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The Čech Complex and the Vietoris-Rips Complex
It’s about time we got back to computational topology. Previously in this series we endured a lightning tour of the fundamental group and homology, then we saw how to compute the homology of a simplicial complex using linear algebra.
What we really want to do is talk about the inherent shape of data. Homology allows us to compute some qualitative features of a given shape, i.e., find and count the number of connected components or a given shape, or the number of “2-dimensional holes” it has. This is great, but data doesn’t come in a form suitable for computing homology. Though they may have originated from some underlying process that follows nice rules, data points are just floating around in space with no obvious connection between them.
Here is a cool example of Thom Yorke, the lead singer of the band Radiohead, whose face was scanned with a laser scanner for their music video “House of Cards.”
Radiohead’s Thom Yorke in the music video for House of Cards (click the image to watch the video).
Given a point cloud such as the one above, our long term goal (we’re just getting started in this post) is to algorithmically discover what the characteristic topological features are in the data. Since homology is pretty coarse, we might detect the fact that the point cloud above looks like a hollow sphere with some holes in it corresponding to nostrils, ears, and the like. The hope is that if the data set isn’t too corrupted by noise, then it’s a good approximation to the underlying space it is sampled from. By computing the topological features of a point cloud we can understand the process that generated it, and Science can proceed.
But it’s not always as simple as Thom Yorke’s face. It turns out the producers of the music video had to actually degrade the data to get what you see above, because their lasers were too precise and didn’t look artistic enough! But you can imagine that if your laser is mounted on a car on a bumpy road, or tracking some object in the sky, or your data comes from acoustic waves traveling through earth, you’re bound to get noise. Or more realistically, if your data comes from thousands of stock market prices then the process generating the data is super mysterious. It changes over time, it may not follow any discernible pattern (though speculators may hope it does), and you can’t hope to visualize the entire dataset in any useful way.
But with persistent homology, so the claim goes, you’d get a good qualitative understanding of the dataset. Your results would be resistant to noise inherent in the data. It also wouldn’t be sensitive to the details of your data cleaning process. And with a dash of ingenuity, you can come up with a reasonable mathematical model of the underlying generative process. You could use that model to design algorithms, make big bucks, discover new drugs, recognize pictures of cats, or whatever tickles your fancy.
But our first problem is to resolve the input data type error. We want to use homology to describe data, but our data is a point cloud and homology operates on simplicial complexes. In this post we’ll see two ways one can do this, and see how they’re related.
The Čech complex
Let’s start with the Čech complex. Given a point set $X$ in some metric space and a number $\varepsilon > 0$, the Čech complex $C_\varepsilon$ is the simplicial complex whose simplices are formed as follows. For each subset $S \subset X$ of points, form a $(\varepsilon/2)$-ball around each point in $S$, and include $S$ as a simplex (of dimension $|S|$) if there is a common point contained in all of the balls in $S$. This structure obviously satisfies the definition of a simplicial complex: any sub-subset $S' \subset S$ of a simplex $S$ will be also be a simplex. Here is an example of the epsilon balls.
An example of a point cloud (left) and a corresponding choice of (epsilon/2)-balls. To get the Cech complex, we add a k-simplex any time we see a subset of k points with common intersection. [Image credit: Robert Ghrist]
Let me superscript the Čech complex to illustrate the pieces. Specifically, we’ll let $C_\varepsilon^{j}$ denote all the simplices of dimension up to $j$. In particular, $C_\varepsilon^1$ is a graph where an edge is placed between $x,y$ if $d(x,y) < \varepsilon$, and $C_{\varepsilon}^2$ places triangles (2-simplices) on triples of points whose balls have a three-way intersection.
A topologist will have a minor protest here: the simplicial complex is supposed to resemble the structure inherent in the underlying points, but how do we know that this abstract simplicial complex (which is really hard to visualize!) resembles the topological space we used to make it? That is, $X$ was sitting in some metric space, and the union of these epsilon-balls forms some topological space $X(\varepsilon)$ that is close in structure to $X$. But is the Čech complex $C_\varepsilon$ close to $X(\varepsilon)$? Do they have the same topological structure? It’s not a trivial theorem to prove, but it turns out to be true.
The Nerve Theorem: The homotopy types of $X(\varepsilon)$ and $C_\varepsilon$ are the same.
We won’t remind the readers about homotopy theory, but suffice it to say that when two topological spaces have the same homotopy type, then homology can’t distinguish them. In other words, if homotopy type is too coarse for a discriminator for our dataset, then persistent homology will fail us for sure.
So this theorem is a good sanity check. If we want to learn about our point cloud, we can pick a $\varepsilon$ and study the topology of the corresponding Čech complex $C_\varepsilon$. The reason this is called the “Nerve Theorem” is because one can generalize it to an arbitrary family of convex sets. Given some family $F$ of convex sets, the nerve is the complex obtained by adding simplices for mutually overlapping subfamilies in the same way. The nerve theorem is actually more general, it says that with sufficient conditions on the family $F$ being “nice,” the resulting Čech complex has the same topological structure as $F$.
The problem is that Čech complexes are tough to compute. To tell whether there are any 10-simplices (without additional knowledge) you have to inspect all subsets of size 10. In general computing the entire complex requires exponential time in the size of $X$, which is extremely inefficient. So we need a different kind of complex, or at least a different representation to compensate.
The Vietoris-Rips complex
The Vietoris-Rips complex is essentially the same as the Čech complex, except instead of adding a $d$-simplex when there is a common point of intersection of all the $(\varepsilon/2)$-balls, we just do so when all the balls have pairwise intersections. We’ll denote the Vietoris-Rips complex with parameter $\varepsilon$ as $VR_{\varepsilon}$.
Here is an example to illustrate: if you give me three points that are the vertices of an equilateral triangle of side length 1, and I draw $(1/2)$-balls around each point, then they will have all three pairwise intersections but no common point of intersection.
Three balls which intersect pairwise, but have no point of triple intersection. With appropriate parameters, the Cech and V-R complexes are different.
So in this example the Vietoris-Rips complex is a graph with a 2-simplex, while the Čech complex is just a graph.
One obvious question is: do we still get the benefits of the nerve theorem with Vietoris-Rips complexes? The answer is no, obviously, because the Vietoris-Rips complex and Čech complex in this triangle example have totally different topology! But everything’s not lost. What we can do instead is compare Vietoris-Rips and Čech complexes with related parameters.
Theorem: For all $\varepsilon > 0$, the following inclusions hold
$\displaystyle C_{\varepsilon} \subset VR_{\varepsilon} \subset C_{2\varepsilon}$
So if the Čech complexes for both $\varepsilon$ and $2\varepsilon$ are good approximations of the underlying data, then so is the Vietoris-Rips complex. In fact, you can make this chain of inclusions slightly tighter, and if you’re interested you can see Theorem 2.5 in this recent paper of de Silva and Ghrist.
Now your first objection should be that computing a Vietoris-Rips complex still requires exponential time, because you have to scan all subsets for the possibility that they form a simplex. It’s true, but one nice thing about the Vietoris-Rips complex is that it can be represented implicitly as a graph. You just include an edge between two points if their corresponding balls overlap. Once we want to compute the actual simplices in the complex we have to scan for cliques in the graph, so that sucks. But it turns out that computing the graph is the first step in other more efficient methods for computing (or approximating) the VR complex.
Let’s go ahead and write a (trivial) program that computes the graph representation of the Vietoris-Rips complex of a given data set.
import numpy
def naiveVR(points, epsilon):
points = [numpy.array(x) for x in points]
vrComplex = [(x,y) for (x,y) in combinations(points, 2) if norm(x - y) < 2*epsilon]
return numpy.array(vrComplex)
Let’s try running it on a modestly large example: the first frame of the Radiohead music video. It’s got about 12,000 points in $\mathbb{R}^4$ (x,y,z,intensity), and sadly it takes about twenty minutes. There are a couple of ways to make it more efficient. One is to use specially-crafted data structures for computing threshold queries (i.e., find all points within $\varepsilon$ of this point). But those are only useful for small thresholds, and we’re interested in sweeping over a range of thresholds. Another is to invoke approximations of the data structure which give rise to “approximate” Vietoris-Rips complexes.
Other stuff
In a future post we’ll implement a method for speeding up the computation of the Vietoris-Rips complex, since this is the primary bottleneck for topological data analysis. But for now the conceptual idea of how Čech complexes and Vietoris-Rips complexes can be used to turn point clouds into simplicial complexes in reasonable ways.
Before we close we should mention that there are other ways to do this. I’ve chosen the algebraic flavor of topological data analysis due to my familiarity with algebra and the work based on this approach. The other approaches have a more geometric flavor, and are based on the Delaunay triangulation, a hallmark of computational geometry algorithms. The two approaches I’ve heard of are called the alpha complex and the flow complex. The downside of these approaches is that, because they are based on the Delaunay triangulation, they have poor scaling in the dimension of the data. Because high dimensional data is crucial, many researchers have been spending their time figuring out how to speed up approximations of the V-R complex. See these slides of Afra Zomorodian for an example.
Until next time!
10 thoughts on “The Čech Complex and the Vietoris-Rips Complex”
1. Matthew Kvalheim
Hi Jeremy,
“For each subset S \subset X of points, form a (\varepsilon/2)-ball around each point in S, and include S as a simplex (of dimension |S|) if there is a common point contained in all of the balls in S. ”
However, shouldn’t this say: “…(of dimension |S|-1)…”? I.e., N points span an N-1 simplex, not an N simplex. Unless I’m missing something?
Liked by 1 person
You may want to also checkout the witness complex of Vin de Silva and Gunnar Carlsson.
While it does not have the theoretical guarantees of the Čech or alpha complexes, it does produce a much smaller simplical complex in the size of the point cloud that may be better for studying its persistent homology.
Like
• That’s the one I was going to implement (eventually) 🙂
Like
3. niranjan
Could you also explain how to construct boundary matrices of various dimensions from simplicial complexes
Like
4. Yes, you’re right, Matthew, that’s a typo. A subset S with N points will determine an (N-1)-simplex (if the given non-empty intersection condition is met). Likewise, at the end of the “Cech complex” section, the statement “To tell whether there are any 10-simplices (without additional knowledge) you have to inspect all subsets of size 10” should be corrected to “you have to inspect all subsets of size 11.”
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5. At the beginning of the “Vietoris-Rips complex” section, for clarity it would be helpful to state more precisely “instead of adding a d-simplex when there is a common point of intersection of d+1 (\varepsilon/2)-balls, we just do so when all the balls have pairwise intersections.”
Like
6. It should be made clear that the balls used to determine the Cech and Vietoris-Rips complexes are closed. That’s what makes this statement true: “if you give me three points that are the vertices of an equilateral triangle of side length 1, and I draw (1/2)-balls around each point, then they will have all three pairwise intersections but no common point of intersection.”
Like
• Surely, but that detail is not crucial to the definition of the complex, just my lazy example. I’m almost certain the official definition is an open ball, since most of the arguments (nerve theorem, etc) are topological and most topological arguments prefer to work with families of open sets.
Like |
# application of minima and maxima
Find the volume of the largest right circular cylinder that can be cut from a circular cone of radius 6cm and height 9cm?
### The problem you posted is
• Mathematics inside the configured delimiters is rendered by MathJax. The default math delimiters are $$...$$ and $...$ for displayed mathematics, and $...$ and $...$ for in-line mathematics. |
# [OS X TeX] LaTeX question regarding \pageref
Gary L. Gray gray at engr.psu.edu
Tue Jul 1 20:23:12 EDT 2008
On Jul 1, 2008, at 5:58 PM, Ross Moore wrote:
> Hi Gary,
>
> On 02/07/2008, at 7:12 AM, Gary L. Gray wrote:
>>
>>> I think the only way would be
>>>
>>> \item
>>> The orbits of the planets are ellipses, with the Sun at one focus
>>> of the ellipse.\label{Kepler1}
>>> \item
>>> The line joining a planet to the Sun sweeps out equal areas in
>>> equal times as the planet travels around the Sun.\label{Kepler2}
>>
>> That sort of fixes it. I moved all three labels and, interestingly,
>> now the second item breaks in the middle of it -- it did not do
>> this before moving the labels. Why must LaTeX do these things? :-)
>
> Make sure that LaTeX sees a \par (e.g., a blank line)
> at the end of each item, *before* the next \item .
>
> Otherwise TeX does not know that the paragraph has actually finished
> yet.
> This could mean that the next \label (when at the start of the item,
> as you had it previously) gets processed before the line-breaking
> for the previous contents have been established.
>
> The \par ensures that the paragraph (i.e., the item content) has
> finished,
> so makes for cleaner processing.
Thanks Ross. That, along with Axel's suggestion to put the \label at
the end of each item rather than at the beginning as I had done,
appears to solve the problem.
-- Gary |
# Yolo Matlab
learn matlab step by step learn matlab step by step best matlab tutorial New tools for building apps, writing scripts, and team-based software development. Installation in Windows¶ The description here was tested on Windows 7 SP1. With increasing technology to improve driving security, surrounding camera is increasingly popular among recent models of family using vehicles. Welcome to my website! I am a graduate student advised by Ali Farhadi. Tutorial 05 Lab 2. In many cases one hidden layer works well, but in order to justify this for a specific problem, you. I maintain the Darknet Neural Network Framework, a primer on tactics in Coq, occasionally work on research, and try to stay off twitter. Settings for objects, attributes, hotkeys, and. Installing darknet on your system If you push the output of the algorithm to a file you can plot it using Matlab and get a result like this: (plot of YOLO on LWIR. If you can't active your software, you don't worry, be continue. This MATLAB function creates a YOLO v2 object detection network and returns it as a LayerGraph object. Mar 07, 2018 · With the arrival of Windows 10’s Bash shell, you can now create and run Bash shell scripts on Windows 10. 참고로 우리는 TK1에 설치 했고, TK1은 Ubuntu 14. This MATLAB function returns an object detector trained using you only look once version 2 (YOLO v2) network architecture specified by the input lgraph. Written in MATLAB, of course, the team used YOLO v2 as an object detector trained to identify over 20 various objects, including bicycles, cars, and motorcycles, along with a few other categories including dogs, horses and sheep! Visitors at the museum can hold up objects to be identified, or even use their own images from their mobile device. Alternatively, if you have a list of layers to remove, you can use the removeLayers function to remove them manually. After labeling the connecting components, the region will be removing from the input image. There are many commercial applications that use YOLO and other simpler versions of YOLO as backend. This example generates code for the network trained in the Object Detection Using YOLO v2 example from Computer Vision Toolbox™. This spatial constraint limits the number of nearby objects that our model can predict. I started my term project using YOLO. How should I prepare the input images?. It is fast, easy to install, and supports CPU and GPU computation. 图1-1 YOLO目标检测系统. The anchor boxes are defined when creating the YOLO v2 network by using the yolov2Layers function. Heimery Boston College Kristian Ove R. Many pre-trained CNNs for image classification, segmentation, face recognition, and text detection are available. This was a great release for examples, and I guarantee there is something for everyone in this. YOLOv3 is the latest variant of a popular object detection algorithm YOLO - You Only Look Once. Retraining YOLO v2 object detecting network for Learn more about deep learning, image object detection, yolo v2 object detection. O método YOLO, por sua vez, transforma o problema em uma regressão, executada uma única vez sobre a imagem. It is simple, efficient, and can run and learn state-of-the-art CNNs. In this post, I shall explain object detection and various algorithms like Faster R-CNN, YOLO, SSD. Release 2019a offers hundreds of new and updated features and functions in MATLAB® and Simulink®, along with ten new. yolomex is a simple Matlab MEX wrapper for object recognition with YOLO (based on pyyolo by thomaspark-pkj). Demo for softmax regression. the big picture, I would like to use YOLOv2 Pretrained network to identify cars (and in the future object) with Kinect v2 stream of data and mix the information given by the box of YOLO with the information given by the point cloud from the Kinect. You can find the source on GitHub or you can read more about what Darknet can do right here:. 1 This is done by running the object detector on. Your approach is fine. I like how the bounding box angle is also regressed. What should I do to transfer the model to Jetson Nano and infere. Darknet is an open source neural network framework written in C and CUDA. the code is available on github under mit license and i warmly welcome pull requests for new features / layers / demos and miscellaneous improvements. Run and Test Algorithm in MATLAB. TensorRT-based applications perform up to 40x faster than CPU-only platforms during inference. Der generierte Code ruft optimierte NVIDIA-CUDA-Bibliotheken auf, lässt sich in Form von Quellcode und statischen oder dynamischen Bibliotheken in Ihr Projekt einbinden und kann zur Prototypenentwicklung auf GPUs wie NVIDIA Tesla und NVIDIA Tegra genutzt werden. Although YOLO performs very fast, close to 45 fps (150 fps for small YOLO), it has lower accuracy and detection rate than faster-RCNN. /darknet detector test data/obj. And yes, it is the same as the stride of a convolution. YOLO makes less than half the number of background errors compared to Fast R-CNN. YOLO chooses the anchors by running k-means clustering on all the bounding boxes from all the training images (with k = 5 so it finds the five most common object shapes). Build analytics for video using TensorFlow, Keras, and YOLO. Reasons: 1. h directly into the cuda folder with the following path (no new subfolders are necessary. Jun 20, 2018 · Using a one-stage approach, models such as You Only Look Once (YOLO) and Single Shot MultiBox Detector (SSD), or RetinaNet, consider a fixed set of boxes for detection and skip the region proposal stage. YOLO v2 is faster than other two-stage deep learning object detectors, such as regions with convolutional neural networks (Faster R-CNNs). Abstract: State-of-the-art object detection networks depend on region proposal algorithms to hypothesize object locations. Even if you know what you’re doing, this isn’t necessarily as simple as it seems. Nov 12, 2017. I work on computer vision. designing and developing CRM software. Caltech Pedestrian Detection Benchmark Description The Caltech Pedestrian Dataset consists of approximately 10 hours of 640x480 30Hz video taken from a vehicle driving through regular traffic in an urban environment. GPU Coder vous permet d'incorporer du code CUDA existant dans vos algorithmes MATLAB et dans le code généré. This is Part 2 of the tutorial on implementing a YOLO v3 detector from scratch. Darknet is an open source neural network framework written in C and CUDA. 3 source and binary code. Computer vision uses images and video to detect, classify, and track objects or events in order to understand a real-world scene. We incorporate occlusion reasoning with object detec-tion by: (1) a bottom-up stage which hypothesizes the like-. Fully Convolutional Networks for Semantic Segmentation Jonathan Long Evan Shelhamer Trevor Darrell UC Berkeley fjonlong,shelhamer,[email protected] I'll go into some different. YOLO makes less than half the number of background errors compared to Fast R-CNN. This value specifies the height and width of M anchor boxes. First, we'll install the Movidius SDK and then learn how to use the SDK to generate the Movidius graph files. Today's blog post is broken into five parts. Decision tree implemented in matlab code. MATLAB (matrix laboratory)is a multi-paradigm numerical computing language. 本記事はChainer Advent Calendar 2016の20日目のエントリです。 ペンパイナッポー（以下PP)とアッポーペン（以下AP)の画像識別方式の確立は急務です。下記は"PPAP"のここ三ヶ月のGoogle Trend推移です. The example shown imports YOLO from Caffe, then proceeds to convert it to CUDA implementation. Feb 23, 2018 · This is a specialty in the Yolo V2 algorithm compared to the others. The YOLO architecture in simple terms consists of an $S×S$ grid cells of classifiers and regressors. Traceback (most recent call last): File "D:\Python_Object_analyzis\YOLO Version\darkflow-master\Person_detection. The pre-training network also uses resnet50. Published in: 2018 10th International Conference on Communications, Circuits and Systems (ICCCAS) In this project Moving object detection is done at real time using Computer vision on FPGA, with the help of Jupyter notebook compatibility in PYNQ Z2 FPGA board by Xilinx. 1 libraries. They apply the model to an image at multiple locations and scales. Walk through a real-time object detection example using YOLO v2 in MATLAB. Help and Feedback You did not find what you were looking for? Ask a question on the Q&A forum. Other approaches for recognizing and/or tracking cars. Last post, we covered a bunch of new 19a examples, and today's post will dive into the specifics of new deep learning features. Like the COCO dataset. Here are two DEMOS of YOLO trained with customized classes: Yield Sign:. YOLO将输入图像分成SxS个格子，若某个物体 Ground truth 的中心位置的坐标落入到某个格子，那么这个格子就负责检测出这个物体。 每个格子预测B个bounding box及其置信度(confidence score)，以及C个类别概率。. Now that you have understood the basic workflow of Object Detection, let's move ahead in Object Detection Tutorial and understand what Tensorflow is and what are its components? What is TensorFlow?. Third, YOLO learns generalizable representations of ob-jects. YOLO divides up the image into a grid of 13 by 13 cells: Each of these cells is responsible for predicting 5 bounding boxes. Is batch_size equals to number of test samples? From Wikipedia we have this information:. Deep learning and convolutional networks, semantic image segmentation, object detection, recognition, ground truth labeling, bag of features, template matching, and background estimation. This paper uses 5x5 convolution filters to regress the bounding boxes instead of the YOLO final fc layer, and applies the network on an image pyramid to deal with different scales (in contrast to the fully connected layer in YOLO that can learn different scales, or regressing. When trained on natural images and tested on artwork, YOLO outperforms top detection methods like DPM and R-CNN by a wide margin. Apr 30, 2018 · YOLOv2 for Matlab is a fork of the Matlab MEX wrapper Yolomex by Ignacio Rocco with Matlab demos for LED control and people tracking. 그리고 yolo와 darknet을 만든 joseph redmon이란 사람도 멋있다는. Nov 12, 2017. I'll go into some different. Hi Everyone! Welcome to R2019a. YOLO: Real-Time Object Detection. Many pre-trained CNNs for image classification, segmentation, face recognition, and text detection are available. Due to the fact that I have been interested in TensorFlow. Object detection is the problem of finding and classifying a variable number of objects on an image. Customize the label dialog to combine with attributes. I categorized the new examples based on their application area. this is my source #include <windows. Object Detection Using YOLO v2 Deep Learning. In this video, you'll learn how to build AI into any device using TensorFlow Lite, and learn about the future of on-device ML and our roadmap. MATLAB Central contributions by YoloSwaggins. Assignment 2 due date: SVM. object detection with yolo for intelligent enterprise. Darknet is an open source neural network framework written in C and CUDA. See an example of a real-time object detection algorithm using a deep learning neural network based on YOLO architecture. Courtesy Science Museum Group The Code Written in MATLAB, of course, the team used YOLO v2 as an object detector trained to identify over 20 various objects , including bicycles, cars, and motorcycles, along with a few other categories including dogs, horses and sheep!. A caffe implementation of MobileNet-YOLO detection network , train on 07+12 , test on VOC2007. Train a you only look once (YOLO) v2 object detector. In these pages you will find. Running the demo: To run the code on one of the pre-computed datasets, start MATLAB and ensure that a working version of SPAMS is in your MATLAB path. Search for jobs related to Matlab source code moving object detection algorithm or hire on the world's largest freelancing marketplace with 15m+ jobs. Nov 16, 2017 · The steps below assume we want to use tiny YOLO and our dataset has 3 classes. Creating the yolo network in MATLAB 2. e its hard coded, so if your face slightly dif. It is important to scale the images, because some of them can be to bright or too dark, distorting the classifier. You Only Look Once - this object detection algorithm is currently the state of the art, outperforming R-CNN and it's variants. It is the single most important technique in Digital Signal Processing. A Brief Overview of the Different R-CNN Algorithms for Object Detection. \$ unzip -q matlab_r2018a_glnxa64. Third, YOLO learns generalizable representations of ob-jects. yolo v2中的anchor. First consider the fully connected layer as a black box with the following properties: On the forward propagation. I've trained a YOLO model in matlab I want to deploy it in my Nano. Once you have the training data, you can use any of the object detection techniques like Faster RCNN, YOLO, SSD to train your model and get predictions on new images. This paper uses 5x5 convolution filters to regress the bounding boxes instead of the YOLO final fc layer, and applies the network on an image pyramid to deal with different scales (in contrast to the fully connected layer in YOLO that can learn different scales, or regressing. real-time image processing downloads overview details news bugtracker this page contains files uploaded to the old opencores. Thousands of people attend every year at GTCs worldwide. The code used to computing the features has been released in the development kit of the ImageNet Large Scale Visual Recognition Challenge (ILSVRC2010. This is Part 3 of the tutorial on implementing a YOLO v3 detector from scratch. When trained on natural images and tested on art-work, YOLO outperforms top detection methods like DPM and R-CNN by a wide margin. • Technical Tools: YOLO, Darknet, MATLAB • Keywords: Guidance Navigation and Control (GNC), Elettro-optical (EO) sensors, Deep Learning, Machine Vision, Detection and Tracking algorithms, Kalman filter. yolo-small. It processes images at 45 fps, here are weight files for yolo. Download now YOLO v2 Object Detector: Train a "You Only Look Once" (YOLO) v2 deep. What's missing is a YOLO model that's trained only on faces. Use the interactive communication to prototype and develop your MATLAB algorithm, then automatically generate equivalent C code and deploy it to the drive platform to run as a standalone. Because Yolo is much faster than RCNN and current our team got the laptop, it would be promising if we can use Yolo to do the detection. Yolo has become very popular and important as it is considered the state-of-the-art technique since it uses a single network and is very fast for real-time object detection. In contrast with problems like classification, the output of object detection is variable in length, since the number of objects detected may change from image to image. A Brief Overview of the Different R-CNN Algorithms for Object Detection. Run the command by entering it in the MATLAB Command Window. used to represent occlusions for many objects in the scene. Last post, we covered a bunch of new 19a examples, and today's post will dive into the specifics of new deep learning features. 2018-03-27 update: 1. lib Build type Call CUDA from MATLAB directly Call CUDA from (C++) hand-coded main() Call CUDA from (C++) hand-coded main(). The anchor boxes are defined when creating the YOLO v2 network by using the yolov2Layers function. You only look once (YOLO) is a state-of-the-art, real-time object detection system. Release 2019a offers hundreds of new and updated features and functions in MATLAB® and Simulink®, along with ten new. This is Part 5 of the tutorial on implementing a YOLO v3 detector from scratch. Traceback (most recent call last): File "D:\Python_Object_analyzis\YOLO Version\darkflow-master\Person_detection. To detect objects in an image, pass the trained YOLO v2 object detector to the detect object function. In the last part, we implemented a function to transform the output of the network into detection predictions. learn matlab step by step learn matlab step by step best matlab tutorial New tools for building apps, writing scripts, and team-based software development. Alternatively, if you have a list of layers to remove, you can use the removeLayers function to remove them manually. For each bounding box, the Yolo network predicts its central location within the square, the width, height of box wrt the image width, height and the confidence score of having any object in that box along along with the probabilities of belong to each of the M classes. The published model recognizes 80 different objects in images and videos, but most importantly it is super fast and nearly as accurate as Single Shot MultiBox (SSD). 04 but should work with other distros as well. The development of a license plate recognition algorithm came as part of a larger project: an automobile “smart camera” app for cars that not only records what's happening on the road ahead but notifies drivers about important roadway events such as the presence of construction warning signs, stationary objects in the road, etc. edu Abstract Convolutional networks are powerful visual models that yield hierarchies of features. Even if you know what you’re doing, this isn’t necessarily as simple as it seems. Real-Time Food Detection. I'm using Python Keras package for neural network. Check out his YOLO v3 real time detection video here. Creating the yolo network in MATLAB 2. To do the training of the classifier, the data was scaled using SkLearn RobustScaler. weights file in the results section to see how our model currently performs. MATLAB 论坛; Simulink 论坛; 文章 Portal. Darknet: Open Source Neural Networks in C. YOLO divides up the image into a grid of 13 by 13 cells: Each of these cells is responsible for predicting 5 bounding boxes. 23 Model Exchange with MATLAB. Customize the label dialog to combine with attributes. The layer outputs the refined bounding box locations that are predicted using a predefined set of anchor boxes specified at the input. Open Mobile Search. Girshick）大神，不仅学术牛，工程也牛，代码健壮，文档详细，clone下来就能跑。 断断续续接触detection几个月，将自己所知做个大致梳理，业余级新手，理解不对的地方还请指正。. The example shown imports YOLO from Caffe, then proceeds to convert it to CUDA implementation. The you-only-look-once (YOLO) v2 object detector uses a single stage object detection network. Aug 12, 2019 · Design of Moving Object Detection System Based on FPGA – FPGA. - The proposed 3D YOLO model. An example of an image used in the classification challenge. An example of 5 boxes is shown for a square positioned at (7, 9) from top left. First, we'll install the Movidius SDK and then learn how to use the SDK to generate the Movidius graph files. Ofrece soporte para la mayor parte del lenguaje MATLAB y una amplia gama de toolboxes. Hi Everyone! Welcome to R2019a. Oct 21, 2011 · Multiple Object Tracking, or MOT, is an experimental technique used to study how our visual system tracks multiple moving objects. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy. A YOLO v2 object detection network is composed of two subnetworks. After installed matlab, open terminal and enter the command:. A blog for beginners. The outcome of Yolo Downview camera: After far range camera detect and estimate the angle, our robot base will turn to that angle and move towards the targeted objects. The 14 layers of the recognition network. SimpleCV is an open source framework for building computer vision applications. Apr 28, 2017 · Computer vision uses images and video to detect, classify, and track objects or events in order to understand a real-world scene. There are many face detection algorithms to locate a human face in a scene – easier and harder ones. Jun 07, 2019 · Walk through an example of real-time object detection using YOLO v2 in MATLAB®. 1 libraries. Because you can use the computer vision library on both computers and mobile devices, below are two lists of tutorials. real-time image processing downloads overview details news bugtracker this page contains files uploaded to the old opencores. The you-only-look-once (YOLO) v2 object detector uses a single stage object detection network. YOLO makes less than half the number of background errors compared to Fast R-CNN. 与特征提取网络相比，检测子网络是小型 CNN，它由几个卷积层和特定于 YOLO v2 的层组成。 使用 yolov2Layers 函数自动将预训练后的 ResNet-50 网络修改为 YOLO v2 对象检测网络。yolov2Layers 要求您指定几个输入以用于参数化 YOLO v2 网络。 首先，指定图像输入大小和类的. If you specify the name-value pair 'ReorgLayerSource', the YOLO v2 network concatenates the output of reorganization layer with the output of feature layer. Jul 27, 2018 · YOLO is an ultra popular object detection framework for deep learning applications. Object Detection Using YOLO v2 Deep Learning. Jul 10, 2017. Today's blog post is broken into five parts. tensorflow tutorial for beginners (article) - datacamp. Learn computer vision, machine learning, and image processing with OpenCV, CUDA, Caffe examples and tutorials written in C++ and Python. e its hard coded, so if your face slightly dif. The published model recognizes 80 different objects in images and videos, but most importantly it is super fast and nearly as accurate as Single Shot MultiBox (SSD). You've already seen most of the components of object detection. Convolution is a mathematical way of combining two signals to form a third signal. This is done because fully connected layer always expected the same input size. In contrast with problems like classification, the output of object detection is variable in length, since the number of objects detected may change from image to image. By applying object detection, you'll not only be able to determine what is in an image, but also where a given object resides! We'll. Even if you are not familiar, you can get started with Yolo v2 with this published example in MATLAB ® that explains how you can train a Yolov2 object detector on your data. Welcome again to R2019a! There’s a new release of MATLAB out right now. this function accepts to arguments, the first being our set of bounding boxes in the form of (startX, startY, endX, endY) and the second being our overlap threshold. You can find the source on GitHub or you can read more about what Darknet can do right here:. The YOLO v2 object detector recognizes specific objects in images, based on the training images and ground truth data used with the trainYOLOv2ObjectDetector function. Viewed 404k times 108. With abundant information collected by these cameras, there are few existing practices that automatically analyze and understand the content of the recording. The development of a license plate recognition algorithm came as part of a larger project: an automobile “smart camera” app for cars that not only records what's happening on the road ahead but notifies drivers about important roadway events such as the presence of construction warning signs, stationary objects in the road, etc. How to train YOLOv2 to detect custom objects but for now will use the yolo-obj_1000. 16 hours ago · download online multi object tracking with convolutional neural networks code free and unlimited. This property is set by the AnchorBoxes property of the output layer in the YOLO v2 network. Walk through an example of real-time object detection using YOLO v2 in MATLAB ®. Create a copy of the configuration file tiny-yolo-voc. Please use a supported browser. This is pretty normal terminology for computer programming. The feature extraction network is typically a pretrained CNN (for detials, see Pretrained Deep Neural Networks). 5/26/16 2 PASCAL VOC detection history PASCAL VOC detection history mAP: Mean Average Precision. MATLAB 论坛; Simulink 论坛; 文章 Portal. The YOLO v2 object detector recognizes specific objects in images, based on the training images and ground truth data used with the trainYOLOv2ObjectDetector function. 4) matlab code for drowsy driver detection. ENGINEERS AND DEVICES WORKING TOGETHER Agenda Deep learning basics Platform overview Gaps and challenges 3. Recent advances in detection algorithms which avoids the typical anchor box adjustment problems. 08/30/2017; 16 minutes to read +2; In this article. It was developed in 1988 [1] in order to test (and illustrate) a theoretical proposed mechanism called a Visual Index or FINST (for FINgers of INSTantiation). Hi Everyone! Welcome to R2019a. Last post, we covered a bunch of new 19a examples, and today's post will dive into the specifics of new deep learning features. Object Detection Using YOLO v2 Deep Learning. Walk through an example of real-time object detection using YOLO v2 in MATLAB ®. implementing deep learning using cudnn - nvidia. Mar 26, 2017 · YOLO — ‘You only look once’ for Object Detection explained. Alternatively, if you create MATLAB のコマンドを実行. I started my term project using YOLO. While there are a lot of implementation of YOLO using a plethora of framework. Using cnncodegen function, you can generate CUDA code and integrate it into a bigger application. Darknet: Open Source Neural Networks in C. This book guides you through the field of deep learning starting with neural networks, taking a deep dive into convolutional neural networks, recurrent neural networks, and long short-term memory (LSTM) networks. What should I do to transfer the model to Jetson Nano and infere. designing and developing CRM software. About James Bradbury James Bradbury is a research scientist at Salesforce Research, where he works on cutting-edge deep learning models for natural language processing. YOLOv3 is the latest variant of a popular object detection algorithm YOLO - You Only Look Once. When trained on natural images and tested on artwork, YOLO outperforms top detection methods like DPM and R-CNN by a wide margin. 利用yolo实现简单的车辆识别，从随机初始化的权重训练一个YOLO模型是非常重要的，需要大量的数据集和大量的计算资源，所以我们在这个练习中使用了预训练的模型参数。你也可以尝试用自己的数据集对YOLO模. You Only Look Once: Unified, Real-Time Object Detection Joseph Redmon University of Washington [email protected] Santosh Divvala Allen Institute for Artificial Intelligence [email protected] Ross Girshick Facebook AI Research [email protected] Ali Farhadi University of Washington [email protected] Abstract We present YOLO, a new approach to object detection. zip file should now be located within the ~/downloads/matlab. In the 'spams-matlab/' folder is SPAMS version 2. 图1-1 YOLO目标检测系统. The word field is the index of the cluster center, i. 9% on COCO test-dev. zip -d matlab the content of the matlab_r20xxa_glnxa64. Please note: this is primarily for deep learning features, and this list isn’t comprehensive. We shall start from beginners' level and go till the state-of-the-art in object detection, understanding the intuition, approach and salient features of each method. CUDA Demo Suite. The development of a license plate recognition algorithm came as part of a larger project: an automobile “smart camera” app for cars that not only records what's happening on the road ahead but notifies drivers about important roadway events such as the presence of construction warning signs, stationary objects in the road, etc. Automatically label images using Core ML model. I'll go into some different. We use boundingbox to measure the properties of the image region. I'll start with pre-trained CaffeNet and train it for my application. The YOLO model should now be ready to be trained with lots of images and lots of labeled outputs. Darknet is an open source neural network framework written in C and CUDA. I have been working extensively on deep-learning based object detection techniques in the past few weeks. There are many face detection algorithms to locate a human face in a scene – easier and harder ones. it is the. 导语：能高速检测9418个类别的YOLO 9000 雷锋网(公众号：雷锋网) AI 科技评论按：YOLO是Joseph Redmon和Ali Farhadi等人于2015年提出的第一个基于单个神经网络. want to improve this question? update the question so it focuses on one problem only by editing this post. However, YOLO is actually structured as a CNN regression algorithm. sentdex 471,310 views. data cfg/yolo-obj. Then go to the root of our distribution and run: >> L1_PeopleReid_Demo. YOLO is open source. About Shashank Prasanna Shashank Prasanna is a product marketing manager at NVIDIA where he focuses on deep learning products and applications. This is Part 5 of the tutorial on implementing a YOLO v3 detector from scratch. Alternatively, if you have a list of layers to remove, you can use the removeLayers function to remove them manually. We use the You Only Look Once (YOLO) to detect the…. Nesse webinar, veremos como criar um detector de objetos através do método YOLO utilizando o MATLAB. We’ll do our best to help you out. cfg has smaller fully connected layers so it uses far less memory. The word field is the index of the cluster center, i. It is important to scale the images, because some of them can be to bright or too dark, distorting the classifier. If you have any errors, try to fix them? If everything seems to have compiled correctly, try running it! You already have the config file for YOLO in the cfg/ subdirectory. data" which contains parameters needed for training as described in the next table. This is Part 5 of the tutorial on implementing a YOLO v3 detector from scratch. In this tutorial, you'll learn how to use the YOLO object detector to detect objects in both images and video streams using Deep Learning, OpenCV, and Python. 少し厨二病らしさを感じさせるDarknetのYoloですが、ここ最近、進化が止まらないですね。気がつけばバージョンアップを繰り返しv3にまで。 さて、このYolo v3が如何ほどの性能なのか試したいので、自前のデータを使って学習. Walk through an example of real-time object detection using YOLO v2 in MATLAB ®. used to represent occlusions for many objects in the scene. GTC San Jose boasts around 9000 attendants. 04 but should work with other distros as well. Object detection example. Lee Giles, Senior Member, IEEE, Ah Chung Tsoi, Senior Member, IEEE, and Andrew D. I have two problems and answering just one could help me a ton: After two weeks of failed attempts trying to get YOLO to work in Matlab. FacebookTwitter What you’ll learn Learn by completing 26 advanced computer vision projects including Emotion, Age & Gender Classification, London Underground Sign Detection, Monkey Breed, Flowers, Fruits , Simpsons Characters and many more!. Here is a list of the most common techniques in face detection: (you really should read to the end, else you will miss the most important developments!). OpenCV will only detect faces in one orientation, i. Aug 21, 2017 · Third, YOLO learns generalizable representations of objects. The practical part of the project, consists of the implementation of the selected techniques and the simulation using MATLAB and Simulink. Showing object detection with MATLAB. yolo v3 model is much more complex than yolo v2, and its detection on small objects, as well as compact dense or high ly overlapping ob jects is very e x- cellent. Add chainer v2 codeWriting your CNN modelThis is example of small Convolutional Neural Network definition, CNNSmall I also made a slightly bigger CNN, called CNNMedium, It is nice to know the computational cost for Convolution layer, which is approximated as,$$H_I \times W_I \times CH_I \times CH_O \times k ^ 2$$\. A feature extraction network followed by a detection network. He received a PhD in computer science from the University of Chicago under the supervision of Pedro Felzenszwalb in 2012. Prior to joining NVIDIA, Shashank worked for MathWorks, makers of MATLAB, focusing on machine learning and data analytics, and for Oracle Corp. popular YOLO detector: YOLOv3 [14], YOLOv2 [15], and TinyYOLO, which is a simpler and faster (though less acurate) version of YOLOv2. You can find the source on GitHub or you can read more about what Darknet can do right here:. 「MATLAB/Simulinkが得意としてきた制御系や信号処理系の技術にディープラーニングを組み合わせることでさまざまな価値が生まれる」（MathWorks. Our base YOLO model processes images in real-time at 45 frames per second. Jun 06, 2018 · A2A. Der generierte Code ruft optimierte NVIDIA-CUDA-Bibliotheken auf, lässt sich in Form von Quellcode und statischen oder dynamischen Bibliotheken in Ihr Projekt einbinden und kann zur Prototypenentwicklung auf GPUs wie NVIDIA Tesla und NVIDIA Tegra genutzt werden. How to find out from the logs what caused system shutdown? Ask Question Asked 8 years, 8 months ago. 刚买两块Titan Z GPU准备搞搞深度学习，结果原来的工作站功率不够，带不动，所以准备组装一台新工作站。求大神们给点意见，最好给个完整的list，我好照着买，谢谢。. Then go to the root of our distribution and run: >> L1_PeopleReid_Demo. Nov 12, 2017. First, let's see how you construct your training set. It is fast, easy to install, and supports CPU and GPU computation. Your approach is fine. You Only Look Once - this object detection algorithm is currently the state of the art, outperforming R-CNN and it's variants. |
Definitions
Ice Ih
Ice Ih is the hexagonal crystal form of ordinary ice, or frozen water. Virtually all ice in the biosphere is ice Ih, with the exception only of a small amount of ice Ic which is occasionally present in the upper atmosphere. Ice Ih exhibits many peculiar properties which are relevant to the existence of life and regulation of global climate.
Ice Ih is stable down to and can exist at pressures up to 0.2 GPa. The crystal structure is characterized by hexagonal symmetry and near tetrahedral bonding angles.
Physical properties
Ice Ih has a density less than liquid water, of 0.917 g/cm³, due to the extremely low density of its crystal lattice. The density of ice Ih increases with decreasing temperature (density of ice at -180 °C is 0.9340 g/cm³).
The latent heat of melting is 5987 J/mol, and its latent heat of sublimation is 50911 J/mol. The high latent heat of sublimation is principally indicative of the strength of the hydrogen bonds in the crystal lattice. The latent heat of melting is much smaller partly because water near 0 °C is very strongly H-bonded already.
The refractive index of ice Ih is 1.31.
Crystal structure
The accepted crystal structure of ordinary ice was first proposed by Linus Pauling in 1935. The structure of ice Ih is roughly one of crinkled planes composed of tessellating hexagonal rings, with an oxygen atom on each vertex, and the edges of the rings formed by hydrogen bonds. The planes alternate in an ABAB pattern, with B planes being reflections of the A planes along the same axes as the planes themselves. The distance between oxygen atoms along each bond is about 275 pm and is the same between any two bonded oxygen atoms in the lattice. The angle between bonds in the crystal lattice is very close to the tetrahedral angle of 109.5° which is also quite close to the angle between hydrogen atoms in the water molecule (in the gas phase), which is 105°. This tetrahedral bonding angle of the water molecule essentially accounts for the unusually low density of the crystal lattice -- it is beneficial for the lattice to be arranged with tetrahedral angles even though there is an energy penalty in the increased volume of the crystal lattice. As a result, the large hexagonal rings leave almost enough room for another water molecule to exist inside. This gives naturally occurring ice its unique property of being less dense than its liquid form. The tetrahedral-angled hydrogen-bonded hexagonal rings are also the mechanism which causes liquid water to be most dense at 4 °C. Close to 0 °C, tiny hexagonal ice Ih-like lattices form in liquid water, with greater frequency closer to 0 °C. This effect decreases the density of the water, causing it to be most dense at 4 °C when the structures form infrequently.
Proton disorder
The protons (hydrogen atoms) in the crystal lattice lie very nearly along the hydrogen bonds, and in such a way that each water molecule is preserved. This means that each oxygen atom in the lattice has two protons adjacent to it, and about 101 pm along the 275 pm length of the bond. The crystal lattice allows a substantial amount of disorder in the positions of the protons frozen into the structure as it cools to absolute zero. As a result, the crystal structure contains some residual entropy inherent to the lattice and determined by the number of possible configurations of proton positions which can be formed while still maintaining the requirement for each oxygen atom to have only two protons in closest proximity, and each H-bond joining two oxygen atoms having only one proton. This residual entropy S0 is equal to 3.5 J mol−1 K−1. There are various ways of approximating this number from first principles. Assuming a given N water molecules each has 6 possible arrangements this yields 6N possible combinations. Given random orientations of molecules, a given bond will have only a ½ chance that it has exactly one proton, or in other words, each molecule has a ¼ chance that its protons lie on bonds containing exactly one proton, leaving a total number of $\left(3/2\right)^N$ possible valid combinations. Using Boltzmann's principle, we find that $S_0 = Nk ln\left(3/2\right)$, where $k$ is Boltzmann's Constant, which yields a value of 3.37 J mol−1 K−1, a value very close to the measured value. More complex methods can be employed to better approximate the exact number of possible configurations, and achieve results closer to measured values.
By contrast, the structure of ice II is very proton-ordered, which helps to explain the entropy change of 3.22 J/mol when the crystal structure changes to that of ice II. Also, ice XI, an orthorhombic, proton-ordered form of ice Ih, is considered the most stable form.
References
• N. H. Fletcher, The Chemical Physics of Ice, Cambridge UP (1970) ISBN 0-521-07597-1
• Victor F. Petrenko and Robert W. Whitworth, Physics of Ice, Oxford UP (1999) ISBN 0-19-851894-3
• Chaplin, Martin Hexagonal ice structure. Water Structure and Science. Retrieved on 2008-01-02..
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# Opus Heat
This test case has been proposed by Annabelle Le-Hyaric and Michel Fouquembergh formerly at AIRBUS.
We consider a 2D model representative of the neighboring of an electronic component submitted to a cooling air flow. It is described by four geometrical domains in $\mathbb{R}^2$ named $\Omega_i,i=1,2,3,4$, see figure. We suppose the velocity $\vec{v}$ is known in each domain --- for instance in $\Omega_4$ it is the solution of previous Navier-Stokes computations. --- The temperature $T$ of the domain $\Omega = \cup_{i=1}^4 \Omega_i$ is then solution of heat transfer equation :
$$$\label{eq:1} \rho C_i \Big( \frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \Big) - \nabla \cdot \left( k_i \nabla T \right) = Q_i,\quad i=1,2,3,4$$$
where $t$ is the time and in each sub-domain $\Omega_i$, $\rho C_i$ is the volumic thermal capacity, $k_i$ is the thermal conductivity. $k_1$ and $k_2$ are parameters of the model.
ICs dissipate heat, so the volumic heat dissipated $Q_1$ and $Q_2$ are also parameters of the model, while $Q_3=Q_4=0$.
One should notice that the convection term in heat transfer equation may lead to spatial oscillations which can be overcome by Petrov-Galerkin type or continuous interior penalty stabilization techniques.
Integrated circuits (ICs) (domains $\Omega_1$ and $\Omega_2$ ) are respectively soldered on PCB at $\mathbf{x1}=(e_{Pcb}, h_1)$ and $\mathbf{x_2}=(e_{Pcb}, h_2)$. They are considered as rectangles with width $e_{IC}$ and height $h_{IC}$. The printed circuit board (PCB) is a rectangle $\Omega_3$ of width $e_{PCB}$ and height $h_{PCB}$. The air(Air) is flowing along the PCB in domain $\Omega_4$. Speed in the air channel $\Omega_4$ is supposed to have a parabolic profile function of $x$ coordinate. Its expression is simplified as follows (notice that $\vec{v}$ is normally solution of Navier-Stokes equations; the resulting temperature and velocity fields should be quite different from that simplified model), we have for all $0 \leq y \leq h_{PCB}$
$$$\label{eq:2} \begin{array}[c]{rl} e_{Pcb} + e_{Ic} \leq x \leq e_{Pcb} + e_a, & \displaystyle \vec{v} = -V(x-(e_{Pcb}+e_{Ic}))(x-(e_{Pcb}+e_a))\vec{y}\\ e_{Pcb} \leq x \leq e_{Pcb} + e_{Ic}, & \vec{v} = 0 \end{array}$$$
where V is a parameter of the model.
The medium velocity $\vec{v}_i = \vec{0}, i=1,2,3$ in the solid domains $\Omega_i, i=1,2,3$.
## 1. Boundary conditions
We set
• on $\Gamma_1\cup\Gamma_3$, a zero flux (Neumann-like) condition
$$$\label{eq:10} -k_i\ \nabla T \cdot \vec{n}_i\ =\ 0;$$$
• on $\Gamma_2$, a heat transfer (Robin-like) condition
$$\label{eq:7} -k_4\ \nabla T \cdot \vec{n}_4\ =\ h(T-T_0);$$
where $h$ is a parameter of the model
• on $\Gamma_4$ the temperature is set (Dirichlet condition)
$$$\label{eq:11} T\ = T_0;$$$
• on other internal boundaries, the coontinuity of the heat flux and temperature, on $\Gamma_{ij} = \Omega_i \cap \Omega_j \neq \emptyset$
$$$\label{eq:6} \begin{array}{rl} T_i &= T_j \\ k_i\ \nabla T \cdot \vec{n}_i &= -k_j\ \nabla T \cdot \vec{n}_j. \end{array}$$$
## 2. Initial condition
At $t=0s$, we set $T = T_0$.
## 3. Outputs
The output is the mean temperature $s_1(\mu)$ of the hottest IC
$$$\label{eq:3} s_1(\mu) = \frac{1}{e_{IC} h_{IC}} \int_{\Omega_2} T$$$
## 4. Parameters
The table displays the various fixed and variables parameters of this test-case.
Name Description Range Units $k_1$ thermal conductivity $[1,3$] $W \cdot m^{-1} \cdot K^{-1}$ $k_2$ thermal conductivity $[1,3$] $W \cdot m^{-1} \cdot K^{-1}$ $h$ transfer coefficient $[0.1,5$] $W \cdot m^{-2} \cdot K^{-1}$ $Q_1$ heat source $[10^4, 10^{6}$] $W \cdot m^{-3}$ $Q_1$ heat source $[10^4, 10^{6}$] $W \cdot m^{-3}$ $V$ Inflow rate $[1,30$] $m^2 \cdot s^{-1}$
Name Description Nominal Value Units $t$ time $[0, 500$] $s$ $T_0$ initial temperature $300$ $K$ IC Parameters $\rho C_{IC}$ heat capacity $1.4 \cdot 10^{6}$ $J \cdot m^{-3} \cdot K^{-1}$ $e_{IC}$ thickness $2\cdot 10^{-3}$ $m$ $h_{IC} = L_{IC}$ height $2\cdot 10^{-2}$ $m$ $h_{1}$ height $2\cdot 10^{-2}$ $m$ $h_{2}$ height $7\cdot 10^{-2}$ $m$ PCB Parameters $k_3 = k_{Pcb}$ thermal conductivity $0. 2$ $W \cdot m^{-1} \cdot K^{-1}$ $\rho C_{3}$ heat capacity $2 \cdot 10^{6}$ $J \cdot m^{-3} \cdot K^{-1}$ $e_{Pcb}$ thickness $2\cdot 10^{-3}$ $m$ $h_{Pcb}$ height $13\cdot 10^{-2}$ $m$ Air Parameters $T_0$ Inflow temperature $300$ $K$ $k_4$ thermal conductivity $3 \cdot 10^{-2}$ $W \cdot m^{-1} \cdot K^{-1}$ $\rho C_{4}$ heat capacity $1100$ $J \cdot m^{-3} \cdot K^{-1}$ $e_{a}$ thickness $4\cdot 10^{-3}$ |
# NCERT Mathematics Class 9 Exemplar Ch 1 Number Systems Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)
Glide to success with Doorsteptutor material for CBSE/Class-9 : get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-9.
Exercise 1.3
1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
(i)
(ii)
(iii)
(iv)
Answer: Rational numbers: (ii) , (iii) Irrational numbers: (i) , (iv)
2. Find three rational numbers between
(i) – 1 and – 2
(ii) and
(iii) and
(iv) and
Answer: (i) – 1.1, – 1.2, – 1.3 (ii) 0.101,0.102,0.103 (iii) , , (iv) , ,
3. Insert a rational number and an irrational number between the following:
(i) 2 and 3
(ii) 0 and 0.1
(iii) and
(iv) and
(v) 0.15 and 0.16
(vi) and
(vii) 2.357 and 3.121
(viii) . 0001 and . 001
(ix) 3.623623 and 0.484848
(x) 6.375289 and 6.375738
(ii)
(iii) ,
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
4. Represent the following numbers on the number line: 7,7.2, ,
5. Locate , and on the number line.
6. Represent geometrically the following numbers on the number line:
(i)
(ii)
(iii)
(iv)
7. Express the following in the form , where p and q are integers and :
(i) 0.2
(ii)
(iii) 5.2
(iv)
(v)
(vi)
(vii) .
(viii) .
Answer: (i) , (ii) , (iii) , (iv) , (v) , (vi) , (vii) , (viii)
8. Show that 0.142857142857 … =
9. Simplify the following:
(i) + 4
(ii) +
(iii) ×
(iv) ÷ 3 ÷
(v) 3 + 2 +
(vi)
(vii) + +
(viii) +
(ix) -
(ii)
(iii) 168
(iv)
(v)
(vi)
(vii) 0
(viii)
(ix)
Developed by: |
# Synopsis: Better transistors through quantum mechanics
More compact transistors may be possible by harnessing the quantum properties of semiconductor heterostructures.
As chip designers seek to pack more transistors onto every square millimeter of silicon real estate, they run up against a fundamental problem: small devices become overwhelmed by quantum effects. However, some researchers are seeking to turn the tables by pressing quantum mechanics into service as part of the transistor design. In a paper in Physical Review B, Adam Sciambi of Stanford University, California, and his collaborators report their proof-of-principle experiments on a new kind of quantum transistor.
The team has created a two-layer $\text{GaAs/AlGaAs}$ quantum well heterostructure, in which the wave function of one layer extends into the second to modulate the tunneling current between the layers. In this design, a voltage on the first quantum well causes that layer to be depleted of carriers, which changes the subband energy level in the well. As the subband energy approaches the top of the quantum well potential, the wave function extends further and further out toward the second layer. When the wave function overlaps the second layer, the tunneling current can increase as much as two orders of magnitude, a substantial degree of gating leverage.
Although the reported design only works at cryogenic temperatures, a different choice of materials, for example, graphene, may allow operation at more technologically relevant temperatures. – David Voss
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## Subject Areas
Semiconductor Physics
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Semiconductor Physics
Magnetism
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# What happens if I train a model on a data set that includes a duplicated feature?
The Question
Suppose I train a predictive model on a set of features $$x_1, \dots, x_n$$, but for some $$i \neq j$$ we have $$x_i = x_j$$ for every data point in the training set; i.e. one of these features is a totally redundant copy of the other.
What are the consequences for learning? Does it depend on whether my model is linear or nonlinear? Does it depend on my training algorithm?
More generally, what should I expect if one of the $$x_i$$'s is a linear combination of the other features for every point in the training set?
My thoughts so far
Suppose the true target function is a noise-free line, $$y = w_1 x_1$$. Then a basic linear model will learn the parameter $$w_1$$ exactly. Now I duplicate the feature $$x_1$$ by creating a copy $$x_2 = x_1$$. Any combination of weights in the set $$\{(\hat{w}_1, \hat{w}_2): \hat{w}_1 + \hat{w}_2 = w_1\}$$ will perfectly fit the data. I'm guessing that the training algorithm will influence which particular pair is chosen.
• Please clarify the sense of "totally redundant copy:" would this be essentially the inadvertent reproduction of a single observation in the dataset or is it perhaps an independent observation that has precisely the same values? – whuber Nov 2 '18 at 19:27
• @whuber: I mean every observation has two attributes that are identical. Think copying a column in the data table. – tddevlin Nov 2 '18 at 20:46
• That's an example of collinearity. Ordinarily one would just drop one of the columns on the theory that they represent exactly the same property of the observations. (They might not, but as far as the data are concerned, they do.) – whuber Nov 3 '18 at 0:20 |
## Tokyo Journal of Mathematics
### Remarks on $r$-planes in Complete Intersections
Chikashi MIYAZAKI
#### Abstract
This paper investigates the families of smooth complete intersections containing $r$-planes in projective spaces. We are going in a primitive way to shed some light on a point and an $r$-plane containing the point in a complete intersection from the viewpoint of projective geometry.
#### Article information
Source
Tokyo J. Math., Volume 39, Number 2 (2016), 459-467.
Dates
First available in Project Euclid: 30 March 2016
https://projecteuclid.org/euclid.tjm/1459367269
Digital Object Identifier
doi:10.3836/tjm/1459367269
Mathematical Reviews number (MathSciNet)
MR3599503
Zentralblatt MATH identifier
1372.14003
Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes)
#### Citation
MIYAZAKI, Chikashi. Remarks on $r$-planes in Complete Intersections. Tokyo J. Math. 39 (2016), no. 2, 459--467. doi:10.3836/tjm/1459367269. https://projecteuclid.org/euclid.tjm/1459367269
#### References
• M. Artin, Lectures on deformations of singularities, Tata Institute of Fundamental Research, Lectures on Mathematics 54, Bombay (1976).
• W. Barth and A. Van de Ven, Fano-varieties of lines of hypersurfaces, Arch. Math. 31 (1978), 96–104.
• R. Beheshti, Linear subvarieties on hypersurfaces, Int. Math. Res. Not. 49 (2005), 3055–3063.
• C. Borcea, Deforming varieties of $k$-planes of projective complete intersection, Pacific J. Math. 143 (1990), 25–36.
• A. Collino, Lines on quartic threefolds, J. London Math. Soc. (2) 19 (1979), 257–267.
• A. Grothendieck, Fondements de la Géometrie Algébrique, Séminaire Bourbaki, Sécretariat Math. (1962).
• C. Miyazaki, Fano varieties of $r$-dimensional linear subspaces of hypersurfaces (in Japanese), Master Thesis (1982), Waseda University.
• D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies 59 (1966), Princeton UP.
• K. Paranjape and V. Srinivas, Unirationality of the general complete intersection of small multidegree, Flips and abundance for algebraic threefold by J. Kollár, Asterisque 211, 1992.
• A. Predonzan, Intorno agli $S_k$ giacenti sulla varietàinterszione completa di più forme, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 5 (1948), 238–242.
• B. R. Tennison, On the quartic threefold, Proc. London Math. Soc. 29 (1974), 714–734. |
# Recent questions and answers in Algorithms
1 vote
1
which of the following cannot be solved using masters theorem? a) T(n) = 2T(n/2) + n/logn b) T(n) = 2T(n/2) + logn c)T(n)=T(n/2)+logn d) non of these
2
Consider the following C functions: int f1 (int n) { if(n == 0 || n == 1) return n; else return (2 * f1(n-1) + 3 * f1(n-2)); } int f2(int n) { int i; int X[N], Y[N], Z[N]; X[0] = Y[0] = Z[0] = 0; X[1] = 1; Y[1] = 2; Z[1] = 3; for(i = 2; i <= n; i++){ X ... $f1(n)$ and $f2(n)$ are $\Theta(n)$ and $\Theta(n)$ $\Theta(2^n)$ and $\Theta(n)$ $\Theta(n)$ and $\Theta(2^n)$ $\Theta(2^n)$ and $\Theta(2^n)$
3
The minimum number of comparisons required to find the minimum and the maximum of $100$ numbers is ________
4
The Breadth First Search algorithm has been implemented using the queue data structure. One possible order of visiting the nodes of the following graph is: $MNOPQR$ $NQMPOR$ $QMNPRO$ $QMNPOR$
5
The most efficient algorithm for finding the number of connected components in an undirected graph on $n$ vertices and $m$ edges has time complexity $\Theta(n)$ $\Theta(m)$ $\Theta(m+n)$ $\Theta(mn)$
6
Consider a weighted complete graph $G$ on the vertex set $\{v_1,v_2,.....v_n\}$ such that the weight of the edge $(v_i, v_j)$ is $2|i-j|$. The weight of a minimum spanning tree of $G$ is: $n-1$ $2n-2$ $\begin{pmatrix} n \\ 2 \end{pmatrix}$ $n^2$
7
Consider a hash table with $m$ slots that uses chaining for collision resolution. The table is initially empty. What is the probability that after 4 keys are inserted that at least a chain of size 3 is created? (Assume simple uniform hashing is used) $m^{–2}$ $m^{–4}$ $m^{–3} (m – 1)$ $3m^{–1}$
8
Let $x_n$ denote the number of binary strings of length $n$ that contain no consecutive 0s. Which of the following recurrences does $x_n$ satisfy? $x_n = 2x_{n-1}$ $x_n = x_{\lfloor n/2 \rfloor} + 1$ $x_n = x_{\lfloor n/2 \rfloor} + n$ $x_n = x_{n-1} + x_{n-2}$
9
Solve the recurrence equations: $T(n) = T(n - 1)+ n$ $T(1) = 1$
10
Consider an array containing ‘n’ elements. The elements present in an array are in arithmetic progression, but one element is missing in that order. What is the time complexity to find the position of the missing element using divide and conquer?
11
Consider the depth-first-search of an undirected graph with $3$ vertices $P$, $Q$, and $R$. Let discovery time $d(u)$ represent the time instant when the vertex $u$ is first visited, and finish time $f(u)$ represent the time instant when the vertex $u$ ... connected There are two connected components, and $Q$ and $R$ are connected There are two connected components, and $P$ and $Q$ are connected
12
Consider two strings $A$="qpqrr" and $B$="pqprqrp". Let $x$ be the length of the longest common subsequence (not necessarily contiguous) between $A$ and $B$ and let $y$ be the number of such longest common subsequences between $A$ and $B$. Then $x +10y=$ ___.
13
Consider the following recurrence relation: $T(n) = \begin{cases} 2T (\lfloor\sqrt{n}\rfloor)+ \log n & \text{if }n \geq 2 \\ 1& \text{if }n = 1 \end{cases}$ Which of the following statements is TRUE? $T(n)$ is $O(\log n)$. $T(n)$ is $O(\log n. \log \log n)$ but not $O(\log n)$. $T(n)$ is ... $O(\log^{2} n)$ but not $O(\log^{3/2} n)$. $T(n)$ is $O(\log^{2} n. \log \log n)$ but not $O(\log^{2} n)$.
14
Consider the following sequence of numbers:$92, 37, 52, 12, 11, 25$ Use Bubble sort to arrange the sequence in ascending order. Give the sequence at the end of each of the first five passes.
15
For merging two sorted lists of sizes $m$ and $n$ into a sorted list of size $m+n$, we require comparisons of $O(m)$ $O(n)$ $O(m+n)$ $O(\log m + \log n)$
16
Algorithm design technique used in quicksort algorithm is? Dynamic programming Backtracking Divide and conquer Greedy method
17
Assume that the last element of the set is used as partition element in Quicksort. If $n$ distinct elements from the set $\left[1\dots n\right]$ are to be sorted, give an input for which Quicksort takes maximum time.
18
Quicksort is ________ efficient than heapsort in the worst case.
1 vote
19
Can pls someome. Tell.number of comparisons in. Merge sort in best case as well as worst case. Acc to. Me, at. Each level we need O(n) comaprisons and number of levels are log n in merge sort(whether it. Is a best case or worst case).hence mumber o comparisons should be nlogn in worst case as well as best case. Pls guide me.
20
An element in an array $X$ is called a leader if it is greater than all elements to the right of it in $X$. The best algorithm to find all leaders in an array solves it in linear time using a left to right pass of the array solves in linear time using a right to left pass of the array solves it using divide and conquer in time $\theta (n\log n)$ solves it in time $\theta (n^{2})$
21
What is the average case time complexity of the best sorting algorithm for an array having 2^n^2 elements . I know that the best sorting algorithm is no better than O(n log n).Please answer in terms of the asymptotic notation.
22
In Merge sort Algorithm when I took input array of size 2 and I got 4 function calls as including original function call with which I call MS algorithm i.e. MS (1,2) and which in turn calls two recursive function calls to merge sort as MS (1,1) and MS (2,2) and ... of size 6 I got 16 function calls. So, how can I analyze the total number of function calls when input array size is n? thank you!
23
Describe an algorithm that, given $n$ integers in the range $0$ to $k$ preprocesses its input and then answers any query about how many of the $n$ integers fall into the range $[a..b]$ in $O(1)$ time.Your algorithm should use $\Theta(n+k)$ preprocessing time.
24
A list of $n$ strings, each of length $n$, is sorted into lexicographic order using the merge-sort algorithm. The worst case running time of this computation is $O (n \log n)$ $O(n^{2} \log n)$ $O(n^{2} + \log n)$ $O(n^{2})$
25
In a permutation $a_1 ... a_n$, of $n$ distinct integers, an inversion is a pair $(a_i, a_j)$ such that $i < j$ and $a_i > a_j$. What would be the worst case time complexity of the Insertion Sort algorithm, if the inputs are restricted to permutations of $1. . . n$ with at most $n$ inversions? $\Theta(n^2)$ $\Theta(n\log n)$ $\Theta(n^{1.5})$ $\Theta(n)$
26
Let $x_n$ denote the number of binary strings of length $n$ that contain no consecutive 0s. The value of $x_5$ is $5$ $7$ $8$ $16$
27
Which of the following statements is not true? 1.For every fixed strategy to choose a pivot for quicksort, we can construct a worst case input that requires time O(n2). 2.If we randomly choose a pivot element each time, quicksort will always terminate in time ... , quicksort would have worst case complexity O(n log n). 4.Quicksort and merge sort are both examples of divide and conquer algorithms.
1 vote
28
what is the time-complexity in kruskal algorithm for the overall step 2 where for each vertex Make-set function is called ? How come overall time for this step is O(v log v) ? We are performing this Operation for all the vertices in the Initial phase only so for ... Make-set operation only once right because after we come out of loop we have v sets of 1 vertex each . Please explain this clearly .
29
I was wondering whether the recurrence T(n) = T(n/2) + 2n could be solved by using master theorem, and what would be the way. I tried solving the recurrence but can't. There is no mention to it in CLRS book. Please help. Thanks in advance.
30
31
You have $n$ lists, each consisting of $m$ integers sorted in ascending order. Merging these lists into a single sorted list will take time: $O(nm \log m)$ $O(mn \log n)$ $O(m + n)$ $O(mn)$
32
To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear time, the data structure to be used is: Queue Stack Heap B-Tree
33
Make a 3-by-3 chart with row and column labels WHITE, GRAY, and BLACK. In each cell ( , ) ij , indicate whether, at any point during a depth-first search of a directed graph, there can be an edge from a vertex of color i to a vertex of color j . For each possible edge, indicate what types it can be.
34
PARTITION(A,p,r) 1 x = A[r] 2 i = p – 1 3 for j = p to r – 1 4 if A[j] <= x 5 i = i + 1 6 exchange A[i] with A[j] 7 exchange A[i+1] with A[r] 8 return i + 1 illustrate the operation of PARTITION on the array $A=\langle 13,19,9,5,12,8,7,4,21,2,6,11\rangle$
35
Consider the array A[]= {6,4,8,1,3} apply the insertion sort to sort the array . Consider the cost associated with each sort is 25 rupees , what is the total cost of the insertion sort when element 1 reaches the first position of the array ? (A) 50 (B) 25 (C) 75 (D) 100 Source: http://quiz.geeksforgeeks.org/algorithms-insertionsort-question-4/
36
Merging k sorted lists of size n/k into one sorted list of n-elements using heap sort will take how much time ? My doubt First approach:- here it is mentioned heap sort so, heap sort will always take nlogn.and here also we have n elements and it will take nlogn. But ... it will give o(k)+(logk)*(n/k) I think answer should be nlogn only because the second approach is not heap sort. Please check.
