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# Chapter 3 METHODOLOGIES

The surveys were carried out in the autumn of 2021, to collect lithological, structural data and RBF rock samples. Geological data during the field activity were recorded through the Fieldmove app, to be processed later in Qgis. Structural data instead were processed with Tectonics FP to produce stereo plots. Twelve oriented samples from the fault zone were collected to document and analyze the microstructures of the RBF rocks that accommodate the displacement of the roof block. The microstructures described in chapter 5 (microscope observations) are those that appear in the samples. Particular care was taken during the sampling and transport of these rocks from the field to the laboratory to avoid disintegration. Given the fractured and brittle nature of these fault rocks, samples were impregnated and stabilized with epoxy resin. In the following chapter, all the methodologies used for the realization of this thesis work will be described in chronological order.

## 3.1 Fieldmove App

Global Positioning System (GPS) devices are commonly used to aid navigation, having grown in popularity over the past decade. Over the last few years, this has extended to smartphones and digital compasses, which are often equipped with GPS functionality. GPS is a valuable aid to navigation during fieldwork, although it is important to keep safety at the forefront. Modern technologies are created to facilitate the task. Knowledge of new programs and mastery of functions is the clue to success in any scope. What could be edited only after field nowadays can be partly automatized in the field, processing the information and thus significantly reducing the field period and simplifying the cameral period.

FieldMove is a map-based digital field mapping app for geological data capture. The app is presented in a map-centric format for use on larger touchscreen tablets. When sufficient data has been collected, you can use the drawing tools that include a virtual cursor for precision drawing to create geological boundaries, fault traces and other linework on your basemap. It is also possible to create simple polygons to show the distribution of different rock types. FieldMove relies on three sensors inside your device, a magnetometer, a gyroscope and an accelerometer. Together, these sensors can be programmed to measure the orientation of planar and linear features in the field. All three sensors are installed as standard in iPhones and iPads, but are not always present in other hardware devices. You should always check your device to ensure that all three sensors are present and that the compass and the clinometer give accurate readings before starting to collect data. You can choose to manually enter data if you do not trust the internal sensors, or if you are using a traditional hand-held compass clinometer. FieldMove supports Mapbox™ online maps and can cache this extensive online map service to work offline in the field. In addition to handling online maps, geo-referenced basemaps in MBTile or GeoTIFF format can be imported and also are available offline when working in the field.
Data can be exported as MOVE, CSV or KMZ files.

## 3.2 Quantum Gis Software (QGIS)

GIS is the acronym for Geographical Information System, often also called the Territorial Information System (SIT), an information system computerized that allows the acquisition, the recording, the analysis, the visualization and the return of information deriving from geographical data. It is a capable computer system to produce, manage and analyze spatial data by associating one or more geographic elements with each geographic element information. Through a GIS program, the user is able to view and superimpose different maps themes of a specific area, ensuring the correspondence of the geographical coordinates of the scale and, therefore, distances. Themes can be images, for example, aerial and satellite photos (called raster data) or drawings, e.g. reference points, contour lines, geological limits (vector files). A GIS stores the position of data using a projection and coordinate system that defines the geographic location of the object. The GIS can also manage data from different systems at the same time projection and reference and carry out conversions between the various systems. Unlike cartography on paper, the scale in a GIS does not indicate the display scale (in fact in whatever GIS it is the various portions of the territory can be enlarged as desired) but highlights the quality of data. Data collected in a GIS system are usually represented through cartograms or tables relating to more or less extensive areas but, unlike other computer systems, there are almost infinite possibilities of use.
In fact, GIS technology integrates 2 systems:

1. the computerized or CAD design system;

2. the relational database DBMS or Data Base Management System;

Thus allowing both to carry out the analysis and research operations of Databases, with the geographical analysis and the typical representation of the CAD. In this way there is a greater yield because it is possible to analyze a geographical area at 360 ° by simultaneously analyzing:

• Geometric data (shape, size, geographical position of the objects);

• Topological data (connection, adjacency and reciprocal relations between objects);

• Information data (numerical or textual data relating to objects)

In this way, it is possible to simultaneously analyze a geographical entity under different aspects in a more interactive, effective, and immediate way. QuantumGIS (QGIS) (www.qgis.org) is an Open Source GIS software (it is, therefore, possible to copy and distribute it in perfect legality) that allows you to view, query, modify cards, create, print and perform simple spatial analyzes. QGIS, used as an interface to the most powerful software GIS Open Source GRASS (http://grass.osgeo.org), allows you to create the most different and complex geographic analysis operations such as spatial modeling and image analysis satellite. The main advantages of QGIS are the fact of being free software, having a user-friendly interface, allowing easy analysis of spatial data, and presenting an architecture that can be adapted to different functions thanks to the use of specific plugins. One of its main capabilities is to allow users to view and overlay vector and raster data in different projections and formats without the need to convert them to internal or common files.

## 3.3 Tectonics FP

TectonicsFP for WindowsTM operating systems is a menu–driven computer program that can produce stereographic plots of orientation data (great circles, poles, lineations). Frequently used statistical procedures like calculating eigenvalues and eigenvectors, calculating mean vector with concentration parameters and confidence cone can be easily performed. Fault data can be plotted in stereographic projection (Angelier and Hoeppener plots). Sorting of datasets into homogeneous subsets and rotation of tectonic data can be performed in interactive two-diagram windows. The paleostress tensor can be calculated from fault data sets using graphical (calculation of kinematic axes and right dihedra method) or mathematical methods (direct inversion or numerical dynamical analysis). The calculations can be checked in dimensionless Mohr diagrams and fluctuation histograms. 32bit Windows application can be run on all Microsoft Windows™-platforms. Program’s General features are:

• Full MDI (multi document interface) application: open as many graphics- and editor-windows as you like

• Developed from geologists for geologists for all cases where directional data have to be processed, with emphasis on brittle tectonics

• Enter data in a database spreadsheet. Exchange data with other spreadsheet or database applications via clipboard

• Use all available Windows™ fonts, colors and line styles and choose from 5 point symbols to customize your plots

• Combine several plots to receive high and appropriate information content

• Export graphics easily via clipboard or file to other Windows™-based applications

• Print your graphics files directly from TectonicsFP with metrical scaling independently of printer used

• Multiuser support: customize your version of TectonicsFP. Continue with your private settings next time

• Number of data is not limited for most plots and depends only on available resources of your computer

• Online-help to give on-screen information

• Short manual for fast “how-to-use”-information

• Full manual with all references for plots.

## 3.4 Microstructural Analysis

Microstructural analysis has been performed by using petrographic microscope on thin sections oriented parallel to the slip direction. The analysis was aimed at the characterisation of the style of deformation (e.g., discontinuous vs. continuous, coaxial vs. non-coaxial), the associated deformation mechanism (e.g., fracturing, veining, cataclastic flow, frictional sliding), and the cross-cutting relationships, and thus a relative sequence of deformation events, among different generations of structures. Particle size distribution and particle geometry analysis have been performed on selected, representative cataclastic textures with the aim to investigate the dominant comminution mechanism. Comminution may reflect the complex and often transient interplay of numerous physical and chemical processes and mechanisms such as cataclastic flow, wear abrasion, fluid-assisted brecciation, volume dilation and collapse (i.e. J´ebrak, 1997 and references therein). Several studies suggest that the activated fragmentation process generally controls the actual particle size distribution, initial size distribution, energy input, finite strain, number of fracturing events, and the confining pressure (e. g., Scholz, 1987; Sammis et al., 1987; Sammis and Biegel, 1989; Blenkinsop, 1991; Storti et al., 2003; Billi, 2005 and 2007). In this study, we have applied the semi-quantitative approach proposed for hydrothermal breccias by J´ebrak (1997) on high-resolution images of the selected textures taken at different magnifications in order to derive fractal parameters for the Particle Size Distribution (Ds) and Morphology (Dr) of the clasts. Parameters Ds and Dr have been estimated by using programs like ImageJ and Matlab.

### 3.4.1 ImageJ

ImageJ is a computer program for digital image processing, designed with an open architecture that provides the possibility of having extensions through small "Java plugins" subprograms and many recordable macros. ImageJ can display, edit, analyze, process, save, and print gray level (8 bit, 16 bit and 32 bit) and color (8 bit and 24 bit) images. It can read many image formats including TIFF, PNG, GIF, JPEG, BMP, DICOM, FITS, as well as some "raw" formats. ImageJ supports image stacks, series of overlapping images (both spatially as "sections" of a three-dimensional body, and temporally as a sequence of two or three-dimensional images, that share a single window, and is multithreaded enabled, so some time-consuming operations can be performed in parallel on multi-CPU hardware. ImageJ can calculate area and pixel value statistics in user-defined selections and segmented objects based on intensity thresholds. It can also measure distances and angles. It can create density histograms and draw profile lines (between defined points). It supports standard image processing functions, such as logical and arithmetic operations between images, brightness and contrast adjustment, convolution, Fourier analysis, sharpening, smoothing, edge recognition and median filtering, mathematical morphology. It also performs geometric transformations such as scaling, rotation and reflection. The program interface is very simple because it has a single toolbar from which it is possible to modify the images and their color scales, transform them, process them as binaries to apply filters, analyze them mathematically and geometrically, and draw and write on them. The way we used this program was for fractal analysis of clasts in section photos in order to obtain Dr parameter. First, load the image to be processed by dragging it into the program window and set the scale by tracing the existing one obtained from the microscope photograph, selecting Analyze / set scale and entering the known distance and the unit of measurement. In this case 500 $$\mu m$$ (Fig. 3.1).

In this way the program associates the dimensions of the single pixel of the image with the relative micrometric dimensions adopted. It is then necessary to use the commands of the Image tool, first transforming the image into 8-bit to remove the colors and make everything in grayscale and then applying a threshold filter that allows the program to identify the individual within the image elements (clasts). Playing with the application bars, we make the clasts black and the background and voids white (Fig. 3.2).

After having adjusted and redefined the edges of the elements and the cracks inside them with the drawing tools, we proceeded by processing the image making it binary, filtering out the noise and opening and closing the binary filters. The image will increasingly resemble the original photo (Fig.3.3).

The image is ready for the geometric analysis, so from Analyze / set measures tool, we check all the measurement parameters that we want to obtain, both in view of the procedure necessary for the fractal analysis, and to obtain a restitution table of the parameters necessary for the calculation of the Ds parameter through Matlab. From the Analyze particles tool, setting the consideration of the edges in the bar, you start the procedure for identifying the calculation areas, which returns a .csv table and an image of the elements actually found by the program (Fig.3.4).

