1-1+1-1\cdots \end{array}}}$
such a medod must awso sum Grandi's series as 1 − 1 + 1 − 1 + ... = 12.[5]
### Cauchy product
In 1891, Ernesto Cesàro expressed hope dat divergent series wouwd be rigorouswy brought into cawcuwus, pointing out, "One awready writes (1 − 1 + 1 − 1 + ...)2 = 1 − 2 + 3 − 4 + ... and asserts dat bof de sides are eqwaw to 14."[6] For Cesàro, dis eqwation was an appwication of a deorem he had pubwished de previous year, which is de first deorem in de history of summabwe divergent series.[7] The detaiws on his summation medod are bewow; de centraw idea is dat 1 − 2 + 3 − 4 + ... is de Cauchy product (discrete convowution) of 1 − 1 + 1 − 1 + ... wif 1 − 1 + 1 − 1 + ....
The Cauchy product of two infinite series is defined even when bof of dem are divergent. In de case where an = bn = (−1)n, de terms of de Cauchy product are given by de finite diagonaw sums
${\dispwaystywe {\begin{array}{rcw}c_{n}&=&\dispwaystywe \sum _{k=0}^{n}a_{k}b_{n-k}=\sum _{k=0}^{n}(-1)^{k}(-1)^{n-k}\\[1em]&=&\dispwaystywe \sum _{k=0}^{n}(-1)^{n}=(-1)^{n}(n+1).\end{array}}}$
The product series is den
${\dispwaystywe \sum _{n=0}^{\infty }(-1)^{n}(n+1)=1-2+3-4+\cdots .}$
Thus a summation medod dat respects de Cauchy product of two series — and assigns to de series 1 − 1 + 1 − 1 + ... de sum 1/2 — wiww awso assign to de series 1 − 2 + 3 − 4 + ... de sum 1/4. Wif de resuwt of de previous section, dis impwies an eqwivawence between summabiwity of 1 − 1 + 1 − 1 + ... and 1 − 2 + 3 − 4 + ... wif medods dat are winear, stabwe, and respect de Cauchy product.
Cesàro's deorem is a subtwe exampwe. The series 1 − 1 + 1 − 1 + ... is Cesàro-summabwe in de weakest sense, cawwed (C, 1)-summabwe, whiwe 1 − 2 + 3 − 4 + ... reqwires a stronger form of Cesàro's deorem,[8] being (C, 2)-summabwe. Since aww forms of Cesàro's deorem are winear and stabwe, de vawues of de sums are as we have cawcuwated.
## Specific medods
### Cesàro and Höwder
Data about de (H, 2) sum of 14
To find de (C, 1) Cesàro sum of 1 − 2 + 3 − 4 + ..., if it exists, one needs to compute de aridmetic means of de partiaw sums of de series. The partiaw sums are:
1, −1, 2, −2, 3, −3, ...,
and de aridmetic means of dese partiaw sums are:
1, 0, 23, 0, 35, 0, 47, ....
This seqwence of means does not converge, so 1 − 2 + 3 − 4 + ... is not Cesàro summabwe.
There are two weww-known generawizations of Cesàro summation: de conceptuawwy simpwer of dese is de seqwence of (H, n) medods for naturaw numbers n. The (H, 1) sum is Cesàro summation, and higher medods repeat de computation of means. Above, de even means converge to 12, whiwe de odd means are aww eqwaw to 0, so de means of de means converge to de average of 0 and 12, namewy 14.[9] So 1 − 2 + 3 − 4 + ... is (H, 2) summabwe to 14.
The "H" stands for Otto Höwder, who first proved in 1882 what madematicians now dink of as de connection between Abew summation and (H, n) summation; 1 − 2 + 3 − 4 + ... was his first exampwe.[10] The fact dat 14 is de (H, 2) sum of 1 − 2 + 3 − 4 + ... guarantees dat it is de Abew sum as weww; dis wiww awso be proved directwy bewow.