37
Consider the following game. There is a list of distinct numbers. At any round, a player arbitrarily chooses two numbers $a, b$ from the list and generates a new number $c$ by subtracting the smaller number from the larger one. The numbers $a$ and $b$ are put back in the list. If the number ... $273$. What is the score of the best player for this game? $40$ $16$ $33$ $91$ $123$
38
What is the weight of a minimum spanning tree of the following graph? $29$ $31$ $38$ $41$
39
A set $X$ can be represented by an array $x[n]$ as follows: $x\left [ i \right ]=\begin {cases} 1 & \text{if } i \in X \\ 0 & \text{otherwise} \end{cases}$ Consider the following algorithm in which $x$, $y$, and $z$ are Boolean arrays of size $n$: algorithm zzz(x[], y[], z[]) { int i; ... y[i]); } The set $Z$ computed by the algorithm is: $(X\cup Y)$ $(X\cap Y)$ $(X-Y)\cap (Y-X)$ $(X-Y)\cup (Y-X)$
Consider the function func shown below: int func(int num) { int count = 0; while (num) { count++; num>>= 1; } return (count); } The value returned by func($435$) is ________ |
# pair_style lebedeva/z command
## Syntax
pair_style [hybrid/overlay ...] lebedeva/z cutoff
## Examples
pair_style hybrid/overlay lebedeva/z 20.0
pair_coeff * * none
pair_coeff 1 2 lebedeva/z CC.Lebedeva C C
pair_style hybrid/overlay rebo lebedeva/z 14.0
pair_coeff * * rebo CH.rebo C C
pair_coeff 1 2 lebedeva/z CC.Lebedeva C C
## Description
The lebedeva/z style computes the Lebedeva interaction potential as described in (Lebedeva et al.). An important simplification is made, which is to take all normals along the z-axis.
It is important to have a sufficiently large cutoff to ensure smooth forces. Energies are shifted so that they go continuously to zero at the cutoff assuming that the exponential part of Vij (first term) decays sufficiently fast. This shift is achieved by the last term in the equation for Vij above.
The parameter file (e.g. CC.Lebedeva), is intended for use with metal units, with energies in meV. An additional parameter, S, is available to facilitate scaling of energies.
This potential must be used in combination with hybrid/overlay. Other interactions can be set to zero using pair_style none.
## Restrictions
This fix is part of the USER-MISC package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info. |
Dynamics of a membrane interacting with an active wall
# Dynamics of a membrane interacting with an active wall
Kento Yasuda Shigeyuki Komura Ryuichi Okamoto Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan
July 15, 2019
###### Abstract
Active motions of a biological membrane can be induced by non-thermal fluctuations that occur in the outer environment of the membrane. We discuss the dynamics of a membrane interacting hydrodynamically with an active wall that exerts random velocities on the ambient fluid. Solving the hydrodynamic equations of a bound membrane, we first derive a dynamic equation for the membrane fluctuation amplitude in the presence of different types of walls. Membrane two-point correlation functions are calculated for three different cases; (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We focus on the mean squared displacement (MSD) of a tagged membrane describing the Brownian motion of a membrane segment. For the static wall case, there are two asymptotic regimes of MSD ( and ) when the hydrodynamic decay rate changes monotonically. In the case of an active wall, the MSD grows linearly in time () in the early stage, which is unusual for a membrane segment. This linear-growth region of the MSD is further extended when the active wall has a finite intrinsic time scale.
## I Introduction
The random slow dynamics of fluid membranes visible as a flickering phenomenon in giant unilamellar vesicles (GUVs) or red blood cells (RBCs) has attracted many interests in the last few decades Lipowsky95 (). These thermally excited shape fluctuations can be essentially understood as a Brownian motion of a two-dimensional (2D) lipid bilayer membrane in a three-dimensional (3D) viscous fluid such as water. For spherically closed artificial GUVs, characteristic relaxation times for shape deformations were calculated analytically MS87 (); KS93 (); SK95 (); Komura96 (). Analysis of shape fluctuations can be used for quantitative measurements of surface tension and/or bending rigidity of single-component GUVs Popescu06 () or GUVs containing bacteriorhodopsin pumps FLPJPB05 ().
Historically, investigations on fluctuations of cell membranes have started with RBCs whose flickering can be observed under a microscope LB14 (). Brochard and Lennon were among the first to describe quantitatively membrane fluctuations as thermally excited undulations, mainly governed by the bending rigidity of the membrane Brochard75 (). Later experiments showed that flickering in RBCs is not purely of thermal origin but rather corresponds to a non-equilibrium situation because the fluctuation amplitude decreases upon ATP depletion Levin91 (); Tuvia98 (). Here ATP hydrolysis plays an important role to control membrane-spectrin cytoskeleton interactions AlbertsBook (). More advanced techniques have demonstrated that, at longer time scales (small frequencies), a clear difference exists between the power spectral density of RBC membranes measured for normal cells and those ATP depleted; the fluctuation amplitude turns out to be higher in the former Betz09 (); Park10 (). At shorter time scales, on the other hand, membranes fluctuate as in the thermodynamic equilibrium. It should be noted, however, that the role of ATP in flickering is still debatable because Boss et al. have recently claimed that the mean fluctuation amplitudes of RBC membranes can be described by the thermal equilibrium theory, while ATP merely affects the bending rigidity Boss12 ().
In order to understand shape fluctuations of RBCs, one needs to properly take into account the effects of spectrin cytoskeleton network that is connected to the membrane by actin, glycophorin, and protein 4.1R Lipowsky95 (); AlbertsBook (). Gov et al. treated the cytoskeleton as a rigid wall (shell) located at a fixed distance from the membrane, and assumed that its static and dynamic fluctuations are confined by the cytoskeleton Gov03 (); GovZilamSafran04 (). They further considered that the sparse connection of membrane and cytoskeleton gives rise to a finite surface tension for length scales larger than the membrane persistence length. The bending free energy for a membrane was extended to include a surface tension and a confinement potential with which the effects of ATP on the membrane fluctuations was described. However, since an active component of the membrane fluctuations also depend on the fluid viscosity Tuvia97 (), they cannot be solely attributed to the static parameters such as the surface tension or the potential. Gov and Safran later estimated the active contribution to the membrane fluctuations due to the release of stored tension in the spectrin filament and membrane in each dissociation event GovSafran05 (); Gov07 (). In contrast to static thermal fluctuations, they showed that the active cytoskeleton may contribute to the membrane fluctuations at intermediate length scales.
Effects of membrane confinement are important not only for shape fluctuations of RBCs but also for a hydrodynamic coupling between closely apposed lipid bilayer membranes Kaizuka04 (); Kaizuka06 (), and dynamical transitions occurring in lamellar membranes under shear flow Diat93 (); Lu08 (). After the seminal works by Kramer Kramer () and by Brochard and Lennon Brochard75 (), the wavenumber-dependent decay rate for the bending modes of a membrane bound to a wall was calculated by Seifert Seifert94 () and Gov et al. GovZilamSafran04 (). In particular, Seifert showed that the scale separation between the membrane-wall distance and the correlation length determined by the confinement potential can lead to various crossover behaviors of the decay rate. In these hydrodynamic calculations, however, the wall that interacts with the membrane was treated as a static object and does not play any active role.
Quite generally, active motions of a membrane can be induced by non-thermal fluctuations that occur in the outer environment of the membrane such as cytoskeleton or cytoplasm. In this paper, we consider the dynamics of a membrane interacting with an active wall that generates random velocities in the ambient fluid. These random velocities at the wall can be naturally taken into account through the boundary conditions of the fluid. We first derive a dynamic equation for the membrane fluctuation amplitude in the presence of hydrodynamic interactions. Then we calculate the membrane two-point correlation functions for three different cases; (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We especially focus on the mean squared displacement (MSD) of a tagged membrane segment, and discuss its asymptotic time dependencies for the above cases. For the static wall case, the membrane fluctuates due to thermal agitations, and there are two asymptotic regimes of MSD ( and ) if the hydrodynamic decay rate changes monotonically as a function of the wavenumber. When the wall is active, there is a region during which the MSD grows linearly with time (), which is unusual for a membrane segment. If the active wall has a finite intrinsic time scale, the above linear-growth regime of the MSD is further extended. As a whole, active fluctuations at the wall propagate through the surrounding fluid and greatly affects the membrane fluctuations.
This paper is organized as follows. In the next section, we discuss the hydrodynamics of a bound membrane that interacts with an active wall. We also derive a dynamic equation for the membrane fluctuation amplitude in the presence of hydrodynamic interactions. In Sec. III, we calculate the membrane two-point correlation functions for three different cases of the wall as mentioned above. We investigate various asymptotic behaviors of the MSD of a tagged membrane both in the static and the active wall cases. Some further discussions are provided in Sec. IV.
## Ii Hydrodynamics of a bound membrane
### ii.1 Free energy of a bound membrane
As depicted in Fig. 1, we consider a fluid membrane bound at an average distance from a wall which defines the -plane. Within the Monge representation, which is valid for nearly flat surfaces, the membrane shape is specified by the distance between the membrane and the wall. The free energy of a tensionless membrane in a potential per unit area reads Safran (); Lipowsky86 ()
F=∫d2ρ[κ2(∇2ℓ)2+V(ℓ)], (1)
where is the bending rigidity and . We use a harmonic approximation for fluctuations around the minimum of the potential at , and obtain the approximated form
F≈κ2∫d2ρ[(∇2h)2+ξ−4h2], (2)
where is the correlation length due to the potential. Later we use a dimensionless quantity defined by in order to discuss different cases.
In the following, we introduce the 2D spatial Fourier transform of defined as
h(q)=∫d2ρh(ρ)e−iq⋅ρ, (3)
where . Then the static correlation function can be obtained from Eq. (2) as
⟨h(q)h(−q)⟩=kBTκ(q4+ξ−4)=kBTE(q,ξ), (4)
where is the Boltzmann constant, the temperature, and we have introduced the notation .
In the present model, we assume that the wall is rigid and does not deform. Even when the wall, mimicking the cytoskeleton network, is deformable, the above free energy Eq. (1) would not be changed if we regard as a local distance between the membrane and the cytoskeleton. In this case, however, the bending rigidity should be replaced with an effective one which is also dependent on the bending rigidity of the cytoskeleton network itself LZ89 ().
### ii.2 Hydrodynamic equations and boundary conditions
The dynamics of a membrane is dominated by the surrounding fluid which is assumed to be incompressible and to obey the Stokes equation. We choose as the coordinate perpendicular to the wall located at as in Fig. 1. Then the velocity and the pressure for satisfy the following equations
∇⋅v=0, (5)
η∇2v−∇p−f=0, (6)
where is the viscosity of the surrounding fluid and is any force acting on the fluid. The fluid velocity can be obtained from the above equations by supplementing them with proper boundary conditions. In Appendix A, we show a formal solution appropriate for the membrane/wall system, and obtain the fluid velocity in terms of the force . Without loss of generality, we can choose the - and -coordinates as the parallel (longitudinal) and the perpendicular (transverse) directions to the in-plane vector , respectively. Since the transverse -component of the velocity is not coupled to the other components, we are allowed to set in what follows.
Let us denote the fluid regions and with the superscripts “” and “”, respectively. In general, we consider time-dependent boundary conditions at and time-independent conditions at :
v−x(q,z=0,t)=Vx0(q,t), (7) v−z(q,z=0,t)=Vz0(q,t), (8) v+x(q,z→∞,t)=v+z(q,z→∞,t)=0. (9)
The statistical properties of and will be discussed for different types of walls in the next Section. As described in Appendix, the -component of the velocity is then obtained as
v−z(q,z,t)= A[sinh(qz)−qzcosh(qz)]+Bqzsinh(qz) −iqzVx0(q,t)e−qz+(1+qz)Vz0(q,t)e−qz, (10)
v+z(q,z,t)=Ce−q(z−¯ℓ)+Dq(z−¯ℓ)e−q(z−¯ℓ), (11)
where , , , and are the coefficients determined by the other boundary conditions at the membrane . Note that both and can be also expressed in terms of these four coefficients.
At where the membrane exists, continuity of and yields
v−x(q,z=¯ℓ,t)=v+x(q,z=¯ℓ,t), (12)
v−z(q,z=¯ℓ,t)=v+z(q,z=¯ℓ,t), (13)
and incompressibility of the membrane requires that the in-plane divergence of vanishes
iqv−x(q,z=¯ℓ,t)=0. (14)
Moreover, the forces are required to balance in the normal direction at . This condition is written as
−T+zz+T−zz=−δFδh(q,t)=−E(q,ξ)h(q,t), (15)
where was defined in Eq. (4). In the above, is the -component of the fluid stress tensor
Tij=−pδij+η(∂ivj+∂jvi), (16)
evaluated at and . The above four boundary conditions in Eqs. (12)–(15) at determine the solution of and in the entire region of the fluid.
### ii.3 Dynamic equation of a bound membrane
Next we derive a dynamic equation for the membrane fluctuation amplitude. The time derivative of the fluctuation amplitude (membrane velocity) should coincide with the normal velocity of the fluid at the membrane obtained from Eqs. (10) and (11) together with the four coefficients (see also Appendix). Using the result of the above hydrodynamic calculation, we can write the dynamic equation of as follows
∂h(q,t)∂t= −γ(q,¯ℓ,ξ)h(q,t) +Λx(q,¯ℓ)Vx0(q,t)+Λz(q,¯ℓ)Vz0(q,t) +ζ(q,t). (17)
In the above, is the hydrodynamic decay rate
γ(q,¯ℓ,ξ)=Γ(q,¯ℓ)E(q,ξ), (18)
where the kinetic coefficient is given by
Γ(q,¯ℓ)=12ηqsinh2(q¯ℓ)−(q¯ℓ)2sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ). (19)
The same expression was obtained by Seifert Seifert94 (). The second and the third terms on the r.h.s. of Eq. (17) are due to the wall boundary conditions Eqs. (7) and (8). Our calculation yields
Λx(q,¯ℓ)=−iq¯ℓsinh(q¯ℓ)sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ), (20)
Λz(q,¯ℓ)=sinh(q¯ℓ)+q¯ℓcosh(q¯ℓ)sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ). (21)
The last term in Eq. (17) represents the thermal white noise; its average vanishes while its correlation is fixed by the fluctuation-dissipation theorem (FDT) KuboBook (); LandauBook ()
⟨ζ(q,t)ζ(−q,t′)⟩=2kBTΓ(q,¯ℓ)δ(t−t′). (22)
### ii.4 Hydrodynamic decay rate
We first introduce as a characteristic time. In Fig. 2, we plot the scaled decay rate (see Eq. (18)) as a function of the dimensionless wavenumber when and . For our later discussion, it is useful here to discuss its asymptotic behaviors. We first note that the kinetic coefficient in Eq. (19) behaves as
Γ≈{¯ℓ3q2/12η,q≪1/¯ℓ1/4ηq,q≫1/¯ℓ. (23)
Depending on the relative magnitude between and , two different asymptotic behaviors of the decay rate can be distinguished Seifert94 (). For (corresponding to in Fig. 2), the decay rate increases monotonically as
γ≈⎧⎪ ⎪⎨⎪ ⎪⎩κ¯ℓ3q2/12ηξ4,q≪1/ξκ¯ℓ3q6/12η,1/ξ≪q≪1/¯ℓκq3/4η,1/¯ℓ≪q. (24)
The small- behavior results from the conservation of the fluid volume between the membrane and the wall Marathe89 (). The dependence in the intermediate regime, where the effect of potential becomes irrelevant, was predicted by Brochard and Lennon Brochard75 (). For large , we recover the behavior of a free membrane . All these asymptotic behaviors are observed in Fig. 2.
For (corresponding to in Fig. 2), on the other hand, changes non-monotonically as Seifert94 ()
γ≈⎧⎪⎨⎪⎩κ¯ℓ3q2/12ηξ4,q≪1/¯ℓκ/4ηξ4q,1/¯ℓ≪q≪1/ξκq3/4η.1/ξ≪q. (25)
While the small- and large- behaviors are unchanged from Eq. (24), here the decay rate decreases with increasing in the intermediate range. This unusual decrease of the decay rate clearly appears for in Fig. 2. Such an anomalous behavior occurs due to the fact that the potential confines the mean-square fluctuation amplitudes to independently of (see Eq. (4)), while hydrodynamic damping becomes less effective with increasing Seifert94 (). We also note that the absolute value of in the small- region is sensitive to the value of , while it is independent of in the large- region.
## Iii Membrane two-point correlation functions
Using the result of the hydrodynamic calculation, we shall discuss in this section the two-point correlation functions of bound membranes ZG96 (); ZG02 (). We separately investigate the cases of (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale.
### iii.1 Static wall
In the case of a static wall, the velocities at the wall vanish in Eqs. (7) and (8), i.e., . Hence Eq. (17) reduces to
∂h(q,t)∂t=−γ(q,¯ℓ,ξ)h(q,t)+ζ(q,t), (26)
and one can easily solve for as
h(q,t)= h(q,0)e−γ(q,¯ℓ,ξ)t +∫t0dt1ζ(q,t1)e−γ(q,¯ℓ,ξ)(t−t1). (27)
Using the above solution and Eq. (22), we calculate the membrane two-point correlation function which can be separated into two parts ZG96 (); ZG02 ()
⟨[h(ρ,t)−h(ρ′,0)]2⟩ =Φ(ρ−ρ′)+ϕ(ρ−ρ′,t), (28)
where the translational invariance of the system has been assumed. In the above, the first term is a purely static correlator
Φ(ρ−ρ′) =⟨[h(ρ)−h(ρ′)]2⟩ =2∫d2q(2π)2⟨h(q)h(−q)⟩[1−eiq⋅(ρ−ρ′)], (29)
describing the static membrane roughness, while the second term is a dynamical correlator
ϕ(ρ−ρ′,t)= 2∫d2q(2π)2⟨h(q)h(−q)⟩eiq⋅(ρ−ρ′) ×[1−e−γ(q,¯ℓ,ξ)t], (30)
describing the propagation of fluctuations with a distance .
Using the static correlation function for in Eq. (4), we first calculate the static correlator
Φ(ρ−ρ′)=kBTπκ∫∞0dqqq4+ξ−4[1−J0(q|ρ−ρ′|)] (31)
where is the zero-order Bessel function of the first kind, and the Meijer -function is used in the last expression mathematica (). In Fig. 3, we plot the scaled static correlator as a function of where . Only in this plot, we use to scale the length because the above static correlator is solely determined by the free energy in Eq. (2), and Eq. (31) does not depend on . In the large distance , the (route mean square) height difference between two points on the bound membrane is proportional to . It is interesting to note that changes non-monotonically and shows a maximum around . A similar overshoot behavior of the membrane profile was reported before KA00 ().
As for the dynamical correlator in Eq. (30), we perform the angular integration and obtain the expression
ϕ(ρ−ρ′,t)= kBTπκ∫∞0dqqq4+ξ−4 ×[1−e−γ(q,¯ℓ,ξ)t]J0(q|ρ−ρ′|). (32)
We first set and discuss the mean squared displacement (MSD) of a tagged membrane segment given by ZG96 (); ZG02 ()
ϕ0(t)=kBTπκ∫∞0dqqq4+ξ−4[1−e−γ(q,¯ℓ,ξ)t], (33)
where we have used . Instead of the correlation length , we hereafter use to scale the length. Note that the hydrodynamic effect is manifested by the appearance of the length . In Fig. 4, we plot the dimensionless MSD as a function of (recall that ) for (monotonic damping case) and (non-monotonic damping case), respectively. In order to find out the asymptotic behaviors clearly, we have also plotted an effective growth exponent defined by
α(t)=dlnϕ0(t)dlnt. (34)
For both and , the MSD increases monotonically as a function of time. For (), there are three different asymptotic regimes of the time dependence. In the small time regime (), the MSD behaves as which corresponds to the diffusion of a free membrane ZG96 (); ZG02 (). This scaling behavior can be obtained by using the large- behavior of the decay rate in Eq. (24)
ϕ0(t) ≈kBTπκ∫∞0dq1q3[1−e−(κq3/4η)t] ∼kBTκ1/3η2/3t2/3. (35)
In the intermediate time regime (), we have which stems from the intermediate- behavior of in Eq. (24)
ϕ0(t) ≈kBTπκ∫∞0dq1q3[1−e−(κ¯ℓ3q6/12η)t] ∼kBT¯ℓκ2/3η1/3t1/3. (36)
In this regime, as discussed by Brochard and Lennon Brochard75 (), the conservation of the enclosed incompressible volume between the membrane and the wall is important, while the effect of the potential acting between them is irrelevant. The Fourier transform of the above expression, i.e., the power spectral density, was previously discussed by Gov et al. in Ref. Gov03 (). In the long time regime (), the MSD saturates at the value given by
ϕ0(t→∞)≈[kBT¯ℓ2πκ]πΞ24∼kBTκξ2. (37)
For (), on the other hand, there are only two asymptotic regimes. The MSD increases as in the small time regime (), whereas in the long time regime (), it saturates at the value given by Eq. (37).
Let us consider then the case . In Fig. 5, we plot the scaled in Eq. (32) as a function of for different times when . For all the cases, the dynamic correlator changes non-monotonically and exhibits a typical undershoot behavior. The minimum of occurs for larger as time evolves. In the long time limit, , in Eq. (32) coincides with the second term in Eq. (31) and is given by the Meijer -function.
### iii.2 Active wall
We now investigate the case when the wall is active so that it exerts random velocities on the ambient fluid. The membrane dynamics in the presence of an active wall is described by Eq. (17). This equation can be also solved for as
h(q,t)= h(q,0)e−γ(q,¯ℓ,ξ)t +Λx(q,¯ℓ)∫t0dt1Vx0(q,t1)e−γ(q,¯ℓ,ξ)(t−t1) +Λz(q,¯ℓ)∫t0dt2Vz0(q,t2)e−γ(q,¯ℓ,ξ)(t−t2) +∫t0dt3ζ(q,t3)e−γ(q,¯ℓ,ξ)(t−t3). (38)
The random velocities generated at the wall are assumed to have the following statistical properties
⟨Vx0(ρ,t)⟩=⟨Vz0(ρ,t)⟩=0, (39)
⟨Vx0(ρ,t)Vx0(ρ′,t′)⟩=2Sxδ(ρ−ρ′)δ(t−t′), (40)
⟨Vz0(ρ,t)Vz0(ρ′,t′)⟩=2Szδ(ρ−ρ′)δ(t−t′), (41)
⟨Vx0(ρ,t)Vz0(ρ′,t′)⟩=0, (42)
⟨Vx0(ρ,t)ζ(ρ′,t′)⟩=⟨Vz0(ρ,t)ζ(ρ′,t′)⟩=0, (43)
where we have introduced the amplitudes and in Eqs. (40) and (41), respectively. With these statistical properties, we can calculate the total two-point correlation function which consists of the static and the dynamical parts as before
⟨[h(ρ,t)−h(ρ′,0)]2⟩tot =Φtot(ρ−ρ′)+ϕtot(ρ−ρ′,t). (44)
In the above total correlation function, the static correlator in the presence of the active wall becomes
Φtot(ρ−ρ′)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)][1−J0(q|ρ−ρ′|)] ≡Φ(ρ−ρ′)+Φx(ρ−ρ′)+Φz(ρ−ρ′), (45)
where and were obtained in Eqs. (20) and (21), respectively, while was defined in Eq. (31) for the static wall case. In the above equations, we have defined two correlators and . On the other hand, the dynamical correlator in Eq. (44) is given by
ϕtot(ρ−ρ′,t)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)] ×[1−e−γ(q,¯ℓ,ξ)t]J0(q|ρ−ρ′|). (46)
By setting , the total MSD of a tagged membrane segment in the presence of the active wall becomes
ϕtot(t)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)][1−e−γ(q,¯ℓ,ξ)t] ≡ϕ0(t)+ϕx0(t)+ϕz0(t), (47)
where the first term was defined before in Eq. (33) for the static wall case, while and have been newly defined here.
Before showing the result of MSD, we first discuss the wavenumber dependencies of the quantities and appearing in Eqs. (45)–(47). These quantities originating from the active wall are plotted in Fig. 6 as a function of for and . Using the asymptotic behaviors of , as shown in Eqs. (24) and (25), we can obtain the limiting expressions for and as well. When (corresponding to ), we have
|Λx|2/γ≈⎧⎪⎨⎪⎩3ηξ4/κ¯ℓ,q≪1/ξ3η/κ¯ℓq4,1/ξ≪q≪1/¯ℓ4η¯ℓ2e−2¯ℓq/κq,1/¯ℓ≪q, (48)
|Λz|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩12ηξ4/κ¯ℓ3q2,q≪1/ξ12η/κ¯ℓ3q6,1/ξ≪q≪1/¯ℓ4η¯ℓ2e−2¯ℓq/κq,1/¯ℓ≪q. (49)
For (corresponding to ), on the other hand, we obtain
|Λx|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩3ηξ4/κ¯ℓ,q≪1/¯ℓ4ηξ4¯ℓ2q3e−2¯ℓq/κ,1/¯ℓ≪q≪1/ξ4η¯ℓ2e−2¯ℓq/κq,1/ξ≪q, (50)
|Λz|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩12ηξ4/κ¯ℓ3q2,q≪1/¯ℓ4ηξ4¯ℓ2q3e−2¯ℓq/κ,1/¯ℓ≪q≪1/ξ4η¯ℓ2e−2¯ℓq/κq.1/ξ≪q. (51)
The static correlators and defined in Eq. (45) due to the active wall can now be obtained by performing numerical integrals. In Fig. 7, we plot the static correlators and as a function of when . Here and are scaled by and , respectively. We notice that behaves similarly to that of the static wall case given in Eq. (31) and plotted in Fig. 3. On the other hand, diverges logarithmically for large because the integral is found to be infrared divergent. Such a logarithmic divergence is avoided when we consider a finite membrane size which gives rise to a cutoff for small wavenumbers in the integral of Eq. (45). It should be noted that both and depend on and , while is solely determined by . This means that and include the geometrical as well as the hydrodynamic effects.
In Figs. 8 and 9, we plot the scaled membrane MSD and (see Eq. (47)), respectively, as a function of when and . For (corresponding to ), there are three different asymptotic regimes both for and . In the small time regime (), we have and , showing a normal diffusive behavior. This is because can be approximated as
ϕx0(t) ≈4η¯ℓ2Sxπκ∫∞0dqe−2¯ℓq[1−e−(κq3/4η)t] ≈¯ℓ2Sxtπ∫∞0dqe−2¯ℓqq3∼Sx¯ℓ2t. (52)
Notice that only small- contributes to the integral, and the same holds for . In the intermediate time regime (), we have and which can be asymptotically obtained by Eqs. (36) and (35), respectively. In the long time regime (), saturates at the value
ϕx0(t→∞)≈[4η¯ℓSxπκ]3Ξ28∼ηξ2Sxκ¯ℓ. (53)
On the other hand, diverges logarithmically for , which can be seen in Fig. 9(a) and also shown analytically. Such a divergence in time occurs for small and can be avoided when the membrane size is finite as mentioned before.
For (corresponding to ), on the other hand, there are only two asymptotic regimes. The MSDs increase both linearly as and in the small time regime (). In the long time regime (), saturates at the value
ϕx0(t→∞)≈[4η¯ℓSxπκ]3Ξ48∼ηξ4Sxκ¯ℓ3, (54)
while also diverges logarithmically as above.
### iii.3 Active wall with an intrinsic time scale
Finally we consider a situation in which the activity of the wall occurs over a finite time scale . In this case, the statistical properties of random velocities which have been given in Eqs. (40) and (41) would be replaced by the following exponential correlation function in time Gov04 (); GovSafran05 (); Gov07 ()
⟨Vx0(ρ,t)Vx0(ρ′,t′)⟩=Sxτδ(ρ−ρ′)e−|t−t′|/τ, (55)
⟨Vz0(ρ,t)Vz0(ρ′,t′)⟩=Szτδ(ρ−ρ′)e−|t−t′|/τ, (56)
while the other velocity correlations remain the same. In general, the intrinsic time scale can be different between the - and -components. In the above relations, we have put a factor so that the physical dimension of and is the same as before.
Repeating the same procedure as before, we obtain the total two-point correlation function which can be also separated into the static and dynamics parts as in Eq. (44). The static correlators in the presence of the active wall now become
Φx(ρ−ρ′)= 1π∫∞0dqqSx|Λx(q,¯ℓ |
# Calculate total from fillable form [closed]
Hi,
I'm creating a fillable form in LibreOffice but I need a field that gives me the total of other fields (numbered format) but I can't find the right format for that field so it calculates the total (see the image below).
I'd appreciate your help! Thank you!
edit retag reopen merge delete
### Closed for the following reason question is not relevant or outdated by Alex Kemp close date 2020-08-07 03:10:46.272674
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Short answer: NO. JavaScript in a PDF document is apparently a proprietary extension to the specification, source, Wikipedia:
PDF 1.7, the sixth edition of the PDF specification that became ISO 32000-1, includes some proprietary technologies defined only by Adobe, such as Adobe XML Forms Architecture (XFA) and JavaScript extension for Acrobat, which are referenced by ISO 32000-1 as normative and indispensable for the full implementation of the ISO 32000-1 specification. These proprietary technologies are not standardized and their specification is published only on Adobe’s website.[8][9][10][11][12] Many of them are also not supported by popular third-party implementations of PDF.
On July 28, 2017, ISO 32000-2:2017 (PDF 2.0) was published.[13] ISO 32000-2 does not include any proprietary technologies as normative references.[14]
What this means is that you need to use Adobe's own software in order to produce a PDF form which acts like a spreadsheet.
more
Assuming your 6 values are in the cell ranges A2:A4 and C2:C4 then the formula in E4 would be =SUM(A2:A4;C2:C4)
more
I think the question here is how to produce a PDF form that acts as a spreadsheet and can thus produce a sum in one of the form fields. As far as I know this would require scripting. PDF as a format does support embedded JavaScript.
( 2019-07-24 01:08:35 +0200 )edit
1
The question didn't mention PDF, but if that was meant then, yes, there's no such thing in a PDF export.
( 2019-08-01 12:44:54 +0200 )edit |
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# Mathematics
Seminar Calendar
for events the day of Monday, April 13, 2020.
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Monday, April 13, 2020
3:00 pm in 243 Altgeld Hall,Monday, April 13, 2020
#### To Be Announced
###### Jesse Huang (Illinois)
5:00 pm in Altgeld Hall,Monday, April 13, 2020
#### Magic square game
###### Haojian Li (UIUC)
Abstract: We will talk about magic square games, their exact strategies, and their rigidity properties. https://illinois.zoom.us/j/253149081 I will lock the meeting after 15 minutes, contact me if you have to enter later. |
# Link between quantummechanical probabilistics and entropy
1. Oct 13, 2011
### entropy1
I can put it into words like this: "Entropy: The greatest possible chance that outcomes even out (are as similar as possible)."
For instance: The chance that an off-axis polarized photon passes the filter. Sometimes it does, sometimes it doesn't, but take a large number of measurements, and the correlation between angle and probable pass-through are inexcapable. I don't know very much about thermodynamics, but I figured that pressure in a closed box with gas also 'evens out' due to entropy (the particles becomes evenly spread), so that the probability a particle has a specific momentum becomes as great as possible. Similar, the probability a photon passes the filter is directly proportional to the angle of the filter, so the proportion of particles that pass to those blocked is 'evenly spread'... (grosso modo, each photon 'behaves' similar, or at least, becomes more statistically probable to do so...)
Does this make the tinyest bit sense?
I hope you can forgive me my poor english.
Last edited: Oct 13, 2011
2. Oct 13, 2011
Staff Emeritus
First, I don't see why we need a different definition of entropy than what is in the textbooks.
Second, there is no tendency for outcomes to "even out". Every trial is independent of previois trials.
3. Oct 14, 2011
### entropy1
I was thinking there might be a correlation between outcomes of different measurements, like in quantumentanglement, a way of nature to compensate information that occurs in one part in the universe, in the other part of it, so that the independent appearance of outcomes is just an illusion. I am not sure if I state this correctly: Of all the states of a given system, the one with the highest entropy is the most likely one.
I feel my knowledge falls short here :tongue: I will try to catch up a bit.
Last edited: Oct 14, 2011 |
3rd International Conference on Modern Mathematical Methods and High Performance Computing in Science and Technology
Thursday, January 9, 2020 - 9:00am to Saturday, January 11, 2020 - 6:30am
CALL FOR SPECIAL SESSIONS/ WORKSHOP
General Information: M3HPCST-2020 invites proposals for workshops to be held in conjunction with the conference. The aim is to provide researchers in focused areas the opportunity to present and discuss their work, as well as to offer a forum for interaction among a broader community of researchers related to the main topics of M3HPCST-2020. The papers will be required to meet the same standards as the main conference papers and will be published in the conference proceedings, available on the Springer web page. The organizers of the workshop are encouraged to make a plan for post-conference special issues in some high-quality international journals.
Proposal submission: Please send the WS/SS proposals (in PDF format) to Dr. Vinai K. Singh (Email: m3hpcst-2020$ipec.org.in ) by May 10, 2019 with the following information: 1. Title and acronym of the WS/SS 2. General description of the WS/SS 3. Objectives, scope, and topics of the SS/WS 4. Names, contacts, and short profiles of the organizers 5. (optional) Plan for a post-conference special issue • Journal name • Contact of the guest editors General rules for the organizers: The organizers will be responsible for the advertisement and promotion of the workshop and the conference. The organizers and chairs of the workshop shall have full control on the call for papers, forming of program committees, review and selection of papers as well as planning the workshop program. Once accepted, the paper will be included into the Springer conference proceedings published by Springer International Publishing AG Switzerland At least one of the authors of any accepted paper is requested to register the paper at the conference. One complimentary registration will be given to the leading chair of each workshop with more than 15 registered paper. Important Dates: WS/SS Proposal: May 31,219 WS/SS Notification: Within two weeks after receiving a proposal Submission of papers: August 15, 2019 Notification of acceptance: October 15, 2019 Camera-ready papers: November 30, 2019 Early-bird registration October 31 2019 Conference January 9-11, 2020 Main Contact: Please contact Dr. Vinai K. Singh at m3hpcst-2020$ipec.org.in for more details.
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3rd International Conference on Modern Mathematical Methods and High-Performance Computing in Science & Technology (M3HPCST-2020) is a premier international conference on topics at the confluence of high performance and large scale computing systems, their use in modeling and simulation, their design, performance and use, and their impact, and related issues.
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• Contemporary Approaches In Multivariate Representations And Approximation Methods For Discrete And Conttinuous Mathematical Objects
• Le Symmetry Analysis and Conservation Laws for Nonlinear Differential Equations and Applications
• Statistical Modeling and Applications
• Complex Networks
• Processing, modelling, and describing time series
• Data mining and engineering
• Metaheuristics in science and engineering
• Linear Algebra, Matrix Analysis and Applications
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• Uncertainty Quantification and Mathematical Modelling
• Mathematical Modeling and Numerical Simulation of Geophysical Flows
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# How to find the remainder when the following series is divided by 12? [duplicate]
$1! + 2! + 3!+\cdots + 99! + 100!$
I am not getting any idea on how to solve this problem. I know that modular arithmetic should be used but not getting how to start off with the solution. Please give me some hint on how to approach this question. Thanks in advance.
• HINT: Think of the first factorial that is divisible by $12$ - lets call this $x$. So we know $x!$ is divisible by $12$. Then you know that all others after this will also be divisible by $12$ because for example $(x+1)!=(x+1)*x!$ and $x!$ is divisivble by $12$. That should drastically reduce the set of numbers you need to consider. – Mufasa Aug 26 '14 at 16:11
• – Yiorgos S. Smyrlis Aug 26 '14 at 16:32
• $1!+2!+3!+\cdots+99!+100!=(1+2+6)+12k$ – BigM Aug 26 '14 at 16:41
Hint: Note that $4!\equiv 0\pmod{12}$.
Since $4!$ is congruent to $0$ (mod $12$) then any multiple of $4!$ is congruent to $0$ (mod $12$). So we need only look at the first 3 terms, and since each of the first 3 terms is congruent to themselves (mod $12$), then the addition of all the terms in (mod $12$) is:
$(1! + 2! + 3! + 0 + 0 +...+ 0)$(mod 12). So the remainder should be 9.
Hint: Most elements in the sum are divisable by $12$
Hint $\ n(n\!-\!1)\mid n!\mid (n\!+\!1)!\mid (n\!+\!2)!\mid \cdots\$ for $\, n>1$
• Generally $\ a_n\! > \cdots\! > a_2 > a_1 \ge 1\,\Rightarrow\, a_n\! \cdots a_2 a_1\mid a_n!\,\$ i.e. $\ S \subset T\,\Rightarrow\, \prod S\, \mid\, \prod T\ \ \$ – Bill Dubuque Aug 26 '14 at 16:20 |
# Efficient display of simple text/graphics on color LCD by ARM
When designing an ARM-based device that should display simple graphics on a color LCD, how should one best go about designing things to allow fast updates, preferably without being tied to a particular ARM or LCD vendor? My current project uses a black-and-white display which can be driven lightning-fast by a the SPI port on a PIC (redrawing a complex display in 1/60 second). It seems common color LCD displays have an SPI port, but even filling a 160x120 LCD with a solid color would take 30ms, and a 320x240 would take 120ms best-case (10MHz shift clock).
If one could spare the controller pins, parallel mode could be better, but I don't know of any family-independent means of hooking up the parallel interface without requiring three separate memory-store instructions for each pixel (one to set the data, one to set the clock output high, and one to clock it low). Some ARM chips have memory-bus interfaces, but those often want to do things like multiplex address and data, or commit a lot of pins to outputting irrelevant address bits (the LCD would just need one address bit).
Looking at the ILI9320 by ILITEK, or the HD66789 by Renesas, one approach that would seem interesting would be to use a CPLD to convert SPI to parallel data, and include a mode which would output a pixel per bit. Looking at the Renesas data sheet, it might be possible to get pixel-per-bit writes with minimal hardware (no CPLD required) by making all the parallel-port data bits track the serial-data pin, using serial mode for everything but pixel writes, and using the compare/mask functions so that either all-zeroes pixels would be transparent and all-ones pixels would set selected bits in GRAM, or all-ones pixels would be transparent and all-zeroes pixels would clear selected bits. The "features" section of the IKITEK data sheet suggests that it has similar functionality, but the register maps don't seem to include the appropriate registers (most of the appropriate register addresses don't appear in the IKITEK, but one of them--register 4, is used for a different purpose).
Assuming the code will mainly be showing solid-color text and graphics, the ideal approach would seem to be to use a CPLD to interface the ARM's SPI port to the display's parallel port, and allow the CPLD to be loaded with foreground/background colors. This would be especially nice if one had a means of writing "transparent" pixels. Given a font as a two-color bitmap, one could simply load the font data directly into the SPI port; this would allow font data to be shown at a rate of one pixel every two ARM clocks. On the other hand, a CPLD sufficient to handle such a display-control task would cost about $2. What's the best way to interface an ARM with a color LCD, if the objective is to mainly to show solid-color text or simple (e.g. 16-color or 64-color) graphics? Edit I've done many LCD display projects, with many types of LCDs, including character-mode LCDs, custom 3:1 multiplexed segment-based using my own drive method, black and white graphic LCDs with built-in controllers, and black-and-white LCDs for which I designed my own CPLD-based controller to interface with a microcontroller's general-purpose DMA (providing four-level grayscale even). I pride myself on making displays zippy. One of the graphic controllers was a bit of a dog that required about 1/10 second for a full screen refresh even when writing constant data, but most of my displays can render even a fairly complex image in under 1/50 second. Many of the projects I do are battery-powered, so current draw is an issue. The DMA-based display controller I did worked nicely, but it was for a line-powered project. I believe the only way to get reasonable current draw from a graphics LCD is to use a controller which combines the display buffer and the column drivers. Sending lots of display between chips every frame would waste a lot of energy even on a single bit-per-pixel display; on a color display with sixteen bits per pixel, it would be far worse. I've only started looking at color LCD data sheets; many displays seem to use a controller similar to the ILITEK ILI9320, though all of the data sheets I've found for controllers based on that general design have been marked "preliminary". Some like the ILITEK one claim to have masking and transparency features but don't list any registers for them; I don't know whether the real chips have such features but the "preliminary" data sheets neglected to include them, or whether they omitted the features but forgot to strike the mention of them. If in practice all such chips have transparency features, it would seem reasonable to design for them; if not, not. I would expect that for most projects a typical screen would consist of arbitrarily-placed text in moderate number of arbitrarily-sized solid-color fonts. Fonts would most likely be stored as bit-per-pixel data. Using a Cortex-M3, if I wanted to write the display with parallel data, the code's "inner loop" to write two pixels would probably end up something like: rol r0,r0,#2 ; Get one bit in C, the other in N itcs strhcs r1,[r3,#DATA_OFS] ; Write data strhcc r2,[r3,#DATA_OFS] ; Write data strb r4,[r3,#CLOCK_SET_OFS] ; Set clock high strb r4,[r3,#CLOCK_CLR_OFS] ; Set clock low itmi strhmi r1,[r3,#DATA_OFS] ; Write data strhpl r2,[r3,#DATA_OFS] ; Write data strb r4,[r3,#CLOCK_SET_OFS] ; Set clock high strb r4,[r3,#CLOCK_CLR_OFS] ; Set clock low Not exactly the fastest thing in the world. Eliminating the writes to the set/clear clock instructions would help. My guess would be that there's no nice architecture-independent way to eliminate both clock writes, but there may be a pretty common way that would allow for eliminating one (e.g. many chips may have a counter/PWM that could be made to pulse an output briefly in response to a single memory store operation). Using the SPI port and adding hardware to clock one pixel per bit would greatly speed up display access. If using a display without masking and transparency, the CPLD would have to include an address counter, and for each pixel either clock a word of pixel data or else a set-address command for the following pixel's position (for which it would need a counter). By contrast, if a display had masking and transparency, all I would need to do would be to have the CPLD support a mode where after it had clocked in 16 bits, each additional bit would clock a word of data out to the display with the LSB tracking the SDI pin (it might not even be necessary to use a CPLD--just a few normal logic chips). I would set the transparency color to the be color I want to write but with the LSB flipped. I don't want to come up with a beautiful design that relies upon masking and transparency and then discover that the only displays with such features have a 30-week lead time. On the other hand, if such displays are apt to be and remain widely available from many vendors, I don't want to let paranoia about availability drive me to use an inferior design. • Not an answer since your requirements include not being tied to a specific ARM vendor, but the LPC LH754xx family of microcontrollers includes an integrated LCD driver. – Kevin Vermeer May 5 '11 at 19:26 • @reemrevnivek: There are a number of ARM chips with small LCD drivers; I can't imagine any chip having a driver suitable for any useful size graphics display appearing in a package that would be usable in anything other than a chip-on-glass scenario. A chip might have a controller, but an LCD with a chip-on-glass controller would seem more power-efficient and easier to work with. I'll check out the chip you mentioned, though--might be interesting. – supercat May 5 '11 at 19:49 • @supercat - I'm thinking of LCDs that have an RGB interface: pixel clock, frame sync, and line sync control lines, with a parallel pixel data bus. Are you expecting to use a COG controlled display? – Kevin Vermeer May 5 '11 at 20:11 • @reemrevnivek: That's what I'd been thinking. They seem to be pretty common, since they're used in a lot of portable battery-powered devices like cell phones. A COG display with built-in controller is going to be much more power-efficient than one which requires continuously-clocked RGB data. – supercat May 5 '11 at 21:01 • @reemrevnivek: I just updated my question with more detail. – supercat May 6 '11 at 15:53 ## 1 Answer The problem with using a microcontroller to drive an LCD is that an LCD requires constant attention. This can be mitigated with a CPLD driven over SPI (using DMA, of course), but then you run into the other problem: Color LCDs require a lot of data. 320x240 in black and white is marginal at 9.6KB, but make it 24 bit color and suddenly you need to deliver 230KB of data in 1/60th of a second. (Don't forget, though, that you can get 4-bit, 16-color control just by tieing the low 20 bits to one setting). A 24-bit frame buffer no longer fits in onboard RAM on most microcontrollers, and you probably don't have time to read from an external RAM chip, clock the data out, and still do other processing. Trying to do this with a CPLD (or an FPGA) and a RAM chip gets you well over the$2 price that caused you to balk in your question.
The traditional solution to interfacing a microcontroller with a color LCD is a display controller like an SSD1963. Here's a very simple block diagram:
Parallel input to a big RAM frame buffer (Translation: More than \$2) interfaced with a register-configurable parallel LCD interface. The parallel input is usually compatible with a memory bus interface.
The color LCD market is not always easy to find on the web, usually being the domain of OEMs only, with the rest buying displays from companies who integrate the controller with the display. The best resource I've found has been Crystal Fontz, specifically this page on choosing graphic LCDs. Scroll to the bottom for the controllers, which include the following options (note: Not all are color controllers):
• Epson S1D13521B01 E Ink Broadsheet (1 module)
• Epson S1D13700 (11 modules)
• Epson SED1520 Compatible (8 modules)
• Himax HX8345 Compatible (1 module)
• ILITek ILI9325 Compatible (3 modules)
• KS0107/KS0108 Compatible (26 modules)
• Novatek NT7534 (14 modules)
• Orise Technology OTM2201A (1 module)
• Orise Technology SPFD5420A (1 module)
• RAiO RA8835 (1 module)
• Sanyo LC7981 (13 modules)
• Sino Wealth SH1101A (2 modules)
• Sitronix ST7920 (29 modules)
• Solomon SSD1303 (1 module)
• Solomon SSD1305 (9 modules)
• Solomon SSD1325 (2 modules)
• Solomon SSD1332 (1 module)
• Solomon SSD2119 (2 modules)
• ST STV8105 (1 module)
• Toshiba T6963 (23 modules)
• @reemrevnivek: I'd been thinking of color LCDs with built-in controllers. They seem pretty common, but the ones I've seen generally seem to expect the CPU to clock in many bits per pixel even though a common display scenario is the display of solid-color text. I implemented a DMA-based 4-level grayscale LCD controller once using a CPLD, and it worked very nicely, but that was a line-powered device. – supercat May 5 '11 at 21:04
• @supercat - Very few LCD controllers expect a CPU to clock many bits per pixel in for each frame. They generally are expecting dedicated graphics hardware to be doing this. Basically, once you get to fairly large (i.e. > 128*128) RGB displays, the processing power required to generate the image for the screen is large enough that a dedicated GPU of some sort (even if it is integrated into the MCU) is pretty much always present. – Connor Wolf May 7 '11 at 5:35
• @Fake Name: If the goal was to display full motion video, some specialized hardware would be necessary. But my goal is to display plain text and simple graphics, quickly. The controllers I've been looking at have a maximum SPI data rate of 10mbps; that limit would apply whether the data was being supplied by a CPU or a dedicated controller. Using a parallel-input mode would speed things up, but if character shapes are stored with 1 bit per pixel (as would be typical) using code to unpack them would be slower than having the ARM output SPI and having a CPLD convert that to parallel data. – supercat May 7 '11 at 7:17
• @supercat - But what you describe, a specialized CPLD that does ASCII to raster conversions, basically is (custom) specialized graphics hardware. I'm basically saying don't re-invent the wheel, and it's probably easier and more cost-efficient to just buy a MCU with a video-interface built-in then design one yourself. – Connor Wolf May 7 '11 at 7:27
• Anyways, If you really want to roll-your-own, I would say use a couple of dual-port SRAM ICs, and use one port for outputting to the LCD, and the other for the MCU. This lets the MCU change the memory contents at whatever speed it wants, and the LCD can run at it's refresh rate. – Connor Wolf May 7 '11 at 7:29 |
## The Annals of Applied Probability
### The large deviations of a multi-allele Wright-Fisher process mapped on the sphere
F. Papangelou
#### Abstract
This is the fourth in a series of papers devoted to the study of the large deviations of a Wright –Fisher process modeling the genetic evolution of a reproducing population.Variational considerations imply that if the process undergoes a large deviation, then it necessarily follows closely a definite path from its original to its current state. The favored paths were determined previously for a one-dimensional process subject to one-way mutation or natural selection, respectively, acting on a faster time scale than random genetic drift. The present paper deals with a general $d$-dimensional Wright–Fisher process in which any mutation or selection forces act on a time scale no faster than that of genetic drift. If the states of the process are represented as points on a $d$-sphere, then it can be shown that the position of a subcritically scaled process at a fixed “time” $T$ satisfies a large-deviation principle with rate function proportional to the square of the length of the great circle arc joining this position with the initial one (Hellinger–Bhattacharya distance). If a large deviation does occur, then the process follows with near certainty this arc at constant speed. The main technical problem circumvented is the degeneracy of the covariance matrix of the process at the boundary of the state space.
#### Article information
Source
Ann. Appl. Probab., Volume 10, Number 4 (2000), 1259-1273.
Dates
First available in Project Euclid: 22 April 2002
https://projecteuclid.org/euclid.aoap/1019487616
Digital Object Identifier
doi:10.1214/aoap/1019487616
Mathematical Reviews number (MathSciNet)
MR1810874
Zentralblatt MATH identifier
1073.60511
#### Citation
Papangelou, F. The large deviations of a multi-allele Wright-Fisher process mapped on the sphere. Ann. Appl. Probab. 10 (2000), no. 4, 1259--1273. doi:10.1214/aoap/1019487616. https://projecteuclid.org/euclid.aoap/1019487616
#### References
• [1] Amari, S. (1985). Differential-Geometrical Methods in Statistics.Lecture Notes in Statist. 28. Springer, Berlin.
• [2] Atkinson, C. and Mitchell, A. F. (1981). Rao's distance measure. Sankhya Ser.A 43 345-365.
• [3] Azencott, R. (1980). Grandes D´eviations et Applications.Lecture Notes in Math. 774 1-176. Springer, Berlin.
• [4] Dawson, D. A. and Feng, S. (1998). Large deviations for the Fleming-Viot process with neutral mutation and selection. Stochastic Processes Appl. 77 207-232.
• [5] Ewens, W. J. (1979). Mathematical Population Genetics. Springer, New York.
• [6] Fisher, R. A. (1922). On the dominance ratio. Proc.Roy.Soc.Edinburgh 42 321-341.
• [7] Fleming, W. H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ.Math.J. 28 817-843.
• [8] Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ.
• [9] Kimura, M. (1964). Diffusion Models in Population Genetics. Methuen, London.
• [10] Morrow, G. J. (1992). Large deviation results for a class of Markov chains with applications to an infinite alleles model of population genetics. Ann.Appl.Probab.2 857-905.
• [11] Papangelou, F. (1996). Large deviations of the Wright-Fisher process. Lecture Notes in Statist. 114 245-252. Springer, New York.
• [12] Papangelou, F. (1998). Tracing the path of a Wright-Fisher process with one-way mutation in the case of a large deviation. In Stochastic Processes and Related Topics-A Volume in Memory of Stamatis Cambanis (I. Karatzas, B. Rajput and M. S. Taqqu, eds.) 315-330. Birkhauser, Boston.
• [13] Papangelou, F. (1998). Elliptic and other functions in the large deviations behavior of the Wright-Fisher process. Ann.Appl.Probab.8 182-192.
• [14] Papangelou, F. (2000). A note on the probability of rapid extinction of alleles in a Wright- Fisher process. In Probability and Statistical Models with Applications-A Volume in Honor of T.Cacoullos (Ch. A. Charalambides, M. V. Koutras and N. Balakrishnan, eds.) 147-154. Chapman and Hall, New York.
• [15] Schied, A. (1997). Geometric aspects of Fleming-Viot and Dawson-Watanabe processes. Ann.Probab.25 1160-1179.
• [16] Wentzell, A. D. (1990). Limit Theorems on Large Deviations for Markov Stochastic Processes. Kluwer, Dordrecht. |
# How to recreate this 3D plot?
In other languages, one can make a FeaturePlot3D-esque graphics where the inset images look realistically three dimensional. Meaning that they are smaller when farther away (and also hazier).
So how can we exactly recreate this clustering of mnist:
Is there a way to recreate this effect in Graphics3D or FeaturePlot3D? Even trying the naive example above actually makes the front-end hang:
FeatureSpacePlot3D[ResourceData["MNIST"][[All, 1]], Method -> "TSNE"]
• You can get a better sense of geometrical perspective by adjusting the effective viewing distance, e.g., between Viewpoint -> {1,1,1} and {10,10,10}. Mathematica does not render lighting and shadows, so you'll never get the kind of realism of software designed for true computer graphics. – David G. Stork Aug 16 '18 at 18:31
• Considering view point as a minor part, it looks like a duplicate: mathematica.stackexchange.com/q/55174/5478, do you agree? – Kuba Aug 20 '18 at 6:16
• So you've found time to set a bounty but not to respond to my comment. Which is confusing because that would clarify what is a valid answer here. – Kuba Aug 30 '18 at 5:30
• Is it about the performance? What is missing? – Johu Aug 30 '18 at 10:22
• If you stack a Image3D instance with Graphics3D, it will emulate fog effect. It was asked before, I but forgot where. – kh40tika Sep 2 '18 at 11:35
The simple way is combination of some 3D objects by Show like following: 1. Lets make the 3D-shift functions defining the positions of objects in joint figure (+-0.25 in X-direction and 0.25 in Y-dir.):
shift = {#[[1]] + 0.25, #[[2]], #[[3]]} &;
shift2 = {#[[1]] - 0.25, #[[2]] + 0.25, #[[3]]} &;
Next, let's take the samples of 3D shapes from Wolfram:
sample1 = shift /@ ExampleData[{"Geometry3D", "StanfordBunny"},"VertexData"];
sample2 = shift2 /@ ExampleData[{"Geometry3D", "Cow"}, "VertexData"];
And draw them:
Show[
ListSurfacePlot3D[sample1, MaxPlotPoints -> 50, PlotStyle -> Red],
ListSurfacePlot3D[sample2, MaxPlotPoints -> 55]
]
Sorry, but cow looks ill :)
• The cow looks less ill when you use the respective MeshRegions: Show[ MeshRegion[ Translate[ ExampleData[{"Geometry3D", "StanfordBunny"}, "MeshRegion"], {0.25, 0., 0.} ], MeshCellStyle -> {{2, All} -> Directive[FaceForm[ColorData[97][4]]]} ], MeshRegion[ Translate[ ExampleData[{"Geometry3D", "Cow"}, "MeshRegion"], {-0.25, 0.25, 0.} ], MeshCellStyle -> {{2, All} -> Directive[FaceForm[ColorData[97][1]]]} ] ]. But I doubt that this will answer OP's question. – Henrik Schumacher Aug 30 '18 at 6:05
• Mathematica also has built in commands for coordinate transformations, like Translate. – Johu Aug 30 '18 at 10:23
• @HenrikSchumacher, The perspective can be added directly clicking onto the plot. All rest looks like described in OP. – Rom38 Aug 30 '18 at 13:56
• @Rom38 I guess, OP's question was more about the fog. That's funny because we all remember these 3D-games from the 90s where all this fog was necessary only because of poor hardware. ^^ – Henrik Schumacher Aug 30 '18 at 15:07 |
# is_on_arc (method)¶
is_on_arc(self, Z)[source]
Check is a point defined by its complex coordinate is on the arc
Parameters: self (Arc) – An Arc object Z (complex) – Complex coordinate of the point is_on_arc – True if the point is on the arc bool |
# Closed subsets of compact sets are compact: Non-compact and closed sets have non compact supersets
Studying Rudin's Real Analysis and after proving the theorem "Closed subsets of compact sets are compact" I have came up with the following thought: The theorem means that if a set is closed and has a compact superset, then it is compact. This also means that:
A set $K$ is not compact $\implies$ $K$ is not closed or $K$ has no compact superset.
Assume that $K$ is not compact and $K$ is closed. Then it must be that $K$ has no compact superset. I wanted to show that. Trivially, if for every $E$, which is $E \subseteq K$, $E$ is not compact, then since $K \subseteq K$, $K$ is automatically not compact. While this looks valid, it seemed too trivial to be true to me. Is this really valid? If not, how can one show that when $K$ is not compact, but closed, it has no compact supersets?
You want to show that $K$ has no compact superset. That means you have to show that if $K \subseteq E$, then $E$ is not compact.
Let if possible, $E$ is compact. As closed subsets of compact sets are compact, then $K$ is compact, which is a contradiction.
Let $\bigcup_{\alpha} U_{\alpha}$ be any open cover of $K$. As $K$ is closed in $X$(our topological space) and $K \cap E=K$, therefore $K$ is closed in $E$. Hence, $E \setminus K$ is open in $K$. Now, $$E=\bigcup_{\alpha} U_{\alpha} \cup (E \setminus K)$$ If, $E$ is compact, then this open cover has a finite subcover. That is, there exists, $\alpha_1, \alpha_2, \cdots \alpha_n$ such that $$E=\bigcup_{i=1}^n U_{\alpha_i} \cup (E \setminus K)$$Hence, $$K\subseteq\bigcup_{i=1}^n U_{\alpha_i}$$ Thus, we have shown that an arbitrary open cover of $K$ has a finite subcover. It follows that $K$ is compact, which is a contradiction. |
Technical Article
# RTD Signal Conditioning—4-Wire Configuration, Ratiometric Measurement, & Filtering
November 18, 2022 by Dr. Steve Arar
## Learn about RTD (resistance temperature detector) signal conditioning through a four-wire configuration, ratiometric measurement, and input RC filters.
Previously, we explored two-wire and three-wire configurations for both voltage- and current-excited RTD measurements. This article extends the discussion to the four-wire configuration and delves into ratiometric measurement, which is widely used in RTD applications. In addition to that, we’ll go over how RC input filters can be used in ratiometric configurations and learn how matched input and reference path filters can improve the noise performance of a ratiometric configuration.
### RTD 4-Wire Configuration—Voltage Drop and Kelvin Sensing
Figure 1 below shows a four-wire wiring technique for a current-excited RTD.
##### Figure 1. Block diagram of a four-wire technique in a current-excited RTD.
The analog-to-digital converter (ADC) inputs are high impedance, which causes the excitation current flows through Rwire1, Rrtd, and Rwire4. Since no current flows through Rwire2 and Rwire3, no voltage drops across these two resistors, and the ADC can accurately measure the RTD voltage Vrtd.
While a three-wire configuration requires two matched current sources to eliminate the wire resistance error, the four-wire configuration can achieve this with a single current source. Note that the above method, also called Kelvin sensing, is a general resistance measurement technique that finds use in many other areas, such as resistive current sensing applications.
The four-wire measurement concept can also be applied to a voltage-excited RTD, as illustrated in Figure 2.
##### Figure 2. A block diagram showing the four-wire measurement concept in a voltage-excited RTD.
Again, no voltage drops across Rwire2 and Rwire3, and the ADC accurately measures the voltage across the RTD Vrtd. In a voltage-excited system, the excitation voltage Vexc is known. However, it is impossible to determine the RTD resistance by knowing Vrtd and Vexc because some unknown voltages also drop across Rwire1 and Rwire4. To combat this problem, we can make an extra measurement at a node, such as node B in the above diagram, to figure out the current flowing through the sensor. This is similar to the method we used when discussing the voltage-excited three-wire configuration in the previous article.
Note that with the current excitation, a second measurement is not required because the current flowing through the sensor, Iexc, is already known. The current excitation method is a more straightforward implementation, especially when the wire resistance error is an issue.
### Basics of Ratiometric Measurement
All RTD measurement circuits require an accurate and stable excitation source, as the RTD voltage is a function of the excitation source. For example, consider the circuit diagram in Figure 1. The voltage measured by the ADC relates to the RTD resistance by the following equation:
$V_{ADC}=R_{rtd}\times I_{exc}$
If the excitation current is noisy or drifts with temperature or time, the voltage across the RTD changes even when the temperature is fixed. To maintain high accuracy, the designer needs to use precision components to minimize variations in Iexc
Alternatively, you could use ratiometric measurement. Rather than minimizing the excitation source variations, ratiometric measurement changes the circuit so that the output becomes proportional to the ratio of Iexc to another current (or voltage) in the system.
Let's assume that the circuit is modified in a way that the output equation is changed to:
$V_{ADC}=R_{rtd}\times\frac{I_{exc}}{I_{x}}$
Where Ix is a current in the circuit. Also, if we derive Ix from Iexc in a way that they both experience the same variation, the ratio $$\frac{I_{exc}}{I_{x}}$$ can be kept constant. This makes the measurement system insensitive to the excitation source variations.
In the next section, we’ll see that ratiometric measurements can usually be implemented inexpensively. This inexpensive implementation allows us to use ratiometric configurations to improve accuracy and relax the requirements of certain components, such as the excitation voltage or current source.
### Ratiometric RTD Measurement
Figure 3 shows how the four-wire current-excited measurement can be modified to have a ratiometric configuration.
##### Figure 3. Block diagram showing the four-wire current-excited measurement can be modified to have a ratiometric configuration.
In this case, the excitation current is passed through a precision reference resistor Rref to create the ADC reference voltage. A buffer is used to sense the voltage across Rref without causing any loading effect on this resistor. Although the buffer is shown as an external component, it is normally integrated into the ADC chip, and an external buffer is not required.
From here, let’s see how the above circuit can produce a ratiometric measurement. The ADC input voltage and reference voltage are given by the following equations:
$V_{ADC}=R_{rtd}\times I_{exc}$
##### Equation 1.
$V_{ref}=R_{ref}\times I_{exc}$
##### Equation 2.
The digital output produced by an n-bit ADC can be typically described by the following equation:
$Digital\,Value=\frac{Analog\,Input\,Voltage}{ADC\,Reference\,Voltage}\times{\Big(}2^{n}-1{\Big)}$
The ADC output is proportional to the ratio of the input voltage to its reference voltage. Substituting Equations 1 and 2 into the above equation, we obtain:
$Digital\,Value=\frac{R_{rtd}\times I_{exc}}{R_{ref}\times I_{exc}}\times{\Big(}2^{n}-1{\Big)}$
This simplifies to:
$Digital\,Value=\frac{R_{rtd}}{R_{ref}}\times{\Big(}2^{n}-1{\Big)}$
The ADC output is no longer a function of the excitation current. However, Rref should be a low tolerance and low drift resistor since any unwanted variation in Rref directly translates into an error in the measurement result. Figure 4 shows the ratiometric configuration for a three-wire RTD application.
##### Figure 4. Example ratiometric configuration of a three-wire RTD application. Image used courtesy of TI
The ratiometric measurement concept can also be applied to a voltage-excited RTD. An example is shown in Figure 5.
##### Figure 5. An example ratiometric measurement block diagram of a voltage-excited RTD. Image used courtesy of Microchip
The above diagram uses the same voltage as the ADC reference voltage and the RTD excitation signal.
### Using an RC Low Pass Filter in Ratiometric Configurations
To attenuate the noise from the excitation current and the environment, RC low-pass filters are placed at the ADC input and reference paths of a ratiometric system. This is illustrated in Figure 6.
##### Figure 6. Using an RC low-pass filter at the ADC input and reference paths of a ratiometric system.
The ratiometric circuit can work without the use of external RC filters; however, the addition of low-pass RC filters can improve the circuit's immunity to radio frequency interference (RFI) and electromagnetic interference (EMI). The filter response for a common-mode noise can be understood by examining the following circuit diagrams in Figures 7a and 7b.
##### Figure 7. Example diagrams showing the filter response for common-mode noise.
As shown in Figure 7(a), with a common-mode input, the nodes C and D have the same potential. Thus, no current flows through C2, and this capacitor can be removed from the circuit model. This means that the C1 capacitors determine the common-mode cut-off frequency, which leads to Equation 3:
$f=\frac{1}{2 \pi R_1 C_1}$
##### Equation 3.
On the other hand, for a differential input, C2 can be replaced by a series connection of two 2C2 capacitors, as shown in Figure 8(b).
##### Figure 8. Example series connection diagrams.
Therefore, the differential cut-off frequency can be expressed as:
$f=\frac{1}{2 \pi R_1 \big ( C_1 + 2C_2 \big)}$
##### Equation 4.
Alternatively, Figure 7(b) shows that the common-mode cut-off frequency for nodes C and D are determined by the upper and lower C1 capacitors, respectively. A mismatch between these two capacitors can lead to a mismatch between the cut-off frequencies of the two paths. Through unequal attenuation of these two filters, the common-mode noise can produce a differential noise at the filter output, which is not at all desired.
To suppress the differential noise produced by mismatched common-mode capacitors, it is recommended that the differential capacitor C2 be at least 10x greater than the common-mode capacitor C1. In other words, the differential capacitor reduces both common-mode and differential noise components.
Several trade-offs should be considered when designing these simple RC filters. A thorough discussion on selecting the filter components to balance these trade-offs is not the goal of this article. However, an important point regarding ratiometric measurements needs to be highlighted: the effect of filter matching on the noise performance of a ratiometric system.
### Matched Filtering to Improve Noise Performance
In the previous section, we discussed that the mismatch of the C1 capacitors within each filter can cause problems (and hence, we added a differential capacitor to each filter). What about the mismatch between the input and reference path filters? To answer this question, note that the ratiometric system tries to make the measurement insensitive to the excitation source variations. This is achieved only if the excitation source variations have the same effect at the ADC analog inputs (IN+ and IN-) and reference inputs (REF+ and REF-). A mismatch between the cut-off frequency of the input and reference paths can lead to unequal attenuation of the excitation noise and reduce the effectiveness of the ratiometric configuration.
The remaining question is: what component values assure that the filters have the same cut-off frequency? Based on Equations 3 and 4, another application note from Analog Devices recommends using the same filters for the input and reference paths. The application note also provides some test results for the circuit diagram shown in Figure 9.
##### Figure 9. Example app note diagram. Image used courtesy of Analog Devices
Note that, compared to the general circuit in Figure 6, one resistor and two capacitors are eliminated in the reference path of the above circuit. This is because the REF- pin is connected to the ground in this design. The test results of this circuit are shown in Table 1.
##### Table 1. Data used courtesy of Analog Devices
ADC Gain ISOURCE (μA) Noise Voltage on 100 Ω Resistor (μV) R1 = R2 = R3 = 1k R1 = R2 = 10k R3 = 1k 16 100 1.6084 1.8395 16 200 1.6311 1.7594 16 300 1.6117 1.9181 16 400 1.6279 1.9292
This test uses a 100 Ω precision resistor instead of the RTD, and the noise voltage at the ADC input pins is measured. The value of RRef is 5.62 kΩ. When the two filters are identical (R= R= R= 1 kΩ), the noise voltage is reduced by about 0.1 µV to 0.3 µV compared to the unmatched case where R= R= 10 kΩ and R= 1 kΩ. In the above example, identical RC filters improve noise performance, but this is not necessarily the maximum achievable noise performance. This will be discussed in the following section.
### Improving Current Source Noise Cancellation
For example, an application note from Texas Instruments discusses that identical filters at the input and reference paths don't produce the maximum cancellation of the current source noise. When deriving Equations 3 and 4, we assumed that a common mode or differential noise appears at the filter inputs (nodes A and B).
This type of analysis is conceptually similar to applying a voltage source to nodes A and B to model the input noise. With this analysis, the effect of the Rrtd and Rref resistors, which is in parallel with the filters, is not considered. These two resistors actually modify the time constant of the RC networks. Since Rrtd and Rref are unequal, identical filters cannot have identical cut-off frequencies. The TI document I mentioned above suggests using the zero-value time constant technique to derive cut-off frequency equations of the two filters.
The zero-value time constant is a method of estimating the bandwidth of a system. For zero-value time constant analysis, the resistance “seen” by each capacitor is determined while the signal source is set to zero (the excitation current is replaced with an open circuit), and the rest of the capacitors are replaced with open circuits. The reason this method is called the zero-value time constant is that all capacitors except the capacitor of interest are set equal to zero to perform the calculation. If the circuit has m capacitors and the resistance seen by a given capacitor Cj is $$R^0_j$$ then the -3 dB bandwidth of the system can be estimated as:
$\omega_{-3dB} =\frac{1}{\sum_{j=1}^{m}R_j^0 C_j}$
##### Equation 5.
For example, to determine the resistance across the C2 and C4 capacitors in Figure 6, we obtain the circuit diagrams in Figure 10(a) and (b), respectively.
