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http://www.physicsforums.com/showthread.php?t=623516	Physics Forums ## Logarithmic spiral 1. The problem statement, all variables and given/known data a logarithmic spiral is given. The center lies on the x-axis. after a turn of 180 degrees counterclockwise I am 86.23m away from the starting point, after 360 degrees I'm 75.41m away from the start (radius, not along the spiral). Where am I after I walked exactly 3km along the spiral trajectory? 2. Relevant equations equation: r=ae^b*theta starting point corresponds to theta=0 3. The attempt at a solution Um.. that's my problem. I never really learnt how to solve logarithmic calculations. Btw, this is not a homework, but part of a puzzle. I got the numbers from previous steps and found the equation online. I understand that r=distance from origin, theta = angle with x-axis, a and b some constants. Right. But I don't see any way of working with this this equation or any of the expressions connected to it I could need a little push in the right direction. Thanks a lot, Martine PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug Recognitions: Science Advisor Quote by martine [b] Um.. that's my problem. I never really learnt how to solve logarithmic calculations. Btw, this is not a homework, but part of a puzzle. I got the numbers from previous steps and found the equation online. I understand that r=distance from origin, theta = angle with x-axis, a and b some constants. Right. But I don't see any way of working with this this equation or any of the expressions connected to it I could need a little push in the right direction. Why don't you just write the two equations (with the known values of "r,theta" you have) and divide one equation by the other to eliminate "a" and solve for "b". Hint1: "b" is negative in this case. After you solve for the parameters "a" and "b" you then have to work out how to find the length of a path along the spiral. Hint2: $dl = \left( \sqrt{r^2 + \left( dr/d\theta \right)^2} \right) d\theta$ Recognitions: Gold Member Science Advisor Staff Emeritus You say "180 degrees" and "360 degrees" but radian measure should be use here. when $\theta= \pi$, r= 86.23 so $86.23= ae^{b\pi}$. When $\theta= 2\pi$, r= 75.42 so $75.41= ae^{2b\pi}$. As uart says, dividing one equation by the other will cancel the 'a's leaving $$\frac{75.41}{86.23}= \frac{e^{2b\pi}}{e^{b\pi}}= e^{b\pi}$$ that is easy to solve for b. Then use that value of b with either of the first equations to find a. For the final problem you will need to integrate the "differential of arc length" that uart gave from 0 to $\theta$ and set it equal to 3000 to solve for $\theta$. Thread Tools \| \| \| \| \|-----------------------------------------\|------------------------------\|---------\| \| Similar Threads for: Logarithmic spiral \| \| \| \| Thread \| Forum \| Replies \| \| \| Differential Geometry \| 2 \| \| \| Special & General Relativity \| 2 \| \| \| Calculus & Beyond Homework \| 0 \| \| \| Calculus & Beyond Homework \| 6 \| \| \| Calculus & Beyond Homework \| 3 \|	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9154030084609985, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/287942/series-convergence-divergence-and-tests-and-sums	Series convergence divergence and tests and sums 1. For which values of $p$ is the series $$\sum_{n=1}^\infty\frac1{(n (1+\ln n)^p)}$$ convergent/divergent? I've been trying to solve this earlier but the best I can come up with is the ratio test and I know that's not the proper way to solve for $p$. 2. Where are each of these series convergent/divergent? 1. $$\sum_{n=1}^\infty\frac1{n^a}-\frac1{(n+1)^a}$$ 2. $$\sum_{n=1}^\infty\ln\left(1+\frac1n\right)$$ Again, I've been trying to do the ratio test but it's not working, mostly because of that extra variable that confuses me. - – Rahul Narain Jan 27 at 7:51 1 Answer Hints: $1.$ Some useful facts: If $p \gt 1$, then $\sum_1^\infty \frac{1}{n^p}$ converges. If this has not already been covered in your course, you can prove it using the Integral Test, or the Cauchy Condensation Test. But it is a very standard sort of series, often called a $p$-series. $\sum_2^\infty \frac{1}{n\ln n}$ diverges. This is done for example using the Integral Test. If I read your expression correctly, these facts and the Comparison Test should settle things. By the way, $\sum_2^\infty \frac{1}{n(\ln n)^p}$ converges if $p\gt 1$, but you won't need this. a) Find the sum of the first two terms. Find the sum of the first three. Find the sum of the first four. Whole lot of cancelling! Now you can write down a simple expression for the sum of the first $n$ terms. Under what conditions on $a$ does this converge to something finite? b) Note that $\ln(1+1/n)=\ln((n+1)/n)=\ln(n+1)-\ln(n)$. Then use the same idea as in a). - You can get a list item with multiple paragraphs to be correctly formatted if you just indent each subsequent paragraph with a space. I can edit that into your answer, if you like. – Rahul Narain Jan 27 at 7:55 Thanks for the offer. The list only has two items, so maybe not worthwhile, but certainly fine with me. I am accustomed to TeX lists, and unfortunately this site only uses TeX for the mathematics. – André Nicolas Jan 27 at 7:59 If the p-series says p>1 is always convergent isnt it the same for 2a?? it seems is should work – user1730308 Jan 28 at 2:41 Certainly these work. But more do. If you followed my hint, you found that the sum of the first $n$ terms is $1-\frac{1}{(n+1)^a}$. If $a\gt 0$, this approaches $1$. And then there is the trivial $a=0$, where we are just adding up $0$'s. But if $a\lt 0$ then the term $\frac{1}{(n+1)^a}$ blows up. So the answer is $a\ge 0$ gives convergence, $a\lt 0$ gives divergence. – André Nicolas Jan 28 at 2:55 Im guessing to find the first few sums you did telescoping? but i got 1/n^a - 1/(n+1)^a.... Am i doing it wrong? Also to get the sum what method would you suggest i use? – user1730308 Jan 28 at 3:24 show 3 more comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9303992986679077, "perplexity_flag": "head"}
http://crypto.stackexchange.com/questions/1793/is-rsa-of-a-random-nonce-with-no-padding-safe	# Is RSA of a random nonce with no padding safe? Consider the following protocol: Bob has a private RSA key $B_{priv}$, and Alice knows the public key $B_{pub}$. Alice wants to send confidential messages to Bob (no integrity intended). To send a message, Alice randomly generates a single-use symmetric key (say, an AES key) $K$, encrypts the message with this symmetric key, and sends this encrypted message alongside with $K$ encrypted with RSA with Bob's public key. Suppose we use no padding for $\textrm{RSA}_{B_{pub}}(K)$ (in other words, to pad with zero bytes). The obvious (to me) problems with a lack of padding are that an attacker can guess the plaintext (not possible here, it's random) and that encrypting the same plaintext twice gives the same result (again, not happening here). What are the known weaknesses of this scheme (or is it safe)? - ## 1 Answer If you pad a 128-bit value $K$ with zeroes to the left, and interpret the value numerically with big-endian convention (as happens in PKCS#1 in general), you end up with a value which is no greater than $2^{128}$. If the public exponent is $e = 3$, then the "encrypted value" is no greater than $2^{768}$, and no actual modular reduction takes place. Consequently, a simple cube root extraction on plain integers reveals $K$. So, that can be quite unsafe. What is safe is the following: • you generate a random integer $x$, between $0$ and $n-1$ ($n$ is the RSA modulus); • you encrypt $x$ by simply raising it to the power $e$ modulo $n$; • the encryption key you will use with AES is $h(x)$ for some hash function $h$. It is still relatively limiting, in that this is safe as long as you do one encryption only of $x$. If you want to "send" your symmetric key $K$ to several recipients (a common model, e.g. for email encryption), then the scheme above does not work, and, more generally, you need some per-encryption randomness inserted (as PKCS#1 does for encryption). - Ouch, defeated by simple math. Hmm, what if I encrypted $K + 2^{\mathop{lg}(n)-1}$? And incidentally, with multiple recipients, does the unpredictability of the padding matter, or is it enough never to apply RSA to the same plaintext with different keys? (Don't worry, I'm asking out of curiosity, I'm not planning to implement anything like this.) – Gilles Feb 2 '12 at 0:02 1 @Gilles: if padding is deterministic, and you encrypt the same $K$ (padded to $\pi(K)$) for three different recipients, with $e = 3$, then you send the value $\pi(K)^3$ modulo $n_1$, $n_2$ and $n_3$. By the Chinese Remainder Theorem, this is sufficient to rebuild $\pi(K)$ modulo $n_1n_2n_3$, and since $\pi(K) \lt n_i$, the modulo disappears. This is the historical justification for using $e = 65537$ instead of $e = 3$, but the real issue is that a deterministic padding for RSA encryption is definitely unsafe. You MUST have a random padding. – Thomas Pornin Feb 2 '12 at 12:34 @ThomasPornin Your argument doesn't show that per-encryption randomness is required. For example you don't show why padding based on `hash(m,e,n)` is a problem. Such a scheme is still deterministic, and doesn't suffer from this specific problem. – CodesInChaos Apr 18 '12 at 13:27 @CodeInChaos: yes, my argument does not show why per-encryption randomness is required, which is why I wrote in my response "more generally". You need per-encryption randomness to avoid exhaustive search attacks on the encrypted text (which may or may not be applicable, depending on context -- it depends whether you are encrypting a meaningful message, or a random session key). – Thomas Pornin Apr 22 '12 at 22:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 34, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9030187726020813, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/80177/scheme-of-dual-numbers-over-a-field/80196	# Scheme of dual numbers over a field Let $k$ be a field and $D:=\operatorname{Spec}(k[t]/(t^2)$ the scheme of dual numbers over $k$. Then what is the fibre product $D \times_k D$ with itself over $k$? In other words, what is $\operatorname{Spec}(k[t]/(t^2) \otimes_k k[t]/(t^2)$ And how do line bundles over this scheme look like? - ## 2 Answers The tensor product $k[t]/(t^2)\otimes k[t]/(t^2)$ is equal to $k[x,y]/(x^2,y^2)$, where $k[x,y]$ is the polynomial ring in the two variables $x,y$ over $k$, together with the natural maps $f,g: k[t]/(t^2)\rightarrow k[x,y]/(x^2,y^2)$, $f(t+(t^2))=x+(x^2)$, $g(t+(t^2))=y+(y^2)$. Proving the universal property is a bit lengthy but straightforward as far as I see. - It is not lengthy, $R[x]/(p) \otimes_R R[y]/(q) = R[x,y]/(p,q)$ is just abstract nonsense (compare hom-functors and use universal properties). – Martin Brandenburg Nov 11 '11 at 9:06 As Hagen explained, the scheme $X=D\times_{k} D$ is the affine scheme associated to the ring $R=k[x,y]/(x^2,y^2)$. Since that ring is local, $all$ vector bundles of any rank on $X$ are trivial, not only line bundles . Indeed vector bundles (or equivalently locally free sheaves $\mathcal F$) on $X$ correspond to finitely generated projective $R$-modules $P$ ( $\mathcal F \leftrightarrow \tilde P$ ) and all projective modules on a local ring are trivial. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 24, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8769290447235107, "perplexity_flag": "head"}
http://lalashan.mcmaster.ca/theobio/worden/index.php/Consensus_Dynamics_Notes	# Consensus Dynamics Notes From Worden ## Historically Historically, this research program starts with this research proposal, which I wrote for an opening at the Santa Fe Institute. ## Lit review Currently this is more like "literature to review"... ### Collective, evolutionary and other search processes • go back to Kauffman and Levin 1987, Towards a general theory of adaptive walks on rugged landscapes and more recent developments from it. • Daniel Grunbaum, Iain Couzin? [1] • Metaheuristics literature ### Distributed consensus research I believe this is somewhat different, but there may be important overlap. Need to read various stuff in the field to get an idea of what's relevant. Some random links: ### Models of social process Suggested by Jordan Ellenberg: • Ben Golub and Matthew Jackson on learning in networks • Adam Elga, How to disagree about how to disagree. Also Consensual Decision-Making Among Epistemic Peers may be relevant. ### Group process • Via Jason Jixuan Hu: The Change Handbook • On Conflict and Consensus: a handbook on Formal Consensus decisionmaking, CT Butler and Amy Rothstein • Tom Atlee's Co-Intelligent Practices list. • "The Five-Fold Path of Productive Meetings" Movies: JD suggests Of Gods and Men. There's also Twelve Angry Men. #### In social movements • Graeber, Direct Action: An Ethnography includes participant observations on consensus meetings. • 2011 "Real democracy" movement • in Greece • June 24 resolutions, for example • Democracy is born in the squares, Christos Gionanopoulos • Greek process, primary document including paragraph-by-paragraph consensus taking • See also these notes, below • in Spain • assemblies in Argentina, Spain, and elsewhere • And, of course, this year's Occupy movement ### Tech support for collective decisions • http://liquidfeedback.org/mission/ (in use by German pirate party) • http://peoplesskype.org/ • http://openassembly.org/ • The SUPERASSEMBLY! • in Lorea (Spanish derivative of Elgg?) • PartyX: "There is some information about the algorithm on our wiki, at http://wiki.partyx.com. There is in particular: a simple video overview: http://www.youtube.com/user/partyxevolition#p/a/u/0/XMvSa0KSzIw; an early beta for a 2-person decision: http://www.influentsalgorithm.com; the general specifications: http://wiki.partyx.com/index.php?title=Influents_Algorithm_Specifications; some theory: http://wiki.partyx.com/index.php?title=Influents_Algorithm_Theory" • http://zelea.com/project/votorola/home.xht • Mako Hill: Selectricity and RubyVote • NYCGA notes on consensus on mailing lists • David Stodolsky's papers - look them up, assess what's relevant And lots more, I think. Also related: • OccupyResearch wiki and listserv • occupy-dev listserv ## Possible avenues ### Distributed computing metaphor What can we learn from people who know a lot about distributed computing? For instance, Google uses many thousands of computers working together to divide up a search request and get the answer incredibly quickly. For instance, imagine all the different cities' Occupy movements trying to make a national- or global-scale decision together. A good process might make the difference between doing it efficiently and effectively, and giving up in frustration. • A good tutorial on distributed computing: http://code.google.com/edu/submissions/mapreduce/listing.html • Plan for failures at every point of the protocol. • cell phones stop working; internet connections fail; point people unreachable; facilitator unavailable; meeting place for coordinators unavailable; some groups of participants take too long to respond; disgruntled factions become uncooperative and disruptive; etc • Parallelization models [2] • Some things are easy to distribute, some are not. • Testing for consensus is no problem. Given a proposal: send it to every subgroup, ask them whether they have a block. If anyone says yes, there's no consensus, and if not there is. (Except for failures... what if someone doesn't answer, etc.) • Distributed logistics may be more difficult... SF has computer servers to spare, NY has lots of donated money, etc., who provides what resources to whom else, can include lots of things that depend on each other. Not easy to decompose into independent pieces, potentially a very hard problem. (see also: [3]) • "Each of these problems represents a point at which multiple threads must communicate with one another, or access a shared resource." Translation: distributing a problem is difficult if the players need to communicate with each other, or if they need to access information or resources that someone else also needs to access while solving the problem. • "Keeping shared state to a minimum reduces total system complexity." Translation: it's best if each subgroup can do its job without needing to know anything about what's going on in the other groups. • Data parallelism and task parallelism. Many questions and tasks decompose into subproblems that can be done in parallel. If a question has independent subquestions, they can be attacked by different people simultaneously, and brought together at the end. But if there's one or more resources that are needed in both subtasks, you can have trouble. "Intelligent task design eliminates as many synchronization points as possible, but some will be inevitable." That is, the point where one team has to stop and get information or a resource from another team is a bottleneck that can cause delays or even deadlock. • Common patterns for parallelism • Master/Workers. Oversight person farms out tasks to others, gets the results back and integrates them. • Producer/Consumers. "Producers" generate tasks to be done, "consumers" step up and do them. This "subcontracting" can continue in a chain with no central integration of information. • Work queues. Tasks to be done are posted to a bulletin-board or similar system, where workers can claim them and return the results without interacting directly with the task's producer. • Note: all these "tasks" may be informational tasks, like evaluating a proposal, generating amendments, etc., rather than tasks like fundraising, outreach, etc. It's not obvious to me how that works in these contexts, but it's intriguing. Or spatial things like coordinating movement of demonstrators, somehow... • Implementation techniques [4] • Is there something to be learned from the "primitive" patterns that distributed programmers have developed? • `fence()`: pause and don't continue until everybody else has also reached this point in the processing. • Mutual exclusion lock: like the bathroom door, close off access to the resource while using it so it only has one user at a time. The idea that one speaker "has the floor" also fits this pattern. • Communication patterns: • one-to-many: broadcast. Facilitator does this. • Sometimes it is useful to have a way any one participant can send a broadcast message. Mic check! • many-to-one: for example, collecting all the results of a vote. See reduce, below. • many-to-many. Conversations that happen between assemblies. Facebook, twitter, texts, emails, etc. Many-to-many communication could conceivably be incorporated into meeting process as well, if it's a useful part of the design. • All-to-all messaging: "Combination of above paradigms; individual processes contribute components to a global message which reaches all group members". Interesting. I guess documents like this are under this heading. That is, not the subject matter of that document, but the creation of that document itself. There may also be ways to construct an all-to-all message without formal consensus (or in that example, voting)? Might be worth digging up examples of how this is done in computers. • MapReduce [5] • Simplifies distributed programming by handling only problems that can be broken down into a map function and a reduce function. Each node (processor... participant... subgroup...) performs the map function on its data, and then all the nodes' results are combined using the reduce function. Each map operation is completely independent of the others. • Example: to find out whether there are any objections to a proposal, the map function, applied to each individual, is "do you have an objection", and the reduce function is simply logical or: there are objections if any of the individuals said yes. • Actually, there's some more complexity in MapReduce, where it aggregates together all the map values for a given key before doing the reduce operation... might or might not be of interest to us • Handles failures: if a node doesn't complete its map task, just ask someone else to do that task. Additional level: if a particular piece of data causes the map to crash repeatedly, don't do that one any more. This can work around bugs in the program! • Thinking of things in terms of map and reduce instead of messy implementation details can free up the imagination and creativity. What group-deliberation tasks are naturally parallel? • Again, testing for consensus fits the MapReduce mold easily. • Collecting concerns, or proposed amendments, can be a MapReduce. For example, break into small groups, brainstorm, and collect the ideas raised by people in your group (map). Then collect and combine all the lists (reduce). • So there could be a series of map/reduce phases: present a proposal and find out whether people like it; generate new proposals; find out how people like those... • If people's focus is good enough, it would be possible to send out multiple propositions for evaluation at the same time, i.e. break into small groups and collect everyone's opinions about each of these 5 proposals... that could be impractical but I can certainly imagine it. • All this becomes less theoretical and more urgent if we imagine a large-scale process involving multiple assemblies in different parts of the continent, all trying to come to agreement together. It might seem ludicrous to even imagine trying for true consensus on such a scale (as opposed to using a 90% supermajority vote, say), but imagine how heartening it would be if we had a good enough process that we could do it and succeed! ### Constraint satisfaction See Consensus Dynamics Notes/Constraint Satisfaction. ### Minimum value question • if the goal is to find something that minimally satisfies all, it's probably best to focus only on