The oder commonwy formuwated generawization of Cesàro summation is de seqwence of (C, n) medods. It has been proven dat (C, n) summation and (H, n) summation awways give de same resuwts, but dey have different historicaw backgrounds. In 1887, Cesàro came cwose to stating de definition of (C, n) summation, but he gave onwy a few exampwes. In particuwar, he summed 1 − 2 + 3 − 4 + ..., to 14 by a medod dat may be rephrased as (C, n) but was not justified as such at de time. He formawwy defined de (C, n) medods in 1890 in order to state his deorem dat de Cauchy product of a (C, n)-summabwe series and a (C, m)-summabwe series is (C, m + n + 1)-summabwe.[11]
### Abew summation
Some partiaws of 1 − 2x + 3x2 + ...; 1/(1 + x)2; and wimits at 1
In a 1749 report, Leonhard Euwer admits dat de series diverges but prepares to sum it anyway:
... when it is said dat de sum of dis series 1 − 2 + 3 − 4 + 5 − 6 etc. is 14, dat must appear paradoxicaw. For by adding 100 terms of dis series, we get −50, however, de sum of 101 terms gives +51, which is qwite different from 14 and becomes stiww greater when one increases de number of terms. But I have awready noticed at a previous time, dat it is necessary to give to de word sum a more extended meaning ...[12]
Euwer proposed a generawization of de word "sum" severaw times. In de case of 1 − 2 + 3 − 4 + ..., his ideas are simiwar to what is now known as Abew summation:
... it is no more doubtfuw dat de sum of dis series 1 − 2 + 3 − 4 + 5 etc. is 14; since it arises from de expansion of de formuwa 1(1+1)2, whose vawue is incontestabwy 14. The idea becomes cwearer by considering de generaw series 1 − 2x + 3x2 − 4x3 + 5x4 − 6x5 + &c. dat arises whiwe expanding de expression 1(1+x)2, which dis series is indeed eqwaw to after we set x = 1.[13]
There are many ways to see dat, at weast for absowute vawues |x| < 1, Euwer is right in dat
${\dispwaystywe 1-2x+3x^{2}-4x^{3}+\cdots ={\frac {1}{(1+x)^{2}}}.}$
One can take de Taywor expansion of de right-hand side, or appwy de formaw wong division process for powynomiaws. Starting from de weft-hand side, one can fowwow de generaw heuristics above and try muwtipwying by (1 + x) twice or sqwaring de geometric series 1 − x + x2 − .... Euwer awso seems to suggest differentiating de watter series term by term.[14]
In de modern view, de series 1 − 2x + 3x2 − 4x3 + ... does not define a function at x = 1, so dat vawue cannot simpwy be substituted into de resuwting expression, uh-hah-hah-hah. Since de function is defined for aww |x| < 1, one can stiww take de wimit as x approaches 1, and dis is de definition of de Abew sum:
${\dispwaystywe \wim _{x\rightarrow 1^{-}}\sum _{n=1}^{\infty }n(-x)^{n-1}=\wim _{x\rightarrow 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.}$
### Euwer and Borew
Euwer summation to 12 − 14. Positive vawues are shown in white, negative vawues are shown in brown, and shifts and cancewwations are shown in green, uh-hah-hah-hah.
Euwer appwied anoder techniqwe to de series: de Euwer transform, one of his own inventions. To compute de Euwer transform, one begins wif de seqwence of positive terms dat makes up de awternating series—in dis case 1, 2, 3, 4, .... The first ewement of dis seqwence is wabewed a0.
Next one needs de seqwence of forward differences among 1, 2, 3, 4, ...; dis is just 1, 1, 1, 1, .... The first ewement of dis seqwence is wabewed Δa0. The Euwer transform awso depends on differences of differences, and higher iterations, but aww de forward differences among 1, 1, 1, 1, ... are 0. The Euwer transform of 1 − 2 + 3 − 4 + ... is den defined as
${\dispwaystywe {\frac {1}{2}}a_{0}-{\frac {1}{4}}\Dewta a_{0}+{\frac {1}{8}}\Dewta ^{2}a_{0}-\cdots ={\frac {1}{2}}-{\frac {1}{4}}.}$
In modern terminowogy, one says dat 1 − 2 + 3 − 4 + ... is Euwer summabwe to 14.