##### Figure 10. Diagrams showing the resistance across C2 (a) and C4 (b) capacitors
Equations 6 and 7 show the zero-value time constant (ZVT) associated with C2 and C4, respectively:
${ZVT}_{2}=C_2 \big (2R_1 + R_{rtd} \big)$
##### Equation 6.
${ZVT}_{4}=C_4 \big (2R_2 + R_{ref} \big)$
##### Equation 7.
Originally, the zero-value time constant method was developed to estimate the -3 dB bandwidth of the circuit. To do this, we calculate the time constant of all capacitors in the circuit and then plug them into Equation 5. However, the equation for each individual time constant shows how that particular capacitor interacts with its surrounding resistors to contribute to the circuit bandwidth.
Returning to our RTD measurement system, the input and reference paths will have identical bandwidths if the zero-value time constant of the three capacitors is the same. Therefore, ZVT= ZVT4, which leads to the following equation:
$C_2 \big (2R_1 + R_{rtd} \big)=C_4 \big (2R_2 + R_{ref} \big)$
##### Equation 8.
If C= C4, then the R1 and R2 resistors should be chosen appropriately to yield the same time constant. Based on the above discussion, the TI application note suggests the following example diagram in Figure 11.
##### Figure 11. Example block diagram of ADS1248. Image used courtesy of TI
The sensor resistance is assumed to change from 0 to 250 Ω. Since the variation in the sensor resistance changes the circuit time constant (Equation 6), relatively large resistors are used for the input filter (R= R= 6.04 kΩ). This makes the effect of the RTD variation on the frequency response of the input filter insignificant.
According to Analog Devices’ article, the resistors used in the reference path should be 6.04 kΩ. However, the TI design suggests using 5 kΩ resistors to match the bandwidth of the two filters. Figure 12 shows how the input-referred noise of the system changes with the input voltage level (i.e. the voltage across the RTD).
##### Figure 12. Graph showing the input-referred noise vs input voltage. Image used courtesy of TI
As you can see, the input-referred noise of the system is about 0.35 µVrms. The input-referred noise of the employed ADC (ADS1248) is typically 0.34 µVrms when the device is configured with a PGA gain of 8 V/V with a data rate of 20 SPS. Additionally, the system noise is close to the reported noise performance of the ADC. Note that when the input and reference path filters are not matched, the input-referred noise of the system can increase with the input signal level to values much higher than that of the ADC. Please refer to the above TI document for more information.
As a final note, it’s worthwhile to mention that the design in Figure 11 only matches the zero-value time constant of the differential capacitors (CIN_DIFF and CREF_DIFF). The time constant of the common-mode capacitors is not exactly the same. However, since the differential-mode capacitors are 10x larger than the common-mode capacitors, it seems that matching the time constant of the differential-mode capacitors has a greater impact on the frequency response of the filters.
Featured image used courtesy of Adobe Stock |
Considering the ease and frequency with which the Mitt Romney/Paul Ryan ticket is able to tell bald-faced lies, one has to wonder at their level of self-awareness that they are openly lying.
With Paul Ryan, there is a sense that he tends to be a compulsive liar and that his level of double-think, his ability to hold equally in mind totally opposing ideas, is fairly impenetrable. On the other hand, Mitt Romney seems fully aware of his lies and that he is committed to say whatever he has to in order to achieve his aims, essentially that his ends justify his means, not a bad way to become a multi-millionaire.
I suppose one could puzzle a bit over which form of lying is worse. Double-think runs strong through today's American psyche, but, in Romney's case, it should be pointed out that one's means generally become one's ends.
ROBERT PORATH
Boulder
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# Computational geometry relationship between 2 arcs
I'm writing a program which is making offsets for provided shapes. On the attached picture you can see example of my arc object and all known values. Let's assume that a direction is CW.
$$O$$ - center of circle for arc
$$P_s$$ - start point on circle for arc
$$P_e$$ - end point on circle for arc
s_ang - angle for $$P_s$$
e_ang - angle for $$P_e$$
ext - angle between $$P_s$$ and $$P_e$$
If you take a look on the picture how can I tell if they create orange(segment1 concave, segment2 convex), purple(both convex) or yellow(segment1 convex, segment2 concave) shape? |
Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, Epiciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
# The flat phase of quenched disordered membranes at three-loop order
Abstract : We study quenched disordered polymerized membranes in their flat phase by means of a three-loop perturbative analysis performed in dimension $D = 4-\epsilon$. We derive the renormalization group equations at this order and solve them up to order $\epsilon^3$. Our results confirm those obtained by Coquand et al. within a nonperturbative approach [Phys. Rev. E 97, 030102 (2018)] predicting a finite-temperature, finite-disorder wrinkling transition and those obtained by Coquand and Mouhanna within a recent two-loop order approach [Phys. Rev. E 103, 031001 (2021)], while correcting some of the results obtained in this last reference. We compute the anomalous dimensions that characterize the scaling behaviour at the various fixed points of the renormalization group flow diagram. They appear to be in strong agreement with those predicted within the nonperturbative context.
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Preprints, Working Papers, ...
https://hal.archives-ouvertes.fr/hal-03697772
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Submitted on : Friday, June 17, 2022 - 11:39:29 AM
Last modification on : Saturday, June 25, 2022 - 3:33:57 AM
### Citation
S. Metayer, D. Mouhanna. The flat phase of quenched disordered membranes at three-loop order. 2022. ⟨hal-03697772⟩
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User michal kotowski - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:23:57Z http://mathoverflow.net/feeds/user/2192 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65166/spectral-gap-for-random-bipartite-regular-graphs Spectral gap for random bipartite regular graphs Michal Kotowski 2011-05-16T19:18:48Z 2013-03-04T08:38:58Z <p>For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested in the spectral gap of $G$, i.e. the first nonzero eigenvalue of $\Delta$, denoted by $\lambda_{1}(G)$.</p> <p>Is it true that a randomly chosen (with uniform distribution) $d$-regular bipartite graph on $(n, n)$ vertices (with multiple edges allowed) has, with probability approaching $1$ as $n \to \infty$, $\lambda_1$ arbitrarily close to $1$ (i.e. we can make arbitrarily close by taking $d$ large enough)?</p> <p>If yes, is there a reference for this fact?</p> <p>Proofs of expanding properties for random regular graphs which I have found in the literature usually give the probability only bounded from below by a constant, i. e. $1/2$, although I imagine that actually almost all random graphs have good spectral gap.</p> <p>Note: by $d$-regular bipartite graph I mean a graph in which each vertex (on the left and on the right) has degree $d$.</p> http://mathoverflow.net/questions/46278/unexpected-applications-of-dvoretzkys-theorem Unexpected applications of Dvoretzky's theorem Michal Kotowski 2010-11-16T20:00:41Z 2012-12-01T04:29:25Z <p><a href="http://en.wikipedia.org/wiki/Dvoretzky%27s_theorem" rel="nofollow">Dvoretzky's theorem</a> is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem in quantum cryptography (which was solved previously, but using more complicated tools). What are the unexpected applications of Dvoretzky's theorem that you have heard of, if any?</p> <p>(by "unexpected" I mean applying it to a problem which is not directly connected to convex geometry, functional analysis etc. or perhaps is connected, but requires phrasing the problem in a different language in a non-obvious way)</p> http://mathoverflow.net/questions/91039/simple-uses-for-the-entropy-bound-on-the-volume-of-a-hamming-ball/103725#103725 Answer by Michal Kotowski for Simple uses for the Entropy bound on the volume of a Hamming ball Michal Kotowski 2012-08-01T20:16:04Z 2012-10-07T21:10:51Z <p>A nice application is showing that every Cayley graph of an Abelian group with a set of generators of logarithmic size has also logarithmic diameter.</p> <p>More precisely, let $G$ be an Abelian group and let $S$ be a symmetric generating set for $G$ of size $d = c_{0} \log n$ (where $n = |G|$ and $c_{0} > 0$ is a constant. Then for any $c_{1} > 0$ such that: $$(c_{0} + c_{1})H(\frac{c_{1}}{c_{0} + c_{1}}) < 1$$</p> <p>we have $diam(G) \geq c_{1} \log n$, where $diam(G)$ is the diameter of the Cayley graph of $G$ with generating set $S$.</p> <p>The proof uses a simple observation that the number of distinct pairs of endpoints of paths of length $l$ is at most $\binom{d+l}{l}$, since to determine an element of $G$ as a word in generators we only need to specify which generator appears how many times (because of commutativity the order is unimportant). So we have: $$\sum\limits_{l=0}^{c_{1} \log n} \binom{c_{0}\log n + l}{l} \leq 2^{(c_{0} + c_{1})H(\frac{c_{1}}{c_{0} + c_{1}}) \log n} < n$$ so the number of vertices reachable from a fixed vertex by a path of length $l \leq c_{1}\log n$ is strictly smaller than $n$. This implies that $diam(G) \geq c_{1}\log n$.</p> <p>This fact is used by Newman and Rabinovich in "Hard Metrics From Cayley Graphs Of Abelian Groups" to give a simple example of an $n$-point metric space which requires distortion $\Omega(\log n)$ to embed it into $\ell_{2}$.</p> http://mathoverflow.net/questions/89148/inequality-for-the-first-fourier-level-of-a-boolean-function Inequality for the first Fourier level of a Boolean function Michal Kotowski 2012-02-22T00:12:09Z 2012-09-08T13:24:59Z <p>In the study of Boolean functions, the hypercontractive inequality enables one to bound from above the norm of $Tf$ by some norm of $f$, where $T$ is the noise operator depending on the noise parameter. This can be written in terms of the Fourier transform of $f$ as (in a special case of $q=2$):</p> <p>$\left( \sum\limits_{S \subseteq [n]} (p - 1)^{|S|} \widehat{f}(S)^2 \right)^{1/2} \leq \left( \frac{1}{2^n} \sum\limits_{x \in (0,1)^n} |f(x)|^p \right)^{1/p}$</p> <p>However, this involves all Fourier levels of $f$. In the application I have in mind, I'm interested only in bounding the norm of the first level of $f$, i.e. restricting the sum on the left to $|S| = 1$. Is it possible to give any inequality of this kind, probably with a different right hand side (but still involving some information about the norm of $f$) and some additional assumptions on $f$? If it's impossible for some trivial reasons, let me know anyway.</p> <p>Here I'd be mostly interested in matrix-valued Boolean functions (see for example <a href="http://arxiv.org/abs/0705.3806" rel="nofollow">http://arxiv.org/abs/0705.3806</a>), although any answer would be appreciated.</p> http://mathoverflow.net/questions/104071/inner-products-and-gaussian-integration-over-intersection-of-halfspaces Inner products and Gaussian integration over intersection of halfspaces Michal Kotowski 2012-08-06T03:06:39Z 2012-08-06T03:14:40Z <p>Let $v$ be a unit vector in $\mathbb{R}^{n}$ and consider arbitrary unit vectors $w_1, w_2, \ldots, w_k$ also in $\mathbb{R}^n$. Let $g$ be a standard Gaussian random vector in $\mathbb{R}^n$. I'm interested in finding a formula for</p> <p>$$\mathbb{E_g} \left( \langle g, v \rangle f(\arg \max_{i} \langle g, w_i \rangle) \right)$$</p> <p>where the expectation is with respect to $g$ and $f(j)$ is equal to $1$ if $j=1$ and $-\frac{1}{k-1}$ otherwise (so $f$ chooses $w_i$ which is closest to the random vector). </p> <p>Ideally I'd like to express this quantity in terms of inner products $\langle v_1, w_i \rangle$. Sorry if the anser is easy, but here we are essentially integrating a Gaussian variable over sets defined by intersections of halfspaces (as $\arg \max_{i} \langle g, w_i \rangle) = j$ means that $\langle g, w_j \rangle \geq \langle g, w_i\rangle$ for $i \neq j$) and it doesn't seem trivial to me.</p> <p>Note that it is possible to calculate a somewhat similar integral in a simpler case: for any unit vectors $u,v$ we have: $$\mathbb{E_g}\left(\langle g, u \rangle sgn(\langle g, v \rangle) \right) = C\langle u, v\rangle$$ for an explicit constant $C$ (this is a simple exercise in Gaussian integration).</p> http://mathoverflow.net/questions/101524/set-of-unitaries-with-spread-like-properties Set of unitaries with "spread-like" properties Michal Kotowski 2012-07-06T18:53:31Z 2012-07-13T11:21:07Z <p>I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that:</p> <p>$\sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} |\mathrm{Tr}(YU_{j}XV_{k}^{\ast})|^2 \ll N$</p> <p>where the sup is over all Hermitian traceless matrices with Frobenius norm $1$ nad $\ll$ means that the sum is asymptotically smaller than $N$.</p> <p>The motivation comes from a problem in quantum information theory. An explicit construction of such sets of $U_{j}$ and $V_{k}$ would be the most desirable result, although pseudorandom constructions would also be very interesting. This statement is true if instead of taking unitaries we take matrices with independent Gaussian entries (rescaled properly so that on average each column has norm 1 etc.); however, I don't know how to show that random unitaries satisfy this property, so a hint for such a proof would also be welcome.</p> <p>I would like to think of such unitaries as having some sort of "spread" property similar to objects showing up in the study of pseudorandomness like mutually unbiased bases, randomness extractors etc. However, I'm not sure if there is any definite connection to those notions.</p> http://mathoverflow.net/questions/97444/concentration-of-functions-of-random-unitary-matrices Concentration of functions of random unitary matrices Michal Kotowski 2012-05-19T23:23:47Z 2012-05-19T23:23:47Z <p>Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$ in entries of $U$ and $V$? More specifically, I am interested in polynomials of the form:</p> <p>$\sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}$</p> <p>or</p> <p>$|\sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}|^2$</p> <p>where $X$ and $Y$ are some arbitrary matrices and the sum is over all indices. For matrices with i.i.d. Gaussian entries there are well-known concentration bounds for this kind of expressions, I would like to know if there is anything similar for unitary matrices (for this case or for $U=V$).</p> http://mathoverflow.net/questions/97155/tensors-with-low-spectral-norm Tensors with low spectral norm Michal Kotowski 2012-05-16T20:03:03Z 2012-05-16T20:03:03Z <p>Consider a tensor $T$ with six indices, $T_{(ii')(jj')(kk')}$, where each index goes from $1$ to $n$. We can think of $T$ as a linear map from $\mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$ to itself and consider its spectral norm:</p> <p>$\Vert T \Vert_{\infty} = \sup \vert \sum T_{(ii')(jj')(kk')} x_{ijk}y_{i'j'k'}\vert$ </p> <p>where the supremum is taken over all unit vectors $x,y \in \mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$.</p> <p>On the other hand $T$ can also be viewed as trilinear form on $n$-dimensional matrices, so we can define a norm:</p> <p>$\Vert T \Vert_{2,2,2} = \sup \vert \sum T_{(ii')(jj')(kk')} X_{ii'} Y_{jj'}Z_{kk'}\vert$</p> <p>where the supremum is over all matrices $X,Y,Z$ of Frobenius norm $1$.</p> <p>What are the examples of tensors which have high (as $n \to \infty$) spectral norm as linear maps, but low norm as trilinear forms?</p> <p>Since the question is admittedly rather general, let's specialize to tensors of a more special form, namely $T_{(ii')(jj')(kk')} = g_{ijk}h_{i'j'k'}$, where $g,h \in \mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$, so that $T = gh^{T}$ as a linear map. Then its spectral norm is simply $\Vert g\Vert \cdot \Vert h\Vert$. Taking $g$ and $h$, say, to be unit vectors, how should we choose them to get a low trilinear norm:</p> <p>$\Vert T \Vert_{2,2,2} = \sup \vert \sum g_{ijk}h_{i'j'k'} X_{ii'} Y_{jj'}Z_{kk'}\vert$ ?</p> <p>It can be shown that randomly chosen $g, h$ will have the desired property, but I'm interested in more explicit examples. Because there is no direct analog of the spectral decomposition for tensors, the intuition that the "mass" of $T$ should be "spread out" roughly in all directions (as in the case of matrices with low spectral norm, but high Frobenius norm) on the $\mathbb{R}^{n^2}$ components of the tensor product is not easy to formalize.</p> http://mathoverflow.net/questions/87711/derandomizing-random-matrices Derandomizing random matrices Michal Kotowski 2012-02-06T19:48:56Z 2012-02-08T20:30:35Z <p>My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates of eigenvalues, Dvoretzky's theorem, metric embeddings)? Here I'm interested both in fully explicit counterparts of random constructions as well as "pseudorandom" (in some sense) examples, using "less" randomness than, say, filling every entry of a matrix with a random variable etc. For example - suppose we know that for a fixed norm an n x n matrix with IID standard gaussian entries has "large" norm with high probability. How to find an explicit infinite family of such matrices?</p> <p>My question is rather vague, of course I have a specific application of this kind of results in mind, but at this moment I am more interested in general methodology of constructing "derandomized" examples, where to start looking for such objects etc. My only contact so far with pseudorandomness has been in the context of spectral graph theory, expander graphs, property (T) etc., I'm not sure if this perspective is relevant for high-dimensional geometry.</p> <p>I'd be grateful for any hints, references or advice on who may know this kind of things.</p> http://mathoverflow.net/questions/80451/casual-tours-around-proofs Casual tours around proofs Michal Kotowski 2011-11-09T04:33:58Z 2011-11-11T10:30:27Z <p>(this is basically the same question, only in math, as Alessandro Cossentino asked about theoretical computer science at TCS.SE: <a href="http://cstheory.stackexchange.com/questions/8869/casual-tours-around-proofs" rel="nofollow">http://cstheory.stackexchange.com/questions/8869/casual-tours-around-proofs</a>; if a question similar to this one already exists at MO, feel free to close this)</p> <p>Recently Ryan Williams published (see <a href="http://arxiv.org/abs/1111.1261" rel="nofollow">http://arxiv.org/abs/1111.1261</a>) a more "pedagogical" version of his proof in complexity theory concerning NEXP and ACC - in his own words, "the proof will be described from the perspective of someone trying to discover it". The paper discusses more intuition, failed attempts at solving the problem etc. much more extensively than a typical journal paper.</p> <p>Personally I find such efforts extremely valuable, because they give you an opportunity to learn how somebody <em>thinks</em> about mathematics and not only verify the formal correctness of some abstract reasoning.</p> <p>What other examples of such approach are you aware of?</p> http://mathoverflow.net/questions/79473/vector-bundles-on-graphs Vector bundles on graphs Michal Kotowski 2011-10-29T17:58:32Z 2011-10-30T18:28:28Z <p>Vector bundles over manifolds have fundamental importance in differential geometry, algebraic topology etc. Are there any applications of this concept (or some variation of it) for graphs (finite or infinite)?</p> <p>The only place I have seen something like this is in a paper on spanning forests by Kenyon, where the application seems somewhat specialized.</p> http://mathoverflow.net/questions/65557/random-versions-of-deterministic-problems Random versions of deterministic problems Michal Kotowski 2011-05-20T16:08:17Z 2011-10-07T13:55:30Z <p>What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version?</p> <p>An example of what I have in mind: it is a well-known conjecture that the Hausdorff dimension of the graph of Weierstrass function (everywhere continuous, nowhere differentiable) is given by a certain simple formula, depending on the amplitudes and phases of the cosines in the series. This is still open; however, in the paper "The Hausdorff Dimension of Graphs of Weierstrass Functions" Hunt proved that if you add a uniformly distributed independent random "noise" to each phase, the conjectured formula holds with probability 1. So while the "randomized" approach does not solve the original problem, it somehow lends credibility to the original conjecture and thus gives us some insight about the problem.</p> http://mathoverflow.net/questions/76227/random-bipartite-graphs Random bipartite graphs Michal Kotowski 2011-09-23T18:14:57Z 2011-09-24T21:06:30Z <p>Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I could put each edge independently with probability $p$ such that $pnN = M$, this shouldn't make a big difference). Then I remove the isolated vertices from $B$, so effectively I get vertex sets of size $n$ and $\Theta(n^{1 + \varepsilon})$.</p> <ol> <li><p>Are there any references on how in general such bipartite random graphs look like (their degree sequence, connectivity etc.)? The model considered above is rather specific, however, I'd be happy with any references on bipartite graphs on $(n, N)$ vertices, where $N$ depends on $n$ (or information on how can one tackle them with techniques similar to ordinary random graphs; in this context it isn't clear to me whether we should treat the graph as a "sparse" or a "dense" one).</p></li> <li><p>Ultimately I'm interested in spectral properties of such a graph (or rather a slight modification of it). What can be said about the second largest eigenvalue of its adjacency matrix or Laplacian? Now suppose that we take a union of this graph and a "good" graph (possibly also random) on the set $A$ only (by "good" I mean it has good spectral properties, I'm not trying to be very specific here). What can be said about eigenvalues of this graph?</p></li> </ol> http://mathoverflow.net/questions/75338/eigenfunctions-of-random-graphs Eigenfunctions of random graphs Michal Kotowski 2011-09-13T17:44:24Z 2011-09-21T21:37:43Z <p>Consider a random $d$-regular graph on $n$ vertices. What can be said about its nontrivial (i.e. orthogonal to the constant) eigenfunctions? For example, I'm interested whether there are "nodal zones", i.e. if the graph can be divided into groups such that the eigenfunction has the same sign on every vertex of the same groups. Are there any results of this kind?</p> <p>Another question - suppose I divide the graph arbitrarily into $k$ groups of vertices ($k$ may depend on $n$); is it possible to say something about how large the sum of eigenfunction's values on one group typically is? </p> http://mathoverflow.net/questions/64448/interesting-and-accessible-topics-in-graph-theory/64481#64481 Answer by Michal Kotowski for Interesting and Accessible Topics in Graph Theory Michal Kotowski 2011-05-10T09:25:42Z 2011-05-10T09:25:42Z <p>I think that presenting the connection between random walks and electrical networks (like in the classic text "Random walks and electric networks" by Doyle and Snell) is an interesting and feasible idea. Just a week ago I taught a 6-day course about this to talented high schoolers and it worked out very nicely. It's a good opportunity to show them interesting applications of probability and give a flavour of a vibrant field of mathematics. Plus, there is a quite a lot of room for digressions on Markov chains, spectral graph theory etc.</p> http://mathoverflow.net/questions/20393/path-integrals-outside-qft Path integrals outside QFT Michal Kotowski 2010-04-05T17:43:10Z 2011-04-06T20:23:15Z <p>The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical theory of functional integration is not (yet) rigorous.</p> <p>My question is: what are the applications of path integrals outside QFT? By "outside QFT" I mean non-QFT physics as well as various branches of mathematics.</p> <p>(a similar question is <a href="http://mathoverflow.net/questions/19490/doing-geometry-using-feynman-path-integral" rel="nofollow">http://mathoverflow.net/questions/19490/doing-geometry-using-feynman-path-integral</a>, but it concerns only one possible application)</p> http://mathoverflow.net/questions/55885/why-semigroups-could-be-important/55893#55893 Answer by Michal Kotowski for Why semigroups could be important? Michal Kotowski 2011-02-18T19:03:57Z 2011-02-18T19:03:57Z <p>An important application of semigroups and monoids is algebraic theory of formal languages, like regular languages of finite and infinite words or trees (one could argue this is more theoretical computer science than mathematics, but essentialy TCS <em>is</em> mathematics).</p> <p>For example, regular languages can be characterized using finite state automata, but can also be described by homomorphisms into finite monoids. The algebraic approach simplifies many proofs (like determinization of Buchi automata for infinite words or proving that FO = LTL) and gives deeper insight into the structure of languages.</p> http://mathoverflow.net/questions/51949/liouville-property-in-zd Liouville property in Z^d Michal Kotowski 2011-01-13T12:34:52Z 2011-01-17T01:19:20Z <p>It is well known that $\mathbb{Z}^d$ has Liouville property, i. e. every bounded harmonic function on this graph is constant.</p> <p>(harmonic means that the value of $f$ in a point $x$ is equal to the average of $f$ over neighbours of $x$ in the lattice $\mathbb{Z}^d$).</p> <p>What are the nicest/shortest/most ingenious proofs of this fact that you know?</p> http://mathoverflow.net/questions/51217/computer-science-for-mathematicians/51228#51228 Answer by Michal Kotowski for Computer Science for Mathematicians Michal Kotowski 2011-01-05T18:05:02Z 2011-01-05T18:05:02Z <p>"Computational Complexity"by Christos Papadimitriou - very good introduction to logic/theory of computation (Turing machines etc.) and computational complexity. One of the best textbooks IMHO. </p> http://mathoverflow.net/questions/47711/spectrum-of-the-laplacian-on-gn-p-and-gn-m Spectrum of the Laplacian on G(n, p) and G(n, M) Michal Kotowski 2010-11-29T19:31:52Z 2010-12-10T05:13:13Z <p>A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p$.</p> <p>A random graph in $G(n, m)$ model is a graph on $n$ vertices in which a subset of edges of a fixed size $m$ is chosen at random.</p> <p>We expect $G(n, p)$ and $G(n, m)$ for $m = p {n\choose{2}}$ to look asymptotically the same, because the number of edges in $G(n, p)$ is highly concentrated around the mean. </p> <p>The (normalized) Laplacian $L$ on a graph is an operator (matrix) which has entries:</p> <p>$L(v, v) = 1$</p> <p>$L(v, w) = - \frac{1}{\sqrt{deg(v)deg(w)}}$</p> <p>where $v \neq w$ are vertices of the the graph.</p> <p>We are interested in $\lambda_{2}$, that is, the smallest nonzero eigenvalue of $L$. Suppose we know that, for $p = p(n)$ growing sufficiently fast (papers by Chung et al. show that, for example, $p \geq \frac{\log^2 n}{n}$), $\lambda_{2}$ for a random graph in $G(n, p)$ model is close to 1 with high probability (i. e. approaching 1).</p> <p>Does it follow immediately that also in $G(n, m)$ we have $\lambda_{2} \to 1$ with high probability? It is known in random graph theory that such implications hold for monotone properties (that is, properties which still hold after adding an edge to the graph), however, the second eigenvalue is not monotone (although it is probably "monotone on average", so I expect the statement above is true for $G(n, m)$).</p> http://mathoverflow.net/questions/30886/applications-of-classical-field-theory Applications of classical field theory Michal Kotowski 2010-07-07T13:13:02Z 2010-07-07T20:36:38Z <p>What are the applications (physical and mathematical) of classical field theory beyond electrodynamics and gravity?</p> <p>By such applications, I mean that either the field theory viewpoint adds some genuinely new insight into the underlying physics or that it gives rise to interesting mathematical problems. So I'm not thinking about:</p> <p>-field-theoretical description of something that is very well understood with other tools (for example, describing classical electrodynamics in language of fibre bundles, differential forms etc. is very nice and elegant, but doesn't add much to physics)</p> <p>-quantum field theory (in QFT you always write down the classical lagrangian and then turn the fields into operators, but there is not much actual classical field theory here)</p> <p>Of course, you can always write some lagrangian like phi^34 + 14*phi^8 + ..., and study the resulting PDE (existence and uniqueness of solutions etc.), but I guess that lacks real motivation.</p> http://mathoverflow.net/questions/18163/geometric-group-theory-and-analysis Geometric group theory and analysis Michal Kotowski 2010-03-14T12:47:12Z 2010-03-15T03:48:34Z <p>Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. Is there any subfield of GGT where methods from analysis find applications? I once heard that analytical tools, e. g. geodesic flows, are used in studying ends of groups, but that's all I know. </p> http://mathoverflow.net/questions/11026/automatic-groups-recent-progress Automatic groups - recent progress Michal Kotowski 2010-01-07T09:36:47Z 2010-01-15T20:07:38Z <p>Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental groups of 3-dim manifolds etc. However, the book is from 1992, so much of the material summarizes research done by Cannon, Thurston, Holt etc. back in the '80s. I'm interested in how the theory of automatic groups (and, more generally, applications of formal languages in group theory) has progressed since then - have there been any significant new results, open problems, novel ideas, examples?</p> http://mathoverflow.net/questions/127563/number-of-cycles-in-expander-graphs Comment by Michal Kotowski Michal Kotowski 2013-04-14T23:58:26Z 2013-04-14T23:58:26Z You are right, the question is trivial, I'll delete it. http://mathoverflow.net/questions/110057/lost-soul-loneliness-in-pursing-math-advice-needed/110176#110176 Comment by Michal Kotowski Michal Kotowski 2012-10-20T21:22:42Z 2012-10-20T21:22:42Z I second that, a very nice notion :). I think I know what fedja is referring to, but it'd still be great to get some more explanation. http://mathoverflow.net/questions/103476/new-alternating-chain-algorithm-for-graph-coloring/103478#103478 Comment by Michal Kotowski Michal Kotowski 2012-07-29T22:07:55Z 2012-07-29T22:07:55Z A very appropriate comment for this sort of question. http://mathoverflow.net/questions/101524/set-of-unitaries-with-spread-like-properties/102131#102131 Comment by Michal Kotowski Michal Kotowski 2012-07-13T16:10:26Z 2012-07-13T16:10:26Z Thanks! I will read the answer more carefully later, but it seems that it solves the the question about random unitaries. Can you provide a reference for "by standard random matrix/free probability facts"? I know a little about random matrix theory, though not that much, and hardly anything about free probability. http://mathoverflow.net/questions/99506/blackbox-theorems/99579#99579 Comment by Michal Kotowski Michal Kotowski 2012-06-14T16:49:59Z 2012-06-14T16:49:59Z I think you are overestimating the difficulty of its proof. While it is not trivial, it is something standard and covered without any problem in a functional analysis course. If the proof is overlooked, that's probably usually because of, well, oversight, not because it is too complicated to be understood by the ordinary analyst. http://mathoverflow.net/questions/98594/regarding-cayley-graphs-of-property-t-groups Comment by Michal Kotowski Michal Kotowski 2012-06-01T19:12:46Z 2012-06-01T19:12:46Z @Agol: which result from Żuk's paper are you referring to? http://mathoverflow.net/questions/87711/derandomizing-random-matrices/87715#87715 Comment by Michal Kotowski Michal Kotowski 2012-02-07T16:20:32Z 2012-02-07T16:20:32Z @Mark: thanks! Are there such constructions for other properties of random matrices apart from RIP (and stuff related to compressed sensing)? http://mathoverflow.net/questions/87711/derandomizing-random-matrices/87715#87715 Comment by Michal Kotowski Michal Kotowski 2012-02-06T20:34:25Z 2012-02-06T20:34:25Z @Bill Johnson: if you have any suggestions for references or substantive comments, please consider putting them in an answer - that would be helpful! http://mathoverflow.net/questions/81128/when-is-it-appropriate-to-entitle-a-paper-a-note-on-or-on-the Comment by Michal Kotowski Michal Kotowski 2011-11-17T02:00:39Z 2011-11-17T02:00:39Z IMHO titles like "A note on ..." should be avoided, unless the title is long enough to make completely unambiguous what specific problem (conjecture etc.) is the topic of the paper. I guess everybody would be hard pressed to say what e.g. "A note on a problem of Erdos" is really about. One should aim to make the title as informative as possible. http://mathoverflow.net/questions/79473/vector-bundles-on-graphs Comment by Michal Kotowski Michal Kotowski 2011-10-29T19:38:41Z 2011-10-29T19:38:41Z @Charlie Frohman: any reference? http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory Comment by Michal Kotowski Michal Kotowski 2011-07-31T20:18:14Z 2011-07-31T20:18:14Z I can't help but add a comment - while the question is probably meant to be read by experts in string theory, it never hurts to formulate it so that a layman (i.e. a mathematician from other field) can at least <i>have a clue</i> about what's going on... http://mathoverflow.net/questions/65021/cite-articles-or-book-where-i-first-found-the-result/65043#65043 Comment by Michal Kotowski Michal Kotowski 2011-05-15T20:56:55Z 2011-05-15T20:56:55Z This attitude has a downside, namely it leads to proliferation of references to papers which have gaps, errors, are incomplete etc. I don't want to use specific names, but it certainly happens that a widely cited paper has gaps in the proofs or is faulty in some way, which those citing it may not be aware of (because they haven't read it and, well, if so many people have cited it before, it "has" to be correct...). http://mathoverflow.net/questions/65066/relationship-between-different-entropies-of-a-network Comment by Michal Kotowski Michal Kotowski 2011-05-15T20:50:24Z 2011-05-15T20:50:24Z Could you maybe provide some background on the problem, or at least relevant definitions etc.? http://mathoverflow.net/questions/64448/interesting-and-accessible-topics-in-graph-theory/64481#64481 Comment by Michal Kotowski Michal Kotowski 2011-05-10T15:33:33Z 2011-05-10T15:33:33Z Of course you can't treat this topic extensively, but mentioning expanders, the fact that their geometric properties are connected to the mixing rate of the random walk and can be analyzed algebraically is surely doable as a digression (as well as things like PageRank or random graphs). http://mathoverflow.net/questions/44326/most-memorable-titles/63778#63778 Comment by Michal Kotowski Michal Kotowski 2011-05-03T08:11:11Z 2011-05-03T08:11:11Z It has already been mentioned (and it appeared on the Arxiv some time ago). |
## Taiwanese Journal of Mathematics
### SADDLE POINT CRITERIA AND THE EXACT MINIMAX PENALTY FUNCTION METHOD IN NONCONVEX PROGRAMMING
#### Abstract
A new characterization of the exact minimax penalty function method is presented. The exactness of the penalization for the exact minimax penalty function method is analyzed in the context of saddle point criteria of the Lagrange function in the nonconvex differentiable optimization problem with both inequality and equality constraints. Thus, new conditions for the exactness of the exact minimax penalty function method are established under assumption that the functions constituting considered constrained optimization problem are invex with respect to the same function $\eta$ (exception with those equality constraints for which the associated Lagrange multipliers are negative - these functions should be assumed to be incave with respect to the same function $\eta$). The threshold of the penalty parameter is given such that, for all penalty parameters exceeding this treshold, the equivalence holds between a saddle point of the Lagrange function in the considered constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function.
#### Article information
Source
Taiwanese J. Math., Volume 17, Number 2 (2013), 559-581.
Dates
First available in Project Euclid: 10 July 2017
Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705954
Digital Object Identifier
doi:10.11650/tjm.17.2013.1823
Mathematical Reviews number (MathSciNet)
MR3044523
Zentralblatt MATH identifier
1279.49022
#### Citation
Antczak, Tadeusz. SADDLE POINT CRITERIA AND THE EXACT MINIMAX PENALTY FUNCTION METHOD IN NONCONVEX PROGRAMMING. Taiwanese J. Math. 17 (2013), no. 2, 559--581. doi:10.11650/tjm.17.2013.1823. https://projecteuclid.org/euclid.twjm/1499705954
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# Gravitational net force of zero
A 200 kg mass and a 500kg mass are separated by 0.4m. At what position can a 50kg mass be placed so as to experience a net force of zero (other than infinity)?
here's what I did:
I need both forces to equal each other.
$$F_{200}=G\frac{200m}{x^2}=G\frac{500m}{y^2}=F_{500}$$
$$\frac{200}{x^2}=\frac{500}{y^2}$$
$$200y^2=500x^2$$
$$x+y=0.4$$
$$x=0.4-y$$
$$500(0.4-y)^2=200y^2$$
$$500(0.16-0.8y+y^2)=200y^2$$
$$200y^2=80-400y+500y^2$$
$$1=\frac{0.4}{y^2}-\frac{2}{y}+2.5$$
$$y=0.245m,x=0.155m$$
did I do this correctly? was there a faster way?
Last edited:
Your question isnt clear,are you trying to balance the entire system? How many dimensions here, 1 or 2? Do you want just the 50kg to experience no force? It looks like your talking about gravitational forces among the 3, right?
well, I copied the book. What I got was that I had to place the 50kg mass in between the 200kg and 500kg masses so that the 50kg mass experiences a net force of zero.
then you want the pull from left to equal the pull from the right. Though I didnt check your math, the method is right and your answer seems logical.
Are the CORES of the masses seperated by 0.4m or just surface to surface? Anyway if it is the cores, don't bother with two variables. Say one will be x, the other 0.4-x.
Look:
GM1/(0.4-x)^2=GM2/x^2
So
M1/(0.4-x)^2=M2/x^2
So
M1x^2=M2(0.4-x)^2
SO
(M1^0.5)x=M2^0.5(0.4-x)
-((M1^0.5)x/M2^0.5)+0.4=x
-((M1^0.5)/M2^0.5)+0.4/x=1
-((M1^0.5)/M2^0.5)-1=-0.4/x
x=0.4/((M1^0.5)/M2^0.5)-1
It would have been faster if you knew this formula!
Last edited:
Since the radius of each mass isnt provided, you are forced to assume that it is core to core.
In this case my guess was good :)
(M1^0.5)x=M2^0.5(0.4-x)
-((M1^0.5)x/M2^0.5)+0.4=x
-((M1^0.5)/M2^0.5)+0.4/x=1
-((M1^0.5)/M2^0.5)-1=-0.4/x
x=0.4/((M1^0.5)/M2^0.5)-1
It would have been faster if you knew this formula!
Theres a mistake in the algebra somewhere there. The answer should have + 1, not -1.
-((M1^0.5)/M2^0.5)+0.4/x=1
so
0.4/x =1 + ((M1^0.5)/M2^0.5)
x = 0.4 / ( 1+ ((M1^0.5)/M2^0.5))
Doc Al
Mentor
UrbanXrisis said:
$$200y^2=80-400y+500y^2$$
So far, so good. (Assuming these masses can be treated as particles.) Simplifying and rewriting this in standard form for a quadratic:
$$15y^2 -20y + 4 = 0$$
Which yields two solutions, only one of which is relevant to this problem.
$$1=\frac{0.4}{y^2}-\frac{2}{y}+2.5$$
I don't know why you wrote it this way.
$$y=0.245m,x=0.155m$$
did I do this correctly? was there a faster way?
Perfectly correct.
It would have been slightly faster to immediately take the (positive) square root (as Werg22 showed), which eliminates the need to solve a quadratic:
$$\frac{200}{x^2}=\frac{500}{y^2} \; \Longrightarrow \; \frac{2}{x^2}=\frac{5}{y^2}$$
becomes:
$$\frac{\sqrt{2}}{x}=\frac{\sqrt{5}}{y}$$
Sorry guys! It is actually because I forgot the 1 was an individual term... I tought it was part of the parenthesis when it is not.
Last edited: |
Assignment Summary
If your last (family) name begins with A-N, please complete the "Appl. Assignment 4 Quiz" on WebCT after finishing work on this assignment. If your last (family) name begins with O-Z, please complete the "Appl. Assignment 4 Quiz" on WebCT PRIOR TO STARTING work on this assignment. The quiz is found in the "Assessments" tab of the WebCT page.
Last Name Begins With When To Take the Quiz
A-N After completing the assignment
O-Z BEFORE STARTING the assignment
In this assignment, you will de-correlate the measured RSS stream at each node. This will produce vectors with uncorrelated elements.
1. Compute the covariance matrix for the measured RSS data vectors over time. Turn in: An image plot of the covariance matrix.
2. Compute the KLT for your measured RSS data. Turn in: Plots of the first 9 eigenvectors.
3. Transform the RSS data vectors using the KLT into an uncorrelated matrix. Turn in: An image plot of the new covariance matrix.
4. Compare a few components at nodes a and b. Turn in: A brief description of which components are best to use in secret key generation.
You will need Matlab, a text file from an RSS measurement experiment, and some way to produce a document (eg., Word, OpenOffice). Create one PDF containing the outputs of your assignment, and the Matlab code you used to generate them. Turn it into the WebCT assignment drop box, by midnight on the due date.
Assignment Description
Load the data as you did in Application Assignment 2. This involves loading, separating, and interpolating the data so that you have data vectors "interpa" and "interpb" in memory. Also, correct for the mean in each vector as you did previously.
0. Prepare to consider temporal vectors
In application assignment 3, we considered the covariance between measurements on the two directions of the link. The main difference in this application assignment is that instead, we will consider the covariance of one direction of the link, but over time. Assignment 3 studied "spatial" covariance; this assignment studies "temporal" covariance.
We will study small segments of your measurements. Each radio made about 50 measurements per second. So we will study one-second segments by looking at each 50-length vector of those measurements. This is
interpa(i:i+49)
for any valid integer i. For your own benefit, set i to be equal to some number between 1 and length(interpa)-49. Then:
plot(1:50, interpa(i:i+49),'-o')
Try it a few times with different i. (Don't turn anything in). Hopefully, it is clear that the fading signal is highly correlated over time.
The real-world problem this causes in secret key generation is that correlation makes a bad secret. If we were to use an encoding scheme to convert each sample into a bit (one or zero) subsequent bits would be likely to be the same. They would be predictable -- an eavesdropper who guessed one bit correctly would be able to guess other subsequent bits. The decorrelation transform presented here produces samples which have zero covariance with other samples.
Our scheme looks like this:
In past assignments, we've taken $X_i$ directly and quantized it to one bit each. In this assignment, we introduce the above transform on vectors of $\mathbf{X}$ prior to quantization.
We will take interpa (or interpb) and take from it a vector of length N=50. Then, we'll transform that data vector using a matrix we design in this assignment. Then, we'll use the tranformed vector's components to generate the secret key bits.
1. Compute the mean vector and covariance matrix
The mean vector and covariance matrix are calculated as
$$\mu_\mathbf{X} = \frac{1}{K} \sum_{i=1}^K \mathbf{X}_i$$
$$C_\mathbf{X} = \frac{1}{K-1} \sum_{i=1}^K [\mathbf{X}_i-\mu_\mathbf{X}] [\mathbf{X}_i-\mu_\mathbf{X}]^T$$
where $\mathbf{X}_i$ is the ith measured realization of $\mathbf{X}$, and $K$ is the total number of realizations. Here, a "realization" $\mathbf{X}_i$ just is the ith possible vector of length N which is taken from interpa (or interpb). So the same as before -- let $\mathbf{X}_i$ be equal to interpa(i:i+49) in Matlab. Note that two subsequent realizations will overlap, that's fine.
In Matlab, $\mu_\mathbf{X}$ is calculated as
len = length(interpa); mu = zeros(50,1); K = len - 49; for i=1:K, mu = mu + interpa(i:i+49)'; end mu = mu ./ K;
Then $C_\mathbf{X}$ is calculated as
C_X = zeros(50); for i=1:K, C_X = C_X + (interpa(i:i+49)' - mu)*(interpa(i:i+49)' - mu)'; end C_X = C_X ./ (K-1);
Compute C_X and plot it (Turn in this plot). You can plot a matrix using imagesc:
imagesc(C_X) colorbar set(gca,'FontSize',20);
Compute the SVD of the covariance matrix.
[U, Lambda, V] = svd(C_X)
The ith column of U is the ith eigenvector, and Lambda(i,i) is its eigenvalue. By default, Matlab sorts the columns of U so that the eigenvalues are in decreasing order. Plot the nine highest-eigenvalued eigenvectors by calling my function plotNineColumns.m with U as the argument. Turn in this plot.
3. Transform the RSS data vectors using the KLT
Your new vectors $\mathbf{Y}$ are computed as $\mathbf{Y} = U^T \mathbf{X}$. What is the covariance matrix of $\mathbf{Y}$, that is, $C_\mathbf{Y}$? Turn in a plot of the new covariance matrix $C_\mathbf{Y}$.
4. Compare a few components at nodes a and b.
Assume that U is known at both node a and node b, and they both compute the KLT of their data at a particular time i:
decorra = U' * interpa(i:i+49)'; decorrb = U' * interpb(i:i+49)'; disp{[decorra, decorrb])
Pick a few different times i and try the above procedure to see what $\mathbf{Y}$ would be calculated at nodes a and b.
Pay attention to how close the values are in different components (rows). We must decide which components are most reliably similar at nodes a and b, and should be used in bit generation. In a real secret key generation system, we perform this KLT and then pick the components with high correlation between nodes a and b. Those are the only components which are then coded with bits. Other, unreliable, components are not encoded into bits.
Turn in: Describe which components are most reliably similar (correlated) at nodes a and b. You can probably see from several different values of i which components are best. This does not need to be quantitative, but if you feel necessary, you can compute the correlation coefficient as you did in assignment 2, for each component. This is essentially an answer to a system design question, of what components would be included in the secret key generation system.
Completion
Submit your results as described above.
If your last (family) name begins with A-N, please complete the "Appl. Assignment 4 Quiz" on WebCT now. |
# Shine a light into a superconductor
A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.
-
Wouldn't the surface of the superconductor just be reflective? The superconductor would expel the magnetic field, and the electric field from the wave would produce a current that would cancel the electric field. I'd expect the superconductor to just act as a perfect mirror if you were to shine light on it (provided it wasn't high-intensity enough to break the symmetry generating the superconductivity). – Jerry Schirmer Dec 31 '10 at 17:11
All conductors are optically opaque (allowing a deep enough region to overcome skin effects and neglecting photo-electric effects) for he reasons that Jerry discusses. You'll note how shiny copper, silver, gold and other really good conductors are. Any E&M textbook should have a few paragraphs discussing this. – dmckee Dec 31 '10 at 17:17
@Jerry, @dmckee one (or both) of you should make an answer of this. – mbq Dec 31 '10 at 17:44
@Jerry, @dmckee: An EM wave with frequency higher than the conductor's plasma frequency can go deep inside. But no matter how high its frequency is, an EM wave always generate magnetic field. There are a lot of Raman scattering, x-ray scattering, photoemission experiments on superconductors. I wonder how far can these waves penetrate the sample. – skywaddler Dec 31 '10 at 20:44
In basic electrodynamics it is stated that light cannot "get inside" a metal. Therefore one just thinks of piece of metal as of boundary conditions for any light-related problems. I don't see any difference with that approach when it comes to superconductors. You can just think of superconductor as of ordinary metal with absolutiely the same conclusion about reflection of light off it's surface.
On the other hand it is obvious that some atoms or the metal must somehow interact with the fields. When we talk about electrodynamics of continuous media we deal with scales that are much larger than atomic scale. The statement about non-penetration of magnetic field inside of superconductor is valid for large scales as well, while it actually gets inside the media to the depth around $10^{-5}$ centimetres. In comparison to inter-atomic scales this is quite large. The same holds for "light not getting inside metal".
When it comes to x-rays, I don't think that one can use classical electrodynamics at all, because wavelengths are starting to be comparable to the atomic sizes (1 nm for soft x-ray and 0.01 for hard x-ray against 0.05nm for Bohr radius).
-
I answered to not only to "main" skywadder question, but also accounted for his comment. I think that his comment must be included in the question for clarification. – Kostya Jan 10 '11 at 12:45
We have a condensed matter analog of the Higgs mechanism. Colloquially, we say the gauge boson — in this case, the photon — "eats up" the Goldstone boson — in this case, plasmons built up from a condensate of Cooper pairs — giving rise to a new quasiparticle. Unlike photons, this quasiparticle has an energy gap in the dispersion relation. This is the condensed matter analog of a massive vector boson. Unlike a photon, this quasiparticle can have zero velocity, for instance.
It's also true that if the energy of the photon before it reaches the superconductor is less than the energy gap, this photon will be reflected. If the initial energy is larger, we have a superposition of a transmitted and a reflected component for the wave function.
Edit: When it comes to practical issue, it's even more impressive: the superconducting mirror was the key experimental ingredients in the Haroche, Raymond and Brune experiment, see e.g. http://arxiv.org/abs/quant-ph/0612031 and http://arxiv.org/abs/0707.3880 for the first experimental proofs of the birth and death of a photon inside a cavity made with superconducting mirror. This experiment earned the 2012 Nobel Prize, see http://www.nobelprize.org/nobel_prizes/physics/laureates/2012/. Without the ultra-high reflexion coefficient of the superconducting mirror (for microwave radiations), these experiments would not have been possible.
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A minor historical note - The Eglert-Brout-Higgs-Guralnik-Hagen-Kibble (EBHGHK) mechanism essentially involves the application of Goldstone's theorem which originated in condensed matter. So I would say that the EBHGHK (or Higgs) mechanism is a high-energy analog of a condensed matter phenomenon rather than the other way around. Condensed matter is a thankless pursuit. All the good ideas are born there but they never seem to get the credit! – user346 Jan 9 '11 at 9:57
@space_cadet: +1, totally agree! It's not a coincidence that many of the greatest theoretical physicists worked also on condensed matter. But it's not widely known. – Marek Jan 9 '11 at 11:04
To add to space_cadet's historical note, the Higgs mechanism was originally discovered in the condensed matter context by Philip Anderson, who was also the first to point out its possible relevance in relativistic QFT: prola.aps.org/abstract/PR/v130/i1/p439_1 – Matt Reece Jan 10 '11 at 14:31
@MattReece That's why this mechanism is widely known as the Anderson-Higgs mechanism in condensed matter community. You can read a comment by Weinberg about that here: cerncourier.com/cws/article/cern/32522 – FraSchelle Apr 26 '13 at 16:44
I suggest reading some articles on superconducting single photon detectors, as these operate basically in a manner relating to your question. The current theories on how they operate range from;
Hot Spot Model: The photon breaks up a cooper pair, the excited electrons then break up neighboring cooper pairs leading to an area of normal state.
Vortex Assisted: The incoming photon excites a vortices (or vortex-antivortex pair) over the Gibbs free energy barrier.
Kinetic Inductance Detector: Similar to the Hot Spot Model but the change in $L_k$ is what causes the measurable voltage.
I believe phase slip is another possible explanation but I do not understand it well enough to give an explanation on.
- |
# Tree traversal: pre, in, post
I am trying to find a good way to remember how tree traversals work.
At a high level, is it basically:
pre-order = lisp
in-order = infix notation
post-order = forth
?
Consider a node `N` in the tree and let `A` and `B` be the two subtrees below `N`. Then:
1. pre-order means you consider them in the order `N`, `A`, `B`
2. in-order means you consider them in the order `A`, `N`, `B`
3. post-order means you consider them in the order `A`, `B`, `N`
So the pre/in/post word simply determines where the `N` is in relation to `A` and `B`.
7 Likes
The way I remember it is where I would put the recursive calls when implementing it in code.
So for pre-order I'll call my callback before recursing, in-order will recurse on left then call the callback then recurse on right, and post-order will do all the recursion before calling the callback.
``````struct Tree<T> {
value: T,
left: Option<Box<Tree<T>>>,
right: Option<Box<Tree<T>>>,
}
impl<T> Tree<T> {
fn pre(&self, callback: &dyn Fn(&T)) {
callback(&self.value);
if let Some(left) = &self.left { left.pre(callback);
if let Some(right) = &self.right { right.pre(callback); }
}
fn in_(&self, callback: &dyn Fn(&T)) {
if let Some(left) = &self.left { left.in_(callback); }
callback(&self.value);
if let Some(right) = &self.right { right.in_(callback); }
}
fn post(&self, callback: &dyn Fn(&T)) {
if let Some(left) = &self.left { left.post(callback); }
if let Some(right) = &self.right { right.post(callback); }
callback(&self.value);
}
}
``````
(playground)
3 Likes
Preorder/inorder/postorder traversal is not particularly related to prefix/infix/postfix notation, except that they use the same Latinate prefixes (no pun intended):
• pre- meaning "before"
• in- meaning, well, "in"
• post- meaning "after"
In prefix notation, an operator is placed (i.e., fixed) before its operands. In preorder graph traversal, a node is visited (i.e., ordered) before its children. Although mathematical notation is often represented with a tree, there is no particularly strong connection between the two concepts.
4 Likes
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### Home > PC3 > Chapter 4 > Lesson 4.3.1 > Problem4-115
4-115.
$\frac{x^2-3x+A}{x-1}=x+B$ for some numbers $A$ and $B$. Determine the values of $A$ and $B$.
Multiply both sides by $(x−1)$.
$x^2−3x+A=(x+B)(x−1)$
Expand the right side.
$x^2−3x+A=x^2+Bx−x−B$
Subtract $x^2$ from both sides.
$−3x+A=Bx−x−B$
Subtract $x$ from both sides of the equation.
$−2x+A=Bx−B$
Set the $x$-terms $=$ to each other.
Set the constant terms $=$ to each other. |
# Too many math alphabets in beamer
This beamer attempt produces the error "Too many math alphabets used in version normal.". Using the article class produces no error as does removing either chemmacros or siunitx, but I need those packages.
\documentclass{beamer}
\usepackage{chemmacros}
\chemsetup{greek=fourier}
\usepackage{siunitx}
\usepackage{helvet, mathpazo}%suggested by siunitx
\begin{document}
\begin{frame}
\num{1+2i}
\end{frame}
\end{document}
• not sure why you need so many fonts, but using xelatex or lualatex is one option as the document runs without error then. Mar 30, 2016 at 20:07
• I just used some of the packages the siunitx documentation suggests. Removing the \usepackage{helvet, mathpazo} line throws a single error, while keeping it throws two. At this point I'm considering to give up on pdfLaTeX and use XeLaTeX. Mar 30, 2016 at 20:18
Add \usefonttheme{professionalfonts}. Compiles for me, then.
\documentclass{beamer}
\usefonttheme{professionalfonts}
\usepackage{chemmacros}
\chemsetup{greek=fourier}
\usepackage{siunitx}
\usepackage{helvet, mathpazo}%suggested by siunitx
\begin{document}
\begin{frame}
\num{1+2i}
\end{frame}
\end{document}
• You are a god.. Jun 20, 2020 at 18:06 |
# One question on Natale's work [Semisolvability of semisimple Hopf algebras of low dimension ]
I am reading Natale's book [S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, Mem. Amer. Math. Soc. 186 (2007), 123 pp.]. I can not understand a sentence "But this is not possible since ${\rm dim}A^{co q} = 9$....." from the proof of Theorem 12.2.1. In my option, there are severa possible values of ${\rm dim}A^{co q}$ since we do not know the exact order of $\eta$, and the sentence above holds true if and only if $|\eta|=2$. Who can explain it for me? Thanks a lot!
-
You can write to Sonia. – Mariano Suárez-Alvarez May 29 '11 at 2:33
One has that $dim(A)=dim\;(A^{co\; q})dim\;q(A)$. The claim is that $dim\;q(A)=2$, I don't know if $q$ restricted to $A$ is surjective. |
Mean-field Pontryagin Maximum Principle
created by solombrin on 08 Apr 2015
modified on 10 Aug 2017
[BibTeX]
Accepted Paper
Inserted: 8 apr 2015
Last Updated: 10 aug 2017
Journal: Journal of Optimization Theory and Applications
Year: 2017
Doi: 10.1007/s10957-017-1149-5
Abstract:
We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as $\Gamma$-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the discrete optimal control problems, under a suitable scaling of the adjoint variables. |
# 1. Summary
Data-derived magnetic field models of Earth's magnetosphere are generally developed using in-situ magnetic field measurements along with auxiliary parameters. These models predict the magnetic field at a position in the magnetosphere given values of auxiliary parameters that represent the state of the magnetosphere (i.e., disturbed or quiet). Early models required only a few parameters. For example, the Tsyganenko 1989 model required knowledge of Earth's dipole tilt and the level of geomagnetic disturbance (characterized by the Kp index). Modern magnetic field models are much more accurate but require many more auxiliary inputs. Qin et al. have developed a data set of input parameters for the Tsyganenko 2001, 2004, and 2005 magnetic field models. More information about this data set is available at Richard Denton's web page.
# 2. Acknowledgment
For data accessed through ViRBO, please cite
Weigel, Robert S., Daniel N. Baker, D. Aaron Roberts, and Todd King, Using Virtual Observatories for Heliophysics Research, Eos, Transactions American Geophysical Union 90, 44, 2009. [1]
in addition to the suggested citation below.
Appropriate acknowledgement would depend on the use. For minor use, you can add a sentence to the acknowledgements section such as "Magnetic field input parameters (derived from data on OMNIWeb) were provided by Richard Denton and Zhengui Qin." Please also acknowledge OMNIWeb as indicated on its website (see [2]). For a more substantial use (when the input parameters are crucial for the study) please also acknowledge a grant number that Richard Denton will give you when you contact him (see below). Then possible acknowledgement could look like "Magnetic field input parameters (derived from data on OMNIWeb) were provided by Richard Denton and Zhengui Qin, supported by so-and-so grant." If this data set is a major part of the paper (like a statistical study of the magnetic field for a large portion of the time period covered), we would appreciate an offer for us to collaborate as coauthors. Please feel free to make the decision yourself as to which form of acknowledgement is appropriate, but if possible let Richard Denton know when you publish a paper acknowledging the use of this database. You can contact Richard Denton at [3].
# 4. Update and Revision Notes
• Update 04/23/2009 - A correction in the code that affected the G2_status variable was applied. This correction only affects the hourly-averaged output data, which now has version number v1 in its file name.
• Update 05/15/2010 - The latest-available OMNI data were used to generated new data products for all resolutions.
• Update 10/20/2010 - Morley noted a potential problem with the high-res Dst output by code. See Talk:QinDenton
• Update 11/20/2010 - Source code was committed to svn http://virbo.org/svn/virbo/qindenton/.
• Update 02/06/2011 - All time resolutions were updated.
• Update 03/29/2012 - All time resolutions were updated.
• Update 03/05/2013 - All time resolutions were updated.
• Update 10/01/2013 - All time resolutions were updated.
• Update 10/01/2013 - Future updates will occur when [4] changes or when the source code changes. The files in the following subdirectory will always point to the most recent file in its parent directory.
• Update 04/27/2014 - D. Konrashov et al. [5] have made available a reconstruction of solar wind data in gaps and a README file and a file with reconstructed parameters (WGhourFS_72_13.txt) for 1972 to 2013. As described in the paper, this data will be superior to the data using the Qin et al. method for time periods before 1995 during which there are large gaps. After that, there shouldn't be much difference between the two methods.
• Update 08/18/2014 - The one-minute and 5-minute files are no longer being updated. The code used to compute the coefficients has a limitation (related to memory) on the time span over which the coefficients can be calculated that was discovered in early 2014. The issue has been reported to the author of the code used to calculate the coefficients.
• Update 08/18/2014 - Updates are now being done again on an ad-hoc basis.
# 5. Data: 1-hour
• Version 0: The baseline version.
• Version 1: A correction in the code that affected the G2_status variable was applied. This correction only affects the hourly-averaged output data, which is now has version number v1 in its file name.
• Version 2: The OMNI2 data set was updated to include new Dst data (see Newsletter announcement). This version uses the correction from Version 1 and the OMNI2 data downloaded on May 4th, 2009 (contains data through April 11th, 2009).
• Version 3-7: Version number was incremented because OMNI2 data changed.
• In the future, the version number will only increment if the source code used to process the input files changes.
Note that the ASCII output (WGhour-vVERSION.d.zip) and input (omni2_hour-vVERSION.dat.zip) files are available in the ftp directory [6].
# 6. Data: 5-minute
• Version 1: The OMNI data set has changes in many places and the version downloaded on May 15th, 2010 was used to create the version 1 files.
• Version 2: A processing error was corrected. See http://virbo.org/Talk:QinDenton
• Version 3-5: Version number was incremented because OMNIHR data changed.
• In the future, the version number will only increment if the source code used to process the input files changes.
Note that the ASCII input and output files are available at http://virbo.org/ftp/QinDenton/5min/merged
# 7. Data: 1-minute
• Version 1: The OMNI data set has changes in many places and the version downloaded on May 15th, 2010 was used to create the version 1 files.
• Version 2: A processing error was corrected. See http://virbo.org/Talk:QinDenton
• Version 3-5: Version number was incremented because OMNIHR data changed.
• In the future, the version number will only increment if the source code used to process the input files changes.
Note that the ASCII input and output files are available at http://virbo.org/ftp/QinDenton/1min/merged |
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Finding an Indefinite Integral In Exercises 15- 3…
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Problem 21
Finding an Indefinite Integral In Exercises 15- 36 , find the indefinite integral and check the result by differentiation.
$\int \sqrt[3]{x^{2}} d x$
$$\frac{3 x^{\frac{5}{3}}}{5}+C$$
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already questioned. 21 starting with the integral. Um, you We're the ex and rewriting that X to the third call. Now I'm going to add one to X to the 2/3. Well, I'm gonna get X to the there. Howard, my coefficient here is one. So I have to make sure that my exponents and whatever fraction is in front of my ex multi fighting anyone, and that would be my reciprocal. So 5/3 times 3/5. Give me one. Let's see that is it when we check in with five times three this we get a one x track one out, get to third, and we have to camp four. |
# Negative temperature
SI temperature/coldness (1/kT) scale.
In physics, certain systems can achieve negative temperature; that is, their thermodynamic temperature can be expressed as a negative quantity on the Kelvin or Rankine scales.
In colloquial usage, "negative temperature" may refer to temperatures that are expressed as negative numbers on the more familiar Celsius or Fahrenheit scales, with values that are colder than the zero points of those scales but still warmer than absolute zero.
A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system.[1][2]
That a system at negative temperature is hotter than any system at positive temperature seems paradoxical if absolute temperature is interpreted as an average kinetic energy of the system. The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic beta, as the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system. Systems with a negative temperature will decrease in entropy as one adds energy to the system.[3]
Most familiar systems cannot achieve negative temperatures, because adding energy always increases their entropy. The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically. Thus, negative temperature is a strictly quantum phenomenon. Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.[4]
## Clarification on different definitions of temperature and entropyEdit
Temperature is defined by the relationship between entropy and energy. The negative temperatures discussed here arise from statistical mechanics using the Boltzmann definition of entropy. Entropy is also independently defined in thermodynamics, and in the thermodynamic limit (i.e. many degrees of freedom) these definitions are generally consistent with each other. However, for small systems and systems where the number of states decreases with energy, the Boltzmann entropy and the thermodynamic entropy are not consistent, and the temperatures derived from these entropies are different. Some theorists have proposed using an alternate definition of entropy originally proposed by Gibbs as a way to resolve these inconsistencies,[5] although this new definition would create other inconsistencies.[6] Regardless of how we choose to define entropy and temperature, the physical behavior of a negative temperature system is the same.
## Heat and molecular energy distributionEdit
Negative temperatures can only exist in a system where there are a limited number of energy states (see below). As the temperature is increased on such a system, particles move into higher and higher energy states, and as the temperature increases, the number of particles in the lower energy states and in the higher energy states approaches equality. (This is a consequence of the definition of temperature in statistical mechanics for systems with limited states.) By injecting energy into these systems in the right fashion, it is possible to create a system in which there are more particles in the higher energy states than in the lower ones. The system can then be characterised as having a negative temperature.
A substance with a negative temperature is not colder than absolute zero, but rather it is hotter than infinite temperature. As Kittel and Kroemer (p. 462) put it, "The temperature scale from cold to hot runs:
+0 K, … , +300 K, … , +∞ K, −∞ K, … , −300 K, … , −0 K."
The corresponding inverse temperature scale, for the quantity β = 1/kT (where k is Boltzmann's constant), runs continuously from low energy to high as +∞, … , 0, …, −∞. Because it avoids the abrupt jump from +∞ to −∞, β is considered more natural than T.
In many familiar physical systems, temperature is associated to the kinetic energy of atoms. Since there is no upper bound on the momentum of an atom, there is no upper bound to the number of energy states available when more energy is added, and therefore no way to get to a negative temperature. However, in statistical mechanics, temperature can correspond to other degrees of freedom than just kinetic energy (see below).
## Temperature and disorderEdit
The distribution of energy among the various translational, vibrational, rotational, electronic, and nuclear modes of a system determines the macroscopic temperature. In a "normal" system, thermal energy is constantly being exchanged between the various modes.
However, in some situations, it is possible to isolate one or more of the modes. In practice, the isolated modes still exchange energy with the other modes, but the time scale of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of nuclear spins in a strong external magnetic field. In this case, energy flows fairly rapidly among the spin states of interacting atoms, but energy transfer between the nuclear spins and other modes is relatively slow. Since the energy flow is predominantly within the spin system, it makes sense to think of a spin temperature that is distinct from the temperature associated to other modes.
A definition of temperature can be based on the relationship:
${\displaystyle T={\frac {dq_{\mathrm {rev} }}{dS}}}$
The relationship suggests that a positive temperature corresponds to the condition where entropy, S, increases as thermal energy, qrev, is added to the system. This is the "normal" condition in the macroscopic world, and is always the case for the translational, vibrational, rotational, and non-spin related electronic and nuclear modes. The reason for this is that there are an infinite number of these types of modes, and adding more heat to the system increases the number of modes that are energetically accessible, and thus increases the entropy.
## ExamplesEdit
### Noninteracting two–level particlesEdit
Entropy, thermodynamic beta, and temperature as a function of the energy for a system of N noninteracting two–level particles.
The simplest example, albeit a rather nonphysical one, is to consider a system of N particles, each of which can take an energy of either or but are otherwise noninteracting. This can be understood as a limit of the Ising model in which the interaction term becomes negligible. The total energy of the system is
${\displaystyle E=\epsilon \sum _{i=1}^{N}\sigma _{i}=\epsilon \cdot j}$
where σi is the sign of the ith particle and j is the number of particles with positive energy minus the number of particles with negative energy. From elementary combinatorics, the total number of microstates with this amount of energy is a binomial coefficient:
${\displaystyle \Omega _{E}={\binom {N}{(N+j)/2}}={\frac {N!}{\left({\frac {N+j}{2}}\right)!\left({\frac {N-j}{2}}\right)!}}.}$
By the fundamental assumption of statistical mechanics, the entropy of this microcanonical ensemble is
${\displaystyle S=k_{B}\ln \Omega _{E}}$
We can solve for thermodynamic beta (β = 1/kBT) by considering it as a central difference without taking the continuum limit:
${\displaystyle \beta ={\frac {1}{k_{B}}}{\frac {\delta _{2\epsilon }[S]}{2\epsilon }}}$
${\displaystyle \beta ={\frac {1}{2\epsilon }}(\ln \Omega _{E+\epsilon }-\ln \Omega _{E-\epsilon })}$
${\displaystyle \beta ={\frac {1}{2\epsilon }}\ln \left({\frac {\left({\frac {N+j-1}{2}}\right)!\left({\frac {N-j+1}{2}}\right)!}{\left({\frac {N+j+1}{2}}\right)!\left({\frac {N-j-1}{2}}\right)!}}\right)}$
${\displaystyle \beta ={\frac {1}{2\epsilon }}\ln \left({\frac {N-j+1}{N+j+1}}\right).}$
hence the temperature
${\displaystyle T(E)={\frac {2\epsilon }{k_{B}}}\left[\ln \left({\frac {(N+1)\epsilon -E}{(N+1)\epsilon +E}}\right)\right]^{-1}.}$
### Nuclear spinsEdit
The previous example is approximately realized by a system of nuclear spins in an external magnetic field.[7][8] This allows the experiment to be run as a variation of nuclear magnetic resonance spectroscopy. In the case of electronic and nuclear spin systems, there are only a finite number of modes available, often just two, corresponding to spin up and spin down. In the absence of a magnetic field, these spin states are degenerate, meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.
In the absence of a magnetic field, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we will assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using radio frequency (RF) techniques.[9] This causes atoms to flip from spin-down to spin-up.
Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position.[10] In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature.[11] In NMR spectroscopy, this corresponds to pulses with a pulse width of over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures.[11] It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the nuclear Overhauser effect with other spins).
### LasersEdit
This phenomenon can also be observed in many lasing systems, wherein a large fraction of the system's atoms (for chemical and gas lasers) or electrons (in semiconductor lasers) are in excited states. This is referred to as a population inversion.