Through the Fractal box counter tool, you can start the fractal analysis procedure, based on the count of the number of times each clast crosses one of the edges of the square grid with an increasingly larger side processed by the software. The result of the analysis will be the plotting of a logarithmic graph that relates the square grid with the number of crossings. The angular coefficient of the regression line of the point analysis represents the parameter Dr (Fig.3.5).

### 3.4.2 Matlab

MATLAB (abbreviation of Matrix Laboratory) is an environment for numerical computation and statistical analysis written in C, which also includes the homonymous programming language created by MathWorks and appears as a programming and numerical computation platform used for analysis data, algorithm development, and modeling by combining an optimized desktop environment for iterative analysis and design processes with a programming language that expresses mathematical operations with matrices and arrays directly. Matlab was used in this data processing phase to obtain the Ds parameter starting from the .csv table, suitably processed and transformed into a .txt text file easily readable by the program, obtained through image analysis with ImageJ. The code (CATAMORPH) gives us the plotting of a power low curve defined on the basis of the diameters and the cumulative frequency of the number of existing clasts with a diameter greater than the corresponding one on the abscissas. By working only on the central part of the curve to eliminate out-of-resolution values because they are too small or too large, a single trend line is identified, whose angular coefficient is precisely the parameter Ds.


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# 2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Ex 5.5

Students can Download Maths Chapter 5 Continuity and Differentiability Ex 5.5 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka 2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Ex 5.5

### 2nd PUC Maths Continuity and Differentiability NCERT Text Book Questions and Answers Ex 5.5

Differentiate the functions given in Exercises 1 to 11 w.r.t. x.

Question 1.
cos x. cos 2x. cos 3x
Answer:
Let y = cos x. cos 2x.cos 3x
log y = log cos x + log cos 2x + log cos 3x

Question 2.
$$\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$$
Answer:

Question 3.
$$(\log x)^{\cos x}$$
Answer:

Question 4.
$$x^{x}-2^{\sin x}$$
Answer:

Question 5.
$$(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$$
Answer:
Let y = (x + 3)2 (x + 4)3 (x + 5)4
log y = 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5)

Question 6.
$$\left(x+\frac{1}{x}\right)^{x}+x^{\left(x+\frac{1}{x}\right)}$$
Answer:

Question 7.
(log x)x + xlogx
Answer:

Question 8.
$$(\sin x)^{x}+\sin ^{-1} \sqrt{x}$$
Answer:

Question 9.
$$\mathbf{x}^{\sin \mathbf{x}}+(\sin \mathbf{x})^{\cos \mathbf{x}}$$
Answer:

Question 10.
$$x^{ xcos\quad x }+\frac { x^{ 2 }+1 }{ x^{ 2 }-1 }$$
Answer:

Question 11.
$$(x \cos x)^{x}+(x \sin x)^{\frac{1}{x}}$$
Answer:
let y = $$(x \cos x)^{x}+(x \sin x)^{\frac{1}{x}}$$
u = (x cos x )x
log u = x log (x cos x)

find $$\frac{\mathrm{d} y}{\mathrm{d} x}$$of the given in exercise 12 to 15

Question 12.
xy + yx = 1
Answer:
Let u + v = 1 where u = xy and v = yx

Question 13.
yx = xy
Answer:
Take log on both sides
x log y = y log x

Question 14.
$$\mathbf{x y}=\mathbf{e}^{(\mathbf{x} \cdot \mathbf{y})}$$
Answer:
log xy = (x-y) log e
log(xy) = (x – y)
diff : on both sides

Question 15.
$$(\cos x)^{y}=(\cos y)^{x}$$
Answer:

Question 16.
Find the derivative of the function given by
f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f'(1).
Answer:
f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)
log f (x) = log (1 + x) + log (1 + x2) + log (1 + x4) + log (1 + x8)

Question 17.
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial
(iii) by logarithmic differentiation.
Do they all give the same answer ?
Answer:

Question 18.
If u, v, and w are function of x, then show that
$$\frac{d}{d x}(u, v, w)=\frac{d u}{d x} v \cdot w+u \cdot \frac{d v}{d x} w+u . v \frac{d w}{d x}$$
in two ways – first by repeated aplication of product rule, second by logarithmic differentiation.
Answer:
Let y = uvw (Logarithmic differentiation) log
log y = log u +log v + low w

error: Content is protected !!

format: plain


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# My ideal web development set up (updated Dec 2020)

Back in 2018, I wrote a post about my web development set up and how I planned to go from a local dev environment to a cloud one. In the end it didn't happen. Two years have since passed, and I've revisited my views on it. Here's how I'd go about building a new web app today.

# Design tool: Figma

I might or might not have a designer on the team. If there isn't and I have to design it myself, my design tool of choice is Figma. Many designers still use other tools like Sketch, Adobe Illustrator and some old school people even use Adobe Photoshop. Yes, I've met such people as recently as last year. But to me, they don't even come close to Figma. I might write in another article why I don't like Sketch nor Adobe Illustrator for UI/UX design. But here's why I like Figma so much:

• Web-based tool. No installation required, I know I'm running the latest version automatically, and it doesn't matter which OS I use.
• Components! Making components and instances of components is a life saver. You can change your base components, and your whole UI will update accordingly.
• Auto-layout is amazing. Aligning things perfectly is a breeze, and everything resizes automatically.
• Supports real-time collaboration and comments. This is great when collaborating with someone else.
• Supports making click-through demos. Also great to show your customer and gather feedback.

# Repository: GitHub

GitHub lets you create an unlimited number of private repositories for free. What's more, their pull request UI is great, and now you can use GitHub Actions to run automated tests on your pull requests. More on that below.

# IDE: Visual Studio Code and soon Codespaces

Visual Studio Code is awesome, full of great plugins and what's more it's free. What more to ask? How about making it web-based so that I could code remotely? I would not have to download the repository onto my computer, set up my local dev environment nor install VSC. Wait, it already exists? Oh right, it's called Codespaces. https://github.com/features/codespaces. Well it's still a closed beta, and I haven't received the invite yet. But once it's out, I'll start using that! How things have changed since 2018.

I'll use the following plugins: ESLint, Prettier. I might write a separate page on how to set up these.

# Web framework: React, Hooks in TypeScript

React with Hooks + TypeScript. I like hooks because they are much more concise than classes, and I like the way they bring related logic together with hooks (see Hooks intro). Also I like TypeScript (see TypeScript in 5 minutes) because it is typed. It helps prevent so many bugs.

No Redux. I've found that for smaller webapps, the added complexity of notions and extra layers is not worth the benefits. Especially when using Firestore, since Firestore provides real-time updates.

To create the project, I use https://create-react-app.dev/. It sets up everything for you. The boilerplate code, example tests, the build scripts... No need to mess with Webpack configs. Also it has an option to use the TypeScript template. That's what I use.

# Code quality: ESLint, Prettier

### ESLint

By default, the project created by https://create-react-app.dev/ doesn't enforce many ESLint rules. I like having those rules around. It ensures consistent practices when working with a team. I'd add the following:

• recommended ESLint rules
• TypeScript rules
• React rules
• React Hook rules
• Jest rules

See my full ESLint config below.

### Prettier

Prettier enforces style. I like that too. With this, you never have to ask a teammate to fix their spacing again. Since Prettier conflicts with some ESLint rules, they have set up special ESLint config to disable the rules in question. For example "prettier/@typescript-eslint" will disable the rules of "plugin:@typescript-eslint/recommended", that would conflict with Prettier rules.

### Final ESLint config

Btw, since the introduction of React's new JSX Transform, you don't need to always import React, so I disable two of the ESLint react rules, as documented here: https://reactjs.org/blog/2020/09/22/introducing-the-new-jsx-transform.html.

"eslintConfig": {
"extends": [
"eslint:recommended",
"plugin:jest/recommended",
"plugin:jest/style",
"plugin:react/recommended",
"plugin:react-hooks/recommended",
"plugin:@typescript-eslint/recommended",
"react-app",
"react-app/jest",
"prettier",
"prettier/@typescript-eslint",
"prettier/react"
],
"rules": {
"react/jsx-uses-react": "off",
"react/react-in-jsx-scope": "off"
}
}

## UI Components: Ant Design or Material UI

Both give a large collection of easy to use components, and are well documented. Ant Design provides more than Material UI, like components to upload files.

# Tests

Unit tests with Jest. Jest is nice. It will re-run tests as you modify your code. It will also compute code coverage. Also they support "snapshots", which is kind of a lightweight way of testing like screenshots. See https://jestjs.io/docs/en/snapshot-testing. For larger projects I might also want to write integration tests with React Testing Library, and Cypress for end to end tests. https://reactjs.org/docs/testing.html#tools

# Services

## Email: Mailgun

There are a few other services out there. I haven't tested them but Mailgun was awesome. Verifying domains, using the API, etc. was seamless, and the quotas are high enough that it's extremely cheap to use.

# Database: Firestore

I love Firestore because it provides a great API that supports real-time updates. Let's say your todo app saves a new todo when the user clicks a button. As soon as the todo is saved, your TodoList component in your app that listens to todos in the database is notified and gets rendered again. No need for Redux's centralized store.

# Server: None or Firestore functions

For simple apps, I don't even need a server. But if I have server-side tasks to do, Firestore functions might do the trick. If I really need heavy logic, I'll use Node.js, but I'll try to stay away from it if possible to reduce maintenance. Cause then I'd have to host it on a VM.

# Hosting: Firebase Hosting

They have a simple enough command line to set up environments, then upload your builds to a given environment. Pricing is good too. https://firebase.googleblog.com/2016/07/deploy-to-multiple-environments-with.html

# Continuous integration: GitHub Actions

GitHub actions are awesome, and you only need to read https://docs.github.com/en/free-pro-team@latest/actions/quickstart and https://docs.github.com/en/free-pro-team@latest/actions/learn-github-actions/introduction-to-github-actions to start writing your own workflow.

GitHub actions will run workflows on certain triggers. Mostly I want to run unit tests, ESLint and Prettier checks whenever a pull request is updated, and code is pushed to the main branch. Here are my workflows verbatim.