the people who aren't satisfied, not the people who are. Devise facilitation strategy to look at that directly, maybe do some math about it? • Actually let's take that one step further. Minimum value hypothesis: if everyone agrees that the goal is to find something that's acceptable to all, then everybody is searching together for a point that's acceptably high on the landscape of minimum fitness values. How useful is it to do a simple search algorithm on that landscape? That is, just optimize the function WorkingWiki messages $f_{\text{min}}(x)=\text{min}\{ f_{i}(x)\text{ for all }i\}$. It would be worth trying this, to compare to other facilitation algorithms. • That is, imagine an especially difficult person who wants to please everyone. If I am that person, then my opinion of any given proposal is equal to the worst opinion anyone else has about it. Now how will I find a proposal that satisfies me? (So the m.v.h. could also be called the "worst-teammate-ever method".) • However, it might be that the team can do better by taking into account the perceptions of people other than the one who's currently blocking. • How do the bumpiness and other characteristics of this landscape relate to the landscapes it's made from? What kind of search process is best for it? • There ought to be analogous reduced landscapes for other objectives - for instance, the Influents algorithm considers both the mean and spread of values for each possibility. • also, this suggests a related question: what is the probability that there's no way to satisfy everyone? does it have a simple, meaningful form? ### Use examples Contributed by elizaBeth Simpson: • It would be very good to illustrate the model with a simple example or two, like for instance a family buying a car, with various considerations, price, size, etc., and different ways of valuing them. • This would also be a really good way to validate the model results, since it's not obvious whether conclusions drawn from the Boolean hypercube model would generalize to an example like this. • Note the pizza example is a response to this. ### Pizza example This has its own page: Consensus Dynamics Notes/Pizza. ### Issues vs. Proposals The simulations I ran in July 2011 were based on the idea of dealing with a series of proposals: • proposal → friendly amendment → amended proposal → friendly amendment → ... → ultimately consense or give up. I think this may have been loosely inspired by my notes from Butler and Rothstein's book on formal consensus process, including this diagram: This diagram begins with a proposal, and ends with a decision whether to adopt the proposal. [In the July modeling I chose to simplify all this pretty drastically by not developing a way to model "concerns" except by having model agents say "I don't like the proposal as it is, let's change it to this similar one." See below for more on "concerns."] I've also had the experience of being in various groups that use some version of formal consensus in which there's a boilerplate agenda something very much like "Introductions; working group reportbacks; possible special presentation; proposals; announcements and closing." That is, the only collective decision making takes the form of receiving a proposal and deciding what to do with it. In my experience in various groups, and I don't know how widespread this is, you're not allowed to make counterproposals while a proposal is on the table, or any other suggestions except in the form of concerns or friendly amendments. A decision has to be reached whether to block, postpone, or adopt the proposal (with or without amendments) before other things, including other proposals, can be discussed. There is a fundamental asymmetry hidden in this: people are always allowed to bring proposals (room on the agenda permitting), while blocking is strongly discouraged. This means that a proposal that's not necessarily an improvement over the status quo for most people can still be adopted, and if there are several courses of action that are all acceptable enough to avoid being blocked, the first one proposed is likely to be adopted even if others are significantly better. [I have had citations about this in the past, but am having trouble finding them.] Also, on a more positive note, in thinking about the general idea of looking for a good solution to a problem - i.e. a mutually acceptable location in the space of possible proposals for a given problem domain - I came to find this idea limiting. Wouldn't it be better in some circumstances to say, "Let's take a certain amount of time to talk about a certain issue, and see whether there's anything we'd like to do about it"? We could start by trying to clarify what we know about the issue, make sure we agree that there's a problem to be addressed, lay out multiple possible proposals and find out what people think about them, and then see whether an answer has emerged, or whether there's a need to pick out particular proposals and discuss them. This seems more like what Tom Atlee describes here, and it's more like this diagram from Peter Gelderloos's book: Like I wrote in my plan for March, I want to reconceive the simulation model so that it can include things like brainstorming and comparing multiple proposals, not just considering a single proposal at a time. Maybe I'll try asking what's the simplest possible extension of the model that can include brainstorming, to see how far it gets me. I'm curious whether that will reveal that brainstorming is at one end of a certain spectrum (I don't know what spectrum, yet), with taking one proposal at a time at the other end, and if so it'll be interesting to see what lies in between the two ends of that spectrum. #### Modeling issues involved in this In the July models, when I was only considering what to do with a model, I had the issue of how to formalize what a concern is. In my model, proposals were just anonymous points in a big space, and you could really just say how much you do or don't like one of them, and propose another one in its place: there's no way to say what you don't like about it. That requires some kind of language for talking about the proposal, or another way to look at it might be a way of naming sets of proposals and not just single points in the space. For instance, when I say, "I don't want to get the onion, mushroom and tomato pizza because I don't want onion", I'm saying that all the possible pizzas that include onion are, uh, off the table - which helps the process along quite a lot because it rules out a big chunk of the search space. In the Gelderloos diagram something analogous is happening earlier in the cycle: first there's a cycle involving brainstorming that produces an "idea", then there's a cycle about "how [the] idea will take shape", which produces a "proposal" that has general support, and then there's a cycle involving concerns and friendly amendments that ends with consensus or failure to reach consensus. It might be safe to say these "proposals" are single points in the search space. They come at the end of a narrowing-down process, and the earlier phases deal with something more general: it seems like an "idea" is a lot like "let's get onions on our pizza", i.e. it corresponds to a whole region of the search space - all the different combinations that include onions - very much like concerns, or their counterparts, the "pros" of "pros and cons". This could be described as working with subsets of the search space, or, probably more usefully, characteristics of proposals that are wanted or unwanted. [As an aside, Gelderloos's three-phase diagram could probably be generalized to a simpler idea of narrowing down without discrete phases, starting with broad brush strokes (is group support behind a particular idea or approach) and then narrowing in more and more until there's an actual concrete plan for what to do. This might be worth exploring.] It seems like I really need a clear way to handle this subset issue. Other issues have been emerging as well, which I might write up separately: what difference does it make that people don't change their point of view from listening to each other; what difference does it make that people don't consider "the good of the group" as well as their own desires, for instance wanting to come to agreement quickly or to avoid conflict; and are people likely to use strategy to get more desirable outcomes for themselves, not just something they're willing to accept. ### Strategy What's the difference between strategy and tactics? I have the idea that consensus meetings' proposals tend to be about tactics, and can get hung up on differences about strategy. Whereas deliberation about strategy takes years or longer, and happens in informal discussions, publications, etc., not meetings. Is this true? • This question may overlap in some way with NVC theory of strategies and underlying needs. ### On Nonviolent Communication and seeking consensus David Montgomery writes: ```Hi Lee, I'm not imagining how something like this would go into a simulation, and you may already know this, but I figured it still wouldn't hurt to share. At a high level, the NVC approach to this is as follows: * Get all the needs on the table. That is, translate whatever strategies are up for different participants into the deeper, more universally held needs, until all participants agree that if these needs were met, they would be satisified. This usually happens as a combination of identification of their own needs by individual participants, and guessing by others. In NVC theory, at the level of needs, there isn't conflict anymore. But there's still a question of coming up with a strategy to address these needs. * NVC holds that when people see each other's humanity, and are connected at the level of needs, a tremendous creativity is released to generate solutions that will meet all or most of the needs. The idea is that needs can be met in a great variety of ways, some of which will be compatible, even when the strategies that have come to mind first are in conflict. More space, more degrees of freedom. It's not much of an algorithm/process. But I think the key insight is actually effective in many situations. David ``` ### Notes from Greece From a friendly source on the Diaspora* network, who wants to be anonymous. source Is this your project? I'd like to make a clarification. Greece (and as far as I know Spain too, but I can be really certain about Greece) didn't use a consensus democracy. There was public speaking before the assembly were you could convince the people about your views and then there was voting on proposals based on these views but the process wasn't repeated ad infinitum. After some proposals were rejected they didn't come up again. source Forgot to say, it didn't use consensus democracy, it used the Athenian direct democracy model. lw Thank you! Do you have links to a good description of the Greeks' process? It sounds somewhat like the 90% supermajority voting that OccupyOakland is using. In the theoretical work I'm most interested in consensus but I'm also interested in the differences and I want to correct my descriptions if needed. lw Also can I just say, I posted this thing just now on facebook, g+, twitter and here, and people click "like" or whatever but only here do I get a serious, meaningful response from people I don't even know source I almost didn't see that you responded, the notification system is still buggy. I'll look if there is anything already officially translated about the process and I'll let you know. Yes, diaspora is quite the intellectual place it seems :) lw Thanks! I'll check back later, I'm off to the plaza, where someone from the MST in Brazil is speaking to the occupiers and then there's a solidarity march for the Egyptian movement... so much great stuff to keep up with! source Unfortunately it appears as though the site that was holding most of the translated texts is down. The structure of the Syntagma sq. public assembly was/is rather complex. There are three main parts. The people that constitute the assembly itself who decide everything and speak, the coordination committee which handles administrative issues, and the thematic groups who deal with specific issues and then present their work to the general assembly and ask for validation through voting. The administrative committee is chosen randomly from the crowd of the assembly or rather from a pool of volunteers, and I think it was supposed to change every 3 or 4 days. It's main purpose was to coordinate the speaking and voting process. To speak to the assembly people had to pick numbered tickets and then numbers were chosen randomly through a lot. Each speaker had about 2 to 5 minutes due to time constraints. To vote on an issue, written proposals were given beforehand to the coordination committee. At some point during each meeting, the coordinator would read the proposals and the people would vote on each one by raising hands. Some times, people would object and the voting would be repeated. Three kinds of votes were allowed, for, against and blanks (white votes). Texts that represented the views of the assembly would be read paragraph by paragraph to the assembly and each paragraph would be voted on. I found a translated example of this process here. If you need more details about the process feel free to tell me. I will continue looking for translations in the mean time. source The random functions are a method taken directly from the ancient Athenian democracy. Public servants and administrators were chosen randomly and didn't have any power at all. They were considered employees of the citizens and they had to give a full account of their actions at the end of their terms. In the modern Athenian assemblies they kept the random method but changed it to fit the current situation. lw This is great, thank you! I would guess that as a stochastic process the random sampling would work quite well to reproduce the decisions that would get made by taking a much longer time to include everybody - but that it doesn't work so well at creating the sense of "buy-in" that comes with actually being included. The voting on each paragraph seems more weird to me - I will have to learn more about it. It doesn't make sense to adopt only some paragraphs of a coherent document, so there must be something different from that going on - like maybe they will go back knowing which paragraphs passed and produce a new document to propose... lw Sometime soon I'll copy these notes over to my research wiki. Do you want to be credited? If so, please tell me what name or handle to use to identify you... source It depends on what part you are talking about. In choosing members for the coordination committee, random sampling is used because it provides the greatest level of equality. With a randomly chosen coordinator you avoid any sort of crony-ism, favoritism etc. People accept this because being a coordinator is (and should be) a burden, not something to actually strive for. In picking speakers, randomness is chosen because there wasn't enough time for everybody and there were too many prospective speakers. The only just system for picking who would speak and who would not (because a lot of people didn't get a chance to speak at the assembly at all) is by drawing lots. The effect on the people, as far as I can tell (subjectively) was to try and make the best use of their time slot, since it was essentially rare to get to speak to the assembly. In actuality, if someone didn't have time to speak to the assembly but had an important thing to say, usually gave a written proposal to the coordinators and it went to a vote immediately. The text that was voted on, changed on the spot by yelling different proposals. The coordinator would put the new proposals in a vote immediately. Sometimes, a text went back for complete redrafting etc. It was a very fluid process and the most importance was placed in maintaining absolute power to the assembly and not the thematic groups. I'd prefer not to be credited, I didn't create up the assembly process and it is in a way my duty as someone who took part in it, to spread the knowledge about it. You could just use a random letter to signify that it is a particular source though. If I find any more data I will send it over. By the way, the repeating of each phrase that happens in Wall street, never happened in Greece. lw Yes, the repeating was a strategy for having to meet without any loudspeakers, then became a symbol of the movement. I definitely want to research the specific processes in the Greek movement (we should obviously look to them to teach us about democracy!), and I'm sure the Spanish activists have made important innovations as well. I would add about randomness that it might provides the greatest level of equality outside of including every single person, which is not an option in a large group because it's too inefficient. source The original Athenian democracy had some very smart safeguards against corruption, resurgence of oligarchy etc. and that was essentially the blueprint for what the modern general assemblies did. A reason for their success is that the Greeks grow up with with a very positive image of the ancient Athenian democracy, due to their schooling. Hence, having an example that worked gives a lot of confidence to those that try to recreate it. Another thing that I personally find very interesting is the Pericles network. The Pericles network was a research project developed at the polytechnic university of Athens and it consisted of decentralized computer nodes that would make direct voting on issues by the people possible. They had a working prototype too but it was rejected by the government in the 90s because they deemed that the "people were not ready for this yet". This was supposed to be an adjunct to parliamentary democracy, but I think it can replace it completely with enough tweaks. If you are interested about this you can visit the homepage although unfortunately, it is all in Greek and google translator may not be of much help due to the vocabulary used. The only document I found in English that describes the system is this. ### Lessons from voting [Contributed by JD] If we assume: • People's opinions don't change through deliberation • Information is exchanged efficiently • The space can be explored efficiently • There is no option agreeable to all • We have to reach a decision We move the problem to the world of classic voting theory (see argument below). Even if we don't make this assumption, voting theory has something to teach us about this problem. #### Argument If we accept the assumptions, then we can assign a voting order (and/or voting weights) for each person across the whole list of possible alternatives. This is exactly the problem that classic voting theory is meant to address. I don't necessarily agree: classic voting theory seeks a way to find the best alternative. When our individuals agree to allow something acceptable but supoptimal, I think we are looking at a different problem. Anyway, in this project, I want to shift the focus from voting to searching in a big space. -LW #### Consequences Arrow's theorem applies: there is no algorithm that produces an uncontroversial optimum in all cases. It might be good to evaluate whether a "Condorcet solution" exists; if not, there may be a need to choose between various complicated heuristics. If an optimum is desired: again, finding an optimum is not the focus of this project. -LW These concerns seem to generalize to the broader problem; we should be on the alert for signatures of possible "voting paradoxes" (or the opposite, "Condorcet solutions") even in the absence of complete information, and we have to think about what heuristics we might be implicitly applying. This could be - "Rock, paper, scissors" situations for instance. How do they affect these processes. -LW ### Progressive stack I'm becoming more interested in what to do when an issue is identified, as opposed to when a proposal is made (a proposal is one thing a group can do when there's an issue). So I'm thinking about the stuff that happens before a proposal is brought up. In that phase there's generally a facilitated discussion, or else informal discussion outside of the frame of the meeting. Facilitated discussion can include small-group breakouts, brainstorming, bringing up more or less structured questions, and various other techniques. It often involves keeping a stack of people waiting to speak, in the interest of fairly allocating the right to be heard. Many groups use a progressive stack, in which people whose voices might often be silenced or underacknowledged are given preferential access to the stack. I find myself wondering whether there's anything to be gained by thinking about the different ways a progressive stack could be designed, and what difference it would make. Computer scientists know a lot about keeping track of items to be dealt with. Generally they are kept in either a stack or a queue. Ironically, the stack we're discussing here is clearly a queue. A stack is "last-in-first-out", like ones mental list of digressions in a conversation; when you finish a digression you return to the most recent thread of the conversation, and from there out to the next most recent, and if you're especially good at it you end up all the way back at the topic you started with. A queue, on the other hand, is the kind of thing we get into at the grocery store, where you start at the back and get to the front after all the earlier people are done: "first-in-first-out". There's a huge body of research on queueing theory, which could conceivably have something to contribute to this question of how to combine justice and efficacy in facilitated meetings. Queueing theory is mostly about commercial problems like how many checkout lines to open at