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# How do I model this expression maximization problem as a Knapsack problem?

Given two arrays of the same length $n$:

$A = \{a_1,a_2,\dots,a_n\}$,

$B = \{b_1,b_2,\dots,b_n\}$,

I have to maximize the following expression:

$$\frac{a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k}}{b_{i_1} + b_{i_2} + b_{i_3}+ \dots + b_{i_k}}$$

Where $k \lt n$.

Given the constraints:

Firstly, when you select a particular $a_i$ you have select it's corresponding $b_i$, and vice versa.

Secondly, you have to select exactly $k$ elements from the arrays. Where $k \lt n$.

Assumptions:

The assumption is that $A$ and $B$ both contain natural numbers only.

What I have tried:

I have tried sorting the arrays using the ratio $\frac{a_n}{b_n}$ in the non-decreasing order, but this gives an incorrect result since it's trying to optimize $\frac{a_1}{b_1} + \frac{a_2}{b_2} + \dots \frac{a_k}{b_k}$. That is different from my desired objective function.

I have also thought of treating the denominator as a knapsack but this does not make sense because there are no weights involved.

Any suggestion on what I can do to model this problem as a knapsack optimization problem?

One approach (not based on the knapsack problem): suppose you are given a threshold $\tau$ and want to check whether you can find a subset such that

$$\frac{a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k}}{b_{i_1} + b_{i_2} + b_{i_3}+ \dots + b_{i_k}} \ge \tau.$$

Could you solve that problem? If you could, you could use binary search on $\tau$.

Note that the above condition is equivalent to

$$a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k} \ge \tau \cdot (b_{i_1} + b_{i_2} + b_{i_3}+ \dots + b_{i_k}),$$

or equivalently,

$$a_{i_1} - \tau b_{i_1} + a_{i_2} - \tau b_{i_2} + a_{i_3} - \tau b_{i_3} + \dots + a_{i_k} - \tau b_{i_k} \ge 0.$$

And that enables us to find a simple way to check for the existence of that subset.

See if that gives you any ideas.

• How would I know what are my particular selections of $k$ by just selecting $\tau$ ? Take for instance a select a value of $\tau$ I still need to know what are the exact elements that comprise my set of $k$ elements. How does binary search help me for that? If the exact $k$ elements were known then maybe, but without it I don't get it how to do it. – ng.newbie Aug 22 '18 at 15:34
• @ng.newbie, Binary search doesn't help you find those elements; you'll have to figure out how to solve that problem. This is your exercise, so I'm leaving something for you to do. Based on your question, I believe that if you spend some time on it you will be capable of finding a solution for that part. So spend some time thinking about that one and see what you can come up with. – D.W. Aug 22 '18 at 18:56
• Okay so you think is it possible to maximize a fraction/ratio by the use of DP in any way? DP can maximize the profit, but I know that is true when the profit function is just where you add the value, ie. an addition function. But what happens when the profit function is changed to something more complex? Like a math expression with multiplication and division. Does Knapsack or any other DP work there? – ng.newbie Aug 24 '18 at 7:16
• @ng.newbie, sorry, I can't understand any of that comment. At this point I'm going to leave it to you to solve your exercise -- I've given you a sketch of an approach that might lead you to a solution, but I'm not going to provide a complete solution here. (And nowhere does my answer say anything about dynamic programming.) I think that's probably all I have to say on the subject -- I'm going to leave it to you to take it the rest of the way there. – D.W. Aug 24 '18 at 7:54
• What I mean was the normal knapsack value function is $max(\sum {v_i})$, what if instead of $v_i$ you get a function that is something like in the question. Will knapsack work in that case? Provided you are not restricted on the items that you can pick. The normal profit function is just where you add the value and check the maximum, but what if it is a fraction/ratio? I hope I could explain my doubt. – ng.newbie Aug 27 '18 at 7:47


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# zbMATH — the first resource for mathematics

A faster FPT algorithm for the maximum agreement forest problem. (English) Zbl 1148.68049
Summary: Given two unrooted, binary trees, $$T_{1}$$ and $$T_{2}$$, leaf labelled bijectively by a set of species $$L$$, the Maximum Agreement Forest (MAF) problem asks to find a minimum cardinality collection $$\mathcal F = \{t_{1},\ldots, t_{k}\}$$ of phylogenetic trees where each element of $$F$$ is a subtree of both $$T_{1}$$ and $$T_{2}$$, the elements of $$F$$ are pairwise disjoint, and the leaf labels for the elements of $$F$$ partition the leaf label set $$L$$. We give an efficient Fixed-Parameter Tractable (FPT) algorithm for the MAF problem, significantly improving on an FPT algorithm given in [B. Allen and M. Steel, Ann. Comb. 5, No. 1, 1–15 (2001; Zbl 0978.05023)]. Whereas the algorithm from [loc. cit.] has a running time of $$O(k^{3k}) + p(|L|)$$, our algorithm runs in time $$O(4^{k} k^{5}) + p(|L|)$$, where $$k$$ bounds the size of the agreement forest and $$p(\cdot)$$ is a low order polynomial.

##### MSC:
 68W05 Nonnumerical algorithms 05C05 Trees 05C85 Graph algorithms (graph-theoretic aspects) 68Q25 Analysis of algorithms and problem complexity 68R10 Graph theory (including graph drawing) in computer science 92D15 Problems related to evolution
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# Solving a limit approaching zero with a complex denominator

We've been given the definition of a derivative as:

$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

We are asked to use this to find the derivative of the function $f(x)=\frac{1}{1-x}$ showing every step.

I can get to here: $$f'(x)=\lim_{h\to0}\frac{1}{h-hx-h^2}-\frac{1}{h-hx}$$

When I try using online equation solvers they just jump straight to the answer and I can't figure out how. Wolfram Alpha's step-by-step solution also doesn't give me any intermediate steps between this and the solution:

$$\frac{1}{(x-1)^2}$$

So, my question is, how do I solve a limit like the one above where every value in the denominator approaches 0?

(I'm guessing I'm just missing some algebraic tricks)

-

Hint: Bring the expression you got to a common denominator. There will be pleasant cancellation.

Remark: It would have made life easier if you had kept the $h$ "outside," that is, looked at $$\frac{1}{h}\left(\frac{1}{1-x-h} -\frac{1}{1-x}\right).$$ Then same common denominator hint, somewhat less messy.

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Thanks! While both the answers helped together, I gave it to you because you were 16 seconds faster :P – Josh Oct 16 '12 at 0:08

$$\dfrac{\dfrac1{(1-x-h)} - \dfrac1{(1-x)}}{h} = \dfrac{\dfrac{(1-x) - (1-x-h)}{(1-x)(1-x-h)}}{h} = \dfrac{h}{h(1-x)(1-x-h)} = \dfrac1{(1-x)(1-x-h)}$$ Now apply the limit.

-
Thanks! Took me a while to figure out what was going on, and then I was like, d'oh, of course you can multiply the denominators across both functions to find get the same denominator on both sides. Kids, this is what happens when you stop practicing your maths for 7 years. – Josh Oct 16 '12 at 0:12


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1. ## Int (lnx)/x^2

Integrate (lnx)/(x^2) dx

(Part of a question to determine convergence or divergence)

2. Originally Posted by jlmills5
Integrate (lnx)/(x^2) dx

(Part of a question to determine convergence or divergence)
Try integration by parts with

$\displaystyle u=\ln{x}$ and

$\displaystyle dv=\frac{1}{x^2}dx$

3. Ah, I see what I did...or rather wasn't doing. I used parts like you said but apparently was not simplifying the answer the way the key does. Thanks.

4. I need to find the limit as b -> infinity for [-(lnx+1)/x] from 1 to b.

How do I do this...do I use L'Hopital's rule for the infinity portion? I'm getting -1 as the answer but this isn't right. Any suggestions?

5. Originally Posted by jlmills5
I need to find the limit as b -> infinity for [-(lnx+1)/x] from 1 to b.

How do I do this...do I use L'Hopital's rule for the infinity portion? I'm getting -1 as the answer but this isn't right. Any suggestions?
should be +1. check your signs.


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## The market for gay relationships

The market for homosexual relationships is an interesting one from the analysis perspective. Like the market for heterosexual relationships, it is a matching market (we are in a relationship if and only if I like you AND you like me). Unlike heterosexual relationships, it is not a “bipartite” market, since both the nominal “buyer” and the “seller” in a transaction will come out (no pun intended) of the same pool (gay people of a particular sex).

The other factor that makes this market interesting (purely from an analysis perspective – it’s bad for the participants) is that there is disapproval at various levels for homosexual relationships. Until today, for example, it was downright criminal to indulge in gay sex in India. Even where it is legal, there is massive social and religious opposition to such relationships (think of the shootout at the gay bar in Florida, for example).

Social disapproval has meant that gays sometimes try to keep their sexuality under wraps. Historically, it has been a common practice for gays to enter into heterosexual marriages, and pursue relationships outside. In fact, there is nothing historical about this – read this excellent piece by Srinath Perur on gays in contemporary hinterland Karnataka, for whom Mohanaswamy, a collection of short stories with a gay protagonist, was a kind of life changer.