## Unit tests

name: unit-tests
on: [push]
jobs:
run-unit-tests:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v2
- uses: actions/setup-node@v2
with:
node-version: "12"
- run: npm install
- run: npm test

## ESLint

name: lint
on: [push]
jobs:
run-linter:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v2
- uses: actions/setup-node@v2
with:
node-version: "12"
- run: npm install
- run: npx eslint src --ext .js,.jsx,.ts,.tsx

## Prettier

name: prettier
on: [push]
jobs:
run-prettier:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v2
- uses: actions/setup-node@v2
with:
node-version: "12"
- run: npm install
- run: npx prettier --check src


text:
Article Contents
Article Contents

# Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities

• We study the asymptotic behavior of weak energy solutions of the following damped hyperbolic equation in a bounded domain $\Omega\subset\R^3$:

$\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$

where $\gamma$ is a positive constant and $\varepsilon>0$ is a small parameter. We do not make any growth restrictions on the nonlinearity $f$ and, consequently, we do not have the uniqueness of weak solutions for this problem.
We prove that the trajectory dynamical system acting on the space of all properly defined weak energy solutions of this equation possesses a global attractor $\mathcal A_\varepsilon^{tr}$ and verify that this attractor consists of global strong regular solutions, if $\varepsilon>0$ is small enough. Moreover, we also establish that, generically, any weak energy solution converges exponentially to the attractor $\mathcal A_\varepsilon^{tr}$.

Mathematics Subject Classification: Primary: 35B40, 35B45, 35L30.

 Citation:


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# COT

Chandra Bergmann

#### The COT function returns the cotangent of a number given in radians.

This function takes a single Number as input. It parses that number in radians and outputs its cotangent.

### Declaration

COT(number) -> cotangent

### Parameters

number (required, type: Number)
The number to calculate the cotangent of. This number is parsed in radians.

### Return Values

cotangent (type: Number)
The cotangent of the given number.

### Examples

The following example returns the cotangent of 45. Note that while the cotangent of 45 degrees is 1, the tangent of 45 radians is 0.6173696237835551, and this is the number the COT function returns:

COT(45) -> 0.6173696237835551

In order to calculate the cotangent of a number of degrees, said number must first be converted into radians using the function RADIANS. The following example returns the cotangent of forty-five degrees by first converting forty-five degrees into radians and then calculating the cotangent of the resulting number. Note that, due to small rounding errors, the number COT returns is not exactly one, but something very close to it. This behavior matches that of most computational calculators and should be sufficient for most practical purposes:

COT(RADIANS(45)) -> 1.0000000000000002

Mathematically, the cotangent of 0 radians is undefined, and so when 0 is given as input for the COT function, it outputs NULL, such as in the following example:

COT(0) -> NULL

### Discussion

Mathematically, the cotangent of an angle is the cosine of that angle multiples by the reciprocal of the sine of that angle. The relationship between the functions COT, COS, and SIN is the same, such that, for some number n:

COT(n) = COS(n)/SIN(n)


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# Compact expression of Maxwell's equations: is there a missing minus sign?

The compact form of Maxwell's equations:

$$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation $$\boldsymbol{\mathcal{J}}=(\bar J, ic\rho)$$. The tensor of the electromagnetic field is given by $$F_{\mu\nu}:=\frac{\partial \mathcal{A}_{\nu}}{\partial X_{\mu}}-\frac{\partial \mathcal{A}_{\mu}}{\partial X_\nu} \tag{2}$$ calculated using the four-potential $$\boldsymbol{\mathcal{A}}=\left(\bar{A}, \dfrac ic \varphi\right)$$.

It is known $$F_{\mu\nu}=-F_{\nu\mu}$$ and with several steps (which I am not reporting) I have proved that: $$$$\sum^4_{\nu=1}\frac{\partial F_{\mu\nu}}{\partial X_\nu}=\mu_0\mathcal{J}_\mu,\quad \mu=1,2,3,4. \tag{3}$$$$ Hence $$(3) \iff (1)$$. Obviously if I exchange the subscripts of the tensor of the electromagnetic field I easily get the minus sign. Infact being $$F_{\mu\nu}=-F_{\nu\mu} \tag{4}:$$ $$$$\sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\mu}=-\mu_0\mathcal{J}_\nu,\quad \nu=1,2,3,4. \tag{5}$$$$ or $$\square \,\boldsymbol{\mathsf{F}} =-\mu_0 \boldsymbol{\mathcal{J}} \tag{6}$$

The (6) and the (1) are the same things or with the minus sign it has another meaning?

• If (3) is correct, then (5) must be wrong (for $\mu_0\mathcal{J}_\nu\ne0$) since the equations differ only by an exchange of the indices, i.e. a relabeling of $\mu\leftrightarrow\nu$ and a sign.
– user178876
Jul 11 '19 at 21:10
• I have successfully erased all memory of $\mu_0$ but equation (3) can be derived very simply from the equations of motion for $A_\mu$. And by exchanging $\mu$ and $\nu$ one gets (5) up to the sign, whose origin I do not understand. if you use the antisymmetry of $F_{\mu\nu}$, then you get an equation with a minus, but where the derivative acts on the first index of $F$.
– user178876
Jul 11 '19 at 21:22
• Oh, please, do not use the $x_4 = ict$. It is so 1909, and we are in 2019... Jul 11 '19 at 22:03
• @Sebastiano you have done - implicitly - everywhere, in your definitions of four-vectors, derivatives etc. The use of $i$'s in the time-like components (and incidentally, having it as $x_4$ rather than $x_0$) is generally considered obsolete in favour of the modern geometrical/metric point of view.
– fqq
Jul 12 '19 at 15:45
• Also, $\square$ usually denotes the D'Alembert operator, which does not map 2-tensors to vectors as would be required for the equation to make sense. I can guess what you mean by $\square \overleftrightarrow F$, but AFAIK it's not standard notation, and since you are dealing with a sign problem I guess definitions are important. 9
– fqq
Jul 12 '19 at 15:46

Starting from your equation (3) we have $$$$\mu_0\mathcal{J}_\mu = \sum^4_{\mu=1}\frac{\partial F_{\mu\nu}}{\partial X_\nu} = -\sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\nu}$$$$ therefore your equation (5) should read $$$$\sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\nu} = - \mu_0\mathcal{J}_\mu$$$$ which can also be written (after renaming indices) $$$$\sum^4_{\mu=1}\frac{\partial F_{\mu\nu}}{\partial X_\mu} = - \mu_0\mathcal{J}_\nu$$$$ therefore your equation (5) has a sign error.

But perhaps what you meant to say is that you do not obtain (5) from (3) but rather you just assert that (5) is what you would get if you started out from a different equation in the first place.

I think the problem here may be that you are not aware of an assumption built into the index-free notation you have adopted in (1), concerning on which index of $$F$$ the differential operator is being employed. There must be a convention, otherwise the equation is ambiguous because $$F$$ is not symmetric.

The general point is that you cannot expect $$\partial_\mu F^{\mu b}$$ to be equal to $$\partial_\mu F^{b \mu}$$ when $$F^{ab} \ne F^{ba}$$, so any equation involving such quantities has to pay attention to whether the derivative is taken on the first or the second index.

In order to derive the inhomogeneous Maxwell equations, you do not need to use the antisymmetry of the field strength tensor. They follow from the equations of motion for $$A_\mu$$. If you start with $$\mathscr{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e J^\mu A_\mu\;,$$ the Euler-Lagrange equations give you $$\partial_\mu F^{\mu\nu}=e J^\nu$$ up to a total derivative term that can be gauged or argued away. If you relabel $$\mu\leftrightarrow\nu$$, you get $$\partial_\nu F^{\nu\mu}=e J^\mu\;.$$ It might very well be that our conventions differ because I do not work with these $$\mu_0$$ oddities. That is, all the above assumes natural units, in which $$c=1$$. In any case, you do not get a sign when renaming the summation indices.

• You used the lagrangian and the CGS :-(. I have understood something :-) but the minus sign how do I make it appear? Jul 11 '19 at 21:54
• @Sebastiano I do not know. There are a lot of convention-dependent signs, such as the sign of e (which is negative in my conventions) or the signature of the metric (I am using $(+,-,-,-)$). All I can say is that you cannot get it by relabelling, Buona notte!
– user178876
Jul 11 '19 at 21:57
• Thank you so much for your wonderful and wonderful cooperation. It's my day here and I don't know if you're sleeping. If it were so good to rest. My most sincere greetings. Jul 12 '19 at 9:01
• Nooooooooooooo, why are you canceling from TeX.SE? Don't tell me it's a joke. You have and will have my support fully. I don't really want to believe it, you are part of my friends and not only for the help you gave me. I don't want to think at all that you are leaving. Actually, I am writing to you with the heart to think about it. I have had to face all kinds of offenses but friends first of all. You are an important person. Jul 16 '19 at 21:59
• No, no, I don't really understand. Why would they suspend you? :-( I don't see anything at all that you've done badly. Please stay. Why do good and competent people always have to flee? I don't want to believe you're leaving. Jul 16 '19 at 22:04


text:
Higher dimensions

1. Jan 31, 2005

nightcleaner

I wanted to bring this forward from another thread because I think we may be getting at an important point regarding dimensionality. The last post in the thread is quoted below:

Consider the three spatial basis lines, x, y, z. If there is velocity in x, then there is Lorenz contraction in x. The y and z basis lines are unaffected by velocity in x.

Real objects are always extended in all three spatial basis lines, as well as in time. For this reason every part of any real object will experience contraction due to velocity. Contraction in x will be distributed uniformly throughout the object. The sphere of the object (obtained by rotating an object through every possible rotation at every possible point in the object) will be flattened in x.

Since real objects are always extended in x, y, z, and t, we may propose the generalization that real objects are always extended in every available basis line. If there are extra dimensions which cannot be expressed using combinations of x,y,z, and t, then every real object will have some extension in the basis line of that dimension.

Now when we speak of dimensions as labled $${x_1, x_2, x_3...x_n}$$ we may presume that x,y, and z are included in the innumeration, so that perhaps x is the same as $$x_1$$, y is the same as $$x_2$$ and so on. Since all the extra dimensions in string theory are said to be spatial, I am presumeing that t is a special case and is not included in the set as an $$x_n$$.

Then since there is no distinction made as to which subscript is to represent which basis line (x could as well be $$x_2$$ and y could be $$x_1$$ we must assume that all members of the set of $$x_n$$ are equivalent, so that each $$x_n$$ is a basis line. Then it follows that if velocity is in the direction of any x, there is contraction in that x, but none of the other members of the set is contracted.

This is to say that, if the above reasoning holds, every subscripted x is perpendicular to every other subscripted x. That is, they form an orthagonal set. We cannot easily imagine an orthagonal set of spatial basis lines with more than three members, but it seems apparent that this is what we mean when we speak of higher dimensionalities.