a given time in a store, and the design of the boarding process for airplanes ("Now boarding all passengers in group 3") to get all the passengers into their seats as quickly as possible. I think it's also important in designing the scheduling strategies that computers use to keep large numbers of programs running at the same time on a single processing chip and taking turns using the hard disk, network card, and so forth. But the theory is general and can be applied to many things. Without going off and reading any of the literature, I'm seeing this as a design question in which we imagine evaluating various candidate designs (you could say anyone belonging to certain groups speaks before anyone who doesn't, or you could reserve a certain number of slots, or someone could propose that actually just keeping a straight stack without making any distinctions works best). You have to evaluate a given design in terms of certain objectives, so what would they be? Making space in which people can be heard when they haven't had the opportunity often is a good in itself, so that can be considered an objective, but it can also be that hearing a greater diversity of perspectives helps the entire group make better decisions, and other advantages or disadvantages might also be proposed. The second issue is that in order to evaluate a design in terms of any objective, you have to have some estimate of what will happen given that design, which means you have to have some scenario of who is present in the meeting, who will speak under what circumstances, and what kind of things they're likely to contribute. The question of who is likely to contribute what under what circumstances seems to be a particularly basic one - what do we know about the idea that some people are less likely to contribute than others in a seemingly "fair" system where everyone is treated equally? How specific can we be about it? How much less likely is one to contribute? Without information like that, it's hard to guess what kind of impact particular interventions will have. Of course, information like that is very hard to come by, so most likely questions like this can only be addressed in very general terms by using hypothetical possibilities and maybe a small number of experiments and studies of actual groups. A well-founded approach to this, if there is one, would probably be to make very simple, qualitative assumptions and see whether there are any very general patterns that hold over a large range of specific situations. ### Concerns Here's an idea, borrowed from the satisfiability problem. Say the problem is to find a proposition that is a combination of various things, like "A and B and not G". Each participant has a set of requirements, for instance, "not C and not D and F and not H", or a more complex combination of possibilities. Now when a proposal is on the table, a given participant can say, "well, I don't like it because I don't want H. How about if we get rid of H and include G instead, because I would like that better." Then the others can evaluate whether they would like that better. So the search space is the same as for the satisfiability problem, and the search heuristic is something like the suggestion there: flip some letters on and off and find a better proposition. But does this really get at the idea of concerns? ... note, a more extended model might have participants have some kind of higher level? Like "I want a certain kind of outcome... that can be satisfied by 'A and C and not K' or by 'B and not D and not E'..." Then the interactions would be more complicated, like you might think your concern is that C and not K are satisfied but not A, but actually a better way is to try to satisfy the other criterion... ### Misc to-do items • effect of pickiness of individuals (utility watermark). • maybe a simpler comparison: • $n$ people working independently vs. • $n$ people each generating a proposal and then voting vs. • $n$ people searching together in some simple way: • Compare mean fitness of outcome, how satisfied each person is. • replicate Scott Page's main result? • Can I generate something similar but different to Page's? Like, look at the benefit of having people with somewhat different valuations, as opposed to the same valuation but different search heuristics • compare generating proposals in meeting vs. between meetings - all-to-all vs peer-to-peer communication • diversity in watermark, ruggedness, inter-individual correlation? • network structure, processes for large groups where all-to-all communication is impractical ## Other ### Notes on formal consensus process, from Butler/Rothstein's book [6] ```In the consensus process, only proposals which intend to accomplish the common purpose are considered. ``` • Present proposal or issue • General discussion • Test for consensus If no consensus: • Identify participants' concerns • Seek resolution for concerns • Questions to clarify concerns • Produce modified proposal • Possible stand-asides or blocks • Test for consensus Possibly: use brainstorming technique to collect concerns, then group them before discussion/resolution begins on the grouped concerns. Possibly: Break out into small groups. ### Another description Another description of the project that I wrote for an email and didn't use: Making a decision together is obviously a fundamental human activity and it may seem perverse to think of it in terms of math. In fact thinking of it in terms of math may risk certain kinds of perversity (in the bad sense) - for instance, the risk of seeing the process as mechanical and losing compassion for the people we interact with, or imagining a need to surrender control to a cadre of technocratic nerds. But I think it's possible to evade those risks and use some powerful tools we have at our disposal to learn something useful. Previously I've done research relating to evolution and emergence of cooperation in ecological communities. These problems have something in common with the business of making a decision together, if you look at it a certain way: there are many possibilities and we want to know how and when the good possibilities get selected. In the case of decision making, it looks like this. There is a group of people facing a problem, issue, or opportunity, and seeking to decide what to do. There is a range of possibilities and the group has to choose one. Generally you don't know what all the possibilities are, so there's a process of exploration as well as evaluation. Different people have different opinions of what's a good choice, and different people come up with different possibilities. This is why they need to talk to each other, to combine their ideas and ask each other what they will accept. Given all that, we wonder what's a good process, that is, what pattern of communication is likely to help the group come up with a decision that is agreeable to as many people as possible. This is partly a question of how to make a choice among a small number of known proposals, and partly a question of how to generate good proposals by building on each other's ideas. It includes how can group members behave to help the whole group's process, and what kind of facilitation gets the good outcomes. ### Formalism Consensus process is used when there is an issue. Maybe an issue is a search problem, like "find an proposal that everyone agrees can address this particular set of desires." • We can model that as "find a location in this search space that satisfies each participant's requirements". • Each person's requirements are different — that is, each person has a different function for evaluating members of the search space. So: say our search space is $\Sigma$, and each person $i$ has a valuation function $\phi _{i}$, so that for a given proposal $x\in\Sigma$, each person either likes or dislikes it: $\phi _{i}(x)\in\mathbb{R}$. Then: what is a concern? • If I'm a participant, and the proposal doesn't make my threshold, the concern is a reason why it doesn't. That requires the fitness function to have some kind of structure, by which I can say improving the proposal in a certain way will satisfy me better. Given what is a concern, what is a way to resolve it? • A new proposal, I guess. Do we connect this to Scott Page's models? In his case we share a fitness function but use different heuristics (which is connected to having different landscape topologies). The heuristics are important. We may want to use that as well. It's pretty much like this: given a proposal $x\in\Sigma$, person $i$ can go from there to a few modifications of the proposal: $\{ y_{1},\ldots,y_{n}\}$. A different person would propose a different set of modifications. We have a round of proposed modifications, and people evaluate each other's suggestions, and if one of them is good, we test that one for consensus and concerns. This is the same as Page's model, I think, but it's different when people have different valuations $\phi _{i}$. That's a model worth investigating. But also what about concerns? ### Notes • effectiveness of Constraints vs Goals as in Larry Richards' papers • Maybe a proposal can be a schema like in John Holland's models - a subspace of the domain like `1011010*******01`, where the `` positions are unspecified. Then a concern might be like, I don't like `10110100010001101`, do you have a better example? This could work well with the SAT framework... ### Model development process? Here's a nice-looking document on how to develop your model: http://www.professorgizzi.org/modelingcomplexity/netlogo/modeling_process.html. Maybe it's useful. I'm thinking of using Netlogo to code up the model. Update: No, I'm not, I wrote it in C++. Step 3, Hypothesis Building: • What is the environment? • Strictly, there isn't one. A meeting room that doesn't need to be modeled. A collection of model people. • Who are the agents? • The people seeking consensus. Some number $N$ of them. They won't arrive or leave during the process (or will they?). • Variables? • Characteristics? • What's this mean? • Rules? • What's this mean? • What are the relationships? • No spatial landscape or network, just an all-to-all interaction. • What are the rules? • Evaluate a proposal by either approving it ($\phi _{i}(x)>\mbox{\it threshold}$) or making one or more counterproposals ($\{ y\in\eta _{i}(x):\phi _{i}(y)>\phi _{i}(x)\}$). If all present approve a proposal, stop. Steps 5 and 6 look like a useful framework for testing and analyzing the model.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 20, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9576535224914551, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/162792/smallest-subfield-containing-transcendental-or-algebraic-element?answertab=votes	# smallest subfield containing transcendental or algebraic element Consider the extension $\mathbb R/ \mathbb Q$. what is the smallest subfield $\mathbb Q(\pi)$ of $\mathbb R$ containing $\mathbb Q$ and $\pi$. I think it is $\mathbb R$. A more general question: for all transcendental elements $a$ in an extension $L/K$, we have that the smallest subfield $K(a)$ of $L$ containing $K$ and $a$ is $L$ itself. But if the element $a$ is algebraic then the smallest subfield $K(a)$ of $L$ containing $K$ and $a$ is the field $K[a]\subset L$ of elements $p(a)$, $p\in K[X]$. Is that correct? Do we have an example in which the element $a$ is algebraic and still the smallest subfield $K(a)$ of $L$ containing $K$ and $a$ is $L$ itself? - 6 $\mathbb Q(\pi) = \{\frac{p(\pi)}{q(\pi)} \mid p, q\in \mathbb Q[X], q \ne 0 \}$. That doesn't equal $\mathbb R$, as it is countable. – martini Jun 25 '12 at 10:56 Dear @martini: I wanted to have a look at your website (jungenschaft...) and was attacked by a virus. Are you aware of that problem? – Georges Elencwajg Jun 25 '12 at 11:04 1 If $a$ is transcendental over $K$ and lies in some extension $L$, then the smallest subfield of $L$ containing $K$ is... just $K(a)$! That is, it consists of rational functions in $a$ with coefficients in $K$. That's it. – Qiaochu Yuan Jun 25 '12 at 11:09 so for transcendental elements, the smallest subfield $K(a)$ containing $K$ and $a$ is the field of fractions, $p(a)/q(a)$, $p,q\in K[X]$ is that a general statement for all transcendental elements? – palio Jun 25 '12 at 11:10 Oh i see now that we have a definition confusion. actually i DEFINE $K(a)$ as the smallest subfield containing $K$ and $a$ and not as the field of rational functions in $a$ with coefficients in $K$ and if i well understand you these two notions coincide when $a$ is transcendantel? – palio Jun 25 '12 at 11:13 show 2 more comments ## 2 Answers Let $K$ be a field and let $L$ be an overfield of $K$. For any $a\in L$, we define $K[a]$ to be the smallest subring of $L$ that contains $K$ and $a$, and we define $K(a)$ to be the smallest subfield of $L$ that contains $K$ and $a$. We have: Theorem. Let $K\subseteq L$ be fields, and let $a\in L$. 1. $K[a] = \{p(a)\mid p(x)\in K[x]\}$. 2. $K(a) = \left.\left\{\frac{p(a)}{q(a)}\;\right\|\; p(x),q(x)\in K[x], q(a)\neq 0\right\}$. 3. If $a$ is algebraic over $K$, then $K[a]=K(a)$, and $K(a)\cong K[x]/\langle f(x)\rangle$, where $f(x)$ is the minimal irreducible of $a$ over $K$. 4. If $a$ is transcendental over $K$, then $K(a)\cong K(x)$, the field of rational functions in one indeterminate over $K$. In particular, note that for $K=\mathbb{Q}$, $L=\mathbb{R}$, and $a=\pi$, we have that $\mathbb{Q}(\pi)\cong\mathbb{Q}(x)$ is countable, and so cannot be isomorphic to $\mathbb{R}$. Proof. 1. The homomorphism $K[x]\to K[a]$ induced by mapping $x\to a$ shows that $\{p(a)\mid p(x)\in K[x]\}$ is contained in $K[a]$. Conversely, the image of this map is a subring of $L$ that contains $K$ and $a$, and so $K[a]$ is contained in the right hand side; this proves 1. 1. $K[a]\subseteq K(a)$, so $K(a)$ contains the field of fractions of $K[a]$, which is the right hand side of 2. Conversely, the right hand side is easily seen to be a field that contains both $K$ and $a$, giving the other inclusion. 3 and 4. If $a$ is transcendental, then the map $K[x]\to K(a)$ is one-to-one, and so induces an embedding $K(x)\hookrightarrow K(a)$. By the description in 2 the map is onto, giving the isomorphism in 4. If $a$ is algebraic, then the kernel of the map $K[x]\to K[a]$ given by evaluation is the ideal generated by the minimal polynomial of $a$ over $K$. This is an irreducible, so $\langle f(x)\rangle$ is a maximal ideal of $K[x]$, hence the quotient $K[x]/\langle f(x)\rangle$ is a field; this field embeds into $K[a]$ and by 1 is onto, showing that $K[a]$ is actually a field (and hence equal to $K(a)$), and giving the isomorphism in 3. $\Box$ Your other assertions are incorrect: Say $L=\mathbb{Q}(x)$, $K=\mathbb{Q}$, and $a=x^2$. Then $a$ is transcendental over $K$, but $K(a)\neq L$; worse, take $L=\mathbb{Q}(x_1,\ldots,x_n)$ and $a=x_1$; then $L$ is a transcendental extension of $K(a)$. As to the final question, note that by the Primitive Element Theorem, if $L$ is any finite separable extension of $K$, then there exists $a\in L$ such that $L=K(a)$; so for example every finite algebraic extension of $\mathbb{Q}$ will have one such $a$. - Most of the question has been handled in the comments, all but the very last bit. If $a$ is algebraic over $K$, and we take $L$ to be $K(a)$, then of course the smallest subfield of $L$ containing $K$ and $a$ is $L$ itself. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 131, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9215857982635498, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/319120/struggling-with-technique-based-mathematics-can-people-relate-to-this-and-wh	Struggling with “technique-based” mathematics, can people relate to this? And what, if anything, can be done about it? I'm a 3rd year undergraduate, majoring in pure mathematics. I've done well in the "proof-based" subjects I've taken, and I think that's because I understand the "rules of the game." That is, predicate logic + how to write a coherent proof. Furthermore, people are explicit about what they mean, stating their premises, quantifying explicitly (for all $x$, there exists $y$ such that...), distinguishing between "A implies B" and "A iff B", etc. This obviously really helps. Recently, however, I've been finding that "technique-based" (as opposed to "proof-based") subjects like complex analysis, vector calculus, differential equations etc. are beginning to frustrate me, and I'm starting to get bad marks, too. It's like when I'm sitting in these lectures, the "logic" of math suddenly becomes opaque. I can never tell what the premises are. I often don't know whether we're trying to show that "A implies B", or whether we're trying to show that "A iff B". Stuff is happening on the board, but the "rules of the game" just aren't clear to me. Does anyone else have a similar problem with "technique-based" math? And if so, what can be done about it? Let me give an example. Below, I've copied some of this problem from Wikipedia, and I have inserted my own thoughts in italics. A separable linear ordinary differential equation of the first order must be homogeneous and has the general form $$(1)\qquad \frac{dy}{dt}+f(t)y=0.$$ where $f(t)$ is some known function. I can't tell if (1) is being taken as a premise or not. We may solve this by separation of variables (moving the $y$ terms to one side and the $t$ terms to the other side). $$(2)\qquad\frac{dy}{y}=-f(t)dt$$ Are you asserting that (2) follows from (1), or are you saying they're logically equivalent? And I still don't know whether equation (1) is a premise, or what our premises are. Since the separation of variables in this case involves dividing by $y$, we must check if the constant function $y=0$ is a solution of the original equation. Trivially, if $y=0$ then $y'=0$, so $y=0$ is actually a solution of the original equation. We note that $y=0$ is not allowed in the transformed equation. Clearly, if $y$ is everywhere zero, then equation (1) holds. But what's all this "we must check" nonsense? Are you trying to say that the statement "the function $y$ is everywhere zero, or equation (2) holds" is logically equivalent to the statement that "equation (1) holds?" If that's what you're meaning, why don't you just say so? If the argument was just laid out in a coherent fashion, nonsense like "we must check" simply wouldn't appear. We solve the transformed equation with the variables already separated by integrating, $$(3) \qquad \mathrm{ln} \,y = \left(-\int f(t)dt\right)+C$$ where $C$ is an arbitrary constant. Are you trying to say that if (2) holds, then there exists $C$ such that (3) holds? Then why don't you just say so? Or maybe you're trying to say that for all $C$, (3) holds iff (2) holds. I honestly can't tell. Well you get the general gist. So my question is, can other people relate to this, and what can be done about it? - The argument looks fine to me (but could clearly be more rigorous). Yes, there are definitions, assumptions and even results taken for granted all jumbled together in the above. OTOH, too much rigor leds to nobody understanding a word. Finding the fine line between both isn't easy. – vonbrand Mar 3 at 3:10 I've always taken technique-based explanations as not necessarily meant to be proofs, but rather to be a little more "intuitive". If something doesn't seem obvious, just try working through the motions of the mathematics and see for yourself whether or not something follows. In your example, for instance, yes, the wording of opening statement here implies that (1) is a premise; because of the algebraic steps taken to get from (1) to (2), yes, you can say that "$y=0$ or (2) iff (1)"; and because integration mandates a constant, yes, (2) implies a $C$ to be solved for by initial values. – algorithmshark Mar 3 at 3:26 4 (0) Take some courses in humanities. (1) That Wikipedia article is not an example of good mathematical writing, not even close. (2) Think about the first time you stood up and walked, would you have preferred to have had a course on anatomy first, before taking the first step? (3) If you were to do mathematics by yourself, as opposed to being instructed, what would you do? Do you stop yourself at each step to make it precise? If so then you will feel uncomfortable in technique-based courses. Just as many students feel uncomfortable in proof-based courses. (4) Don't let rigor consume you. – Maesumi Mar 3 at 6:16 6 Answers I can actually relate fairly well to this. Please don't take this the wrong way, but my main advice would be: Stop worrying! By this I don't mean that you should not be concerned about your grades getting worse. But while it appears that you have trained yourself to solid, rigorous thinking in proof-making and such, you have not yet learned to simply apply techniques, and worry later. You need to learn to: (a) live with imprecision, (b) accept temporary uncertainty, and (c) focus, at least for a while, on doing simple exercise you might find dull, over and over. I hope this doesn't strike you as a silly self-help program, and a pat on the back that all will be well. This isn't easy, and, when much younger, I found myself in rather a similar situation. In my math undergrad studies, everything was proofs. And then I studied in France and was blown away by the skill of the average student to simply get stuff done, and to do it fast, and well (I had similar experiences with theoretical physicists everywhere). But while the French educational elite system tends to produce such students, it's also more than anything just a result of having been forced to do the same series majorization over, and over, and over again (to the point that some of my co-students had to recover from mental breakdowns); and in the case of physicists, of having grown up in a world of approximation. But at my old school the saying was also that the best