Organising a market for an item that is illegal, or otherwise frowned upon, is difficult, since people don’t want to be found participating in it. If I were a gay man looking for a partner, for example, I couldn’t go around openly looking for one if I didn’t want my family to know that I’m gay. So the first task would have been discovery – “safe spaces” where I would be happy to expose my sexuality, and where I could also meet potential partners.

When demand and supply exist, buyers and sellers will find a way to meet each other, though often at high cost. One such “way” for homosexual people has been the gay bar. Though not explicitly advertised, such bars act as focal points (I have a chapter on focal points in my book) for gay people.

They also act as an “anti focal point” (a topic I HAVEN’T covered in the book, for a change!) for heterosexual people who want to stay away because they don’t want to be hit on by gay people (thus reducing market congestion – another topic I cover in my book). Similarly other cultural activities have acted as focal points for gay people to get together and meet each other.

Like in heterosexual relationship markets (this is the link to a sample chapter from the book), the advent of dating apps has revolutionised gay dating, as apps such as Tinder and Grindr have provided safe spaces where gays can look for relationships “from the comfort of their homes”. There are studies that show that Grindr has changed the nature of relationships among gay men, and how these apps have “saved lives” in places such as India where homosexuality was criminal until today.

Today’s Indian Supreme Court ruling will have a massive positive impact on gay relationships in India. For starters, there are still millions of people in the closet – while apps such as Tinder and Grindr allowed more people to participate in these markets (since this could be done without really “coming out”), that gay sex was a criminal act would have led to some people to err on the side of caution (and deprive themselves of the chance of a relationship). Gay people who were worried about criminality, but not that much about social sanctions, will now be more willing to come out, leading to an increase in the market size.

Barring congestion (when “bad counterparties” prevent you from finding “good counterparties”),  the likelihood of finding a match in a market is generally proportional to the number of possible counterparties. Since gay relationship markets are not bipartite, we can say that the likelihood of finding a good match varies by the square of the number of market participants (and this brings in the Indian Prime Minister’s infamous 2ab term). In other words, it not only allows the people now coming into the market to find relationships, but it also allows existing players to find better relationships.

Then, there is the second order effect. Decriminalisation will mean that more people will come out of the closet, which will mean more people will find homosexuality to  be “normal” leading to better social mores (to take a personal example, I used to use the word “gay” as a pejorative (to mean “uncool”) until I encountered my first openly gay acquaintance – someone with whom I share on online mailing list). And as social attitudes towards homosexuality change, it will lead to more people coming out of the closet, setting off a virtuous cycle of acceptance of homosexuality.

In other words, today’s decision by the Indian Supreme Court is likely to set off a massive virtuous cycle in the liquidity of the market for homosexual relationships in India!

PS: It is a year since my first book was published, so we are running a promotional offer where you can buy the Kindle version for one dollar (or Rs. 70).

## Information Technology and Large Cities

In my book Between the buyer and the seller, officially released exactly a year ago, I have a chapter on cities. In that I explain why industry clusters form, and certain cities or regions become hubs for certain types of industries.

In that, I spoke about the software industry in California’s Silicon Valley, and in Bangalore. I also mentioned how the Industrial Revolution wasn’t evenly distributed around England, and how it was clustered around textile hubs such as Birmingham and Manchester. I also used that chapter to talk about the problem with government-mandated special economic zones (this podcast with Amit Varma can help you understand the last point).

Back when Silicon Valley was still silicon valley (basically a semiconductor and hardware hub), it wasn’t as concentrated a hub as it is today. It was still fairly common for semiconductor companies to base themselves away from the valley. With the “new silicon valley” and the tech startup scene, though, there is no escaping the valley. It is almost an unwritten rule in US Tech startup circles that if you want to be successful with a tech startup, you better be in the valley.

And this is for good reason, as I explain in the book – Silicon Valley is where the ecosystem for successfully running a tech startup already exists, including access to skilled employees, subcontractors and investors, not to speak of a captive market. This, however, has meant that Silicon Valley is now overcrowded in many respects, with rents being sky high (reflected in high salaries), freeways jammed and other infrastructure under stress.

In fact, it is not just the silicon valley that has got crushed under the weight of being a tech hub – other “secondary hubs” such as Seattle (which also have a few tech majors, and where startups put off by the cost of the valley set up) are seeing their quality of life go down. The traffic and infrastructure woes in Bangalore are also rather similar.

So why is it that information technology has led to hubs that are much larger than historical hubs (based on other industries)? The simple answer lies in investment, or the lack of it.

Setting up an information technology company is “cheap” in terms of the investment in capital expenditure. No land needs to be bought, no plants need to be constructed and no machinery needs to be bought. All one needs is an office space (for which rent is paid monthly), and a set of employees (who again get paid a monthly salary). Even IT infrastructure (such as computing power and storage and communication) can be leased, and paid for periodically.

This implies that there is nothing that stops a startup company from locating itself in one of the existing hubs. This way, the company can avail all the benefits of being in the hub (supplier and customer infrastructure, employee pool, quality of life for employees and investors) without a high upfront cost.

Contrast this to “hard” industries that require manufacturing, where the benefits of being located in hubs is similar but the costs are far higher. As a hub develops, land gets expensive, which puts off further investors from locating themselves in the hub. This puts a natural limit on the size of the hubs, and if you think about it, large cities from earlier era were all “multi-purpose cities”, serving as hubs for several unrelated industries.

With information technology, though, the only impediment to the growth of a hub is the decreasing quality of life, information regarding which gets transmitted in indirect means such as higher rentals and commute times, and poor health. This indirect transmission of costs to investors results in friction, which means information technology hubs will grow larger before they stop growing. And as they go through this process, the quality of life of the hub’s residents suffers!

## Ratings revisited

Sometimes I get a bit narcissistic, and check how my book is doing. I log on to the seller portal to see how many copies have been sold. I go to the Amazon page and see what are the other books that people who have bought my book are buying (on the US store it’s Ray Dalio’s Principles, as of now. On the UK and India stores, Sidin’s Bombay Fever is the beer to my book’s diapers).

And then I check if there are new reviews of my book. When friends write them, they notify me, so it’s easy to track. What I discover when I visit my Amazon page are the reviews written by people I don’t know. And so far, most of them have been good.

So today was one of those narcissistic days, and I was initially a bit disappointed to see a new four-star review. I started wondering what this person found wrong with my book. And then I read through the review and found it to be wholly positive.

A quick conversation with the wife followed, and she pointed out that this reviewer perhaps reserves five stars for the exceptional. And then my mind went back to this topic that I’d blogged about way back in 2015 – about rating systems.

The “4.8” score that Amazon gives as an average of all the ratings on my book so far is a rather crude measure – since one reviewer’s 4* rating might differ significantly from another reviewer’s.

For example, my “default rating” for a book might be 5/5, with 4/5 reserved for books I don’t like and 3/5 for atrocious books. On the other hand, you might use the “full scale” and use 3/5 as your average rating, giving 4 for books you really like and very rarely giving a 5.

By simply taking an arithmetic average of ratings, it is possible to overstate the quality of a product that has for whatever reason been rated mostly by people with high default ratings (such a correlation is plausible). Similarly a low average rating for a product might mask the fact that it was rated by people who inherently give low ratings.

As I argue in the penultimate chapter of my book (or maybe the chapter before that – it’s been a while since I finished it), one way that platforms foster transactions is by increasing information flow between the buyer and the seller (this is one thing I’ve gotten good at – plugging my book’s name in random sentences), and one way to do this is by sharing reviews and ratings.

From this perspective, for a platform’s judgment on a product or seller (usually it’s the seller, but for products such as AirBnb, information about buyers also matters) to be credible, it is important that they be aggregated in the right manner.

One way to do this is to use some kind of a Z-score (relative to other ratings that the rater has given) and then come up with a normalised rating. But then this needs to be readjusted for the quality of the other items that this rater has rated. So you can think of some kind of a Singular Value Decomposition you can perform on ratings to find out the “true value” of a product (ok this is an achievement – using a linear algebra reference given how badly I suck in the topic).

I mean – it need not be THAT complicated, but the basic point is that it is important that platforms aggregate ratings in the right manner in order to convey accurate information about counterparties.

## Platform as a platform

This afternoon, as I was getting off the tube, I looked at the railway platform, and wondered how it compared to “platforms” as we now know in the context of “platform economics“. For those of you under a rock, platform economics talks about the economics of “platforms” that bring together two sides of a market to interact.

In that sense, Uber is a platform connecting drivers to passengers. Ebay is a platform connecting buyers and sellers of used goods. Paypal is a platform connecting people who want to pay and those who want to receive payment. And so forth (these are all textbook examples nowadays).

So is the railway platform a platform? And if not, is it correct that we refer to entities that run two-sided markets as platforms (arguably, the most intuitive meaning of the word “platform” in the last hundred or so years has been in the railway context)? These were some of the questions I grappled with as I walked along the length of the platform at Ealing Broadway.