The Euwer summabiwity impwies anoder kind of summabiwity as weww. Representing 1 − 2 + 3 − 4 + ... as
${\dispwaystywe \sum _{k=0}^{\infty }a_{k}=\sum _{k=0}^{\infty }(-1)^{k}(k+1),}$
one has de rewated everywhere-convergent series
${\dispwaystywe a(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}(k+1)x^{k+1}}{(k+1)!}}=x\sum _{k=0}^{\infty }{\frac {(-x)^{k}}{k!}}=e^{-x}x.}$
The Borew sum of 1 − 2 + 3 − 4 + ... is derefore[15]
${\dispwaystywe \int _{0}^{\infty }e^{-x}a(x)\,dx=\int _{0}^{\infty }e^{-2x}x\,dx=-{\frac {\partiaw }{\partiaw \beta }}{\bigg |}_{2}\int _{0}^{\infty }e^{-\beta x}\,dx=-{\frac {\partiaw }{\partiaw \beta }}{\bigg |}_{2}\beta ^{-1}={\frac {1}{4}}.}$
### Separation of scawes
Saichev and Woyczyński arrive at 1 − 2 + 3 − 4 + ... = 14 by appwying onwy two physicaw principwes: infinitesimaw rewaxation and separation of scawes. To be precise, dese principwes wead dem to define a broad famiwy of "φ-summation medods", aww of which sum de series to 14:
• If φ(x) is a function whose first and second derivatives are continuous and integrabwe over (0, ∞), such dat φ(0) = 1 and de wimits of φ(x) and xφ(x) at +∞ are bof 0, den[16]
${\dispwaystywe \wim _{\dewta \rightarrow 0}\sum _{m=0}^{\infty }(-1)^{m}(m+1)\varphi (\dewta m)={\frac {1}{4}}.}$
This resuwt generawizes Abew summation, which is recovered by wetting φ(x) = exp(−x). The generaw statement can be proved by pairing up de terms in de series over m and converting de expression into a Riemann integraw. For de watter step, de corresponding proof for 1 − 1 + 1 − 1 + ... appwies de mean vawue deorem, but here one needs de stronger Lagrange form of Taywor's deorem.
## Generawization
Excerpt from p.233 of de E212 — Institutiones cawcuwi differentiawis cum eius usu in anawysi finitorum ac doctrina serierum. Euwer sums simiwar series, ca. 1755.
The dreefowd Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 3 + 6 − 10 + ..., de awternating series of trianguwar numbers; its Abew and Euwer sum is 18.[17] The fourfowd Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 4 + 10 − 20 + ..., de awternating series of tetrahedraw numbers, whose Abew sum is 116.
Anoder generawization of 1 − 2 + 3 − 4 + ... in a swightwy different direction is de series 1 − 2n + 3n − 4n + ... for oder vawues of n. For positive integers n, dese series have de fowwowing Abew sums:[18]
${\dispwaystywe 1-2^{n}+3^{n}-\cdots ={\frac {2^{n+1}-1}{n+1}}B_{n+1}}$
where Bn are de Bernouwwi numbers. For even n, dis reduces to
${\dispwaystywe 1-2^{2k}+3^{2k}-\cdots =0.}$
This wast sum became an object of particuwar ridicuwe by Niews Henrik Abew in 1826:
Divergent series are on de whowe deviw's work, and it is a shame dat one dares to found any proof on dem. One can get out of dem what one wants if one uses dem, and it is dey which have made so much unhappiness and so many paradoxes. Can one dink of anyding more appawwing dan to say dat
0 = 1 − 2n + 3n − 4n + etc.