mathematicians were theoretical physicists. From what you write, you seem well-equipped to handle what you face. But where you see an equation and wonder about where it fits into a big scheme, others will just solve it. So stop that. It takes much repetition to get there. If you are not naturally the type for this, get yourself, eg, some Schaums's Outline, or GRE math preparation material. The stuff covered there is largely dull and repetitive, but provides you with much training. So it also takes time and effort now; but if you put this in, and learn to not over-think everything, I think your chances are good to change results fairly fast. From how you describe yourself, it will probably take longer to genuinely internalize this than to raise your grades again. If you need extra motivation, even for successful proof-making you need to learn to deal with the gaps. A (simple) paper might result from some lines written down that you feel might be true, and you trust yourself you will fill in the details later. You don't worry about those for now. My main thesis paper had a gap in the middle I couldn't solve for 2 years. I kept writing, and eventually by brainstorming with someone much better than me we saw how it could be done. This isn't so dissimilar to your problem: solve the ODE; then, later when you have time, think about it, and read some theory. You should also keep in mind that manipulations like the above were what the Leibniz' and Bernoullis etc did on a regular basis, often without having a proof that would live up to today's standards. Deep insights can derive from mastering simple techniques, so throw yourself behind those in the near future. Good luck. - I would not call equation (1) a "premise," it is a type of equation that is being given a name. The logic behind manipulating differential equations (and corresponding lack of "iff" vs "implies") is no different from that of manipulating ordinary algebraic equations in your intermediate/"college" algebra that you might have taken in high school. Sometimes the manipulation is reversible, in which case there is a tacit iff underneath (say, adding $1$ to both sides), and sometimes it is not reversible (like squaring both sides), in which case there is a tacit implies. Keeping track of the logic is, for those experienced, and in some areas of math, less enlightening and less challenging than the bigger task of finding the manipulations required to do what is desired, which is why it is omitted with the frequency that it is. At any rate, the only thing you need to do is consider what type of manipulation is being done and whether or not it is reversible. Of course, another carry-over from high school algebra: when you're solving an equation, you are indeed taking it as a given. If you want to solve $x^2-2x-3=0$, you assume by hypothesis that it is a true statement about some number $x$, then find a chain of implications which tells you what $x$ can be. If you complete the square first and then isolate $x$, you will end up doing a square root, which will introduce $\pm$ signs. Note that $x=\pm{\rm blah}$ has the meaning of $x\in\{+\rm blah,-blah\}$. Or in propositional logic (ish), we'd say $(x-1)^2=4\implies (x-1=2\vee x-1=-2)$, and so on. You will notice that in going from $(1)$ to $(2)$, we had to divide by $y$. This is not possible when $y=0$, so implicitly we have bifurcated into two cases: when $y=0$, and when $y\ne0$. Oftentimes people think hastily; we might have first thought to divide, and then on second thought realized that this isn't always possible and a case where it isn't possible needs to be checked separately. The style and arrangement of mathematical discussion is determined not exclusively by cold logic considerations, but rather is also developed so as to illustrate and mirror natural human thought processes. As with any sort of discussion people have with each other. Just because you can't make sense out of something, does not make something nonsense, by the way. It is, though, a tad too hastily written. If you're writing and have the opportunity to revise, it is generally a good idea to move thoughts around so that they follow along the thought process of someone just being introduced to material, rather than stream-of-consciousness it out. In my opinion, the damage is nonetheless small. The issue you're having is not being able to detect the framework behind these sort of problem-solving tasks, and the framework is very basic which most prepared students are familiar with. As I have said a couple times now, the idea of applying manipulations, either reversible or irreversible, and splitting off into multiple cases based on when certain manipulations are applicable or not, is something that goes all the way back to high school algebra. The reason the author does not explicitly say some things is that they are the sort of things that very widely and very typically go without saying in mathematical talk. The therapy for this is to get into the mindset of problem-solving, not logic. For problem-solving, especially in introductory differential equations (which reads mostly like a large grab-bag of tricks, I think), achieving your goal will involve symbol-pushing, so it is much like chess where you need to move things around according to certain rules in order to achieve one of a number of desired forms. Intermediate steps become present, like "isolate $y$'s" or "reduce the order of derivatives present," or "group like terms" etc. With practice, you can attach the logic to the moves afterwards after you have found the moves you need or want. - A few quick thoughts on your problem: 1. If you're finding that you can't intuitively follow what is going on, you might just not have enough examples in your head. Instead of trying to understand what is being proved in a "technique course" to use your phraseology, just apply the technique several times (as many times as necessary!) and you will probably get an intuitve understanding of what is happening 2. Often, there is much theoretical precision that could be used, but isn't because it is too complicated at the moment. In undergraduate differential equations, there really isn't much technique that can be used compared to algebra, because much of the theorem-proof stuff is complicated analysis. Again, concentrate on doing tons of examples. Differential equations does branch off into many subdisciplines later and some of it is rather nicely theoretical, like microlocal analysis and D-modules 3. Point (2) also holds in a different vein for complex analysis. Some of analysis isn't as "structured" in some sense compared to algebra. However, technique-type courses are more to get an intuitive feel for objects so that you can later apply more structural techniques to them. Sometimes you just have to get your hands dirty so that later when you do learn the theory you'll have a good idea of what to expect. 4. If you find a subject not up to your standards of rigorous precision, that's fine; different people need varying amounts of rigour to keep them comfortable. Personally, I find it tremendously difficult to understand imprecise statements. This is a good opportunity for you: rephrase the statements more precisely. If you can't figure out how, write down something precise that you think might be true and then see if you can prove it. If it looks hard, ask the instructor if you did this part correctly. 5. In case I didn't say it clearly: do more examples. - I think I can relate a little bit, being a physicist who finds some problems presented in physics texts are a bit opaque while pure mathematics tends to be much more precise about statements and definitions. Taking more pure math has given me greater insight into the physics problems I work with. I think one way to help overcome this issue is simply by exposure. Let me try to illuminate what's going on with this wikipedia problem. Admittedly, I'm not well-versed in formal logic, so I can't evaluate what is and is not a "premise." Still, I will try to clarify what's going on. First, let us assume there exists a function $y$ obeying the differential equation (1), and we must construct a means to recover $y(t)$ given $f(t)$. If $y(t) \neq 0$ for all $t$ in the domain of $y$, then equation (2) is equivalent to equation (1). Equation (2) is equivalent to equation (3), for all possible constants $C$. This recovers $y(t)$ whenever $y(t) \neq 0$ for all $t$. (*Note that equation (3) doesn't actually finish recovering $y(t)$. You need to get rid of the logarithm, but this is such a trivial step that the writer clearly didn't even consider it worth getting into.) If $y(t) = 0$ for some $t = t_0$, then equation (2) is not equivalent to equation (1). However, $y(t_0) = 0$ and equation (1) together imply that $dy/dt = 0$. To be honest, I find a lot of the logic here incomplete; they want to jump straight to considering $y(t) = 0$ everywhere without considering $y(t_0) = 0$ for only an isolated point $t_0$. I would argue that a point $t_0$ such that $y(t_0) = 0$ but with $y(t) \neq 0$ in a neighborhood around $t_0$ can be partitioned: you can solve for $y(t)$ to the left and right of $t_0$ by equation (2), and if you know $y=0$ at $t_0$, you therefore know $y(t)$ everywhere in the interval. Only then can you finally consider the case that $y=0$ everywhere, which has a trivial solution and does obey the conditions of (1). The essence of the approach here is to take (1) as given and then consider a set of mutually exclusive and exhaustive cases, each of which admits different avenues toward reconstructing $y(t)$ in terms of $f(t)$. It's key that the cases considered are collectively exhaustive--they must cover all possibilities. One of the things that hung you up here is that "we must check" business. There, the writer had to consider the case $y=0$ everywhere, and he pedantically chose to verify that this case was consistent with the original equation (1). It trivially was, in this case, but it's nevertheless common to consider cases that may or may not be consistent with the original problem--perhaps because it is simpler not to exclude such solutions until a later time. Part of the difficulty here may be that in proofs, you often know the answer you're supposed to arrive at--there is a clear goal that you must achieve, and the focus is on the logical consistency between steps. Here, the focus was on following logical steps to construct a solution for $y(t)$--or, equivalently, to recover it given only that $y(t)$ obeyed equation (1) and that $f(t)$ was known information. Here, part of the general solution technique was to break down possibilities for $y(t)$ into a set of cases, each of which was easier to analyze individually and construct a solution from on its own than when considered as a whole. - I can see your confusion and frustration but this is something that you should work on getting used to and becoming able to translate into your preferred style. Historically, most mathematics was (and in a lot of fields still is) done in an informal style. It was only in the early 20th century that there was any kind of convincing logical foundation for mathematics and there was a lot of amazing (pure and applied) mathematics done before that. This is especially true in calculus: there are a number of operations that are justified here by theorems that aren't stated because it's assumed you know them. You might be frustrated by this but the point is that the operations were true before they were proved to levels of current rigor and abstraction. An intuitive meaning of derivatives as rates of change and belief that the notation works allows you to work efficiently and then you can go back and check everything works precisely. As a pure mathematician you should concentrate more than most on fixed definitions and rigorous theorems, while also realizing that the main use of calculus like this is to solve real world problems. As to the specifics of the problem (and these are only my interpretations, the point of this is to find all solutions to an equation, which doesn't have to be stated as a theorem): 1. We are trying to find all solutions to a differential equation. If you wish you can state this as Theorem: For all continuous $f\colon \mathbb{R}\to\mathbb{R}$ and $F\colon \mathbb{R}\to\mathbb{R}$ with $F'=f$, differentiable $y\colon \mathbb{R}\to\mathbb{R}$ satisfies $$\frac{dy}{dt}+f(t)y=0$$ if and only if $y=Ae^{-F(t)}$ with $A\in \mathbb{R}$. 2. There is a theorem that says if $y\neq 0$ that any solution to (1) is also a solution to (2) with integral signs. This theorem that justifies 'separation of variables' might be in your textbook/lecture notes or you might have to prove it yourself: it's basically the chain rule. Once you have that together with the fact that $y=0$ is a solution then you can see that any solution to (1) must be $0$ or a solution to (2). 3. There's more definitions and propositions that say all the solutions of (2) are of the form (3). Actually there really should be some absolute value signs around the $y$ as you see in example 2 of separation of variables. Solving this modified (3) gives solutions $y=\pm Ae^{-F(t)}$ where $A=e^C$ for any $C\in \mathbb{R}$ so this together with the $y=0$ solution gives the desired theorem. Going through all this has reminded me that converting calculus type informal reasoning to precise mathematics is slightly tricky (my discussion above is still lacking some details I think, corrections welcome) and is a worthwhile exercise. So you should do this as necessary but also try to sometimes understand the material in the way it's presented in a slightly more intuitive way. More random comments: • If you want to state this as a theorem you need to know the answer. It seems easier to just find the answer and then stop writing. • Thinking about algebraic equations might give you some understanding of what people mean by differential equations. Solving something like $\sqrt{x+2}=-x$ would proceed by assuming the equation is true, getting somewhere and then checking if the solutions work. Try to think about this as implications, subsets etc and that might help before getting into calculus also. - The premise being given there is that a separable differential equation must be homogenous and have a certain form, which is given. The rest of the discussion then applies to that kind of differential equation. The nonsense about "we must check" has to do with that differential equations are problems which, like many other kinds of problems, are hard to solve, but easy to verify. In solving differential equations, you can form hypotheses: "such and such a function is a solution". Then you plug in that function into the differential equation and see if it preserves the truth value of that equation. In "tool based math", the tools have underlying proofs, but it's possible to just accept the tools and skip to using them. The big conceptual problem in that variable separation method isn't the division by $y$, but how the differential term $dt$ is being brought over to the other side as if it were an ordinary variable serving as an ordinary denominator. You can try to do same thing using Newton's notation for derivatives rather than those of Liebniz. Then we arrive at the following, and you will see that the Wikipedia's example is skipping a lot of steps: $$y'(t) + f(t)y(t) = 0$$ $$\frac{y'(t)}{y(t)} = -f(t)$$ We can try to integrate both sides. Before we do that, let us make a little digression. The derivative of the natural logarithm is: $$\ln'(x) = 1/x$$ What is the derivative of $ln(y(x))$? For that we need the tool known as the chain rule. $$(f\circ g)'(t) = f'(g(t))g'(t)$$ Applying that to $ln(y(x))$ we get: $$\frac{y'(x)}{y(x)}$$ So you can see where this has been headed. Back to the problem. We want to integrate both sides of: $$\frac{y'(t)}{y(t)} = -f(t)$$ $$\int\frac{y'(t)}{y(t)}dt = \int-f(t)dt$$ Thanks to our little digression, we now know recognize the left side none other than the derivative of $\ln(y(t))$, and so we can continue. You can see that the Wikipedia simply pulled out the solution as a rabbit out of a hat without explaining anything: $$\ln(y(t)) = \int-f(t)dt + C$$ Both indefinite integrals have some constant; the constants can merge into one. The next step is to get $y(t)$ by itself by raising $e$ to both sides: $$y(t) = e^{\int-f(t)dt + C}$$ So you can see, we can add a little more rigor by making it clear at every step that $y$ is actually a function of $t$, and not sweeping its derivative $y'(t)$ under the rug with the Liebniz notation. The goal of solving the differential equation is to determine $y(t)$. This is what is meant by separation of variables. It's actually a separation of functions: getting $f(x)$ and $y(x)$ on opposite sides of the equation, and isolating $y(x)$ as an expression purely based on $f(x)$. - 1 – user50229 Mar 4 at 14:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 99, "mathjax_display_tex": 13, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9697792530059814, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/104244/if-c-bx-separable-then-x-is-a-compact-metrizable-space	# If $C_b(X)$ separable then $X$ is a compact metrizable space I am very familiar with the proof of the following statement: If $X$ is a compact Hausdorff space such that the Banach algebra $C(X)$ is separable, then $X$ is metrizable. Can this be used to prove a more generalized version of this statement with the set $C_b(X)$ of all continuous bounded functions on $X$? Namely, if $X$ is a completely regular Hausdorff space such that $C_{b}(X)$ is separable, then $X$ is a compact metrizable space. - ## 1 Answer $C_b(X)$ is isomorphic and homeomorphic to $C(\beta X)$, where $\beta X$ is the Čech-Stone compactification of $X$, which exists, as $X$ is completely regular and Hausdorff. The isomorphism is of course given by mapping a bounded continuous real-valued function $f$, which has a compact codomain, essentially, to its (unique) Čech-Stone extension to $\beta X$. This also preserves the sup norm, as we can use $[\inf{f},\sup{f}]$ as the codomain of both $f$ and its extension. So it's an isometry between the function spaces. Now, if the former is separable, so is the latter, and then we can apply your theorem to conclude that $\beta X$ is metrizable, but this only happens if $X$ was already compact metrizable (and thus $X = \beta X$). See this question for that. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9431952238082886, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/307015/non-linear-diophantine-equation-in-three-variables	# Non Linear Diophantine Equation in Three Variables Find all positive integer solution to $abc-2=a+b+c$. - ## 2 Answers None of the variables can be greater than $4$, and at least two of them have to be greater than $1$. Also, by looking at the equation modulo 2, you can not have exactly one of them being odd. Finally, this equation is symmetric in its variables, which means I just have to say which three numbers I pick, not which one I assign to which variable. A brute force tactic doesn't take too long from here. First we go with all the variables being odd. There are only two cases conforming to the list of demands in the last paragraph, and that is $(3, 3, 1)$ and $(3, 3, 3)$. The first set does solve the equation, the second one doesn't. Next is two of them being odd. Setting the even number to $4$ makes the left side too large, so we need the even number to be $2$. The only set left is then $(2, 3, 1)$, which does not solve the equation. Lastly, the case of all the digits even. There are four of these sets, depending on how many $2$s and $4$s we use. The sets $(4, 4, 4)$, $(4, 4, 2)$ and $(4, 2, 2)$ don't lead to a solution, but the set $(2, 2, 2)$ does. All in all we have 2 sets of solution values, which can be distributed over the variables in a total of four different ways. Edit It has been made known to me, thanks to Ross Pure, that I was a bit hasty to exclude the case of one variable being equal to $5$. So there is a set $(5, 2, 1)$, which generates another 6 solutions, depending on which number you give to which variable. - I can't comment but in response to Arthur's answer: it's not so that the variables can't be greater than 4; (5,2,1) works. - You're right, I somehow calculated that to yield $8 = 7$ and discarded it without a second thought. – Arthur Feb 18 at 12:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9730771780014038, "perplexity_flag": "head"}
