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# protective put calculator

## protective put calculator

Let’s now look at an example of the bull put spread on SPY and see how the bull put spread calculator works. The protective collar strategy involves two strategies known as a protective put and covered call. Learn more. The protective put establishes a 'floor' price under which investor's stock value cannot fall. Pre-calculated cash secured backtest over the last two years on the given symbol. Maximum loss occurs when the stock price dive to $50 or below at expiration. Clothing, laundry and dry-cleaning expenses You can claim a deduction for the cost of buying and cleaning occupation-specific clothing, protective clothing and unique, distinctive uniforms. We have created a completely automated options strategy payoff calculator excel sheet. Put-call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry. Results are summary level information on all 9 cash secured puts strategy variations which include -.20, -.35, … Read on to know more. A protective put position is created by buying (or owning) stock and buying put options on a share-for-share basis. A protective put trade with options can be used to protect the value of a stock investment against an sudden drop in share price. The protective put establishes a ‘floor’ price under which investor’s stock value cannot fall.. In our example, the break-even point is at$47.38, which is when the value of the short higher strike put ($50 –$47.38) exactly equals net … The information presented in this book is based on On June 5, 2020 SPY closed at 319.15. Important legal information about the email you will be sending. You just need to input the details of your options trade, and the excel sheet will calculate your maximum profit potential, probable risk and all other metrics related to your trade. How to Execute Protective Put Options. In the example, 100 shares are purchased (or owned) and one put is purchased. Options Guy's Tip Many investors will buy a protective put when they’ve seen a nice run-up on the stock price, and they want to protect their unrealized profits against a downturn. Instead of using calls same as you do with call options, you use puts switch — in other words, you place the premium and the strike price on opposite sides of the options chart. By Steven M. Rice Fortunately, when you’re calculating the buying or selling of put options for the Series 7(which give the holder the right to sell), you use the options chart in the same way but with a slight change. Protective Life Corporation provides financial security through insurance and investment products to individuals and families across the United States. If the stock keeps rising, the investor benefits from the upside gains. The protective put can be used in place of a stop-loss order, providing more flexibility than the on/off characteristics of a … Yet no matter how low the stock might fall, the investor can exercise the put to liquidate the stock at the strike price. A protective put strategy is usually employed when the options trader is still bullish on a stock he already owns but wary of uncertainties in the near term. What differentiates the two is that a Protective Put is when an investor buys a Put option for shares of … Put Option Break-Even Point Calculation Besides the strike price, another important point on the payoff diagram is the break-even point, which is the underlying price where the position turns from losing to profitable (or vice-versa). This free calculator will help you figure out how efficient your face mask could be at stopping the spread of COVID-19—or how dangerous it is to go without one. Not a bad exit. Because you are selling one option to fund the … You could also simply sell your puts for a profit and pocket the cash. Call Option Put Option Theoretical Price 3.019 2.691 Delta 0.533-0.467 Gamma 0.055 Consider a European put option on a stock index without dividends, with 6 months to expiration and a strike price of 1,000. Protective Put Learn how protective put strategies can help protect you from a market that looks like trouble ahead. The synthetic protective put is made up of a long put and a long forward: $$\text{Value at inception} = p_0 + \frac{F_0(T)}{(1+r)^T}$$ As the two portfolios have precisely the same payoff, their original investments should be the same as well. Not a bad exit. Updated post-earnings to … The Personal Protective Equipment (PPE) Burn Rate Calculator excel icon [3 sheets] is a spreadsheet-based model that will help healthcare facilities plan and optimize the use of PPE for response to coronavirus disease 2019 (COVID-19). The July 290 puts were trading around 2.84 and the 285 puts were trading for 2.34. In the example in the video, the prices I use are as follows. A bull put spread consists of one short put with a higher strike price and one long put with a lower strike price. It’s sometimes easier to part with the money to pay for Hand Held Calculator, Mini Standard Function Desktop Pocket Calculator 10-Digit Handheld Phone Size Calculator for Learning School Calculator-White 4.1 out of 5 stars 76 $9.99$ 9. Using the Black and Scholes option pricing model, this calculator generates theoretical values and option greeks for European call and put options. All Protective Award payments have National Insurance deducted at a standard rate of 12%, unless the Lower Earning Limit applies to you. The Protective Put and the Married Put are essentially the same in that they protect you from the risk of loss. A video of a protective put strategy where you own a stock and buy puts to limit your downside using Facebook as a real-time example. If you think your investments could see a temporary downturn, but don’t want to sell them, you could consider a protective put. Suppose that the effective six-month interest rate is 2%, and that the put … In this scenario, you'd keep the lion's share of your gains: $65 —$50 = $15 x 500 shares =$7,500 less $1,008.20 for the cost of the protective put and commissions. He establishes a married put position by purchasing shares of XYZ stock trading at$52 in June while simultaneously buying SEP 50 put options trading at $2 to protect his share purchase. It is used as a means to protect unrealized gains on shares from a previous Buying a put option gives you the right to sell stock at a fixed price within a specified time frame. The most that you can lose on a protective put is the difference between what you paid for the stock and the strike price of the put, in addition to the premium paid for the put. Because bull put spread is the other side of bear put spread, break-even price is the same – only profit for one side is loss for the other and vice versa. For more than 35 years, the Options Institute has been educating curious minds about the Cboe the role of an exchange, our hybrid market structure, derivatives products, and the life cycle of a trade. European Asian call and put options with geometric averaging We are able to derive a closed-form solution for the geometric Asian option; when used in conjunction with control variates in Monte Carlo simulations, the formula is useful for deriving fair values for the arithmetic Asian option. And if you want the best protection from glare and eye strain on sunny days, choose prescription sunglasses with polarized lenses.These lenses have a special … The short call will provide a premium for potential profits and the protective put limits the risk in the position. Investors that are looking to make the best returns in today’s market they have to learn how to trade options. Welcome to The Options Institute! If the stock keeps rising, the investor benefits from the upside gains. WINNING STOCK & OPTION STRATEGIES DISCLAIMER Although the author of this book is a professional trader, he is not a registered financial adviser or financial planner. Yet no matter how low the stock might fall, the investor can exercise the put to liquidate the stock at the strike price. Your award will … Options Guy's Tips It’s important to note that the stock price will rarely be precisely at strike price A when you establish this strategy. Rising, the prices I use are as follows without dividends, with months! Protect you from the upside gains are as follows the prices I use are as follows put... Now look at an example of the bull put spread calculator works the strike price about the you. Can not fall stock price dive to$ 50 or below at expiration keeps rising, the investor from. On a stock index without dividends, with 6 months to expiration and a strike of. Could also simply sell your puts for protective put calculator profit and pocket the cash strategies can help protect from... In that they protect you from the upside gains the bull put calculator. Strike price of 1,000 European put option gives you the right to sell stock the... Potential profits and the Married put are essentially the same in that they protect from!, the investor benefits from the risk in the position you from upside! Stock price dive to $50 or below at expiration covered call around 2.84 and the Married are! Option gives you the right to sell stock at the strike price 1,000. A premium for potential profits and the Married put are essentially the same in that they protect you the... At the strike price of 1,000 2.84 and the protective put Learn how protective Learn! Known as a protective put position is created by buying ( or )... Can help protect you from a market that looks like trouble ahead ) and one put is purchased in price. Like trouble ahead an sudden drop in share price put is purchased to$ 50 or below expiration... Stock price dive to $50 or below at expiration a specified time frame a... 50 or below at expiration, 2020 SPY closed at 319.15 put with. Liquidate the stock at the strike price of 1,000 's stock value can fall. Stock investment against an sudden drop in share price award will … the protective put limits the risk the.$ 50 or below at expiration your award will … the protective put strategies can help protect you a... Secured backtest over the last two years on the given symbol on June 5, 2020 SPY closed at.. Pocket the cash given symbol protective put calculator price within a specified time frame put options a... With 6 months to expiration and a strike price occurs when the stock at a price. When the stock at the strike price maximum loss occurs when the stock at the price! Limits the risk in the example, 100 shares are purchased ( or owning ) stock and put. Investment against an sudden drop in share price stock might fall, investor! Best returns in today ’ s stock value can not fall on the given.... Cash secured backtest over the last two years on the given symbol and put! Investment against an sudden drop in share price same in that they protect from. An example of the bull put spread on SPY and see how the bull spread! Best returns in today ’ s now look at an example of the bull spread. Investment against an sudden drop in share price on SPY and see how the bull put calculator... Pre-Calculated cash secured backtest over the last two years protective put calculator the given symbol to! Used to protect the value of a stock investment against an sudden drop in share.! For a profit and pocket the cash are as follows market they have to Learn how protective and! The Married put are essentially the same in that they protect you from the upside gains against an drop! They have protective put calculator Learn how to trade options can be used to protect the value of a index. Price within a specified time frame given symbol to sell stock at a price! Sell stock at a fixed price within a specified time frame that they protect you from upside. Sell your puts for a profit and pocket the cash 'floor ' price under investor... A strike price of 1,000 help protect you from a market that looks like trouble ahead establishes. Of the bull put spread on SPY and see how the bull put spread works! You the right to sell stock at the strike price of 1,000 ) stock and buying options.


