text:
## Friday, September 30, 2011

### Is there a case for supervision of alternative investment funds? A new working paper

The task of financial regulation can be broken up into consumer protection (where we worry about small consumers being cheated by financial firms), prudential regulation (where we worry about the possibility of bankruptcy of one financial firm) and systemic risk regulation (where we worry about the procyclicality of financial regulation). Everything that we do in financial regulation must be motivated by one of these three issues.

In the class of fund management mechanisms, there is one interesting special case: the alternative investment management mechanisms' which include hedge funds, private equity funds, venture capital, etc. The defining feature of these is that each customer places a large sum of money under the control of the fund manager. A typical value for the minimum ticket size is $1 million. Once this is done, it is no longer possible to argue that the investor is a small consumer who might be cheated by the fund manager. A person who places atleast$1 million with a fund manager has the capability and resources to protect his own interests. Hence, the mainstream strategy utilised all over the world has been to leave these fund managers completely unregulated.

Indeed, there has been a healthy competitive tension between these investment vehicles (which are unregulated) versus mutual funds (which are regulated). Large customers have the choice between going with mutual funds, where the cost of regulation is suffered, or going to an alternative investment mechanism where this cost is not suffered. If these customers feel the gains from regulation are not justified, they have the choice of walking away and not incurring the costs.

The world over, there are debates brewing about the need for hedge funds to begin disclosing regular information on performance, positions and counterparties to regulatory authorities. For example, the SEC recently proposed a rule requiring U.S.-based hedge funds to report such information to a new financial stability panel established under the Dodd-Frank Act. Unsurprisingly, hedge funds argued against this proposal, citing concerns that the government regulator responsible for collecting the reports could not guarantee that their contents would not eventually be made public.

In a recent paper, my coauthors Andrew J. Patton and Michael Streatfield and I examine one element of the relationship between a hedge fund and its customers: disclosure about returns. The paper is titled The reliability of voluntary disclosures: Evidence from hedge funds.

Hedge funds are notoriously protective of their proprietary trading models and positions, and generally disclose only limited information, even to their own investors. However they do voluntarily report their monthly returns and assets under management to a wider audience through one or more publicly available databases. These databases are widely used by researchers, current and prospective investors, and the media.

Our paper examines the reliability of these voluntary disclosures by hedge funds, by tracking snapshots of these hedge fund databases captured at different points in time between 2007 and 2011. In each vintage of these databases, hedge funds provide their entire historical records (rather than just the new performance information since the previous vintage). Using these data, we detect that older performance records of hedge funds are revised as a matter of course. Nearly 40% of the 18,000 or so hedge funds in our sample revise their previous returns at least once over the vintages that we consider.

We then categorize hedge funds in real-time into revising and non-revising funds, and find that on average revising funds significantly underperform non-revising funds, and have a higher risk of experiencing large negative returns. This suggests that mandatory, audited disclosures by hedge funds, such as those proposed by the SEC earlier this year, would be beneficial to investors and help to prevent such negative outcomes.

SEBI has recently put out a request for comments on a proposed strategy for regulation and supervision of alternative investment vehicles. Our paper can help in thinking about the issues faced in this field on the consumer protection, and analysing the policy choices faced there. While there is much merit to the mainstream strategy of leaving this industry unregulated, our paper suggests that a small dose of supervision, focusing on basic hygiene and motivated by consumer protection, may help.

## Tuesday, September 27, 2011

### Two seminars at NIPFP

Seminars by Sanjay Banerji (today) and Tarun Ramadorai (Thursday).

## Friday, September 23, 2011

### What in the world is happening to the rupee?

The INR/USD rate is now nudging Rs.50 to the dollar. This is a big move over a short period: a depreciation of 12.1 per cent over the 84 days from 1 July till 23 September.

### What fluctuations of the INR/USD can we reasonably expect?

After the rupee became a float, so far, it has had average volatility of roughly 9 per cent annualised. Roughly speaking, this means that over a one year horizon, the movement over a year would range between -18 per cent and +18 percent, with a 95 per cent probability. More extreme movements would happen with a 5 per cent probability.

Over a period of 84 days, roughly speaking, we'd have expected this 95 per cent range to run from -8.6 per cent to +8.6 per cent. Compared with that, a 12.1 per cent move is a bit unusual.

It's only a bit unusual because the historical volatility of the INR/USD, in the period of the float, was rather low. The USD/EUR rate, which is perhaps the world's most liquid market, has had an annualised volatility from January 1999 onwards of 10.3 per cent. The INR/USD has got to surely be more volatile than this, given the inferior liquidity of the INR and given the greater macroeconomic volatility in India. Hence, I think we should consider the 9 per cent vol, that was seen in the early days of the float, as relatively unusual. The future will most likely hold bigger values for this vol.

The implied volatility of the INR/USD at the NSE has reared up to values like 14 per cent annualised. That sounds more sensible to me.

We tend to do wrong by focusing too much on the bilateral INR/USD rate. In the recent days of distress, as fear has resurged, people have taken money out of everything under the sun and put it into US Treasury bills. This has given a strong dollar at the expense of essentially every other currency. Here's the picture for the INR, against the four major currencies of the world, from 1 July till 22 September:

 1 July 22 Sep. Depreciation (per cent) USD 44.585 48.821 9.50 EUR 64.804 66.103 2.00 JPY 0.553 0.636 15.01 GBP 71.720 75.481 5.24

The picture of the rupee is much more complex than that implied by simply watching the bilateral rupee/dollar rate.

### Can RBI block such a large depreciation?

Let's think through the steps which would follow if RBI tried to sell dollars in trying to prop up the INR:

• Global trading in the INR stands at roughly $75 billion a day. If you want to manipulate this market, you need a big stick. Small trades will do nothing. If preventing INR depreciation is the goal, RBI has to go into this with trades of$2 to $5 billion a day, with the willingness to stick it out for the long run. With reserves of$281 billion, there is not much hope here. Specifically, if RBI sells $80 billion in reserves, the market will see that. They will know that further rupee defence is now going to be hard (since$200 billion of reserves is starting to look like a small hoard), and speculators across the world will start betting that RBI's defence of the rupee will fail.
• Reserve money is only $275 billion. For each$27.5 billion that RBI sells, reserve money drops by 10%. At a difficult time like this, a sharp and sudden monetary tightening will be an unpleasant side effect of defending the rupee. (This trading can be sterilised, but that has its own problems. I just want to emphasise that selling reserves is not easy and is not a free lunch).
• The rational speculator knows that the exchange rate will eventually find its level. When RBI prevents a large INR depreciation today, they are giving a free lunch to the speculator, who would take a bet that INR would depreciate in the future. Specifically, it would be efficient for domestic and foreign investors to dump assets in India, take money out at (say) Rs.45 to the dollar which is the artificial price, wait for the gradual depreciation to Rs.50 to the dollar, and come back into India to buy back the same assets. This trade generates 11% returns over a short period and is thus very attractive. In other words, a defence of the rupee would trigger off an asset price collapse in India.

Meddling in the affairs of the currency market is thus highly ill-advised for a central bank.

### Should RBI try to block INR depreciation, even if they could?

Let us play a thought experiment where RBI had $2810 billion, i.e. 10x larger than what's with us today. In that case, RBI could play in the currency market, selling$2 to \$5 billion a day for a year without serious distress. Is this a good idea?

I would argue that this is not a good idea. When times are bad, the rupee should depreciate. This drives up the profit rates of all Indian tradeables firms and thus bolsters the economy.

Under a floating rate, in good times, the INR appreciates (which pulls back the exuberance of tradeables) and in bad times, the INR depreciates (which fuels profits and thus the physical investment in tradeables). This is arguably the only element of stabilisation in Indian macroeconomic policy.

### RBI is playing this mostly right

From early 2007 onwards, the INR has been quite flexible. In particular, after early 2009, RBI's trading on the market has tailed off. There have been a few months with minor amounts of trading by RBI. This trading has mystified me, since these small trades can do nothing to influence the price. In practice, the INR has been a float.

A floating exchange rate is exactly the right stance for difficult times like this. In bad times, the best thing that can happen for India is a big INR depreciation, thus bolstering the tradeables sector.

Let's evaluate an alternative policy platform: To peg the INR in normal times but to let go in difficult times. Is this feasible? Yes. But this is very disruptive: if economic agents have been given an implicit promise that the INR will not move, then the large move (which will surely come) would cause pain. It is far better to stay out of the market all the time, and create a trustworthy structure of expectations in the minds of economic agents about what the future holds.

We had a large depreciation in the crisis of 2008, and that served India well. In similar fashion, we should welcome the INR depreciation that is accompanying global gloom.

The only element of RBI policy where I have a major disagreement is communication. RBI has never used the words floating exchange rate. RBI needs to clearly communicate to the economy that the rupee is now a market determined exchange rate, and RBI is no longer in the business of trading in this market. There is greater clarity of thought at RBI as compared with the quality of communciation; the speech writing still suffers from twinges of 1960s economics.

### What is the collateral damage of a large INR depreciation?

There are three things that go wrong alongside a big INR depreciation:

1. Firms who have unhedged foreign currency borrowing get hurt, because they have to pay back more than anticipated. A person who borrowed Rs.100 (in unhedged USD) has to pay back Rs.110, owing to the 10 per cent INR depreciation. The stock market is doing a fine job of identifying these firms and beating down their stock prices.
Of crucial importance is the fact that from early 2009 onwards, the INR had already moved to a float with a 9 per cent annualised vol. So CEOs and CFOs knew that the INR/USD rate was going to fluctuate. They were not lulled into complacence thinking that the exchange rate was going to be stable. By avoiding this moral hazard associated with pegged exchange rates, RBI's decision to float in early 2009 laid a good foundation for the structure of firm borrowing as of July 2011.
When a country has a pegged exchange rate, you tend to see a big buildup of unhedged currency exposure on corporate balance sheets. When the big depreciation comes, the big businessmen then queue up to the central bank begging for defence of the LCY. Prevention is better than cure: It is far better to have high exchange rate volatility all along, so that firms do not undertake such risks, and the toxic political economy does not come into play.
2. With an INR depreciation, tradeables become costlier. On one hand, this bolsters the profitability of tradeables firms, and thus their investment plans. But at the same time, this feeds into inflation. In recent months, tradeables inflation has been sleeping while non-tradeables have contributed to the high CPI-IW inflation. We will now see a resurgence of tradeables inflation. This will exacerbate the inflation crisis. RBI will need to stay on the project of raising rates in order to combat this inflation.
3. The government's subsidy program with petroleum products and fertilisers gets costlier when the INR depreciates. So India's fiscal crisis gets a bit worse when the INR depreciates.