However if we relax the rule that orthagonal axis must be at 90 degrees to each other, we can imagine more than three axis in ordinary space. We can build a representative structure containing more than three axis, with the rule intact that each axis represents an independent dimension, which does not experience contraction when motion occurs along one of the other basis lines.

In fact, we can represent higher dimensionalities on a two dimensional surface by drawing a circle and dividing the circumference into 2n points, where n is the number of dimensions required. For a three dimensional space, this results in the familiar figure called a hexagram, in which the cord of the endpoints is equal to the radius of the circle.

In this scheme, motion along one axis will result in contraction along that axis, but no contraction along any other axis. If motion could be carried to the extreme, the axis would contract to a point, and the circle would divided into a figure shaped like the infinity sign. If we were carrying out this operation on a model in three spatial dimensions, the result would be a torus.

This action would be equivalent to adding a hole to a topological figure, or to punching a hole through the spacetime fabric. Of course we are forbidden in practical terms from doing so with any real object.

There is however another way to analyse the geometry of higher dimensions. I would like to go on from here to explore the possiblitiy that time is not a special case but is to be included in the $$x_n$$ set, which makes rational sense, since if there is motion along one of the basis lines, time must be extended along that basis. In fact, time must be extended along any basis line which can accomodate the sense of motion. This is in concordance with the principle of spacetime equivalence due to Einstein and Minkowski. Then we must consider each axis in x as containing within it an axis in t.

I hope to go on with this idea in another post. Any comments are welcome, especially those directed to challenging this line of reason.

Thanks,

Richard T. Harbaugh,
Nightcleaner

2. Jan 31, 2005

Staff Emeritus
Good idea. I have skipped the copy of my last post to address your concerns freshly.

Only if the velocity is ONLY along x. The real thing that determines is the velocity vector. If that mixes x, y,and z then there will be proportionate contractions in x, y, and z. This is where the full Lorentz matrices come in.

OK.

This is correct. If the compacted diensions are real, then we extend into them and move through them.

You are absolutely right about x, y, and z. The t coordinate is represented as $$x_0$$

Right on. Well expressed, too.

Right again! Have you considered teaching this stuff?

An example of doing this is the "tesseract" models which represent a (euclidean) four-cube. The four edges meeting at a corner are represented with three of them at right angles to each other and the fourth heading off at an acute angle. It's important to note that although such representations are useful for visualizing some relations they inevitable distort other relations. Generally speaking it's topological features that survive and metric ones that are obliterated.

Yes I suppose you could do this. I suspect if you work around with it you will find it hard to use it for details. See my comment above.

Well following up your statement that the contraction is uniform across the body (I think we are both assuming a sphere) it would become a flat disk in 3-space, and a "flat hyperball" in n-space, i.e. it would be a full ball in the (n-1)-space orthogonal to the direction of the relative velocity. I think this is a case where your reductive dimensional model leads you astray.

This is based on your torus, which is not what happens. See above.

Close but not quite. The accurate statement is that the direction of a real relative velocity is "timelike". This means the velocity vector is not orthogonal to the time dimension but has a component along it, in addition to its spatial component. So this vector has to point in the cone between the set of v=c vectors and the time axis. The fact that every physical velocity has a non-zero time component is the reason that time and space get mixed up in the Lorentz transformations. You are accustomed to see length contraction and time dilation as separate formulas, but they are each special cases of the true formula. Suppose the velocity vector (relative to an observer) has extension t along the $$x_0$$-axis and x along the $$x_1$$-axis. Then the dimensions experienced by the observer (they are denoted by t' and x') are given by

$$x' = \gamma x - \gamma\beta ct$$
$$t' = -\gamma\beta cx + \gamma t$$

where $$\beta = \frac{v}{c} and \gamma = \frac {1}{\sqrt{1-\beta^2}}$$

Last edited: Jan 31, 2005
3. Jan 31, 2005

nightcleaner

I am expecting company today and tomorrow (they will have to take universal precautions to prevent transfer of the virus which is visiting my house also) so will have small time for now to consider this post. Your comments are very welcome also, and much less discomforting than the hundred day cough, which is going around here.

I am in fact driven to teach this stuff, which is why most of my close friends groan whenever the topic comes up. They have all patiently suffered being driven into glazed states of semi-coma by my extemporaneous lectures. However in the past two years I have had more success, and even on occassion been specifically invited over to talk about physics! The magnitude of this victory can only be revealed when you know that most of my friends, honored and beloved as they are, are not physicers, nor even scientists, but farmers, laborers, loggers, and small freeholders. A rather anarchous group, we sometimes refer to ourselves as technicolor "amish," because of a preference for communal self reliance and animal husbandry, not in any religious context.

I have chores to do, but will look forward eagerly to my next opportunity to work with the formulae and corrections to my text which you have given. Thanks! (And yes, I was imagining velocity only in x, a condition which may turn out to be an ideal oversimplification. If all motion is curved, then there is no way to move in only one direction!

Be well,

Richard T. Harbaugh

4. Jan 31, 2005

Staff Emeritus
Yes I know the eye-glazed look. My daughter got it yesterday when I described our exchanges here. We can love and admire people who sadly don't share our interests.

5. Feb 3, 2005

nightcleaner

Hi.

The sky has cleared after many days of thick cloud, and the stars are very bright. Pegasus is on the low horizon, and Cassiopea not much higher. I have a couple of things on my mind tonight.

First, there is prescience. Cassiopea is a great summation sign in the heavens, and Pegasus, the great square, is a Greek delta, the sign of difference, change. Between them, like a full colon, the eyes of Draco stare:

Code (Text):

A
*                                *
*          *
*             *                        *
*          *
*                                 *
B
Andromeda, our neighbor galaxy, sits at A.

The delta, or great square, is the sign of difference, and might be thought of in mathematical symbols as we use them today either as the minus sign, for subtraction, or as the cross multiplication sign, X. The Summa sign of Cassiopea means addition, and Draco, between them, is the indicator for a ratio. What is the ratio of the difference to the sum? And is that not a strange and beautiful symmetry? One wonders what might be found inthe region of B.

But enough dreaming.

I want to think about the light cone for a moment. What is the light cone, physically?

Well, we consider an event, let us say it is a flash of light in an otherwise empty space. It has an origin (my friend Guadalupe has thought a lot about origins, and I am wondering what he would think about this? Is he awake tonight?) at a place. This means that we could think of the first instant of the flash of light as a very small region of space which contains the light, and which is contained itself in a very large region in which the light is not present.

As time passes, the light expands. It starts out in a very small region of the available space, and ends up filling the available space. In between the beginning and the end, it smoothly increases in size. We may wish to examine this process piecewise along a line in time. In order to do so, we lay out a line of time as if it were a line in space, and then consider the expansion in the other two available spacelike dimensions.

Two spacelike dimensions makes a surface, and since we have already said that the space is otherwise empty, we may assume that the surface is Euclidian, a flat plane, like an unbounded piece of paper. Well we don't need to consider infinity just yet, so it is ok to think of it as a piece of paper with bounds, that is edges, if we so wish. Let us just say that the surface is large enough to contain the flash of light from its beginning to its ending. And, as my friend Guadelupe would tell you, everything which has a beginning must also end.

Since we have laid out a line of time along one of the available spacelike dimensions, and we are considering the expansion in the other two available dimensions, the expansion at any instant looks like a circle on the piece of paper. Now we need to stack up really a lot of pieces of paper, in order, to see the cone. Let the origin of the light be a very small circle, and the next page a very sligntly larger circle, and the next page a slightly larger circle, and so on, and after a few billion billion pages, we see the cone developing.

There are several conditions we must fullfull to see the cone develop. There is the substitution of a time line for a line in space. There is the division of the time line into small equal increments, the thickness of the pages. There is the origin, which must be held in the same place from page to page. And we must remember that the page, flat as it is, is only part of the space in which the flash occurs, since we have deleted the third dimension in space in order to see the development of the cone in time. We see on each page a two dimensional circle, but we must remember that really the circle represents a three dimensional sphere, just that we have flattened it to see it stack up as a cone over time.

Ok, I believe these are the necessary and sufficient conditions for a light cone. Now let us discuss the origin for a moment, the first page.

We have assumed that there is a flash of light in an otherwise empty space. The flash of light, however, also must have some conditions, if it is to represent conditions as we actually find them in our space and time. Where did the light come from? How intense is the light? How long does the light flash last?

The light had to come from somewhere. Maybe it is comeing out of a little hole in a heated box. But then, it would not give us a very pretty light cone, since it would be going in one direction in space but not in all the other directions. The substance of the box would prevent the light from traveling in the direction opposite to the hole. Unless, of course, we add some more special conditions. We might lay out our time line starting at the hole in the box and traveling outward in time in the same direction that the light travels, so we preserve our light cone by choosing a special direction.

But these conditions are somewhat unsatisfactory. There is that clunky box, and why is it hot anyway, and how do we know in advance which way to stack the papers up to catch the cone as it is emitted? Before the flash, we can't know where the hole is, and after the flash, if it is a short flash, it is gone and we can never catch up to it, since it is light and nothing travels as fast as light. We have assumed some problems with our conditions of origin, if we assume a black box with a little hole, and we see right away that the presense of the box violates our stipulation that the flash occurs in otherwise empty space.

Maybe the flash is a virtual photon escaping from near the surface of a black hole, while its partner gets swallowed up. This is called Hawking radiation, for anyone reading this who wants to know more about virtual photons near black holes. But then we have more problems. Clearly a black hole is not otherwise empty space. In fact, we might even suggest that a black hole is just a blace box with a hole in it, turned inside out. Well, that is not very satisfactory either then.

Maybe the flash is a proton decay in a tank of heavy water deep under the Tower Sudan mine. That is not really empty space, but it is very regular space, which is maybe not much different from space if it were empty. Anyway the water is transparent to the light so as far as the light is concerned it is as if it were empty. Then we surround the tank with photon detectors and catch the light as it tries to escape the empty space....well. So much for the flat stack of papers. The detectors have to be curved around the tank, and once the photon goes into a detector, it is ended. We can't really know what happened to the photon between its beginning and its ending. In fact, since it is a single photon, and has a direction, it doesn't make a nice circle on the paper at all, only a little point.

It seems we are having problems finding conditions that realistically approach those we imagined for our light cone. Well nevermind. It is still a good idea, and there are many good ideas we can imagine that we can't exactly reproduce in the laboratory. That is why we have thought experiments, and it seems this discussion is turning into one of those.