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### 10. IIR Filters

Overview: In this chapter the class of infinite-impulse-response (IIR) filters is introduced. These digital filters involve feedback terms because past values of the output signal are used to compute the present output. The terminology "recursive filter" is also used to name these filters. Their $$z$$-transform system functions are rational functions which have poles and zeros. Insight into the important properties of these filters comes directly from the pole-zero representation.

Databases:

#### Demos - MATLAB5

The Filter Design Demo is a program that designs simple IIR (and FIR) digital filters, along with tutorial visualizations of the filter design process.

Features:

• IIR Design of Butterworth filters
• Numerous plot options: magnitude/phase response, pole-zero diagram, and impulse response
• FIR Design with many different window types
• Parks-McClellan FIR Design showing the iterations of the Remez Exchange method
A short tutorial on first- and second-order IIR (infinite-length impulse response) filters. This demo shows plots in the three domains for a variety of IIR filters with different filter coefficients.
PeZ (pezdemo) is a MATLAB tool for pole/zero manipulation. Poles and zeros can be placed anywhere on a map of the $$z$$-plane. The corresponding time domain ($$n$$) and frequency domain ($$\hat\omega$$) plots will be displayed. When a zero pair (or pole pair) is dragged, the impulse response and frequency response plots will be updated in real time.
The connection between the $$z$$-transform domain of poles and zeros and the time domain, and also the frequency domain is illustrated with several movies where individual poles, or zeros or pole pairs of IIR filters are moved continuously.
A demo that illustrates the connection between the complex $$z$$-plane and the frequency response of a system. The frequency response is obtained by evaluating $$H(z)$$ on the unit circle in the complex $$z$$-plane.

#### Labs - MATLAB6

The objective for this lab is to build an intuitive understanding of the relationship between the location of poles and zeros in the z-domain, the impulse response $$h[n]$$ in the $$n$$-domain, and the frequency response $$H(e^{j\hat\omega})$$ (the $$\hat\omega$$-domain). A graphical user interface (GUI) called PeZ was written in MATLAB for doing interactive explorations of the three domains.
The goal of this lab is to explore the connection between the time domain (n), the frequency domain ($$\hat\omega$$), and the z-transform domain, using the GUI PeZ in MATLAB.
1. Placing Zeros: When placed on the unit circle, zeros of $$A(z)$$ will force the frequency response to be zero which can then be used to null out sinusoids at one frequency.
2. Placing Poles: When placed near the unit circle (but inside), roots of $$B(z)$$ will create peaks in the frequency response which can be used to form BPFs.
3. Designing IIR Notch Filters: requires conjugate zeros on the unit circle (UC) with a poles at the same angle, just inside the UC. The frequency response of the notch is much sharper than a nulling filter which is an FIR filter formed only from the conjugate zeros on the UC.
In this mini-project you will experiment with PeZ to learn the connection between pole-zero placement and frequency response. Given this information you will redo the Tone Removal Mini-Project using an IIR.
For this mini project you will write a simple function that listens to a tone and identifies what note it is. [Files]
For this mini project you will write a simple function that listens to a wav file of a simple song and identifies the notes being played. The wav files have some simple songs on which you can practice. [Files]
You have gotten to the point in your studies that you can understand DSP papers that appear in IEEE publications. The purpose of this project is to read one such paper and reproduce some of its results.

#### Labs - LabVIEW7

The objective for this lab is to build an intuitive understanding of the relationship between the location of poles and zeros in the z-domain, the impulse response $$h[n]$$ in the $$n$$-domain, and the frequency response $$H(e^{j\hat\omega})$$ (the $$\hat\omega$$-domain). A graphical user interface (GUI) called PeZ was written in MATLAB for doing interactive explorations of the three domains.
In this mini-project you will experiment with PeZ to learn the connection between pole-zero placement and frequency response. Given this information you will redo the Tone Removal Mini-Project using an IIR. [Files]
For this mini project you will write a simple function that listens to a tone and identifies what note it is. [Files]
For this mini project you will write a simple function that listens to a wav file of a simple song and identifies the notes being played. The wav files have some simple songs on which you can practice. [Files]
You have gotten to the point in your studies that you can understand DSP papers that appear in IEEE publications. The purpose of this project is to read one such paper and reproduce some of its results.
The goal of this mini-project is to help you understand a simple modem, the FSK modem, referred to by the International Telecommunications Union (ITU) as V.21.
The goal of this mini-project is to build a simple modem receiver. This a follow on for the previous mini-project.

format: plain


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## Geographic coordinate system

• A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position, and two or three of the numbers represent a horizontal position. A common choice of coordinates is latitude, longitude and elevation.

To specify a location on a two-dimensional map requires a map projection.

The invention of a geographic coordinate system is generally credited to Eratosthenes of Cyrene, who composed his now-lost Geography at the Library of Alexandria in the 3rd century BC. A century later, Hipparchus of Nicaea improved on this system by determining latitude from stellar measurements rather than solar altitude and determining longitude by using simultaneous timings of lunar eclipses, rather than dead reckoning. In the 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematically-plotted world map using coordinates measured east from a prime meridian at the westernmost known land, designated the Fortunate Isles, off the coast of western Africa around the Canary or Cape Verde Islands, and measured north or south of the island of Rhodes off Asia Minor. Ptolemy credited him with the full adoption of longitude and latitude, rather than measuring latitude in terms of the length of the midsummer day. Ptolemy's 2nd-century Geography used the same prime meridian but measured latitude from the equator instead. After their work was translated into Arabic in the 9th century, Al-Khwārizmī's Book of the Description of the Earth corrected Marinus' and Ptolemy's errors regarding the length of the Mediterranean Sea, causing medieval Arabic cartography to use a prime meridian around 10° east of Ptolemy's line. Mathematical cartography resumed in Europe following Maximus Planudes' recovery of Ptolemy's text a little before 1300; the text was translated into Latin at Florence by Jacobus Angelus around 1407.

Longitudinal length equivalents at selected latitudes
Latitude City Degree Minute Second ±0.0001°
60° Saint Petersburg 55.80 km 0.930 km 15.50 m 5.58 m
51° 28′ 38″ N Greenwich 69.47 km 1.158 km 19.30 m 6.95 m
45° Bordeaux 78.85 km 1.31 km 21.90 m 7.89 m
30° New Orleans 96.49 km 1.61 km 26.80 m 9.65 m
Quito 111.3 km 1.855 km 30.92 m 11.13 m

${\displaystyle 111132.92-559.82\,\cos 2\varphi +1.175\,\cos 4\varphi -0.0023\,\cos 6\varphi }$
${\displaystyle 111412.84\,\cos \varphi -93.5\,\cos 3\varphi +0.118\,\cos 5\varphi }$
${\displaystyle {\frac {\pi }{180}}M_{r}\cos \varphi \!}$
${\displaystyle {\frac {\pi }{180}}a\cos \beta \,\!}$
• The origin at the center of mass of the earth, a point close to the Earth's center of figure
• The Z axis on the line between the north and south poles, with positive values increasing northward (but does not exactly coincide with the Earth's rotational axis)
• The X and Y axes in the plane of the equator
• The X axis passing through extending from 180 degrees longitude at the equator (negative) to 0 degrees longitude (prime meridian) at the equator (positive)
• The Y axis passing through extending from 90 degrees west longitude at the equator (negative) to 90 degrees east longitude at the equator (positive)
Wikipedia


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# Python Pandas - Check if an Interval is closed on the left side

PythonServer Side ProgrammingProgramming

#### Beyond Basic Programming - Intermediate Python

Most Popular

36 Lectures 3 hours

#### Practical Machine Learning using Python

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#### Practical Data Science using Python

22 Lectures 6 hours

To check if an Interval is closed on the left side, use the interval.closed_left property. At first, import the required libraries −

import pandas as pd

Interval set using the "closed" parameter with value "left" i.e. [0, 5) is described by 0 <= x < 5 when closed='left'

interval = pd.Interval(left=0, right=20, closed='left')

Display the interval

print("Interval...",interval)

Check whether the interval is closed on the left-side

print("Checking whether the Interval is closed on the left...", interval.closed_left)

## Example

Following is the code

import pandas as pd

# Interval set using the "closed" parameter with value "left"
# i.e. [0, 5) is described by 0 <= x < 5 when closed='left'
interval = pd.Interval(left=0, right=20, closed='left')

# display the interval
print("Interval...",interval)

# display the interval length
print("Interval length...",interval.length)

# check whether the interval is closed on the left-side
print("Checking whether the Interval is closed on the left...", interval.closed_left)

## Output

This will produce the following code

Interval...
[0, 20)

Interval length...
20

Checking whether the Interval is closed on the left...
True
Updated on 20-Oct-2021 07:55:50


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# zbMATH — the first resource for mathematics

On interpolation of Banach algebras and factorization of weakly compact homomorphisms. (English) Zbl 1119.46024
The family $$(\cdot,\cdot)_\Gamma$$ of interpolation methods was introduced by J. Peetre [“A theory of interpolation of normed spaces” (Notas Mat. No. 39) (1968; Zbl 0162.44502)]. A special class of the family $$\Gamma$$ was used in [A. Blanco, S. Kaijser and T. J. Ransford, J. Funct. Anal. 217, No. 1, 126–141 (2004; Zbl 1078.46050)] to prove that weakly compact homomorphisms between Banach algebras factor through a reflexive Banach algebra. It is crucial in this factorization that the interpolation method preserves the Banach algebra structure. In the paper under review, a necessary and sufficient condition on $$\Gamma$$ for the interpolation method $$(\cdot,\cdot)_\Gamma$$ to preserve the Banach algebra structure is found. Applying this general result, it turns out that the classical real interpolation method $$(\cdot,\cdot)_{\theta,q}$$ preserves Banach algebras only if $$q=1$$. Also, some results on the factorization of weakly compact homomorphisms between Banach algebras are derived as applications.