For those of you who’re not in the know, I’ve written a book on market design. The Takshashila Institution is publishing it, and the book should be out fairly soon (manuscript is complete, but there’s still plenty to do). In that book, I have a chapter on taxi marketplaces such as Uber/Lyft/Ola, and how they’ve transformed the efficiency of the taxi market. Before I introduce these characters, though, I draw the history of the taxi market.

In that, I talk about taxi stands. Taxi stands work in the way of Thomas Schelling’s focal points. Passengers go there because they know empty taxis will go there. Taxi drivers looking for passengers go there because they know passengers looking for taxis will go there. This way, rather than waiting at a random place looking for either a passenger or a ride, going to the taxi stand is rational. And in that sense, taxi stands are a platforms.

In a way, railway platforms are platform in the same sense. Think of a train that wants to pick up passengers, and passengers who want to travel on a train. If there were no designated pick up points, trains would stop at random places, which passengers would have to guess. While engine drivers could see passengers waiting by the side, stopping at random places might have meant that the train would have had to go empty.

From this perspective, railway platforms act as platforms – they are focal points where trains and passengers come together. Passengers wait there because they know trains stop there, and vice versa. And helpfully, there is an actual physical platform that elevates passengers to the height of the train door so they can get on and off easily!

Isn’t this a wonderful way to have complicated a rather simple concept?

## Market depth, pricing and subsidies

A few days back I had written about how startups should determine how much to subsidise their customers during the growth phase – subsidise to the extent of the long-term price. If you subsidise too much initially, elasticity might hit you when you eventually have to raise prices, and that can set you back.

The problem is in determining what this long-term sustainable price will be. In “one-sided markets” where the company manufactures or assembles stuff and sells it on, it is relatively easy, since the costs are well known. The problem lies in two-sided markets, where the long-term sustainable price is a function of the long-term sustainable volume.

A “bug” of any market is transaction costs, and this is especially the case in a two-sided market. If you are a taxi driver on Ola or Uber platform, the time you need to wait for the next ride or distance you travel to pick up your next customer are transaction costs. And the more “liquid” the market (more customers and more drivers), the lesser these transaction costs, and the more the money you make.

In other words, the denser a market, the lower the price required to match demand and supply, with the savings coming out of savings in transaction costs.

So if you are a two-sided market, the long-term sustainable price on your platform is a function of how big your market will be, and so in order to determine how much to subsidise (which is a function of long-term sustainable price), you need to be able to forecast how big the market will be. And subsidise accordingly.

It is well possible that overly optimistic founders might be too bullish about the eventual size of their platform, and this can lead to subsidising to an extent greater than the extent dictated by the long term market size. And some data points from the Indian “marketplace industry” show that this has possibly happened in India.

Having remained credit card only for a long time now, Uber has started accepting cash payments – in order to attract customers who are not comfortable transacting money online. This belated opening shows that Uber perhaps didn’t hit the numbers they had hoped to, using their traditional credit card / wallet model.

Uber has problems on the driver side, too, with an increasing number of its drivers turning out to be rather rude (this is anecdata from several sources, I must confess), refusing rides, fighting with passengers, etc. Competitor Ola has started buying cars and loaning them to drivers, perhaps indicating that the driver side of the market hasn’t grown to their expectations. They are all indicative of overestimation of market size, and an attempt to somehow hit that size rather than operating at the lower equilibrium.

So an additional risk in running marketplaces is that if you overestimate market size, you might end up overdoing the subsidies that you provide to build up the market. And at some point in time you have to roll back those subsidies, which might lead to shrinkage of the market and a possible death spiral.

Now apply this model to your favourite marketplace, and tell me what you think of them.

## Barriers to entry in cab aggregation

The news that Reliance might be getting into the cab aggregation game got me thinking about the barriers to entry in this business. Considering that it is fundamentally an unregulated industry, or rather an industry where players actively flout regulations, the regulatory barrier is not there.

Consequently, anyone who is able and willing to make the investment and set up the infrastructure will be able to enter the industry. The more important barrier to entry, however, is scale.

Recently I was talking to an Uber driver who had recently switched from TaxiForSure. The latter, he said had lost “liquidity” over the last couple of months (after the Ola takeover), with customers and drivers deserting the service successively in a vicious cycle. Given that cab aggregation is a two-sided market, with prominent cross-sided network effects (number of customers depends on number of cabs and vice versa), it is not possible to do business if you are small, and it takes scale.

For this reason, for a new player to enter the cab aggregation business, it takes significant investments. The cost of acquisition for drivers and passengers is still quite high, and this has to be borne by the new player. Given that a significant number of drivers have to be initially attracted, it takes deep pockets to be able to come in.

Industry players were probably banking on the fact that with the industry already seeing consolidation (when Ola bought TaxiForSure), Venture Capitalists might stop funding newer businesses in this segment, and for that reason Uber and Ola might have a free rein. Ola had even stopped subsidising passengers in the meantime, reasoning (correctly for the time) that with their only competition being Uber they might charge market rates.

From this perspective it is significant that the new player who is entering is an industrial powerhouse with both deep pockets and with a reputation of getting their way around in terms of regulation. The first ensures that they can make the requisite investment (without resorting to VC money) and the second gives the hope that the industry might get around the regulatory troubles it’s been facing so far.

I once again go back to this excellent blog post by Deepak Shenoy on the cab aggregation industry. He had mentioned that what Uber and Ola are doing is to lay down the groundwork for a new sector and more efficient urban transport services. That they may not survive but the ecosystem they create will continue to thrive and add value to urban transport. Reliance’s entry into this sector is a step in making this sector more sustainable.

Will I switch once they launch? Depends upon the quality of service. Currently I’m loyal to Uber primarily because of that factor, but if their service drops and Reliance can offer better service I will have no hesitation in switching.

The ET article linked above talks about drivers cribbing about falling incentives by Uber and Ola. It will be interesting to see how the market plays out once the market stabilises and incentives hit long-run market rates (at which aggregators need to make a profit). A number of drivers have invested in cabs now looking at the short-term profits at hand, but these will surely drop with incentives as the industry stabilises.

Reliance’s entry into cab aggregation is also ominous to other “new” sectors that have shown a semblance of settling down after exuberant VC activity – in the hope that VCs will stop funding that sector and hence competition won’t grow. After the entry into cab aggregation, I won’t be surprised if Reliance Retail were to move into online retail and do a good job of it. The likes of Flipkart beware.

## Pipes, Platforms, the Internet and Zero Rating

My friend Sangeet Paul Chaudary, who runs Platform Thinking Labs, likes to describe the world in terms of “pipes” and “platforms”. One of the themes of his work is that we are moving away from a situation of “dumb pipes”, which simply connect things without intelligence, to that of “smart platforms”. Read the entire Wired piece (liked above) to appreciate it fully.

So I was reading this excellent paper on Two-Sided Markets by Jean-Charles Rochet and Jean Tirole (both associated with the Toulouse School of Economics) earlier today, and I found their definition of two-sided markets (the same as platform business) striking. This is something I’d struggled with in the past (I admit to saying things like “every market is two-sided. There’s a buyer and a seller”), especially given the buzzword status accorded to the phrase, but it is unlikely I’ll struggle again. The paper says:

A necessary condition for a market to be two-sided is that the Coase theorem does not apply to the relation between the two sides of the markets: The gain from trade between the two parties generated by the interaction depends only on the total charge levied by the platform, and so in a Coase (1960) world the price structure is neutral.

This is an absolutely brilliant way to define two-sided markets. The paper elaborates:

Definition 1: Consider a platform charging per-interaction charges $a^B$ and $a^S$ to the buyer and seller sides. The market for interactions between the two sides is one-sided if the volume V of transactions realized on the platform depends only on the aggregate price level

$a=a^B +a^S$

i.e., is insensitive to reallocations of this total price a between the buyer and the seller. If by contrast V varies with $a^B$ while a is kept constant, the market is said to be two-sided.

So for a market to be two-sided, i.e. for it to be intermediated by an “intelligent platform” rather than a “dumb pipe”, the volume of transactions should depend not only on the sum of prices paid by the buyer and seller, but on each price independently.

The “traditional” neutral internet, by this definition, is a platform. The amount of content I consume on Youtube, for example, is a function of my internet plan – the agreement between my internet service provider and me on how much I get charged as a function of what I consume. It doesn’t depend on the total cost of transmitting that content from Youtube to me. In other words, I don’t care what Youtube pays its internet service provider for the content it streams. Transaction costs (large number of small transactions) also mean that it is not practically possible for Youtube to subsidise my use of their service in this model.

Note that if buyers and sellers on a platform can make deals “on the side”, it ceases to be a platform, for now only the total price charged to the two matters (side deals can take care of any “adjustments”). The reason this can’t take place for a Youtube like scenario is that you have a large number of small transactions, accounting for which imposes massive transaction costs.

The example that Rochet and Tirole take while explaining this concept in their paper is very interesting (note that the paper was written in 2004):

…As the variable charge for outgoing traffic increases, websites would like to pass this cost increase through to the users who request content downloads…

..an increase in their cost of Internet traffic could induce websites that post content for the convenience of other users or that are cash-strapped, to not produce or else reduce the amount of content posted on the web, as they are unable to pass the cost increase onto the other side.