where n is a positive number. Here's someding to waugh at, friends.[19]
Cesàro's teacher, Eugène Charwes Catawan, awso disparaged divergent series. Under Catawan's infwuence, Cesàro initiawwy referred to de "conventionaw formuwas" for 1 − 2n + 3n − 4n + ... as "absurd eqwawities", and in 1883 Cesàro expressed a typicaw view of de time dat de formuwas were fawse but stiww somehow formawwy usefuw. Finawwy, in his 1890 Sur wa muwtipwication des séries, Cesàro took a modern approach starting from definitions.[20]
The series are awso studied for non-integer vawues of n; dese make up de Dirichwet eta function. Part of Euwer's motivation for studying series rewated to 1 − 2 + 3 − 4 + ... was de functionaw eqwation of de eta function, which weads directwy to de functionaw eqwation of de Riemann zeta function. Euwer had awready become famous for finding de vawues of dese functions at positive even integers (incwuding de Basew probwem), and he was attempting to find de vawues at de positive odd integers (incwuding Apéry's constant) as weww, a probwem dat remains ewusive today. The eta function in particuwar is easier to deaw wif by Euwer's medods because its Dirichwet series is Abew summabwe everywhere; de zeta function's Dirichwet series is much harder to sum where it diverges.[21] For exampwe, de counterpart of 1 − 2 + 3 − 4 + ... in de zeta function is de non-awternating series 1 + 2 + 3 + 4 + ..., which has deep appwications in modern physics but reqwires much stronger medods to sum.
## References
1. ^ Hardy p.8
2. ^ Beaws p.23
3. ^ Hardy (p.6) presents dis derivation in conjunction wif evawuation of Grandi's series 1 − 1 + 1 − 1 + ....
4. ^ Hardy p.6
5. ^ Hardy p.6
6. ^ Ferraro, p.130.
7. ^ Hardy, p.8.
8. ^ Hardy, p.3; Weidwich, pp.52–55.
9. ^ Hardy, p.9. For de fuww detaiws of de cawcuwation, see Weidwich, pp.17–18.
10. ^ Ferraro, p.118; Tucciarone, p.10. Ferraro criticizes Tucciarone's expwanation (p.7) of how Höwder himsewf dought of de generaw resuwt, but de two audors' expwanations of Höwder's treatment of 1 − 2 + 3 − 4 + ... are simiwar.
11. ^ Ferraro, pp.123–128.
12. ^ Euwer et aw., p. 2. Awdough de paper was written in 1749, it was not pubwished untiw 1768.
13. ^ Euwer et aw., pp. 3, 25.
14. ^ For exampwe, Lavine (p. 23) advocates wong division but does not carry it out; Vretbwad (p.231) cawcuwates de Cauchy product. Euwer's advice is vague; see Euwer et aw., pp. 3, 26. John Baez even suggests a category-deoretic medod invowving muwtipwy pointed sets and de qwantum harmonic osciwwator. Baez, John C. Euwer's Proof That 1 + 2 + 3 + ... = −1/12 (PDF). Archived 2017-10-13 at de Wayback Machine maf.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.
15. ^ Weidwich p. 59
16. ^ Saichev and Woyczyński, pp.260–264.
17. ^ Kwine, p.313.
18. ^ Hardy, p.3; Knopp, p.491.
19. ^ Grattan-Guinness, p.80. See Markushevich, p.48, for a different transwation from de originaw French; de tone remains de same.
20. ^ Ferraro, pp.120–128.
21. ^ Euwer et aw., pp.20–25.
## Footnotes
• Beaws, Richard (2004). Anawysis: An Introduction. Cambridge UP. ISBN 978-0-521-60047-7.
• Davis, Harry F. (May 1989). Fourier Series and Ordogonaw Functions. Dover. ISBN 978-0-486-65973-2.
• Euwer, Leonhard; Wiwwis, Lucas; Oswer, Thomas J. (2006). "Transwation wif notes of Euwer's paper: Remarks on a beautifuw rewation between direct as weww as reciprocaw power series". The Euwer Archive. Retrieved 2007-03-22. Originawwy pubwished as Euwer, Leonhard (1768). "Remarqwes sur un beau rapport entre wes séries des puissances tant directes qwe réciproqwes". Mémoires de w'Académie des Sciences de Berwin. 17: 83–106.
• Ferraro, Giovanni (June 1999). "The First Modern Definition of de Sum of a Divergent Series: An Aspect of de Rise of 20f Century Madematics". Archive for History of Exact Sciences. 54 (2): 101–135. doi:10.1007/s004070050036.
• Grattan-Guinness, Ivor (1970). The devewopment of de foundations of madematicaw anawysis from Euwer to Riemann. MIT Press. ISBN 978-0-262-07034-8.