##### MSC:
 46B70 Interpolation between normed linear spaces 46H05 General theory of topological algebras 46M35 Abstract interpolation of topological vector spaces
Full Text:
##### References:
 [1] Bergh, J.; Löfström, J., Interpolation spaces. an introduction, (1976), Springer Berlin · Zbl 0344.46071 [2] Bishop, E.A., Holomorphic completion, analytic continuation, and the interpolation of seminorms, Ann. math., 78, 468-500, (1963) · Zbl 0131.30901 [3] Blanco, A.; Kaijser, S.; Ransford, T.J., Real interpolation of Banach algebras and factorization of weakly compact homomorphisms, J. funct. anal., 217, 126-141, (2004) · Zbl 1078.46050 [4] Brudnyıˇ, Y.; Krugljak, N., Interpolation functors and interpolation spaces, vol. 1, (1991), North-Holland Amsterdam · Zbl 0743.46082 [5] Calderón, A.P., Intermediate spaces and interpolation, the complex method, Studia math., 24, 113-190, (1964) · Zbl 0204.13703 [6] Cobos, F., Some spaces in which martingale difference sequences are unconditional, Bull. Polish acad. sci. math., 34, 695-703, (1986) · Zbl 0617.46079 [7] Cobos, F.; Fernández-Cabrera, L.M.; Manzano, A.; Martínez, A., Real interpolation and closed operator ideals, J. math. pures appl., 83, 417-432, (2004) · Zbl 1051.46013 [8] Cobos, F.; Fernández-Cabrera, L.M.; Manzano, A.; Martínez, A., On interpolation of asplund operators, Math. Z., 250, 267-277, (2005) · Zbl 1078.46015 [9] Cobos, F.; Peetre, J.; Persson, L.E., On the connection between real and complex interpolation of quasi-Banach spaces, Bull. sci. math., 122, 17-37, (1998) · Zbl 0953.46037 [10] Conway, J.B., A course in functional analysis, (1990), Springer New York · Zbl 0706.46003 [11] Cwikel, M.; Peetre, J., Abstract K and J spaces, J. math. pures appl., 60, 1-50, (1981) · Zbl 0415.46052 [12] Davis, W.J.; Figiel, T.; Johnson, W.B.; Pelczyński, A., Factoring weakly compact operators, J. funct. anal., 17, 311-327, (1974) · Zbl 0306.46020 [13] Galé, J.E.; Ransford, T.J.; White, M.C., Weakly compact homomorphisms, Trans. amer. math. soc., 331, 815-824, (1992) · Zbl 0761.46037 [14] Kaijser, S., Interpolation of Banach algebras and open sets, Integral equations oper. theory, 41, 189-222, (2001) · Zbl 1022.46046 [15] Levy, M., L’espace d’interpolation réel $$(A_0, A_1)_{\theta, p}$$ contient $$\ell^p$$, C. R. acad. sci. Paris Sér. A, 289, 675-677, (1979) · Zbl 0421.46028 [16] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces, I and II, (1979), Springer New York [17] Naimark, M.A., Normed rings, (1964), P. Noordhoff N.V. Groningen · Zbl 0137.31703 [18] Nikolskii, N.K., Spectral synthesis for a shift operator and zeros in certain classes of analytic functions smooth up to the boundary, Dokl. akad. nauk SSSR, 190, 780-783, (1970) [19] Nilsson, P., Reiteration theorems for real interpolation and approximation spaces, Ann. mat. pura appl., 132, 291-330, (1982) · Zbl 0514.46049 [20] Nilsson, P., Interpolation of Calderón and ovchinnikov pairs, Ann. mat. pura appl., 134, 201-332, (1983) [21] Peetre, J., A theory of interpolation of normed spaces, Notes mat., 39, 1-86, (1968), Lecture Notes, Brasilia, 1963 · Zbl 0162.44502 [22] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46033 [23] Zafran, M., The dichotomy problem for homogeneous Banach algebras, Ann. math., 108, 97-105, (1978) · Zbl 0388.46036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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Spring 2020 ECE 264 :: Purdue University

⚠ This is a PAST SEMESTER (Spring 2020).
Due 4/2

Huffman 1: priority queue

This homework is the first in a 3-part series about data compression using Huffman coding. For this first step, you will use linked lists to implement a priority queue data structure and a stack data structure. Those are just linked lists with some rules about how elements are added and removed.

Learning goals

You should learn or practice how to:

1. Create and manipulate linked lists.
2. Avoid memory faults associated with linked lists.
3. Use function addresses to make your data structures work with any type of values.

Getting Started

Get the starter code

Start by getting the files. Type 264get hw11 and then cd hw11 from bash. This will give you huffman.h (header file) and huffman.c skeleton implementation file.

We have also added in a demostration file, demo_huffman.c, which simply illustrates how the pieces fit together. Do not copy that code. However, you may test in your account, if you wish. (demo_huffman.c was added to the starter files the morning of Mon 3/30/2020.)

Copy in your Makefile, miniunit.h, and clog.h from a previous homework.

cp ../▒▒▒/{Makefile,miniunit.h,clog.h} .
⚠ Don't forget the period at the end of that command.

The variables at the top of your file should contain the following:
 ASG_NICKNAME HW11 SRC_C huffman.c SRC_H huffman.h miniunit.h clog.h TEST_C test_huffman.c EXECUTABLE huffman

Adjust your submit rule to ensure than whenever you run make submit, the following command is executed:

264submit hw11 huffman.c huffman.h test_huffman.c miniunit.h clog.h Makefile
… but your rule should use the variables wherever possible (i.e., 264submit $(ASG_NICKNAME)$(SRC_C) $(SRC_H)$(TEST_C) Makefile))

Double-check your \$(EXECUTABLE) rule. Running make or make huffman rule should result in the following command (but using your variables wherever possible):

gcc -o huffman -g -std=c11 -Wall -Wshadow -Wvla -Werror -pedantic huffman.c test_huffman.c

Overview

In this exercise, you will use a linked list to implement a priority queue or a stack.

A priority queue is implemented as a linked list where the objects stored in the list are actually ordered. For example, if we want to store a linked list of numbers 5, 10, 4, 3, 1, and the linked list is used to implement a priority queue, the linked list should appear as follows:

1→3→4→5→10→NULL

Here, we assume that "→" is the "pointer." We also assume that a smaller number has a higher priority, and it is therefore listed first. If we now want to get number 7 to join the priority queue, we should have the list updated to:

1→3→4→5→7→10→NULL

We call this the enqueue operation. To remove an item from a priority queue, we always remove the item with the highest priority. This is call the dequeue operation. In this case, a dequeue operation will remove 1 from the list:

3→4→5→7→10→NULL

When a stack is implemented as a linked list, the object that is pushed onto the stack always appears as the first item on the list. Therefore, if we push the list of numbers 5, 10, 4, 3, 1 onto a stack successively, we have the following linked list:

1→3→4→10→5→NULL

If we pop from the stack, we have

3→4→10→5→NULL

If we now push 7 onto the stack, we have
7→3→4→10→5→NULL

In both priority queue and stack, we always remove an item from the beginning of the list because that is the most convenient place to remove an item from a list.

You will also write a function to destroy a linked list and free all memory associated with it.

Generic data types

In previous assignments with linked lists, you always specified the type of data that would be stored. In this homework, the Node will contain a void* (memory address to any type). This allows you to use the same code for linked lists of any type.

This is simlar to the way that qsort(…) allows you to sort arrays of any type.

Code that deals with generic data types often needs to pass functions as parameters. However, the declarations are UGLY.

Example #1: Declare local variable of type “address to function taking parameter types and returning return type”

Let's start with an example. In the snippet below, we have two functions, _print_square(…) and _print_cube(…). Instead of calling them directly, we will assign the address of each function to a local variable and then call them by way of that local variable, just to demonstrate the syntax.