http://nrich.maths.org/1866/note?nomenu=1	## 'Take Three from Five' printed from http://nrich.maths.org/ ### Why do this problem? This problem looks like a number task, possibly revision about multiples, but it becomes a question about establishing why something can never happen, and creating a real proof of this. When it comes, the proof often feels powerful, satisfying and complete, and students leave feeling they have achieved something. ### Possible approach It would be very useful for students to work on What Numbers Can We Make? before attempting this problem. Introduce the problem the way Charlie does in the video, by inviting students to suggest sets of five whole numbers, circle three of them that add up to a multiple of three, and write down the total. Don't say anything - let students work out what is special about the sum of the numbers you select. Suggest that if they know what is going on they may like to choose $5$ numbers that stop you achieving your aim. At some stage check that they all know what is going on. Challenge them to offer five numbers that don't include three that add up to a multiple of $3$. Allow them time to work on the problem in pairs or small groups, and suggest that they write any sets they find up on the board. Students may enjoy spotting errors among the suggestions on the board. Allow negative numbers, as long as they will allow you negative multiples of $3$ (and zero). At some stage there may be mutterings that it's impossible. A possible response might be: "Well if you think it's impossible, there must be a reason. If you can find a reason then we'll be sure." Once they have had sufficient thinking time, bring the class together to share ideas. If it hasn't emerged, share with students Charlie's representation from What Numbers Can We Make? All numbers fall into one of these 3 categories: Type A (multiple of $3$) Type B (of the form $3n+1$) Type C (of the form $3n+2$) We have found that trying to use algebraic expressions as above, is tricky, students often end up with n having two or more values at once. Students are unlikely to know the notation of modular arithmetic, but the crosses notation above is sufficient for the context, and it suggests a geometrical image that students can use in explaining their ideas. "Which combinations of A, B and C give a multiple of three?" "Can you find examples in our list on the board where you gave me one of those combinations?" A few minutes later... "Great, then all you have to do is find a combination of As, Bs and Cs that doesn't include AAA, BBB, CCC or ABC!" Later still... "It's impossible! All the combinations will include AAA, BBB, CCC or ABC!" "OK, but can you prove it? Can you convince me that it's impossible?" ### Possible extension What Numbers Can We Make Now? is a suitable follow-up task. A challenging extension: You can guarantee being able to get a multiple of $2$ when you select $2$ from $3$. You can guarantee being able to get a multiple of $3$ when you select $3$ from $5$. Can you guarantee being able to get a multiple of $4$ when you select $4$ from $7$? ### Possible support Select sets of $3$ numbers. Your sets will always include two numbers that add up to an even number. Why?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9540449380874634, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/71272/height-of-ordered-set	## Height of ordered set ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) In all "constructive" fixed-point theorems for functions on ordered sets that I am aware of, where the fixed point is obtained as the limit of a stationary increasing transfinite sequence, it is always an ordinal of cardinality greater than the cardinality of the set that is used in the proof. While this is of course sufficient, it seems like a real overkill. What you really need is the least ordinal that cannot be orderly embedded in the ordered set. Motivated by the finite case, I would be inclined to call that ordinal the height of the ordered set, even though the latter need not be well founded. Is there any standard terminology or work on this? - ## 1 Answer EDIT SUMMARY: The first version of the answer only talked about linearly ordered sets; prompted by Ricky Demer's inquiry, I modified item 5, and added items 6 and 7. The second paragraph of (6) is the newest addition. I have never seen the concept introduced in the question studied systematically, but it is a very natural concept, and one can say a few sensible things about it. For a poset $P$. let $o(P)$ be the least ordinal that cannot be order-isomorphically embedded into a partially ordered set $P$ (indeed $P$ can be even a quasi-order; i.e., ordered by a relation required only to be reflexive and transitive). 1. (Warm-up). For every ordinal $\alpha$, $o(\alpha) = \alpha + 1$. 2. $o(\Bbb{Q}) = \omega_1$ (the least uncountable ordinal), thanks to a theorem of Cantor, which states that every countable linear order can be embedded inside $\Bbb{Q}$. 3. $o(\Bbb{R}) = \omega_1$. This follows from (2) and the fact that any collection of pairwise disjoint open intervals of the real line is at most countable (by the density of rationals in the real line). 4. For every natural number $n$, let $\Bbb{R}^n$ be the n-fold lexicographically ordered Cartesian power of the real line. Then $o(\Bbb{R}^n)=\omega_1$ for each $n$. This follows from (3) using induction on $n$. A theorem of Harrington and Shelah vastly generalizes this this to : for every analytic linear order $L$, $o(L)\leq\omega_1$. 5. Let $L$ be a linear order of cardinality greater than $2^{\aleph_0}$, then $o(L) > \omega_{1}$, or $o(L^{}) > \omega_{1}$, where $L^{}$ is the reverse of $L$; by the Erdős–Rado theorem. More generally, if $P$ is a poset of cardinality greater than $2^{\kappa}$ for some infinite cardinal $\kappa$, then either $o(P) > \kappa^{+}$, or $o(P^{}) > \kappa^{+}$, or $P$ contains a subset of cardinality $\kappa^+$ of pairwise incomparable elements. 6. Let $A\subset_ B$ be defined as "$A$\$B$ is finite". Then $o(\cal{P}(\omega), \subset_) > \omega_1$, by a diagonal argument plus transfinite induction (a more sophisticated version of this argument produces a "Hausdorff gap"). By a classical theorem of Parovicenko, every Boolean algebra of cardinality at most $\aleph_1$ can be embedded into $(\cal{P}(\omega), \subset_)$. This implies that indeed $o(\cal{P}(\omega), \subset_) \geq \omega_2$. In the presence of the continuum hypothesis $o(\cal{P}(\omega), \subset_) = \omega_2$. 7. A similar result to (6) is true for the set of functions from $\omega$ to itself, ordered by eventual dominance. Hardy (the number theorist) was among the people who noticed that there is a sequence of functions from $\omega$ to itself of order-type $\omega_1$; and he was quite impressed by this discovery. This list is obviously woefully incomplete, but should give a rough idea that many results of (combinatorial) set theory can be couched in terms of $o(P)$. - 1 Why restrict to a linear order? (for 1-4) – Ricky Demer Jul 26 2011 at 2:29 Thanks for the comments Ricky; I modified my answer a bit to make it more general. – Ali Enayat Jul 26 2011 at 5:10 $(\omega_1)^2 \leq o(2^{\omega},\subseteq_)$, by partitioning $\omega$ into a countable well-ordered number of infinite pieces and embedding that many copies of $\omega_1$ in a chain. If CH, then $o(2^{\omega},\subseteq_) \leq \omega_2$ since $\|2^{\omega}\| = \omega_1 < \omega_2$. – Ricky Demer Jul 26 2011 at 7:29 In item 5, there should be a third option, namely that $o$ of the reversed order is big. For example, $P$ could be a long anti-well-order. (Apologies for the awkward formulation, caused by a French keyboard.) – Andreas Blass Jul 26 2011 at 14:26 @Andreas: you are right, thanks, I will fix it. – Ali Enayat Jul 26 2011 at 15:22 show 3 more comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 48, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9263657331466675, "perplexity_flag": "head"}
http://mathoverflow.net/questions/4648?sort=newest	## When to pick a basis? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the diagonal elements, but many mathematicians would never consider such a definition since it presupposes a choice of basis. For someone working on algorithms, however, this might be a very natural perspective. What are the advantages and disadvantages to choosing a specific basis? Are there any situations where the "right" proof requires choosing a basis? (I mean a proof with the most clarity and insight -- this is subjective, of course.) What about the opposite situation, where the right proof never picks a basis? Or is it the case that one can very generally argue that any proof done in one manner can be easily translated to the other setting? Are there examples of proofs where the only known proof relies on choosing a basis? - 5 "For example, one might define the trace of a matrix to be the sum of the diagonal elements, but many mathematicians would never consider such a definition since it presupposes a choice of basis." This <b> is </b> the standard definition of trace. I am not aware of a single mathematician who would <i> never </i> consider such a definition. – Gil Kalai Nov 8 2009 at 17:42 15 "I am not aware of a single mathematician who would <i> never </i> consider such a definition." Except Bourbaki, of course :) There the trace, as a function from endomorphisms of a finite dimensional $k$-vector space $V$ to $k$ is defined as the composition of the canonical isomorphism $End(V) \cong V \otimes V^$ followed by the dual pairing $V \otimes V^ \to k$. You've got to admit that it has its charm :) – José Figueroa-O'Farrill Nov 8 2009 at 17:55 2 Bourbaki may be better than anyone else at hiding bases, but they are surely in there somewhere. For instance, how do you know that $V^$ is non-trivial? – Greg Kuperberg Nov 8 2009 at 17:59 4 It's probably not worth saying, but the issue with the definition given in terms of sum of the diagonal entries is that it's not at all clear that such a definition is independent of basis, whereas the dual pairing, if it exists, is totally canonical. – Qiaochu Yuan Nov 8 2009 at 19:05 4 I don't think pedagogically you should introduce the determinant in order to define the trace. Traces can be defined when determinants aren't. – Qiaochu Yuan Nov 9 2009 at 14:31 show 6 more comments ## 9 Answers One answer to your question is already hinted at in the question. At the level of algorithms, basis-independent vector spaces don't really exist. If you want to compute a linear map $L:V \to W$, then you're not really computing anything unless both $V$ and $W$ have a basis. This is a useful reminder in our area, quantum computation, that has come up in discussion with one of my students. In that context, a quantum algorithm might compute $L$ as a unitary operator between Hilbert spaces $V$ and $W$. But the Hilbert spaces have to be implemented in qubits, which then imply a computational basis. So again, nothing is being computed unless both Hilbert spaces have distinguished orthonormal bases. The reminder is perhaps more useful quantumly than classically, since serious quantum computers don't yet exist. On the other hand, when proving a basis-independent theorem, it is almost never enlightening (for me at least) to choose bases for vector spaces. The reason has to do with data typing: It is better to write formulas in such a way that the two sides of an incorrect equation are unlikely to even be of the same type. In algebra, there is a trend towards using bases as sparingly as possible. For instance, there is widespread use of direct sum decompositions and tensor decompositions as a way to have partial bases. I think that your question about examples of proofs can't have an explicit answer. No basis-independent result needs a basis, and yet all of them do. If you have a reason to break down and choose a basis, it means that the basis-independent formalism is incomplete. On the other hand, anything that is used to build that formalism (like the definition of determinant and trace and the fact that they are basis-independent) needs a basis. There is an exception to the point about algorithms. A symbolic mathematics package can have a category-theoretic layer in which vector spaces don't have bases. In fact, defining objects in categories is a big part of the interest in modern symbolic math packages such as Magma and SAGE. - 3 I really like your "data typing" explanation of how basis avoidance helps. – Andrew Critch Nov 8 2009 at 22:07 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. If we restrict ourselves to the field of linear algebra, my personal point of view, which I do not want to force on anybody, is that one should never use bases, matrices, or coordinates. The main reason is that you lose geometric intuition whenever you introduce a basis, and geometric intuition in linear algebra is extremely important for me, not only in definitions but also in theorems and their proofs. When I learned linear algebra I made sure I understood the geometric meaning of every single definition, theorem, and proof. For example, an element of a vector space is a vector or a 1-dimensional subspace with an oriented metric, an element of the dual vector space is a hyperplane with an oriented metric on its “complement”, i.e., the factor space, an element of the exterior algebra is a formal sum of vector subspaces (dimension equals degree) equipped with oriented metrics, an element of the exterior algebra of the dual space is a formal sum of vector subspaces (codimension equals degree) with an oriented metric on their “complement”, i.e., the factor space, the exterior product of two elements of the exterior algebra is the direct sum of the corresponding spaces (or zero if they have nontrivial intersection) with the obvious choice of an oriented metric, the inner product of an element of the exterior algebra and an element of the dual exterior algebra is the intersection or the sum (depends on the type of the inner product) of the corresponding subspaces with the obvious choice of an oriented metric, Hodge star is a particular case of the previous construction (if you have a subspace with an oriented metric and also an oriented metric on the entire space then you can canonically produce an oriented metric on the “complement”, i.e., the factor space), trace and determinant also have an obvious geometric meaning in this framework etc. etc. etc. All of this is fully rigorous and all theorems and their proofs become trivial once you have a geometric intuition for all definitions, and you don't need any bases, coordinates, or matrices, even when you prove something. Ironically, the best source for geometric intuition in linear algebra for me was Bourbaki's Algebra, which is often blamed for its abstractness. Actually it is the only source known to me that (indirectly) explains the geometric meaning of exterior algebra (please tell me if you know other sources). I badly want to see a sufficiently advanced textbook on linear algebra that at least includes all the notions mentioned above (and many others, of course) and satisfies the following two conditions: (1) It explains the geometric meaning of every single definition, theorem, and proof (or states them in such a way that their geometric meaning is evident); (2) It never uses bases, coordinates, or matrices and does not even define these notions. - I think this isn't a tenable position to take unless you also talk about trace diagrams, which are also independently interesting. – Qiaochu Yuan Nov 10 2009 at 20:44 2 Well, one can use either trace diagrams or the usual algebraic notation, whatever is more convenient. I do not see how this affects the validity of my point. – Dmitri Pavlov Nov 11 2009 at 17:15 3 This answer is great :) – Jan Weidner Apr 22 2010 at 21:36 Have you looked at any of the books on Geometric Algebra/Calculus by Hestenes and his collaborators. They try and do all of analytic geometry and linear algebra using Clifford Algebras to do coordinate free computations. Hestenes even wrote a book on classical mechanics from this perspective. – Justin Hilburn Jan 25 2011 at 3:51 Just to throw an idea out (where it will no doubt sink forlornly) - the first proof I was ever shown that the Fourier transform on L^1(R) has a unique continuous extension to a unitary operator on L^2(R) was done by checking on appropriate eigenfunctions (i.e., a basis was chosen for L^2(R)). None of this gainsays the remarks above about avoiding a choice of basis; I would only say (as I think people already have hinted) that when judiciously chosen, bases can be rather useful. (I also get the impression in the study of classical Banach spaces that, in addition to the general coordinate-free principles of linear functional analysis, you really have to hack around with bases.) - Although the first definition of "finite dimensional" is usually "there is a finite basis", this isn't the only way to characterise finite dimensional vector spaces and often a different way to characterise them can lead to a more elegant statement and proof of the theorem under consideration. 1. A vector space is finite dimensional if it is isomorphic to some Euclidean space. This is quite close to the notion of a basis and it is obvious that choosing such an isomorphism is tantamount to choosing a basis. However, it explains one of the roles of bases as explained in Greg's answer: to make an abstract vector space look like Euclidean space (and thus also make abstract linear transformations look like matrices). 2. There's Todd Trimble's definition in this question which relates finite dimensionality to duality. 3. A definition that doesn't use a "there exists" property (which implies that at some point you might want to make a choice) starts in the category of locally convex topological vector spaces, wherein a LCTVS is finite dimensional if it is a nuclear Banach space. This is particularly relevant to the definition of trace, since a space $V$ is nuclear if every continuous linear map $V \to E$, where $E$ is a Banach space, is trace class. Thus if $V$ is nuclear and Banach, every continuous linear map $V \to V$ must admit a trace. 4. A vector space is finite dimensional if its exterior algebra has finite grading. Moreover, it has dimension $n$ if $\Lambda^n V$ is 1-dimensional. Thus we only need to know what 1-dimensional means for this to work. In so far as defining trace is concerned, if one accepts that there is a way of defining determinants that doesn't involve defining bases (say, by using the top exterior power) then one can equally well define trace by differentiating the determinant: $$\frac{\det(I + tA) - 1}{t} \to \operatorname{Tr} A$$ Basically, choosing a basis is evil and should only be done when no-one is watching you and with proper precautions. More seriously, my answer to the original question "when to choose a basis" is: 1. When you need to do a computation (as Greg says) 2. When you want to convince yourself that a particular result is true before setting about the task of finding an elegant proof thereof. Edit: I've thought of two more reasons to choose a basis: 1. When the question is already evil. 2. To avoid complicated convergence issues in Hilbert spaces: basically (pardon the pun), it's really easy to see when a sequence in which the terms are pairwise orthogonal converges so orthonormal bases (and orthonormal families) allow one to separate out the messy convergence from the elegant geometry. - I agree with Elizabeth's answer and Brian Conrad's philosophy: avoid bases in theorem statements if possible, and use them sparingly for proofs. More generally, when a definition of something says "something exists" (like a finite basis!), then at some point in your theory you'll essentially have to "choose" one of those things in order to complete a proof. The definition of "finite-dimensional" means "a finite basis exists", so there's really no way around it. To illustrate this, we could work with "finite length as a k-module" as an alternative equivalent definition of finite dimensional vector space, but this just means "A finite maximal chain of vector subspaces exists," and what you find is that somewhere early in the foundations you have to "choose" such a chain in order complete a proof. Edit: I'm not suggesting here that there are no equivalent characterizations of finite dimensional vector spaces; rather, I'm claiming that proving some of the properties of finite-dimensional vector spaces will involve the existence of "choices" in some way or another (as a trivial example, the property of having a finite basis). Of course making this claim rigorous and proving it would be a lot of work, but unfortunately I think the same is true for its negation. - 3 I strongly disagree with the last paragraph, especially with the “there's really no way around it” part. There are many ways to define finite-dimensionality without ever mentioning bases. One can say that the canonical embedding V → V* is an isomorphism. Equivalently one can say that a vector space is finite-dimensional if it is fully dualizable (i.e., there are maps V ⊗ V* → k and k → V* ⊗ V with the obvious properties). And one does not need to “choose” anything because all maps are completely canonical. – Dmitri Pavlov Nov 10 2009 at 18:01 Are you sure about that? Although the properties you mention are provably equivalent to having a basis, certainly not every consequence of having a finite basis follows formally from these abstract definitions without choosing a basis (or doing something extremely similar to it). An easy example is the consequence of having a finite basis itself. – Andrew Critch Nov 10 2009 at 20:20 Your last example explicitly mentions bases and sounds a bit like a tautology to me. How you can choose a (finite) basis without choosing a (finite) basis? If one interprets your question as referring to the proof of the fact that the commutative rig of isomorphism classes of finite-dimensional vector spaces is isomorphic to the rig of natural numbers, then this can be done without using bases. – Dmitri Pavlov Nov 11 2009 at 20:34 There is also an interesting fact that the free vector space functor from the category of finite sets with matrices as morphisms to the category of finite-dimensional vector spaces is essentially surjective and fully faithful. However, to construct the inverse functor we must use some form of the axiom of choice. (I think that the existence of the inverse functor is actually equivalent to some weak form of the axiom of choice.) Thus if one interprets your question this way then we can prove that we must make some arbitrary choice (i.e., the choice of a basis for every vector space). – Dmitri Pavlov Nov 11 2009 at 20:40 1 Existence of basis for arbitrary vector space is equivalent to the axiom of choice, the proof can be found in Herrlich's book. I think it might be adapted to this case, but I am not sure about this. One must also be careful to distinguish different variants of the category of matrices. For example, the free vector space functor defined on the category of natural numbers and matrices is essentially surjective and fully faithful, and the existence of its inverse implies the axiom of choice for families of finite sets. I