• Hardy, G. H. (1949). Divergent Series. Cwarendon Press. xvi+396. ISBN 978-0-8218-2649-2. LCCN 49005496. MR 0030620. OCLC 808787. 2nd Ed. pubwished by Chewsea Pub. Co., 1991. LCCN 91-75377. ISBN 0-8284-0334-1.
• Kwine, Morris (November 1983). "Euwer and Infinite Series". Madematics Magazine. 56 (5): 307–314. CiteSeerX 10.1.1.639.6923. doi:10.2307/2690371. JSTOR 2690371.
• Lavine, Shaughan (1994). Understanding de Infinite. Harvard UP. ISBN 978-0-674-92096-5.
• Markusevič, Aweksej Ivanovič (1967). Series: fundamentaw concepts wif historicaw exposition (Engwish transwation of 3rd revised edition (1961) in Russian ed.). Dewhi, India: Hindustan Pub. Corp. p. 176. LCCN sa68017528. OCLC 729238507. Audor awso known as A. I. Markushevich and Awekseï Ivanovitch Markouchevitch. Awso pubwished in Boston, Mass by Heaf wif OCLC 474456247. Additionawwy, OCLC 208730, OCLC 487226828.
• Saichev, A.I. & Woyczyński, W.A. (1996). Distributions in de Physicaw and Engineering Sciences, Vowume 1. Birkhaüser. ISBN 978-0-8176-3924-2.
• Tucciarone, John (January 1973). "The devewopment of de deory of summabwe divergent series from 1880 to 1925". Archive for History of Exact Sciences. 10 (1–2): 1–40. doi:10.1007/BF00343405.
• Vretbwad, Anders (2003). Fourier Anawysis and Its Appwications. Springer. ISBN 978-0-387-00836-3.
• Weidwich, John E. (June 1950). Summabiwity medods for divergent series. Stanford M.S. deses. OCLC 38624384.
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2019-07-20 09:36:58
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http://clay6.com/qa/34269/what-happens-when-electrolysis-of-alpha-aminoacid-is-done-at-isoelectric-po
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Browse Questions
# What happens when electrolysis of alpha aminoacid is done at isoelectric point?
$\begin{array}{1 1}(a)\;\text{At isoelectric point, alpha amino acid migrate towards cathode when electric field is applied}\\(b)\;\text{At isoelectric point, alpha amino acid migrate towards anode when electric field is applied}\\(c)\;\text{At isoelectric point, alpha amino acid do not migrate when electric field is applied}\\(d)\;\text{Electrolysis of alpha aminoacid is impossible}\end{array}$
Can you answer this question?
At isoelectric point, alpha amino acid do not migrate when electric field is applied.
Hence (c) is the correct answer.
answered Mar 26, 2014
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2016-10-27 16:55:12
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https://socratic.org/questions/how-do-you-factor-5a-20#266075
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# How do you factor 5a - 20 ?
May 15, 2016
$5 \left(a - 4\right)$
#### Explanation:
We begin with $5 a - 20$. The first thing I noticed with this problem is that both $5$ and $20$ are divisible by $5$. That means that we can factor out a $5$ from both components, which leaves us with $5 \left(a - 4\right)$.
There, that's it. $5 \left(a - 4\right)$ cannot be factored any further.
May 15, 2016
$5 \left(a - 4\right)$
#### Explanation:
To factor this expression, tell yourself
"I need a number (or variable) that 'go into' both 5a and -20.
Well, 5 is a factor of 5 and -20.
So 5 go into 5a (1a times).
That sounds confusing, but remember, 5 go into 5 one time, but in algebra, you don't normally write the number 1 in coefficient, so it's just 'a'. 5 can't go into 'a' because a is just a single variable.
5 can go into 20, 4 times.
So now our final answer will be 5(a-4).
Hopefully this make sense!
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2021-09-22 01:56:04
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http://math.stackexchange.com/questions/545563/positive-solutions-for-squarefree-diophantine-equation
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# Positive solutions for “squarefree” diophantine equation
I would like to find solutions in positive integers for diophantine equations having no variable squared. (And having some other limitations, but I will not consider them now.) Take, for example, $abcd-3bcd+2abc+ad-3=0$.