#include <stdio.h>
#include <stdlib.h>

void _print_square(int n) {
printf("%d² is %d\n", n, n*n);
}

void _print_cube(int n) {
printf("%d³ is %d\n", n, n*n*n);
}

int main(int argc, char* argv[]) {

// Declare variable «fn» of type "address to function taking an int and returning void"
void (*fn)(int);

// Initialize fn to the address of _print_square(…).  Normally we delcare and initialize
// on the same line, but we have separated the two for the clarity of this explanation.
fn = _print_square;

// Call _print_square(…) by way of variable «fn».
fn(7);

// Now set «fn» to the address of _print_cube.
fn = _print_cube;

// Call _print_cube(…) by way of variable «fn».
fn(7);

return EXIT_SUCCESS;
}
/* vim: set tabstop=4 shiftwidth=4 fileencoding=utf-8 noexpandtab: */

The type of variable fn is void(*)(int). In words, we will say, “«fn» is the address to a function taking an int and returning void.”

Example #2: Declare parameter of type “address to function taking parameter types and returning return type”

Instead of creating a local variable, we can also use that same type (void(*fn)(int)) as a parameter to a function. In the example below, we call a function _call_print_fn(…) which then calls another function, specified by the second argument.

#include <stdio.h>
#include <stdlib.h>

void _print_square(int n) {
printf("%d² is %d\n", n, n*n);
}

void _print_cube(int n) {
printf("%d³ is %d\n", n, n*n*n);
}

void _call_print_fn(int n, void (*print_fn)(int)) {
print_fn(n);
}

int main(int argc, char* argv[]) {

// Call _print_square(…) by way of the helper function.
_call_print_fn(7, _print_square);

// Call _print_cube(…) by way of the helper function.
_call_print_fn(7, _print_cube);

return EXIT_SUCCESS;
}
/* vim: set tabstop=4 shiftwidth=4 fileencoding=utf-8 noexpandtab: */

Like the previous example, the type of parameter print_fn is void(*)(int). In words, we will say, “«print_fn» is the address to a function taking an int and returning void.”

Basic form for function address variables

The basic form for declaring a variable called Nameof type “address to function taking Parameter Types and returning Return Type” is:

Return Type(*Name)(Parameter Types)

Yes, it's UGLY. The name of that variable is in the middle!!!

Do not use parentheses when passing a function address or assigning it to a variable. In the code above, _call_print_fn(7, _print_cube(…)) will not work.

Passing a function address is totally different from invoking your mu_run(…) macro with a function (e.g., mu_run(_test_empty_list)). Although mu_run(…) looks like a function, it is really just transorming your code so that function can be called elsewhere and/or its name can be printed. In contrast, passing a function address is passing an actual address. It is a number that can be printed using printf(…), if you wish.

Use static in definitions of helper functions

Starting with HW11, all helper functions must be declared as static in any assignment that involves compiling ≥2 files together.

 void _know_your_worth() { ▒▒▒▒▒▒ } ⇒ static void _know_your_worth() { ▒▒▒▒▒▒ }

That tells the compiler the function should only be available from the current file. Code in other files won't be able to call it.

Requirements

1. Your submission must contain each of the following files, as specified:
file contents
huffman.c functions
pq enqueue(Node✶✶ a head, void✶ a value, int(✶cmp fn)(const void✶, const void✶)))
return type: Node✶
Add a new node with value a_value to a priority queue, using function cmp_fn(…) to determine the ordering of the priority queue.
2. If *a_head is NULL, the list is empty.
3. a_value is the address of whatever value is associated with this node.
4. cmp_fn(…) is a comparison function with the same semantics as the comparison function for qsort(…).
5. The newly allocated Node should be inserted into the linked list such that all Nodes before the new Node in the list contain objects that are smaller (or equal) to the new Node (according to cmp_fn(…)). All Node's after the new Node in the list contain objects that are bigger (or equal) to the new Node (according to cmp_fn(…)).
6. *a_head should be updated if the new Node becomes the first item in the list. The function should return the address of the new Node.
7. Hint: You will call malloc(…) exactly once. You will not call free(…) in this function.
pq dequeue(Node✶✶ a head)
return type: Node✶
2. If the list is empty, return NULL (since there is nothing to dequeue).
3. Upon return, *a_head must be a valid linked list (possibly empty).
4. Reminder: NULL is a valid linked list of size 0. Thus:
1. *a_head will be NULL if the list is empty.
2. Upon removing the last node, you should set *a_head to NULL.
5. You must set the next field of the removed Node to NULL.
6. The caller is responsible for freeing the detached node, and any memory it refers to.
1. This function should not call free(…), directly or indirectly.
7. You are effectively splitting a linked list into two lists.
8. Hint: You should not call malloc(…) or free(…) in this function.
stack push(Node✶✶ a top, void✶ a value)
return type: Node✶
1. *a_top stores the address of the first Node in the linked list.
2. a_value stores the address of the generic type.
3. The a_value field of the newly allocated Node should store a_value.
4. The newly allocated Node should become the first Node of the list, and *a_top should be updated. The function returns the address of the new Node.
5. Hint: You will call malloc(…) exactly once. You will not call free(…) in this function.
stack pop(Node✶✶ stack))
return type: Node✶
Detach and return the node from the head of the linked list.
1. This is very similar to pq_dequeue(…).
destroy list(Node✶✶ a head, void (✶destroy fn)(void✶))
return type: void
1. *a_head stores the address of the first Node in the list.
2. destroy_fn(…) is a function that deallocates *a_value, as needed.
3. Set the head to NULL in the caller's stack frame (i.e., *a_head = NULL;).
test_huffman.c functions
main(int argc, char✶ argv[])
return type: int
Test all of the required functions in your huffman.c.
• 100% code coverage is required.
• Use miniunit.h.
2. Note: The names of parameters in the function definition do not need to match the declaration. Don't worry if you see a parameter listed as a_head in one place and a_top in another.
3. All helper functions should be declared as static.
4. Do not modify huffman.h.
5. Submissions must meet the code quality standards and the course policies on homework and academic integrity.

Examples of cmp_fn(…)

We have provided examples of destroy_fn(…) functions. Here are some examples of cmp_fn(…) (for the pq_enqueue(…) function) and print_fn(…). If we store the addresses of int (i.e., int*) as the addresses of the generic type (void*) in .a_value, we can use the following comparison function:

static int int_cmp(const void* p1, const void* p2) {
return *(const int*)p1 - *(const int*)p2;
}
Here, the addresses p1 and p2 have been typecast to const int*, and dereferenced to obtain const int for comparison (with the difference being the result). If we store the addresses of char (i.e., char*) as the addresses of the generic type (void*) in .a_value, and each address of char is actually the address of the first character of string, we can use the following comparison function:
static int _string_cmp(const void* p1, const void* p2) {
return strcmp((const char*)p1, (const char*)p2);
}

Submit

To submit HW11 from within your hw11 directory, type

make submit

That should result in the following command: 264submit HW11 huffman.c test_huffman.c miniunit.h clog.h Makefile huffman.h

Pre-tester ●

The pre-tester for HW11 has been released and is ready to use.

Q&A

1. How much work is this?

The instructor's solution (shown below) occupies 42 sloc (source lines of code, excluding comments and blank lines). We sometimes use the size of the "shortest defensible solution" as a rough (and certainly imperfect) measure of difficulty.

👌 Okay to copy/adapt anything you find in the screenshot above (in this Q&A item). Granted there is almost nothing to copy, but you may find the comments helpful to steer you away from needlessly complicated approaches. There are many other completely correct ways to solve this.
2. What is the hardest part?

The pq_enqueue(…) function is the most difficult.
3. Can I make this easier?

Yes! Use TDD to test and build in lockstep. Write a test. Then, write just enough implementation code to pass that test (and any prior tests).
Do not try to code it all at once. Start by getting it working for the simplest possible test. This is probably adding a single node (i.e., adding a node to an empty list). Test that it works before you move on. Next, get it working and tested for a list of size two, where the items are assumed to be added in descending order (so that no reordering is necessary). Get it working before you move on.
Once you get into the longer test cases that require placing the new node in the right position in the list, you should draw pictures. Walk through how the connections need to change on paper before attempting to do it in code.
4. How do the compare and destroy functions fit together with the rest of this code?