am not sure how to adapt this construction to finite sets and matrices. – Dmitri Pavlov Nov 12 2009 at 14:42 show 1 more comment In my opinion, there is absolutely nothing wrong with picking a basis whenever it exists and using it makes the proof more understandable. I personally would prefer a proof that uses an arbitrarily (or conveniently) chosen basis to a proof that avoids bases at the cost of raising the level of abstraction to the sky. The definition of a finite-dimensional space in the post Reid Barton referred to can send almost any linear algebra student running into the night screaming. We can enjoy the "independent formalism" and even find it "enlightening" but for a huge group of finite-dimensional linear algebra users out there the "basis dependent" considerations are the most understandable ones. - 3 The problem is, most linear algebra students think only in terms of bases. Bases are useful, but can also be a crutch. Stressing basis independent thinking discourages being stuck in one paradigm. – Richard Dore Nov 10 2009 at 19:21 Let K be a field and V a K-vector space of infinite dimension (some infinite cardinal). Then the dual V* of V has dimension [which is much bigger than , and in particular proves that V* is not isomorphic to V]. This is stated by Bourbaki in his Algebra I, Chapters 1-3 , Exercise 3 for Chapter 2 §7, page 400 (the reference is to Springer's English translation), where the result is attributed to Erdös-Kaplansky. In the hints to this exercise, Bourbaki makes heavy use of bases (but what is dimension anyway ?) and this might be relevant to Steve's question. - Brian Conrad has a handout (pdf) in which he talks about tensorial maps. In it he notes that one should construct maps independently of bases, but that in order to prove the properties of such maps it made sense to choose bases or spanning sets. I think this is generally applicable: it seems to me that picking bases should be the last part of your work on a problem, and that it mostly comes in at the level of computation. Bases provide a useful structure to a vector space that enables one to start somewhere, and proofs can be easier to do with them. But if you choose a basis too early on, you have to carry it around for the whole problem, and you might have to show how it transforms. Perhaps you could come up with ideas and proofs using bases, and then edit them to show what's really going on at the level of maps? In class we recently constructed the determinant of a linear transformation $T:V\rightarrow V$ over and $n$-dimensional vector space V, and to do this we defined the exterior power and used the fact that $T$ became multiplication by a scalar in $\wedge^n(V)$. To be sure, we showed that given a basis $v_1, ...v_n$ of $V$, one would make the single basis element $v_1\wedge\ldots\wedge v_n$ of $\wedge^n(V)$, and used this to give the combinatorial formula for determinant. But the properties of determinant are invariant under change of basis, so we didn't prove them using a basis. - It's a good point that you can prove facts in tensor algebra without using bases. I personally think though that most people need bases to work out examples when you introduce tensors, dual space, etc. Hard to build intuition if you don't have a specific example in mind. – Ilya Nikokoshev Nov 8 2009 at 19:55 Yes, that's what I intended to suggest: use bases to work out ideas, then edit the proofs. – Elizabeth S. Q. Goodman Nov 9 2009 at 4:54 One example when you should "choose" a basis for your proofs is when there is an obvious choice of basis, e.g., group algebras and path algebras (paths on a Bratteli diagram give an orthonormal basis for a Hilbert space). The subfactor equivalent to a basis is called a Pimsner-Popa basis. As of yet, there is no way to define the canonical planar algebra associated to a subfactor without choosing a basis (even though the result is independent of the choice). Another example of the "right proof" requiring picking a basis is Michael Burns' proof that the rotation is periodic on the relative commutants of a finite index $II_1$-subfactor. There is a way to show this basis independently (see Planar Algebras I, arXiv:math/9909027, pages 84-85), but Burns' treatment is more elegant. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 28, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9394463300704956, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/tagged/computational-complexity+functions	# Tagged Questions 2answers 34 views ### How can we denote the following function in terms of big-O notation? I have got a function and want to denote it in terms of bigO notation. f(n) = log4n+n(1/3). Is this function O(n)? ( here is the multiplication) Thanks for your help 2answers 32 views ### The growth rate of the functions with respect to each other There are two functions , for example $f(n)=3\sqrt{n}$, and $g(n)=\log n$. Which one dominates, in other words, is $f(n)=O(g(n))$ or $f(n)= \Omega(g(n))$? Thank you. 2answers 45 views ### For what $f(n)$ does $O(f(n) \log n)=O(\log\log n)$? $k=f(n)$. Given $O(k \log_2 n)$, what function $f$ of $n$ would be needed for it to equal $O(\log_2 \log_2 n)$? (where $k \in n \in \mathbb{Z}^+$) 2answers 96 views ### Möbius function help Given some large random integer k, how much longer would it take to determine the primality of k, then to calculate mobius(k), and how much longer would it take to factor k, then to calculate ... 0answers 10 views ### On the computational complexity of plugging in numbers into general expressions to obtain special ones There are many expressions, which can be considered straight generalizations of others. I'm motivated by values of integral expressions specifically, for example there is \int_0^\infty e^{-a ... 3answers 43 views ### Computational complexity proof I would like to know how to prove the following: $2^n \in O(n!)$ I know that I have to show that for a constant C, we have $2^n \leq C*n!$ Right?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9301978945732117, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/257857/divide-40-to-4-parts-such-that-every-number-from-1-40-can-be-realized?answertab=oldest	# Divide $40$ to $4$ parts such that every number from $1-40$ can be realized How can you divide $40$ to $4$ parts such that every number from $1-40$ can be realized just by adding or subtracting those $4$ parts? - 2 – Marc van Leeuwen Dec 13 '12 at 12:41 ## 2 Answers All integers from 1 to 40 can be expressed by adding or subtracting 1,3,9 and 27 in such a way that each number is used at most once, and it should be either added or subtracted. - You mean, "each number is used at most once". – TonyK Dec 13 '12 at 13:25 Oops. That's true. – Aneesh Karthik C Dec 29 '12 at 7:04 My answer is the same as Aneesh, but I'll try to give some informal intuition/justification of the answer. Think of it this way: What numbers would you have needed if you could only add them atmost once? So, you could say that the coefficients of each of these numbers would be either 0 or 1. This translates to the binary number system, and you would require 1, 2, 4, 8, 16, 32 to generate all numbers from 1 to 40 (all powers of 2 less than 40). Now look at your question. Going on similar lines, the appropriate coefficients would be -1, 0 and 1. You can treat it as a base-3 (ternary) number system, and you would need all the powers of 3 (less than 40) to generate all numbers from 1-40, namely: 1, 3, 9 and 27. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9403878450393677, "perplexity_flag": "head"}
http://mathhelpforum.com/advanced-algebra/111983-need-help-solving-equations-matrix.html	# Thread: 1. ## Need help solving equations from this matrix I have a matrix 1 0 5 0 0 0 1 1 5 0 0 0 0 0 1 Which gives the three equations: X1 + 5X3 = 0 X2 + X3 + 5X4 = 0 4X5 = 0 Now in my notes it says that you can solve it to get: X1 = 2X3 + 0X4 X2 = 6X3 + 2X4 X3 = 1X3 + 0X4 X4 = 0X3 + 1X4 X5 = 0X3 + 0X4 Could someone help explain to me how you go from the first equations to the second, for the life of me i can't figure it out 2. I've been looking at this for a while myself..! I am no genius, but are you sure you are looking on the right notes? How can $x_1 + 5x_3 = 0$ go to $x_1 = 2x_3$, unless $x_3 = 0$ I guess? 3. Yeah i just double checked the notes, that's exactly what it says. And there not my notes either there the lecturers. Can't understand what's happening and he dosen't explain it anywhere, it's to do with finding the complete solution to this system of linear equations. Where these two are matricies. 4 2 1 3 6 ......4 0 3 3 1 0.x.=..6 3 2 4 1 5 ......2 over $Z7$ Firstly he applies GJ to the whole lot in one matrix to get: 1 0 5 0 0 3 0 1 1 5 0 2 0 0 0 0 1 2 Then he splits the matrix into to and gets: ......3 ......2 G....0 ......0 ......2 1 0 5 0 0 0 1 1 5 0....=....0 0 0 0 0 1 Then he solves to get the equations listed above 4. Originally Posted by BigBoss22 I have a matrix 1 0 5 0 0 0 1 1 5 0 0 0 0 0 1 Which gives the three equations: X1 + 5X3 = 0 X2 + X3 + 5X4 = 0 4X5 = 0 Now in my notes it says that you can solve it to get: X1 = 2X3 + 0X4 From the first equation above, X1= -5X3, not 2X3 X2 = 6X3 + 2X4 and X2= -X3- 5X4, not 6X3+ 2X4 X3 = 1X3 + 0X4 X4 = 0X3 + 1X4 X5 = 0X3 + 0X4 The last equation in the set clearly says that X5= 0. Could someone help explain to me how you go from the first equations to the second, for the life of me i can't figure it out #### Search Tags View Tag Cloud Copyright © 2005-2013 Math Help Forum. All rights reserved.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9127296805381775, "perplexity_flag": "middle"}
http://mathhelpforum.com/advanced-applied-math/4154-tennis-ball-example-coefficient-restitution.html	# Thread: 1. ## tennis ball example/coefficient of restitution a tennis ball is hit from a height h above the ground with a speed V and at an angle alpha to the horizontal. it hits a wall at a horizontal distance d away . Air resistance is negligible i was told to find expressions for the height and velocity of the ball for which i did shown below Vx = Vocos(alpha) Where : Vo is the initial velocity alpha is the launch angle Vx is the x component of velocity Vy = Voysin(alpha) - gt From the pythagorean theorem V = sqrt(Vx^2+Vy^2) X = Vocos(alpha)t Y = Vosin(alpha)t - 1/2gt^2 Assuming that i started at (x,y) = (0,0) i will then end upat the point (D,h) that was the first part of the quesiton...but the part im stuck on is ...auusuming that the coefficient of restitution between the tennis ball and the wall is e, i need to write down an expression for te the velocity of the ball immediately after impact......is there anyone that can help ???thankz 2. Originally Posted by dopi a tennis ball is hit from a height h above the ground with a speed V and at an angle alpha to the horizontal. it hits a wall at a horizontal distance d away . Air resistance is negligible i was told to find expressions for the height and velocity of the ball for which i did shown below Vx = Vocos(alpha) Where : Vo is the initial velocity alpha is the launch angle Vx is the x component of velocity Vy = Voysin(alpha) - gt From the pythagorean theorem V = sqrt(Vx^2+Vy^2) X = Vocos(alpha)t Y = Vosin(alpha)t - 1/2gt^2 Assuming that i started at (x,y) = (0,0) i will then end upat the point (D,h) that was the first part of the quesiton...but the part im stuck on is ...auusuming that the coefficient of restitution between the tennis ball and the wall is e, i need to write down an expression for te the velocity of the ball immediately after impact......is there anyone that can help ???thankz I think it will end up at (D,-h). Let $t_1$ be the total time of flight. You can use your equation:Y = Vosin(alpha)t - 1/2gt^2 If t= $t_1$, then Y=-h. Rest you can do yourself. Keep Smiling Malay 3. ## expression/coefficient of restittuon Originally Posted by malaygoel I think it will end up at (D,-h). Let $t_1$ be the total time of flight. You can use your equation:Y = Vosin(alpha)t - 1/2gt^2 If t= $t_1$, then Y=-h. Rest you can do yourself. Keep Smiling Malay i dont understand how that will help me to find an expression for the velocity of the ball immediately after the impact, when the coefficient of restitution between the ball and the wass is e.???? thankz 4. Originally Posted by dopi a tennis ball is hit from a height h above the ground with a speed V and at an angle alpha to the horizontal. it hits a wall at a horizontal distance d away . Air resistance is negligible i was told to find expressions for the height and velocity of the ball for which i did shown below Vx = Vocos(alpha) Where : Vo is the initial velocity alpha is the launch angle Vx is the x component of velocity X = Vocos(alpha)t You have done good work, I will use two of your equations(see quote). To find the velocity after impact, we need to know two things: coefficient of restitution(which we know) velocity before impact(which is our focus now) Velocity before impact can be divided into two components: horizontal component(which you know $V_ocos\alpha$) vertical component--- for it you need to know the total time of flight[t1](since it depends on t) You could find t1 using the equation: X = Vocos(alpha)*t you get $t_1=\frac{D}{V_0cos\alpha}$ I hope you have got your answer. Feel free to say. Keep Smiling Malay 5. Originally Posted by dopi i dont understand how that will help me to find an expression for the velocity of the ball immediately after the impact, when the coefficient of restitution between the ball and the wass is e.???? thankz The definition for the coefficient of restitution is: $KE_f = eKE_0$ So if you know the KE before collision (KE0) you can find the KE after collision (KEf). This also works for multibody collisions where the KE's in the formula are the total KE for the system. By the way, you say you are looking for the velocity after collision. That isn't correct. You can only get the speed after collision using this equation. What's going to happen to the velocity is anyone's guess because we don't know what effect the collision will have on the individual components of velocity. (The only example I can think of where you CAN get the final velocity is if the collision is "head-on." Then the velocity after impact will be directly opposite the velocity before impact.) -Dan 6. Originally Posted by topsquark The definition for the coefficient of restitution is: $KE_f = eKE_0$ Not according to my sources, which seem to think it is the ratio of the after to before velocity component normal surface. RonL 7. Originally Posted by CaptainBlack Not according to my sources, which seem to think it is the ratio of the after to before velocity component normal surface. RonL (sigh) Different books, different definitions. Gotta love it. -Dan 8. Heelo Topsquark! You said: $KE_f=eKE_o$ this implies that e is the ratio of squares of final and initial velocities. Do you want to say that? Keep Smiling Malay 9. Originally Posted by malaygoel Heelo Topsquark! You said: $KE_f=eKE_o$ this implies that e is the ratio of squares of final and initial velocities. Do you want to say that? Keep Smiling Malay Well, that's how I learned it anyway, I think. (Sigh) I went back to the book I thought I had gotten it from (Serway's intro Physics book) and couldn't even find it! Seeing as I can't find it in my Graduate Mechanics book either, I obviously learned it from some out of book notes. I could simply be remembering it wrong. -Dan 10. Originally Posted by CaptainBlack Not according to my sources, which seem to think it is the ratio of the after to before velocity component normal surface. RonL I propose we do an experiment. Let us throw a ball gently on the patio so it bounces and has forward motion. If Topsquark is right the ball will stop bouncing with no residual velocity along the patio. If I am right it will continue rolling along the patio with approximately the same speed as the original horizontal component of the velocity (some loss to friction and conversion of translational to rotational energy), yes? I've done this experiment, and so know the answer RonL PS my spell checker wants to change TopSquark to pipsqeak! 11. Originally Posted by CaptainBlack I propose we do an experiment. Let us throw a ball gently on the patio so it bounces and has forward motion. If Topsquark is right the ball will stop bouncing with no residual velocity along the patio. If I am right it will continue rolling along the patio with approximately the same speed as the original horizontal component of the velocity (some loss to friction and conversion of translational to rotational energy), yes? I've done this experiment, and so know the answer RonL PS my spell checker wants to change TopSquark to pipsqeak! Good point. ("pipsqueak" Shakes his head in despair!) -Dan 12. Originally Posted by CaptainBlack I've done this experiment, and so know the answer I haven't performed the experiment, but let me know the answer. Malay 13. Originally Posted by malaygoel I haven't performed the experiment, but let me know the answer. Malay The point is, using my definition, the ball would eventually stop even over a frictionless surface. (e is always less than 1 in reality) If the ball had a horizontal component of velocity Netwon's 1st says it should always be in motion. A contradiction, so my definition can't be correct. -Dan 14. Originally Posted by topsquark The point is, using my definition, the ball would eventually stop even over a frictionless surface. (e is always less than 1 in reality) If the ball had a horizontal component of velocity Netwon's 1st says it should always be in motion. A contradiction, so my definition can't be correct. -Dan It means that coefficient of restitution gives the ratios of final and initial normal components of velocities. Right??? Malay 15. Originally Posted by malaygoel It means that coefficient of restitution gives the ratios of final and initial normal components of velocities. Right??? Malay Yes. According to CaptainBlack (and I no longer have any reason to doubt his definiton) the coefficient of restitution is a ratio between the final and initial "vertical" components of the velocity. (Vertical referring to a normal to the surface of impact.) -Dan	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9278186559677124, "perplexity_flag": "middle"}
http://stats.stackexchange.com/questions/25386/power-of-chi-square-test-for-goodness-of-fit-as-function-of-sample-size	Power of chi-square test for goodness-of-fit as function of sample size I would like to calculate the power of a Chi-Square test for goodness-of-fit as a function of sample size for a specified alpha-value (say 0.01). Specifically, I am referring to power as the probability that the test will correctly reject the null hypothesis. In most places I look, I can find only vague references that this can be done and/or am referred to software. It seems to me that this should not be difficult. The specific question I would like to be able to answer (without using some sort of opaque software) follows: Given an $\alpha$ value, for what sample size, $n$, can I expect a particular power, $\beta$? Even better, how might I be able to calculate an ROC? - 1 Answer Does R fall into your idea of opaque software? If you are interested in this sort of calculation I would strongly recommend you use a stats package of some sort - and probably R in particular. In R, package pwr provides the function pwr.chisq.test which answers your question for you. It isn't quite as simple as saying "for each sample size, what is the power of my test?" because as well as sample size, there is the question of the size of the effect in the underlying population you are inferring to. eg if there is a massive effect, then even a very small sample has a high power. As the effect gets smaller, you need a bigger sample size for the same power. The documentation for pwr.chisq.test refers to Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum. Also, a quick google search comes up with this reference (lecture 25), which shows that under the alternative hypothesis the test statistic has (asymptotically) a non-central Chi square distribution and provides a way to estimate the non-centrality parameter for a given alternative hypothesis. - Ideally, what I would like is a good reference. I do not want to use software without a very solid fundamental understanding of what I'm doing. – cyrus1.618 Mar 29 '12 at 16:51 I understand that effect size plays into it as well. My issue is that I can't find a satisfactory explanation for this calculation that does not at some point simply fall back and refer to software. It seems to me that this is something that should not be too difficult. I suppose this begs a further question: why is it so difficult to find a good explanation? Is it because it is considered too elementary? Thanks! – cyrus1.618 Mar 29 '12 at 16:54 ok, I've added some references – Peter Ellis Mar 29 '12 at 18:41 Thanks for the references. The url reference is tangentially related (my distributions are not multinomial) - but it does illustrate an approach. Upon first glance in Cohen - it appears as if as soon as the text gets to power analysis of the chi-square test, it immediately jumps to tables without explanation of the derivation of the tables. I'll have to give a more thorough read through. Thanks for the references! – cyrus1.618 Mar 29 '12 at 23:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9450452923774719, "perplexity_flag": "head"}