It is known that $xy+xz+yz=C$ has positive solutions except for few special values of $C$ and proof of this is quite complicated. So, this is hard question in general. But sometimes an answer might be easy to find or to prove that there is no solutions.
For our example, we see that $abcd > -3bcd+2abc+ad+3$ at least when $a,b,c,d > |2+1+3|=6$ -- i.e. $abcd$ "dominates" value. So we have at most $6^4$ new equations, just let every variable have values from 1 to 6. And for example when $b=2$ we have $2acd-6cd+4ac+ad-3=0$. Now $2acd$ dominates, and we can take recursive approach. Of course we should check that GCD of coefficients divides constant.
In C++ one can have recursive template. Then compiler might be able to optimize code, because every level of recursion would be own function with at least some values for loops being constants. And if we might have overflow, we can first look for solutions mod $2^{32}$ with normal unsigned ints, and only for those possible solutions check if they are real solutions with some arbitrary length interger library.
EDIT: Example happened to be "too easy". Here is another one: $abcd-35bcd-7acd-5abd-3abc+1$. Now I can't see how to factor parts like suggested on first answer.
But is there better approach? Or ready code for this?
-
If you call $e$ the product $bc$, the equation is written as $ade+2ae+ad=3+3de$.
Now study the following cases:
(1). $3\mid{a}$; in this case the equation has no solution.
[Let $a=3k$, hence $k(de+2e+d)=de+1$. But we always have $k(de+2e+d)>de+1$ as $k,e,g\geq1$.]
(2). $a=1$; in this case the equation has no solution.
[In this case $2e+d=2de+3$. Therefore $2de-2e-d+3=0$, hence $2(de-e-d+1)+d+1=0$ and $2(d-1)(e-1)+d+1=0$, impossible.]
(3). $a=2$. There are only two solutions.
[In this case $2(de+2e+d)=3(de+1)$, hence $4e+2d=de+3$. Therefore $d,e$ are odd. Let $d=2x+1$ and $e=2y+1$, $x,y\geq0$, then $3y+x+1=2xy$. Since $y(2x-3)=x+1$, then $2x-3\mid{x+1}$. In particular $2x-3\leq{x+1}$, and $x\leq4$. The solutions are $(x,y)=(4,1),(2,3)$, hence $(d,e)=(9,3),(5,7)$.]
(4). $3\not\mid{a}$ and $a>2$; in this case the equation has no solution.
[In this case $a(de+2e+d)=3(de+1)$, hence $3\mid{de+2e+d}$. And $3\mid{d}$ if, and only if, $3\mid{e}$. If $d,e\not\equiv{0}\pmod{3}$, then the only possibility is $d\equiv{2}\pmod{3}$ and $e\equiv{1}\pmod{3}$.
(4.1). If $d,e\equiv{0}\pmod{3}$, let $d=3x$, $e=3y$, hence $a(de+2e+d)=3(de+1)$; $a(3^2xy+2\times3y+3x)=3(3^2xy+1)$ and $a(3xy+2y+x)=3^2xy+1$, which impossible as $a>3$, $x,y\geq1$.
(4.2). If $d,e\not\equiv{0}\pmod{3}$, let $d\equiv{2}\pmod{3}$ and $e\equiv{1}\pmod{3}$. Let $d=3x+2$, $e=3y+1$, hence $a(de+2e+d)=3(de+1)$; $a((3x+2)(3y+1)+2(3y+1)+3x+2)=3((3x+2)(3y+1)+1)$ and $a(3xy+4y+2x+2)=3(3xy+2y+x+1)$; which is impossible as $a>3$, $x,y\geq1$.]
Therefore the positive integer solutions to the original equation are: $(a,b,c,d)=(2,3,1,9),(2,1,3,9),(2,1,7,5)$ and $(2,7,1,5)$.
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In this answer $ae$ should be $2ae$. But idea is good. So I should always test if I can split equation to left and right side so that at least on side could be factorized. I must think about how to do this in general. – Jori Mäntysalo Oct 31 '13 at 14:38
I made another example where I can't use your approach. – Jori Mäntysalo Nov 4 '13 at 9:06
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2015-10-10 03:52:37
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