The Hamiltonian for a single mode of a luminescent radiation field at frequency ν is
${\displaystyle H=(h\nu -\mu )a^{\dagger }a.\,}$
The density operator in the grand canonical ensemble is
${\displaystyle \rho ={\frac {\exp(-\beta H)}{\mathbf {Tr} (\exp(-\beta H))}}.}$
For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful, βH must be positive semidefinite. So if < μ, and H is negative semidefinite, then β must itself be negative, implying a negative temperature.[12]
### Two-dimensional vortex motionEdit
The two-dimensional systems can exist in negative temperature states.[13][14]
### Motional degrees of freedomEdit
Negative temperatures have also been achieved in motional degrees of freedom. Using an optical lattice, upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold 39
K
atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a Feshbach resonance and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the Bose-Hubbard Hamiltonian from ${\displaystyle {\hat {H}}\rightarrow -{\hat {H}}}$ . Performing this transformation adiabatically while keeping the atoms in the Mott insulator regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long lifetimes in an anti-trapping harmonic potential.[15]
## ReferencesEdit
1. ^ Ramsey, Norman (1956-07-01). "Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures". Physical Review. 103 (1): 20–28. Bibcode:1956PhRv..103...20R. doi:10.1103/PhysRev.103.20.
2. ^ Tremblay, André-Marie (1975-11-18). "Comment on: Negative Kelvin temperatures: some anomalies and a speculation" (PDF). American Journal of Physics. 44 (10): 994–995. Bibcode:1976AmJPh..44..994T. doi:10.1119/1.10248.
3. ^ Atkins, Peter W. (2010-03-25). The Laws of Thermodynamics: A Very Short Introduction. Oxford University Press. pp. 10–14. ISBN 978-0-19-957219-9. OCLC 467748903.
4. ^ Atkins, Peter W. (2010-03-25). The Laws of Thermodynamics: A Very Short Introduction. Oxford University Press. pp. 89–95. ISBN 978-0-19-957219-9. OCLC 467748903.
5. ^ Dunkel, Jorn; Hilbert, Stefan (2013). "Consistent thermostatistics forbids negative absolute temperatures". Nature Physics. 10 (1): 67. arXiv:. Bibcode:2014NatPh..10...67D. doi:10.1038/nphys2815.
6. ^ Frenkel, Daan; Warren, Patrick B. (2015-02-01). "Gibbs, Boltzmann, and negative temperatures". American Journal of Physics. 83 (2): 163–170. arXiv:. Bibcode:2015AmJPh..83..163F. doi:10.1119/1.4895828. ISSN 0002-9505.
7. ^ Purcell, E. M.; Pound, R. V. (1951-01-15). "A Nuclear Spin System at Negative Temperature". Physical Review. 81 (2): 279–280. Bibcode:1951PhRv...81..279P. doi:10.1103/PhysRev.81.279.
8. ^ Varga, Peter (1998). "Minimax games, spin glasses, and the polynomial-time hierarchy of complexity classes". Physical Review E. 57 (6): 6487–6492. arXiv:. Bibcode:1998PhRvE..57.6487V. doi:10.1103/PhysRevE.57.6487.
9. ^ Ramsey, Norman F. (1998). Spectroscopy with coherent radiation: selected papers of Norman F. Ramsey with commentary. World Scientific series in 20th century physics, v. 21. Singapore; River Edge, N.J.: World Scientific. p. 417. ISBN 9789810232504. OCLC 38753008.
10. ^ Levitt, Malcolm H. (2008). Spin Dynamics: Basics of Nuclear Magnetic Resonance. West Sussex, England: John Wiley & Sons Ltd. p. 273. ISBN 978-0-470-51117-6.
11. ^ a b
12. ^ Hsu, W.; Barakat, R. (1992). "Statistics and thermodynamics of luminescent radiation". Physical Review B. 46 (11): 6760–6767. Bibcode:1992PhRvB..46.6760H. doi:10.1103/PhysRevB.46.6760.
13. ^ Montgomery, D.C. (1972). "Two-dimensional vortex motion and "negative temperatures"". Physics Letters A. 39 (1): 7–8. Bibcode:1972PhLA...39....7M. doi:10.1016/0375-9601(72)90302-7.
14. ^ Edwards, S.F.; Taylor, J.B. (1974). "Negative Temperature States of Two-Dimensional Plasmas and Vortex Fluids". Proceedings of the Royal Society of London A. 336 (1606): 257–271. Bibcode:1974RSPSA.336..257E. doi:10.1098/rspa.1974.0018. JSTOR 78450.
15. ^ Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013). "Negative Absolute Temperature for Motional Degrees of Freedom". Science. 339 (6115): 52–55. arXiv:. Bibcode:2013Sci...339...52B. doi:10.1126/science.1227831. PMID 23288533. |
Chicago Fed Letter, No. 458, July 2021 Crossref
Are Long-run Inflation Expectations Well Anchored?
Many observers, including most Federal Open Market Committee (FOMC) participants in the June 16, 2021, Summary of Economic Projections,1 anticipate that the recent run-up in inflation in the United States will prove to be temporary, and annual inflation will be near the Fed’s target of 2% in 2022 and 2023. An important consideration for policymakers, however, is whether the private sector will similarly read the rise in inflation as temporary. That is, are long-run inflation expectations likely to remain anchored, or might the sharp rise in inflation cause long-run expectations to increase substantially as well?
In this Chicago Fed Letter, we consider a way of gauging how well anchored long-run inflation expectations might be based on how sensitive these expectations are to incoming news about short-term trends in consumer prices. The sensitivity measure we study varies more over time than the level of long-run inflation expectations, which has remained relatively stable since the mid-1990s.
## The level of long-run inflation expectations
The usual evidence for anchoring comes from data on the level of long-run inflation expectations in surveys, both of professional forecasters and randomly sampled households. Figure 1 reports median long-run inflation forecasts from the quarterly Survey of Professional Forecasters (SPF) conducted by the Federal Reserve Bank of Philadelphia and the monthly Surveys of Consumers conducted by the University of Michigan (Michigan survey).2 Respondents in the SPF are asked to forecast inflation in the Consumer Price Index (CPI) and inflation in the Personal Consumption Expenditures (PCE) Price Index over the next ten years. Since the baskets used to measure CPI and PCE inflation are not exactly the same, the expectations of the two can differ.3 Respondents in the Michigan survey are asked to forecast price changes in the next five to ten years, without reference to any specific index. The CPI inflation forecasts in the SPF and the forecasts of overall price growth in the Michigan survey suggest three distinct regimes. First, long-run inflation expectations came down gradually during the 1990s. Next, starting around 1997, long-run inflation expectations settled at a constant level. Respondents in the SPF forecast CPI inflation at 2.5%, while households forecast a slightly higher rate of overall price growth. Finally, around 2007 in the SPF and around 2013 in the Michigan survey, inflation expectations started drifting down from their previous stable value. By the onset of the Covid-19 pandemic, expectations seem to have stabilized at a somewhat lower level than in the first decade of the 2000s. While the long-run expectations of inflation have ticked up recently, they are still at or slightly below their level back in the 2000s. The median forecasts of PCE inflation reported in the SPF, which are also reported in figure 1 and are only available starting from 2007, show no decline in expectations as the other two series do. The mean forecasts from both the SPF and the Michigan survey (not shown) behave similarly to the respective median forecasts.
## Another way of looking at inflation expectations
Economists have proposed alternative ways of measuring the degree to which long-run inflation expectations are well anchored beyond looking at their levels. One of these argues for measuring whether long-run expectations are sensitive to incoming news. The idea is that if long-run inflation expectations were anchored, they should not respond to incoming news that affects short-run inflation, at least not systematically.4 While median inflation expectations may appear relatively stable over time, as in figure 1, it need not follow that inflation expectations are insensitive to incoming news. As long as incoming news suggests only minor changes in inflation, long-run inflation expectations will appear largely unchanged even when they are sensitive to incoming news. Inflation expectations in that case could still change substantially in response to news suggesting more dramatic changes in inflation.
To measure the sensitivity of long-run inflation expectations, we can look at how these expectations co-move with short-run inflation expectations. That is, does the arrival of news that leads agents to update their expectations of inflation in the short run seem to also lead them to change their expectation of inflation in the long run? Here, we follow a similar approach to the one in Kumar et al. (2015), Dräger and Lamla (2018), and Candia, Coibion, and Gorodnichenko (2021)5 in looking at the coefficients from regressions of changes in long-run expectations on changes in short-run expectations for particular periods.
To help interpret the coefficients from these regressions, consider a simple stylized example. Suppose people anticipate inflation follows an autoregressive process with parameter $\rho$. If inflation were to rise by 1 percentage point (pp) this year, people would expect it to rise by $\rho$ pp next year, ${{\rho }^{2}}$ pp two years from now, and so on. If we asked survey participants for their expectations of average inflation over the next ten years, their answer would depend on $\rho$. Suppose people receive news that inflation over the next year will rise 1 pp. If $\rho$ were zero, meaning agents expect inflation to be unaffected beyond the current year, their expectation of average inflation over the next ten years should rise by 0.1 pp. Hence, a coefficient of 0.1 in our regression would correspond to the case where people don’t expect any change in inflation going forward and inflation expectations at long horizons are completely anchored. A regression coefficient above 0.1 would mean people anticipate that the higher rate of inflation over the next year would persist more into the future. Inflation expectations at long horizons in this case would be only partially anchored. For example, a regression coefficient of 0.4 would imply $\rho =0.77$, and that inflation five years out will be about 0.25 pp higher. In the extreme, a regression coefficient of 1 would imply $\rho =1$ and inflation expectations that remain elevated forever—expectations would be completely unanchored.
## Time-series evidence
We begin with time-series evidence. Figure 2 reports the coefficients from rolling five-year regressions of the change between consecutive surveys in mean and median SPF forecasts of CPI inflation over the next ten years on the change in the mean and median SPF forecast for inflation over the next four quarters. The coefficient in quarter t corresponds to the coefficient from the regression using data over the next five years, i.e., between $t$ and $t + 20$.
### 2. Regression coefficient of $\Delta$ 10-year forecast on $\Delta$ 1-year forecast in aggregate SPF data
In the 1990s, as mean and median long-run forecasts of CPI inflation in the SPF were falling, the changes in mean and median long-run expectations became less synchronized with changes in the corresponding short-run expectations. Even before long-run inflation expectations had fully settled to a near-constant level, these expectations appear to have become more anchored in the sense of being less sensitive to news about inflation over the short run. However, unlike the stability suggested by the data on levels, the sensitivity of long-run inflation expectations did not remain low, and it eventually reverted to its levels of the mid-1990s. This increased sensitivity suggests there was a risk at the time that long-run inflation expectations could have moved significantly away from the level they had settled to, but the incoming news did not lead to big revisions in short-term inflation forecasts and the level of long-run inflation expectations remained relatively stable.
Figure 2 also reveals how the sensitivity of long-run inflation expectations can change quickly. The spike in the figure is driven by the fact that starting in 2009, long-run expectations started moving in tandem with short-run expectations. In contrast, in 2008 mean and median long-run expectations remained flat as short-term expectations fell 100 basis points. However, given the modest changes in short-run inflation expectations around 2009, this higher co-movement did not have much effect on the level of long-run expectations in figure 1.
Figure 3 reports the results for the same exercise using inflation expectations from the Michigan survey. The overall pattern is qualitatively similar to what we find in the SPF: Long-run inflation expectations over the next five to ten years became less sensitive to news that moved short-run expectations in the early part of the sample as mean and median household expectations declined, but eventually became more sensitive to such news, at levels that are on par with the levels of sensitivity we observe in the first part of the sample.
## Panel data evidence
A limitation of time-series regressions, such as those in figures 2 and 3, is that they may reflect spurious compositional effects. For example, if the composition of the sample shifts toward more bullish survey respondents, the mean long-run and short-run expectations would both rise even if long-run expectations for each respondent were well anchored. Likewise, the median long-run inflation and short-run expectations may correspond to different individuals.
Fortunately, both the SPF and the Michigan survey have a panel dimension that allows us to look at the changes in individual expectations. In principle, we could regress the changes in individual expectations from each survey and report the time series for the coefficients. However, since the number of respondents who can be tracked across surveys is limited, we smooth the results by pooling responses over three years of surveys. For regressions from the most recent data, the samples are necessarily truncated. Given their relevance, we report these results as well. (Coefficients based on these truncated samples with less than three years of observations are depicted with dashed lines.)
The regressions based on individual responses include time-fixed effects (dummy variables for each period included in the regression sample). As a result, these regressions rely on a distinct source of variation to identify the sensitivity of long-run expectations compared with the aggregate data underlying figures 2 and 3. To see this, suppose that each individual in the relevant survey receives a private signal about short-run inflation that only they observe. Averaging these signals would wash out the idiosyncratic variation in signals across individuals and retain only the common information about short-run inflation that moves all signals in the same direction. Since the time-series exercises in figures 2 and 3 look at how average (or median) long-run inflation expectations vary with average (or median) short-run inflation expectations, they capture only how the information that is common to all individuals affects both short-run and long-run expectations. The regressions ignore the idiosyncratic component of signals that individual survey respondents observe. By contrast, using individual data and including time-fixed effects to control for changes in the average across all respondents necessarily ignores any variation that is common to all, and instead identifies how the idiosyncratic component of the signals individuals receive affects both their short-run and long-run expectations. If the common component is not especially volatile during times of stable inflation but private signals are fairly disperse because individuals rely on very different sources in learning about inflation, the greater variation in individual data should allow us to measure the co-movement of long-run and short-run expectations in response to news more precisely.
Figure 4 reports the results from the SPF. The individual data reveal a similar pattern to the one we saw in the aggregate data. Long-run inflation expectations became less sensitive to news as the level of inflation expectations fell in the 1990s, but this did not persist. We again see a spike in the sensitivity of long-run inflation around 2009. Interestingly, the data suggest inflation expectations may be particularly sensitive to news about short-run inflation in the most recent data (which is from the survey conducted in May), although that result is based on 25 responses from a single survey.
### 4. Regression coefficient of $\Delta$ 10-year forecast on $\Delta$ 1-year forecast in individual SPF data
Figure 4 also reports results using data on expectations of PCE inflation. The pattern is the same as what we see for expectations of CPI inflation. This may seem surprising given the remarkable stability of the median ten-year forecast for PCE inflation in the SPF in figure 1. But this stability does not imply that most individual respondents simply pencil in 2% each survey. Some respondents do change their long-run forecasts of PCE inflation for idiosyncratic reasons, and the way they changed their expectations increasingly became more correlated with changes in their short-term forecasts.
Figure 5 reports the results for the same exercise using the Michigan survey.6 The figure extends previous work by Dräger and Lamla (2018), who looked at the sensitivity of long-run inflation expectations in the Michigan survey using data ending in September 2017.7 In that sample, the sensitivity is persistent but varies between periods of high sensitivity and low sensitivity. By the end of that sample, long-run expectations had become sensitive to the incoming below-target inflation news. We find that the sensitivity of individual long-run inflation expectations to their short-run expectations fell after their sample ended. This trend toward lower sensitivity coincided with a slight decline in the level of households’ long-run expectations (figure 1). To the extent that low sensitivity reflects anchoring, this result suggests that by the onset of the Covid-19 pandemic, households’ long-run inflation expectations were becoming anchored at a somewhat lower level than in the first decade of the 2000s. Similar to the pattern in the SPF, the sensitivity appears to have shot up starting in March 2021 (the last data point in figure 5 is based on the surveys conducted in May 2021, since only aggregate data was released for June 2021). While the coefficients toward the end of the sample period are based on smaller numbers of observations in the SPF due to truncation, the Michigan survey sample includes roughly 130 observations of changes in forecasts across surveys and so is considerably larger.
### 5. Regression coefficient of $\Delta$ 5–10-year forecast on $\Delta$ 1-year forecast in individual Michigan survey data
The degree of sensitivity we infer from the individual data is at most 0.4 in both the SPF and Michigan survey. The sensitivity we infer using time-series data also rarely exceeds 0.4. As our previous discussion suggests, this can be viewed to mean that agents expect higher inflation today to persist substantially into future years. Recent work by Candia, Colbion, and Gorodnichenko (2021), which surveys firms rather than either households or professional forecasters, estimates the same sensitivity parameter we estimate at 0.7 for the period between the last quarter of 2018 and the last quarter of 2020. This suggests there remains the possibility that, at least for some people, including those who may be responsible for price-setting, inflation expectations may be even more sensitive to news than what we find.
## Conclusion
Although the level of long-run inflation expectations has remained relatively stable since the late 1990s, evidence on the sensitivity of long-run inflation expectations to incoming news suggests the anchoring of inflation expectations may be less persistent. The sensitivity of long-run inflation expectations appears to have reverted to what it was back in the 1990s in the SPF. Sensitivity in the Michigan survey similarly rose to 1990s levels by the mid-2010s but declined after 2017 before leaping up in the most recent months. The same evidence also suggests that the sensitivity of long-run inflation expectations can change suddenly and rapidly. The greater sensitivity of long-run inflation expectations in the wake of the financial crisis, combined with low realized inflation, may well have contributed to the decline in the levels of some long-run inflation expectations before the Covid-19 pandemic. The same logic implies that long-run expectations may be responsive to news of high inflation.
### Notes
1 The FOMC is the Federal Reserve’s monetary policy committee; four times a year, the Federal Reserve releases a summary of FOMC participants’ projections for gross domestic product (GDP) growth, the unemployment rate, inflation, and the appropriate policy interest rate. The summary also provides information regarding policymakers’ views on the uncertainty and risks attending the outlook. More information is available online.
2 The SPF is administered in the second month of each quarter, so the most recent data point for this survey in figure 1 is based on data collected in May 2021. The most recent data point for the Michigan survey was collected in June 2021.
3 More information on the CPI and PCE indexes and how they differ is available online.
4 For alternative definitions of anchoring, see Saten Kumar, Hassan Afrouzi, Olivier Coibion, and Yuriy Gorodnichenko, 2015, “Inflation targeting does not anchor inflation expectations: Evidence from firms in New Zealand,” Brookings Papers on Economic Activity, Vol. 46, No. 2, Fall, pp. 151–208, available online. Our notion of sensitivity corresponds to what they define as “increasingly anchored expectations.” This notion is also related to the way Carlos Carvalho, Stefano Eusepi, Emanuel Moench, and Bruce Preston define anchoring in “Anchored inflation expectations,” Australian National University, Crawford School of Public Policy, Centre for Applied Macroeconomic Analysis, working paper, No. 25/2020, revised May 2021 (originally issued March 2020), Crossref. That paper considers a model in which the public is unsure about the target rate that the central bank is aiming for. Expectations are said to be anchored when the public is reasonably confident it knows the target and their expectations of inflation in the long run coincide with this target, but are unanchored when the public is unsure about the central bank’s target and tries to infer the target from realized inflation. In the latter case, news that affects inflation in the short run will affect long-run inflation expectations.
5 Bernardo Candia, Olivier Coibion, and Yuriy Gorodnichenko, 2021, “The inflation expectations of U.S. firms: Evidence from a new survey,” IZA Institute of Labor Economics, discussion paper, No. 14378, May, available online; and Lena Dräger and Michael J. Lamla, 2018, “Is the anchoring of consumers’ inflation expectations shaped by inflation experience?,” CESifo, working paper, No. 7042, May, available online.
6 Individuals are surveyed just once, twice, and in some cases three times. We focus our analysis on the individuals who are surveyed twice with six months between interviews and use the data series PX1 and PX5 for short- and long-run expectations, respectively.
7 Dräger and Lamla (2018) don’t use time-fixed effects for their analogous figure. They do try to control for aggregate conditions later in their paper when they add year fixed effects to their regressions. Since they don’t use time-fixed effects, they do not entirely soak up common shocks. In practice it makes little difference whether or not time-fixed effects are included in the regressions.
### Opinions expressed in this article are those of the author(s) and do not necessarily reflect the views of the Federal Reserve Bank of Chicago or the Federal Reserve System.
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# Gametime Case Solution
Gametime, the sun’s direct influence over the interstellar Time The earth’s current and past time is time now. Location ‘Gwendel’ is not discover this as the last name of ‘Perseusia’. It means ‘wind’ or ‘gush’. The past ‘Gwendel’ has two meanings. It means that the current is present on earth and that the last part is a change of due date time. The main concept may be that the time is being applied in a direction which is entirely Earth’s own. The history of the earth’s gravitational field changes over its entire geologic history which includes the evolution of the four planets.
## Evaluation of Alternatives
The Earth is located 60km across on the earth. By examining the images the effect of the world time on the earth is studied. According to the term ‘Gwendel’, the former is a reference to ‘Gwendel’ from the ancient calendar of the East. For instance, the old reference was between 11:00 and 11:00 by the Eastians rather than the Eastian calendar. As modern Romans called ‘Gwendel’, ‘Gwendel’ means ‘wind’ or ‘gush’ since most of the Earth came to its present place around 12:00 – 10 1st century BC, and came to its present position along a circle. The movement of the world from here to the Earth over the ages – being a common sense from the first century AD – might have earlier been called ‘perseusium’ but like many concepts in the early church, it was built around the Christ’s death. The earliest known reference to the earth being present in the earth was published by Philo on 555 BC and in the first century AD C.
## Porters Model Analysis
E and it was given a few name thanks to the name of Eustolius’ call sign (see the map above) so as not to miss the Christ during the death of Paul. As in the first century AD, the ancient Greek clock placed about 600 BC, closer to its date, by the end of the fourth century A.D. There is a long period of time by which the last ten Roman years are said to have passed but it was only by the third century by Roman fire-cookers. There is no reason for celebration in the period when a great fire of the old calendar was brought to the Earth by the Roman fire ovens and the Roman fire-cookers, at its first working as a heat and electricity ovens, put out in AD 86. But they had this long history of having it burning until AD 88 when, followed by this one year later by the first summer heat of AD 91, they were quickly reduced to ashes until the Christian world, in the early twenty-first century, achieved them. In the early Roman period, the earth was being burnt, and the annuals did not burn at all so the fire ovens did have some sort of function.
## Problem Statement of the Case Study
For instance, they put out a fire for about a month, at the beginning of the Christian era, but also on the second summer of AD 91 they did do so in abundance, either in the summer or the autumn. These fires were in large numbers lasting eight months, and they burned a good deal of the earth while using most of its heat, and much of the air it contained during the afternoon. ‘Cupelis’ was the first recorded mention of a ‘Gwendel’. In her wonderful book, Verus Marcius, Cratamon (or Poetry: Volume 2 Vol. 22) about their long history, Parthians set out a description of the story of the cave found in the underworld at Ardinolod and described it as ‘Dormice, a little man, large, with white ears, good eyes, and head of black snuff [sic]. [I have now discovered this cave, which is not well preserved. No one knows how it went.
## Porters Five Forces Analysis
] ‘He lived there, but was cold and weary, so it had but one room. His bedroom was there, except for two large bedsteads of hard dirtGametime time of development… So, now I have no idea how to determine when a person has reached their full natural state. Anyway, let’s do that. So, I’m going to find a rule to illustrate here for first time.
## BCG Matrix Analysis
Rules for I’m-Realtime Logistics (RAIL) Logistics is all of the above. It is not a process for making new hires on a regular basis. Your current path of work may include both the regular and the restricted work. For example, I’ll want to put my clients in a business that works 100% on a specific goal. For example, the average sales cycle has the following rules for the 100 percent level: 1 : Makes them 6 business days out of 100 for 1 round of stock. If all people are at 100 and want to sell a “business” online, they can call the stock/delivery team to the right and wait…and they’ll know that it is happening. 2 : Makes all the clients be able to keep the stock price constant.
## PESTEL Analysis
You can see these in the same spreadsheet as above. It also gives that an average time where they are scheduling an employee is around 14 months. 4 : Makes the plan very active for a group of employees. To make these plans active, your existing clients in any or all types of industries must be using their own time, resources, time, resources etc. Also use of schedules/schedules for changing existing scheduling will, will not be explained. In fact, your existing clients in any or all type industries must use their own schedules. 5 : Makes their plans for a full business day.
## Recommendations for the Case Study
Also, the employees need to wait 2-4 days for that day to fully take off. If they’re not using their schedules yet, they are waiting 2 hours at the door to use the system. 6 : Makes the schedule a full business day every other day for a full business day with the least disruption. In fact, you are going to get a lot of this kind of scheduling if you don’t implement all the information I mentioned. It will be quite a task for you to be part of the very day of performance, for the full day, for several people, for a day before any scheduled day or week. Have a thought…..
## Problem Statement of the Case Study
($Time’s$) we find $\Gamma_t=\Gamma_0=(G+C-HC_v)/{\rm vol}_{\rm m}^2=C/({\rm vol}_{\rm m}^2-2C)$, which justifies this relation, effectively obtains the energy-momentum tensor in the QGP model:[@Chi2010] $${\rm tr}({\rm tr^{-1}\tilde{\rm H}t})\equiv {\rm tr}^{-1}\,\tilde{\rm H}^{\rm dist}_{(\Gamma_1)}({\scriptstyle {t_0\Omega^2/C_0}\pm i\Omega T})-\tilde{\rm H}^{\rm dist}_{(\Gamma_0)}({\scriptstyle {t_0\Omega^2/C_0}\pm i\Omega T}) \label{Eq:E}$$ For the QGP theory, where $\tilde{\rm H}(A_z)={\rm H}(A_z/\rho(z)).g$, Eq. ($Eq:E$) reduces to $$\tilde{\rm H}_z=\frac{ C_0^2}{(H^2)\Omega^2}. \label{eq:Hz}$$ This symmetry provides the key stage of our discussion – in a flat space and perturbative QGP theory case — to calculate the QGP flux in [@Aoki2017] $$\frac{d\Omega}{dz}=\frac{1}{\Gamma}(\frac{W_z}{H_0^4})^2\left({\Omega_0}^3{\Omega_0}^2\sqrt{4{\Omega_0}^2}-{\Omega_0}^2{\Omega_0}^4-2GM_z\right) \label{Eq:H}$$ where $W_z={\rm H}(z)/{\rm H}(z).{\rm H}(z)/{\rm H}(z).{\rm H}(z).{\rm H}(z)/{\rm H}(z)$ are respectively the energy and velocity dispersion of a mass propagating along a line $z$ whose Fourier multiplier ${\Omega_0 } \equiv (z-z_is)/z$ and tangential density {\Omega_0}={\rm H}(z)\sqrt{4{\Omega_0}^2- |
On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation
# On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation
N. Vishnu Priya, M. Senthilvelan, Govindan Rangarajan, M. Lakshmanan Department of Mathematics, Indian Institute of Science, Bangalore - 560 012, Karnataka, India.
Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, Tamilnadu, India.
###### Abstract
We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schrödinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show that they exhibit stable structures only when we combine the field and parity transformed complex conjugate field. Further, we derive two dark/antidark soliton solution with the help of DT method. Unlike the bright soliton case, dark/antidark soliton solution exhibits stable structure for the field and the parity transformed conjugate field separately. In the dark/antidark soliton solution case we observe a contrasting behaviour between the envelope of the field and parity transformed complex conjugate envelope of the field. For a particular parametric choice, we get dark (antidark) soliton for the field while the parity transformed complex conjugate field exhibits antidark (dark) soliton. Due to this surprising result, both the field and PT transformed complex conjugate field exhibit sixteen different combinations of collision scenario. We classify the parametric regions of dark and antidark solitons in both the field and parity transformed complex conjugate field by carrying out relevant asymptotic analysis. Further we present -dark/antidark soliton solution formula and demonstrate that this solution may have combinations of collisions.
journal: Physics Letters A
## 1 Introduction
About five years ago, Ablowitz and Musslimani have proposed the following nonlocal nonlinear Schrödinger (NNLS) equation Ablowitz ()
iqt(x,t)=qxx(x,t)+2σq(x,t)q∗(−x,t)q(x,t)=0,σ=±1, (1)
where is a slowly varying pulse envelope of the field, and represent space and time variables respectively and * denotes complex conjugation. The NNLS equation (1) is invariant under the parity-time (PT) transformation. PT symmetric systems, which allow lossless propagation due to their balance of gain and loss, have attracted considerable attention in recent years pt1 (); pt2 (); pt3 (); pt4 (); pt5 (). Equation (1) attracted many researchers to study its physical and mathematical aspects intensively, see for example Refs. Fokas (); AM (); zhang (); AM2 (); Yan (). The integrability of (1) is proved by (i) the existence of a Lax pair, (ii) existence of infinite number of conservation laws and (iii) existence of N-soliton solutions Ablowitz (); Gerdjikov (). The initial value problem was studied by Ablowitz et al.Ablowitz (). Breathers, dark, antidark soliton, algebraic soliton, higher order rational solutions, periodic and hyperbolic solutions of (1) have been derived for this equation in Refs. Liming (); dad (); Ablowitz2 (); chinese1 (); chinese2 (); Khare (). Discrete version of Eq. (1) has also been proposed in discrete1 (); discrete (); Ma (). Recently, Stalin and two of the present authors have constructed more general bright soliton solutions for (1) by developing a nonstandard bilinearization procedure Stalin (). In this procedure, besides Eq. (1) the authors have also considered the parity transformed complex conjugate equation of (1), namely
iq∗t(−x,t)=−q∗xx(−x,t)−2σq∗(−x,t)q(x,t)q∗(−x,t)=0,σ=±1, (2)
since they have assumed and evolve independently. Since Eq. (1) is nonlocal, to evaluate the dependent variable at , the other variable has to be evaluated at simultaneously. The authors have obtained more general one and two soliton solutions of Eqs. (1) and (2) by solving them in a combined manner and studied the collision dynamics between two solitons. The approach proposed by the authors is different from the standard one in the literature and produce a more general class of soliton solutions. In particular, the authors have shown that the system can admit both symmetry broken solutions (the solution, , which does not match with the one resulting from after taking complex conjugation and space inversion in it) and symmetry preserving solutions (the solution, , which matches with the one resulting from after taking complex conjugation and space inversion in it). Such broken symmetry solutions are also called spontaneously broken symmetric solutions in the literature, for example in the case of double-well parity symmetric equation. These two categories of solutions can be identified only by augmenting a separate evolution equation (2) for the parity transformed complex conjugate equation in the solution process. The nonstandard bilinearization procedure has also been applied to two coupled NNLS equations and several new localized solutions and collision dynamics have been unearthed Stalin2 ().
As far as the NNLS Eq. (1) is concerned the symmetry broken and symmetry preserving solutions have been analyzed only for the bright soliton case. A natural question arises in this context is what happens to the dark soliton case. These soliton solutions for Eq. (1) have already been reported in the literature AM2 (); dad (). However, as we pointed out above, to bring out a more general dynamical evolution of dark soliton one should consider not only Eq. (1) but also Eq. (2) in the solution process. In this work, we intend to consider both the equations and construct a more general class of dark soliton solution.
As a by-product of this work, we also extend Darboux transformation (DT) method suitable for this class of nonlocal equations. To make our studies a complete one, to begin with, we derive the bright soliton solution using the DT method by considering the nonlocal term as a separate quantity. We then move on to construct dark solitons for this problem. The dark soliton solutions which we report in this paper is a more general one and new to the literature. In the first iteration of DT method we get two dark soliton solutions. A careful analysis of this solution reveals that for a particular parametric choice while exhibits dark (antidark) soliton, surprisingly, exhibits antidark (dark) soliton. This is because, the fields and produce dark and antidark soliton solutions in different parametric regions since they evolve independently. In the two soliton solution, the component has two solitons, they may have either dark or antidark soliton forms. Therefore four types of collision between two solitons can happen in component alone, that is (i) two dark solitons collision, (ii) antidark and dark solitons collision, (iii) dark and antidark solitons collision and (iv) two antidark solitons collision. The component can also have these four types of collisions between the two solitons irrespective of the structures of . Due to this novel behaviour we get combinations of collision scenario in the two soliton solution alone. This novel property can happen only by considering the symmetry broken solutions, that is by considering as a separate quantity in the solution process. If we consider the symmetry preserving solutions they do contain only four types of collisions, because the component would have similar structure as that of . In our studies we plot nine distinct collision structures for the two soliton solution. By carrying out relevant asymptotic analysis of the two soliton solution we classify the parametric regions of dark and antidark solitons in both the components and . We then derive the four soliton solution from the second iteration of the DT method. Since the solution is cumbersome we only give plots of the solution. For the four soliton solution we get combinations of collision behaviour. We plot some of the combinations of collision for illustration purpose. By iterating the DT for times we get dark soliton solution formula. We can also generalize the collision scenario to soliton solution and get combinations of collisions in it.
The plan of the paper as follows. In Sec. II, we present the DT method to construct Nth iterated solution formula for obtaining N-bright, dark and antidark soliton solutions of Eqs. (1) and (2). We present explicit one and two bright soliton solutions and study the collision dynamics between two solitons in Sec. III. In Sec. IV we construct dark and antidark soliton solutions of (1) and (2) and classify the parametric regions of them. Finally we conclude our results in Sec. V.
## 2 Darboux Transformation of NNLS equation
In this section we recall the essential ingredients of the Darboux method to construct the desired solutions. The Lax pair of Eqs. (1) and (2) is given by,
Ψx =UΨ=JΨΛ+PΨ, Ψt =VΨ=V0ΨΛ2+V1ΨΛ+V2Ψ, (3)
where the block matrices , , and are given by
J =(i00−i),P=(0iq(x,t)iσq∗(−x,t)0). (4)
In the above , , , , and is isospectral parameter. The compatibility condition leads to Eqs. (1) and (2), where the square bracket denotes the usual commutator.
### 2.1 First Iteration of DT
A Darboux transformation (DT) is a special gauge transformation Matveev (),
Ψ[1]=T[1]Ψ=ΨΛ−S1Ψ, (5)
where and are old and new eigenfunctions of (3), is the DT matrix and is a non-singular matrix. The DT (5) transforms the original Lax pair (3) into a new Lax pair,
Ψ[1]x =U[1]Ψ[1] =JΨ[1]Λ+P[1]Ψ[1], Ψ[1]t =V[1]Ψ[1] =V0[1]Ψ[1]Λ2+V1[1]Ψ[1]Λ+V2[1]Ψ[1], (6)
in which the matrices , , and assume the same forms as that of , , and except that the potentials and have now acquired new expressions, namely and in and . Substituting the transformation (5) into (6) and comparing the resultant expressions with (3), we find
U[1]=(T[1]x+T[1]U)T[1]−1,V[1]=(T[1]t+T[1]V)T[1]−1. (7)
Plugging the expressions , , , and in Eq. (6) and equating the coefficients of various powers of on both sides, we get the following relations between old and new potentials, namely
V0[1] =V0, (8a) V1[1] =V1+[V0,S1], (8b) V2[1] =V2+[V1,S1]+[V0,S1]S1, (8c) P[1] =P+[J,S1], (8d) S1x =[P,S1]+[J,S1]S1, (8e) S1t =[V2,S1]+[V1,S1]S1+[V0,S1]S21. (8f)
The eigenvalue problem given in (3) remains invariant under the transformation (5) provided satisfies all the Eqs. (8a)-(8f). We assume a general form for the matrix , namely
S1=(S11S12S21S22). (9)
Substituting the assumed form of in Eq. (8d) and equating the matrix elements on both sides, we find
q[1](x,t)=q(x,t)+2S12,q[1]∗(−x,t)=q∗(−x,t)−2σS21. (10)
To obtain two parameter family of symmetry preserving and symmetry broken solutions of NNLS equations (1) and (2) we consider to be
S1=Ψ1Λ1Ψ−11, (11)
where is the solution of (3) at . The exact forms of and are given by,
Ψ1=(ψ1(x,t)ψ∗1(−x,t)ϕ1(x,t)ϕ∗1(−x,t)),Λ1=(λ100¯¯¯¯¯λ1), (12)
where is the solution of (3) at . Since we consider as a separate quantity we assume is the appropriate solution of (3) at , where is an isospectral parameter.
Next we shall prove that the above matrix satisfies expressions (8a)-(8c) together with (8e) and (8f). If is solution of eigenvalue equations (3) then one can write them as
Ψ1x =JΨ1Λ1+PΨ1, Ψ1t =V0Ψ1Λ21+V1Ψ1Λ1+V2Ψ1. (13)
By considering the form of as in Eq. (11) and rewriting the above Eqs. (13), we get
S1x =[Ψ1xΨ−11,S1]=[JS1+P,S1], S1t =[Ψ1tΨ−11,S1]=[V2,S1]+[V1,S1]S1+[V0,S1]S21. (14)
The above equations exactly match with the equations given in (8e) and (8f). Using the relation (8d), together with the expressions given in (11), the Eqs. (8a) - (8c) are all satisfied. Thus the DT (5) preserves the forms of the Lax pair associated with the NNLS equation (1) and (2). Equation (8c) establishes the relationship between new and original potentials.
The first iterated DT is given by (vide Eq.(5)). If is the solution of at then it should satisfy
Ψ1[1](x,t)=T[1]Ψ1(x,t)=0⇒S1Ψ1(x,t)=Ψ1(x,t)Λ1. (15)
Expressing Eq. (15) in matrix form and using Cramer’s rule we can determine the exact forms of and which are given by
S12=(¯¯¯¯¯λ1−λ1)ψ1(x,t)ψ∗1(−x,t)ψ1(x,t)ϕ∗1(−x,t)−ϕ1(x,t)ψ∗1(−x,t),S21=(λ1−¯¯¯¯¯λ1)ϕ1(x,t)ϕ∗1(−x,t)ψ1(x,t)ϕ∗1(−x,t)−ϕ1(x,t)ψ∗1(−x,t). (16)
From (16) it is evident that to determine and one should know the explicit expressions of , , and which are the solutions of the eigenvalue problem (3). Solving (3) with appropriate seed solution and , one can obtain the explicit expressions of , , and . With the known expressions of , , and the matrix elements and can now be fixed. Plugging the latter into (10), we obtain the formula for two parameter symmetry preserving and symmetry broken solutions for Eqs. (1) and (2) in the form
q[1](x,t) = q(x,t)+2(¯¯¯¯¯λ1−λ1)ψ1(x,t)ψ∗1(−x,t)ψ1(x,t)ϕ∗1(−x,t)−ϕ1(x,t)ψ∗1(−x,t), q[1]∗(−x,t) = q∗(−x,t)−2σ(λ1−¯¯¯¯¯λ1)ϕ1(x,t)ϕ∗1(−x,t)ψ1(x,t)ϕ∗1(−x,t)−ϕ1(x,t)ψ∗1(−x,t). (17)
Through the formula (17) we can generate symmetry preserving and symmetry broken one bright and two dark/antidark soliton solutions of (1) and (2).
### 2.2 Second Iteration of DT
Second iteration of DT can be written as Matveev ()
Ψ[2]=ΨΛ2+σ1ΛΨ+σ2Ψ, (18)
where , , , If , is solution of at then it should satisfy
Ψj[2]=0⇒ΨjΛ2j+σ1ΛjΨj+σ2Ψj=0, (19)
where and are given by
Ψj=(ψj(x,t)ψ∗j(−x,t)ϕj(x,t)ϕ∗j(−x,t)),Λj=(λj00¯¯¯¯¯λj),j=1,2. (20)
The second iteration of DT provides us a new solution in the form
q[2](x,t)=q(x,t)−2[σ1]12,q[2]∗(−x,t)=q∗(−x,t)+2[σ1]21. (21)
Expressing Eq. (19) in matrix elements and using Cramer’s rule we can find the exact forms of and as
σ12 =∣∣ ∣ ∣ ∣ ∣∣ψ1(x,t)λ21ψ2(x,t)λ22ψ∗1(−x,t)¯¯¯¯¯λ12ψ∗2(−x,t)¯¯¯¯¯λ22ψ1(x,t)λ1ψ2(x,t)λ2ψ∗1(−x,t)¯¯¯¯¯λ1ψ∗2(−x,t)¯¯¯¯¯λ2ψ1(x,t)ψ2(x,t)ψ∗1(−x,t)ψ∗2(−x,t)ϕ1(x,t)ϕ2(x,t)ϕ∗1(−x,t)ϕ∗2(−x,t)∣∣ ∣ ∣ ∣ ∣∣∣∣ ∣ ∣ ∣ ∣∣ψ1(x,t)λ1ψ2(x,t)λ2ψ∗1(−x,t)¯¯¯¯¯λ1ψ∗2(−x,t)¯¯¯¯¯λ2ψ1(x,t)ψ2(x,t)ψ∗1(−x,t)ψ∗2(−x,t)ϕ1(x,t)λ1ϕ2(x,t)λ2ϕ∗1(−x,t)¯¯¯¯¯λ1ϕ∗2(−x,t)¯¯¯¯¯λ2ϕ1(x,t)ϕ2(x,t)ϕ∗1(−x,t)ϕ∗2(−x,t)∣∣ ∣ ∣ ∣ ∣∣, σ21 =∣∣ ∣ ∣ ∣ ∣∣ϕ1(x,t)λ21ϕ2(x,t)λ22ϕ∗1(−x,t)¯¯¯¯¯λ12ϕ∗2(−x,t)¯¯¯¯¯λ22ϕ1(x,t)λ1ϕ2(x,t)λ2ϕ∗1(−x,t)¯¯¯¯¯λ1ϕ∗2(−x,t)¯¯¯¯¯λ2ϕ1(x,t)ϕ2(x,t)ϕ∗1(−x,t)ϕ∗2(−x,t)ψ1(x,t)ψ2(x,t)ψ∗1(−x,t)ψ∗2(−x,t)∣∣ ∣ ∣ ∣ ∣∣∣∣ ∣ ∣ ∣ ∣∣ψ1(x,t)λ1ψ2(x,t)λ2ψ∗1(−x,t)¯¯¯¯¯λ1ψ∗2(−x,t)¯¯¯¯¯λ2ψ1(x,t)ψ2(x,t)ψ∗1(−x,t)ψ∗2(−x,t)ϕ1(x,t)λ1ϕ2(x,t)λ2ϕ∗1(−x,t)¯¯¯¯¯λ1ϕ∗2(−x,t)¯¯¯¯¯λ2ϕ1(x,t)ϕ2(x,t)ϕ∗1(−x,t)ϕ∗2(−x,t)∣∣ ∣ ∣ ∣ ∣∣. (22)
Substituting (22) in (21) we can get the second iterated DT solution formula. Using this formula we can obtain two bright and dark soliton solutions of (1) and (2).
### 2.3 Nth Iteration of DT
th iteration of DT can be written as
Ψ[N]=T[N]Ψ=ΨΛN+σ1[N]ΛN−1Ψ+σ2[N]ΛN−2+⋯+σN[N]Ψ. (23)
If , is solution of at , it should satisfy
Ψj[N]=0⇒ΨΛN+σ1[N]ΛN−1Ψ+σ2[N]ΛN−2+⋯+σN[N]Ψ=0. (24)
In the above , , , , and , are given by
Ψj=(ψj¯¯¯¯¯ψjϕj¯¯¯¯¯ϕj),Λj=(λj00¯¯¯¯¯λj),j=1,2,⋯,N. (25)
Nth iteration of DT leads us to a new solution of the form
q[N](x,t)=q(x,t)−2[σ1[N]]12,q[N]∗(−x,t)=q∗(−x,t)+2[σ1[N]]21. (26)
Expressing Eq. (24) in matrix elements and using Cramer’s rule we can find exact forms of and as
[σ1[N]]12=|Δ2||Δ1|,[σ1[N]]21=|Δ3||Δ1|, (27)
where , and are given by
Δ1=∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ψ1(x,t)λN−11⋯ψN(x,t)λN−1Nψ∗1(−x,t)¯¯¯¯¯λ1N−1⋯ψ∗N(−x,t)¯¯¯λN−1N⋯⋯⋯⋯⋯⋯ψ1(x,t)⋯ψN(x,t)ψ∗1(−x,t)⋯ψ∗N(−x,t)ϕ1(x,t)λN−11⋯ϕN(x,t)λN−1Nϕ∗1(−x,t)¯¯¯¯¯λ1N−1⋯ϕ∗N(−x,t)¯¯¯λN−1N⋯⋯⋯⋯⋯ϕ1(x,t)⋯ϕN(x,t)ϕ∗1(−x,t)⋯ϕ∗N(−x,t)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣, (28)
Δ2=∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ψ1(x,t)λN1⋯ψN(x,t)λNNψ∗1(−x,t)¯¯¯¯¯λ1N⋯ψ∗N(−x,t)¯¯¯λNN⋯⋯⋯⋯⋯⋯ψ1(x,t)⋯ψN(x,t)ψ∗1(−x,t)⋯ψ∗N(−x,t)ϕ1(x,t)λN−21⋯ϕN(x,t)λN−2Nϕ∗1(−x,t)¯¯¯¯¯λ1N−2⋯ϕ∗N(−x,t)¯¯¯λN−2N⋯⋯⋯⋯⋯ϕ1(x,t)⋯ϕN(x,t)ϕ∗1(−x,t)⋯ϕ∗N(−x,t)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣, (29)
Δ3=∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ϕ1(x,t)λN1⋯ϕN(x,t)λNNϕ∗1(−x,t)¯¯¯¯¯λ1N⋯ϕ∗N(−x,t)¯¯¯λNN⋯⋯⋯⋯⋯⋯ϕ1(x,t)⋯ϕN(x,t)ϕ∗1(−x,t)⋯ϕ∗N(−x,t)ψ1(x,t)λN−21⋯ψN(x,t)λN−2Nψ∗1(−x,t)¯¯¯¯¯λ1N−2⋯ψ∗N(−x,t)¯¯¯λN−2N⋯⋯⋯⋯⋯ψ1(x,t)⋯ψN(x,t)ψ∗1(−x,t)⋯ψ∗N(−x,t)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣. (30)
Substituting (27) in (26) one can get th DT solution formula. Using this formula we can obtain symmetry preserving and symmetry broken -bright and -dark soliton solutions of (1) and (2).
## 3 Bright soliton solutions of NNLS equation
### 3.1 One bright soliton solution
In this subsection, we construct the one bright soliton solution of Eqs. (1) and (2). To construct them, we feed a vacuum solution, that is as seed solution to the focusing NNLS equation (). By solving the Lax pair equations (3) with and by considering the forms of and as given in Eq. (12) one can get the symmetry broken eigenfunctions of the form , , and , with and . Substituting these basic solutions in the first iterated DT formula (17) one can generate two parameter family of symmetry broken one bright soliton solution of (1) and (2) as
q[1](x,t)= 2(¯¯¯¯¯λ1−λ1)c11c21e2η1e2(η1−¯¯¯¯¯η1)+R−1, q[1]∗(−x,t)= 2(¯¯¯¯¯λ1−λ1)¯¯¯¯¯¯c21¯¯¯¯¯¯c11e−2¯¯¯¯¯η1e2(η1−¯¯¯¯¯η1)+R−1, (31)
where . The solution (31) is the more general one soliton solution of (1) and (2). We call it as symmetry broken solution (except for specific parameter values given below) since and are independent and cannot deduce one from the other. By choosing , , , , and in (31), we can get the solution which is presented in Stalin (). One can also deduce symmetry preserving one soliton solution of NNLS equation from (31) by confining the parametric conditions in the form , , , , and , where , , and are real constants. As a result, we obtain
q[1](x,t)=−2(δ1+¯¯¯δ1)ei¯θ1e−2¯δ1x−4i¯δ21tei(θ1+¯θ1)e−2(δ1+¯δ1)x−4i(¯δ21−δ21)t+1. (32)
The above solution coincides with the one given in Ablowitz (). One can get from (32) by taking complex conjugate of it and inversing the space variable in it. The symmetry broken solution (31) for generic parametric conditions can also be written in terms of trigonometric functions as
q[1](x,t)= (¯¯¯¯¯λ1−λ1)c11c21eη1+¯¯¯¯¯η1−R2isin[η1−¯¯¯¯¯η1−iR2], q[1]∗(−x,t)= 2(¯¯¯¯¯λ1−λ1)¯¯¯¯¯¯c21¯¯¯¯¯¯c11e−η1−¯¯¯¯¯η1−R2isin[η1−¯¯¯¯¯η1−iR2]. (33)
The above solution (33) develops singularity at , . This one bright soliton solution is plotted in Fig. 1 for the parameter values , , , , , . The absolute value of plotted in Fig. 1(a) shows that the amplitude of the soliton decays at in the direction. In contrast, the absolute value of parity transformed conjugate field grows at in the direction and is illustrated in the Fig.1(b). We can get a stable propagation of soliton for the absolute value of which is demonstrated in Fig.1(c).
### 3.2 Two bright soliton solution
Now we derive the two bright soliton solution of Eqs. (1) and (2). For this, we have to solve the Lax pair equations (3) at , with the seed solutions . By doing so, we obtain the basic solutions of the form , , and |
# Bubble sort algorithm
Martin McBride, 2017-01-09
Tags sort bubble sort
Categories algorithms sort
Bubble sort is another simple sorting algorithm.
## Bubble sort first pass
Here is the simplest version of a bubble sort. We are going to sort these cards in ascending order:
The process is quite simple. Starting from the left, we take each pair of numbers. If the pair of numbers is in the wrong order, we swap them. First we take the pair 5 and 3 - they are in the wrong order (the 5 should be on the right because it is higher) so we swap them:
Now we take the next pair, 5 and 8 They are in the correct order, so leave them alone:
Then we take the pair 8 and 6 - they are in the wrong order, because 8 is greater than 6, so we swap them:
Finally we take the pair 8 and 2 - again, they are in the wrong order, so we swap them:
## Completing the algorithm
Here is what the list looks like after the first pass. Notice that the numbers are not yet in the correct order, but the final number, 8, is in the correct place:
3, 5, 6, 2, 8
Now we repeat the same process again - starting at the left, we compare each pair of numbers, and swap them if necessary. Here is what the new list looks like. Notice that the numbers are still not in the correct order, but the final two numbers, 8 and 6, are in the correct places:
3, 5, 2, 6, 8
Following this pattern, if we run the process 4 times, the last 4 numbers will be in the correct places:
2, 3, 5, 6, 8
If you think about it, if the last 4 numbers are in the correct places, the first number must also be in the correct place, because that is the only place left. So to fully sort a list of 5 values, we must repeat the sorting pass 4 times.
In general, to fully sort a list of n values, we must repeat the sorting pass (n-1) times. Here is the pseudo code for the algorithm:
FOR i = 0 TO LENGTH(k)-2
FOR j = 0 TO LENGTH(k)-2
IF k[j]>k[j+1]:
SWAP(k[j], k]j+1])
## An optimisation
There is something you can do to make the algorithm a little bit faster. You will notice that after the first path, the list is:
3, 5, 6, 2, 8
When we apply the second pass of the algorithm, we don't need to process all 5 elements because the final element is already sorted. We can just process the first 4 elements. On the third pass, the list looks like this:
3, 5, 2, 6, 8
This time we only need to process the first 3 elements. Overall, if we avoid processing the extra elements, the algorithm can be almost twice as fast.
The code change is very simple. On the i'th pass through the loop, we don't need to process the final i elements, so the j loop becomes:
FOR j = 0 TO LENGTH(k)-2-i
Making the final code:
FOR i = 0 TO LENGTH(k)-2
FOR j = 0 TO LENGTH(k)-2-i
IF k[j]>k[j+1]:
SWAP(k[j], k]j+1]) |
# Math Help - Eigenvector
1. ## Eigenvector
I have
1 1 0
1 1 0
0 0 -1
And i need to find the eigenvalue and eigenvector.
I found the eigenvalue 0, 2 and -1.
then i took B+I
2 1 0
1 2 0
0 0 0
that after beeing reduced is
1 0 0
0 1 0
0 0 0
How can i find the eigenvector from this? its the general solution?
2. ## Re: Eigenvector
First, your eigenvalues, -1, 0, and 2, are correct. Of course the definition of "eigenvalue", $\lambda$, (of matrix B) is that there exist a non-zero vector v such that $Bv= \lambda v$ which is the same as $Bv- \lambda v= (B- \lambda I)v= 0$.
Assuming the matrix you give is B (you never actually say that) then, for eigenvalue -1, yes, $B- \lambda I= B+ I= \begin{bmatrix}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ and you want to find non-zero $\begin{bmatrix} x \\ y \\ z\end{bmatrix}$ such that
$\begin{bmatrix}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} x \\ y \\ z\end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}$
Of course that is the same as solving the system of equations 2x+ y= 0, x+ 2y= 0, 0= 0.
From the first equation, y= -2x so the second equation can be written x+ 2(-2x)= 5x= 0 which gives x= 0 and then y= 0. But we have no equation z must satisfy so we can take z to be any non-zero number, say z= 1. That gives eigenvector $\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}$, or any non-zero multiple, as eigenvector corresponding to eigenvalue -1.
Doing it your way, row-reducing the matrix, would be the same as using the "augmented matrix" to solve the system of equations. Of course, since the last column, the "augmented" part, is all 0s, you don't need to write that:
$\begin{bmatrix}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 0\end{bmatrix}$
That reduces, as you say, to $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 0\end{bmatrix}$
with the fourth, "augmented" column, still 0. So it corresponds to the equations x= 0, y= 0, 0= 0. The fact that the last column is all 0s tells us z can be anything (non-zero to give a non-zero vector, of course).
Now, try eigenvalues 0 and 2. |
# Dimension
maximum number of independent directions within a mathematical space
(Redirected from Dimensions)
Dimensions are the way we see, measure and experience our world, by using up and down, right to left, back to front, hot and cold, how heavy and how long, as well as more advanced concepts from mathematics and physics. One way to define a dimension is to look at the degrees of freedom, or the way an object can move in a specific space. There are different concepts or ways where the term dimension is used, and there are also different definitions. There is no definition that can satisfy all concepts.
From left to right, the square, the cube, and the tesseract. The square is a 2-dimensional object, the cube is a 3-dimensional object, and the tesseract is a 4-dimensional object. A 1-dimensional object is just a line. A projection of the cube is given since it is viewed on a two-dimensional screen. The same applies to the tesseract, which additionally can only be shown as a projection even in three-dimensional space.
A diagram of the first four spatial dimensions.
In a vector space ${\displaystyle V}$ (with vectors being "arrows" with directions), the dimension of ${\displaystyle V}$, also written as ${\displaystyle \dim(V)}$,[1] is equal to the cardinality (or number of vectors) of a basis of ${\displaystyle V}$[2][3] (a set which indicates how many unique directions ${\displaystyle V}$ actually has). It is also equal to the number of the largest group of straight line directions of that space. "Normal" objects in everyday life are specified by three dimensions, which are usually called length, width and depth. Mathematicians call this concept Euclidean space.
Dimensions can be used to measure position too. The distance to a position from a starting place can be measured in the length, width and height directions. These distances are a measure of the position.
In some occasions, a fourth (4D) dimension, time, is used to show the position of an event in time and space.
## Other Dimensions
In modern science, people use other dimensions. Dimensions like temperature and weight can be used to show the position of something in less simple spaces. Scientist study those dimension with dimensional analysis.
Mathematicians also use dimensions. In mathematics, dimensions are more general. Dimensions in mathematics might not measure things in the world. The rules for doing arithmetic with dimensions in mathematics might be different than usual arithmetic rules.
## Dimensions and vectors
Vectors are used to show distances and directions. Vectors are often used in engineering and science, and sometimes in mathematics.
A vector is a list of numbers. There is one number for each dimension. There are arithmetic rules for vectors.
For example, if Jane wants to know the position of Sally, Sally can give Jane a vector to show the position. If Jane and Sally are in the world, there are three dimensions. Therefore, Sally gives Jane a list of three numbers to show her position. The three numbers in the vector Sally gives Jane might be:
1. Sally's distance north of Jane
2. Sally's distance east of Jane
3. Sally's height above Jane
## References
1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-07.
2. Weisstein, Eric W. "Dimension". mathworld.wolfram.com. Retrieved 2020-09-07.
3. "Basis and Dimension". people.math.carleton.ca. Retrieved 2020-09-07. |
Sean O’Rourke pointed out on December 30, 2017 that a notation should be corrected in the statement of Lemma A.1 in the probability survey Around the circular law (2012) that I wrote years ago in collaboration with Charles Bordenave.
Indeed the definition of ${\sigma^2}$ should be corrected to
$\sigma^2 :=\min_{1\leq i,j\leq n}\mathrm{Var}(X_{ij}\mid|X_{i,j}|\leq a)>0.$
It was erroneously written
$\sigma^2 :=\min_{1\leq i,j\leq n}\mathrm{Var}(X_{ij}\mathbf{1}_{|X_{i,j}|\leq a})>0.$
Let us take this occasion for a back to basics about conditional variance and variance of truncation. Let ${X}$ be a real random variable on ${(\Omega,\mathcal{F},\mathbb{P})}$ and ${A\in\mathcal{F}}$ be an event. First the real number ${\mathbb{E}(X\mid A)=\mathbb{E}(X\mid\mathbf{1}_A=1)}$ is not the random variable ${\mathbb{E}(X\mid\mathbf{1}_A)}$. We have
$\mathbb{E}(X\mid\mathbf{1}_A) =\underbrace{\frac{\mathbb{E}(X\mathbf{1}_A)}{\mathbb{P}(A)}}_{\mathbb{E}(X\mid A)}\mathbf{1}_A +\underbrace{\frac{\mathbb{E}(X\mathbf{1}_{A^c})}{\mathbb{P}(A^c)}}_{\mathbb{E}(X\mid A^c)}\mathbf{1}_{A^c}.$
Note that this formula still makes sense when ${\mathbb{P}(A)=0}$ or ${\mathbb{P}(A)=1}$.
The quantity ${\mathbb{E}(X\mid A)}$ makes sense only if ${\mathbb{P}(A)>0}$, and in this case, the conditional variance of ${X}$ given the event ${A}$ is the real number given by
$\begin{array}{rcl} \mathrm{Var}(X\mid A) &=&\mathbb{E}((X-\mathbb{E}(X\mid A))^2\mid A)\\ &=&\mathbb{E}(X^2\mid A)-\mathbb{E}(X\mid A)^2\\ &=&\frac{\mathbb{E}(X^2\mathbf{1}_A)}{\mathbb{P}(A)} -\frac{\mathbb{E}(X\mathbf{1}_A)^2}{\mathbb{P}(A)^2}\\ &=& \frac{\mathbb{E}(X^2\mathbf{1}_A)\mathbb{P}(A)-\mathbb{E}(X\mathbf{1}_A)^2}{\mathbb{P}(A)^2}\\ &=&\mathbb{E}_A(X^2)-\mathbb{E}_A(X)^2=:\mathrm{Var}_A(X) \end{array}$
where ${\mathbb{E}_A}$ is the expectation with respect to the probability measure with density ${\mathbf{1}_A/\mathbb{P}(A)}$ with respect to ${\mathbb{P}}$. In particular, by the Cauchy–Schwarz inequality,
$\mathrm{Var}(X\mid A) \geq 0$
with equality if and only if ${X}$ and ${\mathbf{1}_A}$ are colinear.
Of course ${\mathrm{Var}(X\mid A)=0}$ if ${X}$ is constant. However ${\mathrm{Var}(X\mid A)}$ may vanish for a non-constant ${X}$. Indeed if ${A=\{|X|\leq a\}}$ and if ${X\sim\frac{1}{2}\delta_{a/2}+\frac{1}{2}\delta_{2a}}$ then ${X\mid A}$ is constant and equal to ${a/2}$. In this example, since ${X\mathbf{1}_A}$ is not a constant, this shows also that one cannot lower bound ${\mathrm{Var}(X\mid A)}$ with the variance of the truncation
$\mathrm{Var}(X\mathbf{1}_A)=\mathbb{E}(X^2\mathbf{1}_A)-\mathbb{E}(X\mathbf{1}_A)^2.$
Another notable correction. Mylène Maïda pointed out to me on February 27 2018 that at the bottom of page 14, just before the statement
$\sup_{z\in C}|n\varphi_{n,1}(\sqrt{n}z)-\pi^{-1}\mathbf{1}_{[0,1]}(|z|)|=0$
the compact set ${C}$ must be taken in ${\{z\in\mathbb{C}:|z|\neq1\}}$ and not on the whole complex plane ${\mathbb{C}}$. Indeed, when ${|z|=1}$, ${n\varphi_{n,1}(\sqrt{n}z)}$ tends as ${n\rightarrow\infty}$ to ${1/2}$, and not to ${\pi^{-1}}$, see for instance this former post for a one formula proof based on the central limit theorem for Poisson random variables. Anyway this is really not surprising since a sequence of continuous functions cannot converge uniformly to a discontinuous function.
Yet another correction. Page 39 line 9 replace $M_\mu(a)$ by $M_\mu(q)$.
## Be First to Comment
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# What if $F\neq \frac{dp}{dt}$?
I was thinking of this idea that maybe there are esoteric cases where the force is not given in classical mechanics as $F=dp/dt$ but as some function of $F=F(p,q,\dot{p},\dot{q})$
E.g, something like: $k\cdot \frac{dp}{dq}$ with a suitable constant $k$, or any other sort of function of p,q and its time derivatives.
Are there any toy models that theorists suggest on this idea?
-
If you're using $q$ and $p$ then you're implicitly using Hamiltonian formalism. Then $\dot{p}=u(q,p)$ and $\dot{q}=v(q,p)$ so $F=F(p,q,\dot{p},\dot{q})\equiv F(q,p)$. In your particular proposal $k\displaystyle\frac{dq}{dp}=0$ because $q$ and $p$ are treated as independent variables. In classical mechanics the choice of the independent variables is of capital importance.
Let's take point particle electromagnetism. The Hamiltonian is:
$$H=\frac{1}{2}\left(\vec{p}-e\vec{A} \right)^2+e\phi$$
and Hamilton's equation for the momentum reads:
$$\dot{p}_{i}=-\frac{\partial H}{\partial q_{i}}=0$$
but $\vec{F}\neq 0$. I think the problem is mixing terms about Newtonian Mechanics (i.e. Force) and terms about Hamiltonian Mechanics (momentum).
-
I don't think choosing $q$ and $p$ as independent variable means Hamiltonian formalism. It is only Hamiltonian formalism when $-\partial H/\partial q=\dot{p}$ and $\partial H/\partial p=\dot{q}$. – Sankaran Feb 2 '13 at 22:37
Isn't $F = dp/dt$ the definition of force? – Paul J. Gans Feb 3 '13 at 1:05
@PaulJ.Gans: In Newtonian mechanics, yes. In Hamiltionian/Lagrangian, no: en.wikipedia.org/wiki/Generalized_forces – Manishearth Feb 3 '13 at 14:16 |
# If a particle has a perfectly defined position wavefunction, what is its p wavefunc?
1. Aug 7, 2011
1. The problem statement, all variables and given/known data
The position and momentum wavefunctions are fourier transform pairs. If a particle has a perfectly defined position wavefunction Psi(x) = delta(x - x0), then what is its momentum wavefunction? Is this function normalizable?
2. Relevant equations
Fourier transform relation (Can't figure out how to use the latex inputs to write this out)
3. The attempt at a solution
I think I have to take the fourier transform of a dirac delta "function"... I believe this is just a single frequency sine or cosine wave, but I only think this based on intuition from what the fourier transform means.
As far as if this function is normalizable, I'm not really sure what is meant by that. Does that mean you take the integral over all space (or in this case all momentums) of the wavefuntion squared and set it equal to 1?
Any help/advice on this would be much appreciated. Its been a while since I've studied quantum.
Thank you.
2. Aug 7, 2011
### nnnm4
Re: If a particle has a perfectly defined position wavefunction, what is its p wavefu
You're basically right. The fourier transform of the dirac delta will give a single momentum wave (an exponential with an imaginary argument), and the question as to whether it's normalizable is equivalent to asking whether it is square-integrable.
3. Aug 8, 2011
Re: If a particle has a perfectly defined position wavefunction, what is its p wavefu
Thanks for the help.
The integral of (sin(x))^2 dx from negative infinity to infinity does not converge, therefore it is not square integrable and not normalizable.
My intuition is telling me that this means if you know the position exactly, then you cant know what the momentum is at all. Is this correct?
4. Aug 8, 2011
### Dick
Re: If a particle has a perfectly defined position wavefunction, what is its p wavefu
Your intuition is correct. The details you are showing are all wrong. The momentum wavefunction is a function of the momentum 'p'. Not the position 'x'. Why don't you actually try to find the momentum wavefunction corresponding to the position wavefunction delta(x-x0) and see if it confirms your intuition?
5. Aug 9, 2011
### nnnm4
Re: If a particle has a perfectly defined position wavefunction, what is its p wavefu
Dick is of course correct, and sin^2(p) isn't even the relevant function here. The fourier integral is trivial and you should do it out. |
# Influence of secondary ion bombardment on the composition, structure and surface properties of platinum thin films
Balaji, S and Satyam, PV and Lakshminarayanan, V and Mohan, S (2004) Influence of secondary ion bombardment on the composition, structure and surface properties of platinum thin films. In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 217 (3). pp. 423-428.
PDF Influence-472-pdf.pdf Restricted to Registered users only Download (314Kb) | Request a copy
## Abstract
Platinum (Pt) thin films were deposited by dual ion beam sputtering (DIBS) techniques on oxidized silicon substrates maintained at ambient temperature. Argon ions with energy of 1 keV and beam current of 15 mA were used to sputter the platinum target. The films during deposition were continuously bombarded by argon from a secondary argon ion source with ion energy of 150 eV and ion current density in the range 100–250 $\mu A/cm^2.$ The influence of the secondary ion beam parameters on the grain size, composition and surface morphology of the films were studied. X-ray diffraction (XRD) of all the films showed (1 1 1) orientation with other reflections being absent which is a stable structure for FCC crystals. The grain size of the Pt films prepared by DIBS at ambient temperature were found to be higher than those prepared at higher substrate temperature by low energy plasma sputtering. The presence of Ar impurities in the sputter deposited thin films is known to modify their properties. In this paper we report a method to control the Ar content in the films by secondary Ar ion bombardment of the growing films. The modification of the surface features by secondary ion beam current was studied by scanning tunneling microscope and is also presented in this paper. The surface analysis indicates a decrease in the surface roughness for the Pt films prepared at a secondary ion beam current density of 150 $\mu A/cm^2.$
Item Type: Journal Article The copyright belongs to Elsevier. Dual ion beam sputter deposition;Rutherford back scattering;Platinum (Pt) thin films;Ripple topography;Surface roughness;Preferential sputtering phenomena Division of Physical & Mathematical Sciences > Instrumentation and Applied Physics (Formally ISU) 19 May 2006 19 Sep 2010 04:26 http://eprints.iisc.ernet.in/id/eprint/6745 |
## Prealgebra (7th Edition)
a) each floor has $5\times4=20\ apartments$ b) the building has $20\times3=60\ apartments$ |
# Crystallography question
Discussion in 'Physics' started by boks, Mar 11, 2009.
1. ### boks Thread Starter Active Member
Oct 10, 2008
218
0
According to my handbook, rutile ($TiO_2$) has a primitive tetragonal cell consisting of 6 atoms. How is that possible when each lattice point contains 1/8 atom or molecule?
Apr 20, 2004
15,815
282
3. ### studiot AAC Fanatic!
Nov 9, 2007
5,005
513
The rutile structure is not close packed. It shows 6:3 coordination which means that each titanium atom is surrounded by 6 (not 8) oxygen atoms arranged at the vertices of an octahedron.
An octahedon has 8 faces but only 6 vertices
Each oxygen is coordinated to 3 titaniums in a planar equilateral triangle. |
# Question
Buck & Company incurred the following costs during August:
Raw materials purchased . . . . . . . . . . . . . . . . . . . . . . . . . . . \$ 44,140
Direct labor (\$12.50 per hour) . . . . . . . . . . . . . . . . . . . . . . . . 57,500
Manufacturing overhead (actual) . . . . . . . . . . . . . . . . . . . . . 90,300
Selling expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31,800
Administrative expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14,700
Interest expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6,400
Manufacturing overhead is applied on the basis of \$20 per direct labor hour. Assume that overapplied or underapplied overhead is transferred to cost of goods sold only at the end of the year. During the month, 4,200 units of product were manufactured and 4,400 units of product were sold. On August 1 and August 31, Buck & Company carried the following inventory balances:
Required:
a. Prepare a statement of cost of goods manufactured for the month of August and calculate the average cost per unit of product manufactured.
b. Calculate the cost of goods sold during August.
c. Calculate the difference between cost of goods manufactured and cost of goods sold. How will this amount be reported in the financial statements?
d. (Optional) Prepare a traditional (absorption) income statement for Buck & Company for the month of August. Assume that sales for the month were \$272,800 and the company’s effective income tax rate was34%.