Well ok. Flat sheet of paper, first one in the stack, the origin. We won't wonder how we knew which way to lay the paper, and we will say that the flash at the origin is of some intensity, so it is composed of many photons going off in all directions at once. Next page, small circle. Next page, slightly larger circle. Next page and so on. Jump to the end of the chapter. Bigger circle. End of next chapter, even bigger circle. Actually, in order to see the light cone forming, we have to make our pages pretty thick. Well, thick in time, not necessarily in space! We can use really thin paper, but we have to remember that light travels fast, so if we want to see the cone in a small enough space to appreciate that it is a cone, we have to imagine that we are taking the circles and sort of spreading them out along the time line, that is, making thick pages. Even though the pages have to be thick in space, so that we can see the cone, they have to represent very thin slices of time, so that we can catch the expansion before it gets away from us.

And it is good that we make the pages thin, in space, because they have to be very wide pages, again because light moves so fast. We need really a lot of pages, and they have to be really wide pages, and so they had better be really thin or we will quickly run out of resources!

Lets skip over all the in-between pages and turn directly to the end of the book. What will we find? Does the circle ever go off the edge of the paper? Does it fade away, losing intensity until we cannot see it any more? Do the pages have to get thicker or thinner to accomodate the scale of time and space as it approaches infinity? Ooops. There's infinity again. I didn't really want to talk about that. But it just keeps on showing up. Nevermind.

Well, we had really a lot of photons to start out with. We had to start with really a lot of photons because we saw that a single photon travels in a single direction and so does not make a nice circle for our light cone. Very well. But we are not going to assume an infinite number (ooops again) of photons! That would be impossible. The universe, our universe anyway, can only contain some limited number of photons, surely?

So as we get to the end of the book, what happens to the circle? Does it eventually divide up into individual points, each point an individual photon? And then if we keep on going, say we decide to follow one arc of the circle because it gets too big to draw the whole thing on a single piece of paper. And then we eventually repeat that process until we are following only one photon, and it is a single line. What happened to the light cone? It seems this brings us right back to the single photon in otherwise empty space. And we have already seen that a single photon in otherwise empty space cannot create a light cone.

Well, I am not trying to say that the idea of a light cone is not an otherwise interesting and useful object. It is nice to be able to talk about angles and velocities and wonder about what lies outside the cone or before the cone or after the cone. Maybe we can just set our pages to a thickness so that the speed of light is represented by fortyfive degrees of angle, and that is pretty cool. Then what lies outside of fortyfive degrees? Why, nothing. After all, nothing can travel faster than light, can it? All right, nevermind tachyons, nevermind ignorons, nevermind Unruh radiation, and for gosh sake, lets not talk about positrons and if they might be traveling backwards in time. All of that is clearly not part of the discussion. Is it?

But, anyway, back at the circle "A" ranch, a bunch of us cowpokes were wondering, well, now, those circles, don't they have to be somewhere? So doesn't there have to be something outside of them? I mean, even if it is at an angle that would have to be greater than the speed of light. I mean, just because Andromeda is some exponent lightyears away, so the light we see is coming from some exponent number of years ago, do we really have to say that something as big as a galaxy that was there so many years ago isn't really there in any meaningful sense, "now", whatever "now" means. I mean it doesn't seem likely that Andromeda has just gone away, leaving her skirts of light trailing behind her? She has to still be there, somehow, somewhere, doesn't she?

Well, you may say, it isn't that Andromeda isn't there any more, it is just that the information we have about Andromeda is a little dated. I mean, we can calculate the speeds of the stars in Andromeda and observe their directions and come to the conclusion that if they were there then, they must be over here a little further now, unless they have been swallowed up by a black hole or gone supernova or something. But there are a lot of stars in Andromeda and it seems unlikely that all of them have gone supernova already (now that would be a flash of light!) or that all of them have been swallowed up by a black hole or something, so Andromeda as a collection of stars must still be there, not?

So outside the circles of our cone of light there must be something there, and even if we cannot know exactly what it is, we can use our powers of observation and our knowlege of physics and our faith in an isotropic, homeostatic universe, to predict with some certainty, with a nod to Heisenburg, that there is a universe, kind of, outside our light cone. And even behind our light cone (uh oh, now I have gone into deep hot water) there is something. Something was there before the flash, it is just that information (light) from before the flash cannot ever catch up to us as we go on turning the pages. In other words, the past is still there! We just can't see it because information from the past cannot catch up to us.

But wait a minute, if information from the past cannot catch up to us, what is memory? If information from the past can't catch up to us, how do we know that there is any past there? Light from nothing in an otherwise empty universe? Ho ho ho, of course not. That was just the conditions we started with so we could see our idea of a light cone, remember?

So, when the learned doctors tell us that nothing exists outside the light cone, much less behind the light cone, we have to remember that they are insane and have stared at the stars too long instead of remembering the importance of cuddling up in nice warm soft blankets here by the campfire. With friends, if possible. Ah friends, the importance of friends.

Now, what was it about that symmetry that was bothering me so much? Nevermind. Pegusus has set, and Andromeda is low on the horizon, hiding behind her skirts, and the sum of all things is looking up, and the dragon is laid out flat, and Spring is surely coming. I have to decide. Should I plant a garden? Or study the stars? My friends are laughing.

Be well, yes, and love,

Richard

Last edited by a moderator: Feb 3, 2005
6. Feb 3, 2005

nightcleaner

I should very much like to plant a garden this spring, and have time to look at the stars. Perhaps I can find a way to do both. My garden friends may be willing to care for the garden if I am away for, say, a month at the height of the growing season. Or maybe my star friends will get a clue and schedule their stargazeing meetings at a season more suitable for gardeners. Well it doesn't really matter. It is unlikely, as A. Zee says, that the fate of democracy hangs in the balance.

Anyway, last night's post was a little rush of ideas and I got a bit loose with throwing papers around and so on. For example there was the idea about a hole in a black box being kind of the inverse of a black hole...that wasn't very well stated. And as for the learned doctors being insane from staring too long at the stars, well, certainly that thought could be taken as ill-considered. I meant no disrespect. I was just doing a little anti-elitist teasing, a kind of tickle under the blankets, and I hope none of the learned doctors will be offended. As it happens, I have worked with certified insane individuals from time to time, and they can actually be good, stimulating company. I have a sort of fondness for crazy people.

I recently re-read "The Man Who Loved Only Numbers" by Paul Hoffman about the stellar mathematician, Paul Erdos. Now there was a case in the vertice. It seems he would have starved to death if people weren't there to prepare his meals. Forget about laundry. It seems amazing, reading his story, that he did not fall into the bands of a nice, quiet asylum. It seems we must take him as a kind of savant, because his daily behavior was extreme, and must have at times bordered on the intolerable to his friends. Lucky for him, he had plenty of freinds, and he spread his peregrinations out fairly thinly among them.

But what I really wanted to talk about was the light cone again, or rather, a sort of inversion of the light cone. Instead of thinking about light radiating out in all directions, we might find it instructive to imagine light falling in from all directions...a condition called Infall. Then we won't have to worry so much about the source of the light...maybe it is big bang thermal energy or something like that. And that annoying question about what happens to the light way out in the outer limits goes away...we know what happens to the light, it falls in until it reaches some central object, and then it stops. And, since we have a central object, it is easy to pin our papers to it and draw our circles. Pretty much any central object will do, we do not have to worry about if it has a hole in it or not.

The only problem seems to be to decide on a location for the time line and to decide on the thickness of the paper slices. The rest pretty much takes care of itself. So let's lay out our time line in, say, that direction. Then the papers are laid out perpendicular to the time line, or we could say, the time line is normal to the surface of the papers. Every page has a little dot at the center of it and that is our collection point. On the first page, just the dot.

Then on the next page, a tiny circle. That circle represents all the points on the page which are close enough to the point for light to reach from them to the point in the time represented by the thickness of the paper. You see it is just the opposite of the light cone idea, in that light is now falling in to the center of the circle rather than going out toward the edges of the paper. One condition, and this is a rather stange one and might bear some thinking. That is, the central point.....it isn't really staying in one place any more, is it? That is, it seems to be moving through the book from page to page, present right there at the center on each and every paper. That is, it is moving through time. Well it has to be, doesn't it? It could hardly be said to exist if it did not move through time. It has to have the quality of duration. But you see that this is a change from the light cone, which after all did not have anything at the center of each page, just the origin of the light on the first page, and after that, emptyness, the innermost reach of the expanding circle. Now we have a little dot. Well it isn't much difference really. We still have the circle, and it is still expanding page to page, as we advance through the book, forming a sort of cone. If you cut out each circle on each page and stacked them up, you would see the cone shape very clearly.

Now we can still adjust the thickness of the paper so that the cone expands outward at fourtyfive degrees if we wish. That is, the fortyfive degree angle still represents the speed of light as it falls inward in our cone. The circle represents the region of everything which can be known by the point in the time since the origin. This is convenient. What is outside the circle? Could be anything. We just don't know.

We don't have to worry about the probable position of Andromeda as we chop our way through the snake, because Andromeda won't be part of the picture for millions of years anyway, and when we do get to Andromeda, as Andromeda crosses the boundary of our Infall cone, we can just make a note of it, and not be concerned about where and if it was somewhere before it fell into our cone. There is no history of Andromeda, no physics known outside the cone, no need to speculate about black holes swallowing up galaxies and so forth. The only thing that matters, the only thing there is, is the information that exists inside the circle. Outside the circle is completely unknown, just as if we were one of Schroedinger's kittens, waiting to be born. (That's an oblique reference to a book I saw at the bookstore, but I havn't read it yet.)

The future is where it belongs, in the unknown and unbounded. I imagine Heisenburg would be pleased.

What is outside the cone? I dunno. We can't know. No one can know. What is behind the cone? Before the cone? Same answer. There be dragons. We can speculate that what is outside the cone must be similar to what is inside the cone, but that is all we can do, and there is the lurking uncertainty. Anything could be out there, and we just havn't encountered it yet. Could be anything. Could be nothing. There is really no reason or grounds to wonder. Something completely against all our known physics could be just beyond the horizon. All the stars could decide to gather in one corner of the sky and spell out "Eat At Betty's Pies." It isn't impossible, even if it is a blatant plug. (Acutally it is the plot gimmick of a story in an old sci-fi magazine from my youth...Analog or Amazing or Strange something or other. I don't remember the author but I hope he or she would be pleased that at least I remembered the gimmick.)