This logic is rooted in high levels of de facto capital account openness. Sometimes, policy analysts think that you can have your cake and eat it too, and try to dodge these arguments by utilising capital controls. This has not worked in India, and the levels of de facto openness have only grown through the years.

### In summary, what should RBI be doing?

RBI should be focused on using the short-term interest rate as a tool to bring CPI-IW inflation under control, without distortions of interest rate policy caused by trying to meddle in the currency market. This should be accompanied by liberalisation of the Bond-Currency-Derivatives Nexus so as to achieve an effective monetary policy transmission. These are the two things that RBI needs to focus on.

India shifted away from government interference in the currency market, from 2007 onwards but particularly after 2009. This is one of the biggest achievements in India's economic liberalisation. This is a bigger issue in economic liberalisation than (say) decontrol of petroleum product prices. The INR is now a market. Nifty and INR are the two most important markets in the economy. It is time for all of us to analyse the INR as we analyse Nifty: as the outcome of a market process.

### Is RBI back to trading the INR?

We don't know. The data only comes out at monthly resolution, with a two month lag. But early signs that would show up would be unusual jumps in the weekly data about reserves, reserve money, etc. Greater transparency from their side would help greatly.

## Monday, September 19, 2011

A nice story about UIDAI, by Lydia Polgreen, in the New York Times.

A new insight into India's north-east states: they are part of a region provisionally named Zomia. An interesting article in the Chronicle of Higher Education by Ruth Hammond. The book.

On 21 April 1956, Jawaharlal Nehru did the first convocation address at IIT, Kharagpur. It's a good read, and it's surprising how much of it makes sense in 2011. E.g.: in the larger context of history, and looking at it in this way it seems to me that at the present moment there is no more exciting place to live in than India. Mind you, I use the word exciting. I did not use the word comfortable or any other soothing word, because India is going to be a hard place to live in. Let there be no mistake about it; there is no room for soft living in India, not much room for leisure, although leisure, occasional leisure is good. But there is any amount of room in India for living the hard, exciting, creative adventure of life. In case you have not yet seen the Steve Jobs commencement speech, it is worth watching.

How civilised: Literature festivals in India, by Abhilasha Ojha in Mint.

A fascinating story from rural India about the differences between boys and girls on mathematics, by Maia Szalavitz in Time magazine.

Who's to blame for India's inflation and India's Inflation Is a Lesson for Fast-Growing Economies by Alex Frangos in the Wall Street Journal.

When do stock futures dominate price discovery? by Nidhi Aggarwal and Susan Thomas, IGIDR working paper, has some surprising results.

Anupama Chandrasekaran and Vidya Padmanabhan, in Mint, on Indian ventures into farming in Ethiopia.

Raghu Dayal in the Business Standard on the huge opportunities in better India-Bangladesh relations.

Mobis Philipose in Mint, on recent developments in SEBI and on currency derivatives trading.

We need a Hazare in the financial sector by Tamal Bandyopadhyay in Mint. N. Sundaresha Subramanian in the Business Standard. Ex-SEBI member to PM: ID leaked, family at grave risk by P. Vaidyanathan Iyer in the Indian Express. CVC to Fin Min: Probe both sides' complaints by Ritu Sarin in the Financial Express. And, reportage in India Today. Spat between Abraham, SEBI, finance ministry gets murkier by Appu Esthose Suresh in Mint. Supreme Court wants petition on SEBI refiled by Nikhil Kanekal and Appu Esthose Suresh in Mint. A first and then a second article on these issues, by R. Jagannathan, on FirstPost. An editorial in the Business Standard. Subhomoy Bhattacharjee in the Financial Express.

R. Jagannathan on post offices as banks (on firstpost). And, you might like this related document.

China's A. Q. Khan problem: an article by Michael Wines in the New York Times.

A great story by Anthony Shadid in the New York Times about being on the run in Syria.

A great article by Paul Berman, in the New Republic, about Islamism.

Why is it so hard to find a suicide bomber these days by Charles Kurzman, in Foreign Policy.

Love and war, by Janine di Giovanni, in the New York Times.

What's next for the dollar? by Martin Feldstein.

Sussane Craig has a great profile, in the New York Times, of how Howard W. Lutnick brought Cantor Fitzgerald back to life after the firm was savaged in the 9/11 attacks.

## Sunday, September 18, 2011

### Market making versus the electronic limit order book

Exchanges in India all operate as electronic limit order book markets. There are no market makers'; there is just a publicly visible limit order book. Anyone is free to supply liquidity, by placing limit orders. The person who places market orders is the consumer of liquidity: he pays market impact cost. [A guide to the jargon].

Prior to the rise of the anonymous limit order book, there used to be a great deal of effort on thinking about the market maker. Market makers played a big role in many old markets. E.g. at the NYSE, the specialist' was obliged to provide liquidity. RBI established primary dealers' thinking that they would provide liquidity.

These market structures involved complicated problems of measuring the liquidity provision by market makers, correctly compensating them, avoiding monopoly power in the hands of the market market, and enforcing against market manipulation by the market maker. The rise of the open electronic order book cut through this Gordian knot.

For many years, there used to be a debate about whether the anonymous open limit order book market (where anyone can provide liquidity) is better or worse than a market maker market (where limit orders can only be placed by one or more market makers). That debate died down in the 1990s with the success of the electronic limit order book.

### Market making on the electronic limit order book

But even on a limit order book, does it make sense to pay one or more market makers to provide liquidity? The public would be free to place limit orders, but one or more market makers would be paid to place limit orders.

The positive argument runs like this. In the life of every contract, at first there is a lack of liquidity as various market participants are reluctant to take the plunge and trade on an illiquid contract. This leads to a chicken and egg problem. Illiquidity inhibits participation, and the lack of participation is illiquidity.

From a regulatory perspectives, exchanges might try to make payments for liquidity provision (or outright turnover) by various underhand means. If that is going to happen, then it is better to have this come out into the open.

But there are also important problems that can come out by going down this route. The resources that an exchange puts into portraying tight spreads or high turnover could potentially be used to improve services for customers. Market participants would make wrong decisions about an investment decision when they see a product as looking liquid on screen, whereas this liquidity is actually artificial: the screen would be falsely portraying liquidity. When exchanges compete on payments to market makers, this can degenerate into a slugfest where the deepest pockets win.

The artificial liquidity pushed by mercenary market makers would tend to lull the exchange into complacence. In the absence of market making, the exchange would run harder to solve problems of market mechanisms and contract design, and to get the word out about the contract.

### Recent developments in India

On 2 June 2011, SEBI chose to move ahead with the specification of a `Liquidity Enhancement Scheme' (LES).

By these rules, LES is applicable for individual stocks where the trading volume on the last 60 days is below 0.1 per cent of the market capitalisation. (How would this be scaled to derivatives such as currency futures, where market capitalisation cannot be defined?) I think this makes sense. The LES would be used to kickstart liquidity when it is abysmal. The moment a small amount of liquidity comes about, the LES would step aside.

Based on these rules, NSE announced a program for market making on the derivatives products recently launched at the exchange: on the S&P 500 and the Down Jones Industrial Average (launched in partnership with the Chicago Mercantile Exchange). These incentives are over and above the absence of charges by the exchange. I was disappointed to see a payment based on mere turnover. This would give the market maker an incentive to do circular trading and thus show a lot of trades. But turnover is not liquidity.

This program came into effect on 15 September. It may matter more in the coming week, given that new contract series start trading from tomorrow.

### Will it matter? How will we know that it mattered?

Derivatives on the S&P 500 and the Dow Jones indexes have gotten off to a surprisingly good start, even though there was no such program. This has perhaps been helped by unusual levels of volatility in the US after the launch of these contracts.

The early days of a contract can be a rollicking ride and even after these time-series fall into place, it will not be easy to tell whether LES was useful in the history of these contracts or not.

Similar thinking is taking place at BSE also: See Will BSE's biggest initiative work? by Mobis Philipose in Mint. The text there -- obligations such as providing two-way continuous quotes within specified parameters for quote size and spread -- sounds good, but here also there are payments per crore of turnover. By and large, the payments being made at BSE look much bigger than those at NSE.

In the case of BSE, if LES is able to lift BSE out of zero market share in derivatives trading, even after the six month period has expired, then it would be a clear proof that the LES helped. So this experiment is unlike that of NSE where it will be hard to evaluate whether or not the LES mattered.

## Wednesday, September 07, 2011

### The reversal of reforms on the New Pension System?

In December 2002, the NDA made a very big move in pension reforms. They decided that from 1/1/2004 onwards, all new staff recruited into the government would be switched out of the traditional defined-benefit pension and instead placed into a new individual-account defined contribution pension system. This was one of the major achievements of the economic reforms of that period. For a conceptual picture of the New Pension System (NPS), see this article, and for a story of that period, see this article.

An essential feature of the NPS was that it was a defined contribution system. India has a long history with getting into trouble with guaranteed returns. UTI's assured return schemes turned into a problem for the exchequer. EPS, run by EPFO, is bankrupt. When pension promises are made, they require peering into many decades into the future and arriving at estimates of longevity and asset returns. In the best of times, it is hard to make such estimates; honest mistakes are possible. In addition, when governance is weak, there are political pressures to make extravagant promises, which will look popular right now but generate staggering costs for the government in the future. As an example, rough calculations show that the implicit pension debt on account of the traditional civil servants pension in India (the one which was replaced by the NPS) stand at roughly 70% of GDP. This is a very big price to pay, for a tiny sliver of the workforce.