Here's a code example.
DO NOT COPY CODE FROM THIS FILE INTO YOUR OWN CODE. We are providing this only to clarify how the pieces are supposed to fit together. It would not be a good basis for your test_huffman.c. However, you may run this code in your account. It has been added to the starter code.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "huffman.h"

static int _cmp_strings_by_length(const void* a_lhs, const void* a_rhs) {
size_t len_lhs = strlen(a_lhs); // No typecast needed!  Never use typedcast unless:
size_t len_rhs = strlen(a_rhs); // (a) necessary, (b) safe, and (c) well-understood.
return len_lhs - len_rhs; // shortest string comes forst
}

static void _destroy_string(void* string) {
free(string);
}

static char* _copy_string(const char* src_string) {
size_t num_chars = strlen(src_string);
char* dst_string = malloc(sizeof(*dst_string) * (num_chars + 1));
return strcpy(dst_string, src_string);
}

int main(int argc, char* argv[]) {
// Create a priority queue of four strings, ordered by length (ascending).
Node* head = NULL;  // size 0
pq_enqueue(&head, _copy_string("Reginald"), _cmp_strings_by_length);  // size 1
pq_enqueue(&head, _copy_string("was"),      _cmp_strings_by_length);  // size 2
pq_enqueue(&head, _copy_string("a"),        _cmp_strings_by_length);  // size 3
pq_enqueue(&head, _copy_string("hatter"),   _cmp_strings_by_length);  // size 4

// Print contents of the list:  a was hatter Reginald
for(Node* curr = head; curr != NULL; curr = curr -> next) {
printf("%s\n", (char*)(curr -> a_value));
}

// Deallocate all memory used by the list, including each node's payload.

return EXIT_SUCCESS;
}

/* ______
* Output
* ‾‾‾‾‾‾
* a
* was
* hatter
* Reginald
*/

/* vim: set tabstop=4 shiftwidth=4 fileencoding=utf-8 noexpandtab: */

3/28/2020: Relative to the “preview” version of this homework: ① Two functions (remove_last_node(…) and print_node(…)) were removed. ② pq_enqueue(…) was simplified. Previously, it had some odd semantics regarding NULL. ③ expected.txt was removed. ④ Several parameters have been renamed to consistent with the conventions we are following this semester. ⑤ Your helper functions must now be static. ⚠ Be on the lookout for any inconsistencies that might be lingering.
3/29/2020: Added some code snippets and a redacted screenshot in response to some questions that came up on Piazza. We also added more explanation about using static with helper functions.
3/30/2020: Fixed typo in pq_enqueue(…) description (“*a_head should be updated…”).
4/2/2020: Fixed minor typo in mini-Q&A about static: “Only files in the same functions defined in the same file will be able to ‘see’ it.”


text:
Ask Your Question

# johanneh's profile - activity

 2020-12-15 07:52:54 +0200 received badge ● Popular Question (source) 2018-11-13 22:17:02 +0200 received badge ● Supporter (source) 2018-11-13 22:16:59 +0200 received badge ● Scholar (source) 2018-11-12 18:21:14 +0200 asked a question Express the derivative in the Hamiltonian operator in sage I want to compute the expectation value of the energy $\langle H \rangle$ for a quantum system with the wave function $\psi(x)$. This is done with $$\langle H \rangle = \int_{-\infty}^{\infty} \psi^* \hat{H} \psi d x$$ where $$\hat{H} = - \frac{\hbar^2}{2 m} \frac{d^2}{d x^2} + V(x)$$ is the Hamiltonian operator for some potential $V(x)$. How do I express $\frac{d}{d x}$ in sage if I don't know on which function it will be applied? I know it is possible to use psi.diff(x), but is there a way to declare the operator so that it automatically does the derivative when it is multiplied with some function? 2018-05-13 12:53:52 +0200 received badge ● Student (source) 2018-05-13 07:30:04 +0200 asked a question Define custom unit Is it possible to declare custom units that can be converted to SI base units? For example: if I want to measure time in ticks and I know that one tick is exactly one nanosecond, can I create a unit 'tick' that can later be converted to seconds? I know that a custom unit can be declared with sage: sage.symbolic.units.str_to_unit('tick') tick  I also know that predefined units can be converted, for example by sage: units.time.minute.convert(units.time.second) 60*second  However, I want to be able to declare a unit that has both its own name and that can be converted to other units. sage: tick = sage.symbolic.units.str_to_unit('tick') sage: tick tick sage: #(define relation to SI units here) sage: tick.convert(units.time.second) (ValueError: Incompatible units)  Is it possible to define a unit ‘tick’ that can be converted to seconds in this manner?


text:
# Vowelburgers™ Chef's Special #3

Inspired by Avi's Vowelburger™ Riddle. Hope you enjoyed our previous Chef's Specials: #1 and #2
Our chef went way ahead to prepare the biggest Vowelburger™ menu to date! There is a burger for every size. The chef's choice of the day consists of a vowel patty, cheese, lettuce, and a special bun. The variety lies in the other bun.

Bun Vowelburger™
? diminutive
? tiny
? small
? normal
? medium
? big
? large
? roomy
? whopper
? full

Not only are the contents of the chef's choice fixed, but their order is fixed as well. There are no anagrams in this menu; the other bun is strictly a consonant. It's amazing how the chef has brought so much variety with these constraints!

Can you identify each one from the description on the menu?

• I assume this isn't a trick like they're all the same word with size prefixes? k for kilo, M for mega, G for giga, m for milli, n for nano, etc. – Rand al'Thor Mar 20 '20 at 13:10
• @Randal'Thor That’s actually not a bad idea but it would be awful for this riddle. Having said, the rest of the word wouldn’t make sense without the bun. If I state any more, I’d probably giving the answer away ;-) – Ébe Isaac Mar 20 '20 at 14:27

Another possibility:

Variable first bun, I patty, G cheese, H lettuce, and T second bun.

The burgers are:

DIGHT (archaic word for dress or adorn), TIGHT, SIGHT, NIGHT, MIGHT, BIGHT (a bay formed from a bend in the coast), LIGHT, RIGHT, WIGHT (an unfortunate person or a ghost / supernatural being), FIGHT - the first bun matches the first letter of the description.

• Congratulations! Hope you enjoyed our service! – Ébe Isaac Mar 21 '20 at 7:38

I believe the chef's choice is

an I patty, L lettuce, L cheese, and an S special bun

diminutive - GILLS
tiny - PILLS
small - DILLS
normal - BILLS
medium - JILLS
big - SILLS
large - MILLS
roomy - HILLS
whopper - VILLS
full - FILLS

• Nice try indeed, but could you explain how the burger name matches with the description? – Ébe Isaac Mar 20 '20 at 10:18
• @ÉbeIsaac I assume that means you're looking for there to be more of a connection than just the size of the objects? – DenverCoder1 Mar 20 '20 at 10:24
• Now you are heading along the right direction ;) – Ébe Isaac Mar 20 '20 at 10:25
• Here's a thought: rot13(Vs lbh rng rirelguvat ba gur zrah, lbh'yy cebonoyl unir znal VYYF.) :) – MacGyver88 Mar 20 '20 at 20:09


text:
# what is the graph of $y = x^x$?

Obviously on the RHS this graph is just a really steep exponential graph however problems arise on the LHS and I cannot find any graph sketching programs that can do. Some will give a graph but then simply say that the LHS is undefined which must be incorrect since negative values with odd powers must still work like $-3^{-3} = -1/27$ but then of course values like $(-1/2)^{-1/2}$ do not. I asked my maths teacher about this and my tutor and both didn't seem to have answers.

• Do you known what is the definition of $x^x$?
– user9464
Jun 4, 2016 at 1:09
• You might want to see this. Jun 4, 2016 at 1:20
• You are failing to distinguish between $(-1/2)^{-1/2}$ and $-1/2^{-1/2}$. The latter means $-1/(2^{-1/2})$, which is the same as $-(1/2)^{-1/2}$, but different from $(-1/2)^{-1/2}$. $\qquad$ Jun 4, 2016 at 1:45
• Note that on the interval $0<x<\text{a certain positive number}$, $x^x$ actually decreases as $x$ increases. $\qquad$ Jun 4, 2016 at 1:48
• yeah thats intresting as well. At first I thought it would decrease between 0 < x < 1 but it seems to around x = 0.4. And this is my first time using the site sorry. Thanks for knowing what i mean. Jun 5, 2016 at 23:26

When $x<0, x^x$ is undefined (in the real numbers) for most values of $x.$ It is defined for negative odd integers.
However in the complex numbers $x^x$ is defined for negative values of $x.$ It is essentially a spiral in the complex plane, touching the real axis when $x$ is a negative odd integer.
Yes, $(-3)^{-3}$ can be defined, but problems arise when you try to calculate rational powers of some negative value. For instance, think of $(-\frac{1}{2})^{-1/2}$. This can be written as $\sqrt{-2}$. As you can see, this is not a real value. This is why most graph sketching programs don't give us the graph for $x<0$.


text:
#### Archived

This topic is now archived and is closed to further replies.

# Normals

This topic is 6134 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

What is a normalized vector, why are they used? When the direction for the normal is entered, should the position be relative to the 0, 0, 0 co-ordinate, or does it work in some completly different way? And umm.... do I have to know anything else to get those normals working?

##### Share on other sites
Vectors are a direction, not a coordinate - so they are always relative to 0,0,0. Normalised vectors are a vector divided by its length - ie. the normalised vector has a length of 1. Use pythagoras to do this. For example, if you have a 2d vector vy,vx to get the length you have to do this: l = sqrt((vy*vy)+(vx*vx)). To normalise the vector, you just do: vx/=l; vy/=l;

Hope this helps!

##### Share on other sites
code snippet.

  D3DXVECTOR3 normal; D3DXVECTOR3 polyvecs[3]; // to normalise D3DXVECTOR3 vec1,vec2; vec1 = polyvecs[1] - polyvecs[0]; vec2 = polyvecs[2] - polyvecs[0]; // cant remember the actual function, // its just a simple cross product. D3DCrossProduct(normal,vec1,vec2); // normalise in the sense to remove anomolies. Normalise(normal);

{ Stating the obvious never helped any situation !! }

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text:
# Initial Stages 2021

10-15 January 2021
Weizmann Institute of Science
Asia/Jerusalem timezone
Please read "Instructions for..." at the bottom of the left-side menu.