So now after all this I have a little suggestion, and I hope you may find it amusing, or even interesting. Here it is. The realm of general relitivity is the light cone. The realm of quantum mechanics is the inverse light cone, may I call it the Infall cone. So the relationship between GR and QM should be an inverse one. That's all. I hope it is useful.

Now there are two ravens in the pine tree outside my wintery window, and the lake and the sky are very very blue, and the shadows on the bending drifts of melting snow are very long, and I get to work tonight dissolving grease with organic solvents and polishing steel with soft rags and aerosols of mineral oils, pays my way, and if the dish doesn't run away with the spoon, I may be back tomorrow with questions about what's a light cone good for anyway?

Be well,

Richard T. Harbaugh,
nightcleaner.

Last edited by a moderator: Feb 3, 2005


text:
# Windows Without Rodents

Posted on 16 September 2010 by Nicolas Wu
Tags: Windows

My preferred set up for developing code is a unix environment, and I run Ubuntu on my home machine, with XMonad as my main interface. I’ve found the combination to be rock solid, and with XMonad I get an efficient way of navigating between the various applications I have open: I rarely have to lift my hands off the keyboard.

Unfortunately, my new workplace requires me to use Windows 7 for all development, and the situation isn’t likely to change. It’s actually been a few years since I have used Windows, and coming back to its clunky plastic interface was quite a shock to me, so I quickly did all I could to make Windows feel a bit more like /home. In the end, I settled on three tools: AutoHotkey, WindowsPager, and WinSplit.

## AutoHotKeys

One of the most important features I seek in a window manager is simple navigation so that I can get around the screen easily without reaching for the mouse. Windows comes with plenty of built in keystrokes that let you navigate between windows. While almost everybody knows about Alt-Tab, I tend to prefer the slightly less famous Alt-Esc, which performs the same function, except doesn’t have a pop up dialog. Being able to rebind keys makes the hard to reach, but really important, functionalities of Windows useful. AutoHotkey, is perfect for this, and here are the settings held in the XMonad.ahk I’ve got on my machine:

#j::Send !+{Escape}
#k::Send !^{Escape}
#d::Send !{F4}

This lets me navigate between currently opened windows by pressing Super-j and Super-k, and to close a currently open window with Super-d. These bindings are modelled after the inteface for Vim, which is my main text editor.

Since I often use the left super key, which activates the Windows program launcher, I disable it using this:

> ~LWin Up::Return

## WindowsPager

Having a single desktop to handle everything is a cluttered and cramped experience. WindowsPager, makes things a bit better by giving you 4 desktops to play with, and works fine right out of the box. This lets you set up different desktop environments based on the tasks you’re doing.

I always have the following setup:

Desktop1 : Development   (vim, visual studio)
Desktop2 : Research      (firefox)
Desktop3 : Execution     (program debugging)
Desktop4 : Communication (email,irc,music)

Using AutoHotKeys, I bind shortcuts to move around by adding the following to my script:

#1::Send #^{Numpad1}

You can move from desktop to desktop by pressing Meta-Ctrl-Numpad1 through to Meta-Ctrl-Numpad4, and you can move a window to another desktop by selecting it and adding Shift to the sequence.

## WinSplit

Having different desktops for different tasks means that there’s plenty of space left for your windows, so that they can be arranged in a clean and consistent way.

WinSplit does this job very well, by allowing windows to be resized from the keyboard into various configurations.

With a couple modifications, everything I need is at the touch of a few simple keystrokes. First, I modify all the WinSplit bindings so that just pressing Super-Numpad1 moves a window to the bottom left corner of the screen, and likewise for all the other numpad keys.

Once this is done, I also bind Super-h and Super-l to allow simple window repositioning. However, since Windows uses Super-l to lock the screen, this feature must first be disabled in the registry. This can be set using AutoHotkey, and adding the following line:

> RegWrite, REG_DWORD, HKEY_CURRENT_USER, Software\Microsoft\Windows\CurrentVersion\Policies\System, DisableLockWorkstation, 1

## Summary

With all these tweaks in place, I noticed my productivity increased significantly, and I spent a lot less time pausing between applications and reaching for the mouse. Two other indispensable tools are Vim, and Vimperator, which make the mouse redundant for almost every task.


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# Observation of $\Lambda_b^0 \to \psi(2S)pK^-$ and $\Lambda_b^0 \to J/\psi \pi^+ \pi^- pK^-$ decays and a measurement of the $\Lambda_b^0$ baryon mass

Abstract : The decays $\Lambda_b^0 \to \psi(2S)pK^-$ and $\Lambda_b^0 \to J/\psi \pi^+ \pi^- pK^-$ are observed in a data sample corresponding to an integrated luminosity of 3fb$^{-1}$, collected in proton-proton collisions at 7 and 8TeV centre-of-mass energies by the LHCb detector. The $\psi(2S)$ mesons are reconstructed through the decay modes $\psi(2S)\to\mu^+\mu^-$ and $\psi(2S)\to J/\psi\pi^+\pi^-$. The branching fractions relative to that of $\Lambda_b^0 \to J/\psi pK^-$ are measured to be \begin{eqnarray*} \frac{\mathcal{B}(\Lambda_b^0 \to \psi(2S) pK^-)} {\mathcal{B}(\Lambda_b^0 \to J/\psi pK^-)} & = & (20.70\pm 0.76\pm 0.46\pm 0.37)\times10^{-2} \,, \frac{\mathcal{B}(\Lambda_b^0 \to J/\psi \pi^+ \pi^- pK^-)} {\mathcal{B}(\Lambda_b^0 \to J/\psi pK^-)} & = & (20.86\pm 0.96\pm 1.34)\times10^{-2} \,, \end{eqnarray*} where the first uncertainties are statistical, the second are systematic and the third is related to the knowledge of $J/\psi$ and $\psi(2S)$ branching fractions. The mass of the $\Lambda_b^0$ baryon is measured to be $M(\Lambda_b^0) = 5619.65 \pm 0.17 \pm 0.17\mathrm{MeV}/c^2,$ where the uncertainties are statistical and systematic.
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-01292928
Contributor : Danielle Cristofol <>
Submitted on : Thursday, March 24, 2016 - 8:31:35 AM
Last modification on : Friday, November 6, 2020 - 11:44:14 AM

### Citation

R. Aaij, C. Abellán Beteta, B. Adeva, M. Adinolfi, A. Affolder, et al.. Observation of $\Lambda_b^0 \to \psi(2S)pK^-$ and $\Lambda_b^0 \to J/\psi \pi^+ \pi^- pK^-$ decays and a measurement of the $\Lambda_b^0$ baryon mass. Journal of High Energy Physics, Springer Verlag (Germany), 2016, 05, pp.132. ⟨10.1007/JHEP05(2016)132⟩. ⟨in2p3-01292928⟩

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Version:

Computes the matrix exponential using the Pade approximation. National Instruments recommends that you use the expmx function instead.

## Syntax

c = expmx_pade(a)
Legacy name: expmdemo1

## a

Square matrix.

## c

Matrix exponential using the Pade approximation of a. c is a square matrix.

C = diag([1, 0, -1])
D-E


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×

# Even numbers expressed as the sum of two primes

Can we prove that the even number greater than 2 can be expressed as the sum of two prime numbers....

Note by Prabhav Bansal
2 years, 1 month ago

Sort by:

This is Goldbach conjecture. No one has proved this conjecture till now. It is million-dollar conjecture.This is the best-known unsolved conjecture. If a person has a correct proof of it, he will be prized 1 million dollars. The proof would be judged by a panel of 6 mathematicians. ( As per the article given in Koshy's book) . Also this is the strong Goldbach conjecture. Go ahead if you know the proof:-)

- 2 years, 1 month ago

As we know after 2 all prime numbers are only odd. & We also know that sum of 2 odd numbers will always be even. Then what what do u want to know exactly???

- 2 years, 1 month ago

But this is the converse. Please don't prove the converse.

- 2 years, 1 month ago

yes

- 2 years, 1 month ago

Can you prove this? I am asking for the proof.

- 2 years, 1 month ago

What have you tried? What are your thoughts?

Staff - 2 years, 1 month ago

I want to ask that can we prove that the even numbers >2 can be expressed as the sum of prime numbers. Eg: 6=3+3 22=17+5

- 2 years, 1 month ago

That is an interesting question. What have you tried? What are your thoughts? Do you think this is always possible? If yes, why? If no, why not?

Staff - 2 years, 1 month ago


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## fast track & slow lane

Posted in Books, Kids, R, Statistics with tags , , , , on September 3, 2022 by xi'an

A riddle from the Riddler where N cars going at random (iid) speeds drive a road with a slow and a fast lane, each car choosing the fast lane iff any of the cars ahead in the slow lane is slowerthan them. With the question of the average number of car convoys.

If there were one single lane, the problem would be to determine how many times a smaller realisation and it has been solved in a much older Riddler. Namely

$\sum_{i=1}^N 1\big/ i$

In the two-lane system, the slow one only gathers cars with speeds lowest than the current speed, i.e. a decreasing sequence of speeds, creating single car convoys.  The fast lane thus captures cars that are above the current minimum in the sequence, which, as it converges to the global minimum at some points, means that all following cars are found in the fast lane. i thought this would bring the number of convoys close to twice the above logarithmic sum (which sealed my destiny during an entrance oral exam 40 years ago!), but there are actually more of them for N large enough , which may be due to the possibility of the slow lane to capture more moderate speed cars in the beginning… The above compares the number of convoys for one lane (in red) and two (in gold), as well as the remarkable fit when regressing these numbers against log(N).

Here is the R code I used for that simulation

convoy=function(N,T=1e5){
for(t in 1:T){
speed=runif(N)
slow=fast=NULL
slow=speed[1]
for(i in 2:N){
if(speed[i]<min(slow))slow=c(slow,speed[i])
else fast=c(fast,speed[i])}
F=F+length(slow)
if(length(fast)>0)F=F+1+sum(fast[-1]<cummin(fast)[-length(fast)])}
return(F/T)}

## [ex?] Paris-Saclay univerXity

Posted in Kids, pictures, University life with tags , , , , , , , , , on November 5, 2017 by xi'an

In the plane to Warwick last Tuesday, I read a fairly long and pessimistic article in Le Monde about the future of the Paris-Saclay university. This debate presumably makes no sense outside French circles, for it relates to the century-old opposition between universities and grandes écoles, these selective engineering and business schools that operate independently from the university structure. In the sense that the selection and the schooling of their students is completely separated, meaning these students may never attend a university program if they choose to do so. But not so independently in terms of hiring part-time professors from universities and sharing resources for their Master programs. And depending on the same agencies (like CNRS) for funding their research program.