The NDA did the unpopular work of switching new recruits out of the defined benefit pensions. But the UPA did not follow through appropriately. At first, many years were lost in hoping that the CPI(M) would come on board the reform. After that, the legal engineering was put into place in order to get an NPS up and running without requiring the legislation. This process was slower than what one might have desired, but it has been making inexorable progress.

But now, a new existential threat seems to have come up : the Parliamentary Standing Committee on Finance seems to be saying that the fundamental idea of the NPS -- defined contributions -- should be scrapped. This would amount to a major reversal of India's economic reforms.

On this subject, see:


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# Linear Equations

### Plotting Points

A linear equation can be graphed by using a table to generate a list of ordered pairs. When all the ordered pairs are plotted, they lie on the same line.

When an equation has two variables, such as $x$ and $y$, the solutions of the equation can be represented by a graph in the coordinate plane. The graph consists of the set of all points $(x,y)$ that make the equation true. A linear equation in two variables is an equation in two variables whose graph is a line. In a linear equation in two variables, each term can be written as a number, such as -2, or a product of a variable and a number, such as $3x$. The variables are not raised to a power, in the denominator of a fraction, or under a radical, such as a square root. For example, $y=3x-2$ is a linear equation, but $y=3x^2-2$ is not.

The most basic method of graphing lines is by plotting points and connecting them. A minimum of two points is required to graph a line, but additional points may be plotted as a check. A table is a good way to organize the values when plotting points to graph a line.

Step-By-Step Example
Using a Table of Ordered Pairs to Graph a Linear Equation
Graph the linear equation:
$y=3x-2$
Step 1

Create a table, and choose a list of $x$-coordinates.

$x$
$y$
$-1$
$0$
$1$
$2$
Step 2

Substitute the chosen coordinates into the equation for $x$, and solve for $y$.

$x$
$y=3x-2$
$y$
$-1$
\begin{aligned}y &= 3(-1)-2\\y &=-5\end{aligned}
$-5$
$0$
\begin{aligned}y &= 3(0)-2\\y &=-2\end{aligned}
$-2$
$1$
\begin{aligned}y &= 3(1)-2\\y &=1\end{aligned}
$1$
$2$
\begin{aligned}y &= 3(2)-2\\y &=4\end{aligned}
$4$
Step 3

Use the table to write a list of ordered pairs.

$x$
$y$
$(x, y)$
$-1$
$-5$
$(-1, -5)$
$0$
$-2$
$(0, -2)$
$1$
$1$
$(1, 1)$
$2$
$4$
$(2, 4)$
Solution
Plot the points, and then connect them with a straight line.

### x- and y-Intercepts

The points where a line crosses the $x$- or $y$-axis are called the intercepts. To find the $x$-intercept, let $y = 0$. To find the $y$-intercept, let $x = 0$.

Another method of graphing a linear equation is to plot the intercepts. Two points determine a line, which means that when two points are plotted, there is only one line that connects them. Any two points can be used to graph a line, but two points that are usually easy to plot are the intercepts. The value of $x$ where a graph touches or crosses the $x$-axis is the x-intercept. It lies on the $x$-axis, so it always has a $y$-coordinate of zero. The value of $y$ where a graph touches or crosses the $y$-axis is the y-intercept. It lies on the $y$-axis, so it always has an $x$-coordinate of zero.

To graph a linear equation by finding the intercepts, first set $x=0$ and solve for $y$. Then set $y=0$ and solve for $x$. Plot the intercepts, and draw the line connecting them.

Step-By-Step Example
Using the Intercepts to Graph a Linear Equation
Graph the linear equation:
$y = -5x + 1$
Step 1

Create a table and insert zero for one $x$-coordinate and one $y$-coordinate.

$x$
$y$
$0$
$0$
Step 2

Substitute zero into the equation for $y$, and solve for $x$. Substitute zero into the equation for $x$, and solve for $y$.

$x$
$y = -5x + 1$
$y$
$0$
\begin{aligned}y &= -5(0)+1\\y &=1\end{aligned}
$1$
$0.2$
\begin{aligned}0 &= -5x+1\\-1 &=-5x\\\frac{-1}{-5} &= x\\0.2&=x\end{aligned}
$0$
Step 3

Use the table to write ordered pairs.

$x$
$y$
$(x, y)$
$0$
$1$
$(0, 1)$
$0.2$
$0$
$(0.2, 0)$
Solution
Plot the points, and then connect them with a straight line.

### Slope of a Line

The steepness of a line can be described by the slope. The slope of a line is the ratio of the change in $y$-values to the change in $x$-values of any two points on the line.
The slope is the ratio of the change in $y$ to the change in $x$ for two points on a line, represented as:
$\frac{\text{Rise}}{\text{Run}}$
• Slope is a measure of the steepness of a line.
• A line with a positive slope goes up from left to right.
• A line with a negative slope goes down from left to right.
• The rise is the vertical change between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: $y_2-y_1$.
• The run is the horizontal change between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: $x_2-x_1$.
• The slope is represented by the variable $m$.
• The formula for the slope of a line through the points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$m=\frac{\text{Rise}}{\text{Run}}=\frac{y_2-y_1}{x_2-x_1}$
Step-By-Step Example
Determine the Slope of a Line
Identify the slope of the line whose graph is shown.
Step 1

Identify two points on the line. Any two points used will result in the same final answer.

Let $(1, 1)$ be the point $(x_1, y_1)$. Then let $(2, 4)$ be the point $(x_2, y_2)$.

Step 2

Identify the rise and run. There are two ways to identify the rise and run between two points.

Method 1: Subtract the coordinates.
\begin{aligned}\text{Rise}&=y_2-y_1\\&=4-1\\&=3\end{aligned}
\begin{aligned}\text{Run}&=x_2-x_1\\&=2-1\\&=1\end{aligned}
Method 2: Count the squares on the grid.

For the rise, count the number of squares vertically from the point $(1,1)$ to the point $(2,4)$. There are 3 squares for the rise.

For the run, count the number of squares horizontally from the point $(1,1)$ to the point $(2,4)$. There is 1 square for the run.
Solution
Determine the slope, or $m$.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{3}{1}\\&=3\end{aligned}
The slope of the line is $m=3$.

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## The higher centers in a group are characteristic

Prove that $Z_k(G)$ is characteristic in $G$ for all $i$.

We begin with a lemma.

Lemma 1: Let $G$ be a group, let $K \leq G$ be characteristic, and let $\varphi$ be an automorphism of $G$. Then the mapping $\psi : G/K \rightarrow G/K$ given by $\psi(aK) = \varphi(a)K$ is an automorphism of $G/K$. Proof: (Well-defined) Suppose $aK = bK$; then $ab^{-1} \in K$. Since $K$ is characteristic, $\varphi(ab^{-1}) = \varphi(a)\varphi(b)^{-1} \in K$, so that $\varphi(a)K = \varphi(b)K$. Thus $\psi(aK) = \psi(bK)$. (Homomorphism) We have $\psi((ab)K) = \varphi(ab)K$ $= (\varphi(a)K)(\varphi(b)K)$ $= \psi(aK)(\psi(bK)$. (Injective) Suppose $\psi(aK) = \psi(bK)$. Then $\varphi(ab^{-1}) \in K$, so that (since $K$ is characteristic) $aK = bK$. (Surjective) Let $aK \in G/K$. Since $\varphi$ is surjective, $aK = \varphi(b)K = \psi(bK)$ for some $b \in G$. $\square$

Lemma 2: Let $G$ be a group. Suppose $K \leq H \leq G$ such that $K \leq G$ is characteristic and $H/K \leq G/K$ is characteristic. Then $H \leq G$ is characteristic. Proof: Let $\varphi \in \mathsf{Aut}(G)$. By Lemma 1, the mapping $\psi : G/K \rightarrow G/K$ given by $\psi(aK) = \varphi(a)K$ is an automorphism of $G/K$. Since $H/K$ is characteristic, we have $\psi[H/K] = H/K$; that is, if $h \in H$, then $\psi(hK) = \varphi(h)K = xK$ for some $x \in H$. Thus $\varphi(h)x^{-1} \in K$, and we have $\varphi(h) \in H$. Thus $\varphi[H] \leq H$. Now suppose $h \in H$. Since $H/K$ is characteristic in $G/K$, we have $hK = \psi(xK) = \varphi(x)K$ for some $x \in H$. Then $\varphi(x^{-1})h = k \in K$. Since $K$ is characteristic in $G$, we have $k = \varphi(y)$ for some $y \in K$. Thus $h = \varphi(xy)$, where $xy \in HK = H$. Thus we have $H \leq \varphi[H]$. Hence $H \leq G$ is characteristic. $\square$

We proceed by induction on $k$.

For the base case, if $k=0$, then $Z_k(G) = 1$. Clearly then $Z_k(G)$ is characteristic in $G$.

For the inductive step, let $k \geq 0$ and suppose that $Z_k(G) \leq G$ is characteristic. Consider $Z_{k+1}(G) \leq G$. By definition, $Z_{k+1}(G)$ is the subgroup of $G$ containing $Z_k(G)$ such that $Z_{k+1}(G)/Z_k(G) = Z(G/Z_k(G))$. Now by definition, $Z_{k+1}(G)/Z_k(G)$ is characteristic in $G/Z_k(G)$. By Lemma 2, $Z_{k+1}$ is characteristic in $G$.

By induction, $Z_k(G)$ is characteristic in $G$ for all $k$.


text:
# Combinatorial circle placement

Suppose we have area bound by $0\leq x\leq 2$,$0\leq y\leq 2$

We choose 17 points, there should be at least two with distance max $d\leq\frac{1}{\sqrt{2}}$

Tried to count area of $A_{17circles}=(\frac{d}{2})^2\pi*17=6.67588438887831$

Then to take square $A_{square}=(\frac{d}{2}+2+\frac{d}{2})^2=7.32842712474619$

There is also each corner to be rounded $A_{removecorners}=d^2-(\frac{d}{2})^2\pi=0.107300918301276$

Which results in $A_{square}-A_{removecorners}=7.22112620644491$

Still $7.22112620644491>6.67588438887831$ so I cant dismiss argument that one is able to fit 17 points so, that $max_{distance}\geq d$

How to dismiss this argument? How to prove that if we put 17 points into 2 by 2 box, some of them bound to be closer than $\frac{1}{\sqrt{2}}$

Hint: Divide the $2\times 2$ square into $16$ equal little squares, by drawing three equally spaced horizontal lines, and three vertical lines. Then use the Pigeonhole Principle.


text:
# Allowing mathematical and homework questions on this site

I'm about to complete my four-year physics degree and want to write briefly on the topic of this site.