Note how nicely this argument mirrors what Indian telecom companies are saying on the Zero Rating issue. That a general increase in cost of internet access for consumers can result in small “poor” consumers to not consume on the internet at all, as they are unable to pass on the cost to the other side!

Fascinating stuff!


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A uniform disc of mass M and radius R is rotating freely about avertical axis with initial angular velocityω0. Then sand is poured onto the disk in athin stream so that it piles up on the disk at the radius r<R. The sand is added at the constant rate μ (mass per unitof time).
a. what is the rate of change of the moment of inertia? Whatis I(t)?
b. At what rate are the angular velocity and the rotationalkinetic energy varying with time at a given instant
c. After what length of time is the rotational kinetic energyreduced to half of its initial value


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## anonymous 5 years ago find the domain and range of the given function y(t)=3t^2-2t+1

1. anonymous

Unless a domain is given explicitly with a function, we assume the domain is the set of all t (here) for which the function is defined (i.e. for which the function 'works'). In this case, you can take any value for t, plug it in, and the function will work. Your domain is therefore,$D=\left\{ t |t \in \mathbb{R} \right\}$ (read, "The set of all t, such that t is an element of the real numbers). The range is the set of all y-values that your function will take. If you plot this function, you'll see that it's a parabola. This parabola will have a minimum value at some point, and then extend upwards to infinity from there. Your range, therefore, won't be all the real numbers, unlike your domain. To find the minimum, you can either complete the square and find the t-value that makes the function minimal, or use calculus to find that the point at which f(t) is minimal occurs at $t=\frac{1}{3}$and so$f(\frac{1}{2})=3(\frac{1}{2})^2-2(\frac{1}{2})+1=\frac{3}{4}$The function minimum is therefore$\frac{3}{4}$The function can take values above this value, and at this value, but not below. So your function's range is$R=\left\{ f(t)|f(t) \in \mathbb{R}, f(t) \gt \frac{3}{4} \right\}$which is read, "The range of f is the set of all values f(t) such that f(t) is real and f(t) is greater than 3/4." If you need clarification on anything, please let me know :)

2. anonymous

thanx but i didnt quite get it though


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# zbMATH — the first resource for mathematics

Extremal moduli varieties of semistable sheaves over $$\mathbb{P}_ 2(\mathbb{C})$$. (Variétés de modules extrémales de faisceaux semi-stables sur $$\mathbb{P}_ 2(\mathbb{C})$$.) (French) Zbl 0755.14005
The main theorem gives a description of some extremal moduli spaces $$M(r,c_ 1,c_ 2)$$ of semistable sheaves over the complex projective plane of rank $$r$$ and with Chern classes $$c_ 1,c_ 2$$. Such a moduli space is called extremal, if its dimension is positive but the dimension of $$M(r,c_ 1,c_ 2-1)$$ is not greater than zero. Since the author needs some vanishing of cohomology groups, he must assume an additional cohomological property of $$M(r,c_ 1,c_ 2)$$, so that the main result is only established for certain extremal moduli spaces.
The description is given as a good quotient of an open set in a vector space by a certain nonreductive algebraic group. The author’s construction is based on the facts that every exceptional bundle over the complex projective plane is determined by its slope and that one can associate to such a bundle a certain triple of exceptional bundles with nice properties [the author and J. LePotier, Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 193-243 (1985; Zbl 0586.14007)]. The author uses his generalized Beilinson spectral sequence for such triples [cf. Math. Ann. 275, 25-48 (1986; Zbl 0578.14013)] to describe the semi-stable sheaves under consideration as cokernels of well-structured morphisms.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D20 Algebraic moduli problems, moduli of vector bundles 14L24 Geometric invariant theory
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##### References:
 [1] Drezet, J.-M., Le Potier, J.: Fibr?s stables et fibr?s exceptionnels sur ?2. Ann. Sc. Norm. Sup?r18, 193-244 (1985) · Zbl 0586.14007 [2] Drezet, J.-M.: Fibr?s exceptionnels et suite spectrale de Beilinson g?n?ralis?e sur ?2(?). Math. Ann.275, 25-48 (1986) · Zbl 0578.14013 · doi:10.1007/BF01458581 [3] Drezet, J.-M.: Fibr?s exceptionnels et vari?t?s de modules de faisceaux semi-stables sur ?2(?). J. Reine Angew. Math.380, 14-58 (1987) · Zbl 0613.14013 · doi:10.1515/crll.1987.380.14 [4] Drezet, J.-M.: Groupe de Picard des vari?t?s de modules de faisceaux semi-stables sur ?2(?). Ann. Inst. Fourier38, 105-168 (1988) · Zbl 0616.14006 [5] Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math.106, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157 [6] Grothendieck, A.: Sur la classification des fibr?s holomorphes sur la sph?re de Riemann. Am. J. Math.79, 121-138 (1957) · Zbl 0079.17001 · doi:10.2307/2372388 [7] Le Potier, J.: Stabilit? et amplitude sur ?2. Vector bundles and differential equations. (Progr. Math, 7, pp. 145-182. Boston: Birkh?user 1980) [8] Le Potier, J.: Fibr?s stables de rang 2 sur ?2(?). Math. Ann.241, 217-256 (1979) · Zbl 0405.14008 · doi:10.1007/BF01421207 [9] Maruyama, M.: Moduli of stable sheaves. II. J. Math. Kyoto Univ.18, 557-614 (1978) · Zbl 0395.14006 [10] Mumford, D., Fogarty, J.: Geometric invariant theory. (Ergeb. Math. Grenzgeb. Bd.34) Berlin Heidelberg New York: Springer 1982 · Zbl 0504.14008 [11] Newstead, P.E.: Introduction to moduli problems and orbit spaces. TIFR (Lect. Notes Math. vol. 51) Berlin Heidelberg New York: Springer 1978 · Zbl 0411.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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HSC Science (Computer Science) 12th Board ExamMaharashtra State Board
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# Solution for Sucrose Decomposes in Acid Solution into Glucose and Fructose According to the First Order Rate Law with T1/2= 3 Hours. What Fraction of the Sample of Sucrose Remains After 8 Hours? - HSC Science (Computer Science) 12th Board Exam - Chemistry

ConceptIntegrated Rate Equations Half-life of a Reaction

#### Question

Sucrose decomposes in acid solution into glucose and fructose according to the first order rate law with t1/2= 3 hours. What fraction of the sample of sucrose remains after 8 hours?

#### Solution

t_(1/2)=3 Hours

Now we know that,

k=0.693/t_(1/2)=0.693/3=0.231 h^(-1)

Put above value in the formula of first order reaction,

k=2.303/t log([R]_0/([R]))=2.303/8log([R]_0/([R]))

So,

log([R]_0/([R]))=(0.231xx8)/2.303=0.8024

Taking antilog on both sides,

[R]_0/([R])=6.3445

[R]/[R]_0=0.158

Fraction of the sample of sucrose remaining after 8 hours = 0.158

Is there an error in this question or solution?

#### APPEARS IN

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Solution Sucrose Decomposes in Acid Solution into Glucose and Fructose According to the First Order Rate Law with T1/2= 3 Hours. What Fraction of the Sample of Sucrose Remains After 8 Hours? Concept: Integrated Rate Equations - Half-life of a Reaction.
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# Energy consumption

Energy consumption is the amount of energy or power used.[1]

## Biology

In the body, energy consumption is part of energy homeostasis. It derived from food energy. Energy consumption in the body is a product of the basal metabolic rate and the physical activity level. The physical activity level are defined for a non-pregnant, non-lactating adult as that person's total energy expenditure (TEE) in a 24-hour period, divided by his or her basal metabolic rate (BMR):[2]

${\displaystyle {\text}={\frac {\text}{\text}}}$

## Demographics

Topics related to energy consumption in a demographic sense are:


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# Schrödinger local and deterministic?

Gerenuk
There have been many QM Interpretation thread, but I haven't found this question answered:

Taking aside the fact that a complex probability amplitude is not something we can picture, is the Schrödinger equation local and deterministic at once?

xepma
Yes, the Schrodinger equation and the evolution of the wavefunction that follows from it is local and unitary. Unitary means that the time evolution of the wavefunction is unique and completely determined by the initial conditions. It is therefore deterministic.

There is, however, a major practical obstruction that prevents us from actually calculating this time evolution for any macroscopic system. This is partically because it is practically impossible to determine the initial state of a macroscopic system. But even if we did know this state or if we are somehow able to finetune it, the time-evolution itself is a many-body problem which is, again, computationably intractable.

We therefore always need to resort to some form of approximation, e.g. a statistical description of the system or ignoring a large number of degrees of freedom. Such a statistical description automatically introduces a degree of 'uncertainty' which manifests itself as a non-determinstic description of the system.

So even if you put the whole measurement problem aside, you still end up with a non-deterministic description of macroscopic systems due to practical limitations.

Gerenuk
So how does that compare to the saying "QM can't be local and deterministic" by Bell's theorem and similar ones?