Anyway, the core message of this article was that the influence of former students from Polytechnique [aka X] in the high administration is such that they can prevent the integration of the different engineer schools on the Saclay plateau into the intended superstructure of the Paris-Saclay university. And turn this somewhat megalomanic Paris-Saclay project initiated by Nicolas Sarkozy into a mere geographical superposition of separate institutions, with very unequal State funding, perpetuating the two speed regime for public higher education… And a ever more confusing international image that will not help an inch moving up the Shanghai ranking (a major reason for Sarkozy pushing this project). Very French indeed!


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# We can make the edges smooth of an curved image by selecting

The anti-alias option from Option bar

The anti-alias option from Layer Effect

None of them


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# How to plot the results of a Kruskal-Wallis or a Welch's ANOVA?

Let's assume that I have two datasets for which I have measured three continuous variables. In both datasets, these variables have been measured on three groups of observations and these groups only contains 5 observations each.
I would like to know if, among these two sets of groups, some tend to have higher values for the measured variables. As my sample sizes are small and as these observations appear to be non-normaly distributed, I would like to use a non-parametric method. Consequently, I use a Kruskal-Wallis test on the first dataset, and Welch's ANOVA on the second one (because some data are heteroscedastic).
My question is what is the appropriate way to plot the results of a Kruskal-Wallis test? And what is the correct way to plot the results of a Welch's ANOVA?

Most people seem to use boxplots for such a purpose. Yet, if I understand correctly, the Kruskal-Wallis test does not exactly compare medians (but mean ranks), and boxplots do not give any information on the means of heteroscedastic samples (and are thus ill-suited for Welch's ANOVA results). So what should I use to plot my results?
On someone's blog, I read about plotting the groups' distributions. But since I only have very small sample sizes, I fear that it would not be very clear and/or informative (of course, I realize that with only 5 observations per group, my tests are not very informative themselves).

• In Chapter 7 of BBR I cover how to generalize the Kruskal-Wallis test to the proportional odds model, and how to use that model to estimate means, medians, exceedance probabilities, and odds ratios. Jan 30, 2020 at 15:51

As you already mentioned the Kruskal-Wallis test is a test of significance based on the ranks. In my opinion however, plotting the ranks isn't really that helpful for the reader in order to understand the underlying response variable. Instead, what I would do is to plot the individual data points (including the median for descriptive purposes) plus the ranks as differently colored points. To make it clear, you could also place the letters indicating significant difference next to those points indicating the ranks. You can also obviously report everything you don't want to plot (e.g. the ranks as separate points, etc.) in a separate table (see example below).

I am not sure which software package you are using but below is an example using R to illustrate what I mentioned above (note: this approach may not look nice if the numerical values of the data points and the ranks are largely different. In that case, I would plot the data points and the significant differences via letters, and report the ranks in a separate table.

### required packages
require(tidyverse)
require(agricolae)
### set seed for reproducibility
set.seed(564)

### subset the PlantGrowth dataset (available in R) to replicate your n=5 scenario
PlantGrowth %>%
group_by(group) %>%
slice(sample(1:5)) -> d_sub

### run Kruskal test from the agricolae package
k <- kruskal(d_sub$$weight, d_sub$$group, console = TRUE)
#>
#> Study: d_sub$$weight ~ d_sub$$group
#> Kruskal-Wallis test's
#> Ties or no Ties
#>
#> Critical Value: 3.290877
#> Degrees of freedom: 2
#> Pvalue Chisq  : 0.192928
#>
#> d_sub$$group, means of the ranks #> #> d_sub.weight r #> ctrl 8.3 5 #> trt1 5.3 5 #> trt2 10.4 5 #> #> Post Hoc Analysis #> #> t-Student: 2.178813 #> Alpha : 0.05 #> Minimum Significant Difference: 5.816519 #> #> Treatments with the same letter are not significantly different. #> #> d_sub$$weight groups
#> trt2         10.4      a
#> ctrl          8.3      a
#> trt1          5.3      a

### create summary table incl. mean rank sums and significant differences letters
(t_comp <- k$$means %>% rownames_to_column(var = "group") %>% rename(weight = d_sub.weight) %>% as_tibble() %>% left_join(as_tibble(k$$groups), by = c("rank" = "d_sub\$weight")))
#> # A tibble: 3 x 11
#>   group weight  rank   std     r   Min   Max   Q25   Q50   Q75 groups
#>   <chr>  <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <fct>
#> 1 ctrl    5.11   8.3 0.788     5  4.17  6.11  4.5   5.18  5.58 a
#> 2 trt1    4.57   5.3 0.851     5  3.59  5.87  4.17  4.41  4.81 a
#> 3 trt2    5.57  10.4 0.446     5  5.12  6.31  5.37  5.5   5.54 a

### create plot with ranks as blue dots and align the letters next to them
d_sub %>%
ggplot(aes(x = group, y = weight)) +
geom_point(color = "grey50", size = 2) +
# add ranks as separate points
geom_point(data = t_comp, aes(x = group, y = rank), col = "blue", size = 3) +
# add median as horizontal line
stat_summary(fun.y = median, geom = "errorbar", aes(ymax = ..y.., ymin = ..y..),
width = .75, col = "red") +


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# Projecting a new entry onto the two largest eigenvectors of a PCA model

I have a bit of problems understanding how PCA and SVD works, as most materials focus on calculating the factors rather than the classification of new entries. In order to provide some context of my questions, let us first consider the following model:

I have some data set $X$ with $p$ features and $n$ rows (i.e. observations). Now suppose I use the first two factors $f_{1}$ and $f_{2}$ to create the following model $y=\beta_{0}+\beta_{1}\cdot f_{1} + \beta_{2}\cdot f_{2}$. Now suppose I get the following row vector $r^{*}$. How can I project $r^{*}$ onto $f_{1}$ and $f_{2}$?

A second question, as far as my understanding goes if we do SVD on matrix $X$ we get: $X=U\Sigma V^{*}$, but which are actually the eigenvectors? And are those eigenvectors the same as what I got with $f_{1}$?

Thank you for helping me understanding.

The model you define is somewhat inaccurate unless $y$ is of dimensions $p \times 1$. Is it?
Having said that assuming the eigenvectors $[f_1 f_2]$ define a $p \times 2$ matrix you simply do the multiplication: $[f_1 f_2]^T r^*$ to get the new scores. This is equivalent of projecting $r^*$ to the axial system defined by $f_1$ and $f_2$.
Both $U$ and $V^*$ are unitary matrices. If the matrix $X$ was symmetric $U$ would equal $V^*$. If you want the eigenvectors that are the same as the ones returned by PCA you are looking to use $V^*$.


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# GATE EE 2010

 Question 1
The value of the quantity P, where $P=\int_{0}^{1}xe^{x}dx$, is equal to
 A 0 B 1 C e D 1/e
Engineering Mathematics   Calculus
Question 1 Explanation:
$P=\int_{0}^{1}xe^x\; dx$
Integrating by parts:
\begin{aligned} \text{Let } \; u&= x,\\ dv&= e^x\; dx\\ du&= dx,\\ v&=\int e^x\; dx=e^x \\ \text{Now, } \int udv&= uv-\int v\; du\\ \therefore \;\;\int x e^x\; dx &=xe^x-\int e^x\; dx \\ &=xe^x-e^x+c \\ \int_{0}^{1} xe^x\; dx&= [xe^x-e^x]_0^1\\ &= (1 \times e^1 -e^1)-(0 \times e^0 -e^0)\\ &= 0-(-1)\\ &= 1 \end{aligned}
 Question 2
Divergence of the three-dimensional radial vector field $\vec{r}$ is
 A 3 B 1/r C $(\hat{i}+\hat{j}+\hat{k})$ D $3(\hat{i}+\hat{j}+\hat{k})$
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} \vec{r} &= x\hat{i}+y\hat{j}+z\hat{k}\\ \text{div } \vec{r}&= \bigtriangledown \cdot \vec{r}\\ &=\left ( \hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z} \right )(x\hat{i}+y\hat{j}+z\hat{k})\\ &=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}\\ &=1+1+1=3 \end{aligned}
 Question 3
The period of the signal $x(t)=8 sin(0.8 \pi t+ \pi/4)$ is
 A 0.4 $\pi$ s B 0.8 $\pi$ s C 1.25 s D 2.5 s
Engineering Mathematics   Complex Variables
Question 3 Explanation:
If period $= T,$ then $x(t+T)=x(t)$
Here, $x(t)= 8 \sin(0.8 \pi t+\pi /4)$
putting $x(t+T)=x(t)$ we get
$8 \sin(0.8\pi (t+T)+\pi /4)=8 \sin (0.8 \pi t+\pi /4)$
$\sin(0.8\pi (t+T)+\pi /4)= \sin (0.8 \pi t+ \pi /4)$
$\sin(\theta _1)=\sin(\theta _2)\Rightarrow \theta _1=\theta _2+2 \pi$
$\therefore \;\; 0.8\pi (t+T)+ \pi /4=0.8 \pi t+ \pi /4+2 \pi$
$0.8\pi T=2 \pi$
$T=2.5 s$
 Question 4
The system represented by the input-output relationship $y(t)=\int_{-\infty }^{5t}x(\tau )d\tau , \; t\gt0$
 A Linear and causal B Linear but not causal C Causal but not linear D Neither liner nor causal
Signals and Systems   Linear Time Invariant Systems
Question 4 Explanation:
Integrator is always a linear system.
\begin{aligned} \text{Since, }y(t)&=\int_{-\infty }^{5t}x(\tau )d\tau \;\;t \gt 0 \\ &\text{For t=1,} \\ y(t)&= \int_{-\infty }^{5}x(\tau )d\tau \end{aligned}
here value at t=1 depends on future values like at t=2,3,... of input x(t).
So, it is a noncausal systems.
 Question 5
The switch in the circuit has been closed for a long time. It is opened at t = 0. At t =$0^+$, the current through the 1 $\mu$F capacitor is
 A 0A B 1A C 1.25A D 5A
Electric Circuits   Transients and Steady State Response
Question 5 Explanation:
As the switch has been closed for a long time, the circuit is in steady state. At steadystate, capacitor is open circuit,