I've spent a lot of time on the physics Stack Exchange site (not on this account) and believe, overall, it's hugely detrimental to the physics cause - more specifically, the general public's aspirations, understanding and enjoyment of the subject.

I constantly see excited learners ask reasonable, enthusiastic questions but are rejected for stupid reasons. What's wrong with a homework-like question? Homework questions are specially written to expand and question one's understanding - it's a totally valid way of learning. If you think someone's trying to cheat, don't answer (or even comment) on the question - the beauty of a website like this is that if people don't respond to crappy questions they won't be popular/seen.

I've seen many, many question written off as one for the maths Stack Exchange site. Yes, the arithmetic/solving is obviously maths - hence the language of physics - but the explanation, the understanding, is obviously physics - don't reject someone for asking how to solve something physical using maths.

Fundamentally, if you spent the same amount of time complaining about someone's question as you did answering it, your impact would be genuinely positive and fantastic. For those who think this site could be some definitive Q&A textbook - it isn't and never will be. Wikipedia will always be wider-reaching and more democratic - and if you want to contribute to a textbook/people's question, go write one.

If you believe in physics you should want it to grow - stop controlling the learning of other less knowledgeable students of the universe - it's not big and you're hampering our progression as a species.

• @Again I agree with this whole post. I also agree with Jeanbaptiste Roux; Especially the maths part. really, actually good post. really. And especially the homework-like questions I really dont understand, if we can't post or answer about those, why do we have a tag about it? so I agree to fully the whole post, respect. Dec 15, 2022 at 11:04
• Yes. which is why I think this is such a great post. truly. it feels like time is more spent on rules and following (strictly following) those rules, like if some question is a BIT wrong, it's closed (because of reason x..y..z..) Instead of just answering, which is.. yea. No offense to those though, but it should be a more, "answer" not "rule following" community. a there is obviously questions that are totally legit-to-be-closed; but it feels like, extreme courage have to be taken to ask. not only research the question itself. Again, respect! wishes from Sweden! Dec 15, 2022 at 11:40
• As mentioned already, this belongs to the meta section. That being said, there you can already find questions regarding the existence of the homework and exercise tag and related problems/discussions. Moreover, keep in mind that this sites exists for quite a long time already and the rules are not some pointless things to annoy you; they are employed for certain reasons you perhaps did not think about yet. Sure, this does not mean they cannot be changed, but they don't have to, only because you (and some others) think so. Dec 15, 2022 at 12:15
• More specifically, it is also reasonable to keep this site "clean" - and to close /downvote questions when appropriate, e.g. when someone simply asks for a solution after posting the screenshot of the exercise. This site is already full of these "spam" questions and it is not the aim of the site to encourage people to post their homework problems here in a manner conflicting the rules. Also, often some personal theories are posted, which is against the rules too and the tools for reacting on these questions is closing and downvoting- this is how this site works. Dec 15, 2022 at 12:26
• @Again Good luck on the Physics Degree as well! Pure respect! really :) Dec 15, 2022 at 12:32
• This has been discussed only about a million times on this site, have you bothered to look at any of those posts before writing yet another thread? I've spent a lot of time on the physics stack exchange (not on this account) Are you that embarrassed by the thread that you don't want to associate yourself with it? Why? Dec 15, 2022 at 20:14
• Although written in a different context, a previous answer of mine physics.meta.stackexchange.com/a/14217/36194 does apply in this case. Dec 16, 2022 at 1:23
• Answering all homework-type questions does absolutely no favors for a lot of askers, who make no effort to answer their own questions, nor do they want to. They just want an answer. They should be trying to learn. In such cases, which make up a good portion of the homework-type questions, we would be doing more harm than good, so thinking that "impact would be genuinely positive and fantastic" is something a lot of members would disagree with. Dec 16, 2022 at 4:10
• @WilliamMartens "I really dont understand, if we can't post or answer about those, why do we have a tag about it?" Because just because a question belongs to that tag does not mean it should be closed. There are ways to make posts about homework and exercises that are on topic for PSE Dec 20, 2022 at 2:47
• @BioPhysicist I know that- but It was kind of like a figure of speech(..) or what to call it; because the reason of the close being "homework is not on topic" which, of course makes the user believe , all homework related things is off topic (even if that is false) so I would probably change the closing reason to be more specific; as it is now it's very broad but thanks for pointing it out , anyway so I could clarify it (here) :) Dec 21, 2022 at 18:56
• @WilliamMartens That's not what the closure message says. Additionally, it's very specific as to the reasoning, and it gives links to find more information, to find useful examples, etc. However, meta is the place to discuss these things; if you have a proposed change to the current closure reason you should suggest it in a new post. Dec 22, 2022 at 0:41
• @BioPhysicist yes, not exactly like that I agree. It was a shorter version so to not make the comment longer; (to give the general point across) anyway, thanks :) Dec 22, 2022 at 10:12
• NOTE! I just have to send this comment as well, if I ever write anything that looks like it is being written in a offensive way I apologize - it isn't Intentional, and it isn't meant as that in any way (so if I do, please point it out! and i will change the format of it writing this comment NOT because someone told me to, but to be sure I don't offend anyone :) please, have a good, continuing corona-free weekend! :) wishes from Sweden! Dec 22, 2022 at 10:15

I'm sorry, but the idea that our site policies are "controlling the learning of other less knowledgeable students of the universe" is just plainly ridiculous. That people on the internet won't volunteer their time to answer your homework question is not "controlling your learning", it's just people not wanting to do your homework.

Indeed, physics.SE could stop accepting new questions altogether tomorrow and it still wouldn't be exerting control over anyone's learning. No one is forced to come here. When we say a question is off-topic that doesn't mean it will be impossible to ever get an answer to it - as you rightly point out, there are other sources on the internet!

Every site needs to have some boundary for what sort of content is acceptable on it. We've apparently chosen to draw this boundary on physics.SE narrower than you would like, but you seem to think that we could just expand that boundary and everything else would stay the same. However, it is not unlikely that changing our policies would alienate some contributors - again, everyone is here voluntarily, because they like to answer questions here, and presumably because they like (or at least can live with) our policies. Getting a lot more questions while having previously prolific answerers leave would certainly not be an improvement for the overall quality of physics education this site provides.

Finally, regardless of what physics.SE chooses to do, we're not "hampering our progression as a species" - that's just pointless hyperbole. The problems our species faces at large will not be solved by a website teaching slightly more people slightly more physics. Please reserve apocalyptic language for the non-zero amount of actual existential threats humanity faces.

• Physics Stack Exchange is one of the (if not the) largest physics question and answer site - of course the way people behave on here impacts a generation of physicists. I'm grateful for anyone willing to answer a question, at any time of the day, for free, within minutes - that's incredible! But, without a doubt, there exists a large group of users that are very quick to reject a question as needing clarity or off-topic or homework or blah blah - it's controlling and I'm willing to bet a bunch of users enjoying policing - that's where the element of control comes from. Dec 15, 2022 at 16:14
• I've written a bit more below, if you're interested. Dec 15, 2022 at 16:15
• @Again sorry but this is completely crazy to the point of disillusion. I must respect that you do not like the policies of this community but surely your criticism are completely out of place given the success of the site. Dec 16, 2022 at 23:03

Personally, I think it is fine for a site to decide what is on-topic and what is off-topic for the site. A different choice wouldn’t have been wrong, but having a specific policy (whatever it is) allows a site to focus and excel in that niche. Other needs outside of that niche are then best met by directing people to the places that already have that focus.

I've seen many, many question written-off as one for the maths stack exchange.

I don’t agree with your point about math questions. Directing people to a better site is not writing off a question, it is sending people to another site with a better focus for that question. They will get better help at a site focused on their specific needs than they would here with a topic that is outside of our focus.

What's wrong with a homework-like question? Homework questions are specially written to expand and question one's understanding - it's a totally valid way of learning.

It is a valid way of learning, but it is not well suited to the Q&A format. What students usually need in homework help is not an answer to the question, but a discussion of the particular conceptual hurdle they are facing. That is more suited to a discussion format.

What I think would be helpful is to point such students to physicsforums.com where homework help is already available. They are already staffed and focused and structured to best help. Similar to how math help is handled, quickly pointing students to physicsforums.com would help better than trying to handle their needs here.

It is better to excel in a specific focus area and point people to other excellent places as needed than to degrade our focus.

• ...but a discussion of the particular conceptual hurdle they are facing. That is more suited to a discussion format. +1 for this line alone. Dec 15, 2022 at 20:15

What's wrong with a homework-like question? Homework questions are specially written to expand and question one's understanding - it's a totally valid way of learning. If you think someone's trying to cheat, don't answer (or even comment) on the question - the beauty of a website like this is that if people don't respond to crappy questions they won't be popular/seen.

This website is not meant to be all things to everybody. You are accusing this community of being actively detrimental to physics because we have a very strict policy regarding homework and homework-like questions, but that policy is supported and enforced by the community. We have collectively decided that we don't want the site to become inundated with check-my-work style questions, which (as Dale points out) are better suited to a discussion forum. And in fact:

If PSE has a policy against explicit homework help but PhysicsForums has an entire section dedicated to homework help for various levels and subjects, why wouldn't you simply go to PhysicsForums instead? Why should this community change its policies to suit what you personally think they should be, especially if other (both free and paid) sites which cater to those desires already exist in the world?

If it's because you prefer this site's Q&A format, or the general quality of the answers received, or the breadth of knowledge of the community as a whole, consider the possibility that these aspects of PSE are inextricably linked to its policies. Speaking as a sample size $$N=1$$, I am not an active user on PhysicsForums precisely because I don't like the relentless back-and-forth of a discussion forum. I am looking for interesting questions with thoughtful, authoritative answers, which is why I am here.