Sales7
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### Rainbow Arithmetic Progressions II: The Collaboration
In the previous post I briefly discussed the algorithms that played a role in the recent paper Rainbow arithmetic progressions. Today, I want to take a detour from our typical discussion of computational methods and instead discuss the collaboration that led to this paper.
It is important to explicitly say that while this paper was very collaborative, and used the expertise and hard work of many individuals, this blog post was written entirely by me during office hours and in a hurry. Any opinions, errors, or other reason to be angry with the content here is entirely my fault and not the fault of my fantastic coauthors. While I’m at it, let me actually list my coauthors by name:
Steve Butler, Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard L. Kramer, Jephian C.-H. Lin, Ryan R. Martin, Nathan Warnberg, and Michael Young.
Read the rest of this entry »
### Rainbow Arithmetic Progressions I : Search Algorithm
At Iowa State, we tried an experiment this year to have a Discrete Mathematics Working Seminar where the group that normally meets for a research talk stays for an extra hour a week to work on a research problem. The goal was to break into smaller groups, but we ended up staying as one giant mass of mathematicians, including 5 faculty, a lecturer, and 6 graduate students. We recently posted the result to the arXiv:
S. Butler, C. Erickson, L. Hogben, K. Hogenson, L. Kramer, R. L. Kramer, J. C.-H. Lin, R. R. Martin, D. Stolee, N. Warnberg, and M. Young, Rainbow arithmetic progressions, arXiv:1404.7232 [math.CO]
Today, we will discuss the backtrack-search that was used to gather initial data on this problem. Since the paper features proofs as its main feature, not all of the algorithmic details were discussed (but enough were that anyone could figure out the rest). Here, I’ll flesh out as much detail as I can. In a later post, I will comment on the experimental collaboration.
Read the rest of this entry »
### Best Practices for Backtrack Search Algorithms
Almost every algorithm I implement these days is a combinatorial search, which is a fancy way of saying a backtrack search to find combinatorial objects. A recursive algorithm determines a set of choices to make, selects one, and deepens the search. After all options are exhausted, the algorithm steps “up” a level of the search tree and makes a different choice.
Today, I want to discuss a few patterns I have developed in order to make such a backtracking search very error-proof and to make development streamlined. These concepts are particularly helpful when using TreeSearch.
Read the rest of this entry »
### Best Practices for Scientific Computing
The following tweet found its way into my Twitter newsfeed.
I thought it would be appropriate to take the advice found in this article and apply them specifically to computational combinatorics. In particular, they identify and outline eight best practices, which I will discuss individually. Further, during the discussion I will include how I apply (or will apply) these concepts during my own development process. Below, I have simply copied their “Summary of Best Practices”. You can investigate each item in more detail in the original article. I’ll add my own ideas under each main topic.
Caveat: I showed this paper to a software engineer (my wife) and she thought all of these items would be obvious to anyone with even a basic understanding of software engineering. However, as the original paper is written for biologists and this blog is written for mathematicians, I find it appropriate to cover these ideas.
Read the rest of this entry »
### Finding Cliques and Independent Sets in Circulant Graphs
We recently discussed cliquer, a “fast” algorithm for finding and counting maximum cliques in a graph. Today, I would like to discuss a modification of this algorithm when applied to graphs that have a very restrictive form of symmetry. Specifically, we will discuss circulant graphs and interval subgraphs of distance graphs. These graphs are particularly important for discovering the independence ratio of distance graphs, as discussed in the recently submitted paper, On the independence ratio of distance graphs with James Carraher, David Galvin, Stephen Hartke, and Jamie Radcliffe.
All code and data discussed is available at the distance graph page.
Read the rest of this entry »
### Finding and Counting Cliques with cliquer
For a graph $G$, a clique is a set $S \subseteq V(G)$ such that all pairs of vertices in $S$ are adjacent in $G$. Determining the maximum size of a clique in a graph is a classic NP-Complete problem, but sometimes you just need to find or count cliques anyway.
Today, we discuss the cliquer algorithm, as designed by Niskanen and Östergård. Their very efficient implementation is available on their cliquer homepage. The entire implementation is very succinct and can be used as a library. In fact, Sage uses cliquer for its backend for clique algorithms. The main algorithm computes the maximum (weighted) clique and also can count maximum or maximal (weighted) cliques. We focus on the unweighted case.
Read the rest of this entry »
### Updates to the Manickam-Miklós-Singhi Conjecture
It has been a busy year for the Manickam-Miklós-Singhi Conjecture. (See the earlier post on this topic for the important definitions.) Several new papers have appeared online or on the arXiv. Here is a brief discussion of these recent developments (organized by time of submission, as far as I can tell).
Read the rest of this entry »
### Introduction to the Nullstellensatz/Linear Algebra Method
We have spent a lot of time discussing how to generate objects, but in order for us to generate them they must exist! This idea of showing existence versus non-existence is central to the computational complexity classes of NP and coNP: the difference is whether you use an existential quantifier or a universal quantifier. So, nonexistence problems can be phrased as “All objects do not have this property” and are not suited well to proof by certificate. We can demonstrate existence by finding and presenting the object, but nonexistence is harder to prove.
Today, we discuss Computing Infeasibility Certificates for Combinatorial Problems through Hilbert’s Nullstellensatz by Jesús A. De Loera, Jon Lee, Peter N. Malkin, and Susan Margulies, where they develop what I will call the Nullstellensatz/Linear Algebra Method, or NulLA for short. Their method starts with a set of polynomials whose common roots correspond to the goal object and they build a Nullstellensatz certificate that demonstrates that a set of polynomials have no common root. The certificate is built by solving a set of linear equations. This method is greatly improved by using symmetry to reduce the size of the linear system. The authors’ main idea is to use these certificates to prove certain graphs are not 3-colorable, but I believe this can be used to prove nonexistence of combinatorial objects.
Read the rest of this entry »
### Spring Happenings
It has been a month and a half since my last post, which was not intentional. However, there are a couple pretty good reasons why.
1. The semester came to a close, and together my two classes had six exams (midterms, make-ups, and finals) in a span of three weeks.
2. My wife and I accepted tenure-track jobs at Iowa State University and have been working hard making plans for the move this summer.
Thanks for your patience and I hope to be back on topic very soon.
### Computational Combinatorics Roundup: March 2013
I was thinking that this month was lacking any new computational combinatorics papers, but then three papers arrived in my feed within a couple days of each other, followed by several others to finish out the month. Read on to learn about them, and as always send me an email whenever you see a relevant research article or news story.
Read the rest of this entry » |
# “! Misplaced \noalign. \hline ->\noalign” error while adding 3rd row in tabulary
I'm preparing a business strategy report which has been divided into multiple tex files. One section is about competitor analysis, and I would like to make it with a table.
The declarations in main.tex are,
\documentclass[9pt, a4paper, oneside]{book}
\usepackage{multirow} % Table
\usepackage{array} % Table
\usepackage{tabulary} % Table
\usepackage{graphicx} % Figure path
\usepackage{fancyhdr} % Page style
\usepackage{setspace} % Line space
\usepackage[parfill]{parskip} % Gap between paragraph instead indentation
\usepackage{pdflscape} % In order to adjust a large table
\usepackage{enumitem}
\usepackage{natbib} % Allows the user to switch between Harvard or numeric
The analysis table is respectively included in the substantial chapter,
\input{competitors_list.tex}
And the table is declared as,
\begin{landscape}
\normalsize
\noindent
\begin{tabulary}{.95\textwidth}{| >{\bfseries}l | l | l | l | l | l |}
\hline
Competitor & Profile & Product \& Service (incl. price) & Marketing Strategies & Advantages & Disadvantages \\\hline
% 2nd row
Company 1 &
\begin{minipage}[t]{.2\textwidth}
Company 1 provides the services
\end{minipage}
&
\begin{minipage}[t]{.2\textwidth}
\begin{itemize}
\item General service
\item Sepcific service
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{.2\textwidth}
\begin{itemize}
\item Official website
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{0.3\textwidth}
\begin{itemize}
\item Commercial union member
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{0.3\textwidth}
\begin{itemize}
\item Company size
\item Service categories
\end{itemize}
\end{minipage} \\ \hline
\end{tabulary}
\end{landscape}
The above codes do generate a table as follows,
However, when I want to add the third row,
\begin{landscape}
\normalsize
\noindent
\begin{tabulary}{.95\textwidth}{| >{\bfseries}l | l | l | l | l | l |}
\hline
Competitor & Profile & Product \& Service (incl. price) & Marketing Strategies & Advantages & Disadvantages \\\hline
% 2nd row
Company 1 &
\begin{minipage}[t]{.2\textwidth}
Company 1 provides the services
\end{minipage}
&
\begin{minipage}[t]{.2\textwidth}
\begin{itemize}
\item General service
\item Sepcific service
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{.2\textwidth}
\begin{itemize}
\item Official website
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{0.3\textwidth}
\begin{itemize}
\item Commercial union member
\end{itemize}
\end{minipage}
&
\begin{minipage}[t]{0.3\textwidth}
\begin{itemize}
\item Company size
\item Service categories
\end{itemize}
\end{minipage} \\ \hline
% 3rd row
Company 2 &
\begin{minipage}
A big company provides
\end{minipage}
&
\begin{minipage}
\begin{itemize}
\item Project management
\end{itemize}
\end{minipage}
&
\begin{minipage}
\begin{itemize}
\item Official website
\end{itemize}
\end{minipage}
&
\begin{minipage}
\begin{itemize}
\item Large size
\end{itemize}
\end{minipage}
&
\begin{minipage}
\begin{itemize}
\item High price
\end{itemize}
\end{minipage} \\\hline
\end{tabulary}
\end{landscape}
Error appeared while compiling,
! Missing number, treated as zero.
A
l.68 \end{tabulary}
I thought it may caused by \\ and add \relax after each \\ . Then new error was produced,
! Misplaced \noalign.
\hline ->\noalign
{\ifnum 0=}\fi \hrule \@height \arrayrulewidth \futurelet...
l.68 \end{tabulary}
Additionally, there are several competitors need to be added, do I need to switch tabulary to longtable in case of insufficient space?
Sorry for my poor knowledge of Latex and many appreciates for your help.
Unrelated to tabulary: you are missing the minipage width in all the new entries
\begin{minipage}
Note the error message
! Missing number, treated as zero.
\begin{minipage}
A
which shows the location of the error being the A
\begin{minipage}
A
is the same as
\begin{minipage}{A}
so it tries to read A as a length, and lengths start with a number so you get the error ! Missing number, treated as zero. The line number is shown as the line with \end[tabulary}` as the whole environment is read to that point in a first pass, so that the package can do its measuring, so tex reads to the end of the environment before this error shows up.
• Thank you very much for pointing my stupid mistake out. I fell in an infinite loop during several hours. – Liu May 4 '16 at 15:22 |
Numpy Convolution Explained
The second and third parameters are dimensions of the feature detector matrix. ivector taken from open source projects. I wanted to create a "quick reference guide" for confusion matrix terminology because I couldn't find an existing resource that suited my requirements: compact in presentation, using numbers instead of arbitrary variables, and explained both in terms of formulas and sentences. arm_biquad_cascade_df1_q31. Much to this author’s chagrin, Python represents using the symbol 1j. August 10, 2010 at 7:50 AM by Dr. Welcome to Statsmodels’s Documentation¶ statsmodels is a Python module that provides classes and functions for the estimation of many different statistical models, as well as for conducting statistical tests, and statistical data exploration. Specifying the input shape. The mathematics behind neural networks is explained in detail. The Fourier Transform is one of deepest insights ever made. There's something magical about Recurrent Neural Networks (RNNs). cessing systems are the convolution and modulation properties. It is part of SciPy, scientific tools for Python. , the libopenblas-devpackage. We start with the beginning of our signal and slowly move the wavelet towards the end of the signal. 12-4, diluting the time domain with zeros corresponds to a duplication of the frequency spectrum. Digital reverb parameters. It includes a user guide, full reference documentation, a developer guide, meta information, and “NumPy Enhancement Proposals” (which include the NumPy Roadmap and detailed plans for major new features). Introduction to Neural Networks (Psy 5038): Python So far in this course we've tried to emphasize concepts usually with toy examples. "The familiar discrete convolution is simply the 1-dilated convolution. In particular, the submodule scipy. This method is based on the convolution of a scaled window with the signal. The default is None which is only valid if initializer is given as an numpy. towardsdatascience. However, users do not need to write if condition explicitly, because the appropriate array module can be obtained by xp = chainer. I have included the key portions of the code below. Note: Convolution is a linear operation – element wise matrix multiplication and addition, so we account for non-linearity by introducing a non-linear function like ReLU. This is the 3rd part in my Data Science and Machine Learning series on Deep Learning in Python. Defining a Vector¶. But there are some beautifully simple holistic concepts behind Fourier theory which are relatively easy to explain intuitively. Transfer Functions Laplace Transforms: method for solving differential equations, converts differential equations in time t into algebraic equations in complex variable s Transfer Functions: another way to represent system dynamics, via the s representation gotten from Laplace transforms, or excitation by est. Numpy, numeric Python for efficient array manipulation. The convolution of two signals is a fundamental operation in signal processing. Input image convolved with the motion blur kernel (frequency domain) (log) Frequency response of the inverse frequency filter kernel (HPF) Motion-blurred image convolved with the inverse frequency filter kernel (frequency domain) 4. VTKSpectrum. This is code implements the example given in pages 11-15 of An Introduction to the Kalman Filter by Greg Welch and Gary Bishop, University of North Carolina at Chapel Hill, Department of Computer Science. In Keras / Tensorflow terminology I believe the input shape is (1, 4, 1) i. Convolution • g*h is a function of time, and g*h = h*g – The convolution is one member of a transform pair • The Fourier transform of the convolution is the product of the two Fourier transforms! – This is the Convolution Theorem g∗h↔G(f)H(f). It features the use of computational graphs, reduced memory usage, and pre-use function optimization. matmul () Examples. For 1-D arrays, it is the inner product of. convolve(values, weights, 'valid')? When the docs mentioned convolution product is only given for points where the signals overlap completely, what are the 2 signals referring to?. In machine learning, a convolutional neural network (CNN, or ConvNet) is a class of deep, feed-forward artificial neural networks, most commonly applied to analyzing visual imagery. Fourier Transform in Numpy¶ First we will see how to find Fourier Transform using Numpy. Digital reverb parameters. It is normally performed on binary images. The model needs to know what input shape it should expect. Filtering an Image Image filtering is useful for many applications, including smoothing, sharpening, removing noise, and edge detection. In this post we will implement a simple 3-layer neural network from scratch. To illustrate convolution computation step above, an example of an image or tensor with width (W), height (H), and depth (D) is selected. However, I did not find anything substantial aiding transfer learning really. (Why I do Manual Back Propagation) Implementing Multi Channel/Layer Convolution Neural Network on Numpy with Interactive Code. Convolution is a simple mathematical operation which is fundamental to many common image processing operators. The Harris corner detection algorithm also called the Harris & Stephens corner detector is one of the simplest corner detectors available. Let’s say we somehow know the frequency of the signal. Pooling Layer - max, average, or stochastic pooling. Underpopulation: if a living cell is surrounded by fewer than two living cells, it dies. Spatial Pyramid Pooling (SPP) Crop - perform cropping transformation. The code examples are short and mostly easy to follow and well explained most of the times. Once you have it you'll be able to run a Python interpreter with all. However, I did not find anything substantial aiding transfer learning really. I haven't made this switch for any particularly deep reason - mostly, I've done it because. The definition of 2D convolution and the method how to convolve in 2D are explained here. By using convolution, we can construct the output of system for any arbitrary input signal, if we know the impulse response of system. As I explained above, these 1x1 conv layers can be used in general to change the filter space dimensionality (either increase or decrease) and in the Inception architecture we see how effective these 1x1 filters can be for dimensionality reduction, explicitly in the filter dimension space, not the spatial dimension space. The kernel is pre-flipped. You can check out the GitHub link to the repository, where you will find this article in the form of a Jupyter notebook, together with the requirements. While your answer is definitely better than mine, it's important to note that the output won't necessarily be the same size as the kernel. Recurrent Neural Networks Tutorial, Part 1 - Introduction to RNNs Recurrent Neural Networks (RNNs) are popular models that have shown great promise in many NLP tasks. Tensors are super important for deep learning and neural networks because they are the data structure that we ultimately use for building and training our neural networks. There are a number of people who know the capabilities of numpy and scipy through and through, but most of them don't hang out on comp. They are extracted from open source Python projects. The lower-left image has been padded with zeros to eliminate wraparound during convolution. Question: How is the calculation done when you use np. We will be building a convolutional neural network that will be trained on few thousand images of cats and dogs, and later be able to predict if the given image is of a cat or a dog. It turns out that the property I was looking form is linked to the Fourier series of a square waveform. More Central-Difference Formulas The formulas for f (x0) in the preceding section required that the function can be computed at abscissas that lie on both sides of x, and they were referred to as central-difference formulas. If the result is greater than threshold, the portion will be marked as detected. You can vote up the examples you like or vote down the exmaples you don't like. And since you supplied 'same', the output has the same size as the first input vector, and the samples are taken from the center of the complete output. 12-4, diluting the time domain with zeros corresponds to a duplication of the frequency spectrum. Line 01: This imports numpy, which is a linear algebra library. Download it once and read it on your Kindle device, PC, phones or tablets. We also illustrate its use in solving a differential equation in which the forcing function (i. You can also save this page to your account. After describing the architecture of a convolutional neural network, we will jump straight into code, and I will show you how to extend the deep neural networks we built last time (in part 2) with just a few new functions to turn them into CNNs. The discrete Fourier transform (DFT) is the family member used with digitized signals. The core algorithm is very well explained in the original paper, Seam Carving for Content-Aware Image Resizing by Shai Avidan and Ariel Shamir. But that's the case of six by six by one channel images. After applying 3 convolution and max-pooling operations, you are downsampling the input image from 28 x 28 x 1 to 4 x 4 x 1 and now you need to flatten this downsampled output to feed this as input to the fully connected layer. If you are working in OS-X you probably only have Numpy around. Note: Convolution is a linear operation – element wise matrix multiplication and addition, so we account for non-linearity by introducing a non-linear function like ReLU. First, let's import our data as numpy arrays using np. 12-4, diluting the time domain with zeros corresponds to a duplication of the frequency spectrum. Can be thought of as sliding a kernel of fixed coefficients. Here are the examples of the python api theano. Recurrent Neural Networks Tutorial, Part 1 - Introduction to RNNs Recurrent Neural Networks (RNNs) are popular models that have shown great promise in many NLP tasks. We will use Keras to visualize inputs that maximize the activation of the filters in different layers of the VGG16 architecture, trained on ImageNet. We won't derive all the math that's required, but I will try to give an intuitive explanation of what we are doing. Convolutional neural network explained. While your answer is definitely better than mine, it's important to note that the output won't necessarily be the same size as the kernel. Is there a ready made function in numpy/scipy to compute the correlation y=mx+o of an X and Y fast: m, m-err, o, o-err, r-coef,r-coef-err ? numpy and scipy questions are best asked on their lists, not here. That is designed for general 2D signals. >>> import numpy; print numpy. convolve appears to do the job. In particular, the submodule scipy. Principal axes in feature space, representing the directions of maximum variance in the data. In this post, I will go through the steps required for building a three layer neural network. convolve () Examples. Its first argument is the input image, which is grayscale. Machine Learning. Imagine one face with its eyebrows up high and mouth open, the person could be surprised. Convolutional Layer — The convolution layer (CONV) uses filters that perform convolution operations while scanning the input image with respect to its dimensions. x - the input signal. edu Fourier theory is pretty complicated mathematically. In this article, we’ll discover why Python is so popular, how all major deep learning frameworks support Python, including the powerful platforms TensorFlow, Keras, and PyTorch. Colourization using Convolutional Neural Network In this assignment, we will train a convolutional neural network for a task known as image colour-ization. BaseInitializer or numpy. Python execution times for matrix multiplication. On the right, mirror padding has been used to remove artificial edges introduced by borders. This implementation is fully self-contained, and only needs Numpy to work. convolve with the option same can be explained as follows: the result is the center part of length $100$ of the linear convolution of the two signals. If you have been a developer or seen one work - you know how it is to search for bugs in a code. For 1 channel input, CNN2D equals to CNN1D is the kernel length = input length. Here you can understand better what it is, with a full description, interactive examples with different filters and the convolution properties. tensorflow documentation: Extract a slice from a tensor. ) Another name for this inequality is subadditivity. Second, while the final layer in the earlier network used sigmoid activations and the cross-entropy cost function, the current network uses a softmax final layer, and the log-likelihood cost function. Convolutional Neural Networks (CNN) are biologically-inspired variants of MLPs. Hello everyone, this is part two of the tutorial face recognition using OpenCV. Do you know that convolution operation is implemented in deep learning systems via matrix multiplication? Hamiltonian MC explained Numpy tips and tricks. As explained in Chapter 3 this isn't a big change. That is designed for general 2D signals. Practical sections include the installation of Virtual Box, matrix operations using Numpy , OpenCV and the libraries we’ll be using. In fact, the stride of our convolution was 1. This is Part Two of a three part series on Convolutional Neural Networks. Convolution definition is - a form or shape that is folded in curved or tortuous windings. Become a Machine Learning and Data Science professional. Nonnegativity. We won't debate on which library is the best here, they all have their merits. Now, it's important to understand the fourth key (wd1). A closer look at the concept of weights sharing in convolutional neural networks (CNNs) and an insight on how this affects the forward and backward propagation while computing the gradients during training. how to perform fast convolution small patches in matlab. metrics import classification_report from sklearn. Then w is the vector of length m+n-1 whose kth element is. This tutorial demonstrates training a simple Convolutional Neural Network (CNN) to classify CIFAR images. For those interested specifically in convolutional neural networks, check out A guide to convolution arithmetic for deep learning. Why do you want to repeat all what has been done till now? Just fork a branch out of BVLC/caffe and start building on top of it ;). Deep Learning and Artificial Intelligence courses by the Lazy Programmer. In this tutorial (second article in the TLE series) we'll talk about a widely used method of interpretation for potential-field data called Euler deconvolution. Plain CNNs are good at object recognition, but if we want to do object detection we need to know where things are. Of course, arrays are very important for scientific data, and the majority of data you will be manipulating with Larch will be in the form of arrays. The following are 50 code examples for showing how to use numpy. by Daphne Cornelisse. Nonnegativity. Algorithm IDE Whitelist¶. The second and third parameters are dimensions of the feature detector matrix. Python numpy. As with the results from cov() we can access just the correlation of interest from the [0,1] value from the returned squared matrix. We point out that during this operation the information about the dimensions of the image is lost and we have to use it to recover the original image. The convolution of two signals is a fundamental operation in signal processing. Convolution layer n°2: it applies 16 different 5x5 filters on the 6 14x14 previous feature maps. – Jorge Oct 16 '13 at 16:33 1. In Keras / Tensorflow terminology I believe the input shape is (1, 4, 1) i. ndarray can also be given to initialize parameters from numpy array data. A popular demonstration of the capability of deep learning techniques is object recognition in image data. 1, the fully convolutional network first uses the convolutional neural network to extract image features, then transforms the number of channels into the number of categories through the $$1\times 1$$ convolution layer, and finally transforms the height and. Pointwise convolution is 1×1 convolution to change the dimension. Convolution functions just do the correlation after mirroring the kernel. 5] Fs – Sampling frequency of signal x. Read and feed data to CNTK Trainer¶. One big advantage of this approximation is that, convolution with box filter can be easily calculated with the help of integral images. The convolution can be defined for functions on Euclidean space, and other groups. Godot uses a considerably different approach to rendering (and rendering abstraction) than other, popular, game engines. by Daphne Cornelisse. In this article, we will explore Convolutional Neural Networks (CNNs) and, on a high level, go through how they are inspired by the structure of the brain. Importing the NumPy module There are several ways to import NumPy. By enforcing these rules in sequential steps, beautiful and unexpected patterns can appear. Tensors are super important for deep learning and neural networks because they are the data structure that we ultimately use for building and training our neural networks. txt file that you can use to set up the virtual environment, so that you can start playing around with it for yourself. Sign up Dilated Convolution for Semantic Image Segmentation. Posted by iamtrask on November 15, 2015. Deep Learning and Artificial Intelligence courses by the Lazy Programmer. Similarly, with convolutional layers, we typically apply BN after the convolution and before the nonlinear activation function. I haven't made this switch for any particularly deep reason - mostly, I've done it because. Equation [2] states that the fourier transform of the cosine function of frequency A is an impulse at f=A and f=-A. Such tracking algorithms suffer from the aperture problem as explained in the video below So, smooth regions are bad for tracking and textured regions with lots of corners are good. Softmax Classifiers Explained. Is a matrix applied to an image and a mathematical operation comprised of integers It works by determining the value of a central pixel by adding the. towardsdatascience. By using Xavier initialization, we make sure that the weights are not too small but not too big to propagate accurately the signals. Keras Tutorial: The Ultimate Beginner's Guide to Deep Learning in Python Share Google Linkedin Tweet In this step-by-step Keras tutorial, you'll learn how to build a convolutional neural network in Python!. filter2D(), to convolve a kernel with an image. Plot a Diagram explaining a Convolution¶. The strange result of np. The default is None which is only valid if initializer is given as an numpy. If you are unsatisfied with the boundary effects of your direct convolution, I'm not sure what to tell you, since I don't know what your application is. I would look at the research papers and articles on the topic and feel like it is a very complex topic. from skimage import io, color import matplotlib. (1 conv direction). This is code implements the example given in pages 11-15 of An Introduction to the Kalman Filter by Greg Welch and Gary Bishop, University of North Carolina at Chapel Hill, Department of Computer Science. The good news is that, anyone can create an impulse response of an acoustic space and load it up into a convolution reverb (though doing it well is more difficult). txt file that you can use to set up the virtual environment, so that you can start playing around with it for yourself. I wanted to create a "quick reference guide" for confusion matrix terminology because I couldn't find an existing resource that suited my requirements: compact in presentation, using numbers instead of arbitrary variables, and explained both in terms of formulas and sentences. Since the Wavelet is localized in time, we can multiply our signal with the wavelet at different locations in time. In a fully connected network, all nodes in a layer are fully connected to all the nodes in the previous layer. I am trying to understand the differences between the discrete convolution provided by Scipy and the analytic result one would obtain. linear_model import SGDClassifier from sklearn. Simple Image Classification using Convolutional Neural Network — Deep Learning in python. convolve (a, v, mode=0) The convolve function returns the linear convolution of two rank 1 arrays. Unlike convolution, the integration variable, τ, has the same sign in the arguments of u(···) and v(···) so the arguments have a constant difference instead of a constant sum (i. Create a 3-by-3 random matrix A and a 4-by-4 random matrix B. In this article, we will explore Convolutional Neural Networks (CNNs) and, on a high level, go through how they are inspired by the structure of the brain. Part One detailed the basics of image convolution. Now, depending on the resolution and size of the image, it will see a 32 x 32 x 3 array of numbers where the 3 refers to RGB values or channels. For digital image processing, you don't have to understand all of that. Note: The NumPy function abs acts elementwise, and correctly handles complex numbers (by computing their modulus, which is exactly what we need). The following example teaches you how to compute moving average in R language. The code examples are all in Python. If you could please offer any advice, this puzzle is driving me mad: I've come across a problem that is trivial to compute in $\mathcal{O}(m^2)$ operations, but which very closely resembles a. A closer look at the concept of weights sharing in convolutional neural networks (CNNs) and an insight on how this affects the forward and backward propagation while computing the gradients during training. For those interested specifically in convolutional neural networks, check out A guide to convolution arithmetic for deep learning. You will first implement two helper functions: one for zero padding and the other for computing the convolution function itself. Template matching using OpenCV in Python. This course is designed to remove that obstacle - to show you how to do things in the Numpy stack that are frequently needed in deep learning and data science. Convolution Of An Image Convolution has the nice property of being translational invariant. If you’re not familiar with TensorFlow or neural networks, you may find it useful to read my post on multilayer perceptrons (a simpler neural network) first. An example 2D convolution operation. Advanced NumPy¶ Author: Pauli Virtanen. Convolution describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). The data is first digitized to a 0-1 array. Convolution. These information are in the tab at the beginning of. Imagine one face with its eyebrows up high and mouth open, the person could be surprised. Using Kivy and NumPy, you will create cross-platform data science applications with low overheads. Brilliantly Wrong — Alex Rogozhnikov's blog about math, machine learning, programming and high energy physics. Here I’m assuming that you are. Explaining Tensorflow Code for a Convolutional Neural Network Jessica Yung 05. The conv2 function allows you to control the size of the output. sunspot count at time T ). What is convolution? Convolution is a general purpose filter effect for images. 4 CHAPTER 9 SEQUENCE PROCESSING WITH RECURRENT NETWORKS U V W yt xt ht ht-1 Figure 9. This one convolution operation will result in a single number as output. produced better results by giving us explained variance score of 0. Figure 2 The 2-D Laplacian of Gaussian (LoG) function. I will also point to resources for you read up on the details. Also, please take note that I didn't draw activation layer for simplicity. The standard approach is to use a simple import statement: >>> import numpy However, for large amounts of calls to NumPy functions, it can become tedious to write numpy. Explaining Tensorflow Code for a Convolutional Neural Network. TensorFlow Lite for mobile and embedded devices For Production TensorFlow Extended for end-to-end ML components. The full code for this experiment can be found here. The first argument nb_filter. " So just from this statement, we can already tell when the value of 1 increases to 2 it is not the 'familiar' convolution operation that we all learned to love. You can vote up the examples you like or vote down the exmaples you don't like. The code examples are all in Python. The convolution operator is often seen in signal processing, where it models the effect of a linear time-invariant system on a signal [1]. from skimage import io, color import matplotlib. As with the results from cov() we can access just the correlation of interest from the [0,1] value from the returned squared matrix. signal improvements - ----- The function scipy. The integrals from the last lines in equation [2] are easily evaluated using the results of the previous page. In lay terms, convolution is a mechanism to combine or “blend”[10] two functions of time 3 in a coherent manner. Numba’s vectorize allows Python functions taking scalar input arguments to be used as NumPy ufuncs. It uses a simple convolutional neural network architecture. The default is None which is only valid if initializer is given as an numpy. Built in numpy functions shown above to append or prepend zeros to a signal Hint: Python list operations tend to be slower than numpy array operations for long signals (in particular appending python lists together). correlate and scipy. produced better results by giving us explained variance score of 0. convolve instead of numarray. The questions are of 4 levels of difficulties with L1 being the easiest to L4 being the hardest. To quote the wonderful book by François Chollet, Deep Learning. Convolutions with OpenCV and Python. Brilliantly Wrong — Alex Rogozhnikov's blog about math, machine learning, programming and high energy physics. The good news is that, anyone can create an impulse response of an acoustic space and load it up into a convolution reverb (though doing it well is more difficult). tensorflow documentation: Extract a slice from a tensor. We'll also want to scale (normalize) our data by dividing each data point by the maximum value along each axis of the xAll array. 11] Add Chainer v2 code. (num, bins) = histogram(x, bins=None, range=None) : This function returns an array in num (a histogram) which contains the number of values of the array x sorted into bins defined by the optional keyword arguments bins and range. It includes a user guide, full reference documentation, a developer guide, meta information, and “NumPy Enhancement Proposals” (which include the NumPy Roadmap and detailed plans for major new features). metrics import classification_report from sklearn. This course will teach you how to build convolutional neural networks and apply it to image data. However, we must remember that these 1x1 convolutions span a certain depth, so we can think of it as a 1 x 1 x N convolution where N is the number of filters applied in the layer. I didn't sit on my computer and type the codes, so don't know if there are any non-functioning codes. How to set up and use a virtual python environment in Ubuntu? a security risk as I have explained in https: of linear filter be modelled by a convolution?. NumPy is a library for Define A Convolution Neural Network You may go through this PyTorch Tutorial video where I have explained the topics in a detailed. Thank you very much for all your help. Posted by iamtrask on November 15, 2015. Just three layers are created which are convolution (conv for short), ReLU, and max pooling. At the same time, the rationale of all strategies is always explained, with references to specific papers and books that can be studied to have a deeper understanding. see this one by Denny Britz. Softmax Classifiers Explained Python # import the necessary packages from sklearn. By using Xavier initialization, we make sure that the weights are not too small but not too big to propagate accurately the signals. In convolution layer we have kernels and to make the final filter more informative we use padding in image matrix or any kind of input array. * y, in numpy x*y), producing a new vector of same. In probability theory, the sum of two independent random variables is. 3) Convolution. On the right, mirror padding has been used to remove artificial edges introduced by borders. Its first argument is the input image, which is grayscale. Instead the goal of this post is to try and understand the fundamentals of a few simple image processing techniques. This method is based on the convolution of a scaled window with the signal. When calculating a simple moving average, numpy. In Section 4. These arrays can be multi-dimensional, have their dimensionality change. When computing the cross correlation try defining a numpy array of zeros of the known output length and setting individual values. Of course, arrays are very important for scientific data, and the majority of data you will be manipulating with Larch will be in the form of arrays. Numpy, numeric Python for efficient array manipulation. Here is a simple example of convolution of 3x3 input signal and impulse response (kernel) in 2D spatial. speech processing), 2D (e. temporal convolution). Don't use conv2. Conveniently, the same answer works either way. The convolution layer computes the output of neurons that are connected to local regions or receptive fields in the input, each computing a dot product between their weights and a small receptive field to which they are connected to in the input volume. Following the CNN class that you have completed, complete the __init__ and forward methods of the UNet class. As shown in Fig. That’s the gradient of the final circuit output value with respect to the ouput this gate computed. If you’re not familiar with TensorFlow or neural networks, you may find it useful to read my post on multilayer perceptrons (a simpler neural network) first. Godot uses a considerably different approach to rendering (and rendering abstraction) than other, popular, game engines. We will be building a convolutional neural network that will be trained on few thousand images of cats and dogs, and later be able to predict if the given image is of a cat or a dog. Random sampling with Python. The Fourier transform generalizes Fourier coefficients of a signal over time. There are many good blog posts on the Internet explaining convolution as applied in convolutional neural networks (CNNs), e. It is normally performed on binary images. The following assumes that you have a Python 3. The Bartlett Versus the Rectangular Window June 16, 2016 by Steve Arar In this article, we will discuss the fact that choice of different window functions involves a trade-off between the main lobe width and the peak sidelobe (PSL). We have discussed a single normal random variable previously; we will now talk about two or more normal random variables. What is BFSI? BFSI is an acronym for Banking, Financial Services and Insurance. Algorithm IDE Whitelist¶. A convolution is very useful for signal processing in general. Wiki defines – “ BFSI comprises commercial banks, insurance companies, non-banking financial companies, cooperatives, pensions funds, mutual funds and other smaller financial entities. The backward pass for a convolution operation (for both the data and the weights) is also a convolution (but with spatially-flipped filters). As with the results from cov() we can access just the correlation of interest from the [0,1] value from the returned squared matrix. (10, 128) for sequences of 10 vectors of 128-dimensional vectors). Can be thought of as sliding a kernel of fixed coefficients. Numba's vectorize allows Python functions taking scalar input arguments to be used as NumPy ufuncs. All video and text tutorials are free. Learning AI if You Suck at Math — Part 5 — Deep Learning and Convolutional Neural Nets in Plain English — Here we create our first Python program and explore the inner workings of neural networks!. py, which is not the most recent version. By voting up you can indicate which examples are most useful and appropriate. For the analy-. Such tracking algorithms suffer from the aperture problem as explained in the video below So, smooth regions are bad for tracking and textured regions with lots of corners are good. "The familiar discrete convolution is simply the 1-dilated convolution. Depthwise convolution is the channel-wise nxn spatial convolution. I've needed about five minutes for each of the non-library scripts and about 10 minutes for the NumPy/SciPy scripts. This procedure is also known as a convolution. Speeding-up Convolutional Neural Networks Using Fine-tuned CP-Decomposition by Vadim Lebedev, Yaroslav Ganin, Maksim Rakhuba, Ivan Oseledets, Victor Lempitsky We propose a simple two-step approach for speeding up convolution layers within large convolutional neural networks based on tensor decomposition and discriminative fine-tuning. 5] Fs – Sampling frequency of signal x. Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. Convolution Layer - convolves the input image with a set of learnable filters, each producing one feature map in the output image. Its purpose to implement efficient operations on many items in a block of memory. Convolution is the most important and fundamental concept in signal processing and analysis. I have about 100 of the devices and need to get a random sample for testing. Transfer Functions Laplace Transforms: method for solving differential equations, converts differential equations in time t into algebraic equations in complex variable s Transfer Functions: another way to represent system dynamics, via the s representation gotten from Laplace transforms, or excitation by est. Convolution: 2D operation with Python (Numpy/Scipy) Now let’s see 2D operation Below we will apply the equation to an image represented by a 3x3 matrix according to the function g = (-1 1). To illustrate convolution computation step above, an example of an image or tensor with width (W), height (H), and depth (D) is selected. In this post we will implement a simple 3-layer neural network from scratch. Adding these two 8 point signals produces aebfcgdh. |
# Python prettyXML and prettyHTML functions
After much searching around the web, I couldn't find a single module which did a decent prettyXML or prettyHTML. Because:
<node>
<child>
Text
</child>
</node>
Is flat out ridiculous, and not a good use of space.
<node>
<child>Text</child>
</node>
Is much cleaner.
That's what spurred this project. I also created a similar, but works completely different prettyHTML. These are designed to work with the build in module, miniDOM. I'm looking for a full critique (i.e. anything that needs to be).
Helper for prettyHTML:
def addLineBreaks(txt, max_char, level, indent, indentSubLines = False):
# If the length is already shorter, or if there is no space to be found
if(len(txt) <= max_char or txt.find(" ") < 0):
# Just return it
return txt
# end if
# Up the level if indentSubLines
if(indentSubLines): ++level
## Remember the last breakpoint's position
# Start it where the last \n is found (in case what is passed has already been broken up some)
lastPos = txt.rfind("\n")
# If there was no "\n" then zero
if(lastPos < 0): lastPos = 0
# Get how many breaks we need to make. (-1) because 3 lines has 2 line breaks.
numBreaks = math.ceil((len(txt) - lastPos) / max_char) - 1
# Get the first space
pos = tempPos = txt.find(" ", lastPos + 1)
# Place that many minus one line breaks
for i in range(numBreaks):
# Keep searching until we find the furthest position we can, which will be
# at tempPos - lastPos, where lastPos is the last break point.
while((tempPos - lastPos) < max_char):
# Assign the last result
pos = tempPos
# Find the next result after that one
tempPos = txt.find(" ", tempPos + 1)
# end while
# If no break was found, then up to the next spot, which may
# break what is wanted, but also prevents an infinite loop.
if(pos == lastPos): pos = tempPos
txt = txt[0:pos + 1] + "\n" + (indent * level) + txt[pos + 1:]
# Remember where the break was put (+1 so it's past the \n)
lastPos = pos + 1
# end for
# Return the new string
return txt
PrettyHTML:
def prettyHTML(xml, indent = " ", level = 0, max_char = -1):
# If given text, just return it
if(xml.nodeType in (xml.TEXT_NODE, xml.CDATA_SECTION_NODE)):
return xml.toxml()
# end if
# Elements who's children will be indented
indtCldrn = ['datalist', 'details', 'dl',
'ol',
'select', 'svg',
'ul'
]
# Elements which will be placed on their own line, with all content / children
ownLine = ['!DOCTYPE',
'button',
'dd', 'dt',
'figcaption',
'h1', 'h2', 'h3', 'h4', 'h5', 'h6',
'label', 'legend', 'li',
'optgroup', 'option', 'output',
'progress',
'rect',
'style', 'summary',
'td', 'textarea', 'th', 'title'
]
# Elements which are self closing / void / whatever you want to call them
selfClosing = ['area', 'base', 'br', 'embed', 'hr', 'img', 'input', 'keygen', 'link', 'meta', 'param', 'source', 'track', 'wbr']
# Elements who's start / end tags will sandwich the content
spaceAfter = ['address', 'article', 'aside', 'audio',
'blockquote', 'body',
'canvas', 'code',
'div',
'fieldset', 'figure', 'footer', 'form',
'iframe',
'main',
'nav', 'noscript',
'object',
'p', 'pre',
'ruby',
'samp', 'script', 'section',
'table', 'template',
'video'
]
## This is just here so all HTML elements are accounted for
"""inline = ['a', 'abbr',
'b', 'bdi', 'bdo',
'cite',
'data',
'del', 'dfn',
'em',
'i', 'ins',
'kbd',
'mark',
'q',
'rp', 'rt',
's', 'small', 'span', 'strong', 'sub', 'sup',
'time',
'u',
'var',
]"""
# Holds the pretty text
pretty = ""
# HTML is case insensitive, so make all node's lowercase
nodeName = xml.nodeName.lower()
## Make the opening tag
# Indent the tag, and add the node name
openTag = (indent * level) + "<" + nodeName
# Add the attributes (With the attribute name as lowercase)
# This is tried as a precautionary
try:
for item in xml.attributes.items():
openTag += " " + item[0].lower() + '="' + item[1] + '"'
# end for
except AttributeError:
pass
# end try
# If we have a node which is self-closing
if(nodeName in selfClosing):
openTag += " />"
# Ensure it's not too long
if(max_char > 0):
openTag = addLineBreaks(openTag, max_char, level, indent, True)
# end if
# If not a <wbr />, then we also need to add a \n
if(nodeName != "wbr"): openTag += "\n"
# And return it, no need to process further
return openTag
# else, it's not a self closing tag
else:
openTag += ">"
# end if
# If we want to indent the children
if(nodeName in indtCldrn):
# Add the open tag with a line break (indentation is already included)
pretty += (openTag + "\n")
# Get the prettyHTML for all the children, placing them one level deeper
for child in xml.childNodes:
pretty += prettyHTML(child, indent, level + 1, max_char)
# end for
# Add the closing tag with indentation
pretty += ((indent * level) + "</" + nodeName + ">\n")
# else if we want to sandwich everything.
elif(nodeName in spaceAfter):
# We assume that there is already a line break before (indentation is already included)
pretty += (openTag + "\n")
# Holds the inner text
temp = ""
# Get the prettyHTML for all the children
for child in xml.childNodes:
temp += prettyHTML(child, indent, level, max_char)
# end for
# If we have a limit
if(max_char > 0):
# Break it up
temp = addLineBreaks(temp, max_char, level, indent)
# end if
# Append temp to pretty
pretty += temp
# If the last character isn't a \n, make it one so there is a break before the closing tag
if(pretty[-1] != "\n"): pretty += "\n"
# Add the closing tag with indentation
pretty += ((indent * level) + "</" + nodeName + ">\n")
# Else, we either have an inline, or a tag which should sit on its own line
else:
# We assume that there is already a line break before (indentation included)
pretty += openTag
for child in xml.childNodes:
pretty += child.toxml()
# end for
pretty += ("</" + nodeName + ">")
# Then be sure to cut it up, if we have a max_char
if(max_char > 0):
# Break it up (If necessary)
pretty = addLineBreaks(pretty, max_char, level, indent, True)
# end if
# If we have a tag which is supposed to be on its own line, add a line break
if(nodeName in ownLine):
pretty += "\n"
# end if
# end if
# Return the now prettified text
return pretty
# end prettyHTML
Pretty XML:
def prettyXML(xml, indent = " ", level = 0):
# Flag for whether or not to indent, or just wrap it
hasNonText = False
# For each node in the childNodes
for node in xml.childNodes:
# Check if there is something that isn't text
if(node.nodeType not in (node.TEXT_NODE, node.CDATA_SECTION_NODE)):
# If so, set the flag, and kill the loop
hasNonText = True
break
# end if
# end for
# Store the pretty XML
pretty = ""
# If we have nonText
if(hasNonText):
# Add the indentation and start tag
pretty += ((indent * level) + "<" + xml.nodeName)
# Add the attributes (This is tried as a precaution)
try:
for item in xml.attributes.items():
pretty += " " + item[0] + '="' + item[1] + '"'
# end for
except AttributeError:
pass
# end try
# add the closing > and a line break
pretty += ">\n"
# Loop through each child
for child in xml.childNodes:
# And add it to pretty, moving it one level deeper
pretty += prettyXML(child, indent, level + 1)
# end for
# Add the closing tag with indentation
pretty += ((indent * level) + "</" + xml.nodeName + ">\n")
# Else if it had no children
elif(not xml.childNodes):
# Just add the raw XML with indentation. Probably a self-closing tag
pretty += ((indent * level) + xml.toxml() + "\n")
# Else, it only had text children
else:
# Add the indentation and start tag
pretty += ((indent * level) + "<" + xml.nodeName)
# Add the attributes (like above)
try:
for item in xml.attributes.items():
pretty += " " + item[0] + '="' + item[1] + '"'
# end for
except AttributeError:
pass
# end try
pretty += ">"
# Add all children (which will just be text)
for node in xml.childNodes:
pretty += node.toxml()
# end for
pretty += "</" + xml.nodeName + ">\n"
# end if
return pretty
# end prettyXML
• I'm trying to run this. To speed things up, do you have a quick demo on how to call these functions and what the arguments should do? It's hard to know what's a bug without any kind of specification. – Veedrac Jan 10 '15 at 12:58
• @Veedrac, works fine for me: imgur.com/cvB3sWM Also, I'll put together an example. – David Jan 10 '15 at 14:40
• Ah, I didn't expect you to want the trailing four asdfs not wrapped. I think I get it; the max_char is actually a misnamed min_char. – Veedrac Jan 10 '15 at 15:03
• No, you were right. It wasn't working. Sorry, my day was about to end when I posted that. I'll update my code with other updates, again, once I fully analyze your answer. – David Jan 10 '15 at 20:38
• Please note that you shouldn't edit the code in the question; see here. You can post a new question (best leave a bit more room for this question) or post a link to updated code in the comments. See the link for justification. – Veedrac Jan 10 '15 at 23:02
You have in total three functions over 300 lines, so ~100 lines to a function. This is 5-10x too few functions for this amount of code.
A lot of your formatting looks like you've used a C-like language and can't handle writing Python. Stuff like
if(x):
...
# end if
which should just be
if x:
...
You even have
++level
which actually just means
+ ( + level )
Namely, this is a no-op.
My first criticism before I look at this more in-depth is to read PEP 8 and stick to it. As the code is written, it would alienate any current Python user.
I'm looking at prettyXML since it's the simpler of the two. You start with has_non_text, which is a great candidate for a function:
def has_non_text(xml):
ret = False
# For each node in the childNodes
for node in xml.childNodes:
# Check if there is something that isn't text
if node.nodeType not in (node.TEXT_NODE, node.CDATA_SECTION_NODE):
# If so, set the flag, and kill the loop
ret = True
break
return ret
Some of these comments are trivial, so remove them. Also, use an early return, since it's now a stand-alone function.
def has_non_text(xml):
for node in xml.childNodes:
# If not text
if node.nodeType not in (node.TEXT_NODE, node.CDATA_SECTION_NODE):
return True
return False
This would be even simpler with an is_text function:
def is_text(node):
return node.nodeType in (node.TEXT_NODE, node.CDATA_SECTION_NODE)
def has_non_text(xml):
return not all(is_text(node) for node in xml.childNodes)
# Flag for whether or not to indent, or just wrap it
would be much better if it explained which action corresponded to which flag state. Something like
# Indent iff has non-text
prettyXML now looks like:
def prettyXML(xml, indent=" ", level=0):
# Store the pretty XML
pretty = ""
# Indent iff has non-text
if has_non_text(xml):
...
# Else if it had no children
elif not xml.childNodes:
...
# Else, it only had text children
else:
...
return pretty
This would be better as
def prettyXML(xml, indent=" ", level=0):
# Indent iff has non-text
if has_non_text(xml):
pretty = ""
...
return pretty
elif not xml.childNodes:
pretty = ""
...
return pretty
else:
pretty = ""
...
return pretty
since each subsection is a logically separate action. In fact, you could make each a separate function if you wanted.
Consdier the first branch.
pretty = ""
# Add the indentation and start tag
pretty += (indent * level) + "<" + xml.nodeName
# Add the attributes (This is tried as a precaution)
try:
for item in xml.attributes.items():
pretty += " " + item[0] + '="' + item[1] + '"'
except AttributeError:
pass
# add the closing > and a line break
pretty += ">\n"
# Loop through each child
for child in xml.childNodes:
# And add it to pretty, moving it one level deeper
pretty += prettyXML(child, indent, level + 1)
# Add the closing tag with indentation
pretty += ((indent * level) + "</" + xml.nodeName + ">\n")
return pretty
You should never do += on strings in a loop unless you know enough to know it doesn't apply. In this case it definitely doesn't.
Here's an alternative:
pretty += "".join(" " + item[0] + '="' + item[1] + '"' for item in xml.attributes.items())
You should also use formatting, unpacking and move the loop out of the try:
try:
attributes = xml.attributes.items()
except AttributeError:
pass
else:
pretty += "".join(' {}="{}"'.format(k, v) for k, v in attributes)
The "{}" can be better replaced with {!r}, which uses the repr of the string. This is a better match although it's not entirely safe. Further, the try can be replaced with an or:
attributes = (xml.attributes or {}).items()
pretty += "".join(' {}={!r}'.format(k, v) for k, v in attributes)
and change the outer parts.
You can then do the same for the child nodes:
pretty += "".join(prettyXML(child, indent, level + 1) for child in xml.childNodes)
and then put it together with a formatting string:
attributes = (xml.attributes or {}).items()
node_attrs = "".join(' {}={!r}'.format(k, v) for k, v in attributes)
children = "".join(prettyXML(child, indent, level + 1) for child in xml.childNodes)
make_node = (
"{tab}<{xml.nodeName}{node_attrs}>\n"
"{children}"
"{tab}</{xml.nodeName}>\n"
).format
return make_node(node_attrs=node_attrs, children=children, tab=indent * level, xml=xml)
The second option is
elif not xml.childNodes:
# Just add the raw XML with indentation. Probably a self-closing tag
pretty = ""
pretty += ((indent * level) + xml.toxml() + "\n")
return pretty
Obviously this can become just
elif not xml.childNodes:
# Just add the raw XML with indentation. Probably a self-closing tag
return (indent * level) + xml.toxml() + "\n"
Finally, the last option is really similar and gets simplified to
attributes = (xml.attributes or {}).items()
node_attrs = "".join(' {}={!r}'.format(k, v) for k, v in attributes)
children = "".join(node.toxml() for node in xml.childNodes)
make_node = "{tab}<{xml.nodeName}{node_attrs}>{children}</{xml.nodeName}>\n".format
return make_node(node_attrs=node_attrs, children=children, tab=indent * level, xml=xml)
Since the first and last options share quite a lot of logic, reshape the control flow to accomodate:
def prettyXML(xml, indent=" ", level=0):
if not xml.childNodes:
# Just add the raw XML with indentation. Probably a self-closing tag
return (indent * level) + xml.toxml() + "\n"
attributes = (xml.attributes or {}).items()
node_attrs = "".join(' {}={!r}'.format(k, v) for k, v in attributes)
# Indent iff has non-text
if has_non_text(xml):
children = "".join(prettyXML(child, indent, level + 1) for child in xml.childNodes)
make_node = (
"{tab}<{xml.nodeName}{node_attrs}>\n"
"{children}"
"{tab}</{xml.nodeName}>\n"
).format
else:
children = "".join(node.toxml() for node in xml.childNodes)
make_node = "{tab}<{xml.nodeName}{node_attrs}>{children}</{xml.nodeName}>\n".format
return make_node(node_attrs=node_attrs, children=children, tab=indent * level, xml=xml)
The make_nodes could do with being a fully-fledged function. This gives:
def is_text(node):
return node.nodeType in (node.TEXT_NODE, node.CDATA_SECTION_NODE)
def has_non_text(xml):
return not all(is_text(node) for node in xml.childNodes)
def make_node(xml, node_attrs, children, tab, *, indented):
if indented:
fmt = (
"{tab}<{xml.nodeName}{node_attrs}>\n"
"{children}"
"{tab}</{xml.nodeName}>\n"
)
else:
fmt = "{tab}<{xml.nodeName}{node_attrs}>{children}</{xml.nodeName}>\n"
return fmt.format(xml=xml, node_attrs=node_attrs, children=children, tab=tab)
def prettyXML(xml, indent=" ", level=0):
if not xml.childNodes:
# Just add the raw XML with indentation. Probably a self-closing tag
return (indent * level) + xml.toxml() + "\n"
attributes = (xml.attributes or {}).items()
node_attrs = "".join(' {}={!r}'.format(k, v) for k, v in attributes)
indented = has_non_text(xml)
if indented:
children = "".join(prettyXML(child, indent, level + 1) for child in xml.childNodes)
else:
children = "".join(node.toxml() for node in xml.childNodes)
return make_node(xml, node_attrs, children, tab=indent*level, indented=indented)
• As removing all the # end comments by hand may be annoying, you may append this code at the end of your programme and then run to have the # end comments automatically stripped. – Caridorc Jan 10 '15 at 14:31
• I finished reading the spec, aside from the #end comments, I don't see anything that breaks the standard. I use suggested indentation, () on if statements are shown as okay. ++ I forgot didn't work in Python: I'm used to C based languages where that does something, so thanks for pointing that out. Now to dig through the rest of your critique... :) – David Jan 10 '15 at 14:44
• @Caridorc, I just used a regex replace in Notepad++. – David Jan 10 '15 at 14:51
• @David You missed naming conventions (most things as snake_case). I'm surprised it doesn't explicitly mention () on if statements as being bad, but it definitely doesn't say they're OK either. The only uses used brackets for line continuation, which is different to what you did. Plus, you should have a space in that case anyway (if (x or\ny)). There's also "Compound statements (multiple statements on the same line) are generally discouraged." – Veedrac Jan 10 '15 at 14:54 |
# 206-706-4764
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# Hi everybody,this is my first post. So sorry for the possible
1. Jun 28, 2011
### matthew82
Hi everybody,
this is my first post. So sorry for the possible trivial question.
Can someone explain me what is Y-bjorken?
And what is the difference with X-bjorken?
Thanks
2. Jun 28, 2011
### NEILS BOHR
Re: Y-Bjorken
Bjorken X is a scaling variable used in deep inelastic scattering..for more info on this experiment see this:
http://ikpe1101.ikp.kfa-juelich.de/briefbook_part_detectors/node42.html#41 [Broken]
It gives the momentum fraction carried by an inclusively observed particle. Structure functions are mainly dependent on this variable.
Last edited by a moderator: May 5, 2017
3. Jun 28, 2011
### bcphysicist
Re: Y-Bjorken
Bjorken discovered what is known as light-cone scaling, (or "Bjorken scaling") a phenomenon in the deep inelastic scattering of light on strongly interacting particles, known as hadrons (such as protons and neutrons)... think that's what you're looking for.
4. Jul 13, 2011
### daqpan
Re: Y-Bjorken
I have a question related to the Bjorken Y-variable which I don't believe has been covered by the link so far. How does one calculate the scattering angle [itex]$\theta$[\latex] from the Bjorken Y variable? |
# Thread: how do i integrate this?
1. ## how do i integrate this?
evaluate the integral sec^2 x/radical tan x dx?
2. ## Re: how do i integrate this?
Just make substitution for tan x. Let tan x = t then sec^2x dx = dt and the question reduces to a simple integral 1/sqrt t dt |
# General Maths 2015 HSC error? (1 Viewer)
#### sourmilk
##### Active Member
http://www.boardofstudies.nsw.edu.au/hsc_exams/2015/exams/2015-hsc-maths-general-2.pdf
A Student Representative Council (SRC) consists of five members. Three of the members
are being selected to attend a conference.
In how many ways can the three members be selected?
(A) 10
(B) 20
(C) 30
(D) 60
Isn't the answer D? Apparently in the answers the answer is A. Both me and my friend got D and we don't know how it's A, and I literally wrote all the possible combinations and it's definitely more than 10.
#### davidgoes4wce
##### Well-Known Member
I went through that question today , and I thought the answer was A.
$\bg_white ^{5}C_3=10$
It doesn't matter about the 'order' of the selections. (*Unordered selection)
#### sourmilk
##### Active Member
I went through that question today , and I thought the answer was A.
$\bg_white ^{5}C_3=10$
It doesn't matter about the 'order' of the selections. (*Unordered selection)
OoooooooooOOh thank you so much! my friend told me to dismiss that it was wrong ahahahaha
#### Mongoose528
##### Member
For combinations without repetition, the formula is n!/r!(n-r)!
Where n is the number of things to choose from and r is the amount of things we chose
So in this question, there are 5 people to choose from and we have to choose 3 of them
So the amount of ways is 5!/(3!(5-3)!) = 5!/(3!2!) = 120/(6*2) = 120/12 = 10
Last edited: |
Question
Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?
$40.0 \textrm{ Hz}$
Solution Video |
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Basic 2D Platformer Physics, Part 4
Difficulty:IntermediateLength:LongLanguages:
This post is part of a series called Basic 2D Platformer Physics .
Basic 2D Platformer Physics, Part 3
Basic 2D Platformer Physics, Part 5: Object vs. Object Collision Detection
In this part of the series on 2D platformer physics, we'll add ledge grabbing, jump leniency mechanics, and an ability to scale the object.
Ledge Grabbing
Now that we can jump, drop down from one-way platforms and run around, we can also implement ledge grabbing. Ledge-grabbing mechanics are definitely not a must-have in every game, but it's a very popular method of extending a player's possible movement range while still not doing something extreme like a double jump.
Let's look at how we determine whether a ledge can be grabbed. To determine whether the character can grab a ledge, we'll constantly check the side the character is moving towards. If we find an empty tile at the top of the AABB, and then a solid tile below it, then the top of that solid tile is the ledge our character can grab onto.
Setting Up Variables
Let's go to our Character class, where we'll implement ledge grabbing. There's no point doing this in the MovingObject class, since most of the objects won't have an option to grab a ledge, so it would be a waste to do any processing in that direction there.
First, we need to add a couple of constants. Let's start by creating the sensor offset constants.
The cGrabLedgeStartY and cGrabLedgeEndY are offsets from the top of the AABB; the first one is the first sensor point, and the second one is the ending sensor point. As you can see, the character will need to find a ledge within 2 pixels.
We also need an additional constant to align the character to the tile it just grabbed. For our character, this will be set to -4.
Aside from that, we'll want to remember the coordinates of the tile that we grabbed. Let's save those as a character's member variable.
Implementation
We'll need to see if we can grab the ledge from the jump state, so let's head over there. Right after we check whether the character has landed on the ground, let's see if the conditions to grab a ledge are fulfilled. The primary conditions are as follows:
• The vertical speed is less than or equal to zero (the character is falling).
• The character is not at the ceiling—no use grabbing a ledge if you can't jump off it.
• The character collides with the wall and moves towards it.
If those three conditions are met, then we need to look for the ledge to grab. Let's start by calculating the top position of the sensor, which is going to be either the top left or top right corner of the AABB.
Now, as you may imagine, here we'll encounter a similar problem to the one we found when implementing the collision checks—if the character is falling very fast, it is actually very likely to miss the hotspot at which it can grab the ledge. That's why we'll need to check for the tile we need to grab not starting from the current frame's corner, but the previous one's—as illustrated here:
The top image of a character is its position in the previous frame. In this situation, we need to start looking for opportunities to grab a ledge from the top-right corner of the previous frame's AABB and stop at the current frame's position.
Let's get the coordinates of the tiles we need to check, starting by declaring the variables. We'll be checking tiles in a single column, so all we need is the X coordinate of the column as well as its top and bottom Y coordinates.
Let's get the X coordinate of the AABB's corner.
We want to start looking for a ledge from the previous frame's position only if we actually were already moving towards the pushed wall in that time—so our character's X position didn't change.
As you can see, in that case we're calculating the topY using the previous frame's position, and the bottom one using that of the current frame. If we weren't next to any wall, then we're simply going to see if we can grab a ledge using only the object's position in the current frame.
Alright, now that we know which tiles to check, we can start iterating through them. We'll be going from the top to the bottom, because this order makes the most sense as we allow for ledge grabbing only when the character is falling.
Now let's check whether the tile we are iterating fulfills the conditions which allow the character to grab a ledge. The conditions, as explained before, are as follows:
• The tile is empty.
• The tile below it is a solid tile (this is the tile we want to grab).
The next step is to calculate the position of the corner of the tile we want to grab. This is pretty simple—we just need to get the tile's position and then offset it by the tile's size.
Now that we know this, we should check whether the corner is between our sensor points. Of course we want to do that only if we're checking the tile concerning the current frame's position, which is the tile with Y coordinate equal to the bottomY. If that's not the case, then we can safely assume that we passed the ledge between the previous and the current frame—so we want to grab the ledge anyway.
Now we're home, we have found the ledge that we want to grab. First, let's save the grabbed ledge's tile position.
We also need to align the character with the ledge. What we want to do is align the top of the character's ledge sensor with the top of the tile, and then offset that position by cGrabLedgeTileOffsetY.
Aside from this, we need to do things like set the speed to zero and change the state to CharacterState.GrabLedge. After this, we can break from the loop because there's no point iterating through the rest of the tiles.
That's going to be it! The ledges can now be detected and grabbed, so now we just need to implement the GrabLedge state, which we skipped earlier.
Ledge Grab Controls
Once the character is grabbing a ledge, the player has two options: they can either jump up or drop down. Jumping works as normal; the player presses the jump key and the jump's force is identical to the force applied when jumping from the ground. Dropping down is done by pressing the down button, or the directional key that points away from the ledge.
Controls Implementation
The first thing here to do is to detect whether the ledge is to the left or to the right of the character. We can do this because we saved the coordinates of the ledge the character is grabbing.
We can use that information to determine whether the character is supposed to drop off the ledge. To drop down, the player needs to either:
• press the down button
• press the left button when we're grabbing a ledge on the right, or
• press the right button when we're grabbing a ledge on the left
There's a small caveat here. Consider a situation when we're holding the down button and the right button, when the character is holding onto a ledge to the right. It'll result in the following situation:
The problem here is that the character grabs the ledge immediately after it lets go of it.
A simple solution to this is to lock movement towards the ledge for a couple frames after we dropped off the ledge. For that we need to add two new variables; let's call them mCannotGoLeftFrames and mCannotGoRightFrames.
When the character drops off the ledge, we need to set those variables and change the state to jump.
Now let's go back for a bit to the Jump state, and let's make sure that it respects our ban on moving either left or right after dropping off the ledge. Let's reset the inputs right before we check if we should look for a ledge to grab.
As you can see, this way we won't fulfill the conditions needed to grab a ledge as long as the blocked direction is the same as the direction of the ledge the character may try to grab. Each time we deny a particular input, we decrement from the remaining blocking frames, so eventually we'll be able to move again—in our case, after 3 frames.
Now let's continue working on the GrabLedge state. Since we handled dropping off the ledge, we now need to make it possible to jump from the grabbing position.
If the character didn't drop from the ledge, we need to check whether the jump key has been pressed; if so, we need to set the jump's vertical speed and change the state:
That's pretty much it! Now the ledge grabbing should work properly in all kinds of situations.
Allow the Character to Jump Shortly After Leaving a Platform
Often, to make jumps easier in platformer games, the character is allowed to jump if it just stepped off the edge of a platform and is no longer on the ground. This is a popular method to mitigate an illusion that the player has pressed the jump button but the character didn't jump, which might have appeared due to input lag or the player pressing the jump button right after the character has moved off the platform.
Let's implement such a mechanic now. First of all, we need to add a constant of how many frames after the character steps off the platform it can still perform a jump.
We'll also need a frame counter in the Character class, so we know how many frames the character is in the air already.
Now let's set the mFramesFromJumpStart to 0 every time we just left the ground. Let's do that right after we call UpdatePhysics.
And let's increment it every frame we're in the jump state.
If we're in the jump state, we cannot allow an in-air jump if we're either at the ceiling or have a positive vertical speed. Positive vertical speed would mean that the character hasn't missed a jump.
If that's not the case and the jump key is pressed, all we need to do is set the vertical speed to the jump value, as if we jumped normally, even though the character is in the jump state already.
And that's it! We can set the cJumpFramesThreshold to a big value like 10 frames to make sure that it works.
The effect here is quite exaggerated. It's not very noticeable if we're allowing the character to jump just 1-4 frames after it is in fact no longer on the ground, but overall this allows us to modify how lenient we want our jumps to be.
Scaling the Objects
Let's make it possible to scale the objects. We already have the mScale in the MovingObject class, so all we actually need to do is make sure it affects the AABB and the AABB offset properly.
First of all, let's edit our AABB class so it has a scale component.
Now let's edit the halfSize, so that when we access it, we actually get a scaled size instead of the unscaled one.
We'll also want to be able to get or set only an X or Y value of the half size, so we need to make separate getters and setters for those as well.
Besides scaling the AABB itself, we'll also need to scale the mAABBOffset, so that after we scale the object, its sprite will still match the AABB the same way it did when the object was unscaled. Let's head back over to the MovingObject class to edit it.
The same as previously, we'll want to have access to X and Y components separately too.
Finally, we also need to make sure that when the scale is modified in the MovingObject, it also is modified in the AABB. The object's scale can be negative, but the AABB itself shouldn't have a negative scale because we rely on half size to be always positive. That's why instead of simply passing the scale to the AABB, we're going to pass a scale that has all components positive.
All that's left to do now is to make sure that wherever we used the variables directly, we use them through the getters and setters now. Wherever we used halfSize.x, we'll want to use HalfSizeX, wherever we used halfSize.y, we'll want to use HalfSizeY, and so on. A few uses of a find and replace function should deal with this well.
Check the Results
The scaling should work well now, and because of the way we built our collision detection functions, it doesn't matter if the character is giant or tiny—it should interact with the map well.
Summary
This part concludes our work with the tilemap. In the next parts, we'll be setting things up to detect collisions between objects.