Mentor
Bell's theorem plus the results of experiments testing it (insofar as one accepts those results and/or the validity of Bell's theorem with respect to those experiments, which is the source of the vigorous arguments here about the subject ) support the statement that "QM can't be local and realistic", which is not the same thing as deterministic.

Gerenuk
Yes sure. And I do not wish to start yet another Bell discussion.

But what's wrong about saying the Schrödinger equation is local and deterministic?
Mathematically it does look so.

Frame Dragger
Yes sure. And I do not wish to start yet another Bell discussion.

But what's wrong about saying the Schrödinger equation is local and deterministic?
Mathematically it does look so.

I believe you said it, "...the fact that a complex probability amplitude is not something we can picture..." is the reason. Well, if it can only exist as math, it's not physics, just math. So, from the perspective of a mathematician... it is as you say. From the perspective of a Physicist... it is too, but it's not useful if it can't be made to do work. Hence all of the rest... so I'd say to answer your question: To avoid confusion.

Gerenuk
I guessed so. Now I'm trying to get some ideas to understand how determinism gets lost.... :)

Frame Dragger
I guessed so. Now I'm trying to get some ideas to understand how determinism gets lost.... :)

See, that's not too hard, because Determinism is lost when we have to calculate positions, velocities, etc... as probabilites. It all comes from the Heisenberg Uncertainty Principle (HUP), now backed up by the CMB surveys.

ManyNames
See, that's not too hard, because Determinism is lost when we have to calculate positions, velocities, etc... as probabilites. It all comes from the Heisenberg Uncertainty Principle (HUP), now backed up by the CMB surveys.

I don't agree that the UP supports an external indeterminism in events; The UP does highlight however our lack of knowledge on a system. Just because there is a lack of knowledge from our behalf should not suggest that the universe is not deterministic.

Frame Dragger
I don't agree that the UP supports an external indeterminism in events; The UP does highlight however our lack of knowledge on a system. Just because there is a lack of knowledge from our behalf should not suggest that the universe is not deterministic.

The CMB would beg to differ, barring a superdeterministic uneven distribution of "stuff" at 360K years post-BB...

Gerenuk
Calculating probabilities is fine. The problem comes in when someone tries to make a theory that works with probabilities alone.

So if ppl wouldn't try to squeeze QM into basic probability theories, then QM would be local, deterministic and even linear?

Maybe some sophisticated ingredient can make even the probabilities logical again.

Frame Dragger
Calculating probabilities is fine. The problem comes in when someone tries to make a theory that works with probabilities alone.

So if ppl wouldn't try to squeeze QM into basic probability theories, then QM would be local, deterministic and even linear?

Maybe some sophisticated ingredient can make even the probabilities logical again.

Time to start building the AI's that can find that... maybe they'll even be nice enough to try and explain it to us!

ManyNames
The CMB would beg to differ, barring a superdeterministic uneven distribution of "stuff" at 360K years post-BB...

No i beg to differ, because the UP is in light of what we can know - its a limitation of knowledge which does not impede determinism.

Frame Dragger
No i beg to differ, because the UP is in light of what we can know - its a limitation of knowledge which does not impede determinism.

Ok... then how is it that something which is a limitation on KNOWLEDGE managed to effect the (should-have-been-EVEN) distribution of "stuff" in the early universe? The HUP explains that nicely, as does SUPERdeterminism. The HUP + Determinism = Horse****.

PhilDSP
Wasn't the fundamental problem that the equation didn't properly model the interaction between particle and wave as de Broglie envisioned? (It rather just models "some wave" of unknown origin and constitution)

Frame Dragger
Wasn't the fundamental problem that the equation didn't properly model the interaction between particle and wave as de Broglie envisioned? (It rather just models "some wave" of unknown origin and constitution)

Yeah... sadly yes... and the Bohmian interpretation replaces that issue with a Pilot wave of "unknown origin and constitution" as you put it so well. Welcome to QM... I need some aspirin.

EDIT: Hence us left with 50-50 chances, or worse, 50-50-1! Never good when you get 101% in a physical theory...

ManyNames
Ok... then how is it that something which is a limitation on KNOWLEDGE managed to effect the (should-have-been-EVEN) distribution of "stuff" in the early universe? The HUP explains that nicely, as does SUPERdeterminism. The HUP + Determinism = Horse****.

You do realize that particles are simply statistical averages right? Physics in general is a statistical theory at best yes? It's statistical because we don't have all the knowledge on a quanum system, but this is because of our lack of knowledge, not because there needs to be an indeterministic world externally of our limited knowledges.

Gold Member
You do realize that particles are simply statistical averages right? Physics in general is a statistical theory at best yes? It's statistical because we don't have all the knowledge on a quanum system, but this is because of our lack of knowledge, not because there needs to be an indeterministic world externally of our limited knowledges.

You keep saying this, but this is generally rejected as a viewpoint. The HUP is not about lack of knowledge, although at one time that was a common belief. It is generally held that particles have attributes only within the context of a measurement.

Frame Dragger
You do realize that particles are simply statistical averages right? Physics in general is a statistical theory at best yes? It's statistical because we don't have all the knowledge on a quanum system, but this is because of our lack of knowledge, not because there needs to be an indeterministic world externally of our limited knowledges.

To paraphrase DrChinese in my own words, representing my own opinion, "No, I don't realize that, because observational data has shown the HUP is a physical law, not merely a statistal event horizon for observers."

eaglelake
There have been many QM Interpretation thread, but I haven't found this question answered:

Taking aside the fact that a complex probability amplitude is not something we can picture, is the Schrödinger equation local and deterministic at once?

Classical determinism: Repeating the same experiment many times always has the same result. Classical mechanics allows us to determine that result.
Quantum determinism: Repeating the same experiment many times yields a unique probability distribution of all possible results. Quantum mechanics allows us to determine that probability distribution.
Quantum mechanics does not predict the experimental result; it is not deterministic in the classical sense.

Locality is a property of the space-time of classical physics. It is classical in nature. The wavefunction (probability amplitude) is defined in a Hilbert space. It seems to me that locality is an issue only if the wavefunction propagates in space-time, as many believe.

In the classical sense, quantum mechanics is neither deterministic nor local.

Pio2001
I guessed so. Now I'm trying to get some ideas to understand how determinism gets lost.... :)

Hello Gerenuk,
The answer to your question is simple and i am surprised that nobody has given it yet.

The quantum theory relies on two processes. One deterministic process, called U, like Unitary, governed by Schrödinger's equation, and a probabilistic process, called R, like Reduction, governed by Born's rule.

Both are needed for the theory to actually work. The loss of determinism occurs inside the R process, which has nothing to do with Schrodinger's equation.

Gerenuk
Hello Gerenuk,
The answer to your question is simple and i am surprised that nobody has given it yet.
I'd be very, very careful with such a statement ;-)
Usually the guys crying out "it's so easy!", don't have the slightest clue what the problem is about.
This observation doesn't apply here, but it is one thing to remember :)

Both are needed for the theory to actually work. The loss of determinism occurs inside the R process, which has nothing to do with Schrodinger's equation.
To my knowledge the R process is ill-defined, so it's hard to use it for arguments. I mean when is an observation an observation? Why don't we consider the human being as quantum objects and thus have U processes only?
And how does this R process lose locality or determinism?
For me it's very important not to just know a keyword, but to really understand where mathematically either locality or determinism is lost. Or why at all some people say it is lost, whereas all the theory seems to be based on local and deterministic concepts?

Hello Gerenuk,
The answer to your question is simple and i am surprised that nobody has given it yet.

The quantum theory relies on two processes. One deterministic process, called U, like Unitary, governed by Schrödinger's equation, and a probabilistic process, called R, like Reduction, governed by Born's rule.

Well, that is no longer agreed upon I think, since decoherence is now a well-established experimental and theoretical phenomenon that shows it is possible to have very rapid processes that proceed in a unitary fashion according to the TDSE, yet produce observations that are consistent with the original "collapse" (or reduction) theories. In fact, you will see the phrase "there is no collapse" thrown around a lot on this forum.

Both are needed for the theory to actually work. The loss of determinism occurs inside the R process, which has nothing to do with Schrodinger's equation.

I would say that it is very much an open question whether or not there is in fact a loss or determinism as you claim.

Pio2001
To my knowledge the R process is ill-defined, so it's hard to use it for arguments. I mean when is an observation an observation? Why don't we consider the human being as quantum objects and thus have U processes only?

Because the R process makes experimental predictions that the U process doesn't. Example, that YOU will get this or that result when you measure a given system in a given way. If you keep the U process only and use it to built a many world interpretation, you loose the definition of "you", and the above experimental prediction is no more defined.

And how does this R process lose locality or determinism?

The R process lacks determinism in its axiomatic definition, and says nothing about locality.

Later, Bell, CHSH, GHZ, and Mermin (excuse me if I forget some), have shown that locality and determinism could not coexist. In a larger context, we can say that locality, determinism and realism can't coexist in quantum mechanics.

Some people however have suggested workarounds. Mark Rubin, for example, in his article about local realism in the Heisenberg picture of operators in the MWI, or JesseM in this forum, with his idea about pasting parallel universe when their future light-cones meet (which is more or less the same idea, as far as I understand). These ideas deserve to be developed. I'm working on JesseM's idea in my spare time.