I've seen many, many question written-off as one for the maths stack exchange. Yes, the arithmetic/solving is obviously maths - hence the language of physics - but the explanation, the understanding, is obviously physics - don't reject someone for asking how to solve something physical using maths.

I am guilty of answering more than a few questions which, objectively, should have been migrated to MSE. First, migration is not writing a question off. Questions which are purely mathematical in nature are being sent to a community of experts in mathematics which is better equipped to answer the question in a clear and authoritative way. To be sure, there are likely many people at PSE who might be able to answer the question - but there are also people at PSE who can tell you how to change a tire or cook pasta. The point is that purely mathematical questions are not our community's focus, so we send such questions to a community that's a better fit. I can't imagine a way in which this is anything but positive for the question asker.

Secondly, if there is physics involved in the understanding then by definition it is not a pure mathematics question. If anything, this community errs on the side of keeping such questions open (in my experience, at least). Asking how many Killing vector fields must exist on a generic pseudo-Riemannian manifold is a purely mathematical question; asking why Killing vector fields correspond to physical symmetries is not.

Fundamentally, if you spent the same amount of time complaining about someone's question as you did answering it, your impact would be genuinely positive and fantastic. For those who think this site could be some definitive Q&A textbook - it isn't and never will be. Wikipedia will always be wider-reaching and more democratic - and if you want to contribute to a textbook/people's question, go write one.

Clearly this site is not what you are looking for - and that's okay. You are well within your rights to want to change things rather than simply leave, but we are equally well within our rights to keep things the same if the community disagrees with your proposed changes.

Lastly, it is of course true that the community moderation does not always align with my views. Some questions are closed which should, in my opinion, remain open, and vice-versa. Sometimes community members make mistakes or vote hastily, and voting is by no means universally consistent (which is certainly true of me). In such cases, I vote to re-open (with edits if necessary); if the community disagrees, then c'est la vie.

Overall, I think the policies as they are currently are more positive than negative. Disagreements are part of being in a community, and I will accept that sometimes there will be no reconciling my opinion with that of a majority of other users.

• @TobiasFünke Indeed, I didn't mean to imply that there was no recourse and I have edited to reflect that. I simply meant to say that disagreements are natural and that my opinion will not always carry the day, but that is not a sign that anything needs to be changed. Dec 15, 2022 at 15:27

If you believe in physics you should want it to grow - stop controlling the learning of other less knowledgeable students of the universe - it's not big and you're hampering our progression as a species.

This is a huge generalization. This specific site, physics.stackechange.com is one of thousands 830,000,000 , when I asked on google "sites physics questions answers" . A number of them are interactive and it is logical that limits to the type of question an answer should be set.

Choose what suits you better.

• “Choose what suits you better”: well said. If you go to a restaurant and don’t like the food offering, go elsewhere… Dec 16, 2022 at 23:10

My intention was to write this and not reply to the answers - that way danger lies. I will say one final thing though, which could have equally belonged to the original post.

The issue of homework-like questions and moving to the maths site are just examples of the point I was trying to make. Of course there are justified, reasonable reasons for these rules, obviously. Many of the responses have reduced the conversation to just site policy chat.

I wrote this out of frustration at the way enthusiastic learners are often pushed aside. If they didn't ask the question in the right way (which is often the tricky part) why not make a few assumptions about what they intended and try to teach them something at least on-topic.

I think discarding questions as homework or belonging to the math site are examples of the widespread sh*tty/controlling attitude that will discourage people from the subject, and that makes me sad. At the very least, my own experience of this website, as someone who loves physics, has been off-putting in the vast majority of cases. That's it. That's my point.

Of course

it's not big and you're hampering our progression as a species.

is hyperbole. But I genuinely believe a swathe of people interested in learning about physics will have been discouraged.

• why is this getting down voted? It is a new member, and posted a pretty good post. and no explanation to why it was downvoted either. Upvoting because of that. at least explain why the downvote. Dec 15, 2022 at 18:55
• @WilliamMartens I imagine downvotes are because this answer doesn't bother trying to answer the question, it's a response to other posts. Dec 15, 2022 at 20:16
• @KyleKanos oh okay; thanks for explaining. I tried to just grab a explanation to why; Dec 15, 2022 at 20:24
• I am active here now for a little bit less than two years, and I don't see how any "enthusiastic learners" are "pushed aside". For sure, the "system" is not perfect, but I don't see any systematic problem at all. I instead see a lot of "do-my-work-questions" or questions where it is clear that the person did not think/ work long enough about/on their question (e.g. did not use google, WIkipedia, searched on this site, checked some book etc.). And IMHO such posts lower the quality of this site a lot and I thus think that one can safely downvote/close these. Dec 15, 2022 at 23:02
• OTOH, you're right in the sense that if the question does not meet the standards but it seems clear that the OP can change the question such that it will be on-topic, then the community should help (help with editing, explain why it is currently off-topic, try to help to formulate a better/more specific question). But as far as I can see, this is done very often and moreover in a polite way. There may be (and probably are) exceptions, where "the community" was not patient enough; but again: I don't see any systematic problem here. Dec 15, 2022 at 23:05
• Related: Why do I need 50 reputation to comment? What can I do instead?. In this case, it could be done by a single accepted suggested edit. Dec 16, 2022 at 18:25
• The company has indicated it wants to do something on the learning front (even the current homework sites may not guarantee any learning. My emphasis): "The users: learners, from beginners to experts" and "Community. Learning, sharing, and growing together".". And at 23 min 38 secs: "Our outcomes are not just a library. But it is actually a community of people writing, consuming, curating, recommending, bringing in outside pieces of content". Dec 16, 2022 at 18:31
• Further, at 24 min 24 secs: "We want our users to not think of themselves ... as consumers, but to think of themselves as learners." Dec 16, 2022 at 18:41
• @WilliamMartens The question says "I've spent a lot of time on the physics Stack Exchange site (not on this account)", so it is not a new user. Dec 17, 2022 at 12:48
• @NorbertSchuch oh okay, yea but in my opinion it doesn't matter - we should always explain downvotes but yes I saw that when you pointed it out (thanks) :) Have a great day on you! wishes from Sweden. Dec 17, 2022 at 13:20
• The irony is that my previous account was deleted for not asking questions in the right way. Dec 17, 2022 at 18:13
• @WilliamMartens NO! Forcing people to explain downvotes will solve nothing, and will only lead to bickering. Even now, we have another user bitching about rules they don't like, even if the policies are clear, so just imagine the bitchfest that would follow from perceptions of unfair downvoting. Most users who want to leave comments with their downvotes do so, and that's good enough. Dec 17, 2022 at 19:23
• @WilliamMartens I fully agree with ZeroTheHero -- my experience from explaining downvotes (which I do when I feel motivated and believe in the good in people) is that in many times, it leads to pointless discussions with people who don't even want to be convinced that they might not be right. Dec 17, 2022 at 19:47
• @ZeroTheHero @NorbertSchuch this is a bit individual , it depends on the person. I understand it will lead to confusions IN SOME situations but if it was ONLY bad to do it, why are we advice to  explain the downvote when , i.e down-voting? I think.. to explain the downvote is good, because lets face it - no one is perfect. nothing is perfect. there WILL be downvotes-with-garbage explanations. but there will also be the opposite. can't we try from both sides? especially on stack sites; even if the explanation is bad; Dec 17, 2022 at 21:37
• "my previous account was deleted for not asking questions in the right way" This is not true. Neither SE employees nor moderators delete accounts for low-quality content. You might get rate limited or suspended for low-quality contributions, but this is not a deletion reason. Please also note that making new accounts to circumvent rate limits is frowned-upon as sockpuppetry and this can lead to your accounts being suspended and/or deleted.
– ACuriousMind Mod
Dec 18, 2022 at 22:41


text:
Browse Questions

# Find the area of the region bounded by $y^2 = 9x,\; x = 2,\; x = 4$ and the $x$ - axis in the first quadrant.

$\begin{array}{1 1} 16-4\sqrt 2sq.units. \\8-4\sqrt 2sq.units. \\ 16-2\sqrt 2sq.units. \\ 8-4\sqrt 3sq.units.\end{array}$

Toolbox:
• To find the area bounded by the curve y=f(x),x-axis and the ordinate x=a,x=b then the required area is given by,
• $A=\int_a^b y\;dx=\int_a^b f(x)\;dx.$
Given curve is $y^2=9x\Rightarrow y=3\sqrt x.$


text:
Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Homemade and Repurposed Lab Gear Select A Forum Fundamentals   » Chemistry in General   » Organic Chemistry   » Reagents and Apparatus Acquisition   » Beginnings   » Responsible Practices   » Miscellaneous   » The Wiki Special topics   » Technochemistry   » Energetic Materials   » Biochemistry   » Radiochemistry   » Computational Models and Techniques   » Prepublication Non-chemistry   » Forum Matters   » Legal and Societal Issues   » Detritus   » Test Forum

Pages:  1  ..  12    14
Author: Subject: Homemade and Repurposed Lab Gear
Morgan
International Hazard

Posts: 1384
Registered: 28-12-2010
Member Is Offline

Mood: No Mood

Here's a fused quartz sphere repurposed into a feeder. It's nice in that after you tap it to get the solution all the way down it stops dripping yet continues to feed unlike most feeders that drip off and on. I have a pinhole in the top of a coke bottle and inverted it will drip constantly every few seconds. Somehow air skirts in around the tiny hole. It's curious how some shapes won't work and others with 10 times as big a hole do. The nature of surface tension, air bubbles, gravity, and partial vacuums makes for some challenging riddles on how liquids flow.
You can make reverse images with it or start a fire too, another use as a lens.

Morgan
International Hazard

Posts: 1384
Registered: 28-12-2010
Member Is Offline

Mood: No Mood

Yesterday I fiddled around in the garage and came across some magnet clamps purchased from Enco and thrift store stands bought a few years back. One stand was from a fireplace utensil set where a stoker, broom, shovel and tongs hung from. Another some lamp base or such and another mystery stand.