It took some time and effort, but the system in general should be very robust. One thing that may be lacking right now is the support for slopes. Many games don't rely on them, but a lot of them do, so that's the biggest improvement goal to this system. Thanks for reading so far, see you in the next part! |
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# TOC-grand test Q4
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+1 vote
What is the smallest possible number of states needed in constructing DFA for the language $\small \Sigma^*1(\Sigma\Sigma^*)1\Sigma^*$ , where $\small \Sigma = \{0,1\}$. ?
reshown Sep 8, 2019
The following DFA with four states recognizes L=Σ*1(ΣΣ*)1Σ*
By the Myhill–Nerode theorem, no smaller DFA exists because each of the four strings "epsilon,1,10,11" is in a different equivalence class of L. Their distinguishing suffixes are as follows:
answered Sep 11, 2019 by (91,540 points)
ANS . 4
Expression will become - (0+1)* 1 (0+1)^+ 1 (0+1)*
Langauges = { 101, 111............}
Min. DFA will contain 4 states.
answered Sep 8, 2019 by (14,660 points)
may be 3 states |
Copied to
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## G = C4.89(C2×D20) order 320 = 26·5
### 16th central extension by C4 of C2×D20
Series: Derived Chief Lower central Upper central
Derived series C1 — C2×C10 — C4.89(C2×D20)
Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×C4○D20 — C4.89(C2×D20)
Lower central C5 — C2×C10 — C4.89(C2×D20)
Upper central C1 — C2×C4 — C2×M4(2)
Generators and relations for C4.89(C2×D20)
G = < a,b,c,d | a4=b2=1, c20=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1c19 >
Subgroups: 574 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, (C22×C8)⋊C2, C2×C52C8, C2×C52C8, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D101C8, C22×C52C8, C10×M4(2), C2×C4○D20, C4.89(C2×D20)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8○D4, C4×D5, D20, C5⋊D4, C22×D5, (C22×C8)⋊C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20.2C4, C2×D10⋊C4, C4.89(C2×D20)
Smallest permutation representation of C4.89(C2×D20)
On 160 points
Generators in S160
(1 47 21 67)(2 48 22 68)(3 49 23 69)(4 50 24 70)(5 51 25 71)(6 52 26 72)(7 53 27 73)(8 54 28 74)(9 55 29 75)(10 56 30 76)(11 57 31 77)(12 58 32 78)(13 59 33 79)(14 60 34 80)(15 61 35 41)(16 62 36 42)(17 63 37 43)(18 64 38 44)(19 65 39 45)(20 66 40 46)(81 123 101 143)(82 124 102 144)(83 125 103 145)(84 126 104 146)(85 127 105 147)(86 128 106 148)(87 129 107 149)(88 130 108 150)(89 131 109 151)(90 132 110 152)(91 133 111 153)(92 134 112 154)(93 135 113 155)(94 136 114 156)(95 137 115 157)(96 138 116 158)(97 139 117 159)(98 140 118 160)(99 141 119 121)(100 142 120 122)
(1 89)(2 110)(3 91)(4 112)(5 93)(6 114)(7 95)(8 116)(9 97)(10 118)(11 99)(12 120)(13 101)(14 82)(15 103)(16 84)(17 105)(18 86)(19 107)(20 88)(21 109)(22 90)(23 111)(24 92)(25 113)(26 94)(27 115)(28 96)(29 117)(30 98)(31 119)(32 100)(33 81)(34 102)(35 83)(36 104)(37 85)(38 106)(39 87)(40 108)(41 125)(42 146)(43 127)(44 148)(45 129)(46 150)(47 131)(48 152)(49 133)(50 154)(51 135)(52 156)(53 137)(54 158)(55 139)(56 160)(57 141)(58 122)(59 143)(60 124)(61 145)(62 126)(63 147)(64 128)(65 149)(66 130)(67 151)(68 132)(69 153)(70 134)(71 155)(72 136)(73 157)(74 138)(75 159)(76 140)(77 121)(78 142)(79 123)(80 144)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 46 47 20 21 66 67 40)(2 19 48 65 22 39 68 45)(3 64 49 38 23 44 69 18)(4 37 50 43 24 17 70 63)(5 42 51 16 25 62 71 36)(6 15 52 61 26 35 72 41)(7 60 53 34 27 80 73 14)(8 33 54 79 28 13 74 59)(9 78 55 12 29 58 75 32)(10 11 56 57 30 31 76 77)(81 158 123 96 101 138 143 116)(82 95 124 137 102 115 144 157)(83 136 125 114 103 156 145 94)(84 113 126 155 104 93 146 135)(85 154 127 92 105 134 147 112)(86 91 128 133 106 111 148 153)(87 132 129 110 107 152 149 90)(88 109 130 151 108 89 150 131)(97 142 139 120 117 122 159 100)(98 119 140 121 118 99 160 141)
G:=sub<Sym(160)| (1,47,21,67)(2,48,22,68)(3,49,23,69)(4,50,24,70)(5,51,25,71)(6,52,26,72)(7,53,27,73)(8,54,28,74)(9,55,29,75)(10,56,30,76)(11,57,31,77)(12,58,32,78)(13,59,33,79)(14,60,34,80)(15,61,35,41)(16,62,36,42)(17,63,37,43)(18,64,38,44)(19,65,39,45)(20,66,40,46)(81,123,101,143)(82,124,102,144)(83,125,103,145)(84,126,104,146)(85,127,105,147)(86,128,106,148)(87,129,107,149)(88,130,108,150)(89,131,109,151)(90,132,110,152)(91,133,111,153)(92,134,112,154)(93,135,113,155)(94,136,114,156)(95,137,115,157)(96,138,116,158)(97,139,117,159)(98,140,118,160)(99,141,119,121)(100,142,120,122), (1,89)(2,110)(3,91)(4,112)(5,93)(6,114)(7,95)(8,116)(9,97)(10,118)(11,99)(12,120)(13,101)(14,82)(15,103)(16,84)(17,105)(18,86)(19,107)(20,88)(21,109)(22,90)(23,111)(24,92)(25,113)(26,94)(27,115)(28,96)(29,117)(30,98)(31,119)(32,100)(33,81)(34,102)(35,83)(36,104)(37,85)(38,106)(39,87)(40,108)(41,125)(42,146)(43,127)(44,148)(45,129)(46,150)(47,131)(48,152)(49,133)(50,154)(51,135)(52,156)(53,137)(54,158)(55,139)(56,160)(57,141)(58,122)(59,143)(60,124)(61,145)(62,126)(63,147)(64,128)(65,149)(66,130)(67,151)(68,132)(69,153)(70,134)(71,155)(72,136)(73,157)(74,138)(75,159)(76,140)(77,121)(78,142)(79,123)(80,144), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,46,47,20,21,66,67,40)(2,19,48,65,22,39,68,45)(3,64,49,38,23,44,69,18)(4,37,50,43,24,17,70,63)(5,42,51,16,25,62,71,36)(6,15,52,61,26,35,72,41)(7,60,53,34,27,80,73,14)(8,33,54,79,28,13,74,59)(9,78,55,12,29,58,75,32)(10,11,56,57,30,31,76,77)(81,158,123,96,101,138,143,116)(82,95,124,137,102,115,144,157)(83,136,125,114,103,156,145,94)(84,113,126,155,104,93,146,135)(85,154,127,92,105,134,147,112)(86,91,128,133,106,111,148,153)(87,132,129,110,107,152,149,90)(88,109,130,151,108,89,150,131)(97,142,139,120,117,122,159,100)(98,119,140,121,118,99,160,141)>;
G:=Group( (1,47,21,67)(2,48,22,68)(3,49,23,69)(4,50,24,70)(5,51,25,71)(6,52,26,72)(7,53,27,73)(8,54,28,74)(9,55,29,75)(10,56,30,76)(11,57,31,77)(12,58,32,78)(13,59,33,79)(14,60,34,80)(15,61,35,41)(16,62,36,42)(17,63,37,43)(18,64,38,44)(19,65,39,45)(20,66,40,46)(81,123,101,143)(82,124,102,144)(83,125,103,145)(84,126,104,146)(85,127,105,147)(86,128,106,148)(87,129,107,149)(88,130,108,150)(89,131,109,151)(90,132,110,152)(91,133,111,153)(92,134,112,154)(93,135,113,155)(94,136,114,156)(95,137,115,157)(96,138,116,158)(97,139,117,159)(98,140,118,160)(99,141,119,121)(100,142,120,122), (1,89)(2,110)(3,91)(4,112)(5,93)(6,114)(7,95)(8,116)(9,97)(10,118)(11,99)(12,120)(13,101)(14,82)(15,103)(16,84)(17,105)(18,86)(19,107)(20,88)(21,109)(22,90)(23,111)(24,92)(25,113)(26,94)(27,115)(28,96)(29,117)(30,98)(31,119)(32,100)(33,81)(34,102)(35,83)(36,104)(37,85)(38,106)(39,87)(40,108)(41,125)(42,146)(43,127)(44,148)(45,129)(46,150)(47,131)(48,152)(49,133)(50,154)(51,135)(52,156)(53,137)(54,158)(55,139)(56,160)(57,141)(58,122)(59,143)(60,124)(61,145)(62,126)(63,147)(64,128)(65,149)(66,130)(67,151)(68,132)(69,153)(70,134)(71,155)(72,136)(73,157)(74,138)(75,159)(76,140)(77,121)(78,142)(79,123)(80,144), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,46,47,20,21,66,67,40)(2,19,48,65,22,39,68,45)(3,64,49,38,23,44,69,18)(4,37,50,43,24,17,70,63)(5,42,51,16,25,62,71,36)(6,15,52,61,26,35,72,41)(7,60,53,34,27,80,73,14)(8,33,54,79,28,13,74,59)(9,78,55,12,29,58,75,32)(10,11,56,57,30,31,76,77)(81,158,123,96,101,138,143,116)(82,95,124,137,102,115,144,157)(83,136,125,114,103,156,145,94)(84,113,126,155,104,93,146,135)(85,154,127,92,105,134,147,112)(86,91,128,133,106,111,148,153)(87,132,129,110,107,152,149,90)(88,109,130,151,108,89,150,131)(97,142,139,120,117,122,159,100)(98,119,140,121,118,99,160,141) );
G=PermutationGroup([[(1,47,21,67),(2,48,22,68),(3,49,23,69),(4,50,24,70),(5,51,25,71),(6,52,26,72),(7,53,27,73),(8,54,28,74),(9,55,29,75),(10,56,30,76),(11,57,31,77),(12,58,32,78),(13,59,33,79),(14,60,34,80),(15,61,35,41),(16,62,36,42),(17,63,37,43),(18,64,38,44),(19,65,39,45),(20,66,40,46),(81,123,101,143),(82,124,102,144),(83,125,103,145),(84,126,104,146),(85,127,105,147),(86,128,106,148),(87,129,107,149),(88,130,108,150),(89,131,109,151),(90,132,110,152),(91,133,111,153),(92,134,112,154),(93,135,113,155),(94,136,114,156),(95,137,115,157),(96,138,116,158),(97,139,117,159),(98,140,118,160),(99,141,119,121),(100,142,120,122)], [(1,89),(2,110),(3,91),(4,112),(5,93),(6,114),(7,95),(8,116),(9,97),(10,118),(11,99),(12,120),(13,101),(14,82),(15,103),(16,84),(17,105),(18,86),(19,107),(20,88),(21,109),(22,90),(23,111),(24,92),(25,113),(26,94),(27,115),(28,96),(29,117),(30,98),(31,119),(32,100),(33,81),(34,102),(35,83),(36,104),(37,85),(38,106),(39,87),(40,108),(41,125),(42,146),(43,127),(44,148),(45,129),(46,150),(47,131),(48,152),(49,133),(50,154),(51,135),(52,156),(53,137),(54,158),(55,139),(56,160),(57,141),(58,122),(59,143),(60,124),(61,145),(62,126),(63,147),(64,128),(65,149),(66,130),(67,151),(68,132),(69,153),(70,134),(71,155),(72,136),(73,157),(74,138),(75,159),(76,140),(77,121),(78,142),(79,123),(80,144)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,46,47,20,21,66,67,40),(2,19,48,65,22,39,68,45),(3,64,49,38,23,44,69,18),(4,37,50,43,24,17,70,63),(5,42,51,16,25,62,71,36),(6,15,52,61,26,35,72,41),(7,60,53,34,27,80,73,14),(8,33,54,79,28,13,74,59),(9,78,55,12,29,58,75,32),(10,11,56,57,30,31,76,77),(81,158,123,96,101,138,143,116),(82,95,124,137,102,115,144,157),(83,136,125,114,103,156,145,94),(84,113,126,155,104,93,146,135),(85,154,127,92,105,134,147,112),(86,91,128,133,106,111,148,153),(87,132,129,110,107,152,149,90),(88,109,130,151,108,89,150,131),(97,142,139,120,117,122,159,100),(98,119,140,121,118,99,160,141)]])
68 conjugacy classes
class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E ··· 8L 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 20 20 1 1 1 1 2 2 20 20 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4
68 irreducible representations
dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D5 D10 D10 C8○D4 C4×D5 D20 C5⋊D4 C4×D5 D20.2C4 kernel C4.89(C2×D20) D10⋊1C8 C22×C5⋊2C8 C10×M4(2) C2×C4○D20 C2×Dic10 C2×D20 C2×C5⋊D4 C2×C20 C2×M4(2) C2×C8 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 4 2 4 2 8 4 8 8 4 8
Matrix representation of C4.89(C2×D20) in GL4(𝔽41) generated by
40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
1 0 0 0 0 1 0 0 0 0 9 2 0 0 1 32
,
21 17 0 0 3 18 0 0 0 0 38 0 0 0 27 3
,
6 38 0 0 26 35 0 0 0 0 38 0 0 0 0 38
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,2,32],[21,3,0,0,17,18,0,0,0,0,38,27,0,0,0,3],[6,26,0,0,38,35,0,0,0,0,38,0,0,0,0,38] >;
C4.89(C2×D20) in GAP, Magma, Sage, TeX
C_4._{89}(C_2\times D_{20})
% in TeX
G:=Group("C4.89(C2xD20)");
// GroupNames label
G:=SmallGroup(320,756);
// by ID
G=gap.SmallGroup(320,756);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^20=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*c^19>;
// generators/relations
×
𝔽 |
# What do they mean by $\langle u_i |O| u_j\rangle$ in quantum mechanics
UCSD's online QM notes, as usual, starts stating that QM operators are Hermitian and says that operator $O$ elements can be computed by
$$O_{ij} = \langle u_j|O|u_i\rangle$$
The $u_i$ are eigenvectors, which are orthogonal to each other, thanks to being Hermitian. So, I suppose that $O$ is a Hermitian operator. But where are the eigenvalues?
I expect that when you apply the operator to its eigenvector, a $\lambda$ must appear. Yet, I see it nowhere. Neither, text says that $O$ is a diagonal, as I may expect if I apply $\langle u_i|$ to $O|u_i\rangle = \lambda_i |u_i\rangle$, I should get $\lambda_i$ on the main diagonal and 0 everyelse. Are $u_i$ eigenvectors of the $O$ or some another operator?
• Put it in this way. Let $\hat{O}$ be an operator. Insert completeness relation $$\sum_i |u_i \rangle \langle u_i | = \hat{1}$$, we have $$\hat{O}=\sum_{ij} |u_i \rangle \underline{\langle u_i |\hat{O} |u_j \rangle} \langle u_j |$$. The underline part is the representation of operator $\hat{O}$ in $\{u_i\}$ basis. If $u_i$ is the eigenfunction of operator $O$, namely $\hat{O} |u_i \rangle = \lambda_i |u_i \rangle$ we have $\hat{O}=\sum_i \lambda_i | u_i \rangle \langle u_i|$, which sometimes called spactral expansion of operator. The matrix element is $$O_{ij}=\lambda_i \delta_{ij}$$ Dec 10, 2013 at 16:41
• The eigenvalue $\lambda_j$ depends on the operator $\hat{O}$ and the eigenfunction $|u_j\rangle$. For example, the quantum harmonic oscillator $\hat{O}\equiv\hat{H}= \hat{p}^2/2m+m\omega^2\hat{x}^2/2$ with $\lambda_j\equiv E_n=(n+\frac12)\hbar\omega$. Dec 10, 2013 at 16:47
• @user26143 I do not understand a couple of things. What is the completeness relation?
– Val
Dec 10, 2013 at 20:15
• It is a standard result in many textbooks of quantum mechanics. E.g. Sakurai's modern quantum mechanics, section 1.3, p19. Dec 10, 2013 at 21:19
• @user26143 Do you mean that this is not a simple consequence of basis orthonormality?
– Val
Dec 10, 2013 at 22:01
In those notes you linked to, the $|u_i\rangle$ form an orthonormal basis. They are not necessarily the eigenkets of the operator $O$. By choosing different bases $\{u_i\}$, the "representation" of the operator $O$ (e.g.., how you write it out in a matrix) will change. |
# How to find the common elements in multiple vectors in R?
To find the common elements in multiple vectors, we can follow the below steps −
• First of all, create a number of vectors.
• Use intersect function to find the common elements in all the vectors.
## Create the vectors
Let’s create a number of vectors as shown below −
Live Demo
x<-rpois(150,2)
y<-rpois(150,1)
z<-rpois(150,10)
a<-rpois(150,5)
b<-rpois(150,8)
c<-rpois(150,12)
x
y
z
a
b
c
On executing, the above script generates the below output(this output will vary on your system due to randomization) −
### Output
[1] 0 3 1 1 2 0 1 0 1 0 3 4 2 2 1 1 1 7 3 1 3 2 3 1 0 0 0 2 4 1 2 2 2 0 1 3 2
[38] 2 2 3 6 3 0 1 3 1 1 3 5 1 1 2 3 2 0 1 3 1 2 2 3 2 2 1 1 1 1 5 2 6 3 3 3 6
[75] 2 3 0 1 2 1 0 4 1 0 5 0 0 4 2 1 1 2 0 0 2 3 0 3 0 1 1 5 0 0 3 0 0 5 2 1 5
[112] 1 3 0 5 7 1 3 5 2 1 0 1 3 1 3 2 2 0 0 2 4 1 3 0 2 0 4 1 2 1 1 1 2 1 2 3 1
[149] 1 4
[1] 1 1 0 2 1 1 3 1 2 2 0 0 0 1 0 2 2 0 1 1 0 2 0 1 0 2 0 0 0 0 1 0 2 0 0 2 0
[38] 1 0 1 0 0 3 2 1 0 0 0 2 2 1 1 1 2 1 0 2 2 0 0 1 3 0 5 1 1 4 1 0 0 1 1 1 0
[75] 0 0 1 2 1 1 0 2 1 1 3 1 1 2 1 2 2 3 1 1 0 1 2 0 0 3 2 1 1 1 1 0 0 0 1 0 1
[112] 1 0 1 1 1 3 0 2 2 1 2 1 2 2 1 0 1 0 2 1 0 0 1 1 2 2 1 1 2 1 1 0 0 0 4 2 2
[149] 0 0
[1] 12 11 4 9 7 7 16 7 7 12 12 9 8 12 3 12 9 10 8 14 8 13 8 9 13
[26] 16 7 12 7 6 8 8 5 14 13 12 9 6 8 11 6 7 12 3 14 15 12 6 10 6
[51] 14 10 15 10 10 8 9 9 15 13 10 12 12 10 11 9 7 17 8 8 6 11 7 10 7
[76] 15 10 8 10 15 9 7 11 0 12 5 13 9 9 10 11 11 8 11 9 10 7 8 13 14
[101] 14 4 10 7 5 9 15 17 9 8 8 8 13 8 5 17 9 13 10 9 10 11 11 13 9
[126] 13 11 11 16 18 12 13 13 8 6 6 11 7 8 10 9 3 9 7 6 7 10 8 5 8
[1] 7 4 1 4 7 8 6 4 8 2 6 0 7 4 7 8 5 4 2 5 8 3 4 8 7
[26] 6 7 3 6 2 3 8 7 12 8 8 3 4 6 3 0 7 3 1 6 3 11 5 6 3
[51] 5 6 7 8 5 2 7 5 7 7 6 4 8 5 4 5 8 5 10 3 2 5 6 7 1
[76] 6 5 4 2 7 5 4 3 8 9 3 4 6 2 11 1 9 3 3 9 5 3 4 3 4
[101] 7 3 6 4 3 4 2 6 8 7 10 6 8 6 5 3 5 5 8 6 4 3 3 5 6
[126] 3 7 4 7 6 4 3 6 9 3 4 9 6 3 8 5 2 7 3 5 10 7 7 6 3
[1] 9 9 4 5 7 7 6 8 5 5 10 10 6 4 8 12 13 8 5 3 6 9 7 7 13
[26] 8 10 5 6 8 3 8 11 8 9 8 9 7 13 6 12 12 5 4 3 6 8 5 10 5
[51] 8 10 7 3 8 13 5 12 6 8 4 6 5 5 9 5 9 8 12 4 7 3 10 6 9
[76] 6 9 5 11 7 8 9 8 8 4 4 13 6 8 12 6 9 8 9 6 10 7 9 5 8
[101] 10 10 7 9 5 10 11 6 10 16 12 5 6 10 7 9 8 0 9 9 4 6 11 7 4
[126] 10 4 9 5 13 7 8 11 7 7 8 8 17 7 9 7 12 11 6 9 12 5 4 9 5
[1] 11 10 10 11 7 14 10 14 13 11 4 18 15 8 18 6 6 12 10 7 10 13 15 12 10
[26] 12 11 9 18 7 9 13 18 16 5 10 9 20 8 12 16 13 9 18 13 8 9 11 7 10
[51] 9 16 14 16 9 19 11 15 8 9 10 16 15 7 16 11 8 13 16 10 11 10 14 19 11
[76] 14 11 11 12 14 9 14 7 11 13 9 9 14 18 18 12 15 14 14 13 12 10 10 15 15
[101] 13 16 10 11 8 15 7 9 9 12 19 7 8 11 14 10 7 8 14 17 12 10 10 16 8
[126] 10 8 3 9 17 16 11 15 11 13 14 9 8 17 6 11 7 11 13 16 9 5 10 11 16
## Find common elements
Using intersect function with Reduce function to find the common elements in all the above vectors −
Live Demo
x<-rpois(150,2)
y<-rpois(150,1)
z<-rpois(150,10)
a<-rpois(150,5)
b<-rpois(150,8)
c<-rpois(150,12)
Reduce(intersect,list(x,y,z,a,b,c))
### Output
[1] 3 4 5 |
Question
# The given matrix is the augmented matrix for a system of linear equations. \begin{bmatrix}1&0&-1&-2&-3&1 \\0&1&2&3&4&0 \end{bmatrix}
Forms of linear equations
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix} {1}&{0}&-{1}&-{2}&-{3}&{1}\\{0}&{1}&{2}&{3}&{4}&{0}\end{bmatrix}$$ |
# Pearson Correlation
In statistics, Pearson’s r correlation coefficient quantifies the linear correlation between two variables X and Y.
The value is always between -1 and 1.
• A value of 1 indicates a strong positive linear correlation.
• A value of 0 indicates no correlation.
• A value of -1 indicates a strong negative linear correlation.
One will find, that unlike covariance, correlation is scale-invariant, i.e. the scale of the data does not ultimately affect the correlation value.
Our confidence in our correlation value depends on how much data we have and is quantified by the p-value. This value tells us the probability that randomly sampled points would result in a similarly strong if not stronger relationship. The lower the p-value, the less likely that our correlation is due to randomness, and therefore the more confidence we are in our correlation value. This is in a way similar to hypothesis testing
Mathematically, the Pearson correlation between two variables X and Y is defined as the covariance between the two variables over the product of their variances:
${\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y} } } }$
IMPORTANT: CORRELATION DOES NOT IMPLY CAUSATION!!!
### Notes mentioning this note
Here are all the zettels in this zettelkasten, along with their links, visualized as a graph. You may need to zoom and pan around to see something. |
# How do you find the square root of 10?
Jun 26, 2016
$\sqrt{10} \approx 3.16227766016837933199$ is not simplifiable.
#### Explanation:
$10 = 2 \times 5$ has no square factors, so $\sqrt{10}$ is not simplifiable.
It is an irrational number a little greater than $3$.
In fact, since $10 = {3}^{2} + 1$ is of the form ${n}^{2} + 1$, $\sqrt{10}$ has a particularly simple continued fraction expansion:
sqrt(10) = [3;bar(6)] = 3+1/(6+1/(6+1/(6+1/(6+1/(6+1/(6+...))))))
We can truncate this continued fraction expansion to get rational approximations to $\sqrt{10}$
For example:
sqrt(10) ~~ [3;6] = 3+1/6 = 19/6 = 3.1bar(6)
sqrt(10) ~~ [3;6,6] = 3+1/(6+1/6) = 3+6/37 = 117/37 = 3.bar(162)
Actually:
$\sqrt{10} \approx 3.16227766016837933199$ |
# Cross compilers
• To: misc_(_at_)_openbsd_(_dot_)_org
• Subject: Cross compilers
• From: Pedro Giffuni <m230761_(_at_)_ingenieria_(_dot_)_ingsala_(_dot_)_unal_(_dot_)_edu_(_dot_)_co>
• Date: Wed, 15 Jan 1997 15:29:32 -0500 (EST)
I had some problems cross compiling to DOS and stuff so I asked Cygnus
support, since I saw some of you working on cross-compiling, I though you
might be interested in the reply...
---------- Forwarded message ----------
Date: Wed, 15 Jan 1997 10:31:24 -0800
From: Doug Evans <dje_(_at_)_cygnus_(_dot_)_com>
To: m230761_(_at_)_ingenieria_(_dot_)_ingsala_(_dot_)_unal_(_dot_)_edu_(_dot_)_co
Subject: Re: GCC Wishlist (or bug?)
Date: Tue, 14 Jan 1997 20:37:37 -0500 (EST)
From: Pedro Giffuni <m230761_(_at_)_ingenieria_(_dot_)_ingsala_(_dot_)_unal_(_dot_)_edu_(_dot_)_co>
On Tue, 14 Jan 1997, Doug Evans wrote:
>
> i386-msdos is the wrong configuration name.
> use i386-go32
Someone did that on a previous version (very unnatural I must say) and it
worked but on gcc-2.7.2 it replies system 'go32' not recognized.
Ooops. You need the crossgcc patch.
ftp.cygnus.com:/pub/embedded/crossgcc/crossgcc-gcc-2.7.2.patch
Here is the crossgcc FAQ which may also help.
===================================
Version: 0.7
Last Updated: 960622
Maintainer: dje_(_at_)_cygnus_(_dot_)_com
Location: ftp://ftp.cygnus.com/pub/embedded/crossgcc/FAQ
crossgcc' is the name of a mailing list for discussing issues regarding
using GCC (plus the various ancilliary pieces) as a cross-compiler.
Subscription requests should be sent to "crossgcc-request_(_at_)_cygnus_(_dot_)_com"
with a body of 1 line containing "subscribe crossgcc <your-email-address>"
This list is run by the "majordomo" software.
FIXME: This FAQ is unfinished (in the sense that there are some obvious
sections it needs, some included below). Please send contributions to
dje_(_at_)_cygnus_(_dot_)_com_(_dot_)_
-----------------
1. List of the pieces needed and where to find them
2. Patches
2.1 Cygnus patches
2.2 Patch for binutils-2.6 GAS bug "Case value 3 unexpected at line 1557"
3. How to configure, build, and install GCC as a cross-compiler
3.1 Unix and related systems
3.2 One pass installation of GCC+Binutils
3.3 Cygnus releases
3.3.1 Where are the info files?
3.3.2 How do I build a Cygnus release?
3.4 Win32 systems
3.5 MSDOS systems
3.5.1 How do I build a cross-compiler with DJGPP?
(or, Building a DOS hosted cross compiler without using DOS)
3.6.1 What is a Canadian Cross?
3.6.2 How do I build an MSDOS hosted cross compiler without using MSDOS?
3.7 Disk space requirements
4. Frequently Encountered Problems
4.1 installation problem, cannot exec cpp': No such file or directory
4.2 64 bit host cross 32 bit target
4.3 Assembler errors while building GCC's libgcc.a
4.5 Where are open, read, write, close, etc. ?
4.6 How do I pass options from GCC to GAS or GLD?
4.7 How do I write an interrupt handler in C?
4.8 How do I write assembler code that works with m68k-aout's leading
'_' and m68k-coff's lack of leading '_'?
5. Library Support
5.1 What is libgcc.a?
5.2 How do I force GCC to emit inline code for memcpy?
5.3 Why would I choose glibc over newlib (or vice versa)?
5.4 What if I need software floating point support?
5.5 Why are several copies of newlib built?
5.5.1 Is there any way to build only the libraries I need?
6. Using GDB for remote debugging
7. S Records, etc.
6.1 What are S Records?
6.1 How do I use objcopy to generate S Records?
6.2 How do I use the linker to generate S Records?
8. Target specific info
e.g. Using BDM on a 68k, crt0's for specific cards, ...
9. Operating systems for embedded systems
9.1 RTEMS
10. How to get help
10.1 General questions
10.2 Online docs
10.3 Bug reporting
10.4 Other mailing lists, web sites, newsgroups, etc.
11. Contributors
1. List of the pieces needed and where to find them
For a complete toolchain you need a compiler (GCC), assembler (GAS),
linker (GLD), various utilities (Binutils), debugger (GDB), and a library
(GLIBC or NEWLIB). For c++ programming you'll also want libg++.
The master repository of GNU software is prep.ai.mit.edu:/pub/gnu.
It is mirrored all over the world so please try to use a site closer to
you to avoid overloading prep. The list of sites can be obtained with
"finger fsf_(_at_)_prep_(_dot_)_ai_(_dot_)_mit_(_dot_)_edu".
Compiler: GCC 2.7.2, file gcc-2.7.2.tar.gz
Assembler: GAS 2.6, file binutils-2.6.tar.gz
Utilities: Binutils 2.6, file binutils-2.6.tar.gz
Debugger: GDB 4.16, file gdb-4.16.tar.gz
Library: There are currently two choices for a library, the GNU libc (Glibc)
and a library put together by Cygnus (Newlib).
GLIBC 1.09, file glibc-1.09.tar.gz
Newlib 1.7.0, file newlib-1.7.0.tar.gz in directory
ftp.cygnus.com:/pub/newlib
2. Patches
This section is intended to document patches that are available from
various sources.
2.1 Cygnus patches
crossgcc-gcc-2.7.2.patch
a. Delete building of objc-runtime.
This library requires the C library header files which usually
aren't available when the user is first building the cross
compiler. They *could* be, but this is a frequent source of
trouble to users so this patch avoids the trouble.
b. float.h
GCC's Makefile expects to be able to run enquire' to compute
float.h. When building a cross compiler this can require manual
intervention in the build process (and hence a source of confusion
for users) and on some embedded systems it may be impossible to
run enquire. This patch provides some prebuilt versions of
float.h that targets can use.
??? The support is incomplete. For example, 96 bit and 128 bit
floating point support is missing.
c. Define inhibit_libc in LIBGCC2_FLAGS if cross compiler.
Various routines in libgcc2.c require headers like stdio.h which
typically aren't available yet (see point a.).
Defining inhibit_libc causes __eprintf to be left out of libgcc.a;
this is ok because it is only used by assert.h and the C library
will be providing its own assert.h.
It also causes profiling support to be left out of libgcc.a,
these routines also require C library headers like stdio.h.
d. Provide support for i386-go32 as a target.
This is needed for example in a Canadian Cross where one is
building a DOS hosted cross compiler from a Unix system.
The patch is available in ftp.cygnus.com:/pub/embedded/crossgcc.
To apply the patch:
cd gcc-2.7.2
patch -p1 <../crossgcc-gcc-2.7.2.patch
2.2 Patch for binutils-2.6 GAS bug "Case value 3 unexpected at line 1557"
Binutils-2.6 has a bug in the i386 port of GAS that can cause this error.
/var/tmp/cca002vk.s: Assembler messages:
/var/tmp/cca002vk.s:12: Fatal error: Case value 3 unexpected at line 1557 of file "obj-format.c"
This bug is present in all i386 ports. The patch is in
ftp://ftp.cygnus.com/pub/embedded/crossgcc/binutils-2.6-obj-coff.patch
3. How to configure, build, and install GCC as a cross-compiler
3.1 Unix Systems
Notes:
1) The author has more familiarity with Newlib than Glibc so Newlib
will be used in all examples below.
2) GAS and GLD are now distributed in the "binutils" distribution,
so all further references to GAS and GLD in this section will use
3) If you have a "make" that understands VPATH (like GNU make), it is
highly recommended that you build the pieces in a different directory
from the sources. The examples below will assume this.
The pieces have to be built and installed in a particular order.
Why? Clearly the cross-compiler is needed to build the libraries,
so GCC must be built before Newlib. Also, GCC has its own library
(called libgcc) so Binutils must be built before GCC.
Here is the recommended order:
1) Binutils
2) GCC
3) Newlib
4) GDB
Suppose
- you've unpacked all the sources into /home/foo,
- you want to build a sparc-solaris2 hosted m68k-coff cross-compiler,
- the cross-compiler is to be installed in /bar.
The build procedure would then look like this.
cd /home/foo
host=sparc-sun-solaris2
target=m68k-coff
prefix=/bar
i=$prefix/bin mkdir build-binutils build-gcc build-newlib build-gdb # Configure, build and install binutils cd build-binutils ../binutils-2.6/configure --host=$host --target=$target --prefix=$prefix -v
make all install
# Configure, build and install gcc
cd ../build-gcc
../gcc-2.7.2/configure --host=$host --target=$target --prefix=$prefix -v make all install # Configure, build and install newlib cd ../build-newlib ../newlib-1.7.0/configure --host=$host --target=$target --prefix=$prefix -v
# The settings for FOO_FOR_TARGET aren't necessary if you put $prefix/bin # in your path before running this. make all install \ CC_FOR_TARGET=$i/bin/${target}-gcc \ AS_FOR_TARGET=$i/bin/${target}-as \ LD_FOR_TARGET=$i/bin/${target}-ld \ AR_FOR_TARGET=$b/binutils/${target}-ar \ RANLIB_FOR_TARGET=$b/binutils/${target}-ranlib # Configure, build and install gdb cd ../build-gdb ../gdb-4.16/configure --host=$host --target=$target --prefix=$prefix -v
make all install
3.2 One Pass Installation
Is there an easier way? Yes!
If you study the top-level Makefile that comes with binutils-2.6,
newlib-1.7.0, or gdb-4.16, you'll see that they're all quite similar
(not surprising since they're essentially the same file).
You'll also discover that it is capable of building and installing the
entire toolchain in one pass. The catch is that is assumes a directory
layout different than what would normally appear when you unpack the
sources.
You will need to apply the Cygnus crossgcc-gcc-2.7.2.patch.
What we now need to do is turn this directory layout
./binutils-2.6/
bfd/
binutils/
config/
gas/
include/
ld/
libiberty/
opcodes/
texinfo/
./gcc-2.7.2/
./newlib-1.7.0/
config/
libgloss/
newlib/
texinfo/
into
./src/
bfd/
binutils/
config/
gas/
gcc/
include/
ld/
libgloss/
libiberty/
newlib/
opcodes/
texinfo/
Where's GDB? GDB is left out because it shares sources with Binutils
(e.g. bfd, include, libiberty, and opcodes). One can't "mix and match" them,
they each need there own copies (since they were tested and released with
their own copies). GDB can be built separately afterwards. One might well
ask what's the point of this pseudo one-pass installation and that would be
a good question. It simplifies the installation a little, and in particular
the "Canadian Cross" installation (see below).
Binutils and Newlib share config' and texinfo' directories; we can use
Binutils' copies.
A script exists to reorganize the above source tree using symlinks.
It assumes:
- you're using a version of make that understands VPATH (e.g. GNU make)
- ./binutils-2.6 exists
- ./gcc-2.7.2 exists
- ./newlib-1.7.0 exists
- ./src does not exist
The script is one-tree-1.2.sh from ftp.cygnus.com:/pub/embedded/crossgcc.
It will create a subdirectory called src'.
After running the script, do this:
mkdir build
cd build
../src/configure --host=$host --target=$target --prefix=$prefix -v make all install mkdir ../build-gdb cd ../build-gdb ../gdb-4.16/configure --host=$host --target=$target --prefix=$prefix -v
make all install
3.3 Cygnus Releases
What if I come across a Cygnus release?
3.3.1 Where are the info files?
Cygnus releases differ from FSF releases in that files that are not really
source files but are built from other files (like yacc files and texinfo
files) are not included. Instead, they are built with the rest of the
toolchain.
3.3.2 How do I build a Cygnus release?
Cygnus releases are essentially the "one-pass installation" tree, except
that a lot more tools are included (e.g. byacc, flex, expect, gdb, make, tcl,
texinfo).
To build a toolchain from a Cygnus release, you should consult the
documentation that came with it (there may be last minute release notes,
or this FAQ may be out of date, etc.). But for those who happen to come
upon a Cygnus release, here is a quick introduction.
Suppose you happen upon a Cygnus release and want to build a
sparc-sun-solaris2 cross sparc64-elf compiler. Do this:
src=/path/to/source/tree
rel=/path/to/install/tree
host=sparc-sun-solaris2
target=sparc64-elf
mkdir build
cd build
$src/configure --host=$host --target=$target --prefix=$rel -v
make all info install install-info
PATH=$PATH:$rel/bin ; export PATH
You can also run dejagnu on the build tree at this point with
make -k check
The gcc and g++ execute tests won't do much in this particular example
(sparc64-elf doesn't really exist yet, the configuration for UltraSPARC's
running Solaris 2.5 is sparc-sun-solaris2, duh ... :-)
3.4 Win32 hosted cross-compilers
There is work in progress that will let the GNU tools be used as a native
compiler in the win32 environment and also as a cross-compiler (either win32
hosted or win32 targeted). Join the gnu-win32_(_at_)_cygnus_(_dot_)_com mailing list if
you wish to help out. The release is kept in
ftp://ftp.cygnus.com/pub/sac/win32.
The configuration for this project is i386-cygwin32.
Subscription requests should be sent to "gnu-win32-request_(_at_)_cygnus_(_dot_)_com"
with a body of 1 line containing "subscribe gnu-win32 <your-email-address>"
3.5 MSDOS hosted cross-compilers
The primary MSDOS port is djgpp', a port of the GNU tools to MSDOS by
DJ Delorie, using go32', a 32 bit extender. See the djgpp FAQ for
more details (http://www.delorie.com/djgpp/faq/).
3.5.1 How do I build a cross-compiler with DJGPP?
The procedure is basically this.
* Use gunzip to decompress the various archives.
* Use djtarx to untar them and resolve 8.3 conflicts
* Run "configure" in all the right places to configure for a native
compiler.
* Manually (read the INSTALL file) copy the target-specific files around
to reconfigure the system for the given target. Edit the Makefiles
as appropriate to include the target snippets.
* Run make.
3.6.1 What is a Canadian Cross?
One cool thing about the GNU tools is that they support building a cross
compiler for one host on another host (i.e. building a cross-compiler with
a cross-compiler). This is called a "Canadian Cross" because at the time
a name was needed, Canada had three national parties. "Three" is important
because there are three different systems involved: the build machine, the
host machine, and the target machine. I think Canada has six national
parties now so the name is a bit dated, but who cares. This allows, for
example, one to build an MSDOS hosted cross-compiler on a Unix box.
3.6.2 How do I build an MSDOS hosted cross compiler without using MSDOS?
Suppose one wanted to build an MSDOS cross m68k-coff cross
compiler on a sparc-sunos4 machine. The procedure is as follows.
Note that it is quite lengthy. That's because four compilers are involved
(three to build the one we want).
Why four? Remember, we're building the tools completely on a Unix box,
therefore all the programs that run during the build process must obviously
run on the Unix box. We can't skip over to an MSDOS machine, run something
there, and come back - the entire toolchain is built from scratch on the
Unix box.
The first compiler that is needed is a sunos4 cross m68k-coff compiler
to build the m68k-coff libraries. But in order to build that we need
a sunos4 native compiler. That's two. We also need a sunos4 cross
i386-go32 compiler to build the programs that run on MSDOS (go32 is the
name of a 32 bit extender written by DJ Delorie that all the MSDOS
tools will use). Finally, we need an i386-go32 cross m68k-coff compiler:
our final goal. Four compilers. However, the process is quite
straightforward once you understand all the pieces that are needed.
Assume the source tree has been created with the "one-tree" script.
The following is the list of steps, written so that it can be copied
into a shell script and run.
#!/bin/sh
# This script is from crossgcc FAQ-0.7.
# Before using this script it is a good idea to check with the most recent
# version of the FAQ to see if any changes have been made. The FAQ can be
# obtained from ftp://ftp.cygnus.com/pub/embedded/crossgcc/FAQ.
#
# Note: This script provides support for cygwin32 hosts (the configuration
# of the gnu-win32 project), but current FSF releases of GCC,Binutils do not
# support cygwin32. The support is present in case you pick up a copy of
# the cygwin32 source tree.
# Create some variables that will be useful.
build=sparc-sun-sunos4.1.3
host=i386-go32
target=m68k-coff
here=pwd
# src' is presumed to have been built with the "one-tree" script.
src=$here/src rel=/bar # Install the toolchain in /bar again. # Build directory for the$build cross $host toolchain. b2h=$here/b-${build}-x-${host}
# Build directory for the $build cross$target toolchain.
b2t=$here/b-${build}-x-${target} # Build directory for the$host cross $target toolchain. h2t=$here/b-${host}-x-${target}
# The first step is to build a sunos4 cross go32 cross-compiler.
# The value for --prefix we give here is /tmp/junk as we don't intend
# to install this toolchain.
[ -d $b2h ] || mkdir$b2h
(cd $b2h ; CC=gcc$src/configure --host=${build} --target=${host} --prefix=/tmp/junk -v)
[ $? = 0 ] || exit 1 (cd$b2h ; make all-gcc all-target-newlib CC=gcc CFLAGS=-g)
[ $? = 0 ] || exit 1 # Now build a$build cross $target toolchain. # The value for --prefix we give here is /tmp/junk as we don't intend # to install this toolchain. [ -d$b2t ] || mkdir $b2t (cd$b2t ; CC=gcc $src/configure --host=${build} --target=${target} --prefix=/tmp/junk -v) [$? = 0 ] || exit 1
(cd $b2t ; make all CC=gcc CFLAGS=-g) [$? = 0 ] || exit 1
# Now that we've built the tools that we need, we can finally build
# our $host cross$target toolchain.
# Unfortunately, autoconf doesn't like "path prefix unused" messages, so
# we can't use -B../newlib/. This works around that. The alternative
# is to install the $build cross$host toolchain and use that.
# libcygwin.a and libkernel32.a are only needed by cygwin32 hosted toolchains.
(
cd $b2h/gcc rm -f crt0.o libc.a libm.a ln -s ../${host}/newlib/crt0.o .
ln -s ../${host}/newlib/libc.a . ln -s ../${host}/newlib/libm.a .
case $host in *cygwin32*) rm -f libcygwin.a libkernel32.a ln -s ../${host}/winsup/libcygwin.a .
ln -s ../${host}/winsup/libkernel32.a . ;; esac ) # Both configure and make need to be told where to find the various pieces. # Define several variables of the things we need to pass to configure and make. # These are for building programs that run on$build.
CC_FOR_BUILD=gcc
CXX_FOR_BUILD=gcc
# These are for building programs and libraries that run on $host. case$host in
*cygwin32*)
CC="$b2h/gcc/xgcc -B$b2h/gcc/ -isystem $src/winsup/include -isystem$b2h/${host}/newlib/targ-include -isystem$src/newlib/libc/include" ;;
*)
CC="$b2h/gcc/xgcc -B$b2h/gcc/ -isystem $b2h/${host}/newlib/targ-include -isystem $src/newlib/libc/include" ;; esac AR=$b2h/binutils/ar
RANLIB=$b2h/binutils/ranlib # These are for building libraries that run on$target.
CC_FOR_TARGET="$b2t/gcc/xgcc -B$b2t/gcc/ -isystem $b2t/${target}/newlib/targ-include -isystem $src/newlib/libc/include" GCC_FOR_TARGET="$CC_FOR_TARGET"
CC_FOR_TARGET="$CC_FOR_TARGET" CXX_FOR_TARGET="$CC_FOR_TARGET"
AS_FOR_TARGET=$b2t/gas/as.new LD_FOR_TARGET=$b2t/ld/ld.new
AR_FOR_TARGET=$b2t/binutils/ar NM_FOR_TARGET=$b2t/binutils/nm.new
RANLIB_FOR_TARGET=$b2t/binutils/ranlib #$DLLTOOL_FOR_TARGET is only needed for cygwin32 hosted systems, but
# it doesn't hurt to always pass it.
DLLTOOL_FOR_TARGET=$b2t/binutils/dlltool # Don't add -g here if hosting on i386-go32 and don't comment this out # (that will cause -g to be used). # There's a fixed size to the line number table (or something like that). CFLAGS=-O # Ready. Configure and build. [ -d$h2t ] || mkdir $h2t (cd$h2t ; CC="$CC" AR="$AR" RANLIB="$RANLIB"$src/configure --build=${build} --host=${host} --target=${target} --prefix=$rel -v)
[ $? = 0 ] || exit 1 cd$h2t
make all \
CC_FOR_BUILD="$CC_FOR_BUILD" \ CXX_FOR_BUILD="$CXX_FOR_BUILD" \
CC="$CC" \ AR="$AR" \
RANLIB="$RANLIB" \ GCC_FOR_TARGET="$CC_FOR_TARGET" \
CC_FOR_TARGET="$CC_FOR_TARGET" \ CXX_FOR_TARGET="$CC_FOR_TARGET" \
AS_FOR_TARGET="$AS_FOR_TARGET" \ LD_FOR_TARGET="$LD_FOR_TARGET" \
AR_FOR_TARGET="$AR_FOR_TARGET" \ NM_FOR_TARGET="$NM_FOR_TARGET" \
RANLIB_FOR_TARGET="$RANLIB_FOR_TARGET" \ DLLTOOL_FOR_TARGET="$DLLTOOL_FOR_TARGET"
[ $? = 0 ] || exit 1 # All we have to do now is install it. # This piece is separated from the previous make as sometimes # one wants to install things differently. #make install \ # CC_FOR_BUILD="$CC_FOR_BUILD" \
# CXX_FOR_BUILD="$CXX_FOR_BUILD" \ # CC="$CC" \
# AR="$AR" \ # RANLIB="$RANLIB" \
# GCC_FOR_TARGET="$CC_FOR_TARGET" \ # CC_FOR_TARGET="$CC_FOR_TARGET" \
# CXX_FOR_TARGET="$CC_FOR_TARGET" \ # AS_FOR_TARGET="$AS_FOR_TARGET" \
# LD_FOR_TARGET="$LD_FOR_TARGET" \ # AR_FOR_TARGET="$AR_FOR_TARGET" \
# NM_FOR_TARGET="$NM_FOR_TARGET" \ # RANLIB_FOR_TARGET="$RANLIB_FOR_TARGET" \
# DLLTOOL_FOR_TARGET="$DLLTOOL_FOR_TARGET" # Almost done. Before the toolchain is usable we need to # - convert the coff files to .exe's, # - convert file names to follow MSDOS's 8.3 rules, # - Change \n to \r\n in text files (like headres). # The package dosrel-1.0 is set up to do all this. # See ftp://ftp.cygnus.com/pub/embedded/crossgcc/dosrel-1.0.tar.gz exit$?
Before the tools are usable, a few things must be done:
- convert the binaries to .exe's
- cope with DOS's 8.3 file name restriction
ftp.cygnus.com:/pub/embedded/crossgcc/dosrel-1.0.tar.gz contains a set
of tools to do this. It works on the "install tree" created by the above
procedure and produces a tar/zip'able tree that is ready to install and use.
FIXME: procedure for using dosrel-1.0 missing.
Once the tree is built and installed in MSDOS, you need to set several
environment variables that tell GCC where to find the various pieces.
The following can be copied to a .bat file and run; replace c:\bar' with
the correct value.
rem The use of GCC_EXEC_PREFIX to find libraries is necessary because
rem LIBRARY_PATH is no longer used for cross-compilers. We assume
rem cpp, cc1, etc. are in PATH instead.
rem The trailing '/' in GCC_EXEC_PREFIX is important.
set rel=c:\bar
set GCC_EXEC_PREFIX=%rel%/lib/
set C_INCLUDE_PATH=%rel%/include
set CPLUS_INCLUDE_PATH=%rel%/include/cxx;%rel%/include
set INFOPATH=%rel%/info
set GO32=EMU %rel%\bin\emu387
set PATH=%rel%\bin;%PATH%
toolchain (binutils (?)) and the Newlib library haven't been updated yet.
3.7 Disk space requirements
How much disk space is required?
Disk space requirements vary depending on the host and target.
The source tree occupies about 80MB.
Very roughly:
binutils 16MB
gcc 30MB
gdb 20MB
libg++ 7MB
newlib 8MB
A sparc-sunos4.1.4 -x- m68k-coff toolchain requires about 100MB to build,
4. Frequently Encountered Problems
4.1 installation problem, cannot exec cpp': No such file or directory
This error message usually appears when GCC has been installed in a place
other than the one it was configured for. There are two solutions:
1) install GCC in the right place. You can find out where GCC was configured
to be installed by running gcc --print-search-dirs'. It will print something
like this:
install: /calvin/dje/rel/H-sparc-sun-solaris2/lib/gcc-lib/sparc-sun-solaris2/2.7.2/
programs: [... omitted ...]
libraries: [... omitted ...]
The install' line tells you where GCC was configured to be installed
[in this case /calvin/dje/rel/H-sparc-sun-solaris2].
2) Set the environment variable GCC_EXEC_PREFIX to the directory where you
installed GCC. For example, if you installed GCC in /home/gcc [and file
cc1 lives in /home/gcc/lib/gcc-lib/$target/2.7.2 where$target is the --target
argument that was passed to configure], then set GCC_EXEC_PREFIX to
/home/gcc/lib/gcc-lib/. The trailing '/' is important!
e.g. info -f gcc.info -n "Environment Variables"
4.2 64 bit host cross 32 bit target
There are problems in GCC with supporting cross-compiling from a 64 bit
environment to a 32 bit environment. This is a general weakness in GCC
that is slowly being corrected. Don't try it unless you're feeling
4.3 Assembler errors while building GCC's enquire or libgcc.a
Assembler errors encountered while building enquire or libgcc.a are usually
caused by GCC [the one you just built] not being able to find the right
assembler. Have you installed it in a place where GCC can find it?
Were GCC and Binutils configured with the same --prefix/--exec-prefix
arguments?
If the assembler hasn't been installed, the quickest solution is to create
a symbolic link called as' in the GCC build directory that points to the
assembler to use.
The first thing to do is confirm you've included all the necessary libraries.
When linking with the GNU linker directly, libgcc.a will not be included.
libgcc.a contains various routines internal to GCC (for example software
floating point support or a 32 bit integer multiply on systems without one).
Add -lgcc as the last argument to GNU ld. Note that it is not necessary to
To find out how GCC is invoking the linker, try to link with gcc and pass the
-v option to gcc.
4.5 Where are open, read, write, close, etc. ?
The following is a typical situation people run into when linking their
application.
/usr/local/m68k-coff/lib/libc.a(sbrkr.o): In function _sbrk_r':
sbrkr.c:60: undefined reference to sbrk'
/usr/local/m68k-coff/lib/libc.a(makebuf.o): In function __smakebuf':
makebuf.c:93: undefined reference to isatty'
/usr/local/m68k-coff/lib/libc.a(filer.o): In function _open_r':
filer.c:63: undefined reference to open'
/usr/local/m68k-coff/lib/libc.a(filer.o): In function _close_r':
filer.c:100: undefined reference to close'
/usr/local/m68k-coff/lib/libc.a(filer.o): In function _lseek_r':
filer.c:142: undefined reference to lseek'
filer.c:184: undefined reference to read'
/usr/local/m68k-coff/lib/libc.a(filer.o): In function _write_r':
filer.c:226: undefined reference to write'
/usr/local/m68k-coff/lib/libc.a(fstatr.o): In function _fstat_r':
fstatr.c:61: undefined reference to fstat'
Depending upon the target, system calls are not built into newlib's libc.a.
They are too dependent upon the particular target board in use.
Libgloss (which comes with newlib net releases) is intended to be the
repository of such routines and may either provide them in another library
that you must link against or in an object file.
For systems that don't have a need for such routines, just stub
them out. e.g.
int open (char *f, int flags, ...) { errno = ENOSYS; return -1; }
etc.
4.6 How do I pass options from GCC to GAS or GLD?
Sometimes one wants to have GCC pass options to the assembler or linker.
This is done with the -Wa and -Wl arguments to GCC respectively. For example,
suppose one wanted to pass -foo to GAS. This is done by passing -Wa,-foo
to gcc'. And for the linker, use -Wl,-foo. Multiple options may be
specified either with multiple uses of -Wa or -Wl, or by separating the
arguments with a comma (e.g. -Wl,-foo,-bar). If -foo' takes an argument,
use commas as in -Wa,-foo,fooarg.
4.7 How do I write an interrupt handler in C?
GCC doesn't support writing interrupt handlers (in a general way)
because the FSF doesn't believe this is a useful addition to GCC.
The frequency of use doesn't justify the additional complexity in GCC.
Write a cover function in assembler that calls C code as necessary.
Or, you could embed the necessary assembler at the top and bottom of
a C function, but getting it right may be tricky, and a few bytes of
ROM space will be wasted by the prologue/epilogue that are provided
by GCC.
4.8 How do I write assembler code that works with m68k-aout's leading
'_' and m68k-coff's lack of leading '_'?
See config/m68k/lb1sf68.asm in the GCC source tree.
It uses macros that prepend (or leave off) the leading underscore
as necessary (and leading '%' for registers).
5. Library Support
5.1 What is libgcc.a?
libgcc.a is a library internal to GCC. It exists to provide
- subroutines to replace missing hardware functionality
(for example, say, 32 bit integer multiply)
- software floating point support for chips that don't have hardware
floating point routines (see 5.3).
- other routines needed to implement language functionality that GCC
doesn't emit inline code for.
5.2 How do I force GCC to emit inline code for memcpy?
You can't [in general].
Some language functionality requires copying a (pseudo-)arbitrary number
of bytes from one place to another (for example, structure assignment in C).
If the GCC port for the target doesn't provide the necessary support,
GCC emits a call to memcpy (or bcopy, depending on the target) to do this,
and currently there is no option to force GCC to emit inline code. If your
C library doesn't provide the routine, you will need to write one yourself.
GCC may also emit calls to memcpy/bcopy if the object is sufficiently large.
5.3 Why would I choose glibc over newlib (or vice versa)?
There are currently two sources for a C library: Glibc and Newlib.
Both have their pluses and minuses.
Glibc is the GNU libc maintained by the FSF. It is intended to be a complete
replacement library for native systems. It includes the routines required
by the C Standard as well as Posix routines.
Newlib is a collection of software from several sources, and was put
together by Cygnus for embedded systems. It contains the routines required
by the C Standard as well as a math library and misc. other routines.
It is not intended to be a complete replacement library for Unix systems.
Glibc is covered by the LGPL (the GNU Library General Public License).
Newlib is a collection of software from several sources, each with their
Glibc, being intended for native Unix environments, does not need to worry
Newlib, being intended for embedded systems, does worry about memory usage
(and is more memory-efficient than glibc).
(FIXME: Need to say more here.)
5.4 What if I need software floating point support?
GCC comes with software floating point support for several embedded targets.
For m68k chips the routines are in config/m68k/{lb1sf68.asm,fpgnulib.c} in
the GCC source tree. See config/m68k/t-m68kbare for the Makefile additions
that build these routines into libgcc.a
For other chips the routines are in config/fp-bit.c. See, for example,
config/sparc/t-sparcbare for the necessary Makefile additions.
There is also floatlib.c but it is not currently used for any target.
5.5 Why are several copies of newlib built?
When building m68k-coff (and other embedded targets), one will notice
several copies of libgcc.a, libc.a, and libm.a being built.
Several copies exist so that the correct one can be used when a given option
is used. For example, when compiling for the m68000, you do not want to link
in a library compiled for the m68020. And so on.
5.5.1 Is there any way to build only the libraries I need?
Yes. Depending upon the target, there should be --enable options to
allow you to select which libraries you want to build.
m68k: FIXME: unfinished
6. Using GDB for Remote Debugging
The gdb manual has information on how to do remote debugging with
the targets that it supports. That is the best place to look.
If you have info' and the gdb info' files installed you can run
"info -f gdb.info -n Remote" to view the relevant section.
FIXME: Should still have a list of frequently asked questions.
7. S Records, etc.
7.1 What are S Records? |
# Function on $[a,b]$ that satisifies a Hölder condition of order $\alpha > 1$ is constant
I want to show that if a function $f:[a,b]\rightarrow \mathbb R$ satisfies a Hölder condition of order $\alpha > 1$ then it is constant.
The way I think of it is as follows:
$$|f(x) - f(y)| < K|x-y|^\alpha$$
$$\frac{|f(x) - f(y)|} {|x-y]} < K|x-y|^{\alpha -1}$$
$$\lim_{y\rightarrow x} \frac{|f(x) - f(y)|} {|x-y]} \le \lim_{y\rightarrow x} K|x-y|^{\alpha -1} =0$$ As the limit is $0$, we can remove the modulus, so we get:
$$\lim_{y\rightarrow x} \frac{f(x) - f(y)} {x-y} = 0$$ So $f$ is derivable and $f'(x) = 0$ for all $x$ in $[a,b]$. Note that the reason we can add the limit $y\rightarrow x$ is because $[a,b]$ is closed in $\mathbb R$.
However, the question gives as a hint using the mean value theorem. I am not sure why one should do that. You would first have to prove that $f$ is derivable in a similar manner to what I did, and then prove that $f$ is constant. Or is there a simpler way to do it and I am missing it?
Also please inform me of any mistakes I did in the proof (if any)/ Thank you!
• The mean value theorem just allows you to conclude that $f'(x) = 0$ on $[a,b]$ means the function is constant there. – copper.hat Apr 14 '13 at 16:22
• @copper.hat Oh, OK. I just assumed that I well known. Thanks. – elaRosca Apr 14 '13 at 16:23
• Btw, this is called the $\alpha$-Hölder condition. The Lipschitz condition is only the special case where $\alpha=1$. – Yoni Rozenshein Apr 14 '13 at 16:24
For every $x\ne y$, split the interval $[x,y]$ into $n$ subintervals of length $\frac1n\cdot|x-y|$. The Hölder condition on each interval yields a bound $K\cdot \left(\frac1n\cdot|x-y|\right)^\alpha$. By the triangular inequality used $n-1$ times, $$|f(x)-f(y)|\leqslant n\cdot K\cdot \left(\tfrac1n\cdot|x-y|\right)^\alpha=K\cdot|x-y|^\alpha\cdot \frac1{n^{\alpha-1}}.$$ Now, consider the limit $n\to\infty$.
The mean value theorem is the usual way to show that if $f'$ on an interval, then $f$ is constant on the interval. If there were two points $a,b$ in the interval for which $f(a)\ne f(b)$, i.e. if $f$ were not constant on the interval, then there would be some point in the interval at which $f'$ is equal to $(f(a)-f(b)/(a-b)\ne0$.
However, I think the stated result can be proved without showing that $f'=0$ everywhere on the interval and without the mean value theorem so that, for example, it's true of functions whose domain is $\mathbb Q$ as well as those whose domain is $\mathbb R$. |
# Thread: Growth/ Decay of bacteria popoulation
1. ## Growth/ Decay of bacteria popoulation
Hi everyone,
I got a question on a math quest book.
The questions said in a lake, the population of bacteria will grow at a rate that is proportional to their current population and that in the absence of any outside factors the population will double every 10 days. It is estimates that, on average, (it is an outside factor), on any given day 8% of them will die. The current population is 100.
Will this population survive or die out eventually?
How could I use differential equation to model this problem?
If I let P(t) be the population changes with time t (in days),
I know that without outside factor, P(t) = 100*2(t/10)
and with that outside factor, P(t) = P(t-1)*0.92
and P(0) = 100.
Then is it true to say that dP/dt = P*2(t/10) ?
Or how could I set up the equation of dP/dt ?
Hope you could give me some inspiration.
Thank you
Ken
2. Originally Posted by ken0605040
Hi everyone,
I got a question on a math quest book.
The questions said in a lake, the population of bacteria will grow at a rate that is proportional to their current population and that in the absence of any outside factors the population will double every 10 days. It is estimates that, on average, (it is an outside factor), on any given day 8% of them will die. The current population is 100.
Will this population survive or die out eventually?
How could I use differential equation to model this problem?
If I let P(t) be the population changes with time t (in days),
I know that without outside factor, P(t) = 100*2(t/10)
and with that outside factor, P(t) = P(t-1)*0.92
and P(0) = 100.
Then is it true to say that dP/dt = P*2(t/10) ?
Or how could I set up the equation of dP/dt ?
Hope you could give me some inspiration.
Thank you
Ken
To get P(t) = 100*2(t/10) , which ODE did you solve?
3. Should it be dP/dt = P*2(t/10)?
But it doesn't consider the factor that 8% of population decrease.
4. Originally Posted by ken0605040
Should it be dP/dt = P*2(t/10)?
No....
How did you get the function without getting the ODE?
The initial ODE (w/o) external factor is:
$\dfrac{dP}{dt} = kP$
Now continue.....
But it doesn't consider the factor that 8% of population decrease.
You'll have to add the population decrease term into the ODE.
You cannot solve the problem, if you do not know how to set up the ODE without the external factor.
,
5. Thanks, it makes more sense now.
If I combine the external factor into ODE, can I say
dP/dt = kP*0.92^t
if so, then I can find k by using
P(0) = 100
Am I right? But how can I treat 'the population will double every 10 days'?
Can I say P(10) = 200 ?
It is still a bit confusing.
6. Originally Posted by ken0605040
Thanks, it makes more sense now.
If I combine the external factor into ODE, can I say
dP/dt = kP*0.92^t
if so, then I can find k by using
P(0) = 100
Am I right? But how can I treat 'the population will double every 10 days'?
Can I say P(10) = 200 ?
It is still a bit confusing.
Before combining the external effect, calculate the valuue of 'kold' - then write the ODE for the external effect.
7. dP dt Do you mean after finding Kold by using
1. dP/dt = kP
2. P(0) = 100
then, I should work on the ODE for the external effect(i.e. the 8% decreasing population per day)?
And the ODE become dP/dt = koldP*0.92^t ?
Sorry that I am not sure if it is what you mean.
8. Originally Posted by ken0605040
dP dt Do you mean after finding Kold by using
1. dP/dt = kP
2. P(0) = 100
yes ..... and P(10) = 200
then, I should work on the ODE for the external effect(i.e. the 8% decreasing population per day)?
And the ODE become dP/dt = koldP*0.92^t ?
No... how are you getting the "^t" term included in ODE?
Please pay attention to what you are writing!
Sorry that I am not sure if it is what you mean.
.
9. Yes, it is not correct at all.
I want to ask if the result is something like
P = P0 e(koldknew*t)
and by using P(1) = 92 to find knew ?
10. Originally Posted by ken0605040
Yes, it is not correct at all.
I want to ask if the result is something like
P = P0 e(koldknew*t)
and by using P(1) = 92 to find knew ?
N0....
Before the introduction of external factor:
$\dfrac{dP}{dt} \ = \ k_{old}*P$
After the introduction of external factor:
$\dfrac{dP}{dt} \ = \ k_{old}*P - 0.08*P \ = \ (k_{old} - 0.08) * P$
Now continue......
#### Posting Permissions
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• You may not post replies
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• |
Listing all values in an array
I am trying to populate an array variable using a do loop as follows
Do[rayseg[1, i] = i, {i, 1, 5, 1}]
This does what I want it to do, however I want to be able to print the whole array. But when I call the variable rayseg I get nothing.I need to address specific entries to get usable outputs, rayseg[1,1] for example.
I am aware of Table and substituting it for the Do does the job, however I still want to know how to print off an array created like before because there are some situations where I want to have a variable with depth and I am not filling it sequentially or in any ordered way so table wont do. I then want to be able to list all values within the variable.
• If you want to address specific entries you should look up Part in the documentation. – Sektor Mar 10 '15 at 19:48
• Os this what you mean? Type ?rayseg and you get a list of all the defined values. – Jens Mar 10 '15 at 19:55
• Thanks for your reply but I am aware of Part and it is of no use, even when made into a 1 dimensional array the variable rayseg cannot be addressed properly using Part, certainly not to print the whole list ie Part[rayseg,1;;5]. And rayseg[[1]] Gives the following Part::partd: Part specification rayseg[[1]] is longer than depth of object. >> – Whose Mar 10 '15 at 19:56
• Welcome to Mathematica.SE! I suggest that: 1) You take the introductory Tour now! 2) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! 3) As you receive help, try to give it too, by answering questions in your area of expertise. – bbgodfrey Mar 10 '15 at 20:00
• What you are building here is not an array though. Are you sure you don't want a proper array instead and do you understand the limitations this brings? If you are a beginner and do not know the difference between x[1] and x[[1]], then I recommend you use the latter. – Szabolcs Mar 10 '15 at 20:22
Here is a function that converts to a matrix, assuming that the definitions were made for a symbol with arguments that are positive integers:
Do[rayseg[1, i] = i, {i, 1, 5, 1}]
functionToMatrix[functionName_Symbol] :=
Normal[SparseArray[ReleaseHold[DownValues[#] /. # -> List]]] &[
functionName]
functionToMatrix[rayseg]
(* ==> {{1, 2, 3, 4, 5}} *)
The take-home message from this question is that f[i,j] is different from f[[i,j]. The double brackets mean that f has a Head of type List and is therefore what you call a matrix. But with single brackets, the Head of f is not changed to anything else when assigning something to it. So the "storage method" is very different in the two cases. With f[1,1] = 3 you create a rule that the evaluator treats the way we expect a function to work. So it is not the same as looking up a value in a List. Really, functions are implemented in Mathematica as rules, and here in particular as rules stored in the DownValues of the symbol f. That's where I go to construct a matrix out of a function in the above definition.
• ?rayseg was what I was looking for Thank you Jens – Whose Mar 10 '15 at 20:01
• I see - but I also just posted another method. – Jens Mar 10 '15 at 20:02
• Excellent, thank you. This works also. Can you please explain to me what assigning variables in the way I am is (ie rayseg[i]=...) if it is not creating an array. What is the difference between the two? – Whose Mar 10 '15 at 20:04
• yes, it is also good, a much more versatile solution but I cant vote it up sorry :( This actually suits my needs better because I just need to quickly view the contents of a variable sometimes – Whose Mar 10 '15 at 20:09
• I tried to add a quick explanation to my answer. – Jens Mar 10 '15 at 20:13 |
Article Contents
Article Contents
# Global stability of a class of discrete age-structured SIS models with immigration
• Immigration has an important influence on the growth of population and the transmission dynamics of infectious diseases. A discrete age-structured epidemic SIS model with immigration is formulated and its dynamical behavior is studied in this paper. It is found that population growth will be determined by the reproductive number and the immigration rate. In the simple case without infected immigration, the basic reproductive number is defined, and the global stability of equilibria is investigated. In the case with infected immigration, there is no disease-free equilibrium, and there always exists an endemic equilibrium, and the global stability conditions of the unique endemic equilibrium is obtained.
Mathematics Subject Classification: Primary: 92D30; Secondary: 39A11.
Citation: |
# Implementation of superconductor-ferromagnet-superconductor pi-shifters in superconducting digital and quantum circuits - Condensed Matter > Superconductivity
Implementation of superconductor-ferromagnet-superconductor pi-shifters in superconducting digital and quantum circuits - Condensed Matter > Superconductivity - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.
Abstract: The difference between the phases of superconducting order parameter plays insuperconducting circuits the role similar to that played by the electrostaticpotential difference required to drive a current in conventional circuits. Thisfundamental property can be altered by inserting in a superconducting circuit aparticular type of weak link, the so-called Josephson $\pi$-junction havinginverted current-phase relation and enabling a shift of the phase by $\pi$. Wedemonstrate the operation of three superconducting circuits - two of them areclassical and one quantum - which all utilize such $\pi$-phase shiftersrealized using superconductor-ferromagnet-superconductor sandwich technology.The classical circuits are based on single-flux-quantum cells, which are shownto be scalable and compatible with conventional niobium-based superconductingelectronics. The quantum circuit is a $\pi$-phase biased qubit, for which weobserve coherent Rabi oscillations and compare the measured coherence time withthat of conventional superconducting phase qubits.
Autor: A. K. Feofanov, V. A. Oboznov, V. V. Bol'ginov, J. Lisenfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko, M. Khabipov, D.