http://mathoverflow.net/questions/72298/polynomial-vector-fields-on-the-3-sphere	## Polynomial Vector Fields on the 3-Sphere ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$. EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of polynomial vector fields on $S^3$ look like? EDIT: my question is really the following. The Lie algebra of the diffeomorphism group of $S^1$ is the Witt algebra. What is the corresponding Lie algebra for $S^3$? (1) What is the diffeomorphism group of the 3-sphere? My reason for asking is that I want to know if there is an analogue of the Witt (=centerless Virasoro) algebra in three dimensions. I am aware of the $W_n$ series in Cartan's classification, but this is not the generalization I am looking for. (1.0) Alternatively/equivalently, I would like to know how to describe "regular" (smooth?) sections of the tangent bundle on $S^3$ - it seems like it might be helpful (to me, at least) to think of the fibers as copies of $sl(2)(\cong su(2))$. It's been a while since I looked at principal fiber bundles, but this definitely reminds me of one. (0) Currently I'm trying to think of all of this stuff in terms of [unit] quaternions. This seems promising to me for a number of reasons, so if you can tell me anything about the above in such terms or point me towards papers/preprints/steles about the above in quaternionic language, that would be fantabulous. (Obligatory "sorry if this is vague to the point of madness" and "sorry if this has been answered previously; I did my best to check the related questions.") - 2 The tangent bundle of $S^3$ is trivial (view $S^3$ as the group of unit quaternions, and use the group law to trivialize the bundle) so the smooth sections are $C^\infty$ maps $S^3\rightarrow\mathbb{R}^3$. – Alain Valette Aug 7 2011 at 19:58 1 I suspected [something like] this. That's why I'm interested in $C^{\infty}$ maps $S^3\rightarrow S^3(\cong\mathbb{R}^3\cup\{\infty\})$. Thanks for the bundle clarification. Very clear. – Daniel Fleisher Aug 7 2011 at 20:08 1 In fact, $S^3$ is a Lie group. One can find three linearly-independent and nonvanishing vector fields, trivializing $TS^3$ and forming a basis for the corresponding Lie algebra, see en.wikipedia.org/wiki/3-sphere – Francesco Polizzi Aug 7 2011 at 20:10 3 I think that I now understand what your question really is. You have the Lie algebra of polynomial vector fields on $S^3$, which I defined in my answer. You'd like to know if there is a way of describing it in a "concrete" way (i.e., in the way the Witt algebra is usually defined: by formulas). So your question could be formulated as: "could someone provide a nice basis of the Lie algebra of polynomial vector fields on $S^3$ in which the Lie bracket can be described by a nice closed formula?". – André Henriques Aug 7 2011 at 21:21 1 Yes, that's pretty much what I want. The fact that I haven't found anything resembling this makes me somewhat pessimistic. – Daniel Fleisher Aug 7 2011 at 22:08 show 3 more comments ## 3 Answers You mention the Witt algebra, which is a dense Lie algebra inside the Lie algebra of smooth vector fields on $S^1$... There is a similar dense Lie algebra inside the Lie algebra of smooth vector fields on $S^3$: the Lie algebra of polynomial vector fields. Since it might not be clear what one means by "polynomial vector fields", let me be precise: these are algebraic vector fields on the complexification $S^3_\mathbb C=SU(2)_\mathbb C=SL(2,\mathbb C)$, subject to the reality condition that says that their restriction to $S^3$ is everywhere tangent to $S^3$. But you also mention the diffeomorphism group of $S^3$. I am not aware of any dense subgroup of $Diff(S^3):=Diff_{smooth}(S^3)$ that would be to $Vect_{polynomial}(S^3)$ the same as $Diff_{smooth}(S^3)$ is to $Vect_{smooth}(S^3)$. - I feel rather stupid - I never meant to say anything about diffeomorphisms. I am interested in the Lie algebra of polynomial vector fields on $S^3$. Editing the post again... (sorry) How does one work with the algebraic vector fields you mentioned - that is, how does one impose the reality condition to get a nice basis for the Lie algebra? – Daniel Fleisher Aug 7 2011 at 20:51 1 You may forget the reality condition that I mentioned in my answer: the Witt algebra is the complexified Lie agebra of vector fields on $S^1$. You seem to care about the complexified Lie algebra of vector fields on $S^3$. – André Henriques Aug 7 2011 at 21:23 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Complementing André's answer, here's another possible definition of polynomial vector fields on $S^3$. You think of $S^3$ as the unit sphere in $\mathbb{R}^4$ and consider polynomial vector fields on $\mathbb{R}^4$ which are tangent to the sphere; that is, which annihilate the function $\sum x_i^2$. One thing to point out, which may or may not be relevant to the applications you have in mind but which I mention since you did mention the Virasoro algebra, is that the structure of the diffeomorphism algebras (or algebras of polynomial vector fields) in dimension greater than 1 is very different than in dimension 1. For example, you don't have a nice decomposition such as the one $$\mathfrak{Vir} = \mathfrak{Vir}^- \oplus \mathfrak{Vir}^0 \oplus \mathfrak{Vir}^+$$ for the Virasoro algebra, and this in turn hinders the construction of positive energy representations. I am aware on some work on this topic in the mathematical physics literature; e.g., Fock space representations of the algebra of diffeomorphisms of the $N$-torus, F Figueirido and E Ramos and also papers by TA Larsson. - 1 @ José: The fact that our two notions of "polynomial vector fields on $S^3$" agree is actually non-trivial. But this is indeed the case. The reason is the following: the variety $S^3_{\mathbb C}=\{(x,y,z,u)\in\mathbb C^4\|x^2+y^2+z^2+u^2=1\}$ is isomorphic to the Lie group $SL(2,\mathbb C)=\{(a,b,c,d)\in\mathbb C^4\|ad-bc=1\}$. – André Henriques Aug 8 2011 at 19:39 The "Smale Conjecture" (a theorem of Hatcher http://www.jstor.org/pss/2007035) says that the natural inclusion $\operatorname{O}(4)\hookrightarrow\operatorname{Diff}(\mathbb S^3)$ is a homotopy equivalence. Perhaps this is along the lines of what you are looking for. - 2 I think I've asked my question[s] terribly. I'm going to edit the initial post. – Daniel Fleisher Aug 7 2011 at 20:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 40, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9464391469955444, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/292719/question-about-normal-distribution?answertab=votes	# question about normal distribution I'm not being able to identify the mathematical pattern used answer this question: Suppose that the length of a speficic kind of snake may be modelled by a normal distribution with mean 50.8 (cm) and standard deviation 0.8. Calculate the length (cm) that is exceeded by only 7% of this specific kind of snake. What I though: • Plot the C.D.F and find the 93th percentile (maybe using z-score)?. is that correct? - ## 1 Answer You're almost there. Find the $\text{z-score}$ of where the CDF is 0.93. In this case, the value is $1.48$. Knowing the Z-score to be $1.48$, perform standardization. You will get, letting $X$ to be the length of a randomly chosen snake, we attempt to find $$P(X>x) = 0.7$$ Which turns out $$1.48 = \frac {x - \mu}{\sigma}$$ so $$x = 1.48(0.8) + {50.8} = 51.98\text {cm}$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9119437336921692, "perplexity_flag": "middle"}
http://mathhelpforum.com/advanced-statistics/192697-find-expected-number-print.html	# find the expected number Printable View • November 25th 2011, 08:16 PM wopashui find the expected number Assume that Y denotes the number of bacteria per cubic centimetre in a particular liquid and that Y has a Poisson distribution with parameter $\lambda$ . Further assume that $\lambda$ varies from location to location according to a gamma distribution with parameters $\alpha$ and $\beta$ , where $\alpha$ is a positive integer. If we randomly select a location, what is the (A) expected number of bacteria per cubic centimetre? (B) standard deviation of the number of bacteria per cubic centimetre? All times are GMT -8. The time now is 03:54 PM.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.815067708492279, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/76063?sort=votes	## Does a bounded real function have an analytic continuation [closed] ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where • $f$ is real-analytic on the open interval $(0,1)$ • $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-C\leq f(x)\leq C$ for $x\in[0,1]$). Is it true that there is a real-analytic continuation of $f$ to the interval $[-\epsilon, 1+\epsilon]$ for some small positive $\epsilon$? If not, what conditions can be added to make it true? Suggestions for books (or other references) where I could have learned to answer this myself would also be appreciated. - 3 I suggest to reask this question on math.stackexchange.com a similar site but with a broader scope. Here I vote to close. – quid Sep 21 2011 at 15:03 1 While others have provided quite smooth counterexamples, I note that your conditions do not even force $f$ to be continuous at the end-points. – Emil Jeřábek Sep 21 2011 at 15:33 @quid I wasn't aware of the difference between math.stackexchange and mathoverflow. Now that I've looked that up, I agree, this question would have been better on math.stackexchange. – Essex Sep 21 2011 at 16:35 Essex, thanks for the response. No problem, this is a frequent phenomenon. – quid Sep 21 2011 at 16:39 ## 2 Answers $f(x) = \sqrt{1-x^2}$ is real analytic on $(-1,1)$, bounded and continuous on $[-1,1]$, but of course not even one-sided differentiable at the endpoints. But then Igor's example is one-sided differentiable of all orders at the endpoint $0$, but still not real analytic in any neighborhood of $0$. - ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. What about $e^{-1/x^2}?$ - @quid: I will probably still be cryptic, but you do agree this answers the question as stated, no? – Igor Rivin Sep 21 2011 at 16:06 @quid: I agree that this is not a research question, but the police work gets really tedious, especially when it takes thirty seconds to answer the question. – Igor Rivin Sep 21 2011 at 16:36 @quid: I probably voted to close about 10 questions today. – Igor Rivin Sep 21 2011 at 16:36 3 This is perhaps more appropriate for a meta thread but: 1. It is not just a question of the time it takes to answer it, it is also a question of teaching people to not expect to receive answers to inappropriate questions. 2. If one decides to answer it anyway then I think this kind of answer is quite appropriate. If it really is after all a research level question then this is a research level answer, the OP should be able to fill in the details. – Torsten Ekedahl Sep 21 2011 at 19:39 2 @IR: ok, let us agree to disagree. @TE: it's not only that the answer is terse, it is imo in a strict sense incomplete (the other one too, btw). How does it address (on whatever level) the second question (for conditions)? Except in such a cryptic way that I'd say 'the obvious ones' is hardly less cryptic. If one wishes to be brief, that's fine; but one could have been almost as brief while being complete. I see no good reason for not doing this. – quid Sep 21 2011 at 20:32 show 2 more comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9575343132019043, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/60451/classifications-of-finite-simple-objects/60460	## Classifications of finite simple objects ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the informal sense of the term, but also answers that take into account a more sophisticated viewpoint (as in this mo question) are welcome, as well as answers that consider reasonable weaker notions of classification (as e.g. this mo question, that asks about a classification of finite simple groups up to finitely many exceptions). The (apperarently incomplete) case of finite commutative rings has already been discussed here. The finite p-groups have been considered here. Also answers/remarks involving the classification of "finite simple objects" of some category (or higher category) are considered in topic (provided that a reasonable definition of "finite" and "simple" is suggested in that context). - Did you mean finite or are you also interested in finite dimensional objects? Some of the answers are over non-finite fields. – Yiftach Barnea Apr 3 2011 at 18:05 1 I think you might enjoy reading en.wikipedia.org/wiki/ADE_classification – Igor Pak Apr 3 2011 at 20:17 ## 7 Answers The classification that has been an inspiration for over a century is the Cartan-Killing classification of finite-dimensional simple complex Lie algebras. This precedes Wedderburn theory. This is surely (?) the most significant classification. - ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. The classification isn't known (to me anyway), but there is an interesting notion of 'simplicity' for finite graphs: A graph homomorphism is a map from vertices to vertices such that adjacent vertices remain adjacent. Two graphs are homomorphism-equivalent if there are homomorphisms in both directions. A graph is said to be a 'core' if all endomorphisms are automorphisms. Every finite graph $G$ maps onto a core, and this core is unique up to isomorphism; moreover, the core can be obtained as an induced subgraph that is the image of an idempotent endomorphism of $G$. So some problems in finite graph theory reduce to problems about cores in the same way that some problems in finite group theory reduce to problems about simple groups. So what are the core graphs? It is easy to see, for instance, that all complete graphs are cores, and that the only bipartite cores are the complete graphs on less than 3 vertices. I know about this from Peter Cameron, who has done a significant amount of work on the subject. - For any field $K$, finite-dimensional simple $K$-algebras (simple in the sense of having no proper non-zero two sided ideals) are famously classified by a theorem of Wedderburn. More generally, simple Artinian rings are classified by the Artin-Wedderburn theorem. Some say that the latter was the beginning of ring theory as a grown up discipline. - Central simple algebras over local and global fields are classified up to Morita equivalence by class field theory. - Somewhat related to Igor Pak's comment is the classification of the finite irreducible Coxeter groups. Of course they are not "simple" as groups, but the irreducibility seems the natural replacement for simplicity; here "irreducible" means that the Coxeter diagram is connected, or equivalently, that the Coxeter system does not split as the direct product of two Coxeter systems. The outcome is the famous list $A_n$, $B_n = C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$, $H_4$, $I_{(n)}$, where the last three items are maybe less well known people only familiar with Lie groups and Lie algebras and/or algebraic groups since they don't survive there. See also http://en.wikipedia.org/wiki/Coxeter_group. - Kurokawa's Zeta functions of categories contains the following definitions. Definition 1: In a category $C$ with a zero object, a simple object is an object $X$ such that, for every object $Y \in C$, $\text{Hom}(X, Y)$ consists only of monomorphisms and zero-morphisms. (I don't know enough to say whether this is equivalent to the nLab definition.) Definition 2: A non-zero object of $C$ is finite if $\text{Hom}(A, A)$ is finite. So there are some easy examples: if $R$ is a commutative ring, then the finite simple objects of $R\text{-Mod}$ are precisely the simple modules $R/m$ where $m$ is a maximal ideal with finite residue field. In particular if $R = \mathbb{Z}$ then the finite simple objects are the modules $\mathbb{Z}/p\mathbb{Z}$. - This more of a generalization than a direct answer, but it may prove interesting and/or useful to the original poster. Universal Algebra has the student look at the lattice of congruences of various algebraic structures. Simple algebras are then algebras with a lattice of two congruences, i.e. no nontrivial congruences, which for groups corresponds to no nontrivial normal subgroups. A related concept is that of subdirectly irreducible algebra. Here the congruence lattice has a unique nontrivial smallest congruence, that is a congruence which is contained in any other congruence on the algebra (except the trivial one induced by an isomorphism). Any simple algebra is subdirectly irreducible. The utility of the latter concept is that any algebra has a representation as a subdirect product (subalgebra of a direct product) of subdirectly irreducible algebras. So when one looks at classes of algebras (of a single similarity type) which are closed under taking direct products and subalgebras (and often isomorphic images of such), one finds the subdirectly irreducible algebras as natural building blocks to form the class. I recall that semilattices and Boolean algebras had nice classifications of finite subdirectly irreducible algebras. I am confident the general algebra literature contains more.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 27, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9305500388145447, "perplexity_flag": "head"}
http://www.msri.org/seminars/19946	# Mathematical Sciences Research Institute Home » Noncommutative Clusters (NAGRT) # Seminar Noncommutative Clusters (NAGRT) May 22, 2013 (03:30pm PDT - 04:30pm PDT) Parent Program: Noncommutative Algebraic Geometry and Representation Theory MSRI: Simons Auditorium Speaker(s) Arkady Berenstein (University of Oregon) Description No Description Abstract/Media Cluster algebras were introduced by Fomin and Zelevinsky in 2001 and have become an important tool in representation theory, higher category theory, and algebraic/Poisson geometry. The goal of my talk (based on a joint paper with V. Retakh) is to introduce totally noncommutative clusters and their mutations, which can be viewed as generalizations of both classical" and quantum cluster structures. Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster algebra, and expect a Noncommutative Laurent Phenomenon to hold in the most of algebras A(X). Our main examples of "cluster groups" G include principal noncommutative tori which we define for any initial exchange matrix B and noncommutative triangulated groups which we define for all oriented surfaces. No Notes/Supplements Uploaded No Videos Uploaded Video No Video Uploaded • Seminar Home	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.874021053314209, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/286473/proving-holders-inequality	# Proving Hölder's Inequality Let $f,g,\alpha:[a,b]\rightarrow \mathbb{R}$ with $\alpha$ increasing and $f,g \in \mathscr{R}(\alpha)$, and $p,q>0$ with $\frac{1}{p}+\frac{1}{q}=1$. Prove that $$\left\|\int_a^b f(x)g(x)d\alpha\right\|\leq \left(\int_a^b \left\|f(x)\right\|^p d\alpha \right)^{1/p} \left(\int_a^b \left\|g(x)\right\|^q d\alpha \right)^{1/q}$$ I am using Young's inequality, which states that for $a,b>0$, $uv\leq \frac{1}{p}u^{p}+\frac{1}{q}v^{q}$. This gets me as far as showing that $$\left\|\int_a^b f(x)g(x)d\alpha\right\|\leq \int\left( \frac {1}{p}\|f(x)\|^p +\frac{1}{q}\|g(x)\|^q\right)d\alpha$$ But here I'm stuck. I'm vaguely thinking that I could use the fact that $\frac {1}{p}\|f(x)\|^p +\frac{1}{q}\|g(x)\|^q$ is a convex combination and so if I do some Jensen's inequality type thing, but I can't figure out a way to make it work out. - – TenaliRaman Jan 25 at 7:02 ## 1 Answer Suppose $\displaystyle\int_a^b \left\|f(x)\right\|^p d\alpha\neq 0$ and $\displaystyle\int_a^b \left\|g(x)\right\|^q d\alpha\neq 0$. Otherwise, if $\displaystyle\int_a^b \left\|f(x)\right\|^p d\alpha=0$, then $f\equiv 0$ a.e. and the Holder's inequality is trivial in this case. Now applying Young's inequality with $u=\displaystyle\frac{\|f(x)\|}{(\int_a^b \left\|f(x)\right\|^p d\alpha)^{\frac{1}{p}}}$ and $v=\displaystyle\frac{\|g(x)\|}{(\int_a^b \left\|g(x)\right\|^q d\alpha)^{\frac{1}{q}}}$, we have $$\frac{\|f(x)\|}{(\int_a^b \left\|f(x)\right\|^p d\alpha)^{\frac{1}{p}}}\cdot\frac{\|g(x)\|}{(\int_a^b \left\|g(x)\right\|^q d\alpha)^{\frac{1}{q}}}\leq\frac{1}{p}\frac{\|f(x)\|^p}{\int_a^b \left\|f(x)\right\|^p d\alpha}+\frac{1}{q}\frac{\|g(x)\|^q}{\int_a^b \left\|g(x)\right\|^q d\alpha}.$$ Integrating it from $a$ to $b$ with respect to $d\alpha$, we obtain $$\frac{\int_a^b\|f(x)\|\|g(x)\|d\alpha}{(\int_a^b \left\|f(x)\right\|^p d\alpha)^{\frac{1}{p}}(\int_a^b \left\|g(x)\right\|^q d\alpha)^{\frac{1}{q}}}\leq\frac{1}{p}+\frac{1}{q}=1$$ which implies that $$\tag{1}\int_a^b\|f(x)\|\|g(x)\|d\alpha\leq\left(\int_a^b \left\|f(x)\right\|^p d\alpha \right)^{1/p} \left(\int_a^b \left\|g(x)\right\|^q d\alpha \right)^{1/q}.$$ Now the inequality which we want to prove follows from $(1)$ and the inequality $$\left\|\int_a^b f(x)g(x)d\alpha\right\|\leq\int_a^b\|f(x)\|\|g(x)\|d\alpha.$$ - I can't see how we can integrate the first line of Young's inequality like that. Can you help me see that a little bit better? – crf Jan 25 at 7:39 wow actually never mind, it's obvious. I just wasn't seeing straight. Thanks very much! – crf Jan 25 at 8:26 P.S., how did anyone come up with that – crf Jan 25 at 8:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 6, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9701120853424072, "perplexity_flag": "head"}