The other parts are an old curtain rod I had with a curl or hook at the end, some V-shaped fixture that held a light bulb, and odd little square beam clamps bought at a habitat for humanity store that accepts 1/4 20 screws in 3 places and a smaller screw in another plus some holes - maybe not all that useful here, but an afterthought.

The last parts are some fixtures or branch pole things for the magnet bases that are all iron or steel except for the knobs that sleeve over nuts underneath. In a way they are reminiscent of ring stand parts. Its curious how the outermost hole tightens first and then the inner, kind of unusual to get used to. The outermost holes are two sizes if you have rods/poles that match the diameters.
https://www.bhphotovideo.com/c/product/1084866-REG/platinum_...

This hodgepodge isn't all that great or ideal, but was a start cobbled together from dissimilar parts just to see what might evolve.
Now to find some fingers of some sort I guess ...

The small 3-legged stand is another object bought at the thrift store and an old ~12 inch diameter glass stoplight lens which is now a little bird bath.

[Edited on 21-9-2019 by Morgan]
earpain
Harmless

Posts: 6
Registered: 11-9-2019
Member Is Offline

 Quote: Originally posted by yobbo II How would you know they are borosilicate? Yob

This is a question I have faced a lot in my days of DIY glasswork.
1. With tubing or bottles/flasks, if you hold the bottle up to the light above you, so that the walls of the bottle are seen along your line of sight, if you will see a green tint, it is soda-lime aka soft glass. If it still looks completely transparent with no color tint, it is likely boro.

Another test that is even more reliable:
Take two similar pieces of glass, one being the glass in question, the other being a piece that you know for sure is borosilicate. When i say similar I mean, ends are both rod with similar diameter, or ends are both tube(mouth) with similar diameter and similar thickness.

Rotate both over a propane/mapp gas or oxy/propane torch and fuse them together properly like you would normally when creating a weld.

If the mystery piece is not borosilicate, basically upon cooling EVERYTHING in or near the joint will crack, pretty dramatically.
wg48temp9
National Hazard

Posts: 296
Registered: 30-12-2018
Member Is Offline

 Quote: Originally posted by Morgan Here's a fused quartz sphere repurposed into a feeder. It's nice in that after you tap it to get the solution all the way down it stops dripping yet continues to feed unlike most feeders that drip off and on. I have a pinhole in the top of a coke bottle and inverted it will drip constantly every few seconds. Somehow air skirts in around the tiny hole. It's curious how some shapes won't work and others with 10 times as big a hole do. The nature of surface tension, air bubbles, gravity, and partial vacuums makes for some challenging riddles on how liquids flow. You can make reverse images with it or start a fire too, another use as a lens.

There should be a law about that, using a fused quartz flask as a bird feeder should be criminal especially for sm members LOL

i am wg48 but not on my usual pc hence the temp handle.

Thank goodness for Fleming and the fungi.
wg48temp9
National Hazard

Posts: 296
Registered: 30-12-2018
Member Is Offline

 Quote: Originally posted by earpain This is a question I have faced a lot in my days of DIY glasswork. 1. With tubing or bottles/flasks, if you hold the bottle up to the light above you, so that the walls of the bottle are seen along your line of sight, if you will see a green tint, it is soda-lime aka soft glass. If it still looks completely transparent with no color tint, it is likely boro.

I have re-purposed bulb growing glasses and coffee beakers because they looked looked clear with no green tint. I placed one of the bulb growing glasses containing a coolish solution on a temperature controlled hot plate (~80C) and it cracked across the base. That suggests its not boro.

A few days ago I purchased some glass coffee mugs from a charity shop. The first one I examined appeared clear but the second identical one had a very distinct green tint.

Expensive bottles of clear spirits tend to not to have significant green tints and decorative glass items tend to be clear almost no green tint.
You can probably be confident (>90%) that if there is a green tint its not boro. But if its clear its very weak evidence that it is boro.

i am wg48 but not on my usual pc hence the temp handle.

Thank goodness for Fleming and the fungi.
Morgan
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My repurposed synthetic ruby rod/hummingbird perch passed the test today. The rod is supported by 2 clear fused quartz tubes with blue silicone tubing and the male ruby-throated hummingbird is sipping from a borosilicate drinking straw. Only when he ruffles his feathers or the sun hits him just right can you see his iridescent ruby throat.
The perch was fashioned after seeing a simple design such as this. I happened to have an odd single red wooden ball in the garage which was fortuitous and the ruby rod bought on eBay years ago.
https://www.bestnest.com/bestnest/RTProduct.asp?SKU=SOE-SEHH...
https://www.sibleyguides.com/2011/09/the-basics-of-iridescen...

[Edited on 28-9-2019 by Morgan]
earpain
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Quote: Originally posted by wg48temp9
 Quote: Originally posted by earpain This is a question I have faced a lot in my days of DIY glasswork. 1. With tubing or bottles/flasks, if you hold the bottle up to the light above you, so that the walls of the bottle are seen along your line of sight, if you will see a green tint, it is soda-lime aka soft glass. If it still looks completely transparent with no color tint, it is likely boro.

I have re-purposed bulb growing glasses and coffee beakers because they looked looked clear with no green tint. I placed one of the bulb growing glasses containing a coolish solution on a temperature controlled hot plate (~80C) and it cracked across the base. That suggests its not boro.

A few days ago I purchased some glass coffee mugs from a charity shop. The first one I examined appeared clear but the second identical one had a very distinct green tint.

Expensive bottles of clear spirits tend to not to have significant green tints and decorative glass items tend to be clear almost no green tint.
You can probably be confident (>90%) that if there is a green tint its not boro. But if its clear its very weak evidence that it is boro.

Hmm, you may be indeed be correct, and I'll have to revise my system.
Candy thermometers. $3-$5 at the food store. There's no kitchen thermometer sold that has such a wide range of temps. I have bought them before, removed the metallic element fused to the glass on the bottom, along with the capillary tube that shows the temperature, and then worked the remaining wide, but very very thin walled tube.

If it is borosilicate, once I'm both flame working and turning it into various devices, or auxillary glassblowing devices(temporary handle, extension tube, etc.) - it is the WORST I've ever worked with. I have flameworked soft/flint glass more reliably than these thermometers.

I suppose there is an application in mind when a glassware is designed. All glass that is not quartz has something added to it mostly just to lower its working temperature, not everyone has Oxy-Acetylene or mega-kilns handy.

Perhaps the non-green tinted glass that cracked on you, cracked for reasons unrelated to what metals or salts were added to the glass 'stew'? Like with candy thermometers. Surely they must technically be borosilicate. Not sure why they are so finicky.
3 general types of clear glass(in terms of material):
soda-lime
borosilicate

And then there's a plethora of structural factors, and considerations for the items application. In general I would discourage chemists to repurpose or DIY glassware unless they also have some proper training in glasswork. Scientific glass is definitely the most versatile and robust.
wg48temp9
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Here is a pic of the two coffee glasses side by side. The green tint of the right hand one is distinct. Of cause although they looked identical and come with identical metal housings (removed for the pic) I am not certain they are. I also swapped the position of the glasses incase the green tint was from a nearby object.

This is weak evidence as I am assuming they are identical as they were purchased together. Apparently the the green color comes from iron impurities in the material used to make the glass. So these must have been made from different batches of material.

This particular green tint seems more emerald green than the dull green tint I usually see in bottle glass.

Here is a pic of two tall vases and a boro 2l measuring cylinder from Fisher Scientific (right lower). All three are about 0.8m tall. The left hand vase is dark with a hint of green, the top vase has a slight dull green tint and the boro cylinder is lighter with almost no color. I assume the vases are soda glass. All three appear clear when viewed from the side.

Sorry the pic is poor quality it was taken under fluorescent lighting in the bathroom having just rinsed off the dust.

PS I got lucky when I won the measuring cylinder on ebay no one else bid on it so it was very cheap. Its very impressive given it is almost a meter long. I am still trying to buy a stopper for it that does not cost more than I paid for it.

[Edited on 10/1/2019 by wg48temp9]

i am wg48 but not on my usual pc hence the temp handle.

Thank goodness for Fleming and the fungi.
Yttrium2
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Anyone know of where to find borosilicate bulbs/tubes?

Besides HID lighting -- for a diy flask
earpain
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 Quote: Originally posted by Yttrium2 Anyone know of where to find borosilicate bulbs/tubes? Besides HID lighting -- for a diy flask

Do NOT buy from Chemistry Equipment vendors.
Do not buy from coffee shops with their new boro glass straw trend(\$5 for 8in tube)
https://www.mountainglass.com/boro-glass-33-coe/clear
or any supplier specializing in customers who are -glass blowers-
Seriously the link above, I challenge anyone to find a cheaper deal. Disregard everything besides the Boro tubing and the ground glass joints(no 24/40 though -/)
Eisenpanzer432
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DavidJR hey I forgot I posted this haha. I painted it and then reduced the size to fit in my hood I’m building. It will be able to fit into slots and clamp or pin in place, so I can remove it if I need to clean or make more room in the hood. It’ll be a while, hopefully not another year lol; I’m still deciding what to make the sash from before I build. I may use plexiglass with bending hinges at two points to give three sections that I can fold up or down to change the height of the sash; I’m limited on vertical space. I’ll have straps on the sides of the plexi to pull tight against caulk or foam to seal the sides of the plexi agains the sides of the hood when it’s folded down. I hope that makes sense; perhaps a picture will be needed when it’s complete.
SWIM
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 Quote: Originally posted by wg48temp9 PS I got lucky when I won the measuring cylinder on ebay no one else bid on it so it was very cheap. Its very impressive given it is almost a meter long. I am still trying to buy a stopper for it that does not cost more than I paid for it. [Edited on 10/1/2019 by wg48temp9]

Sometimes big adaptors are cheaper than big stoppers.
I've got a few items where I use like a 24/40 to 55/50 adaptor with a 24/40 stopper in the top instead of a 55/50 stopper.

Less then Ideal, but better than nothing.

What size are you after?

EDIT: On another front, I believe that testing the index of refraction might be a good easy way to find boro glass.
A solution with the same index of refraction as boro would make it easy.
Just put glass in solution and see if it disappears.