For me it's very important not to just know a keyword, but to really understand where mathematically either locality or determinism is lost. Or why at all some people say it is lost, whereas all the theory seems to be based on local and deterministic concepts?

They are lost when you violate Bell's inequality in an EPR-like experiment. No modelization of the experiment have been given yet that
1) Describe what happens in terms of realistic objects
2) Predicts the violation of the inequality by means of the above description

Well, that is no longer agreed upon I think, since decoherence is now a well-established experimental and theoretical phenomenon that shows it is possible to have very rapid processes that proceed in a unitary fashion according to the TDSE, yet produce observations that are consistent with the original "collapse" (or reduction) theories.

Consistent yes, but with not as much predictive power. They do not predict the violation of the inequality without completing decoherence with the last part of the R process, which consists in picking one of the possible results out of many, in an undeterministic way.

I would say that it is very much an open question whether or not there is in fact a loss or determinism as you claim.

I don't disagree, but Gerenuk's question was simple, and I gave the simple answer, from which we can go on and start further discussions

Gerenuk
Because the R process makes experimental predictions that the U process doesn't. Example, that YOU will get this or that result when you measure a given system in a given way.
I don't think this R process idea is a satisfactory explanation. And the many attempts for interpretations probably share the same view. It's not well defined when someone is measuring and when he isn't and what reality means.

Later, Bell, CHSH, GHZ, and Mermin (excuse me if I forget some), have shown that locality and determinism could not coexist. In a larger context, we can say that locality, determinism and realism can't coexist in quantum mechanics.
I do not want to discuss their work. I'll go through it later, but I know they all make their own hidden assumptions. Anyway:

If I let the universe run for a long time governed by the Schrödinger equation, and if I make one measurement in the end, then everything was a local and deterministic U process to the very end and I can extract probabilities from this l&d process? Right?

And then someone else comes along and says, I'm only a stupid quantum process and he is the real observer and waiting for an even longer time than me before he does the measurement. So in his theory everything was l&d an even longer time?!

It seem everything is l&d at all times. (unless you insist on removing the wavefunction and introduce real probabilities)

Pio2001
I don't think this R process idea is a satisfactory explanation.

It depends on the explanation you're looking for.

If you want an explanation about the presence of probabilities in the theory in spite of Shrödinger's equation, then the R process is a good answer.

If you want an explanation of reality, then the R process shows severe limitations.

If I let the universe run for a long time governed by the Schrödinger equation, and if I make one measurement in the end, then everything was a local and deterministic U process to the very end and I can extract probabilities from this l&d process? Right?

I'm not sure... If we try to modelize Alice and Bob's EPR experiment this way, we'll run into their decoherence into two different preferred basis. If we simplify their states enough, we don't have the information necessary to get the final probabilities giving the inequality violation. We must not simplify, but keep the initial entanglement scattered into the billions of particles of their environment.
I don't know exactly how it works, nor even if the description of the system in terms of density matrixes holds enough information in order to predict Alice and Bob's correlations.

And if you describe them as wave vector, it doesn't work, because you'd have to consider them as, for example, half-spin particle in order to derive the right probabilities, which they are not : Alice with her detector turned rightside, summed with Alice with her detector turned leftside does not equal Alice with her detector turned upside, because these three state vectors are orthogonal (they're three eigenstates of the position).
However, making this summation is mandatory in order to get the inequality violated !

Don't think in terms of Schrödinger's cats. You can describe them in local and deterministic ways.
Think about Alice and Bob measuring half-spin particles across alpha and beta directions and getting a correlation of -cos(beta-alpha). That's much trickier.

Pio2001
...or maybe Alice's summation holds true thanks to constructive interferences scattered through the environment's microscopic state ?

Pio2001
nor even if the description of the system in terms of density matrixes holds enough information in order to predict Alice and Bob's correlations.

I'm not used to the density matrix formalism, but after a second thought, I think that it holds the necessary information. However, it is non-local.

I have already tried something like you suggest : keep the U evolution until the end and get the probabilities when Alice and Bob meet.
The problem is that in order to have the right probabilities, you need Alice and Bob to be themselves elementary particles with half spin properties, with the problem mentionned above.
Another solution is to have them decohered before they meet, but you then need to associate amplitudes to their decohered copies in order to predict the inequality violation, and in this case, the amplitude of Bob's copies must depend on the angle chosen by Alice and vice versa ! Which is non-local. Otherwise, the inequality is not violated.

I eventually found a solution, relying on the many-world formalism. It can be summarized in three additional postulates.
1)When a system splits into many worlds, its copies keep track of the wave function that caused it to split.
2)When a system splits into many worlds, it also keeps track of a unique label that makes it distinguishable from any other similar system.

-> One split = one wave function = one label = many worlds (each one described by a wave function).

3)When some local parts of a split system meet, its copies recognize each other by means of the label given in postulate 2. Then the probabilities for a local copy to meet with one or the other copy from another place is given by the usual born rule applied to the initial wave function that the copies carry with them according to postulate 1.

This way I think it works. I have still to write the whole calculus.

Gerenuk
First, I'd like to see mathematically how locality is lost. It's the R process? Is it possible to write it down, so that it looks non-local from the mathematical view?

I'm not sure I've understood your explanation fully. Why can't I say B knows about A's complete state and her measurement apparatus completely, and thus is able to model all of A's measurement from an external view with U processes only?

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Pio2001
Hello Gerenuk,
I can do that. I'll try tonight (CET) with entangled photons. They are more interesting than spin 1/2 particles, because they are destroyed during the measurment.

Gerenuk
That'd be great. I just want to mention that I've seen Bell's-inequality-like proofs, but they use a different model/language, so that I still don't know at which point locality is lost. Ideally I'd like to see something which states why the mathematical form of the R process is non-local. Maybe $f(x)=\sum_{x'}g(x,x')$? Whatever the correct definition of non-local is :)

Gold Member
There have been many QM Interpretation thread, but I haven't found this question answered:

Taking aside the fact that a complex probability amplitude is not something we can picture, is the Schrödinger equation local and deterministic at once?
Schrodinger equation is certainly deterministic, there is no doubt on it. However, the question whether it is also local is more subtle. Since wave function lives in the configuration space and not in the ordinary 3+1 space, what does it even MEAN to be "local"?

However, the locality of the Hamiltonian (which is an important ingredient of the Schrodinger equation) is a well defined concept. So we can say that Schrodinger equation is "local" in the sense that realistic quantum systems are based on a local Hamiltonian. On the other hand, we can also say that the Schrodinger equation is "nonlocal" in the sense that the quantity it describes - the wave function - is not a local object. An even more precise terminology is not that the wave function is "nonlocal", but that it is NONSEPARABLE.

Gerenuk
On the other hand, we can also say that the Schrodinger equation is "nonlocal" in the sense that the quantity it describes - the wave function - is not a local object. An even more precise terminology is not that the wave function is "nonlocal", but that it is NONSEPARABLE.
Can you write this down mathematically? I believe in physics you should never trust words alone unless both parties know exactly which mathematics stands behind it. Because once I learned the maths for some advanced topics I started noticing how much rubbish me and other people were talking before, when we knew only words, i.e. popular science.

Edit: preferably using non-locality and not non-separability. Unless you can find a very good proof to draw the connection.

Naty1
Post # 7 by the OP
"Now I'm trying to get some ideas to understand how determinism gets lost."

Here is how Roger Penrose describes the issue, and it adds a bit of detail to Pio's post #21 above:
(from his talk commemorating Stephen Hawkings 60th birthday)

Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described.... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrodinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story..... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level.....quantum state reduction is non deterministic, time-asymmetric and non local....The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers....an essential ingredient of the Schrodinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"....in fact the key to keeping them sitting there is quantum linearity...

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yoda jedi
Consistent yes, but with not as much predictive power. They do not predict the violation of the inequality without completing decoherence with the last part of the R process, which consists in picking one of the possible results out of many, in an undeterministic way.

right, an adscititious "collapse" unlike of intrisic.
because superpositions persist, MWI is deterministic.

Because the R process makes experimental predictions that the U process doesn't. Example, that YOU will get this or that result when you measure a given system in a given way1 If you keep the U process only and use it to built a many world interpretation2, you loose the definition of "you", and the above experimental prediction is no more defined.

1.-Forced.
2.-or you can reduce R.


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## Topic outline

• ### PHP FILES & IO

• Lesson

We can store all data on web server by two methods

1) By using MYSQL

2) By using files, it can be quicker and more convenient to directly access files on the hard disk.

Here in this tutorial, we are going to see briefly about second method.

OPENING A FILE

A better method to open files is with the fopen() function. This function gives you more options than the readfile() function.

The following code illustrates opening a file

<!DOCTYPE html>

<html>

<body>

<?php

$myfile = fopen("abc.txt", "r") or die("Unable to open file!"); echo fread($myfile, filesize("abc.txt"));

fclose($myfile); ?> </body> </html> The fread() function reads from an open file. The first parameter of fread() contains the name of the file to read from and the second parameter specifies the maximum number of bytes to read. The following PHP code reads the "webdictionary.txt" file to the end: fread($myfile, filesize(“abc.txt"));

WRITE TO FILE

The fwrite() function is used to write to a file.