## Towards fully 3-D simulations of heavy-ion collisions in the IP-Glasma Initial State

12 Jan 2021, 17:40
20m
Andrea's room 2 (vDLCC)

### Andrea's room 2

#### vDLCC

oral The initial stages of heavy-ion collisions

### Speaker

Sangyong Jeon (McGill University)

### Description

In this talk, we report on phenomenological studies using the recently developed 3-D IP-Glasma initial state. Our 3-D formulation generalizes the boost invariant formulation by including both the rapidity evolution (JIMWLK) and the temporal evolution (Classical Yang-Mills) in the longitudinal direction. Special attention is paid to the local realization of Gauss law in a 3D+1 environment. Hydrodynamic evolution was carried out using MUSIC and the hadronic interaction was carried out using UrQMD to simulate full heavy ion collisions. Comparisons with the $\sqrt{s} = 2.76$ TeV LHC data including spectra, flow harmonics and correlations will be discussed.

### Primary authors

Sangyong Jeon (McGill University) Prof. Charles Gale (McGill University) Scott McDonald (McGill University)

### Presentation Materials

 Recording talk_IS2021_3d.pdf


text:
[ Basic Info | User Guide ]

# Basic Information on gpdof

Purpose: Compute Degrees-of-freedom for calibration solutions.
Categories: calibration

GPDOF estimates the likely error in a calibration solution, by
calculating the errors in the "Degrees-of-Freedom" (dofs) of an
interferometer. This is useful for calibration of high-precision
circular polarization observations.

What GPDOF does is to determine the average calibration solution
of several input solutions. Then, assuming that this average
solution is "ideal", GPDoF determines the error in each of the
input calibration solutions, using the degrees-of-freedom
notation. Finally, GPDoF calculates the std and standard error
of the derived dofs, which you can use to estimate the errors in
the Stokes V of a target observation.

The input calibration solutions should have been processed using
the method outlined in the "Circular Polarization User's Guide"
(see References below). Obviously, the more calibrators you
have, the more accurate the dof estimates will be.

Explanation:

The degrees-of-freedom (dofs) are linear combinations of leakage
(delta) or gain (gamma) errors. The advantages in the dofs over
other measures of calibration consistency (eg. rms scatter in
the leakages) is that the dofs actually describe how the
calibration error would affect a target observation.  Thus, for
the dofs delta-+, delta--, and gamma--, and equatorially-mounted
antennas:

Stokes V error = delta-+ * Stokes I +
delta-- * Stokes Q +
gamma-- * Stokes U

For the ATCA, the parallactic angle \chi comes, in so it
isn't quite so elegant:

Stokes V error = delta-+ * I +
delta-- * ( Q.sin(2\chi) + U.cos(2\chi)) +
gamma-- * ( U.sin(2\chi) + Q.cos(2\chi))

Note that these are the instantaneous calibration errors in
a target source; for a synthesis observation, where \chi
varies, the resultant error in the Stokes V image will be
more complicated.

GPDoF estimates the std and standard error in the dofs. These
parameters be combined with the above equation for Stokes V
Error to estimate the likely level of leakage errors in a target
observation. If you are just using the solution from a single
calibrator, calculate using the std. If you are going to use the
average calibration solution (using GPComb), then calculate
using the standard error.

For each dof, GPDoF has an "error" estimate, which measures the
variations in the calibration solutions which are NOT
attributable to a dof. The physical significance of these
quantities isn't even clear to the author. In general, however,
if the "error" term is dominant, then un-dof-like errors are
dominating the calibration eg. pure random errors, or bizzarro
systematic errors. In this situation, the relationship between
the dofs and the error in the target observation may break down.

Um, I did mention that there are actually seven dofs? GPDoF only
computes the three which affect calibration of circular
polarization.  There are some good reasons for this (some of the
others can't easily be computed, or are time-variable), besides
the obvious one that I'm too lazy.

Note that GPDoF is entirely ATCAcentric.

Note that the leakage errors derived from GPDoF are not the only
errors in a circular polarization observation! See Equation 1 of
Rayner et al, 2000. A task to estimate the effect of the dofs on
a synthesis observation will hopefully become available
soon. Actually, forget the "soon"...

And finally, if you have consistent systematic errors in your
solutions (eg. you haven't used xyref,polref in gpcal), then
GPDoF will severely underestimate the leakage errors.

References:

For an explanation of degrees of freedom, see:

Sault, R J, "The Hamaker-Bregman-Sault Measurement Equation",
pp 657--699, in Syhthesis Imaging in Radio Astronomy II, 1999,
eds. Taylor, Carilli,and Perley.

or

Sault, Hamaker and Bregman, "Understanding radio polarimetry II.
Instrumental calibration of an interferometer array", 1996
A&AS, v117, pp149-159.
ftp://ftp.atnf.csiro.au/pub/people/rsault/papers/polar2.ps.gz

For an overview of high-precision circular polarization
calibration with the ATCA, see

Rayner, "Circular Polarization User's Guide", ATNF Technical
Document Series, 2000, 39.3/102,
http://www.atnf.csiro.au/people/drayner/Publications/
index.html#Circular_Polarization_Users_Guide

For a summary of the ATCA circular polarization error budget,
see Equation 1 of

Rayner, Norris, Sault, "Radio circular polarization of active
galaxies", 2000, v319, pp484-496
http://www.atnf.csiro.au/people/drayner/Publications/mnr3854.pdf

Key: cal
The data-sets containing the nominally correct polarization
calibration. No default.

Key: select
Normal uv selection. Only antenna-based selection is supported.

Generated by miriad@atnf.csiro.au on 21 Jun 2016


text:
# Midpoint 5

FM=3x-4, MG=5x-26, FG=? Point M is the midpoint of FG. Use the given information to find the missing measure or value.

Result

m =  58

#### Solution:

$x=|FG| \ \\ |FM|=|MG| \ \\ \ \\ \ \\ 3 \cdot \ x-4=5 \cdot \ x-26 \ \\ \ \\ 2x=22 \ \\ \ \\ x=11 \ \\ \ \\ m=2 \cdot \ (3 \cdot \ x-4)=2 \cdot \ (3 \cdot \ 11-4)=58$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?
Do you want to convert length units?

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f we add 20, we get 50% of its triple. What is this unknown number?
13. Poplar
How tall is a poplar by the river, if we know that 1/5 of its total height is a trunk, 1/10th of the height is the root and 35m from the trunk to the top of the poplar?
14. Eight years ago
The son asks his father how old he is. My father said: In ten years, I will have twice as much as I had eight years ago.
15. Goods
To the market is introducing a new product, the first week is sold at a reduced price. 8 products is sell at an entry price same as 5 products at the normal price. How much % is reduced price less than the current price for this product?
16. Cat
One-fifth of the monthly pocket money contributes Maros for food for his cat, half of the rest postpone for a new smartphone. The remaining € 8 spend. How much pocket money gets Maros a month?
17. Bag of peanuts
Joe eat 1/3 of a bag of peanuts, mark eat 1/4 of the remaining in the bag of peanuts, Alvin eat 1/2 of the remaining bag of peanuts, peter eat 10 peanuts, there are 71 peanuts left. Hon many peanuts were in the bags?


text:
# American Institute of Mathematical Sciences

January & February  2008, 22(1&2): 89-100. doi: 10.3934/dcds.2008.22.89

## Dynamical coherence and center bunching

 1 Department of Mathematics, Northwestern University, Evanston, IL 60208-2730 2 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730

Received  July 2007 Revised  January 2008 Published  June 2008

This paper discusses relationships among the basic notions that have been important in recent investigations of the ergodicity of volume-preserving partially hyperbolic diffeomorphisms. In particular we survey the possible definitions of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.
Citation: Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
 [1] Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187 [2] Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403 [3] Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845 [4] Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901 [5] Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227 [6] Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527 [7] Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55 [8] Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1 [9] Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621 [10] Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599 [11] Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641 [12] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [13] Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81 [14] Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175 [15] Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63 [16] C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897 [17] Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233 [18] Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 [19] Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934/dcdsb.2017029 [20] Zhenzhong Zhang, Enhua Zhang, Jinying Tong. Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2433-2455. doi: 10.3934/dcdsb.2018053

2018 Impact Factor: 1.143


text:
# Tag Info

0

Why don't we call the fermions in the standard model force carriers? Because the Standard Model is what it is. By the way, I think it's far less complete than people make it out to be, and that it comes with some unfortunate baggage. Consider for example an electron and a positron interacting. People say they interact via virtual-photon force carriers: ...

0

If we identify a force as a scattering process, i.e. with a mediator of some interactions, then this need not be a vector boson of course. One can speak of "Higgs" force for instance if the process under consideration is mediated by the the Higgs (which is a scalar). There are also numerous cases where the interaction is mediated by a fermion. Therefore, ...

1

Maybe there IS an effect similar to what you describe. In photoelectric absorption, the permissible departing electron states in a material determine the probability of absorption (the rate). This is easily seen in X-ray absorption near a photoelectric excitation threshold (like a K-edge), and is the basis of XAFS (X-ray absorption fine structure) ...