[Edited on 20-12-2019 by SWIM]

Do you want red P?
Eat lots of beets
Morgan
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Quote: Originally posted by wg48temp9
 Quote: Originally posted by earpain This is a question I have faced a lot in my days of DIY glasswork. 1. With tubing or bottles/flasks, if you hold the bottle up to the light above you, so that the walls of the bottle are seen along your line of sight, if you will see a green tint, it is soda-lime aka soft glass. If it still looks completely transparent with no color tint, it is likely boro.

I have re-purposed bulb growing glasses and coffee beakers because they looked looked clear with no green tint. I placed one of the bulb growing glasses containing a coolish solution on a temperature controlled hot plate (~80C) and it cracked across the base. That suggests its not boro.

A few days ago I purchased some glass coffee mugs from a charity shop. The first one I examined appeared clear but the second identical one had a very distinct green tint.

Expensive bottles of clear spirits tend to not to have significant green tints and decorative glass items tend to be clear almost no green tint.
You can probably be confident (>90%) that if there is a green tint its not boro. But if its clear its very weak evidence that it is boro.

Maybe of interest to test green tinted and clear samples using this method to see what comes about.
Index of Refraction - disappearing glassware demonstration // Homemade Science with Bruce Yeany

"You can make Pyrex glass disappear by immersing it in glycerin or mineral oil. However, mineral oil comes in different weights, and each variety has a different index of refraction. To match the index of refraction of Pyrex glass, you’ll need a mixture of mineral oils of different weights. To create the proper mixture, place a Pyrex glass object into a large glass beaker and pour in enough heavy mineral oil to submerge it partially. Slowly add light mineral oil and stir. Watch the glass object as you pour. Most Pyrex glass will disappear when the mixture is two parts heavy mineral oil to one part light mineral oil. Notice the swirling refraction patterns as you mix the oils."
https://www.exploratorium.edu/snacks/disappearing-glass-rods
wg48temp9
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Quote: Originally posted by SWIM
 Quote: Originally posted by wg48temp9 PS I got lucky when I won the measuring cylinder on ebay no one else bid on it so it was very cheap. Its very impressive given it is almost a meter long. I am still trying to buy a stopper for it that does not cost more than I paid for it. [Edited on 10/1/2019 by wg48temp9]

Sometimes big adaptors are cheaper than big stoppers.
I've got a few items where I use like a 24/40 to 55/50 adaptor with a 24/40 stopper in the top instead of a 55/50 stopper.

Less then Ideal, but better than nothing.

What size are you after?

EDIT: On another front, I believe that testing the index of refraction might be a good easy way to find boro glass.
A solution with the same index of refraction as boro would make it easy.
Just put glass in solution and see if it disappears.
[Edited on 20-12-2019 by SWIM]

Its a 45/50. There are adapters on ebay that are 45/50 but they cost about £50. I have now won some blocks of PTFE so when I get round to fixing my old lathe I will turn a stopper and adaptor for for it. Not that I have a particular use for it at this time.

Yes a matched index may work for a small object but how do you test a 2l measuring cylinder and will it have the discrimination? Pyrex is 1.47 window glass is 1.52.

Perhaps a type of refractometer can be used on flat-ish area of one side
of the glass object but the cheap reflectometers are for use with liquids.

i am wg48 but not on my usual pc hence the temp handle.

Thank goodness for Fleming and the fungi.
Morgan
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That does seem difficult to submerge that large object. Maybe just an edge of the flange at the base would be enough to reveal a disappearance. And still as you say the numbers are close to window glass. I was reading quartz is close to borosilicate too,
coming in at 1.458 to boro's 1.474 refractive index.

I bought an oil candle lamp and was wondering what it was made of, the copy referred to it as a crystal oil lamp. Not caring enough yet to test it with oil and having to wash it out I unconventionally heat stressed it enough to burn a spot on a violet colored silicone pad underneath it, right in the bottom center where the direct flame pushes the alcohol off to the sides. In that test it ran well over a minute. While not conclusive it was a positive outcome and you could feel the heat radiating from it after it had nearly run dry.
Having bought small new borosilicate bottles and vials, and the oil lamp for example there is a certain look to the glass, as if it has a kind of glossiness or ultra smoothness to it, maybe like a glazed appearance. Of course this might just be my subjective imagination or that other glasses could emulate the effect.

The candle oil lamp, just shaking water out of it takes some time to empty it, the hole a little over 8 mm. This was just a first test.

As a safety aside, on the Princess oil lamp on an eBay offering, there was some literature or warnings that came with it.

"Use only lamp oil. Do not use gasoline, alcohol, or solvents as fuel in this or any other oil lamp. Failure to follow these instructions may cause fire or explosion which could result in property damage or serious personal injury from cuts or burns."
The lamp is a thin glass, very much like the cylindrical borosilicate food storage jars with bamboo lids sold on eBay. But maybe it's lead crystal.

[Edited on 20-12-2019 by Morgan]
Morgan
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Tidbits on thermal expansion properties of glass - scroll down for each type of glass.

Also maybe of interest
"Thermal conductivity of glasses typically varies between 0.5 – 1.5 W/(m K), the highest being for fused silica and lowest for lead glass. The thermal conductivity measurement is complicated because heat is also transferred internally by radiation. The thermal conductivity is non-linear, increasing with temperature. And, as for any processed material, the exact compound and the processing parameters affect the thermal properties significantly, causing the literature values to vary widely."

Glass: A Group Of Familiar Materials With Varying Properties
https://www.electronics-cooling.com/2003/02/glass-a-group-of...
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text:
## Stream: general

### Topic: updating the nth element of a list

#### Scott Morrison (Nov 30 2018 at 01:11):

Do we have something equivalent to

def patch_nth {α : Type} (f : α → α) : ℕ → list α → list α
| _ []           := []
| 0 (h :: t)     := f h :: t
| (n+1) (h :: t) := h :: patch_nth n t

in mathlib?

#### Scott Morrison (Nov 30 2018 at 01:12):

I've found a few times that it's really painful for update the nth element, because you have to deal with nth returning an option, even when you know it's there.

#### Scott Morrison (Nov 30 2018 at 01:12):

A slight variation that is even more useful:

def opatch_nth {α : Type} (f : α → option α) : ℕ → list α → list α
| _ []           := []
| 0 (h :: t)     := match f h with
| (some e) := e :: t
| none     := t
end
| (n+1) (h :: t) := h :: opatch_nth n t

#### Kenny Lau (Nov 30 2018 at 01:13):

I've found a few times that it's really painful for update the nth element, because you have to deal with nth returning an option, even when you know it's there.

... did you say it's painful to update the nth element?

#### Scott Morrison (Nov 30 2018 at 01:13):

Of course there is update_nth

#### Kenny Lau (Nov 30 2018 at 01:14):

also what is it with your meta?

#### Scott Morrison (Nov 30 2018 at 01:14):

Oh, yeah, those metas are completely unnecessary :-) Just habit, as I was in a whole file where most things were meta.

#### Scott Morrison (Nov 30 2018 at 01:15):

But to use update_nth, you need to use nth earlier to get out the existing element to modify.

#### Scott Morrison (Nov 30 2018 at 01:15):

The typical example here is that I have a list of some structure, and I want to modify a single field of the nth element.

#### Scott Morrison (Nov 30 2018 at 01:16):

In this case the function f can be λ s, { f := x, .. s}.

#### Mario Carneiro (Nov 30 2018 at 03:15):

did you say you want to modify the nth element?

#### Reid Barton (Nov 30 2018 at 03:29):

Did somebody say "lens"?

#### Scott Morrison (Nov 30 2018 at 06:22):

Thanks. And yes, let's have more lens. :-)

#### Keeley Hoek (Dec 01 2018 at 05:29):

Wait, isn't the syntax to declare functions in lean meta def ....? ;)

Last updated: May 14 2021 at 12:18 UTC


text:
# Modifying tree Graphs

Please consider a tree graph. There is one unique path connecting any two vertices.

However, I wonder how to address the following question:

Starting from a generic tree, is there an algorithmic way to connect any two nodes of the tree with $M$ unique paths such that no two paths contain the same edge? How can the total number of edges in the modified graph be kept a minimum?

I trying to answer this from a network-engineer's perspective. Any help is appreciated.


text:
Definitions

# Electromagnetic spectrum

The electromagnetic (EM) spectrum is the range of all possible electromagnetic radiation frequencies. The "electromagnetic spectrum" (usually just spectrum) of an object is the characteristic distribution of electromagnetic radiation from that particular object.

The electromagnetic spectrum extends from below the frequencies used for modern radio (at the long-wavelength end) through gamma radiation (at the short-wavelength end), covering wavelengths from thousands of kilometres down to a fraction the size of an atom. It is thought that the short wavelength limit is in the vicinity of the Planck length, and the long wavelength limit is the size of the universe itself (see physical cosmology), although in principle the spectrum is infinite and continuous.

## Range of the spectrum

The spectrum covers EM wave energy having wavelengths from thousands of meters down to fractions of the size of an atom. Frequencies of 30 Hz and below can be produced by and are important in the study of certain stellar nebulae and frequencies as high as 2.9 * 1027 Hz have been detected from astrophysical sources.

Electromagnetic energy at a particular wavelength λ (in vacuum) has an associated frequency f and photon energy E. Thus, the electromagnetic spectrum may be expressed equally well in terms of any of these three quantities. They are related by the equations:

$c = ,!$frequency x wavelength or $lambda = frac\left\{c\right\}\left\{f\right\} ,!$ and $E=hf ,!$ or $E=frac\left\{hc\right\}\left\{lambda\right\} ,!$

Where $c=299,792,458,!$m/s (speed of light) and $h,!$ is Planck's constant, $\left(h approx 6.626069 cdot 10^\left\{-34\right\} mbox\left\{J\right\} cdot mbox\left\{s\right\} approx 4.13567 mathrm\left\{mu\right\} mbox\left\{eV\right\}/mbox\left\{GHz\right\}\right)$.

So, high-frequency electromagnetic waves have a short wavelength and high energy; low-frequency waves have a long wavelength and low energy.