Using KVL,
$5-I-4I=0 I=1A$
$V_c(0^-)=4 \times 1=4V$
As the voltage across capacitorcan not change abruptly,
So, $V_c(0^+)=V_c(0^-)=4V$
Circuit at $t=0^+$

Current through capacitor at $t=0^+$
$I_c(0^+)=\frac{4}{4}=1A$
 Question 6
The second harmonic component of the periodic waveform given in the figure has an amplitude of
 A 0 B 1 C $2/\pi$ D $\sqrt{5}$
Signals and Systems   Fourier Series
Question 6 Explanation:
The given signal is odd as wel as having half wave symmetry.
So, it has only sine terms with odd harmonics. So, for second harmonic term amplitude =0.
 Question 7
As shown in the figure, a 1 $\Omega$ resistance is connected across a source that has a load line v+i=100. The current through the resistance is
 A 25A B 50A C 100A D 200A
Electric Circuits   Basics
Question 7 Explanation:
A resistor has linear characteristics
i.e. $V=Ri$
$\Rightarrow \;\; V=i$
Load line,
$V+i=100$
$i+i=100$
current through resistance,
$i=\frac{100}{2}=50A$
 Question 8
A wattmeter is connected as shown in figure. The wattmeter reads.
 A Zero always B Total power consumed by $Z_1 \; and \; Z_2$ C Power consumed by $Z_1$ D Power consumed by $Z_2$
Electrical and Electronic Measurements   Measurement of Energy and Power
Question 8 Explanation:

Potential coil draws negligible current. So current through $Z_1$ and $Z_2$ is same.
current through current coil $=I_{cc}=I$
Voltage across potential coil $=V_{pc}$
Voltage across $Z_1=V_{pc}=V$
Wattmeter reads power consumed by $Z_2$, as voltage across potential coil = Voltage across $Z_2$
Current through current coil = Current through $Z_2$.
 Question 9
An ammeter has a current range of 0-5 A, and its internal resistance is 0.2 $\Omega$. In order to change the range to 0-25 A, we need to add a resistance of
 A 0.8 $\Omega$ in series with the meter B 1.0 $\Omega$ in series with the meter C 0.04 $\Omega$ in parallel with the meter D 0.05 $\Omega$ in parallel with the meter
Electrical and Electronic Measurements   Galvanometers, Voltmeters and Ammeters
Question 9 Explanation:
To extend the range of ammeter, a resistance $R_{sh}$ is connected across the meter.

$I_m=$ full scale deflection current=5A
$I=25A$
Multiplying power $=m=\frac{I}{I_m}=\frac{25}{5}=5$
$I_m=\frac{IR_{sh}}{R_{sh}+R_m}$
$\frac{I}{I_m}=\frac{R_{sh}+R_m}{R_{sh}}$
$m=1+\frac{R_m}{R_{sh}}$
$R_{sh}=\frac{R_m}{m-1}=\frac{0.2}{5-1}=0.05$
 Question 10
As shown in the figure, a negative feedback system has an amplifier of gain 100 with $\pm$10% tolerance in the forward path, and an attenuator of value 9/100 in the feedback path. The overall system gain is approximately :
 A $10\pm 1$% B $10\pm 2$% C $10\pm 5$% D $10\pm 10$%
Control Systems   Feedback Characteristics of Control Systems
Question 10 Explanation:
$G=100\pm 10$%
$\frac{\Delta G}{G}=10$% or 0.1
$H=\frac{9}{100}$
Overall gain,
$T=\frac{G}{1+GH}$
$T=\frac{100}{1+100 \times \frac{9}{100}}=10$
Sensitivity w.r.t. $'G'=\frac{1}{1+GH} \times 10$%
$\;\;\;=\frac{1}{1+10 \times 1}\times 10$%
$\;\;\;=\frac{10}{11}\cong 1$%
There are 10 questions to complete.


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# 5.8: Mixed Inconsistence (Ethipramine and Imipramine)

## Ethipramine

Tricyclic anti – depressants have been used widely for symptoms of frequency, urgency, urge incontinence and especially nocturia for many years. Although grade 1 evidence justifying their use is lacking, many patients are satisfied with the results. Ethipramine is inexpensive and widely available, with a multitude of effects – and side effects.
Its actions are anticholinergic in nature, with an adrenergic effect on the bladder neck. Theoretically at least, this makes it ideal for mixed incontinence, but its side – effects are often troublesome. It causes cardiac conduction defects and this has caused the WHO to warn against its use. Dry mouth and drowsiness are the most bothersome side effects, limiting its use. The drug is available in 10mg and 25mg tablets, and the usual starting dose is 10mg in the mornings, with 25mg or 50mg at night. The soporific effect of ethipramine may be used to advantage, allowing increased evening dosage. Contra – indications are as for other anti – cholinergics. If clinicians prescribe ethipramine, they must be aware of its cardiac effects especially in elderly women.

## Imipramine

The use of imipramine is parallel to that of ethipramine – with the proviso that it remains untested as a pure anticholinergic for use in incontinence. Imipramine is primarily, with amytriptyline, an antidepressant, and its useful anticholinergic effects are purely fortuitous. Clinicians must be aware that these agents are of limited use as niche agents, and that ethipramine is perhaps more clinically useful.
Pharmcotherapy remains the mainstay of therapy for the treatment of OAB, and the contemporary literature shows that antimuscarinic agents are used as a first line therapy for OAB. To gain a better understanding of the overall benefits of OAB treatment, it is critical that RCTs use validated instruments to assess HRQL and to relate these changes to changes in OAB symptoms. The International Continence Society advocates the use of HRQL measures in clinical research has provided increasing evidence for the HRQL benefits conferred by effective OAB treatments.
The future emphasis of work in this field must also incorporate patient – perceived outcomes using existing tools to assess bother and QOL.

This page titled 5.8: Mixed Inconsistence (Ethipramine and Imipramine) is shared under a not declared license and was authored, remixed, and/or curated by Stephen Jeffery and Peter de Jong via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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# OrbitalSpace¶

class psi4.core.OrbitalSpace

Bases: pybind11_builtins.pybind11_object

Methods Summary

 C(self) MO coefficient matrix, AO->MO or SO->MO transformation matrix basisset(self) The AO basis set used to create C build_cabs_space(orb_space, ri_space, linear_tol) Given two spaces, it projects out one space from the other and returns the new spaces The first argument (orb_space) is the space to project out. build_ri_space(molecule, obs_key, aux_key, …) Given two basis sets, it merges the basis sets and then constructs an orthogonalized space with the same span. dim(self) MO dimensions evals(self) Corresponding eigenvalues of the C matrix id(self) Unique identifier integral(self) The integral factory used to create C name(self) Name of the orbital space nirrep(self) Returns number of irreps print_out(self) Print information about the orbital space to the output file

Methods Documentation

C(self: psi4.core.OrbitalSpace) → psi4.core.Matrix

MO coefficient matrix, AO->MO or SO->MO transformation matrix

basisset(self: psi4.core.OrbitalSpace) → psi::BasisSet

The AO basis set used to create C

build_cabs_space(orb_space: psi4.core.OrbitalSpace, ri_space: psi4.core.OrbitalSpace, linear_tol: float) → psi4.core.OrbitalSpace

Given two spaces, it projects out one space from the other and returns the new spaces The first argument (orb_space) is the space to project out. The returned space will be orthogonal to this The second argument (ri_space) is the space that is being projected on. The returned space = this space - orb_space The third argument is the tolerance for linear dependencies

build_ri_space(molecule: psi::Molecule, obs_key: str, aux_key: str, lindep_tol: float) → psi4.core.OrbitalSpace

Given two basis sets, it merges the basis sets and then constructs an orthogonalized space with the same span. Linearly dependent orbitals are thrown out. The first argument, molecule, is the molecule to construct the basis for The second argument, obs_key, is the option keyword for orbital basis set ‘BASIS’ The third argument, aux_key, is the option keyword for auxiliery basis set ‘DF_BASIS_MP2’ The fourth argument, lindep_tol, is the tolerance for linear dependencies

dim(self: psi4.core.OrbitalSpace) → psi4.core.Dimension

MO dimensions

evals(self: psi4.core.OrbitalSpace) → psi4.core.Vector

Corresponding eigenvalues of the C matrix

id(self: psi4.core.OrbitalSpace) → str

Unique identifier

integral(self: psi4.core.OrbitalSpace) → psi4.core.IntegralFactory

The integral factory used to create C

name(self: psi4.core.OrbitalSpace) → str

Name of the orbital space

nirrep(self: psi4.core.OrbitalSpace) → int

Returns number of irreps

print_out(self: psi4.core.OrbitalSpace) → None

Print information about the orbital space to the output file

C(self: psi4.core.OrbitalSpace) → psi4.core.Matrix

MO coefficient matrix, AO->MO or SO->MO transformation matrix

basisset(self: psi4.core.OrbitalSpace) → psi::BasisSet

The AO basis set used to create C

build_cabs_space(orb_space: psi4.core.OrbitalSpace, ri_space: psi4.core.OrbitalSpace, linear_tol: float) → psi4.core.OrbitalSpace

Given two spaces, it projects out one space from the other and returns the new spaces The first argument (orb_space) is the space to project out. The returned space will be orthogonal to this The second argument (ri_space) is the space that is being projected on. The returned space = this space - orb_space The third argument is the tolerance for linear dependencies

build_ri_space(molecule: psi::Molecule, obs_key: str, aux_key: str, lindep_tol: float) → psi4.core.OrbitalSpace

Given two basis sets, it merges the basis sets and then constructs an orthogonalized space with the same span. Linearly dependent orbitals are thrown out. The first argument, molecule, is the molecule to construct the basis for The second argument, obs_key, is the option keyword for orbital basis set ‘BASIS’ The third argument, aux_key, is the option keyword for auxiliery basis set ‘DF_BASIS_MP2’ The fourth argument, lindep_tol, is the tolerance for linear dependencies

dim(self: psi4.core.OrbitalSpace) → psi4.core.Dimension

MO dimensions

evals(self: psi4.core.OrbitalSpace) → psi4.core.Vector

Corresponding eigenvalues of the C matrix

id(self: psi4.core.OrbitalSpace) → str

Unique identifier

integral(self: psi4.core.OrbitalSpace) → psi4.core.IntegralFactory

The integral factory used to create C

name(self: psi4.core.OrbitalSpace) → str

Name of the orbital space

nirrep(self: psi4.core.OrbitalSpace) → int

Returns number of irreps

print_out(self: psi4.core.OrbitalSpace) → None

Print information about the orbital space to the output file