The first parameter of fwrite() contains the name of the file to write to and the second parameter is the string to be written..

The following code illustrates to write a file in PHP

<!DOCTYPE html>

<html>

<body>

<?php

$myfile = fopen("newfile.txt", "w","r") or die("Unable to open file!");$txt = "John Doe\n";

fwrite($myfile,$txt);

$txt = "Jane Doe\n"; fwrite($myfile, $txt); echo fread($myfile, filesize("newfile.txt"));

fclose($myfile); ?> </body> </html> CHECKING WHETHER FILE EXISTS To determine whether a file already exists, you can use the file_exists function, which returns either TRUE or FALSE, and is used like this: if (file_exists("testfile.txt")) echo "File exists"; The following code illustrates to create in PHP <?php // testfile.php$fh = fopen("testfile.txt", 'w') or die("Failed to create file");

$text = <<<_END Line 1 Line 2 Line 3 _END; fwrite($fh, $text) or die("Could not write to file"); fclose($fh);

echo "File 'testfile.txt' written successfully";

?>

When we run this in a browser, all being well, you will receive the message File 'testfile.txt' written successfully. If you receive an error message, your hard disk may be full or, more likely, you may not have permission to create or write to the file, in which case you should modify the attributes of the destination folder according to your operating system.

This simple example shows the sequence that all file handling takes:

1. Always start by opening the file. You do this through a call to fopen.

2. Then you can call other functions; here we write to the file (fwrite), but you can also read from an existing file (fread or fgets) and do other things.

3. Finish by closing the file (fclose).

There are several different parameter that can be used for reading, writing and appending a file

 MODE ACTION DESCRIPTION r Read file from start Open for reading only, place the file pointer at the beginning of the file r+ Read file from start and allow writing Open for reading and writing, place the file pointer at the beginning of the file w Write file from start and truncate file Open for writing only w+ Write file from start, truncate file and allow reading Open for reading and writing a Append to file end Open for writing only a+ Append to file end and allow reading Open for reading and writing

APPEND A FILE IN PHP:

The following code illustrates how to append a file in PHP:

<!DOCTYPE html>

<html>

<body>

<?php

$myFile = "testFile.txt";$fh = fopen($myFile, 'a') or die("can't open file");$stringData = "New Stuff 1\n";

fwrite($fh,$stringData);

$stringData = "New Stuff 2\n"; fwrite($fh, $stringData); fclose($fh);

?>

</body>

</html>

COPYING A FILE IN PHP

PHP copy function used to create clone of test file in PHP.

The following is the sample code to copy a file in PHP

<?php

// copyfile.php

copy('testfile.txt', 'testfile2.txt') or die("Could not copy file");

echo "File successfully copied to 'testfile2.txt'";

?>

DELETING A FILE

By using unlink() function ,we will remove  a file from our system.

The following sample code help to delete a file

<?php

// deletefile.php

echo "Could not delete file";

else

echo "File 'testfile2.new' successfully deleted";

?>

file_get_content() function is used for reading the entire file without using a file handler.

The following code illustrates how to read a file by using function file_get-content()

<?php

echo "<pre>";  // Enables display of line feeds

echo file_get_contents("testfile.txt");

echo "</pre>"; // Terminates pre tag

?>


text:
0

# When you multiply a radical number with another radical number do the radical signs cancel each other out?

Wiki User

2011-03-08 18:47:31

Not necessarily. If it is the same radical number, then the signs cancel out. Radical 5 times radical 5 equals 5. But if they are different, then you multiply the numbers and leave them under the radical sign.

Wiki User

2011-03-08 18:47:31
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text:
Statistical significance from an iterated count

Question in brief: I am attempting to determine whether statistical significance can be measured merely from counts generated from iterating the same procedure, but where the probability of success is unknown both longitudinally and cross-sectionally. If so, is there an analytic solution? If not, can you provide guidance on a numerical solution?

Question in full: The following experiment has been conducted. There are 33 children on a sports team, and once a day a coach is tasked with placing each child into one of two groups: group one, which is the group that will actually play the match, and group two, which consists of those children who will be sent home. The coach's objective is to win the match, and he can do that by selecting as many (or as few) children as he desires (i.e., $$[0,33]$$, where a selection of 0 would indicate a forfeit, though empirically this never occurred; also, the game doesn't require a minimum number of participants). This task is performed each day of the school year (175 iterations). We can assume (for reasons unnecessary to expound upon here) that the coach does not learn; i.e., each time he performs this task it is independent of past attempts. Further, the same 33 children are always decided upon and the children's perceived value remains the same each day (i.e., the children do not improve, etc.).

Thus there are two things we have measured: (1) how many children the coach selects in each iteration (i.e., the sum of each "Iteration n" column below), and (2) the number of times each child is selected (i.e., the "Sum of count" column below). The table below shows a sample of the collected data:

Child ID | Iteration 1 | Iteration 2 | ... | Sum of count
----------------------------------------------------------
01       | 0           | 0           | ... | 0
02       | 0           | 1           | ... | 28
03       | 1           | 1           | ... | 175
04       | 0           | 0           | ... | 0
.        | .           | .           | ... | .
.        | .           | .           | ... | .
.        | .           | .           | ... | .
33       | 1           | 1           | ... | 172

where if the child was selected in that iteration he/she received a 1, otherwise a 0. I want to know whether, for example, child 02 was selected a statistically-significant number of times (i.e., is 28 significant?).

Thoughts: Because the selection of a child into group one is a binary event ("yes" or "no"), I thought I could compare each child's count against the binomial CDF. But this doesn't work because I don't know ex-ante the probability of a child being selected; empirically we see that somewhere around only 3 to 6 children are generally selected (meaning around 27 to 30 are placed into group two in each iteration). I think this is effectively because each child is not independent of the others: some are "fast runners," some are "the big guy," etc., and thus the coach is selecting the best from a smaller group of categories. But this is not required in the setup, and I don't know which category the coach will care about on a particular iteration. I thought a Poisson distribution might be appropriate, but I don't know ex-ante lambda (the average number of children selected in an iteration). There are some similarities to the Mann-Whitney U test, but in the present setup there is no rank: group one or group two is all I have. And so on.


text:
# Total curvature

This curve has total curvature 6π, and index/turning number 3, though it only has winding number 2 about p.

In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:

$\int_a^b k(s)\,ds.$

The total curvature of a closed curve is always an integer multiple of 2π, called the index of the curve, or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.

## Comparison to surfaces

This relationship between a local geometric invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem.

## Invariance

According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.

By contrast, winding number about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.

## Generalizations

A closed polygonal chain, with total curvature 2π.

A finite generalization is that the exterior angles of a triangle, or more generally any simple polygon, add up to 360° = 2π radians, corresponding to a turning number of 1. More generally, polygonal chains that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.

The total curvature of a curve γ in a higher dimensional Euclidean space (equipped with its arclength parameterization) can be obtained by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in n-dimensional space is

$\int_a^b \left|\gamma''(s)\right|\sgn \kappa_{n-1}(s)\,ds$

where κn−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function.

The minimum total curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2π for the unknot, but by the Fary–Milnor theorem it is at least 4π for any other knot.


text:
Evlan
SSL
Last updated at 2005/06/06 18:45:51 PDT by Temporal

If you intend to use the Evlan command client to connect to a server over the internet, you must set up SSL support for proper security. Without SSL, an attacker with the ability to sniff network packets could use a replay attack to gain access to your Evlan VM.

Evlan does not have SSL support built-in. This is because the only well-tested open source library available for doing this, OpenSSL, has the absolute worst programming interface ever known to man. Instead, Evlan relies on Stunnel, a convenient program which accepts SSL connections, decrypts them, and forwards them unencrypted to other local services. Using Stunnel, SSL support can be added to Evlan totally transparently. Not a single line of code is needed in the Evlan VM to handle it.

First, download and install Stunnel on the machine on which you will be running the Evlan VM. You will also need to generate a private key and a certificate. The details of this are beyond the scope of this document.

Edit your stunnel.conf to look something like this:

cert = my-certificate.crt
key = my-private-key.key
options = SINGLE_DH_USE
session = 300

[evlan]
accept = 51310
connect = 127.0.0.1:51311

Edit the Evlan VM's config file (in modules/evlan.org/internal/Config.evlan) to accept connections on "127.0.0.1:51311". Now run both. Congratulations, you now have SSL support.

There is still another hurdle to clear, unfortunately. evlan-client uses the SSL library that comes with Java. By default, this library will refuse to connect to servers if it cannot verify their key with a trusted root. This means that if you are using a self-signed key, or if your SSL certificate is signed by someone not on Java's very limited list of trusted roots, you will be unable to connect.