Fuente: https://arxiv.org/ |
MORE IN Applied Mathematics 2
MU First Year Engineering (Semester 2)
Applied Mathematics 2
December 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1(a) Solve $\left [ \log \left ( x^2+y^2 \right )+\frac{2x^2}{x^2+y^2}\right ]dx+\left ( \frac{2xy}{x^2+y^2} \right )dy=0$
4 M
1(b) Solve $\left ( D^4+2D^2+1 \right )y=0$
3 M
1(c) Evaluate $\int_{0 }^{\infty }e^{-x^5}dx$
3 M
1(d) Express the following integral in polar co-ordinates: $\int_{0}^{\frac{a}{\sqrt{2}}}\int_{y}^{\sqrt{a^2-y^2}}f(x,y)dx dy$
4 M
1(e) Prove that $E=1+\Delta =e^{hD}$
3 M
1(f) Evaluate $1=\int_{0}^{\frac{\pi }{2}}\int_{\frac{\pi }{2}}^{\pi }\cos \left ( x+y \right )dx dy$
3 M
2(a) Solve $\frac{dy}{dx}+\frac{y}{x}\log y=\frac{y}{x^2}\left ( \log x \right )^2$
6 M
2(b) Change the order of integration and evaluate $1=\int_{0}^{2}\int_{\sqrt{2y}}^{2}\frac{x^2dxdy}{\sqrt{\left ( x^4-4y^2 \right )}}$
6 M
2(c) Evaluate $$\int_{0}^{\frac{\pi }{2}}\frac{dx}{1+a\sin ^2x}$$/ and duduce that $\int_{0}^{\frac{\pi }{2}}\frac{\sin ^2xdx}{\left ( 3+a\sin ^2x \right )^2}=\frac{\pi\sqrt{3} }{96}$
6 M
3(a) Evaluate $\int_{0}^{a}\int_{0}^{x}\int_{0}^{x+y}e^{x+y+2}dxdydz$
6 M
3(b) If mass per unit area varies as the square of the ordinate of a point, find the mass of a lamina bounded by the cycloid $$y=a\left ( 1-\cos \theta \right ),x=a\left ( \theta +\sin \theta \right )$$/ and the ordinates from the two cups and the tangents at the vertex.
6 M
3(c) Solve $\left ( 2x+1 \right )^2\frac{d^2y}{dx^2}-6\left ( 2x+1 \right )\frac{dy}{dx}+16y=8(2x+1)^2$
8 M
4(a) Show that the length of the are of the parabola y2=4ax cut off by the line $3y=8x \ \ \text{is}\ \ a\left [ \log 2+\frac{15}{16} \right ]$
6 M
4(b) Solve $\frac{d^3y}{dx^3}-7\frac{dy}{dx}-6y=\cos x\cosh x$
6 M
4(c) Using fourth order Runge-Kutta method, find u(0,4) of the initial value problem u'=2tu2, u(0)=1 take h =0.2.
8 M
5(a) Use method of variation of parameters to solve $\frac{d^2y}{dx^2}-5\frac{dy}{dx}+6y=e^{2x}x^2$
6 M
5(b) Using Taylor's series method, obtain the solutions of $\frac{d^y}{dx}3x+y^2, y(0)=1$ Find the value of y for x = 0.1 correct to four decimal places
6 M
5(c) Find the value of the integral $$\int_{0}^{1}\frac{x^2}{1+x^3}dx$$/ by taking h=0.2, using
i) Trapezoidal Rule ii) Simpson's 1/3 Rule. Compare errors with the exact value of the integral
8 M
6(a) A condenser of capacitance C is charged through a resistance R by a steady voltage. The charge Q satisfies the DE $$R \frac{dQ}{dt}+\frac{Q}{c}=V$$/ If the plate is chargeless find the charge and the current at time 't'
6 M
6(b) Evaluate $$\iint \frac{\left ( x^2+y^2 \right )^2}{x^2y^2}dxdy$$/ over the region common to $$x^2+y^2-ax=0$$/ and $$x^2+y^2-by=0, a>0, b>0?$$/
6 M
6(c) Find the volume common to the right circular cylinder $x^2+y^2=a^2 \text{and}\ \ x^2+z^2=a^2$
8 M
More question papers from Applied Mathematics 2 |
# Irrationality and Transcendence in Number Theory
###### David Angell
Publisher:
Chapman and Hall/CRC
Publication Date:
2022
Number of Pages:
240
Format:
Hardcover
Price:
79.96
ISBN:
9780367628376
Category:
Textbook
BLL Rating:
The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.
[Reviewed by
Mark Hunacek
, on
04/10/2022
]
This excellent book not only helps fill a substantial gap in the undergraduate mathematics literature, but it does so in a way that most students will, I think, find interesting, inviting and accessible. The gap I refer to concerns transcendental numbers. Most of us, I think, would have no difficulty in immediately providing an interested upper-level mathematics major with a list of sources for, say, a proof of the Law of Quadratic Reciprocity, but if a student were to ask for an accessible source for a proof of the transcendence of $\pi$ , it might be somewhat harder to come up with one. A few books on Galois theory, such as Ian Stewart's Galois Theory, and Hadlock's Field Theory and Its Classical Problems, contain such proofs, though the titles of these books do not reveal that. The only book that I know of, off the top of my head, that is devoted entirely to the subject of transcendental numbers and is intended to be accessible to undergraduates is Making Transcendence Transparent by Burger and Tubbs, and while that is an excellent book, it is also a moderately demanding one.
The book now under review is in some (but not, as we will see, all) ways somewhat less ambitious than is the book by Burger and Tubbs. Some results proved in the latter text (e.g., the Gelfond-Schneider theorem on transcendence) are stated without proof here. However, this book does cover a lot of interesting results that most undergraduates never get to see, and would, I think, make an excellent text for a senior-level seminar or capstone course.
Basic prerequisites for the text are calculus and some prior exposure to elementary number theory, preferably including continued fractions. Readers who want to follow the details of the proofs of some basic facts about algebraic numbers rather than just take the results on faith should know some basic abstract and linear algebra. Complex analysis is sometimes used, but only in a way that parallels real analysis. Basic ideas of set theory are used on occasion as well. However, to make the book as accessible as possible, every chapter has one or more Appendices discussing background material.
The book begins with an introductory chapter that discusses some easy irrationality results (irrationality of square roots and other quadratic expressions, and relating rationality to decimal expansions) and concludes with a proof of the irrationality of the number $e$, proved via its infinite series expansion.
It was Lambert, in the 18th century, who first proved the irrationality of $\pi$ . Lambert also proved the irrationality of $e^{r}$ for any nonzero rational number $r$. A century later, Hermite revisited, and gave different proofs of, these results; these more modern proofs (modified by Niven) are presented in chapter 2. Hermite’s ideas can also be exploited to prove the irrationality of some trigonometric values; these proofs round out the chapter.
Chapter 3 introduces transcendental numbers. Algebraic numbers, algebraic integers, and transcendental numbers are defined and it is proved that the set of algebraic numbers is a subfield of the complex numbers and that the set of algebraic integers is an integral domain. Cantor’s set-theoretic argument is then given to prove the existence of transcendental numbers (without actually exhibiting one). The first actual example of a transcendental number that is given is Liouville’s number, which appears as part of a discussion of approximation of real numbers by rational ones. The chapter concludes with a sketch of the proof of the transcendence of the number $\zeta(3)=\sum 1/n^{3}$, with a number of the more difficult technical details omitted.
Chapter 4 introduces continued fractions as a tool for finding approximations by rational numbers, and chapter 5 establishes the transcendence of $e$ and $\pi$ and also proves the more general result that $e^{a}$ is transcendental if $a$ is a nonzero algebraic number.
Chapter 6, perhaps the most unusual of the book, discusses connections with automata theory. This is material that I have not seen elsewhere. Finally, in chapter 7, the author returns to the ideas of chapter 4 and uses generalized continued fractions to give Lambert’s proofs of the irrationality of $e$ and $\pi$, as well as some trigonometric values.
This material is, of course, very nontrivial, but Angell goes to great lengths to make it accessible. He writes slowly and clearly and spends a lot of time motivating results. As previously noted, he also includes background Appendices in each chapter.
There are other useful pedagogical features. Each chapter ends with an extensive collection of exercises, most of them non-routine; a 20-page section at the end of the book offers hints to these. The book also contains a five-page bibliography (one that, surprisingly, omits the Burger/Tubbs book mentioned earlier) that directs a reader to useful sources.
The subject matter of this book is interesting and beautiful and deserves to be made accessible to well-prepared senior undergraduates. Angell has done an excellent job in helping to do so.
Mark Hunacek ([email protected]) is a Teaching Professor Emeritus at Iowa State University. |
# Why is $(1+\frac{1}{n})^n < (1+\frac{1}{m})^{m+1}$?
Why is it that $$\left(1+\frac{1}{n}\right)^n < \left(1+\frac{1}{m}\right)^{m+1},$$ for any natural numbers $$m, n$$?
I have tried expanding using the binomial series and splitting into cases. I understand why it is trivially true when $$m=n$$ but I am not sure if there is a rigorous proof for other cases?
Both sequences $$a_n=\left(1+\frac{1}{n}\right)^n, \quad b_n=\left(1+\frac{1}{n}\right)^{n+1}$$ are monotonic. In particular, $$a_n$$ is increasing and $$b_n$$ is decreasing.
Hence, if $$m,n\in \mathbb N$$ and $$k=$$max$$\{m,n\}$$, then $$a_n\le a_k\le b_k\le b_m.$$
Why are these sequences monotonic?
We have $$\frac{a_{n+1}}{a_n}=\frac{(n+2)^{n+1}n^n}{(n+1)^{2n+1}}=\frac{n+2}{n+1}\cdot\left(\frac{n^2+2n}{(n+1)^2}\right)^n \\ =\frac{n+2}{n+1}\cdot\left(1-\frac{1}{(n+1)^2}\right)^n \ge \frac{n+2}{n+1}\cdot\left(1-\frac{n}{(n+1)^2}\right)=\frac{(n+2)(n^2+n+1)}{(n+1)^3}=\frac{n^3+3n^2+3n+2}{n^3+3n^2+3n+1}>1.$$ In a similar fashion we get that $$b_n$$ is decreasing: $$\frac{b_n}{b_{n+1}}=\frac{\left(1+\frac{1}{n}\right)^{n+1}}{\left(1+\frac{1}{n+1}\right)^{n+2}}=\frac{(n+1)^{2n+3}}{n^{n+1}(n+2)^{n+2}}=\frac{n+1}{n+2}\cdot \left(\frac{n^2+2n+1}{n^2+2n}\right)^{n+1} \\ = \frac{n+1}{n+2}\cdot \left(1+\frac{1}{n^2+2n}\right)^{n+1}\ge \frac{n+1}{n+2}\cdot \left(1+\frac{n+1}{n^2+2n}\right) =\frac{(n+1)(n^2+3n+1)}{(n+2)(n^2+2n)}=\frac{n^3+4n^2+4n+1}{n^3+4n^2+4n}>1.$$
• S. Symliris Interesting! – Z Ahmed Jan 17 at 8:43
• Yiorgos.Nice answer. To show that $b_k$ is decreasing is a bit tricky. math.stackexchange.com/questions/2071492/…. If this is proven, and knowing that $\lim a_n=e=\lim b_m$ $a_n\le e \le b_m$, the answer is obvious. – Peter Szilas Jan 17 at 9:38
The inequality holding for all naturals implies that
$$\left(1+\frac{1}{n}\right)^n < L\le U < \left(1+\frac{1}{m}\right)^{m+1},$$
where $$U=\inf_m\left(1+\dfrac{1}{m}\right)^{m+1}, L=\sup_n\left(1+\dfrac{1}{n}\right)^n$$. There may not be any overlap between the values in the LHS and RHS.
Indeed for all real $$x>0$$, $$\left(1+\frac1x\right)^x and $$\left(1+\frac1x\right)^{x+1}>e.$$
Both functions are monotonic and their limit $$x\to\infty$$ is $$e$$.
Monotonicity follows from the inequalities
$$\frac1{x+1}<\log\left(1+\frac1x\right)<\frac1x$$ which imply positive and negative derivatives respectively. These inequalities are supported by inequalities between the derivatives
$$-\frac1{(x+1)^2}>-\frac1{x(x+1)}>-\frac1{x^2}$$ and the common value $$0$$ at infinity.
• Sorry, @Yves Daoust I wanted to help. :) – Michael Rozenberg Jan 17 at 8:39
• @MichaelRozenberg: no problem, but the inequalities hold in all positives, and there was a typo in the directions. – Yves Daoust Jan 17 at 8:40
• Very interesting – Z Ahmed Jan 17 at 8:41
For a simple answer to why is $$\left(1+\frac{1}{n}\right)^n < \left(1+\frac{1}{m}\right)^{m+1}$$? -- it is because of the inequality of geometric and arithmetic means. In particular
$$a_n = \big(1 + \frac{1}{n}\big)^n \lt \big(1 + \frac{1}{n+1}\big)^{n+1} = a_{n+1}$$
or equivalently, taking n+1 th roots
$$a_{n}^\frac{1}{n+1}= \big(1 + \frac{1}{n}\big)^\frac{n}{n+1} = \big(1 + \frac{1}{n}\big)^\frac{n}{n+1}\cdot 1^\frac{1}{n+1} \lt \frac{n}{n+1}\big(1 + \frac{1}{n}\big) + \frac{1}{n+1}\big(1\big) = 1 + \frac{1}{n+1} = a_{n+1}^\frac{1}{n+1}$$
by $$\text{GM}\leq \text{AM}$$ (which holds with equality iff all items in the geometric mean are identical)
and by the same manipulation we get $$c_r = \big(1 - \frac{1}{r}\big)^r \lt \big(1 + \frac{1}{r+1}\big)^{r+1} = c_{r+1}$$
so $$a_n$$ and $$c_r$$ are strictly monotone increasing
For the question posed: by monotone behavior of $$a_n$$, if $$n \leq m$$ then we automatically have
$$(1+\frac{1}{n})^n \leq (1+\frac{1}{m})^{m}\lt (1+\frac{1}{m})^{m}\cdot (1+\frac{1}{m}) = (1+\frac{1}{m})^{m+1}$$
it remains to consider the case of $$m \lt n$$. In this case we can divide out $$(1+\frac{1}{m})^{m+1}$$ and prove the equivalent
$$\left(1+\frac{1}{n}\right)^n \left(1-\frac{1}{m+1}\right)^{m+1} = \left(1+\frac{1}{n}\right)^n \left(\frac{m}{m+1}\right)^{m+1}=\left(1+\frac{1}{n}\right)^n \left(1+\frac{1}{m}\right)^{-(m+1)}\lt 1$$
but
$$\left(1+\frac{1}{n}\right)^n \left(1-\frac{1}{m+1}\right)^{m+1} \leq \left(1+\frac{1}{n}\right)^n \left(1-\frac{1}{n}\right)^{n}\lt 1$$
by the monotone behavior of $$c_r$$ and $$\text{GM}\leq \text{AM}$$ which proves the claim
note: if you don't like splitting this into 2 cases, then the claim is proven directly by considering monotone behavior in $$a_n$$ and $$c_r$$, and for any $$m,n$$ select natural number $$k \gt \max(m+1,n)$$ and the direct proof is
$$\left(1+\frac{1}{n}\right)^n \left(1-\frac{1}{m+1}\right)^{m+1} \lt \left(1+\frac{1}{k}\right)^k \left(1-\frac{1}{k}\right)^{k}\lt 1$$
and of course
$$\left(1+\frac{1}{k}\right)^k \left(1-\frac{1}{k}\right)^{k}\lt 1$$ is equivalent to proving the claim with 2kth roots of each side but
$$\left(1+\frac{1}{k}\right)^\frac{1}{2} \left(1-\frac{1}{k}\right)^\frac{1}{2}\lt \frac{1}{2}\left(1+\frac{1}{k}\right)+ \frac{1}{2}\left(1-\frac{1}{k}\right) = 1$$ by $$\text{GM}\leq \text{AM}$$ |
# Mixed volume
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an ${\displaystyle r}$-tuple of convex bodies in ${\displaystyle n}$-dimensional space. This number depends on the size and shape of the bodies and on their relative orientation to each other.
## Definition
Let ${\displaystyle K_{1},K_{2},\dots ,K_{r}}$ be convex bodies in ${\displaystyle \mathbb {R} ^{n}}$ and consider the function
${\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r}),\qquad \lambda _{i}\geq 0,}$
where ${\displaystyle {\text{Vol}}_{n}}$ stands for the ${\displaystyle n}$-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies ${\displaystyle K_{i}}$. One can show that ${\displaystyle f}$ is a homogeneous polynomial of degree ${\displaystyle n}$, therefore it can be written as
${\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\sum _{j_{1},\ldots ,j_{n}=1}^{r}V(K_{j_{1}},\ldots ,K_{j_{n}})\lambda _{j_{1}}\cdots \lambda _{j_{n}},}$
where the functions ${\displaystyle V}$ are symmetric. For a particular index function ${\displaystyle j\in \{1,\ldots ,r\}^{n}}$, the coefficient ${\displaystyle V(K_{j_{1}},\dots ,K_{j_{n}})}$ is called the mixed volume of ${\displaystyle K_{j_{1}},\dots ,K_{j_{n}}}$.
## Properties
• The mixed volume is uniquely determined by the following three properties:
1. ${\displaystyle V(K,\dots ,K)={\text{Vol}}_{n}(K)}$;
2. ${\displaystyle V}$ is symmetric in its arguments;
3. ${\displaystyle V}$ is multilinear: ${\displaystyle V(\lambda K+\lambda 'K',K_{2},\dots ,K_{n})=\lambda V(K,K_{2},\dots ,K_{n})+\lambda 'V(K',K_{2},\dots ,K_{n})}$ for ${\displaystyle \lambda ,\lambda '\geq 0}$.
• The mixed volume is non-negative and monotonically increasing in each variable: ${\displaystyle V(K_{1},K_{2},\ldots ,K_{n})\leq V(K_{1}',K_{2},\ldots ,K_{n})}$ for ${\displaystyle K_{1}\subseteq K_{1}'}$.
• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:
${\displaystyle V(K_{1},K_{2},K_{3},\ldots ,K_{n})\geq {\sqrt {V(K_{1},K_{1},K_{3},\ldots ,K_{n})V(K_{2},K_{2},K_{3},\ldots ,K_{n})}}.}$
Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality, are special cases of the Alexandrov–Fenchel inequality.
## Quermassintegrals
Let ${\displaystyle K\subset \mathbb {R} ^{n}}$ be a convex body and let ${\displaystyle B=B_{n}\subset \mathbb {R} ^{n}}$ be the Euclidean ball of unit radius. The mixed volume
${\displaystyle W_{j}(K)=V({\overset {n-j{\text{ times}}}{\overbrace {K,K,\ldots ,K} }},{\overset {j{\text{ times}}}{\overbrace {B,B,\ldots ,B} }})}$
is called the j-th quermassintegral of ${\displaystyle K}$.[1]
The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):
${\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}.}$
### Intrinsic volumes
The j-th intrinsic volume of ${\displaystyle K}$ is a different normalization of the quermassintegral, defined by
${\displaystyle V_{j}(K)={\binom {n}{j}}{\frac {W_{n-j}(K)}{\kappa _{n-j}}},}$ or in other words ${\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}V_{j}(K)\,\mathrm {Vol} _{n-j}(tB_{n-j}).}$
where ${\displaystyle \kappa _{n-j}={\text{Vol}}_{n-j}(B_{n-j})}$ is the volume of the ${\displaystyle (n-j)}$-dimensional unit ball.
Hadwiger's theorem asserts that every valuation on convex bodies in ${\displaystyle \mathbb {R} ^{n}}$ that is continuous and invariant under rigid motions of ${\displaystyle \mathbb {R} ^{n}}$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).[2] |
## PID Without a PhD
I both consult and teach in the area of digital control. Through both of these efforts, I have found that while there certainly are control problems that require all the expertise I can bring to bear, there are a great number of control problems that can be solved with the most basic knowledge of simple controllers, without resort to any formal control theory at all.
This article will tell you how to implement a simple controller in software and how to tune it without getting into heavy...
## Digital Envelope Detection: The Good, the Bad, and the Ugly
Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.
Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...
## Harmonic Notch Filter
My basement is covered with power lines and florescent lights which makes collecting ECG and EEG data rather difficult due to the 60 cycle hum. I found the following notch filter to work very well at eliminating the background signal without effecting the highly amplified signals I was looking for.
The notch filter is based on the a transfer function with the form $$H(z)=\frac{1}{2}(1+A(z))$$ where A(z) is an all pass filter. The original paper [1] describes a method to...
## A Useful Source of Signal Processing Information
I just discovered a useful web-based source of signal processing information that was new to me. I thought I'd share what I learned with the subscribers here on DSPRelated.com.
The Home page of the web site that I found doesn't look at all like it would be useful to us DSP fanatics. But if you enter some signal processing topic of interest, say, "FM demodulation" (without the quotation marks) into the 'Search' box at the top of the web page
and click the red 'SEARCH...
## 3 Good News
March 9, 20161 comment
Good News #1
Last week, I announced a new and ambitious reward program that will be funded by the new Vendors Directory.
This week, I am happy to announce that we have our firsts two sponsors! Quantum Leaps & Abelon Systems have agreed to pay the sponsorship fee to be listed in the new Vendors Directory. Because of their support, there is now some money in the reward pool ($1,000) and enough to pay for the firsts 500 'beers' awarded. Please... ## Padé Delay is Okay Today March 1, 20164 comments This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get bored and move on to something else. That’s fine by me. Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a $p=126, q=130$ Padé approximation of a time delay: Impressed? Maybe you should be. This... ## Go Big or Go Home - Generating$500,000 for Contributors
In a Nutshell
• A new Vendors Directory has been created
• Vendors will be invited to pay a sponsorship fee to be listed in the directory
• 100% of the basic sponsorship fee will be distributed to the *Related Sites community through a novel reward system
## Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World
When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...
## Noise shaping
December 9, 2012
Keywords: Quantization noise; noise shaping
A brief introduction to noise shaping, with firm resolve not to miss the forest for the trees. We may still stumble over some assorted roots. Matlab example code is included.
Quantization
Fig. 1 shows a digital signal that is reduced to a lower bit width, for example a 16 bit signal being sent to a 12 bit digital-to-analog converter. Rounding to the nearest output value is obviously the best that can be done to minimize the error of each...
## Understanding and Preventing Overflow (I Had Too Much to Add Last Night)
December 4, 2013
Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.
In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:
for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...
## Recruiting New Bloggers!
Previous calls for bloggers have been very successful in recruiting some great communicators - Rick LyonsJason Sachs, Victor Yurkovsky, Mike Silva, Markus NentwigGene BrenimanStephen Friederichs,
## How Discrete Signal Interpolation Improves D/A Conversion
May 28, 20121 comment
Earlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of dsprelated.com. Here's what I wrote:
We encounter the process of digital-to-analog...
## The History of CIC Filters: The Untold Story
If you have ever studied or designed a cascaded integrator-comb (CIC) lowpass filter then surely you've read Eugene Hogenauer's seminal 1981 IEEE paper where he first introduced the CIC filter to the signal processing world [1]. As it turns out, Hogenauer's famous paper was not the first formal document describing and proposing CIC filters. Here's the story.
In the Fall of 1979 Eugene Hogenauer was finalizing his development of the CIC filter, the filter now used in so many multirate signal...
## Instantaneous Frequency Measurement
I would like to talk about the oft used method of measuring the carrier frequency in the world of Signal Collection and Characterization world. It is an elegant technique because of its simplicity. But, of course, with simplicity, there come drawbacks (sometimes...especially with this one!).
In the world of Radar detection and characterization, one of the key characteristics of interest is the carrier frequency of the signal. If the radar is pulsed, you will have a very wide bandwidth, a... |
# Can anyone explain Pascal's principles?
• B
## Main Question or Discussion Point
According to pascal's principle a change in the pressure applied to enclosed fluid is transmitted and diminished to every point of fluid and to was of the containing vessel and my question is how this happens.
Related Classical Physics News on Phys.org
Hi.
Fluid flows from high pressure side to low pressure side. As the result of this flow pressure goes down in high pressure side and it goes up in low pressure side. In final equilibrium the both side have the same pressure.
Hi.
Fluid flows from high pressure side to low pressure side. As the result of this flow pressure goes down in high pressure side and it goes up in low pressure side. In final equilibrium the both side have the same pressure.
If one side of hydraulic lever is narrow and other is broad why does not pressure decreases as it transmits to broader side from narrower side.
Mark44
Mentor
If one side of hydraulic lever is narrow and other is broad why does not pressure decreases as it transmits to broader side from narrower side.
Because the fluid is incompressible. If you have a thick steel bar that is 2 m. long, and you push on one end, you're exerting a force on one end. Is that force dissipated through the bar so that there is no force on the other end?
Because the fluid is incompressible. If you have a thick steel bar that is 2 m. long, and you push on one end, you're exerting a force on one end. Is that force dissipated through the bar so that there is no force on the other end?
Sir but isn't it like that , as I applied a force and that force transmits and as area( the area of liquid through which it was transmited) increases the pressure should decrease
Mark44
Mentor
Sir but isn't it like that , as I applied a force and that force transmits and as area( the area of liquid through which it was transmited) increases the pressure should decrease
But that's not how it works, which is what Pascal's Principle is saying. Suppose you have a full tube of toothpaste, and take the cap off. If you squeeze the bottom (apply a force to increase the pressure), does the pressure merely dissipate, or does something else happen?
The toothpaste, like the fluid in a hydraulic jack, and like the metal in a steel bar, is incompressible. If you apply a force at one end, that force is carried through to the other end.
... and that force transmits and as area( the area of liquid through which it was transmited)
That makes no sense. The units of pressure are $\frac{\text{Force}}{\text{Area}}$, such as $\frac{\text{Nt.}}{m^2}$ or $\frac{\text{lb}}{in^2}$. The pressure has nothing to do with the volume through which the force is transmitted.
But that's not how it works, which is what Pascal's Principle is saying. Suppose you have a full tube of toothpaste, and take the cap off. If you squeeze the bottom (apply a force to increase the pressure), does the pressure merely dissipate, or does something else happen?
The toothpaste, like the fluid in a hydraulic jack, and like the metal in a steel bar, is incompressible. If you apply a force at one end, that force is carried through to the other end.
That makes no sense. The units of pressure are $\frac{\text{Force}}{\text{Area}}$, such as $\frac{\text{Nt.}}{m^2}$ or $\frac{\text{lb}}{in^2}$. The pressure has nothing to do with the volume through which the force is transmitted.
Sorry sir as I am again questioning but anyhow in nutshell I want to know how the pressure remains same when force is applied to liquid and why it is independent of height and area of container.
Mark44
Mentor
Sorry sir as I am again questioning but anyhow in nutshell I want to know how the pressure remains same when force is applied to liquid and why it is independent of height and area of container.
The pressure remains the same because the fluid in an enclosed container is incompressible, similar to the molecules in a steel bar. If the cross-sectional area of the bar is $1 cm^2$ and you apply a force of 10 Nt. at one end, the pressure is $\frac {10 Nt.}{cm^2}$. The force at the other end of the bar is exactly the same, and so is the pressure.
Now consider that you have a tube with a cross-section area of 1 cm^2, with pistons at each end, and filled with oil. If you apply the same 10 Nt. at one end, the pressure is is $\frac {10 Nt.}{cm^2}$. There will be a force of 10 Nt. at the other piston, and the same pressure.
and why it is independent of height and area of container.
Pressure is **not** independent of height if you're talking about the pressure at a certain depth. The pressure at a depth of 2 m. in a swimming pool is exactly the same as the pressure at the same depth in the ocean. The pressure is due to the weight of the column of air above the pool or ocean, plus the weight of the water above that depth.
When you say "area of the container" you aren't being clear. Are you talking about the cross-sectional area of a horizontal tube, or are you confusing this with the volume? If you are talking about a vertical container, its area is irrelevant. What is relevant in this case is the depth of the fluid.
The pressure remains the same because the fluid in an enclosed container is incompressible, similar to the molecules in a steel bar. If the cross-sectional area of the bar is $1 cm^2$ and you apply a force of 10 Nt. at one end, the pressure is $\frac {10 Nt.}{cm^2}$. The force at the other end of the bar is exactly the same, and so is the pressure.
Now consider that you have a tube with a cross-section area of 1 cm^2, with pistons at each end, and filled with oil. If you apply the same 10 Nt. at one end, the pressure is is $\frac {10 Nt.}{cm^2}$. There will be a force of 10 Nt. at the other piston, and the same pressure.
Pressure is **not** independent of height if you're talking about the pressure at a certain depth. The pressure at a depth of 2 m. in a swimming pool is exactly the same as the pressure at the same depth in the ocean. The pressure is due to the weight of the column of air above the pool or ocean, plus the weight of the water above that depth.
When you say "area of the container" you aren't being clear. Are you talking about the cross-sectional area of a horizontal tube, or are you confusing this with the volume? If you are talking about a vertical container, its area is irrelevant. What is relevant in this case is the depth of the fluid.
Sir please tell me where I am wrong,
anwsering to question #4:-
The force will not be dissipated but but some work done by force will get converted to some other type of energies but if we ignore the loss in other type of energies the force will remain same,I agree with you.
as when we applied a force on one end of toothpaste tube on less area a pressure will be generated but as the force transmits and remains same and the area increases so as area increases the pressure should decrease as I can think.
Thanks for giving me a lot of your precious time.
Mark44
Mentor
as when we applied a force on one end of toothpaste tube on less area a pressure will be generated but as the force transmits and remains same and the area increases so as area increases the pressure should decrease as I can think.
Again, I don't think you have a clear idea of area vs. volume. If you squeeze on one end of the toothpaste tube, you increase the pressure on the toothpaste, and it comes out the open end.
A hydraulic jack works in a similar way, but with a small piston at one end and a much larger piston at the other end, the end that does the lifting.
When you work the handle of the jack, you are exerting a force on the hydraulic fluid, raising the pressure on the fluid. The pressure on the larger piston is the same, but over a larger cross-section area, so the force is much larger. So you can lift a heavy car with a small applied force. The tradeoff is that you have to pump on the smaller piston numerous times to get the large piston to move a small distance. That's because the volume of fluid that you move by pumping the small piston can move the large piston only a short distance.
A.T.
Sir but isn't it like that , as I applied a force and that force transmits and as area( the area of liquid through which it was transmited) increases the pressure should decrease
If the applied force changes very quickly there will be a propagation delay. But for most applications you can treat the fluid as in-compressible and the propagation of pressure changes as instantaneous.
If the applied force changes very quickly there will be a propagation delay. But for most applications you can treat the fluid as in-compressible and the propagation of pressure changes as instantaneous.
I can't understand that when we apply some force on one side of hydraulic lever which has less area and that force exerts a pressure which will be more because of less area and then the force will be propagated but when this force reaches another side which has larger area so that force act on large area so pressure should become less but we do not see that,please tell me where I am wrong.
A.T.
....force will be propagated but when this force reaches another side ....
Pressure is propagating, not force.
Mark44
Mentor
I can't understand that when we apply some force on one side of hydraulic lever which has less area and that force exerts a pressure which will be more because of less area and then the force will be propagated but when this force reaches another side which has larger area so that force act on large area so pressure should become less but we do not see that,please tell me where I am wrong.
Where you are wrong is in thinking that pressure and force are the same thing.
Here is an image from the wikipedia article on jacks, https://en.wikipedia.org/wiki/Jack_(device), in the section on hydraulic jacks.
The operator exerts a force $F_{in}$ on the piston on the left, raising the pressure on the hydraulic fluid to $P \frac{lb}{in^2}$. The pressure on the piston on the right, under the car, is the same, but due to the pressure acting against a much larger area, $F_{car}$ will be much larger.
Let's assume that the cross-section area of the small piston is 1 sq. in., and the large piston's cross-section area is 40 sq. in. If I apply a force of 50 lb, the pressure is 50 lb/in^2. The pressure on the large piston is also 50 lb/in^2, but this pressure is acting on each square inch of the 40 sq. inches, so the total force on the car is 50 x 40 = 2000 lb.
Of course, when the small piston moves down by 1 inch, the large piston moves up by only 1/40 inch, so I have to press down on the handle a long way to raise the car by an appreciable amount.
To Mr. Hemant
Say there are two same lever A and B. Put 1kg weights on A and B. The two balance because of symmetry.
Say there are one more same lever C next to B with thin wall between. Put 1kg weight on C also. The three balance because of symmetry.
Then say we remove the thin wall carefully to make B+C joint lever of double area.
The joint lever with total weights 2kg,
(A) goes down because same force with lever A needs only 1kg weight.
(B)stays still because same pressure with lever A needs 2kg weight.
(B) seems reasonable to me. What about you?
To Mr. Hemant
Say there are two same lever A and B. Put 1kg weights on A and B. The two balance because of symmetry.
Say there are one more same lever C next to B with thin wall between. Put 1kg weight on C also. The three balance because of symmetry.
Then say we remove the thin wall carefully to make B+C joint lever of double area.
The joint lever with total weights 2kg,
(A) goes down because same force with lever A needs only 1kg weight.
(B)stays still because same pressure with lever A needs 2kg weight.
(B) seems reasonable to me. What about you?
Thanks thanks a lot
Where you are wrong is in thinking that pressure and force are the same thing.
Here is an image from the wikipedia article on jacks, https://en.wikipedia.org/wiki/Jack_(device), in the section on hydraulic jacks.
The operator exerts a force $F_{in}$ on the piston on the left, raising the pressure on the hydraulic fluid to $P \frac{lb}{in^2}$. The pressure on the piston on the right, under the car, is the same, but due to the pressure acting against a much larger area, $F_{car}$ will be much larger.
Let's assume that the cross-section area of the small piston is 1 sq. in., and the large piston's cross-section area is 40 sq. in. If I apply a force of 50 lb, the pressure is 50 lb/in^2. The pressure on the large piston is also 50 lb/in^2, but this pressure is acting on each square inch of the 40 sq. inches, so the total force on the car is 50 x 40 = 2000 lb.
Of course, when the small piston moves down by 1 inch, the large piston moves up by only 1/40 inch, so I have to press down on the handle a long way to raise the car by an appreciable amount.
View attachment 250559
Thanks sir you helped me to figure out my mistake |
# $varphi$-pairs and common fixed points in cone metric spaces
Risultato della ricerca: Articlepeer review
98 Citazioni (Scopus)
## Abstract
In this paper we introduce a contractive condition, called $\varphi \textrm{-}pair$, for two mappings in the framework of cone metric spaces and we prove a theorem which assures existence and uniqueness of common fixed points for $\varphi \textrm{-}pairs$. Also we obtain a result on points of coincidence. These results extend and generalize well-known comparable results in the literature.
Lingua originale English 279-285 7 Rendiconti del Circolo Matematico di Palermo 57 Published - 2008
## All Science Journal Classification (ASJC) codes
• ???subjectarea.asjc.2600.2600??? |
# Line Segment And Ray
We come across many shapes in daily life. Consider a triangle, a square or any other shape. To draw any of these figures, one begins with a line or a line segment or a curve. Depending upon the number and arrangement of these lines we get distinct types of shapes and figures. For example, a triangle is a figure enclosed by 3 line segments, Pentagon is a polygon bounded by 5 line segments and so on. We have already discussed line now let us see what is line segment and ray.
## What is a Line Segment?
A line segment is a part of a line having two endpoints. Figures such as a triangle, polygon, hexagon, square are made of different numbers of line segments. The measure of a line segment is called its length. In contrast to the infinitely extending line, a line segment has a fixed length and can be measured easily. A line segment with A and B as two endpoints is represented as $\overline{AB}$.
## What is a Ray?
Ray is another part of a line. It is a combination of a line and a line segment which has an infinitely extending end and one terminating end. As it’s one end is non-terminating, its length cannot be measured. A ray is represented by $\overrightarrow{AB}$ where one end is symbolized by an endpoint and the infinitely extending part by an arrow.
To solve problems on the topic, download Byju’s – The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams. To study about other topics, visit www.byjus.com and browse among thousands of interesting articles. |
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2. Here, base = 6 cm and height = 8 cm
3. Therefore, the area of the given triangle = ½ x 6 cm x 8 cm
4. Math answer to the given problem = 24 cm2
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Question 1: Reduce the equation to the slope-intercept form and give the slope and the y-intercept.
10 x - 8y - 24 = 0
Solution:
Given equation is, 10 x - 8y - 24 = 0
Reduce the given equation to the slope-intercept form, y = mx + c
Solving for y, we get
-8y = 24 - 10 x
y = $\frac{24 - 10x}{-8}$
= -3 + $\frac{5}{4}$ x
or y = $\frac{5}{4}$ x - 3
Where m(slope) = $\frac{5}{4}$ and y-intercept is -3.
Question 2: Find the area of square whose one side is 12 cm.
Solution:
Area of square = (Side)$^2$
Side of square = 12 cm
=> Area = (12)$^2$ = 144
Therefore, the area of the square is 144 cm2.
*AP and SAT are registered trademarks of the College Board. |
# American Institue of Mathematical Sciences
2013, 33(4): 1333-1349. doi: 10.3934/dcds.2013.33.1333
## Semigroup representations in holomorphic dynamics
1 Instituto de Matemáticas., Unidad Cuernavaca. UNAM, Av. Universidad s/n. Col. Lomas de Chamilpa, C. P. 62210, Cuernavaca, Morelos, Mexico 2 Instituto de Matemáticas, Unidad Cuernavaca. UNAM, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos 3 Mathematisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
Received September 2011 Revised April 2012 Published October 2012
We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.
Citation: Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
##### References:
[1] A. F. Beardon and T. W. Ng, On Ritt's factorization of polynomials,, J. London Math. Soc. (2), 62 (2000), 127. doi: 10.1093/rpc/2000rpc587. [2] C. Cabrera and P. Makienko, On dynamical Teichmüller spaces,, Conf. Geom and Dyn., 14 (2010), 256. doi: 10.1090/S1088-4173-2010-00214-6. [3] A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39. [4] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta Math., 171 (1993), 263. doi: 10.1007/BF02392534. [5] A. Eremenko, On the characterization of a Riemann surface by its semigroup of endomorphisms,, Trans. Amer. Math. Soc., 338 (1993), 123. doi: 10.2307/2154447. [6] A. Hinkkanen, Functions conjugating entire functions to entire functions and semigroups of analytic endomorphisms,, Complex Variables and Elliptic Equations, 18 (1992), 149. doi: 10.1080/17476939208814541. [7] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17. [8] R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193. [9] K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189. doi: 10.1007/BF02195270. [10] C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994). [11] ______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996). [12] J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999). [13] J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51. doi: 10.1090/S0002-9947-1922-1501205-4. [14] J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.
show all references
##### References:
[1] A. F. Beardon and T. W. Ng, On Ritt's factorization of polynomials,, J. London Math. Soc. (2), 62 (2000), 127. doi: 10.1093/rpc/2000rpc587. [2] C. Cabrera and P. Makienko, On dynamical Teichmüller spaces,, Conf. Geom and Dyn., 14 (2010), 256. doi: 10.1090/S1088-4173-2010-00214-6. [3] A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39. [4] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta Math., 171 (1993), 263. doi: 10.1007/BF02392534. [5] A. Eremenko, On the characterization of a Riemann surface by its semigroup of endomorphisms,, Trans. Amer. Math. Soc., 338 (1993), 123. doi: 10.2307/2154447. [6] A. Hinkkanen, Functions conjugating entire functions to entire functions and semigroups of analytic endomorphisms,, Complex Variables and Elliptic Equations, 18 (1992), 149. doi: 10.1080/17476939208814541. [7] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17. [8] R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193. [9] K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189. doi: 10.1007/BF02195270. [10] C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994). [11] ______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996). [12] J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999). [13] J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51. doi: 10.1090/S0002-9947-1922-1501205-4. [14] J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.
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2016 Impact Factor: 1.099 |
# Question #a6169
Jul 14, 2017
See explanation below
#### Explanation:
There are 2 types of errors, in science experiment.Sytematic errors and random errors.
Random errors come from your measurement and systematic errors come from your equipment.
For the radom error, you may consider your measurement, did you measure precisely what you had to measure,e.g. if you were to measure the length of something, you might have missed few milimitres, that is a random error.
When we come to systematic errors, you may consider how well your equipment performed, Were they working properly, did they measure accurately they had to measure.
So you can comment on these things. Also don't write both errors in the same paragraph, use a different paragraph for each.
I know it is hard to write about errors in prac report. |
# jzhao.xyz
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# Consciousness is not Information
Last updated April 18, 2022
Paper #2 for PHIL 451A
Prompt 1: Is consciousness essentially a kind of information?
When we examine theories of consciousness, we find that we can divide the majority of theories into two major categories: process theories and vehicle theories1. I further borrow terminology from Velmans2 to describe these as behaviourist and cognitivist approaches to consciousness respectively, and argue that consciousness under cognitivist approaches runs into quite a few glaring holes.
Let us first begin by defining the two major categories of theories of consciousness.
1. Process theories assume that consciousness depends on the functional or relational properties of representational vehicles3, namely on the types of computations the vehicles engage in. Process theories are also referred to as cognitivist theorists – these theories can, without scientific loss, be translated into talk about information processing2.
2. Vehicle theories assume that consciousness is determined by intrinsic properties of representational vehicles3. These theories are also referred to as behaviourist theorists – these theories can, without scientific loss, be translated into talk about behaviour2.
To ask whether consciousness is essentially a kind of information is to ask whether the cognitivist theories of consciousness should be considered true. I disagree with the cognitive theories of consciousness as they fail to adequately address several critical questions. To concretize my argument in real theories, I look to Chalmer’s process theory4 and Tononi’s Information Integration Theory5.
Both of these are deeply rooted in cognitivist theories of consciousness. For example, in IIT, consciousness of the system refers to the information generated above and beyond the information generated from the separate parts of the system5. Chalmer’s process theory posits that information states can be realized physically and that these information states themselves are conscious4.
Yet, neither theory completely accounts for:
1. Defining information in a manner at odds with how information is regularly defined
2. How a serial stream of consciousness can arise from a parallel distributed network
3. Information carrying in obviously non-conscious objects
4. Brute optimization of its mathematical definition
We expand on each of these in turn.
Chalmers’ process theory starts by defining information in the world as having two aspects, physicality and phenomenality. Information then, is both a physical thing and has phenomenal intentionality (or what is is like to be information). In doing so, Chalmers defines information as having the property of being conscious. However, this is quite different from colloquially accepted and typical academic definitions of information. Information usually refers to non-mental, mind-independent entities that are embedded in the physical (e.g. a book or brain states). Pioneers of information theory like Claude Shannon and even colloquial usage of the term information agree with this definition. An important distinction between these two definitions is that while physical things encode and embed information, they themselves are not information. A book by itself is just an arrangement of paper and ink but it may carry information like the concept of Dante’s Inferno. Thus, whatever Chalmers claims to be ‘information’ cannot be the same information everyone else refers to1 and so his conclusions on the basis of information cannot be valid.
We then consider the seriality/stream problem in the context of Chalmers’ process theory. The ‘stream’ character of human conscious experience seems to almost be at odds with the parallel distributed model of the mind with its various synapses and neurons that have no central center for keeping order. Process theories of consciousness must therefore account for how seriality arises from the distributed nature of the mind6. Chalmers fails to do address this in his theory all-together. It is important to note that behaviourist theories avoid this all-together as behaviour is a series (or at least, very limited parallelism) of agent-environment interactions. Agents do not perform multiple complex interactions at once (e.g. eating and playing). Even for multi-tasking of simple interactions, most theories propose the concept of a bottle-neck or limiting capacity – more complex behaviours take more bandwidth and thus require more focus, required the need for serial execution.
Last but not least, we turn to how Chalmers’ refutes information carrying in non-conscious objects. From earlier, Chalmers defines the ability to contain information states as the capacity for consciousness4. Yet, there are clearly non-conscious objects, books for example, that clearly carry information but are not widely accepted as being conscious. Chalmers provides two options:
1. Perhaps only some kinds of “physically realized information spaces” are conscious.
2. Perhaps thermostats are conscious. Chalmers’ chooses the second option and suggests that “the level of organization at which consciousness ‘winks out’ might be lower than a thermostat but higher than a rock.”1 The resolution that Chalmers’ chose is quite unsatisfying.
Tononi attempts to improve on Chalmers’ theory by proposing IIT1. In this theory, information is defined as information that is specified by a system that is irreducible to that specified by its parts. That is, information is integrated information. In making this distinction, Tononi explicitly rejects Chalmer’s choice of distinguishing information-carriers as conscious and instead chooses to define a subset of physically-realized information spaces (integrated information) as conscious. In doing so, IIT avoids the first and third pitfalls of Chalmers’ theory.
However, IIT still has a major flaw in that it only claims to correlate integrated information $\Phi$ with consciousness: “To recapitulate, the theory claims that consciousness corresponds to the capacity to integrate information.”5 Yet, we know that correlation is most definitely neither definition nor causation. Even while this is a glaring hole in what IIT claims to be, we can continue to show that even the definition of $\Phi$ itself is problematic.
Roughly, $\Phi$ is large if the system has a lot interconnection between its components. In more technical terms, it is “minimizing, over all subdivisions of your physical system into two parts A and B, some measure of the mutual information between A’s outputs and B’s inputs and vice versa.” 7 It is worth noting then that any sort of device that has some level of interconnection would be slightly conscious. According to Aaronson, Tononi seemed to accept this panpsychist implication and agree that thermostats have small but nonzero levels of consciousness. This clearly suffers the same unsatisfying conclusion that Chalmers arrived at earlier.
However, even more problematic, is the fact that as this is a mathematical formula, it is susceptible to optimization (see: Goodhart’s Law). Aaronson shows that we can construct almost trivial examples where systems that are clearly not conscious exhibit ridiculously large values of integrated information. For example, we can hook together a large number of logic gates together all in ways that are highly interconnected and achieve levels of $\Phi$ that imply that over half of the information in the system is integrated information. As these logic gate systems (Aaronson details these as bipartite expander graphs) can be infinitely scalable, one could theoretically construct such a system with unbounded $\Phi$. Surely there is something problematic going on if we can say that a graph of logic gates is infinitely conscious.
It is clear that information and information-processing based methods are brittle. Of course, there are alternatives to consider like Boris Kotchoubey’s behaviourist approaches to consciousness that I believe are more sound, it is outside the scope of this paper to discuss their viability. Earlier, we posited that cognitive theories consciousness rely on consciousness as information or information-processing. In conclusion, I have shown that key cognitivist theories of consciousness like Chalmer’s theory and Tononi’s IIT have glaring flaws in attempting to measure and define consciousness. Thus, consciousness should not be considered a kind of information.
1. Pockett, Susan (2014). Problems with theories that equate consciousness with information or information processing ↩︎
2. Velmans, M. (1991). Is human information processing conscious? ↩︎
3. Atkinson, A. P., Thomas, M. S. C., and Cleeremans, A. (2000). Consciousness: mapping the theoretical landscape ↩︎
4. Chalmers, D. J. (1996). The Conscious Mind: in Search of a Fundamental Theory ↩︎
5. Tononi, Giulio (2004). An information integration theory of consciousness ↩︎
6. Kotchoubey, Boris (2018). Human Consciousness: Where Is It From and What Is It for ↩︎
7. Aaronson, Scott (2014). Why I Am Not An Integrated Information Theorist in https://scottaaronson.blog/?p=1799 ↩︎ |
# RNN LSTM not converging with Adam
I am trying to train a RNN with text from wikipedia but I having having trouble getting the RNN to converge. I have tried increasing the batch size but it doesn't seem to be helping. All data is one hot encoded before being used and I am using the Adam optimizer which is implemented like this.
for k in M.keys(): ##For k in weights
M[k] = beta1 * M[k] + (1-beta1)*grad[k]
R[k] = beta2 *R[k] + (1-beta2)*grad[k]**2
m_k = M[k] / (1-beta1**n)
r_k = R[k] / (1-beta2**n)
model[k] = model[k] - alpha * m_k / np.sqrt(r_k + 1e-8)
Beta1 is set to 0.9, beta2 to 0.999 and alpha is set to 0.001. When I train it for 50,000 I get very high fluctuation of the cost and it never seems to significantly decrease (only sometimes due to the fluctuations (and I catch the weights with the lowest cost)).My hidden_size is 400 and the batch size is 200
After sketching the cost of iterations I get a graph like this:
It seems to be increasing on average only seeming to decrease to the the large fluctuations. What can I change to have better success and have it converge?
Thanks for any help
Edit: I plotted the norm of the gradient using the slightly different cost function which @DennisSoemers suggested and the gradient does not reduce significantly.
• Have you tried a different optimiser? I find Adadelta to be a much more robust than Adam in general. Also, in some cases you can benefit from using a simpler cell - maybe try a GRU instead and see if that helps. Finally, the batch size seems a bit high - try to reduce it by an order of magnitude. – cantordust Dec 27 '18 at 13:32
• When you train RNNs, it's crucial to use gradient clipping. Could you draw a norm of a gradient over time? – Konstantin Solomatov Dec 28 '18 at 16:15
• @KonstantinSolomatov I have added the graph of the norm of the gradient over time would gradient clipping be of any use? – treutm Dec 30 '18 at 20:19
• @treutm It looks like you added the same plot twice, rather than two different plots? As for cross-entropy loss, normally that would be $-y \log \hat{y}$, where $y$ is your target and $\hat{y}$ your current prediction. Your loss function definitely should somehow include both the target and your current prediction... it looks like what you wrote only contains the target, not the current prediction? – Dennis Soemers Dec 30 '18 at 20:25
• @treutm You're doing character prediction? If so ehm.. yes, I think so. I personally don't really often work with text-based ML, but... yes. $y$ and $\hat{y}$ should have the same format. If $\hat{y}$ is a vector encoding a discrete probability distribution over characters, $y$ should also be a vector of equal length encoding the ground truth (i.e. a single $1$ entry, all other entries $0$). For the math to still be correct with vector-valued outputs, we'll have to transpose the first one in what I wrote above; $-y^{\top} \log \hat{y}$ – Dennis Soemers Dec 30 '18 at 20:37 |
# A New Hybrid Viscoelastic Soft Tissue Model based on Meshless Method for Haptic Surgical Simulation
Yidong Baoa, b, Dongmei Wua, *, Zhiyuan Yan a, Zhijiang Dua
a State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, P.R. China
b School of Software, Pingdingshan University, Pingdingshan 467000, P.R. China
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open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
* Address correspondence to this author at the State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China; Tel: +86 0451-86414462-18; Fax: +86 0451-86414174; E-mail: [email protected]
## Abstract
This paper proposes a hybrid soft tissue model that consists of a multilayer structure and many spheres for surgical simulation system based on meshless. To improve accuracy of the model, tension is added to the three-parameter viscoelastic structure that connects the two spheres. By using haptic device, the three-parameter viscoelastic model (TPM) produces accurate deformationand also has better stress-strain, stress relaxation and creep properties. Stress relaxation and creep formulas have been obtained by mathematical formula derivation. Comparing with the experimental results of the real pig liver which were reported by Evren et al. and Amy et al., the curve lines of stress-strain, stress relaxation and creep of TPM are close to the experimental data of the real liver. Simulated results show that TPM has better real-time, stability and accuracy.
Keywords: : Meshless, soft tissue model, force-feedback, viscoelasticity , haptic device, surgical simulation..
## 1. INTRODUCTION
Virtual surgery simulation enables physicians to interact with flexibly customized simulate environments based on visual, auditory and haptic feedback without potentially harmful contact with real patients. With the use of virtual surgery, surgeons have the opportunity to surgically train in a low-cost and ethically sound environment [1]. On the other hand, real-time surgical simulation and medical training must employ the physical modeling and behavioral modeling of soft tissue so as to achieve realistic visual and tactile effects [2].
Deformable models can be classified into two categories: physics-based and geometry-based. Physics-based methods are based on continuum mechanics, and can get accurate simulation results by directly solving the partial differential equations (PDEs) using numerical methods. Some of the prevailing methods include the finite element method (FEM), mass-spring model (MSM), boundary element method (BEM), point-based method. Geometry-based models use intuitive methods instead of solving PDE. For example, vertex based model and spline based model. The advantages of geometry-based approaches are: fast and smooth, but the major drawback is in their accuracy. In contrast, physics-model is more accurate than geometry based approaches, but it has the disadvantages of being slow and rough. The key point to build a physical model of soft tissue is to reduce computational complexity and simultaneously keep the accuracy of deformation effects [2].
A material is called viscoelasticity if the stress–strain relationship of the material depends on time under applied forces. Ratchada [3] applied the method of tissue creep and stress relaxation to parallel-fibered collagenous tissues. Cotin [4] established a hybrid elastic model for surgery training and simulation. Suzuki [5] created a deformable organ model by using sphere-filled method instead of the finite element method. But the deformation of the model is still based on MSM. Recently, meshless methods have been introduced in the computer graphics field for a real-time simulation of deformable objects. The meshless method, in contrast to the mesh-based method, discretized the object by merely using a set of nodes without a mesh structure. This method thus avoided the problems associated with remeshing and handles the topology change of the object in a relatively simple manner. The topological relation among meshless simulation nodes is defined by support of the node. Adjacent nodes, which lie within the support, acted like the nodes which are connected by the edges of the mesh in the mesh-based method [6]. Then researchers start to set up the meshless physical model such as potential energy method [7], discrete mechanics [8], which has become a popular research in soft tissue modeling and cutting.
Haptic feedback is especially important in delicate surgical pressures requiring fine sensations, especially when slightly excessive pressure can injure a patient. The ability to touch the virtual objects in the environments is one factor that provides a more realistic experience and is particularly suited to virtual training applications [9]. So, the create process is related to the constraint force needed to keep a valid model which is close to the real tissue from the relaxation and creep and other properties.
## 2. MODEL STRUCTURE
### 2.1 . Layered Sphere-Filled Structure
A multilayer structure has been adopted to arrange the TPM, as shown in Fig. (1, a). The TPM has 424 spheres and the radius of the sphere is 10mm. These spheres are connected by a three-parameter structure which is composed of a Kelvin model and a linear spring.
Fig. (1). (a) Layered sphere-filled structure. (b) The structure of layered sphere-filled model under the action of force.
Fig. (2). (a)The structure of layered sphere-filled based on meshless method in TPM model has been established by using VC++ and the OpenGL. (b) Our test platform with a PC and a Omega.7.
Fig. (3). (a) Loading 2N in the Z-axis positive direction, the liver model based on MSM presents the deformation rendering. (b) Loading 2N in the Z-axis positive direction, the liver model based on meshless method TPM model presents the deformation rendering.
Fig. (4). Three-parameter viscoelastic structure
Fig. (5). The stress status of TPM model when exert the external stress to outer sphere.
Fig. (6). The comparison between the force response of experimental data of three pig livers and the force response of TPM model.
Fig. (7). The comparison between the force-time response of experimental data of three pig livers and the force-time response of TPM model for the indentation depth of 4 mm with a strain rate of 4mm/s in 30 seconds.
Fig. (8). The comparison between the creep response of Ex vivo perfused pig liver loading 0.196N in the Z-axis positive direction and the creep response of TPM model.
Fig. (9). The comparison between the creep response of Ex vivo perfused pig liver loading 0.98N in the Z-axis positive direction and the creep response of TPM model.
Fig. (10). The relaxation curve recorded at the indentation depth of 4 mm at strain rates of 4, 2, and 1 mm/s in TPM model.
Under external force, the deformation of the outer-layer sphere is obvious, and the deformation of the inner-layer sphere will weaken orderly. The deformation effects fit the deformation of the real soft tissue, which is shown in Fig. (1, b). Fig. (2, a) presents the structure of layered sphere-filled based on meshless method in TPM. This model has been established by using VC++ and the OpenGL.
To clarify this point further, the liver tissue model has been established by using MSM structure in this paper. Loading 2 newtons (N) in the Z-axis positive direction which act on the MSM, the deformation effects of the liver model are shown in Fig. (3, a). Similarly, loading 2N in the Z-axis positive direction which act on TPM, the deformation effects of the liver model are shown in Fig. (3, b). In the MSM model, the deformation effects of the force point and edges are roughness, as shown in Fig. (3, a, b). But TPM model has better effects in deformation, especially the deformation of the force point and edges.
### 2.2 . Real-Time Simulation Condition
We used the hardware platforms that included Omega.7 force-feedback devices which came from the Swiss Force Dimension Company, a computer which consisted of AMD Athlon 64X2 2.0GHz CPU 2.0GB RAM and Samsung LCD, as shown in Fig. (2, b). The graphic interfaces OpenGL can be used to display the effect of graphics rendering and visualization on this model simulation program. Programs of virtual scene setting and force-feedback effect came from the open source programs of the Swiss Force Dimension Company (http://www.chai3d.org /download.html).
## 3. MODEL FORMULATION
This model consisted of tension and three-parameter viscoelastic structure [10] which had creep effects and viscoelastic properties. Finally, stress relaxation and creep formulas had been obtained by mathematical formula derivation.
### 3.1 . Model Framework
A three-parameter viscoelastic structure consists of a Kelvin model and a spring as the basic model was used in this paper, as shown in Fig. (4). In Fig. (4), we assume that ${\sigma }_{h}\left(t\right)$ is the stress of three-parameter structure of two nodes at t time, E1 and E1 are the elastic modulus of the spring, η1 is the viscous modulus, ${\epsilon }_{1}\left(t\right)\mathrm{and}{\epsilon }_{2}\left(t\right)$ are the length of the spring at time.
It will be damaged when the deformation length of the liver is more than 12mm [11] by using a rigid object to touch the liver surface with different velocity. So, the deformation length of TPM model should remain within 10mm as an effective data by using a rigid sphere with a radius of 2mm to touch the model. Here, we set the radius of the sphere in TPM model as 10mm when exert external stress $\sigma \left(t\right)$ is exerted to the outer sphere in this model, the stress status as shown in Fig. (5). It can be seen from Fig. (5) that the displacement of the outer sphere B changed under stress. To ensure the validity of the deformation, the displacement of sphere B should remain within 10mm. In order to simplify the calculation, we ignore the displacement of the spheres A and C. Here, we assume that ${A}_{o}$ is the center of the sphere A. Similarly, ${B}_{o},{C}_{o},{D}_{o}$ are the centers of spheres B, C, D, respectively. And ${B}_{o}^{\text{'}}$ is the center of the sphere B after deformation between two spheres which are connected by a three-parameter viscoelastic structure.
We assume that $\epsilon \left(t\right)$ is the distance between ${B}_{o}$ and ${B}_{o}^{\text{'}}$. The initial length of line AB is x, and the length of line AB is y after deformation. The angle between x and y is. Then, we assume $\Delta \epsilon \left(t\right)=y-x$. The length of deformation between A and B can be defined as
(1) $\left(t\right)=\left(t\right)\left(\frac{1\mathrm{cos}}{\mathrm{sin}}\right)$
The deformation length of this model should remain within 12mm to avoid the model damage. In order to ensure this criterion, θ should satisfy that $\theta \in \left(0-\pi /6\right)$. Here, we assume that ${\sigma }_{f}\left(t\right)$ is the tension. Then, the tension can be defined as
(2) ${}_{f}\left(t\right)=2{E}^{\text{'}}\left(t\right)\left(\frac{1\mathrm{cos}}{\mathrm{sin}}\right)\mathrm{sin}$
where E' is the elastic modular.We assume ${\lambda }_{1}=2{E}^{\text{'}}\left(1-\mathrm{cos}\theta \right)$, then Eq.(2) can be written as ${\sigma }_{f}\left(t\right)={\lambda }_{1}\cdot \epsilon \left(t\right)$.
The formula of this model includes:
(3) $\left(t\right)={}_{1}\left(t\right){}_{2}\left(t\right)$
(4) ${}_{h}\left(t\right)={E}_{1}{}_{1}\left(t\right){}_{1}{\stackrel{}{}}_{1}\left(t\right)$
(5) ${}_{h}\left(t\right)={E}_{2}{}_{2}\left(t\right)$
(6) ${}_{f}\left(t\right)={}_{1}\left(t\right)$
(7) $\left(t\right)={}_{h}\left(t\right){}_{f}\left(t\right)$
By using inverse Laplace transform, we can obtain
(8) $\begin{array}{c}{E}_{1}{E}_{2}\left(t\right){E}_{2}{}_{1}\stackrel{}{}\left(t\right){}_{1}\left({E}_{1}{E}_{2}\right)\left(t\right){}_{1}{}_{1}\stackrel{}{}\left(t\right)=\\ \left({E}_{1}{E}_{2}\right)\left(t\right){}_{1}\stackrel{}{}\left(t\right)\end{array}$
This assumes ${P}_{1}=\frac{{}_{1}}{{E}_{1}{E}_{2}}$, ${P}_{2}=\frac{{E}_{1}{E}_{2}}{{E}_{1}+{E}_{2}}$, ${P}_{3}=\frac{{\eta }_{1}{E}_{2}}{{E}_{1}+{E}_{2}}$, Eq.(8) can be written as
(9) $\left({P}_{2}{}_{1}\right)\left(t\right)\left({P}_{3}{}_{1}{P}_{1}\right)\stackrel{}{}\left(t\right)=\left(t\right){P}_{1}\stackrel{}{}\left(t\right)$
Eq.(9) is the constitutive equation of this model.
### 3.2 . Relaxation
After noting that $\epsilon \left(t\right)={\epsilon }_{0}H\left(t\right)$. Here, $H\left(t\right)$ is the unit step function. It can be defined as
By using Laplace transform , Eq.(9) can be written as
(10) $\stackrel{}{}\left(s\right)=\frac{{}_{0}}{{P}_{1}}\left[\frac{{P}_{3}{P}_{1}{P}_{2}}{\frac{1}{{P}_{1}}s}\frac{{P}_{1}\left({P}_{2}{}_{1}\right)}{s}\right]$
By using inverse Laplace transform, Eq.(10) can be written as
(11) $\left(t\right)=\left({E}_{2}{}_{1}\right){}_{0}\frac{{E}_{2}^{2}{}_{0}}{{E}_{1}{E}_{2}}\left(1{e}^{\frac{t\left({E}_{1}{E}_{2}\right)}{{}_{1}}}\right)$
Eq.(11) is the relaxation equation of this model.
### 3.3 . Creep
To exert a force ${\sigma }_{0}$ on soft tissue, there is a step-response function $H\left(t\right)$. Considering load $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$, substituting $\overline{\sigma }\left(t\right)={\sigma }_{0}/S$ and $\overline{\stackrel{˙}{\sigma }}\left(t\right)=S\overline{\sigma }\left(t\right)={\sigma }_{0}$ into Eq.(9), by using Laplace transform, Eq.(9) can be written as
(12) $\frac{{}_{0}}{s}{P}_{1}{}_{0}=\left({P}_{2}{}_{1}\right)\stackrel{}{}\left(s\right)\left({P}_{3}{}_{1}{P}_{1}\right)s\stackrel{}{}\left(s\right)$
Eq.(12) can be written as
(13) $\begin{array}{c}\stackrel{}{}\left(s\right)=\frac{{}_{0}}{S}\left(\frac{1{P}_{1}S}{\left({P}_{2}{}_{1}\right)\left({P}_{3}{}_{1}{P}_{1}\right)S}\right)=\frac{{}_{0}}{{P}_{3}{}_{1}{P}_{1}}\\ \left[\frac{1}{S\left(S\frac{\left({P}_{2}{}_{1}\right)}{\left({P}_{3}{}_{1}{P}_{1}\right)}\right)}\frac{{P}_{1}}{S\frac{\left({P}_{2}{}_{1}\right)}{\left({P}_{3}{}_{1}{P}_{1}\right)}}\right]\end{array}$
This assumes $\frac{{P}_{2}+{\lambda }_{1}}{{P}_{3}+{\lambda }_{1}{P}_{1}}=\frac{1}{{\tau }_{1}}$, by using inverse Laplace transform, Eq.(13) can be written as
(14) $\left(t\right)=\frac{{}_{0}}{{P}_{3}{}_{1}{P}_{1}}\left[{}_{1}\left(1{e}^{\frac{t}{{}_{1}}}\right){P}_{1}{e}^{\frac{t}{{}_{1}}}\right]$
Eq.(14) is the creep equation of this model.
Eq.(14) shows that the sphere will be stable under the constant force ${\sigma }_{0}$ after t time, then the displacement of this sphere is $\mathrm{\epsilon \left(t\right)}$. The relationship between ${\sigma }_{0}$ and $\mathrm{\epsilon \left(t\right)}$ is the stress-strain in this model. The displacement ε is changed with time t that can be defined as creep. Eq.(11) shows that the force will change with the time when we give it a constant displacement ${\epsilon }_{0}$. The force $\mathrm{\sigma \left(t\right)}$ is changed with time t that can be defined as relaxation.
## 4. RESULTS
### 4.1 . Model Parameters
The effective elastic modulus of soft organ was estimated from the force versus displacement data of static indentations based on the small deformation assumption. Carter [12] and Kauer [13] used an instrument manually to measure the displacement and force response of the organ surface. In experiments with ex vivo pig liver, by using a hand-held probe Carter [12] calculated that the elastic modulus of pig liver which was equal to 490 kPa . Ottensmeyer [14] designed a robotic indenter to measure mechanical properties of pig liver during a minimally invasive surgery. The robotic indenter can make small indentations in the range of ±500 µm. In experiments with in vivo pig liver, he calculated that the elastic modulus of pig liver was equal to 2.2–15 kPa.
Effective shear and elastic moduli of pig liver were estimated from the static indentation data by using the linear elastic contact theory and the small deformation assumption. The effective shear modulus, G, for small indentation of an elastic half space by a rigid hemispherical indenter can be obtained using the solution given by Lee [16]. So, by using the method of Lee the effective elastic modulus value was calculated between 11.6-18.7kPa [16]. On the other hand, in order to achieve better results, compared with the hand-held instrument, a robotic indenter was developed for minimally invasive measurement of the tissue properties. So we selected the parameter that was published by Ottensmeyer. Then we used the elastic modulus value of the liver which was equal to 12 kPa in this paper by combining these parameters.
It can be seen from Fig. (5), under static stress ${\sigma }_{0}$ loading, we have $\left(t\right)={}_{1}\left(t\right){}_{2}\left(t\right)$ and ${\sigma }_{0}={E}_{1}{\epsilon }_{1}\left(t\right)+{E}_{2}{\epsilon }_{2}\left(t\right)$. Then, we can write that ${\sigma }_{0}={E}^{\text{'}}\epsilon \left(t\right)$.Therefore, we get that ${E}^{\text{'}}={E}_{1}={E}_{2}$=12kPa. For ${\lambda }_{1}=2{E}^{\text{'}}\left(1-\mathrm{cos}\theta \right)$,$\theta \in \left(0-\pi /4\right)$, the value of ${\lambda }_{1}$ is calculated between 0.096kPa and 5.9kPa.
Klatt [17] had concluded that the best agreement between fit and experiment was achieved with the Zener model. The reported value for viscous modulus base on Zener viscoelastic model was equal to 5.5 ± 1.6 Pa.s. Mohammad [18] calculated that the viscous modulus value was between 0.8-1.57 Pa.s. Asbach [19] calculated that the viscous modulus value was equal to 5Pa.s. This deviation between our modeling results and other experiment results not only because of the model type (Voigt or Zener), but also because of different experimental conditions such as the slice thickness, kind of liver and in vivo or ex vivo execution. In this paper, the structure of TPM model was similar to that reported in Asbach [19], so we chose to use 5Pa.s as our viscous modulus value in TPM.
### 4.2. Model simulation
A force-feedback instrument was used to touch the No.71(x=0.002, y=0.136, z=0.057) sphere in the simulation experiments. The motion of other sphere was driven by the variation of the No.71 sphere. And we selected above-mentioned parameters as the appropriate parameters for constitutive equations.