Whenever light waves (and other electromagnetic waves) exist in a medium (matter), their wavelength is decreased. Wavelengths of electromagnetic radiation, no matter what medium they are traveling through, are usually quoted in terms of the vacuum wavelength , although this is not always explicitly stated.

Generally, EM radiation is classified by coiled wavelength into radio wave, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays and gamma rays.

The behavior of EM radiation depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. When EM radiation interacts with single atoms and molecules, its behavior also depends on the amount of energy per quantum it carries. Electromagnetic radiation can be divided into octaves — as sound waves are.

Spectroscopy can detect a much wider region of the EM spectrum than the visible range of 400 nm to 700 nm. A common laboratory spectroscope can detect wavelengths from 2 nm to 2500 nm. Detailed information about the physical properties of objects, gases, or even stars can be obtained from this type of device. It is widely used in astrophysics. For example, many hydrogen atoms emit a radio wave photon which has a wavelength of 21.12 cm.

While the classification scheme is generally accurate, in reality there is often some overlap between neighboring types of electromagnetic energy. For example, SLF radio waves at 60 Hz may be received and studied by astronomers, or may be ducted along wires as electric power. Also, some low-energy gamma rays actually have a longer wavelength than some high-energy X-rays. This is possible because "gamma ray" is the name given to the photons generated from nuclear decay or other nuclear and subnuclear processes, whereas X-rays on the other hand are generated by electronic transitions involving highly energetic inner electrons. Therefore the distinction between gamma ray and X-ray is related to the radiation source rather than the radiation wavelength. Generally, nuclear transitions are much more energetic than electronic transitions, so usually, gamma-rays are more energetic than X-rays. However, there are a few low-energy nuclear transitions (e.g. the 14.4 keV nuclear transition of Fe-57) that produce gamma rays that are less energetic than some of the higher energy X-rays.

Radio waves generally are utilized by antennas of appropriate size (according to the principle of resonance), with wavelengths ranging from hundreds of meters to about one millimeter. They are used for transmission of data, via modulation. Television, mobile phones, MRI, wireless networking and amateur radio all use radio waves.

Radio waves can be made to carry information by varying a combination of the amplitude, frequency and phase of the wave within a frequency band and the use of the radio spectrum is regulated by many governments through frequency allocation. When EM radiation impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of that conductor by exciting the electrons of the conducting material. This effect (the skin effect) is used in antennas. EM radiation may also cause certain molecules to absorb energy and thus to heat up, thus causing thermal effects and sometimes burns; this is exploited in microwave ovens.

### Microwaves

The super high frequency (SHF) and extremely high frequency (EHF) of microwaves come next up the frequency scale. Microwaves are waves which are typically short enough to employ tubular metal waveguides of reasonable diameter. Microwave energy is produced with klystron and magnetron tubes, and with solid state diodes such as Gunn and IMPATT devices. Microwaves are absorbed by molecules that have a dipole moment in liquids. In a microwave oven, this effect is used to heat food. Low-intensity microwave radiation is used in Wi-Fi, although this is at intensity levels unable to cause thermal heating.

Volumetric heating, as used by microwaves, transfer energy through the material electro-magnetically, not as a thermal heat flux. The benefit of this is a more uniform heating and reduced heating time; microwaves can heat material in less than 1% of the time of conventional heating methods.

When active, the average microwave oven is powerful enough to cause interference at close range with poorly shielded electromagnetic fields such as those found in mobile medical devices and cheap consumer electronics.

Terahertz radiation is a region of the spectrum between far infrared and microwaves. Until recently, the range was rarely studied and few sources existed for microwave energy at the high end of the band (sub-millimetre waves or so-called terahertz waves), but applications such as imaging and communications are now appearing. Scientists are also looking to apply terahertz technology in the armed forces, where high frequency waves might be directed at enemy troops to incapacitate their electronic equipment.

The infrared part of the electromagnetic spectrum covers the range from roughly 300 GHz (1 mm) to 400 THz (750 nm). It can be divided into three parts:

• Far-infrared, from 300 GHz (1 mm) to 30 THz (10 μm). The lower part of this range may also be called microwaves. This radiation is typically absorbed by so-called rotational modes in gas-phase molecules, by molecular motions in liquids, and by phonons in solids. The water in the Earth's atmosphere absorbs so strongly in this range that it renders the atmosphere effectively opaque. However, there are certain wavelength ranges ("windows") within the opaque range which allow partial transmission, and can be used for astronomy. The wavelength range from approximately 200 μm up to a few mm is often referred to as "sub-millimetre" in astronomy, reserving far infrared for wavelengths below 200 μm.
• Mid-infrared, from 30 to 120 THz (10 to 2.5 μm). Hot objects (black-body radiators) can radiate strongly in this range. It is absorbed by molecular vibrations, where the different atoms in a molecule vibrate around their equilibrium positions. This range is sometimes called the fingerprint region since the mid-infrared absorption spectrum of a compound is very specific for that compound.
• Near-infrared, from 120 to 400 THz (2,500 to 750 nm). Physical processes that are relevant for this range are similar to those for visible light.

Above infrared in frequency comes visible light. This is the range in which the sun and stars similar to it emit most of their radiation. It is probably not a coincidence that the human eye is sensitive to the wavelengths that the sun emits most strongly. Visible light (and near-infrared light) is typically absorbed and emitted by electrons in molecules and atoms that move from one energy level to another. The light we see with our eyes is really a very small portion of the electromagnetic spectrum. A rainbow shows the optical (visible) part of the electromagnetic spectrum; infrared (if you could see it) would be located just beyond the red side of the rainbow with ultraviolet appearing just beyond the violet end.

EM radiation with a wavelength between approximately 400 nm and 700 nm is detected by the human eye and perceived as visible light. Other wavelengths, especially near infrared (longer than 700 nm) and ultraviolet (shorter than 400 nm) are also sometimes referred to as light, especially when the visibility to humans is not relevant.

If radiation having a frequency in the visible region of the EM spectrum reflects off of an object, say, a bowl of fruit, and then strikes our eyes, this results in our visual perception of the scene. Our brain's visual system processes the multitude of reflected frequencies into different shades and hues, and through this not-entirely-understood psychophysical phenomenon, most people perceive a bowl of fruit.

At most wavelengths, however, the information carried by electromagnetic radiation is not directly detected by human senses. Natural sources produce EM radiation across the spectrum, and our technology can also manipulate a broad range of wavelengths. Optical fiber transmits light which, although not suitable for direct viewing, can carry data that can be translated into sound or an image. The coding used in such data is similar to that used with radio waves.

### Ultraviolet light

Next in frequency comes ultraviolet (UV). This is radiation whose wavelength is shorter than the violet end of the visible spectrum.

Being very energetic, UV can break chemical bonds, making molecules unusually reactive or ionizing them, in general changing their mutual behavior. Sunburn, for example, is caused by the disruptive effects of UV radiation on skin cells, which can even cause skin cancer, if the radiation damages the complex DNA molecules in the cells (UV radiation is a proven mutagen). The Sun emits a large amount of UV radiation, which could quickly turn Earth into a barren desert; however, most of it is absorbed by the atmosphere's ozone layer before reaching the surface.

### X-rays

After UV come X-rays. Hard X-rays have shorter wavelengths than soft X-rays. X-rays are used for seeing through some things and not others, as well as for high-energy physics and astronomy. Neutron stars and accretion disks around black holes emit X-rays, which enable us to study them.

X-rays will pass through most substances, and this makes them useful in medicine and industry. X-rays are given off by stars, and strongly by some types of nebulae. An X-ray machine works by firing a beam of electrons at a "target". If the electrons are fired with enough energy, X-rays will be produced.

### Gamma rays

After hard X-rays come gamma rays, which were discovered by Paul Ulrich Villard in 1900. These are the most energetic photons having no defined lower limit to their wavelength. They are useful to astronomers in the study of high energy objects or regions and find a use with physicists thanks to their penetrative ability and their production from radioisotopes. The wavelength of gamma rays can be measured with high accuracy by means of Compton scattering.

Note that there are no precisely defined boundaries between the bands of the electromagnetic spectrum. Radiation of some types have a mixture of the properties of those in two regions of the spectrum. For example, red light resembles infrared radiation in that it can resonate some chemical bonds.


text:
# $RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter XIII of May's book, Equivariant Homotopy and Equivariant Cohomology). One impetus for introducing this definition is that it allows one to properly formulate an equivariant analogue of the suspension isomorphism. Indeed, if $E_G^*$ is an $RO(G)$-graded theory with reduced counterpart $\tilde{E}_G^*$, then we have natural $\mathbb{Z}$-module isomorphisms $$\tilde{E}_G^{\alpha}(X)\cong\tilde{E}_G^{\alpha+V}(\Sigma^V(X)).$$ Here, $\alpha\in RO(G)$, $V$ is a real orthogonal $G$-module, $X$ is a $G$-space, and $\Sigma^V(X)$ is its equivariant reduced suspension with respect to $V$.

To recover an underlying $\mathbb{Z}$-grading, one identifies a non-negative integer $n$ with the trivial $G$-module $[\mathbb{R}^n]\in RO(G)$ so that $$\tilde{E}_G^n(X):=\tilde{E}_G^{[\mathbb{R}^n]}(X).$$ The usual suspension isomorphism then determines the negative grading components of $\tilde{E}_G^*$.

$\textbf{Question:}$ Suppose that $V$ is a finite-dimensional orthogonal $G$-representation. What is the relationship between $E_G^{n+V}(\text{pt})$ and $E_G^{n+\dim(V)}(\text{pt})$?

I suspect this is well-known and understood. Yet, I am having difficulty finding references that directly address this issue. I would therefore appreciate any and all advice concerning references.

-

There is no relationship in general. If $E$ is a module over the $G$-equivariant $MU$ spectrum then $E_G^{n+V}(\text{pt})$ is naturally isomorphic to $E^{n+\text{dim}_{\mathbb{R}}(V)}(\text{pt})$ for all complex representations $V$. The same works for real representations if $E$ is a module over the equivariant $MO$.