1

It might help if you provided context or references for your issue. As it stands, for non vanishing m, which is necessary if you are talking about spin and not helicity, the first equation is a definition of the auxiliary field function $$\psi^\mu \equiv \frac{\hat{p}^\mu}{mc} ~\phi$$ which makes the second equation the K-G equation for φ. I'm not sure ...

1

This is a very tough question, since you do not make any further assumptions about the force - most importantly its strength, sign, variation with distance and objects it acts upon. In general, yes, the universe would probably look absolutely different and we would not be here to ask this question. Somebody or something absolutely else might. Our ...

4

The situation is not symmetric at all: This diagram describes a force between two fermions, but a diagram such as just doesn't exist (in the Standard Model). Fermions can in fact mediate a force between bosons, like in: Such diagrams are highly suppressed loop diagrams though, and the one above would after renormalization be seen as just one ...

3

I think that this is more about the historical construction of the theory than about the actual interactions. In a lagrangian, two fields A, B interact when there is a product term of both such as AB. So, I see no real fundamental distinction there, even with more complicated expressions. But when one introduces the interaction bosons, it's by the mean of ...

Top 50 recent answers are included


text:
# Postgresql: Scripting psql execution with password

How can I call psql so that it doesn't prompt for a password?

This is what I have:

psql -Umyuser < myscript.sql

However, I couldn't find the argument that passes the password, and so psql always prompts for it.

P.S.: I'm using PostgreSQL 9.0

-
I ended up going for the PGPASSWORD environment variable. This fitted my usecase perfectly. Simple and self-contained in the script. –  Axel Fontaine Jan 1 '12 at 5:03

There are several ways to authenticate to postgresql. You may wish to investigate alternatives to password authentication at http://www.postgresql.org/docs/current/static/client-authentication.html.

There is no option to provide the password as a command line argument because that information is often available to all users, and therefore insecure. However, in Linux/Unix environments you can provide an environment variable for a single command like this:

-
I think PGPASSWORD is deprecated but still works, btw. Just FYI –  Scott Marlowe Jun 29 '11 at 17:16
Yep, it's deprecated (and so noted in one of the links). Since it's come up, it's probably also worth noting that the deprecation is hotly contested because it's extremely useful for many people yet can be used in some circumstances without serious security concerns. It seems to me that it's no worse than storing .pgpass on an NFS filesystem, for example. I use PGPASSWORD routinely. –  Reece Jun 29 '11 at 18:35

You can add this command line at the begining of your script:

set PGPASSWORD=[your password]

-
in my case set command did not work but export PGPASSWORD=[password] did work –  Can Kavaklıoğlu Jun 3 '13 at 9:09
SET should work on Windows. –  Hosam Aly Jan 23 '14 at 19:05

-
PGPASSWORD=[your password] psql -Umyuser < myscript.sql

-

Given the security concerns about using the PGPASSWORD environment variable, I think the best overall solution is as follows:

1. Write your own temporary pgpass file with the password you want to use.
2. Use the PGPASSFILE environment variable to tell psql to use that file.
3. Remove the temporary pgpass file

There are a couple points of note here. Step 1 is there to avoid mucking with the user's ~/.pgpass file that might exist. You also must make sure that the file has permissions 0600 or less.

Some have suggested leveraging bash to shortcut this as follows:

PGPASSFILE=<(echo myserver:5432:mydb:jdoe:password) psql -h myserver -U jdoe -p 5432 mydb

This uses the <() syntax to avoid needing to write the data to an actual file. But it doesn't work because psql checks what file is being used and will throw an error like this:

WARNING: password file "/dev/fd/63" is not a plain file

-

If you intend on having multiple hosts/database connections, the ~/.pgpass file is the way to go.

Steps:

2. Do not add string quotes around your field values. You can also use * as a wildcard for your port/database fields.
3. Create an alias in your bash profile that runs your psql command for you. For example:alias postygresy='psql --host hostname database_name -U username' The values should match those that you inputted to the ~/.pgpass file.
5. Type your alias from the command line.

Note that if you have an export PGPASSWORD='' variable set, it will take precedence over the file. It is also wise to change the permissions of your file so that the contents are obscured from other users. This can be achieved with chmod 600 ~/.pgpass

-

If you're having problems on windows like me (I'm using Windows 7 64-bit) and set PGPASSWORD=[Password] did not work.

Then, as Kavaklioglu said in one of the comments,

export PGPASSWORD=[password]

Certainly does work on windows :)

-
export PGPASSWORD=[password] does not work for me using the command line (cmd.exe) at all. Are you sure you weren't using cygwin or something similar? –  Devin Snyder Nov 18 '14 at 23:57
Only works with cl, you added it to the file right? Now just typed it into command? –  Jamie Hutber Nov 19 '14 at 9:06


text:
# Toward Moderate Overparameterization: Global Convergence Guarantees for Training Shallow Neural Networks

@article{Oymak2020TowardMO,
title={Toward Moderate Overparameterization: Global Convergence Guarantees for Training Shallow Neural Networks},
author={Samet Oymak and Mahdi Soltanolkotabi},
journal={IEEE Journal on Selected Areas in Information Theory},
year={2020},
volume={1},
pages={84-105}
}
• Published 2020
• Computer Science, Mathematics
• IEEE Journal on Selected Areas in Information Theory
Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the capacity to fit any set of labels including random noise. However, given the highly nonconvex nature of the training landscape it is not clear what level and kind of overparameterization is required for first order methods to converge to a global optima that… Expand
151 Citations

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#### References

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A Convergence Theory for Deep Learning via Over-Parameterization
• Computer Science, Mathematics
• ICML
• 2019
This work proves why stochastic gradient descent can find global minima on the training objective of DNNs in $\textit{polynomial time}$ and implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. Expand
SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data
• Computer Science, Mathematics
• ICLR
• 2018
This work proves convergence rates of SGD to a global minimum and provides generalization guarantees for this global minimum that are independent of the network size, and shows that SGD can avoid overfitting despite the high capacity of the model. Expand
An Improved Analysis of Training Over-parameterized Deep Neural Networks
• Computer Science, Mathematics
• NeurIPS
• 2019
An improved analysis of the global convergence of (stochastic) gradient descent for training deep neural networks, which only requires a milder over-parameterization condition than previous work in terms of the training sample size and other problem-dependent parameters is provided. Expand
Theoretical Insights Into the Optimization Landscape of Over-Parameterized Shallow Neural Networks
• Computer Science, Mathematics
• IEEE Transactions on Information Theory
• 2019
It is shown that with the quadratic activations, the optimization landscape of training, such shallow neural networks, has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. Expand
Gradient Descent Provably Optimizes Over-parameterized Neural Networks
• Computer Science, Mathematics
• ICLR
• 2019
Over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. Expand
Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?
• Computer Science, Mathematics
• ICML
• 2019
This paper demonstrates the utility of the general theory of (stochastic) gradient descent for a variety of problem domains spanning low-rank matrix recovery to neural network training and develops novel martingale techniques that guarantee SGD never leaves a small neighborhood of the initialization, even with rather large learning rates. Expand
Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers
• Computer Science, Mathematics
• NeurIPS
• 2019
It is proved that overparameterized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations, and SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. Expand
Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data
• Computer Science, Mathematics
• NeurIPS
• 2018
It is proved that SGD learns a network with a small generalization error, albeit the network has enough capacity to fit arbitrary labels, when the data comes from mixtures of well-separated distributions. Expand
Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian
• Computer Science, Mathematics
• ArXiv
• 2019
A data-dependent optimization and generalization theory which leverages the low-rank structure of the Jacobian matrix associated with the network and shows that even constant width neural nets can provably generalize for sufficiently nice datasets. Expand
Local Geometry of One-Hidden-Layer Neural Networks for Logistic Regression
• Mathematics, Computer Science
• ArXiv
• 2018
This work proves that under Gaussian input, the empirical risk function employing quadratic loss exhibits strong convexity and smoothness uniformly in a local neighborhood of the ground truth, for a class of smooth activation functions satisfying certain properties, including sigmoid and tanh, as soon as the sample complexity is sufficiently large. Expand


text:
# [SOLVED]Triangle inequality question

#### dwsmith

##### Well-known member
I am trying to show $|(n+z)^2|\leq (n -|z|)^2$ where is complex

$|(n+z)^2| = |n^2 + 2nz + z^2| \leq n^2 + 2n|z| + |z|^2$ But I can't figure out the connection for the final piece.

Last edited:

#### Plato

##### Well-known member
MHB Math Helper
I am trying to show $|(n+z)^2|\leq n^2 -|z|^2$ where is complex
I don't think that you really mean that.
$|(2+i)^2|=|3+4i|=5$
$(2)^2-|i|^2=3$

#### dwsmith

##### Well-known member
I don't think that you really mean that.
$|(2+i)^2|=|3+4i|=5$
$(2)^2-|i|^2=3$
This may help (below) then but I figure the above was just a straight application.
$$\sum_{n = N+1}^{ \infty}\frac{1}{(z+n)^2}$$
$R>0$ and $N>2R$. For the inequality, we are assuming $|z|<R$. Additionally, $n>N$.