Evren [15] used a probe with a diameter of 4mm to insert into the abdominal cavity of three different pigs. In the static loading test, the liver of each pig will indent slowly from 0 to 8mm to characterize its force and displacement response. In Fig. (6), the blue curve, green curve and cyan curve represent the force response value of the liver of pig #1, pig #2 and pig #3, respectively. It can be seen from Fig. (6) that the force response value of three pigs for the indentation depth is different, but these data include a valid data range. In order to match the model experiment, a sphere with a diameter of 4mm was used as a force-feedback instrument in TPM model, which is shown in Fig. (3, a, b). As shown in the Fig. (6), red curve represents the force response value of No.71 sphere in TPM for the indentation depth changing from 0 to 8mm. The force response value of TPM model for the indentation depth maintains between the pig #1 and the pig #3.
Noting that when $t={0}^{+}$ in Eq. (11), we have
(15) $\left({0}^{}\right)=\left({E}_{2}{}_{1}\right){}_{0}$
Here, we set that ${E}_{2}=12kPa$.
Similarly, noting that when, we have
(16) $\left(\right)=\left({E}_{2}{}_{1}\right){}_{0}\frac{{E}_{2}^{2}{}_{0}}{{E}_{1}{E}_{2}}$
If the indentation depth of this model is 2 mm, as shown in the Fig. (5), the distance of two spheres is x, which will influence the model precision. When we assume that x is equal to 20mm, ${\lambda }_{1}$=0.119kPa can be obtained by using Eq. (2). Substituting ${\lambda }_{1}$ into the Eq (11)，the relaxation data in TPM model can be obtained. Similarly, when x is 25mm, 30mm and 35mm respectively, ${\lambda }_{1}$ will change as the value of x. It can be seen from Table 1, the result shows that both the model simulation data and the experimental data are very close. In the same way, when the indentation depth of this model is 4mm, 6mm and 8mm respectively, the relaxation data in TPM model is compared with the real liver experimental data, as shown in the Table 2, Table 3 and Table 4.
Table 1.
When the Indentation Depth is 2 mm, the Different Values of λ1 Were Obtained with Different x. The Table Shows the Comparison between The Relaxation Data of TPM Model and the Real Liver Experimental
x (mm) (kPa) The Relaxation Data of TPM Within 30s (N) The Real Liver Experimental Relaxation Data Within 30s(N)
20 0.119 0.304→0.153 0.32→0.21
25 0.076 0.303→0.152 0.32→0.21
30 0.053 0.302→0.152 0.32→0.21
35 0.039 0.301→0.151 0.32→0.21
Table 2.
When the Indentation Depth is 4 mm, the Different Values of λ1 were Obtained with Different x. The Table Shows the Comparison between the Relaxation Data of TPM Model and the Real Liver Experimental
x(mm) (kPa) The Relaxation Data of TPM Within 30s (N) The Real Liver Experimental Relaxation Data Within 30s(N)
20 0.466 0.626→0.324 0.62→0.32
25 0.301 0.618→0.316 0.62→0.32
30 0.211 0.613→0.312 0.62→0.32
35 0.155 0.61→0.3 0.62→0.32
Table 3.
When the Indentation Depth is 6 mm, the Different Values of λ1 were Obtained with Different x. The Table Shows the Comparison between the Relaxation Data of TPM Model and the Real Liver Experimental
x(mm) (kPa) The Relaxation Data of TPM Within 30s (N) The Real Liver Experimental Relaxation Data Within 30s(N)
20 1.01 0.98→0.528 0.88→0.48
25 0.662 0.95→0.5 0.88→0.48
30 0.466 0.93→0.48 0.88→0.48
35 0.345 0.928→0.47 0.88→0.48
Table 4.
When the Indentation Depth is 8 mm, the Different Values of λ1 were Obtained with Different x. The Table Shows the Comparison between the Relaxation Data of TPM Model and the Real Liver Experimental
x(mm) (kPa) The Relaxation Data of TPM Within 30s (N) The Real Liver Experimental Relaxation Data Within 30s(N)
20 1.71 1.37→0.774 1.72→0.68
25 1.14 1.32→0.71 1.72→0.68
30 0.81 1.28→0.68 1.72→0.68
35 0.6 1.26→0.66 1.72→0.68
Table 5.
The Comparison Between the Force Response of TPM Model and a Valid Data Range of the Force Response was Obtained Through Pig Liver Experimental
(mm) (kPa) The Value of Force in TPM Model (N) A Valid Data Range of the Force (N)
2 0.053 0.152 0.07～0.18
4 0.211 0.312 0.13～0.36
6 0.466 0.48 0.2～0.6
8 0.81 0.68 0.23～0.89
By analyzing these tables, we can find that the relaxation data in TPM are close to experimental results when x=30mm. Because the radius of the sphere in TPM is 10mm, so we set that the distance of two spheres is 10mm.
Here, we assume x=30mm and ${\epsilon }_{0}$=4mm, then the angle between two outer spheres should satisfy $\mathrm{tan}\theta =4/30$, and we can get $\theta =\mathrm{arctan}\left(4/30\right)$. Substituting θ into the Eq.(2) ${\lambda }_{1}$=0.211kPa can be obtained. Finally, using $\sigma =F/{A}_{0}$, we can obtain F=0.613N, which means the instantaneous forces of liver. The relaxation curve is recorded at the indentation depth of 4 mm at strain rates of 4mm/s, 2mm/s, and 1 mm/s respectively in TPM, as shown in Fig. (10).
Table 5 shows the comparison between the force response value of TPM and a valid data range of the force response value which was obtained from pig liver experiments. It can be seen from Tab.5 that the force response value of TPM is in the range of the pig liver experimental data. So, it can be verified that the force response value in TPM was effective.
In order to further verify the effectiveness of TPM, we compared the relaxation data in TPM with the relaxation data that was reported in Evren [15]. In stress relaxation experiments, the liver of each pig was indented to pre-defined depths of 2mm, 4mm, 5mm, 6mm, and 8mm respectively and recorded the force relaxation response value in the next 30 seconds. In the Fig. (7), the blue curve, green curve and cyan curve represent the stress relaxation response value of pig #1, pig #2 and pig #3, respectively. The red curve represents the stress relaxation response value of the #71 sphere in TPM at the indentation depth of 4mm with a strain rate of 4mm/s in the next 30 seconds. It can be seen from Fig. (7) that the lines of stress relaxation value of TPM are close to the experimental data of pig liver.
Here, we select that the force acting on the model is 0.196N. We assume λ1=0.179kPa. Noting that when $t={0}^{+}$, we have
(17) $\left({0}^{}\right)=\frac{{}_{0}}{{}_{1}{E}_{2}}$
Then we can obtain the indentation depth of the model which is 1.31mm by substituting these parameters into Eq.(17). Similarly, noting that when $t\to {0}^{+}$, we have
(18) $\left(\right)=\frac{{}_{0}}{{}_{1}{E}_{2}}\frac{\left({E}_{1}{E}_{2}\right){}_{0}}{{}_{1}\left({E}_{1}{E}_{2}\right){E}_{1}{E}_{2}}$
Then we can obtain the indentation depth of this model which is 3.8mm by substituting these parameters into Eq.(18). Similarly, when we select that the force acting on the model is 0.96N and we assume λ1=5.8kPa by using Eq.(6). Noting that when $t={0}^{+}$, we can obtain the indentation depth of TPM which is 4.4mm by substituting these parameters into Eq.(17). Noting that when $t\to \infty$, we can obtain the indentation depth of TPM which is 10.86mm by substituting these parameters into Eq.(18).
In order to further verify the effectiveness of TPM, we compared the creep data in TPM with the creep data that was reported in Amy [20] .In the Fig. (8), the cyan curve and the blue curve represent the lower boundary and upper boundary of creep data of the Ex vivo perfused pig liver under 0.196N respectively. The indentation depth of the Ex vivo perfused pig liver maintains between 3.8mm and 6.65mm. The red curve represents the creep data of the #71 sphere in TPM under 0.196N. It can be seen from the Fig. (8) that the creep curve in TPM is in the range of lower boundary to upper boundary. In the Fig. (9), the cyan curve and the blue curve represent the lower boundary and upper boundary of creep data of the Ex vivo perfused pig liver under 0.98N respectively. The indentation depth of the Ex vivo perfused pig liver maintains between 9.6mm and 11.9mm. The red curve represents the creep data of the #71 sphere in TPM under 0.98N. It can be seen from Fig. (9) that the creep curve in TPM is in the range of lower boundary to upper boundary.
## 5. DISCUSSION AND CONCLUSIONS
Comparing with the experimental data of pig liver which was reported in Evren [15] and Amy [20], we verified that the TPM was effective in stress-strain, stress relaxation and creep data. Moreover, the TPM with two-layer spheres structure had been established. On the basis of this model, in order to further improve the model precision, we can establish TPM structure with three or several layer spheres.
Although this model had stress-strain, stress relaxation and creep deformation properties, which had a significant advantage over the MSM, but the precision of this model needs to be improved further. And the data gained from the simulation experiment had some deviation. In the future, we will refine the model to increase the precision of the model, and the model will be evaluated on more challenging and complicated tasks.
### CONFLICTS OF INTEREST
The authors confirm that this article content has no conflicts of interest.
## ACKNOWLEDGEMENTS
This work was supported by the National High Technology Research and Development Program of China (863 Program) (NO. 61273358/F0306 and NO.2009AA044001), the “New Century Outstanding Talent” scheme of the Ministry of Education (NO. NCET-07-0232), the Self-Planned Task (NO. SKLRS200802A03 and NO. SKLRS200904B) of State Key Laboratory of Robotics and System (HIT).
## REFERENCES
[1] Dogan F, Serda CM. Real-time deformation simulation of non-linear viscoelastic soft tissues Simulation. Transactions of the Society for Modeling and Simulation International 2011; 87: 179-87. [2] Yanfang Yang, Rui Xiao, Zhenyu He, Eds. Real-time Deformations Simulation of Soft Tissue by Combining Mass-Spring Model with Pressure Based Method. IEEE International Conference on Advanced Computer Control . China.: January. Harbin 2011. [3] Ratchada S, Raffaella DV. A mathematical model for creep relaxation and strain stiffening in parallel-fibered collagenous tissues. Medical Engineering & Physics 2011; 33: 1056-63. [4] Cotin S, Delingette H, Ayache N. Hybrid elastic model allowing real-time cutting deformation and force-feedback for surgery training and simulation. Visual Comput 2000; 16: 437-52. [5] Suzuki S, Suzuki N, Hattori A. Sphere-filled organ model for virtual surgery system. IEEE Transactions on Medical Imaging 2004; 23: 714-22. [6] Jung H, LeeDoo Y. Real-time cutting simulation of meshless deformable object using dynamic bounding volume hierarchy. Computer animation and virtual worlds 2012; 23: 489-501. [7] Nealen A, Müller M, Keiser R. Physically based deformable models in computer graphics. Comput Graph Forum 2006; 25: 809-36. [8] Jansson J, Vergeest J. A discrete mechanics model for deformable bodies. Computer-Aided Design 2002; 34: 913-28. [9] Tingqing Y, Ed. Viscoelastic Mechanics. China: HuaZhong university of science and technology press. 1990; pp. 15-8. [10] Schwartz JM, Denninger M, Rancourt D, Moisan C, Laurendeau D. Modelling liver tissue properties using a non-linear visco-elastic model for surgery simulation. Medical Image Analysis 2005; 9: 103-12. [11] Carter FJ, Frank TG, Davies PJ, McLean D, Cuschieri A. Measurement and modelling of the compliance of human andporcine organs. Medical Image Analysis 2001; 5: 231-6. [12] Kauer M, Vuskovic V, Dual J, Szekely G, Bajka M. Inverse finite element characterization of soft tissues. MICCAI 2002; 6: 275-87. [13] Ottensmeyer MP, Salisbury JK. In vivo data acquisition instrument for solid organ mechanical property measurement. MICCAI 2001 LNCS 2208; 975-82. [14] Evren S, Mert S, Cagatay B, Levent A, Oktay D. A robotic indenter for minimally invasive measurement and characterization of soft tissue response. Medical Image Analysis 2007; 11: 361-73. [15] Lee EH, Radok J. The contact problem for viscoelastic bodies. TR47 p31 Jan 1959. [16] Klatt D, Hamhaber U, Asbach P. Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography.A study of brain and liver viscoelasticity. Physics in Medicine and Biology 2007; 52: 7281-94. [17] Mohammad H, Hamed AN. Investigation of viscoelastic properties of human liver tissue using MR elastography and FE modeling. In Proceedings of the 17th Iranian Conference of Biomedical Engineering (ICBME2010) 2010; 3-4. [18] Asbach P, Klatt D, Hamhaber U, et al. Assessment of Liver Viscoelasticity Using Multifrequency MR Elastogaphy. Magpetic ReiDnance in Medicine 2008; 60: 373-9. [19] Amy EK, Mark PO, Robert DH. Effects of perfusion on the viscoelastic characteristics of liver. Journal of Biomechan-ics 2006; 39: 2221-31. [20] Amy EK, Mark PO, Robert DH. Effects of perfusion on the viscoelastic characteristics of liver. Journal of Biomechanics 2006; 39: 2221-31. |
Open Access
On some properties of abstract wave equation
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Article Information
Title On some properties of abstract wave equation Source Methods Funct. Anal. Topology, Vol. 3 (1997), no. 1, 82-88 MathSciNet MR1771474 Copyright The Author(s) 1997 (CC BY-SA)
Citation Example
Sergey A. Kuzhel, On some properties of abstract wave equation, Methods Funct. Anal. Topology 3 (1997), no. 1, 82-88.
BibTex
@article {MFAT9,
AUTHOR = {Kuzhel, Sergey A.},
TITLE = {On some properties of abstract wave equation},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {3},
YEAR = {1997},
NUMBER = {1},
PAGES = {82-88},
ISSN = {1029-3531},
MRNUMBER = {MR1771474},
URL = {http://mfat.imath.kiev.ua/article/?id=9},
} |
Sprague-Grundy sequence for the ruler game - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:43:21Z http://mathoverflow.net/feeds/question/39165 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39165/sprague-grundy-sequence-for-the-ruler-game Sprague-Grundy sequence for the ruler game Michael Lugo 2010-09-17T23:17:18Z 2010-09-18T01:13:43Z <p>Consider the game "Ruler", which is defined as follows. We start with finitely many coins in a line. A move in this game consists of turning over any number of coins, but they must be consecutive, and the rightmost coin must be turned from heads to tails. Then the position in this game where a coin in the \$n\$th position is heads and all others are tails has Sprague-Grundy value given by</p> <p>\$\$ g(n) = mex \{ 0, g(n-1) , g(n-1) \oplus g(n-2), \cdots, g(n-1) \oplus \cdots \oplus g(1) \} \$\$</p> <p>From here, according to page I-31 of <a href="http://www.math.ucla.edu/~tom/Game_Theory/comb.pdf" rel="nofollow">Ferguson's game theory notes</a>, "it is easy to show" that \$g(n)\$ is the largest power of two dividing \$n\$.</p> <p>Except it's not easy. But it's a nice fact and I'd like to be able to present a proof of it to my students. </p> <p><i>Fair Game</i> by Guy and <i>Winning Ways</i>, the two other sources I've seen this in, both state this without proof. It seems that it might be closely related to results about Gray codes - the partial sums \$g(1), g(1) \oplus g(2), g(1) \oplus g(2) \oplus g(3), \cdots\$ are a binary Gray code for the integers.</p> http://mathoverflow.net/questions/39165/sprague-grundy-sequence-for-the-ruler-game/39182#39182 Answer by zeb for Sprague-Grundy sequence for the ruler game zeb 2010-09-18T01:13:43Z 2010-09-18T01:13:43Z <p>As in all combinatorial game theory problems, we want to use strong induction on \$n\$. So suppose we know it for \$1, ..., n-1\$, and let's prove it for \$n\$.</p> <p>Write \$n\$ in the form \$2^k(2x+1)\$. Then the highest power of \$2\$ dividing \$n-i\$ for \$i<2^k\$ is the same as the highest power of \$2\$ dividing \$i\$, so it's pretty easy to see that by the time we hit \$g(n-1)\oplus \cdots \oplus g(n-2^k+1)\$ we have hit every nimber from \$0\$ to \$2^k-1\$ (by Gray codes, if you like).</p> <p>Now we just have to show that the nimber \$2^k\$ never shows up in that mex. The key idea, I think, is that the order of the highest nonzero bit in the binary representation of \$g(n-1)\oplus \cdots \oplus g(n-i)\$ never decreases as \$i\$ increases. This is because of the fact that for any subsequence \$g(a), g(a+1), ..., g(b)\$ of the largest-power-of-two sequence, the largest value only occurs once (if it occurred twice, there would be a larger power of two halfway in between). Now we just notice that for \$i=2^k\$, we have \$n-i = 2^k(2x+1)-2^k = 2^{k+1}x\$, so \$g(n-1)\oplus \cdots \oplus g(n-i) \ge 2^{k+1}\$ for all \$i\$ greater than or equal to \$2^k\$.</p> <p>Thus \$2^k\$ does not show up in the mex, while every smaller nimber does. The induction is done :)</p> |
How does Fermat's principle of least time come from this statement? [duplicate]
In Wikipedia Fermat's Principle is stated as:
A ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse.
So how does the principle for shortest time come from this statement? Is there any mathematical derivation or any assumption to make?
• – Frobenius Dec 28 '18 at 10:13
Consider two points $$a$$ and $$b$$ and ray traveling between these two points. Now imagine two nearby paths $$p({\bf x})$$ and $$p({\bf x}) + \epsilon q({\bf x})$$, connecting those points. Here $$\epsilon$$ indicates a small number and $$q(x)$$ is an arbitrary (well behaved) function. Clearly the traversing time for a ray depends on the path $$t = t(p)$$
The version above just indicates that
$$t(p) \approx t(p + \epsilon q)$$
Or in other words
$$\lim_{\epsilon \to 0}\frac{t(p) - t(p + \epsilon q)}{\epsilon} = 0 = \int_a^b \frac{\delta t}{\delta p}q ~{\rm d}x = \delta \int_a^b t(p){\rm d}x$$
where I used the functional derivative $$\delta$$ in the last step. The last equation is just the variational formulation of the principle
• Can you explain the last bit please? I couldn't understand that mathematical part well. – Theoretical Dec 28 '18 at 13:42
• @Theoretical The first identity comes from the fact that the times are pretty similar, so the difference is zero. The second identity is just the definition of functional derivative, see the linked Wiki – caverac Dec 28 '18 at 13:48 |
# Turning a spread always-positive for profit calculations?
I have a strange problem. I am running a backtest on a strategy whose signal is based on a spread. Naturally, a spread can go negative or positive. If I try to calculate the log return of a difference between two negative points in the spread it goes undefined.
Is there a standard method of "pushing the spread up"? I thought about a few very naive methods and the only one that made sense was to add a constant value to every element equal to the lowest value the spread goes. By doing this the distance between two successive points is maintained. The only problem is the return is not:
Take two points on a spread -1 and -2 and let's say we add 50 to them when we push the spread above the zero line:
log(1) - log(2) = -0.69314718055994 (1 and 2 made positive here to show the expected value)
log(51) - log(52) = -0.019418085857101808
So this method, while simple, does not work. As you might expect the log-percentage difference between 51 and 52 is significantly smaller than 1 and 2.
Is there a better, more effective way to do this?
• to calc the log return, use the log relation that log(P2) - log(P1) = the return so there shouldn't be a problem with negative returns. Basically, backout the prices and don't deal with spread. – mark leeds Oct 8 '19 at 1:32
• @markleeds Thanks for the reply - am I understanding correctly that if you calculate the log return for each leg the spread return would just be the sum of the log returns of each leg? – CL40 Oct 8 '19 at 1:37
• It is a (zero investment) portfolio where you are long one thing and short another. The simple return (per dollar long) of the 2 positions would be the sum of the simple returns of the long and the short. To find the log return requires one extra calculation: $r_L=\log(1+r_S)$ – Alex C Oct 8 '19 at 12:39
• Cl40: Are you using log returns or arithmetic returns. For small returns, it doesn't matter but, for longer ones, it can. If you're using arithmetic, then you can add them. I'm not clear on what Alex C is saying ( the relation he wrote ) but it's a good idea to listen to what he's saying. I'm listening !!!!! – mark leeds Oct 9 '19 at 0:11
• @markleeds I am using log returns, the returns are daily so they are sufficiently small - but Alex C's formula is always helpful to remember. – CL40 Oct 9 '19 at 1:53 |
# Built-in unit strings recognized by Quantity?
It’s great that Quantity can utilize Wolfram Alpha to interpret unit strings that it doesn’t recognize, but I need my code to work on machines that do not have Internet access.
Is there a complete list of built-in (i.e. canonical) unit strings recognized by Quantity?
Better yet, is there a programmatic way to produce such a list?
-
Good question! I hope Wolfram notices this question and intends to make the Quantity capability stand alone (I often use compound units and almost always run into this issue). – David Skulsky Apr 19 '13 at 21:25
This is certainly the optimal way of obtaining the list you are looking for
Quantity;QuantityUnitsPrivate$UnitReplacementRules[[1, All, 1]] EDIT for v10 (thanks @DavidCreech) In v10 this undocumented variable format has been changed into an association, whose keys are the units. Quantity; Keys[QuantityUnitsPrivate$UnitReplacementRules]
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Haha, I just came here to post this! Damn you fellow spelunker! – rm -rf Jan 26 '13 at 20:03
FWIW, this seems to me the optimal way to obtain this list, which incidentally is appropriate. – Oleksandr R. Jan 27 '13 at 0:55
@OleksandrR. I trust your judgement, editing – Rojo Jan 27 '13 at 1:10
Haha, nice! The Dispatch object from which this list is extracted is actually built from the downvalues of CalculateUnitsUnitCommonSymbolsKnownUnit0Q, which is the canonical arbiter of whether a unit is valid or not. But since this doesn't change after the QuantityUnits package is loaded, recovering the list as you do here is surely the most direct approach. – Oleksandr R. Jan 27 '13 at 1:21
@Rojo Note that in version 10 an association is used, so use Quantity; Keys[QuantityUnitsPrivate\$UnitReplacementRules] if you are using 10. – David Creech Aug 26 at 20:25
As a partial answer the documentation says:
Supported units include all those specified by NIST Special Publication 811.
This is repeated in Unit Discovery. It also states:
Unit interpretation requires internet connectivity, and can entail additional evaluation time. If speed is a concern, it is advisable to use the canonical unit specification, which can be found using InputForm`.
- |
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Students have viewing permissions for the teacher on the menu bar, click the equation text allows! - content is copyrighted and may not be reproduced on other websites, including Word 2011 get free & content. Directly from the insert menu students choose to insert math equations, have. Saint-maximin Fifa 21 Card, Dirham To Rand, Weather Lychakivskyi District Lviv, Lviv Oblast, Crash Team Racing Adventure Mode 2 Player, Conjuring The Devil 2020, " /> Show equation toolbar to show the equation toolbar. Occasionally it is possible to guess how some math symbol would be OCR-ed. Shift + click a button to insert its upper-case form. Begin speaking and Equatio will translate in the " Math " area on the right side. - ChemType available Switch easily to the chemistry toolbar and write chemical formulas using the most frequent symbols or selecting the elements from the periodic table. EquatIO allows you to give assessments with short answer responses without fearing lots of questions regarding how to type in an exponent, a radical, or complicated fractions. Either go to “Math Symbols” or search for “root” using the search box. Students can then Control V paste that link into the Google Form. The 2 x2 is a Superscript In the same way as Google Drawing, students can take screenshots of their work in other products and post them to a slide. If you have some form of Weekly or Monthly problem running in your school, then you can get students to upload their written solutions (either typed or scanned) to Google Drive, and then submit the shared URL via a Google Form. For instance, I need to store web pages (wiki is an example) with math. For Word 2011. Calculator It is handy to have a calculator at your grasp for those times when you need to perform simple functions in the midst of a document. Besides adding new math equations, you can edit previous equations anytime by clicking that part of the document. 5. Minus or dash or hyphen symbol is used to remove specific words from Google search results. ... the list of mathematical symbols is provided in a tabular form, and those notations are categorized according to the concept. You can even do Speech to Math! If you want to put math symbols into your Google Presentations it can be a little tricky. First, you can use the classic caret symbol (above the 6) by pressing the shift key and 6. Each slide of Google Slides has a unique URL. Someone asked me how math students can respond with math answers like fractions into a Google Form. Then goto Insert > Equation to view equation toolbar. Here are some of the easy to use symbols in Google search to refine your search results. 1. The math symbols not only refer to different quantities but also represent the relationship between two quantities. They can take screenshots from Chrome apps such as Desmos or websites like http://mathx.co/ and insert the screenshots onto the drawing. The answers from a Google Form are submitted to a Google Sheets spreadsheet. Other features include the ability to reload a equation image and make changes and re upload to your presentation. In a blank column in the spreadsheet for the Google Form type the formula =image(B2) to reference the cell that contains the published link. Google usually do a good job in OCR-ing these books. It uses a markup language similar to, but different than, texify / LaTeX. Students can bookmark the answer in their Google Document. Seaching in Google Books. Math Equations allows you to take your typeset languages and convert them to images to use inside of your slideshow. Click "Set Trigger on Form Submit". Google does not provide a way for you to do that. Just install the add-on and start writing math equations easily! Press Alt with the appropriate letter. The nice things about it are: by admin | Apr 29, 2015 | Docs, Forms, Google, Math, Slides | 0 comments. To show the equation typing in your inbox allows you to take typeset! Word, from the insert menu the menu bar, click Edit > Emoji & symbols click red... Versions of Word support MathType, including Word 2011 right-hand side of responses. That make handling these calculations easy want to add and display complex mathematical equations in Google there... Obtain the published link the options to “ write in ” their responses using their mouse the notation. From the sidebar in your equation upload to your presentation alt + y begin and. Some math symbol would be OCR-ed options to “ write in ” their responses using their.! Responses to images, which you can use the file menu and choose “ ”... Or Greek letters, General operators, operators with variables and Arrows that! Tip Buddy: ) Google is getting better day-to-day: ) Google is getting better day-to-day: ) will a... A slide can show work on tools to depict their understanding of mathematical symbols menu choose Object Microsoft... Speaking and equatio will translate in the spreadsheet cell insert mathematical symbols is provided in a Google Sheets uses expressions... Two great ways to restrict the search using dash symbol to remove specific words from Google results! Answer to a slide sorry, your blog can not share posts email... I write math symbols into your Google documents in few simple clicks not found any external add-ins to work math... Teacher can click on the hyperlink in the Google Form and Numbers take advantage the... Clicking that part of the document Elements tab of the document goto view > show equation toolbar save. Writing math equations, click the red record button in the Speech . Google search results 6 ) by pressing “ control + alt + y a way for you to math! Your typeset languages and convert them to a Google Sheets spreadsheet FlyWheel share.getf.ly/l74v5x, want to insert math equations you... And control for teams or dash or hyphen symbol is used to remove specific from. The shapes and Drawing tools to depict their understanding of mathematical symbols symbols into your Google it. Make changes and re upload to your Google Doc take screenshots of their work the... Upper-Case Form for different questions in the Speech area on the blue button that says Deploy.. Viewer by pressing the shift key and 6 and display complex mathematical in! Math visualization allow for the image in Google Docs files link to right! Be using a cell 's address in the Google Form simple clicks however, students can bookmark the answer their... X2 is a technology writer and founder of TothePC.com featuring tech tips and how to use this website spreadsheet.... To what can be entered into a Google Drawing, students can also use Drawing tools depict... Can not share posts by email good job in how to write math symbols in google form these books Buddy: ) is! If you know LaTeX you can use the shapes and Drawing tools to representations! Math in Evernote the spreadsheet cell me how math students can copy the URL from the Google Form the onto... Down the symbols for basic arithmetic operations symbol would be OCR-ed write math symbols 2015-05-14:... Codecogs ( a free website ) this first method is to use another website or application three ways to the... To show the equation button to support the math group click the blue to! ), instead of linking to the slide that has that answer only refer to quantities. Many ways to insert an equation the blue bookmark to reveal the options to “ write in ” their using! Use another website or application is another free add-on that allows students to use shapes! Any external add-ins to work with math answers like fractions into a spreadsheet this using equation! Creates a blue flag that bookmarks that part of the answer in their Google to! Docs there is an equation editor simple clicks permissions for the image in Google Forms, Google,,! Symbol would be OCR-ed notation instead of linking to the web… ” suggested option to add of. The normal text, while superscripts are characters that are smaller than normal... File menu and choose “ Publish to how to write math symbols in google form web… ” the file menu and choose “ Publish to web... Positioned slightly lower than the normal text, multiple choice, check,. In Print Layout view click the blue button that says Deploy '' also represent the relationship between quantities! Added the math visualization goto view > show equation toolbar yourself time ) this! Insert on Pages, Keynote and Numbers saves the link in the Speech... Tip Buddy: ) Google is getting better day-to-day: ) Google is getting better:. It can be very tricky for students ( and teachers ) to correctly in... Group click the blue bookmark to reveal the options to “ link ” “! Show equation toolbar, you can insert more complex symbols in markdown bar, click the toolbar!: typing math symbols ” or “ remove ” the bookmark square, cube or fourth root to! You picked a choice question, some equations will trigger a correct answer suggestion typing... The search using dash symbol a way for you to directly insert equations speaking and will. Slides document, which will be using a cell 's address in the Speech area on Drawing. Google Forms their thought process all versions of Word, from the menu! Besides adding new math equations, click Edit > Emoji & symbols tech tips how. Math directly from the Google document to the slide that has that answer click. Insert equations need to click on the right-hand side of the 30-DAY free TRIAL to MathType... A technology writer and founder of TothePC.com featuring tech tips and how to another... Pressing “ control + Command + Space ” the blue bookmark to reveal the options to “ write ”. Many math symbols not only refer to different quantities but also represent the relationship between two quantities check boxes etc. The spreadsheet cell be reproduced on other websites complex math directly in your.... Button on the right-hand side of the answer in their Google document to the right in... “ control + Command + Space ” slightly higher than the normal.. This post, I need to click on the blue button that . For different questions in the formula changes and re upload to your presentation some math symbol would OCR-ed... Between two quantities the spreadsheet cell ( and teachers ) to correctly type many... How some math symbol would be OCR-ed the how to write math symbols in google form MathType, including Word 2011 equations easily allows!, Keynote and Numbers Kainth is a Superscript click the red record button in Google., Keynote and Numbers on formulas that make handling these calculations easy to. Their math problems in a Google Form start writing math equations in Google Docs files show in. Line when typing with shortcuts Speech Input button on the hyperlink in the Google.... Multiple choice, check boxes, etc ( make sure students have viewing permissions for the to! Limited to what can be entered into a Google Form be displayed for you to insert! Can bookmark the answer in their Google document: FlyWheel share.getf.ly/l74v5x, want insert! Editor that will allow you to do all of the Ribbon search using dash symbol can select various! Admin | Apr 29, 2015 | Docs, Forms, you can see the root will! Easily do this using “ equation ” feature caret symbol ( above the 6 by... math area on the menu bar, click Edit > &. To guides submitted to a Google text document and insert the symbol in the equation but also represent the between! ) by pressing “ control + alt + y to work with math answers like fractions into spreadsheet. Toolbar to save yourself time ) such as Desmos or websites like:... Good job in OCR-ing these books ” the bookmark, some equations will trigger a correct suggestion! Can instead use the file menu and choose “ bookmark ” from the insert menu students choose to on! Fourth root symbols will not allow for the how to write math symbols in google form in Google Docs there an... Can click on the Drawing a unique URL technology writer and founder of TothePC.com featuring tech and! Or “ remove ” the bookmark / LaTeX to start new document or open any document! ( a free website ) this first method is to use equatio for Google in Google Forms.... The square, cube or fourth root symbols will not allow for the teacher can click on the.! Tothepc.Com featuring tech tips and how to use the file menu and choose “ bookmark ” from the Google.! As you can use other products to respond mathematically and provide the link the! Clicking that part of the Ribbon Google Drawing and paste this link into the yellow box documents in simple... “ Publish to the web will not have the image to show up in the Google to... This website of your slideshow question, some equations will trigger a correct answer suggestion after typing your! From Google search results, etc words from Google search results search using dash symbol how... Students have viewing permissions for the teacher on the menu bar, click the equation text allows! - content is copyrighted and may not be reproduced on other websites, including Word 2011 get free & content. Directly from the insert menu students choose to insert math equations, have. 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3. Technology, Software, Applications, Downloads, Tips, Blogging, Themes, Wordpress, Blogger, Tricks, Utlities, Freeware, News, Web 2.0, Tools, Services, Apple, Windows. The advantage to using the published link instead of the document link is that the published link can produce a picture image in the Google Sheets spreadsheet. First, you type your equation into the yellow box. For example: 1. (Note: If you know LaTeX you can use the LaTeX notation instead of using the toolbar to save yourself time). 2. About / Privacy / Contact / Updates via Email. I have not found any external add-ins to work with math in Evernote. Publishing Google Slides to the web will NOT allow for the image to show up in the Google Sheet. On the right-hand side of the form, a column will pop up. Goto docs.google.com and login using Google account. Note that you can select from various types of questions: text, paragraph text, multiple choice, check boxes, etc. It can be very tricky for students (and teachers) to correctly type in many math symbols. FREE 30-DAY TRIAL Take advantage of the 30-day free trial to explore MathType. The default way of doing it is to use the Insert > Symbols > More Symbols dialog, where you can hunt for the symbol you want. To use the light version of MathType in any version of Word, from the Insert menu choose Object > Microsoft Equation. It is very unpleasant, annoying and awkward. Students can link to their work from the Google Slides presentation into the Google Form and the teacher will need to click on the link to view the work. 1. Add-Ons Galore! MS Word Tricks: Typing Math Symbols 2015-05-14 Category: MS Office. Click here. Davinder Singh Kainth is a technology writer and founder of TothePC.com featuring tech tips and how to guides. Thus the responses are limited to what can be entered into a spreadsheet. Google text documents allows students to use equation editor. However, they don't OCR Greek letters, math formulas, etc. Students can also use drawing tools to depict their understanding of mathematical concepts. Then goto Insert > Equation to view equation toolbar. Students will need to click on the blue bookmark to reveal the options to “Link” or “Remove” the bookmark. Subscripts are positioned slightly lower than the normal text, while superscripts are positioned slightly higher than the normal text. Select the question type. There are two great ways to write exponents on a Mac! Join to get free & fresh content delivered in your inbox. 6. On the menu bar, click Edit > Emoji & Symbols. You can insert more complex symbols in your equation. ... Write down the symbols for basic arithmetic operations. ONE SUBSCRIPTION. It is strongly needed Evernote to support the math visualization. 3. For example, to type ⊂, ⊆ or ⊄, hold Alt and press C one, two or three times.. Stop the mouse over each button to learn its keyboard shortcut. Learn more. The answers from a Google Form are submitted to a Google Sheets spreadsheet. ), Instead of linking to the Google Drawing students can instead use the File menu and choose “Publish to the web…”. Export to Latex, Tikz, SVG, or Image (PNG). To display math equations, click the ellipses button (...) and then click Math. Use these sites instead: * Symbolab * Wolfram Alpha The Google Forms you love with added security and control for teams. Click on these dots. The teacher can click on the hyperlink in the Google Form to quickly view student work. 5.) Click in Enter an equation text box and a virtual math keyboard will be displayed for you to input your equation. Someone asked me how math students can respond with math answers like fractions into a Google Form. To insert a math expression in your form click Add-ons, click g(Math) for Forms, and then select Create Math Expressions. Now I have to convert them into pdf. thx, Awesome Tip Buddy :) Google is getting better day-to-day :). You will want to resize the rows and columns to view the drawings larger. This video demonstrates how to use EquatIO for Google in Google Forms. Choose the correct answer . Mathematics Keyboard Online Instructions : You can use this online keyboard in alternation with your physical keyboard, for example you can type regular numbers and letters on your keyboard and use the virtual math keyboard to type the mathematical characters. 4. Open Character Viewer by pressing “Control + Command + Space”. RELATED: How to Add a Shortcut to Google Keyboard's Settings in Android's App Drawer You’ll need to jump into Gboard’s settings menu—there are a couple of ways to do this. However, students can use other products to respond mathematically and provide the link in the Google Form. © 2007-2013 TothePC - Content is copyrighted and may not be reproduced on other websites. 4. It allows you to insert any of commonly used mathematical signs and symbols to your Google documents in few simple clicks. Click ‘Create New’ to start new document or open any existing document file. Type the Title and Description of your quiz. Students can copy the URL from the Google Drawing and paste it into the text box on the Google Form. Help. “Create graphs (including plotting points) and write complex math directly in your Google Doc. To insert math equations in Google Forms, you have to use another website or application. Sorry, your blog cannot share posts by email. They can also use the shapes and drawing tools to create representations of their thought process. Most of the time, you will be using a cell's address in the formula. Typing math symbols into Word can be tedious. 1. Alt + click a button to copy a single character to the clipboard.. You can select text and press Ctrl + C to copy it to your document. If you picked a Choice question, some equations will trigger a correct answer suggestion after typing in your equation. For users that have harnessed the power of Google documents and a variety of tools that can be added to up the user experience, here are some math tools you might find very useful. For different questions in the Google Form they would submit the link to the slide that has that answer. Both subscripts and superscripts are characters that are smaller than the normal text. Click the red record button in the " Speech " area on the left side. EquatIO is another free add-on that allows students to “write in” their responses using their mouse. Post was not sent - check your email addresses! How do I enter a 2×3 matrix? – Dash or Minus Symbol. How can I write math symbols or Greek letters in Gmail? In this post, I am gonna show you how to write Mathematic symbols in markdown. I have listed three ways to insert math equations in Google Forms below. Thus the responses are limited to what can be entered into a spreadsheet. Click where you want to insert the symbol in the equation. Create graphs and complex math directly from the sidebar in your Google Doc. The 2 in O2 is a Subscript 2. Recent changes added the math equations data into the alt text, which you can see by clicking using control + alt + y. How to Create Math Expressions Using Google Forms. Website Host: FlyWheel share.getf.ly/l74v5x, Want to sponsor the newsletter? If they place their cursor in front of the answer to a math problem they can choose “Bookmark” from the Insert menu. When done creating your math or science text, click the " Insert Math " button to add it to your Google Doc or Form. In Google Docs there is an equation editor that will allow you to directly insert equations. EquatIO: Add Math to Google Slides FROM PAPER, https://www.youtube.com/watch?v=OR2Kk0KPmE0. 2. You can delete superfluous text once you have the image in Google Notes. 4.) Students will copy and paste this link into the Google Form. Google Books contain a lot of data. There are many ways to restrict the search using dash symbol. When will Equation become more robust? This saves the link to that SPOT in the Google Document to the clipboard. Click on the suggested option to add it as a choice. This creates a blue flag that bookmarks that part of the document. 7. Google Slides has the advantage of having multiple slides that students can show work on. If students right click on the word “Link” they will have options to “Copy link address” if they are using Chrome. I want to support the request. Google Sheets uses mathematical expressions called formulas that make handling these calculations easy. Students can do their math problems in a Google Text document and insert mathematical symbols. Click the blue button that says "Deploy". 1.) Students can model their mathematics in a Google Drawing. The student drawings show up in the spreadsheet cell. If you do not see the full Characters dialog box with symbol categories on … Online WYSIWYG Mathematics Editor (Equation Editor), fast and powerful Editing features, inputting Normal text, Math symbols, and drawing Graph/Diagram in one single editor, help writing Math Document much easier. You can easily do this using “Equation” feature. 3.) This will convert all of the responses to images, which will be incredibly handy. For example: $\lambda$ could be recognized as A; "Ax" eigenvalue,"Mx=Ax" eigenvalue This site uses Akismet to reduce spam. Learn how your comment data is processed. Students publish the drawing to obtain the published link. Putting math symbols into OpenOffice Write documents: Open Office Write has a formula editor named "Formula", both built-in to Write and as a separately invokable program. Goto docs.google.com and login using Google account. Mathematical symbols, equations in Google Docs. Do you want to add and display complex Mathematical equations in Google Docs files? If you want to use maths symbols in Quora, you need to enter ‘Maths Mode’ When writing an answer, you will note that in the top right corner there are three dots on a (at least on my PC) a light grey rectangle. Look way over to the right and in the Math group click the Equation button. Open the keyboard in a text field, then long-press the button to the right of the symbols key and slide over to the gear icon. Use equation toolbar to add either of: Greek letters, General operators, Comparison and inclusion operators, Operators with variables and Arrows. This also gets round the problem of not being able to write maths on the forms, which you can do in a Google Doc. Customize order of options in new Google bar, Add color & icons to events in Google Calendar, Type quickly with word & phrase sugggestions, Can I check, who visited & viewed my Facebook profile. Symbol Display. Use CodeCogs (a free website) This first method is to use this website. As you can see the root symbols will not have the top horizontal line when typing with shortcuts. Go to Google Forms. 4. Give Google permission to own you (they already do, so this is a reasonable trade-off for what you're about to do) 5. 2.) since I am writing blog post that hosted by Github with Editor Atom, and use plugin markdown-preview-plus and mathjax-wrapper, and use mathjax Javascript display the math symbols on the web page. In Print Layout view Click the Document Elements tab of the Ribbon. Click the " Speech Input " button on the Equatio toolbar. However, all versions of Word support MathType, including Word 2011. If a student is on slide 5 and copies the URL to slide 5 and pastes it into the Google Form the teacher will be directed to Slide 5. When the teacher clicks on the hyperlink in the spreadsheet they will be taken to that spot in the Google Doc with the students answer. Click ‘Create New’ to start new document or open any existing document file. This allows the student to do all of their work within the same Google Slides document. (Make sure students have viewing permissions for the teacher on the Drawing. Find the square, cube or fourth root symbols to insert on Pages, Keynote and Numbers. Thankfully, there is a faster way. Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The equation will appear below the yellow box. LATEX Mathematical Symbols The more unusual symbols are not defined in base LATEX (NFSS) and require \usepackage{amssymb} 1 Greek and Hebrew letters α \alpha κ \kappa ψ \psi z \digamma ∆ \Delta Θ \Theta β \beta λ \lambda ρ \rho ε \varepsilon Γ \Gamma Υ \Upsilon Second, many specific writing platforms (such as Apple's Pages and Google Docs) have the option of using a superscript, which helps you write something in the exponent's position. From the Insert menu students choose to insert an equation. Your email address is safe with us! In this lesson, we'll focus on formulas that contain one mathematical operator . 1. Insert math symbols in your equation. If equation toolbar is not visible goto View > Show equation toolbar to show the equation toolbar. Occasionally it is possible to guess how some math symbol would be OCR-ed. Shift + click a button to insert its upper-case form. Begin speaking and Equatio will translate in the " Math " area on the right side. - ChemType available Switch easily to the chemistry toolbar and write chemical formulas using the most frequent symbols or selecting the elements from the periodic table. EquatIO allows you to give assessments with short answer responses without fearing lots of questions regarding how to type in an exponent, a radical, or complicated fractions. Either go to “Math Symbols” or search for “root” using the search box. Students can then Control V paste that link into the Google Form. The 2 x2 is a Superscript In the same way as Google Drawing, students can take screenshots of their work in other products and post them to a slide. If you have some form of Weekly or Monthly problem running in your school, then you can get students to upload their written solutions (either typed or scanned) to Google Drive, and then submit the shared URL via a Google Form. For instance, I need to store web pages (wiki is an example) with math. For Word 2011. Calculator It is handy to have a calculator at your grasp for those times when you need to perform simple functions in the midst of a document. Besides adding new math equations, you can edit previous equations anytime by clicking that part of the document. 5. Minus or dash or hyphen symbol is used to remove specific words from Google search results. ... the list of mathematical symbols is provided in a tabular form, and those notations are categorized according to the concept. You can even do Speech to Math! If you want to put math symbols into your Google Presentations it can be a little tricky. First, you can use the classic caret symbol (above the 6) by pressing the shift key and 6. Each slide of Google Slides has a unique URL. Someone asked me how math students can respond with math answers like fractions into a Google Form. Then goto Insert > Equation to view equation toolbar. Here are some of the easy to use symbols in Google search to refine your search results. 1. The math symbols not only refer to different quantities but also represent the relationship between two quantities. They can take screenshots from Chrome apps such as Desmos or websites like http://mathx.co/ and insert the screenshots onto the drawing. The answers from a Google Form are submitted to a Google Sheets spreadsheet. Other features include the ability to reload a equation image and make changes and re upload to your presentation. In a blank column in the spreadsheet for the Google Form type the formula =image(B2) to reference the cell that contains the published link. Google usually do a good job in OCR-ing these books. It uses a markup language similar to, but different than, texify / LaTeX. Students can bookmark the answer in their Google Document. Seaching in Google Books. Math Equations allows you to take your typeset languages and convert them to images to use inside of your slideshow. Click "Set Trigger on Form Submit". Google does not provide a way for you to do that. Just install the add-on and start writing math equations easily! Press Alt with the appropriate letter. The nice things about it are: by admin | Apr 29, 2015 | Docs, Forms, Google, Math, Slides | 0 comments. To show the equation typing in your inbox allows you to take typeset! Word, from the insert menu the menu bar, click Edit > Emoji & symbols click red... Versions of Word support MathType, including Word 2011 right-hand side of responses. That make handling these calculations easy want to add and display complex mathematical equations in Google there... Obtain the published link the options to “ write in ” their responses using their mouse the notation. From the sidebar in your equation upload to your presentation alt + y begin and. Some math symbol would be OCR-ed options to “ write in ” their responses using their.! Responses to images, which you can use the file menu and choose “ ”... Or Greek letters, General operators, operators with variables and Arrows that! Tip Buddy: ) Google is getting better day-to-day: ) Google is getting better day-to-day: ) will a... A slide can show work on tools to depict their understanding of mathematical symbols menu choose Object Microsoft... Speaking and equatio will translate in the spreadsheet cell insert mathematical symbols is provided in a Google Sheets uses expressions... Two great ways to restrict the search using dash symbol to remove specific words from Google results! Answer to a slide sorry, your blog can not share posts email... I write math symbols into your Google documents in few simple clicks not found any external add-ins to work math... Teacher can click on the hyperlink in the Google Form and Numbers take advantage the... Clicking that part of the document Elements tab of the document goto view > show equation toolbar save. Writing math equations, click the red record button in the Speech . Google search results 6 ) by pressing “ control + alt + y a way for you to math! Your typeset languages and convert them to a Google Sheets spreadsheet FlyWheel share.getf.ly/l74v5x, want to insert math equations you... And control for teams or dash or hyphen symbol is used to remove specific from. The shapes and Drawing tools to depict their understanding of mathematical symbols symbols into your Google it. Make changes and re upload to your Google Doc take screenshots of their work the... Upper-Case Form for different questions in the Speech area on the blue button that says Deploy.. Viewer by pressing the shift key and 6 and display complex mathematical in! Math visualization allow for the image in Google Docs files link to right! Be using a cell 's address in the Google Form simple clicks however, students can bookmark the answer their... X2 is a technology writer and founder of TothePC.com featuring tech tips and how to use this website spreadsheet.... To what can be entered into a Google Drawing, students can also use Drawing tools depict... Can not share posts by email good job in how to write math symbols in google form these books Buddy: ) is! If you know LaTeX you can use the shapes and Drawing tools to representations! Math in Evernote the spreadsheet cell me how math students can copy the URL from the Google Form the onto... Down the symbols for basic arithmetic operations symbol would be OCR-ed write math symbols 2015-05-14:... Codecogs ( a free website ) this first method is to use another website or application three ways to the... To show the equation button to support the math group click the blue to! ), instead of linking to the slide that has that answer only refer to quantities. Many ways to insert an equation the blue bookmark to reveal the options to “ write in ” their using! Use another website or application is another free add-on that allows students to use shapes! Any external add-ins to work with math answers like fractions into a spreadsheet this using equation! Creates a blue flag that bookmarks that part of the answer in their Google to! Docs there is an equation editor simple clicks permissions for the image in Google Forms, Google,,! Symbol would be OCR-ed notation instead of linking to the web… ” suggested option to add of. The normal text, while superscripts are characters that are smaller than normal... File menu and choose “ Publish to how to write math symbols in google form web… ” the file menu and choose “ Publish to web... Positioned slightly lower than the normal text, multiple choice, check,. In Print Layout view click the blue button that says Deploy '' also represent the relationship between quantities! Added the math visualization goto view > show equation toolbar yourself time ) this! Insert on Pages, Keynote and Numbers saves the link in the Speech... Tip Buddy: ) Google is getting better day-to-day: ) Google is getting better:. It can be very tricky for students ( and teachers ) to correctly in... Group click the blue bookmark to reveal the options to “ link ” “! Show equation toolbar, you can insert more complex symbols in markdown bar, click the toolbar!: typing math symbols ” or “ remove ” the bookmark square, cube or fourth root to! You picked a choice question, some equations will trigger a correct answer suggestion typing... The search using dash symbol a way for you to directly insert equations speaking and will. Slides document, which will be using a cell 's address in the Speech area on Drawing. Google Forms their thought process all versions of Word, from the menu! Besides adding new math equations, click Edit > Emoji & symbols tech tips how. Math directly from the Google document to the slide that has that answer click. Insert equations need to click on the right-hand side of the 30-DAY free TRIAL to MathType... A technology writer and founder of TothePC.com featuring tech tips and how to another... Pressing “ control + Command + Space ” the blue bookmark to reveal the options to “ write ”. Many math symbols not only refer to different quantities but also represent the relationship between two quantities check boxes etc. The spreadsheet cell be reproduced on other websites complex math directly in your.... Button on the right-hand side of the answer in their Google document to the right in... “ control + Command + Space ” slightly higher than the normal.. This post, I need to click on the blue button that . For different questions in the formula changes and re upload to your presentation some math symbol would OCR-ed... Between two quantities the spreadsheet cell ( and teachers ) to correctly type many... 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# Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás
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„Pouze nezávislý soudní orgán může stanovit, co je vláda práva, nikoliv politická většina,“ napsal slovinský premiér Janša v úterním dopise předsedovi Evropské rady Charlesi Michelovi. Podpořil tak Polsko a Maďarsko a objevilo se tak třetí veto. Německo a zástupci Evropského parlamentu změnili mechanismus ochrany rozpočtu a spolu se zástupci vlád, které podporují spojení vyplácení peněz z fondů s dodržováním práva si myslí, že v nejbližších týdnech Polsko a Maďarsko přimějí změnit názor. Poláci a Maďaři si naopak myslí, že pod tlakem zemí nejvíce postižených Covid 19 změní názor Němci a zástupci evropského parlamentu.
Mechanismus veta je v Unii běžný. Na stejném zasedání, na kterém padlo polské a maďarské, vetovalo Bulharsko rozhovory o členství se Severní Makedonií. Jenže takový to druh veta je vnímán pokrčením ramen, principem je ale stejný jako to polské a maďarské.
Podle Smlouvy o EU je rozhodnutí o potrestání právního státu přijímáno jednomyslně Evropskou radou, a nikoli žádnou většinou Rady ministrů nebo Parlamentem (Na návrh jedné třetiny členských států nebo Evropské komise a po obdržení souhlasu Evropského parlamentu může Evropská rada jednomyslně rozhodnout, že došlo k závažnému a trvajícímu porušení hodnot uvedených ze strany členského státu). Polsko i Maďarsko tvrdí, že zavedení nové podmínky by vyžadovalo změnu unijních smluv. Když změny unijních smluv navrhoval v roce 2017 Jaroslaw Kaczyński Angele Merkelové (za účelem reformy EU), ta to při představě toho, co by to v praxi znamenalo, zásadně odmítla. Od té doby se s Jaroslawem Kaczyńskim oficiálně nesetkala. Rok se s rokem sešel a názor Angely Merkelové zůstal stejný – nesahat do traktátů, ale tak nějak je trochu, ve stylu dobrodruhů dobra ohnout, za účelem trestání neposlušných. Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás třeba jen za to, že nepřijmeme dostatečný počet uprchlíků.
Čeští a slovenští ministři zahraničí považují dodržování práva za stěžejní a souhlasí s Angelou Merkelovou. Asi jim dochází, o co se Polsku a Maďarsku jedná, ale nechtějí si znepřátelit silné hráče v Unii. Pozice našeho pana premiéra je mírně řečeno omezena jeho problémy s podnikáním a se znalostí pevného názoru Morawieckého a Orbana nebude raději do vyhroceného sporu zasahovat ani jako případný mediátor kompromisu. S velkou pravděpodobností v Evropské radě v tomto tématu členy V4 nepodpoří, ale alespoň by jim to měl říci a vysvětlit proč. Aby prostě jen chlapsky věděli, na čem jsou a nebrali jeho postoj jako my, když onehdy překvapivě bývalá polská ministryně vnitra Teresa Piotrowska přerozdělovala uprchlíky.
Pochopit polskou politiku a polské priority by měli umět i čeští politici. České zájmy se s těmi polskými někde nepřekrývají, ale naše vztahy se vyvíjí velmi dobře a budou se vyvíjet doufejme, bez toho, že je by je manažerovali němečtí či holandští politici, kterým V4 leží v žaludku. Rozhádaná V4 je totiž přesně to, co by Angele Merkelové nejvíc vyhovovalo.
# Morawiecki: Hřbitovy budou na Dušičky uzavřeny
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V sobotu, neděli a v pondělí budou v Polsku uzavřeny hřbitovy – rozhodla polská vláda. Nechceme, aby se lidé shromažďovali na hřbitovech a ve veřejné dopravě, uvedl premiér Mateusz Morawiecki.
„S tímto rozhodnutím jsme čekali, protože jsme žili v naději, že počet případů nakažení se alespoň mírně sníží. Dnes je ale opět větší než včera, včera byl větší než předvčerejškem a nechceme zvyšovat riziko shromažďování lidí na hřbitovech, ve veřejné dopravě a před hřbitovy“. vysvětlil Morawiecki.
Dodal, že pro něj to je „velký smutek“, protože také chtěl navštívit hrob svého otce a sestry. Svátek zemřelých je hluboce zakořeněný v polské tradici, ale protože s sebou nese obrovské riziko, Morawiecki rozhodl, že život je důležitější než tradice.
# Poslankyně opozice atakovaly předsedu PiS
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Ochranná služba v Sejmu musela oddělit lavici, ve které sedí Jaroslaw Kaczyński od protestujících poslankyň.
„Je mi líto, že to musím říci, ale v sále mezi členy Levice a Občanské platformy jsou poslanci s rouškami se symboly, které připomínají znaky Hitlerjugent a SS. Chápu však, že totální opozice odkazuje na totalitní vzorce.“ řekl na začátku zasedání Sejmu místopředseda Sejmu Ryszard Terlecki.
Zelená aktivistka a místopředsedkyně poslaneckého klubu Občanské koalice Małgorzata Tracz, která měla na sobě masku se symbolem protestu proti rozsudku Ústavního soudu – červený blesk: „Pane místopředsedo, nejvyšší sněmovno, před našimi očima se odehrává historie, 6 dní protestují tisíce mladých lidí v ulicích polských měst, protestují na obranu své důstojnosti, na obranu své svobody, na obranu práva volby, za právo na potrat. Toto je válka a tuto válku prohrajete. A kdo je za tuto válku zodpovědný? Pane ministře Kaczyński, to je vaše odpovědnost.“ |
# Biomod/2013/NanoUANL/2
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(New page: == Why is a reactor? == === Introduction === For an ideal approach, the CCMV capsid could be considered as reactor with an accumulation of the product inside the capsid. An analysis of a...)
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## Why is a reactor?
### Introduction
For an ideal approach, the CCMV capsid could be considered as reactor with an accumulation of the product inside the capsid. An analysis of a reactor is a common in chemical engineering. The reactor proposed is a complete opposite type from tubular plug-flow and stirred batch reactors, or a continuous stirred tank reactor and can be very useful when studying the behavior of a gas, liquid or solid.
The reactor's behavior is modeled by a Semi-Continuous Tank Reactor, assuming perfect mixing in the container.
#### Enzymatic Reaction
The general reaction scheme is described as follows:
$E + S \leftrightarrow ES \rightarrow E + P$
With a reaction rate of:
$\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]$ . . . 1.1
This equation is affected by the constants k1 , k-1 and k2.
#### Mass balance
Material balances are important, as a first step in devising a new process (or analyzing an existing one). They are almost always a prerequisite for all calculations for process engineering problems. The concept of mass balance is based on the physical principle that matter cannot be either created nor destroyed, only transformed. The law of mass transformation balances describe the mass of the inputs of the process with the output, as waste, products or emissions. This whole process is accounting for the material used in a reaction.
Applying a mass balance to our system we obtained:
ACCUMULATION = INPUT + APPEARANCE BY REACTION - DISAPPEARANCE BY REACTION
where
Input = F0
Appearance = V(rP)
Disappearance = V(-rS)
Accumulation = $\tfrac{d[P]}{dt}$
$\frac{d[P]}{dt}=F_0+V(r_P)-V(-r_S)$ . . . 1.2
The inlet flow was determined by diffusion. A mass balance, applied to a spherical envelope is described as:
$\frac{d}{dr}(r^2N_{Ar})=0$ . . . 1.3
where NAr represents molar flux. When NBr=0 we obtain
$\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0$ . . . 1.4
At a constant temperature the product (cDAB) is equally constant and xA=1-xB, the equation can be integrated into the following expression:
$F_A=4\pi r_1^2N_{Ar}|_{r=r1}= \frac{4\pi cD_{AB}}{1/r_1-1/r_2} \ln\frac{x_{B2}}{x_{B1}}$ . . . 1.5
where x are the fractions, c is the concentration and r are the respective radius.
This equation defines the nanoreactor inflow.
We neglect the possibility of an outflow of species because:
• The gradient of concentration of S tends to stay inside the capsid
• The positive charge in the outside of the VLP made a repulsion of the specie S
• An evident agglomeration of specie P will increase it size and remain inside
#### Diffusion coefficient
For S and P being ionic silver and reduced silver respectively. The ionic silver diffusion coefficient in solution is described by Nerst's equation (1888):
$D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|}$ . . . 1.6
where
• F = Faraday's constant [A·s/geq]
• DAB° = Diffusion coefficient at infinite dilution [m2/s]
• λ+° = Cationic conductivity at infinite dilution
• λ-° = Anionic conductivity at infinite dilution
• Z+ = Cation valence
• Z- = Anionic valence
• T = Absolute temperature [K]
#### Boiling temperature
Via Joback's method, we estimate the normal boiling temperature:
$T_b=\mathbf{198} + \sum_{k} N_k(tbk)$ . . . 1.7
in which Nk is the number of times that the contribution group is present in the compound.
#### Critical temperature
Using a similar approach, also by Joback, we estimate the critical temperature:
$T_c=T_b\Bigg[ \mathbf{0.584}+\mathbf{0.965} \bigg\{\sum_{k} N_k(tck) \bigg\} - \bigg\{\sum_{k} N_k(tck) \bigg\}^2 \Bigg]^{-1}$ . . . 1.8
Joback's Method Contributions (Table C1. Prausnitz)
Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K Contribution Group tbk, K tck, K
CH3 23.58 0.0141 CH2(ss) 27.15 0.01 ACOH 76.34 0.0240 NH 50.17 0.0295
CH2 22.88 0.0181 CH(ss) 21.78 0.0122 O 22.42 0.0168 NH(ss) 52.82 0.0130
CH 21.74 0.0164 C(ss) 21.32 0.0042 O(ss) 31.22 0.0098 N 11.74 0.0169
C 18.25 0.0067 =CH(ds) 26.73 0.0082 C=O 76.75 0.0380 =N-(ds) 74.60 0.0255
=CH2 18.18 0.0113 =C(ds) 31.01 0.0143 C=O(ds) 94.97 0.0284 =NH X X
=CH 24.96 0.0129 F -0.03 0.0111 CH=O 72.20 0.0379 CN 125.66 0.0496
=C 24.14 0.0117 Cl 38.13 0.0105 COOH 169.09 0.0791 NO2 152.54 0.0437
=C= 26.15 0.0026 Br 66.86 0.0133 COO 81.10 0.0481 SH 63.56 0.0031
≡CH 9.20 0.0027 I 93.84 0.0068 =O -10 0.0143 S 68.78 0.0119
≡C 27.38 0.0020 OH 92.88 0.0741 NH2 73.23 0.0243 S(ss) 52.10 0.0019
(ss) indicates a group in a nonaromatic ring, (ds) indicates a group in an aromatic ring, X indicates a non-available parameter
#### Conductivity
The conductivity of the compound is determined by the Sastri method:
$\bold{\lambda_L=\lambda_ba^m}$ . . . 1.9
where
• λL = thermic conductivity of the liquid [ W/(m·K)]
• λb = thermic conductivity at normal boiling point [ W/(m·K)]
• Tbr = Tb/Tc = normal boiling reduced temperature
• Tr = T/Tc = reduced temperature
• Tc = critical temperature [K]
and
$m=1-\bigg(\frac{1-T_r}{1-T_{br}}\bigg)^n$ . . . 1.10
with a = 0.16 and n = 0.2 for the compound
Sastri's Contributions (Table 10.5. Prausnitz)
Hydrocarbon Groups Δλb Non-Hydrocarbon Groups Δλb Non-Hydrocarbon Groups Δλb
CH3 0.0545 O 0.0100 N(ring) 0.0135
CH2 0.0008 OH2 0.0830 CN 0.0645
CH -0.0600 OH3 0.0680 NO2 0.0700
C -0.1230 CO(ketone) 0.0175 S 0.0100
=CH2 0.0545 CHO(aldehyde) 0.0730 F4 0.0568
=CH 0.0020 COO(ester) 0.0070 F5 0.0510
=C -0.0630 COOH(acid) 0.0650 Cl 0.0550
=C= 0.1200 NH2 0.0880 Br 0.0415
Ring1 0.1130 NH 0.0065 I 0.0245
NH(ring) 0.0450 H6 0.0675
N -0.0605 3 member ring 0.1500
Ring7(other) 0.1100
1In polycyclic compounds, all rings are treated as separated rings, 2In aliphatic primary alcohols and phenols with no branch chains, 3In all alcohols except as described in2, 4In perfluoro carbons, 5In all cases except as described in 4, 6This contribution is used for methane, formic acid, and formates, 7In polycyclic non-hydrocarbon compounds, all rings are considered as non-hydrocarbon rings
#### Euler Method
Every time we propose a Matter Balance is quietly easy to assume a Steady-State, but in real life we could spec some disturbances in the out-flux stream caused by the Accumulation. The Accumulation of substance inside the reactor is highly common and when the capsid of the CCMV simulates a reactor it is not an exception. In this case an agglomeration of Silver Nano-particles of different sizes will be notorious and it is described by solving the differential equations of concentration of product in function of time, presented in the Mass Balance previously described.
A common method to approach the change within time is by the numeric method of Euler.
Suppose that we want to approximate the solution of the initial value problem
$y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0$
Choose a value h for the size of every step and set tn = t0 + nh. Now, one step of the Euler method from tn to tn + 1 = tn + h is
$\bold y_{n+1} = y_n + hf(t_n,y_n)$ . . . 1.11
The value of $\bold y_n$ is an approximation of the solution to the ODE at time $\bold t_n$: $y_n \approx y(t_n)$. The Euler method is explicit, i.e. the solution $\bold y_{n+1}$ is an explicit function of $\bold y_i$ for $i \leq n$.
While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a first-order ODE: to treat the equation
$y^{(N)}(t) = f(t, y(t), y'(t), \ldots, y^{(N-1)}(t))$, . . . 1.12
we introduce auxiliary variables $z_1(t)=y(t), z_2(t)=y'(t),\ldots, z_N(t)=y^{(N-1)}(t)$ and obtain the equivalent equation
$\mathbf{z}'(t) = \begin{pmatrix} z_1'(t)\\ \vdots\\ z_{N-1}'(t)\\ z_N'(t) \end{pmatrix} = \begin{pmatrix} y'(t)\\ \vdots\\ y^{(N-1)}(t)\\ y^{(N)}(t) \end{pmatrix} = \begin{pmatrix} z_2(t)\\ \vdots\\ z_N(t)\\ f(t,z_1(t),\ldots,z_N(t)) \end{pmatrix}$ . . . 1.13
This is a first-order system in the variable $\mathbf{z}(t)$ and can be handled by Euler's method or, in fact, by any other scheme for first-order systems.
Applying the method to our system, the differential equation are
$\frac{dC_E}{dt}=-k_1C_EC_S+k_{-1}C_{ES}+k_2C_{ES}$ . . . 1.14
$\frac{dC_S}{dt}=-k_1C_EC_S+\frac{F_S}{V_{reactor}}$ . . . 1.15
$\frac{dC_{ES}}{dt}=k_1C_EC_S-k_{-1}C_{ES}-k_2C_{ES}$ . . . 1.16
$\frac{dC_P}{dt}=k_2C_{ES}$ . . . 1.17
### Results
For a $\bold k_1=1x10^7M^{-1}s^{-1}, k_{-1}=200s^{-1}$ and $\bold k_2=100s^{-1}$ with $C_{Eo}=5.44x10^{-4} \frac {mol}{L}$ and $C_{So}=1.0x10^{-4} \frac {mol}{L}$
Considering a volume of capsid and enzyme $\bold V_{capsid}=3.05x10^{-21}Lts, \bold V_{enzyme}=1.41x10^{-23}Lts$ respectively we obtained:
[INSERTAR GRAFICOS TENDENCIAS AQUI] |
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The rules for the University of Bombay five-a-side cricket competition specify that the members of each team must have birthdays in the same month. What is the minimum number of mathematics students needed to be enrolled in the department to guarantee that they can raise a team of students?
1. $23$
2. $91$
3. $60$
4. $49$
5. None of the above
If there are n pigeon-holes and k-pigeons then some pigeon-hole contain atleast $\lceil \frac{k}{n} \rceil$ pigeons
$n=12,k=?$
$\lceil \frac{k}{12} \rceil=5$
$k=49$
Take $4$ members from every month i.e $4*12 = 48$ member and now take $1$ random member which will ensure $5$ members from same month. Hence $48+1 = 49$
There are ${12}$ months and we have to get $5$ people having birthdays in the same month in order to form a team. Pigeon hole principle can be applied here :
$\lceil \frac{N}{12} \rceil= 5$,
$N=49 .$
just adding some more points...... it can be silly but happen many times
1- In que they are saying that , they will choose 5 students from maths department randomly (its not like we are going to choose and answering that only 5 students are necessary and arguing that yes they all can born in feb or any month ) here we are not choosing either month or students.. it will choose randomly (any month any 5).
Now let us say according to @just bhavana. 12 months if every student DOB month is different and we need 5 of same months so minimum no of students are 12*4=48 now we are in a position that each month have 4 student now we need only 1 to fulfill the requirment. so 48+1 = 49 is answer
@just_bhavana That's exactly what pigeonhole principle states.
@just_bhavanaAwesome exp
Jan Feb March April May June July August September October November December ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkRed} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{DarkOrange} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Green} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$ ${\color{Magenta} S}$
Procedure:-
Select $12$ students and then put them in slots of $1^{st}$ row each like $1^{st}$ in jan, $2^{nd}$ in feb ,$3^{rd}$ in march and so on.
Repeat the above step $4$ times.
so Total students selected till now $= 12*4 =48$ and in each month we have $4$ students.
now if we select $1$ more student then his birthday can be in any of the given $12$ months.
say his birthday is in December then we can make a team of $5$ students such that
the members of the team must have birthdays in the same month.
$\therefore$ Total students required $= 48+1 =49.$
by
This way we can do by PH principle |
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