The solution is to add your certificate to Java's trusted list. Unfortunately, as with everything else in Java, the methods for doing this are a complete mess, poorly documented, and unintuitive. On Windows, the Java control panel applet has a GUI for managing certificates, but adding your certificate to it will not help. After an epic battle, I managed to get Java to recognize my certificate using obscure command-line utilities. Type the following all on one line:

keytool -import -alias MyCertificate -file my-certificate.crt
-keystore c:\path\to\JRE\lib\security\cacerts
-storetype jks

If this doesn't help, try searching your Java directories for other files called "cacerts", and just add your certificate to every one you find. Hopefully you'll hit the right one eventually.

Note that when Java refuses to make an SSL connection because of this problem, it does not print a happy error message. Instead, it throws an obscure internal exception.

Hopefully some Java guru will read this and help me update the command client's code so that it doesn't require a signed certificate. Until then, this is all I can tell you.

You can use Stunnel to add SSL support to your Evlan-based web server, too. Just add something like this to stunnel.conf:

[https]
accept = 443
connect = 127.0.0.1:80
TIMEOUTclose = 0

(The "TIMEOUTclose" option is to work around a bug present in IE and Firefox, which don't properly respond to requests to close the SSL stream.)

Now, when you get to the step in the web server setup guide where you create the authenticator, use "https://" rather than "http://" in the URL. Users will now use SSL to log in, then drop to unencrypted mode for normal operation. (You could also encourage users to always use SSL if you prefer.)

Unfortunately, there is a down side of using such a tunnel. Users will now be able to mask their IP address by connecting to your site via SSL, since the web server will see all the connections as coming from the local Stunnel daemon. evlan.org implements a hack to fix this, but the hack is too much of a hack to be worth releasing publicly. Consider investigating the module evlan.org/io/TunnelListener if you like.


text:
# Graph r=1+cos(x)

r=1+cos(x)
Rewrite the expression as cos(x)+1.
cos(x)+1
Use the form acos(bx-c)+d to find the variables used to find the amplitude, period, phase shift, and vertical shift.
a=1
b=1
c=0
d=1
Find the amplitude |a|.
Amplitude: 1
Find the period using the formula 2π|b|.
The period of the function can be calculated using 2π|b|.
Period: 2π|b|
Replace b with 1 in the formula for period.
Period: 2π|1|
Solve the equation.
The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.
Period: 2π1
Divide 2π by 1.
Period: 2π
Period: 2π
Period: 2π
Find the phase shift using the formula cb.
The phase shift of the function can be calculated from cb.
Phase Shift: cb
Replace the values of c and b in the equation for phase shift.
Phase Shift: 01
Divide 0 by 1.
Phase Shift: 0
Phase Shift: 0
Find the vertical shift d.
Vertical Shift: 1
List the properties of the trigonometric function.
Amplitude: 1
Period: 2π
Phase Shift: 0 (0 to the right)
Vertical Shift: 1
Select a few points to graph.
Find the point at x=0.
Replace the variable x with 0 in the expression.
f(0)=1+cos(0)
Simplify the result.
The exact value of cos(0) is 1.
f(0)=1+1
f(0)=2
2
2
2
Find the point at x=π2.
Replace the variable x with π2 in the expression.
f(π2)=1+cos(π2)
Simplify the result.
The exact value of cos(π2) is 0.
f(π2)=1+0
f(π2)=1
1
1
1
Find the point at x=π.
Replace the variable x with π in the expression.
f(π)=1+cos(π)
Simplify the result.
Simplify each term.
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
f(π)=1-cos(0)
The exact value of cos(0) is 1.
f(π)=1-1⋅1
Multiply -1 by 1.
f(π)=1-1
f(π)=1-1
Subtract 1 from 1.
f(π)=0
0
0
0
Find the point at x=3π2.
Replace the variable x with 3π2 in the expression.
f(3π2)=1+cos(3π2)
Simplify the result.
Simplify each term.
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
f(3π2)=1+cos(π2)
The exact value of cos(π2) is 0.
f(3π2)=1+0
f(3π2)=1+0
f(3π2)=1
1
1
1
Find the point at x=2π.
Replace the variable x with 2π in the expression.
f(2π)=1+cos(2π)
Simplify the result.
Simplify each term.
2π is a full rotation so replace with 0.
f(2π)=1+cos(0)
The exact value of cos(0) is 1.
f(2π)=1+1
f(2π)=1+1
f(2π)=2
2
2
2
List the points in a table.
xf(x)02π21π03π212π2
xf(x)02π21π03π212π2
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude: 1
Period: 2π
Phase Shift: 0 (0 to the right)
Vertical Shift: 1
xf(x)02π21π03π212π2
Graph r=1+cos(x)


text:
# Computing $\text{Tor}$ for modules over a PID

This is essentially an exercise in Sze-Tsen Hu's "Introduction to Homological Algebra", page 143.

Let $R$ be a PID and consider two $R$-modules $X$ and $Y$. Let $S$ denote the subset of the Cartesian product $$W=X\times Y\times Z$$ consisting of all elements $(x,y,r)\in W$ such that $rx=0$ in $X$ and $ry=0$ in $Y$. Let $F(S)$ denote the free $R$ module generated by the set $S$ and let $M$ denote the smallest submodule of $F(S)$ containing all elements of the forms:$$(x+x',y,r)-(x,y,r)-(x',y,r),$$ $$(x,y+y'r)-(x,y,r)-(x,y',r),$$ $$(x,y,sr)-s.(x,y,r),$$ $$(x,y,sr)-s.(x,y,r),$$ whenever every triple in parenthesis is an element in $S$. Prove$$\text{Tor}(X,Y)\simeq F(S)/M$$ Then deduce that $\text{Tor}(X,Y)=0$ if $X$ or $Y$ is torsion free.

My attempt: Consider the short exact sequence $$0\rightarrow \text{ker}\phi \rightarrow F(X) \rightarrow X\rightarrow 0$$ where $F(X)\rightarrow X$ is given by $\phi$, the linear map from $F(X)$ to $X$ sending $x$ to $x$, and $\text{ker}\phi \rightarrow F(X)$ is given by inclusion, say $\iota$. Note that $\text{ker}\phi$ is also free since we are working with modules over a PID. Taking tensor products with $Y$ gives $$0\rightarrow \text{ker}\phi\otimes Y \rightarrow F(X)\otimes Y \rightarrow X\otimes Y\rightarrow 0$$ and we only need to compute $\text{ker}(\iota \otimes \text{id})$, so assume that $\sum m_j \otimes y_j \in \text{ker}(\iota \otimes \text{id})$ where $j \in J$ for some finite index set $J$ and $m_j \in \text{ker}(\phi), y_j \in Y$ for each $j$.

Since $m_j \in \text{ker}(\phi)$, when written in the form $m_j=\sum r_{ij} .x_i$ (here, $I$ is also some finite index set which does not depend on $j$, and the dot indicates that the scalar multiplication if formal) we must have $0=\sum r_{ij} x_i$, in $X$, for each $i$.

Now the fact that $\sum m_j \otimes y_j \in \text{ker}(\iota \otimes \text{id})$ is equivalnet to $$0=\sum _{j \in J} \left( \sum _{i \in I} r_{ij}.x_i \otimes y_j \right)=\sum _{i \in I} \left( \sum _{j \in J} (r_{ij}.x_i) \otimes y_j \right)=\sum _{i \in I} \left( x_i \otimes \sum _{j \in J} r_{ij}y_j \right)$$ in $F(X) \otimes Y$, but considering the canonical isomorphisms $$F(X) \otimes Y \simeq \left( \bigoplus_{x \in X} R.x \right) \otimes Y \simeq \bigoplus_{x \in X} (R.x \otimes Y) \simeq \bigoplus _{x \in X} Y ,$$ shows that this happens iff $$0=\sum _{j \in J} r_{ij}y_j$$ for each $i \in I$, so considering the case when $\vert I \vert=\vert J \vert=1$ shows that $\text{ker}(\iota \otimes \text{id})$ contains elements in $F(X)$ of the form $(r.x \otimes y)$ for every $(x,y,r) \in S$, and this correspondence from $S$ to $\text{ker}(\iota \otimes \text{id})$ gives a linear map from $F(S)$ to $\text{ker}(\iota \otimes \text{id})$, whose kernel clearly contains $M$, so we actually have a linear map from $F(X)/M$ to $\text{ker}(\iota \otimes \text{id})$, say $\psi$.

Now the only obstacle left is to show that $\psi$ is an isomorphism, and this is the part I'm stuck. I'm sure that this is the right way to solve this problem, but I can't make any more progress showing that $\psi$ is an isomorphism. I tried to show that $\psi$ is both injective and surjective, but failed. I tried to construct the inverse of $\psi$, but failed. The major problem was showing the surjectivity of $\psi$. Once the surjectivity have been established, the inverse of $\psi$ can be constructed... hopefully.

Any advice on proving this point would be a great help :D. Any other approaches on this exercise are also welcome.

PS1. Originally the exercise asks for the case $R=\mathbb{Z}$, but the next exercise asks for a generalization of this to PIDs, so in overall it's the same.

PS2. Can this result be used to prove the fact that for modules over a PID, "torsion-free" imples "flat"? So that two concepts "finitely generated and flat" and "free of finite rank" are equivalent?