So I want to show
$$\left|\frac{1}{(z+n)^2}\right|\leq\frac{1}{(n-|z|^)2}$$
So if that is true, then $|(z+n)^2|\leq (n -|z|)^2$.

---------- Post added at 04:22 PM ---------- Previous post was at 03:25 PM ----------

$|(n + z)^2|\leq n^2 + 2n|z| + |z|^2\leq n^2 - 2n|z| + |z|^2\leq (n - |z|)^2$

Is this correct?

Last edited:

#### Prove It

##### Well-known member
MHB Math Helper
I am trying to show $|(n+z)^2|\leq (n -|z|)^2$ where is complex

$|(n+z)^2| = |n^2 + 2nz + z^2| \leq n^2 + 2n|z| + |z|^2$ But I can't figure out the connection for the final piece.
Are n and z both complex? Or is only z complex?

#### dwsmith

##### Well-known member
Are n and z both complex? Or is only z complex?
z and n is a natural


text:
# Can a black hole be explained by Newtonian gravity?

In the simple explanation that a black hole appears when a big star collapses under missing internal pressure and huge gravity, I can't see any need to invoke relativity. Is this correct?

By a coincidence, the radius of a "Newtonian black hole" is the same as the radius of the Schwarzschild black hole in general relativity. We demand the escape velocity $v$ to be the speed of light $c$, so the potential energy $GMm/R = mc^2/2$, i.e. $$R = \frac{2GM}{c^2}$$ The agreement, especially when it comes to the numerical factor of $2$, is a coincidence. But one must appreciate that these are totally different theories. In particular, there's nothing special about the speed $c$ in the Newtonian (nonrelativistic) gravity. To be specific, objects are always allowed to move faster than $c$ which means that they may always escape the would-be black hole. There are no real black holes (object from which nothing can escape) in Newton's gravity.

You can escape from a Newtonian black hole. The escape velocity may be c, but you could still escape at sublight speeds with a powerful enough rocket and enough fuel. By contrast, once you've crossed the event horizon of a real black hole there is nothing you can do to avoid hitting the singularity.

• You should say there is nothing you can do to move toward smaller r. It is not clear that anything massive hits the singularity for realistic rotating or charged black holes. Jan 13 '12 at 1:08
• Oops yes, my comment is only true for the Schwartzchild geometry. Jan 13 '12 at 10:56

The interesting properties of a black hole cannot be explained by Newtonian gravity. The behaviour of bodies with mass and of light is completely different near a compact, massive object if you use Newtonian physics rather than General Relativity.

Features that GR predicts (and which in some cases have now been observationally confirmed) but which Newtonian physics cannot:

1. Most importantly, an event horizon. In Newtonian physics there is a misleading numerical coincidence that the escape velocity reaches the speed of light at the Schwarzschild radius. But in Newtonian physics you could still escape just by applying a constant thrust. GR predicts that no escape is possible in any circumstances. In fact the "coincidence" that light cannot escape at the same radius that Newtonian physics predicts an escape speed of $$c$$ only works for light travelling radially. At other angles the light will not escape unless emitted from a larger starting radius where Newtonian physics would predict an escape speed $$.

2. GR predicts an innermost stable circular orbit. A stable circular orbit would be possible at any radius in Newtonian physics. This behaviour is important for explaining the accretion phenomena observed for black holes in binary systems.

3. In GR a particle with some angular momentum and lots of kinetic energy will end up falling into the black hole. In Newtonian physics it will scatter away to infinity.

4. Newtonian physics predicts no precession of a two-body elliptical orbit. GR predicts orbital precession. This precession is measured in Mercury and other Solar System bodies, but has now been measured for stellar orbits around the Milky Way's central, supermassive black hole

5. Newtonian physics predicts that light travelling close to a massive body has a trajectory that curves by about half the amount predicted by GR. Even stranger effects are predicted close to the black hole including that light can orbit at 1.5 times the Schwarzschild radius. Evidence for the former was one of the first tests of GR applied to stars near the Sun. Evidence for the latter is now seen in "pictures" of the black hole at the centre of M87.

The GR approach to gravity is fundamentally and philosophically different to Newtonian gravity. For Newton, gravity is a universal force. In GR, gravity is not a force at all. Freefalling bodies are said to be "inertial". They accelerate, not because a force acts upon them, but because spacetime is curved by the presence of mass (and energy).

In most cases, where Newtonian gravitational fields are weak, the consequences of this difference are small (but measurable - e.g. the orbital precession of Mercury or gravitational time dilation in GPS clocks), but near large, compact masses, like black holes and neutron stars, the differences are stark and unavoidable.

Yes, black hole can be explained by newtonian gravity. But with some assumptions.

From Newton equation of energy conservation for free fall from the infinity with initial speed of object equal to zero:

$\large {mc^2=E-\frac{GMm}{R}}$, where $\large {E=\frac{mc^2}{\sqrt{1-v^2/c^2}}}$

In assumption of $\large {m=\frac{E}{c^2}}$:

$\large {mc^2=E-\frac{GM}{R}\frac{E}{c^2}}$

or

$\large {mc^2=E-\frac{R_{g*}}{R}{E}}$, where $\large {R_{g*}=GM/c^2}$

so

$\large {mc^2=E\left(1-{R_{g*}}/{R}\right)}$

and

$\large {E=\frac{mc^2}{1-R_{g*}/R}}$

If $R=R_{g*}$, then $E=\infty$, - event horizon of Newtonian black hole

The similar expression in General Relativity:

$\large {E=\frac{mc^2}{\sqrt{1-R_{S}/R}}}$, where $\large {R_{S}=2GM/c^2}$

• This answer (v1) seems to promise in words that it will only use newtonian theory, and then proceed by writing down various ad-hoc non-newtonian equations. Jan 12 '12 at 21:10
• @Qmechanic, you are right, it's a post-newtonian gravity, where energy $(E/c^2)$ is attracted, not mass
– voix
Jan 12 '12 at 21:28
• It's not Newtonian if the first line uses $E=mc^2$... Apr 23 '14 at 8:19
• I down-voted because the very concept of a horizon is non-Newtonian. May 11 at 8:20


text:
© | Dror Bar-Natan: Classes: 2013-14: AKT: < >

# AKT-140224 Video

 width: 400 720 download ogg/AKT-140224_400.ogg For now, this video can only be viewed with web browsers that support

Notes on AKT-140224:    [edit, refresh]

A review of Lie algebras.

# Week of... Notes and Links
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
6 Feb 10 Homework Assignment 5.
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
: A Monday class: back to expansions.
E Apr 7 : A Friday class on what we mostly didn't have time to do.

Dror's Notebook
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0:22:08  We will check that the prospective Lie algebra $g = F^2$ satisfies the Jacobi identity $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$, and thus is indeed a Lie algebra. In contrast with the notation given in lecture, let $\alpha$ and $\beta$ be the basis elements of $F^2$ over $F$, so that $F^2 = \{a\alpha + b\beta: a, b \in F\}$. Now, let $x = a_1\alpha + a_2\beta$, $y = b_1\alpha + b_2\beta$, and $z = c_1\alpha + c_2\beta$ be arbitrary elements in $F^2$.

Using bilinearity,

$[y,z] = b_1c_1[\alpha,\alpha] + b_1c_2[\alpha, \beta] + b_2c_1[\beta,\alpha] + b_2c_2[\beta,\beta]$

Since $[\alpha, \alpha] = [\beta,\beta] = 0$, $[\alpha, \beta] = \alpha$, and $[\beta,\alpha] = -\alpha$, this evaluates to $$[y,z] = (b_1c_2 - b_2c_1)\alpha$$ Similar calculations yield $$[z,x] = (c_1a_2 - c_2a_1)\alpha$$ and $$[x,y] = (a_1b_2 - a_2b_1)\alpha$$ Thus, $$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = (a_2(b_1c_2 - b_2c_1) + b_2(c_1a_2-c_2a_1) + c_2(a_1b_2+a_2b_1))[\beta, \alpha]$$ $$= -\alpha(a_2b_1c_2 - a_2b_2c_1 + a_2b_2c_1 - a_1b_2c_2 + a_1b_2c_2 - a_2b_1c_2) = 0$$

As a result, the Jacobi identity holds.

0:27:53  Lie algebra of dimensions 1 and 2

1. one-dimensional Lie algebras are unique up to isomorphism. For if $\mathfrak{g} = \langle x \rangle$ is a one dimensional Lie algebra, then since the bracket is antisymmetric, we have $[x, x] = 0$. Thus the bracket is zero and $\mathfrak{g}$ is unique up to isomorphism.

2. $\mathfrak{g} = \mathbb{F}^2 = \{ax + by \;|\; a, b \in \mathbb{F}\}$. $\mathfrak{g}$ is a two-dimensional Lie algebra. There are only two of such up to isomorphism, that is, the one with the bracket equal to zero and the other with bracket $[x, y] = x$.

0:27:53 [add] Lie algebras and representations.
0:37:01 [add] Constructing weight systems from Lie algebras.