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http://math.stackexchange.com/questions/20810/geometric-argument-for-van-kampen	# geometric argument for van-kampen? I've seen Van-Kampen's theorem presented algebraically many times; and although it provides a useful method of calculation; I don't have a very clear picture for "why" it should be true. Does anyone know of a more visual argument; or even an example that makes the it easier to see? - 2 Can you visualize the case where the intersection is simply connected? – Qiaochu Yuan Feb 7 '11 at 10:48 Have you read the proof? The surjectivity part gives you the geometrical insight. – Gabriel Furstenheim Feb 7 '11 at 15:15 I think the proof in Hatcher is quite geometric/visual. – Soarer Feb 7 '11 at 15:55 ## 2 Answers I'm only going to try to give some intuition why the Seifert-van Kampen Theorem is geometrically reasonable, rather than attempt a geometric proof (which would be well beyond my abilities, I fear). The Seifert-van Kampen Theorem ("van Kampen" was the last name, so it should not be hyphenated) says that if $X=U\cup V$, with $U$ and $V$ arcwise connected open subsets, and $U\cap V$ is nonempty and arcwise connected, then choosing a basepoint $x_0\in U\cap V$, then $$\pi(X) \cong \pi(U)_{\pi(U\cap V)}\pi(V);$$ that is, the fundamental group of $X$ is the free amalgamated product of the fundamental groups of $U$ and $V$, amalgamated over the subgroup corresponding to the fundamental group of $U\cap V$. Intuitively: it should be clear that $\pi(X)$ contains subgroups isomorphic to $\pi(U)$ and $\pi(V)$, and since $X=U\cup V$, with both open, it should be reasonable that $\pi(X)$ is generated by these two subgroups: if you have an arbitrary loop in $X$ that meanders between $U$ and $V$, you can try decomposing it into an equivalent loop that alternates being entirely in $U$ and entirely in $V$. Also, $\pi(V)$ and $\pi(U)$ will each have subgroups corresponding to $\pi(U\cap V)$. So the real question is why this should be the only interaction between $\pi(U)$ and $\pi(V)$ (that is, why you get the free product, and not just some quotient of it). A relation would correspond to some loop in $X$ which goes through $U$, then $V$, then $U$, then $V$, etc., and is homotopy equivalent to the trivial loop. Imagine that you find such a loop, and pick one with the smallest possible number of "crossings" from one set to the other which satisfies this. You can imagine setting up your deformation so that it first "shrinks" the "last" part, until it is completely contained in $U\cap V$ (since this is open and that is where the basepoint is). But that would give you a new loop which is also equivalent to the identity, but with a fewer number of "crossings" (since you can call the portion in $U\cap V$ part of $U$ or part of $V$, as convenient). This tells you that the smallest possible number of crossings has to be $0$; that is, the entire thing was contained inside $U\cap V$ in the first place. So you should not expect any relations between the $\pi(U)$ and $\pi(V)$, except those that come from identifying their common subgroup $\pi(U\cap V)$. This is precisely the free product with amalgamation. A good place to try to get some intuition is the case where the intersection is simply connected, like Qiaochu suggests in the comments; I find bouquets of circles particularly fertile ground, but then I'm a group theorist, so this is almost the only kind of fundamental group I play with on a semi-regular basis. - I will say it less acurately than Arturo and sure enough too much roughly. Consider it as a first approxiation to the theorem. The way I see Seifert-Van Kampen theorem is like a sophisticated version of the dimension of the sum of vector subspaces: $$\mathrm{dim} (F + G) = \mathrm{dim} F + \mathrm{dim} G - \mathrm{dim} (F\cap G) \ .$$ Because, forgetting torsion, the fundamental group counts the number of "holes" (non contractible loops, generators of $\pi_1$) in your space $X$. Right? So, if you have $X = U \cup V$, then the number of holes in $X$ must be equal to the number of holes in $U$ plus the number of holes in $V$..., minus the number of holes in $U\cap V$, because these ones you have already counted them twice. Hence, in order to count all the holes in $U$ plus those in $V$, you take the free product $\pi_1 (U) \pi_1(V)$, but you must "identify" the holes "shared" by $U$ and $V$. This is the reason for the amalgamated product $\pi_1(U) *_{\pi_1(U\cap V)} \pi_1(V)$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 51, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9504430890083313, "perplexity_flag": "head"}
http://www.physicsforums.com/showthread.php?t=236524	Physics Forums ## Explicit expressions for creation/annihilation operator of the free scalar field I've been trying to work my way through some of my lecture notes, and have hit this snag. (n.b. I use $k_0 \equiv +\sqrt{\vec{k}^2 + m^2}$) We have $$a(q) = \int d^3 x e^{iqx} \{ q_0 \phi(x) + i \pi(x) \}$$ $$a^{\dagger}(q) = \int d^3 x e^{-iqx} \{ q_0 \phi(x) - i \pi(x) \}$$ To calculate the commutation relation between these operators, we simply multiply them out as required, and substitute the canonical commutation relation between fields and their conjugate momenta. I work through the relatively tedious steps and get $$[a(q),a(p)] = \int d^3 x d^3 y e^{i(qx-py)} \delta^3(\vec{x} - \vec{y}) (q_0 - p_0)$$ $$= \int d^3 x e^{i(q-p)x} (q_0 - p_0)$$ $$= \int d^3 x e^{i(q_0-p_0)x^0} e^{i(\vec{q}-\vec{p})\cdot\vec{x}} (q_0 - p_0)$$ In my notes, the next step is to replace $\vec{q}$ with $\vec{p}$ and so get 0. However, if we integrate over x, surely we are left with a loose delta function outside an integral, which would mean that $[a(q),a(p)] = 0 \Leftarrow q=p$ which I know is wrong. Can anyone explain that last step? Any textbooks I've seen assume this is trivial and just go on to state the commutation relation between the creation/annihilation operators rather than calculating it. PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus Never mind. After doing the integral, I got $$(q_0 - p_0) \delta^3(\vec{q}-\vec{p}) e^{i(q_0-p_0)t}$$ which conspires to be zero when the delta function is non-zero because of the first term in brackets, and is zero everywhere else because of the delta function. Recognitions: Science Advisor Correct! Thread Tools \| \| \| \| \|-------------------------------------------------------------------------------------------------------\|----------------------------------------\|---------\| \| Similar Threads for: Explicit expressions for creation/annihilation operator of the free scalar field \| \| \| \| Thread \| Forum \| Replies \| \| \| Beyond the Standard Model \| 4 \| \| \| High Energy, Nuclear, Particle Physics \| 23 \| \| \| Quantum Physics \| 16 \| \| \| General Physics \| 3 \| \| \| Advanced Physics Homework \| 5 \|	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 6, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.898821234703064, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/21998?sort=newest	## Is there a nice way to characterise the derived equivalence induced by a flop? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers. Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \rightarrow X$. It has been proved that there then exists an equivalence $\Phi : D^b(Y^+) \rightarrow D^b(Y)$. Is there a way to understand this (or any other) equivalence explicitly? I've heard there is a way to find an equivalence by considering a common resolution of $Y$ and $Y^+$ and then using derived pullback and pushforward, is it true? I am mostly interested in what happens to sheaves on $Y^+$ supported on the exceptional locus. Thanks. - 2 "I've heard there is a way to find an equivalence by considering a common resolution of Y and Y+ and then using derived pullback and pushforward, is it true?" True. That's all it is. – VA Apr 21 2010 at 3:46 ## 3 Answers As always, it depends on what you think "explicitly" means. It's a Fourier-Mukai transform; see, for example, Van den Bergh and Hille's expository article. It can also be explained in terms of so-called non-commutative crepant resolutions, see Van den Bergh. - Thanks, this expository article seems very useful! – babubba Apr 22 2010 at 22:48 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. For threefolds an equivalence can be constructed as pull back and push forward through any common resolution. It is true. On the other hand, it is not true for general flops in higher dimensions. - There's a very explicit characterization of the derived equivalence -- in fact this is how Bridgeland constructs the flop (a really gorgeous idea IMHO). Namely you can build a very simple t-structure on D(Y) by a "tilting" procedure, and then the moduli of point objects is the flop Y^+. I forget the exact details but you do a tilt along the curve you want to contract, so that "perverse point sheaves" are just points away from this curve and are perverse coherent sheaves (in this case rank two complexes with H^0 being a line bundle and H^1 being torsion I think? the paper is great, so easy to find the precise statements). The basic idea being that any derived equivalence (appropriately construed) can be characterized by a universal sheaf on the product, which you can interpret as saying the Y^+ will be a moduli of a particular family of objects in the derived category of Y -- so to build Y^+ you just need to say which family of objects (and check some conditions). - What's the name of this paper? (or enough of the name to find it) – Peter Samuelson Apr 21 2010 at 14:53 David, nice answer. I edited the capitals, I hope you don't mind. – Hailong Dao Apr 21 2010 at 14:57 I think David is referring to the paper I linked in the question. @David: thanks for clearing some things up! – babubba Apr 22 2010 at 22:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9368141293525696, "perplexity_flag": "middle"}
http://unapologetic.wordpress.com/2010/06/24/	# The Unapologetic Mathematician ## Hahn Decompositions Given a signed measure $\mu$ on a measurable space $(X,\mathcal{S})$, we can use it to break up the space into two pieces. One of them will contribute positive measure, while the other will contribute negative measure. First: some preliminary definitions. We call a set $E\subseteq X$ “positive” (with respect to $\mu$) if for every measurable $F\in\mathcal{S}$ the intersection $E\cap F$ is measurable, and $\mu(E\cap F)\geq0$. That is, it’s not just that $E$ has positive measure, but every measurable part of $E$ has positive measure. Similarly, we say that $E$ is “negative” if for every measurable $F$ the intersection $E\cap F$ is measurable, and $\mu(E\cap F)\leq0$. For example, the empty set is both positive and negative. It should be clear from these definitions that the difference of two negative sets is negative, and any disjoint countable union of negative sets is negative, and (thus) any countable union at all of negative sets is negative. Now, for every signed measure $\mu$ there is a “Hahn decomposition” of $X$. That is, there are two disjoint sets $A$ and $B$, with $A$ positive and $B$ negative with respect to $\mu$, and whose union is all of $X$. We’ll assume that $-\infty<\mu(E)\leq\infty$ , but if $\mu$ takes the value $-\infty$ (and not $\infty$) the modifications aren’t difficult. We write $\beta=\inf\mu(B)$, taking the infimum over all measurable negative sets $B$. We must be able to find a sequence $\{B_i\}$ of measurable negative sets so that the limit of the $\mu(B_i)$ is $\beta$ — just pick $B_i$ so that $\beta\leq\mu(B_i)<\beta+\frac{1}{i}$ — and we can pick the sequence to be monotonic, with $B_i\subseteq B_{i+1}$. If we define $B$ as the union — the limit — of this sequence, then we must have $\mu(B)=\beta$. The measurable negative set $B$ has minimal measure $\mu(B)$. Now we pick $A=X\setminus B$, and we must show that $A$ is positive. If it wasn’t, there would be a measurable subset $E_0\subseteq A$ with $\mu(E_0)<0$. This $E_0$ cannot itself be negative, or else $B\uplus E_0$ would be negative and we’d have $\mu(B\uplus E_0)=\mu(B)+\mu(E_0)<\mu(B)$, contradicting the minimality of $\mu(B)$. So $E_0$ must contain some subsets of positive measure. We let $k_1$ be the smallest positive integer so that $E_0$ contains a subset $E_1\subseteq E_0$ with $\mu(E_1)\geq\frac{1}{k_1}$. Then observe that $\displaystyle\mu(E_0\setminus E_1)=\mu(E_0)-\mu(E_1)\leq\mu(E_0)-\frac{1}{k_1}<0$ So everything we just said about $E_0$ holds as well for $E_0\setminus E_1$. We let $k_2$ be the smallest positive integer so that $E_0\setminus E_1$ contains a subset $E_2\subseteq E_0\setminus E_1$ with $\mu(E_2)\geq\frac{1}{k_2}$. And so on we go until in the limit we’re left with $\displaystyle F_0=E_0\setminus\biguplus\limits_{i=1}^\infty E_i$ after taking out all the sets $E_i$. Since $-\infty<\mu(E_0)<0$, the measure of $E_0$ is finite, and so the measure of any subset of $E_0$ must be finite as well. Thus the limits of the $\frac{1}{k_n}$ must be zero, so that the measure of the countable disjoint union of all the $E_n$ can converge. And so any remaining measurable set $F$ that can fit into $F_0$ must have $\mu(F)\leq0$. That is, $F_0$ must be a measurable negative set disjoint from $B$. But we must have $\displaystyle\mu(F_0)=\mu(E_0)-\sum\limits_{i=1}^\infty\mu(E_i)\leq\mu(E_0)<0$ which contradicts the minimality of $\mu(B)$ just like $E_0$ would have if it had been a negative set. And thus the assumption that $\mu(E_0)<0$ is untenable, and so every measurable subset of $A$ has positive measure. Posted by John Armstrong \| Analysis, Measure Theory \| 6 Comments ## About this weblog This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”). I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now. • ## Feedback Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 74, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9448443055152893, "perplexity_flag": "head"}
http://physics.stackexchange.com/questions/27662/which-cfts-have-ads-cft-duals?answertab=active	# Which CFTs have AdS/CFT duals? The AdS/CFT correspondence states that string theory in an asymptotically anti-De Sitter spacetime can be exactly described as a CFT on the boundary of this spacetime. Is the converse true? Does any CFT in a suitable number of spacetime dimensions have an AdS/CFT dual? If no, can we characterize the CFTs which have such a dual? - ## 2 Answers The answer is not known, but many believe it is: "Yes, every CFT has an AdS dual." However, whether the AdS dual is weakly-coupled and has low curvature -- in other words whether it's easy to do calculations with it -- is a different question entirely. We expect, based on well-understood examples (like $\mathcal N=4$ SYM dual to Type IIB strings on $\mathrm{AdS}_5 \times S^5$), that the following is true: • For the AdS dual to be weakly-coupled, the CFT must have a large gauge group. • For the AdS curvature scale to be small (so that effective field theory is a good approximation), the CFT must be strongly-coupled. In well-understood examples, the CFT has an exactly marginal coupling which when taken to infinity decouples stringy states from the bulk spectrum. By contrast, at weak CFT coupling, the AdS dual description would involve an infinite number of fields and standard EFT methods would not apply. (This doesn't necessarily mean calculations are impossible: we would just need to better understand string theories in AdS -- something which is actively being worked on.) As far as I know, appropriate conditions for CFTs without exactly marginal couplings to have good AdS EFTs are not known. Also, well-understood AdS/CFT dual pairs where the CFT violates one or both of the above conditions are scarce. - 5 Welcome David. This is a good answer, but perhaps the criteria could be phrased in CFT language, not assuming auxiliary structures like gauge invariance. The first criterion is the CFT having a large central charge, the second is the existence of a gap in the spectrum of conformal dimensions, so there are few operators with dimensions of order one, and most operators have large dimension of order $\sqrt{\lambda}$. – user566 Nov 6 '11 at 0:23 You're right -- those are better ways to state the above conditions. – davidsd Nov 6 '11 at 17:20 1 Regarding weak coupling, it seems like what one really wants is that the CFT is "almost factorized" -- that is, there is a notion of single-trace and multi-trace operators, and a small parameter that controls deviations from mean field theory. A large central charge reflects this, but does it imply it? – davidsd Nov 6 '11 at 17:31 It depends if you are shooting for a weakly coupled string dual, or just a gravity dual. In my mind the specifics of large N counting has to do with the former, where you can think about large N as the classical limit, and large $\lambda$ as the low curvature limit. For an M-theory dual you have just have one parameter (usually some flux) controlling the size of the geometry. It must be that large flux gives factorization, or clustering (maybe by virtue of having a gap in the spectrum of conformal dimensions) but is unrelated to deviations from mean field theory (which are never small). – user566 Nov 6 '11 at 17:48 A recent work on this: http://arxiv.org/abs/1101.4163 I hope davidsd or Moshe can clarify on what they meant by 'mean field theory' (and deviations from it) in large-N CFT. - This is an old question, but since someone edited it, I'll take the opportunity to comment. 'Mean field theory' in CFT means that all operators factorize through Wick's theorem. (In other words, the theory is "almost free" or "almost Gaussian".) We know that this is true for free fields (i.e of dimension 1 in 4D), but you can define a CFT by choosing some field dimension $[\varphi] = \Delta$ and letting all correlation functions be exactly those predicted by Wick's theorem. – Vibert Jan 16 at 11:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9440358877182007, "perplexity_flag": "middle"}
http://www.haskell.org/haskellwiki/index.php?title=Sinc_function&diff=2628&oldid=2541	# Sinc function ### From HaskellWiki (Difference between revisions) \| \| \| \| \| \|---------\|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\|---------\|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\| \| \| \| \| \| \| Line 1: \| \| Line 1: \| \| \| \| == Sinc function == \| \| == Sinc function == \| \| \| \| \| \| \| - \| The sinc function <math>sin(x)/x</math> is a useful function that is a little tricky to use because it becomes 0/0 as x tends to 0. Here is an implementation taken from the [http://www.boost.org/boost/math/special_functions/sinc.hpp Boost] library. \| + \| The sinc function <math>\frac{sin(x)}{x}</math> is a useful function that is a little tricky to use because it becomes 0/0 as x tends to 0. Here is an implementation taken from the [http://www.boost.org/boost/math/special_functions/sinc.hpp Boost] library. \| \| \| \| \| \| \| \| <pre> \| \| <pre> \| ## Sinc function The sinc function $\frac{sin(x)}{x}$ is a useful function that is a little tricky to use because it becomes 0/0 as x tends to 0. Here is an implementation taken from the Boost library. ```epsilon :: RealFloat a => a epsilon = encodeFloat 1 (fromIntegral $ 1-floatDigits epsilon) {- Boosted from Boost http://www.boost.org/boost/math/special_functions/sinc.hpp -} sinc :: (RealFloat a) => a -> a sinc x \| (abs x) >= taylor_n_bound = (sin x)/x \| otherwise = 1 - (x^2/6) + (x^4/120) where taylor_n_bound = sqrt $ sqrt epsilon ```	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8599498867988586, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/62997?sort=newest	## Parallelizability of exotic structure ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) I'd like to discuss a little bit about the problem I asked Diarmuid Crowley that whether all smooth structures on $S^7$ are parallelizable. He first came to using semi-characteristic but then came up with trivial answer which was : Because by chance we know $\pi_7(BO(7))=0$ so all smooth structures are parallelizable. But still I am curious to knpw why smooth structure does not play role here. Can we have trivial bundle in category of topological bundles but not in smooth category? - 2 Related: mathoverflow.net/questions/58131/… – Daniel Litt Apr 26 2011 at 3:54 5 On smooth manifolds there is no difference between smooth and topological vector bundles (up to isomorphism). This is seen by smoothing, the approximation of continuous sections by smooth ones. – Torsten Ekedahl Apr 26 2011 at 3:55 1 The question you actually asked is not, I think, the question you really want to be answered. The tangent bundles on two homeomorphic spaces with different smooth structures are possibly different and it sounds like you want to know if there's ever a case where they're actually different. As Johannes showed in the question linked above, for the case of spheres the answer is 'no.' – Dylan Wilson Apr 26 2011 at 13:59 ## 1 Answer See Torsten's comment above for an answer to the question you asked... And here's an answer to a question you didn't ask: Suppose you have homeomorphic manifolds $M^n$ and $\widetilde{M}^n$ that have distinct differentiably structures but are both stably parallelizable (i.e. they embed into a high-dimensional Euclidean space with a trivial normal bundle.) From the same paper that Hatcher cited in his answer to this question, namelt "Vector fields on $\pi$-manifolds" by Bredon and Kosinski, we have the following result: Define a number $\chi^(M^n)$ to be $\frac{1}{2}\chi(M^n)$ if $n$ is even and $\sum_{i=0}^rH_{i}(M, Z_2)$ (mod 2) for $n = 2r+1$. Then $M^n$ is parallelizable if $n=1,3,7$ or if $1-\chi^(M) = 1$. (Is this the 'semi-characteristic' you mentioned in your answer?) In particular, since $\chi^*$ is clearly a topological invariant, $M^n$ is parallelizable if and only if $\widetilde{M}^n$ is. This takes care of the stably parallelizable case... I don't know anything about the general case. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.938158392906189, "perplexity_flag": "head"}
http://en.wikipedia.org/wiki/Heath%e2%80%93Jarrow%e2%80%93Morton_framework	# Heath–Jarrow–Morton framework The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curve – instantaneous forward rate curve in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) Model of forward rates [1]:394. For direct modeling of simple forward rates the Brace–Gatarek–Musiela Model represents an example. The HJM framework originates from the work of David Heath, Robert A. Jarrow and Andrew Morton in the late 1980s, especially Bond pricing and the term structure of interest rates: a new methodology (1987) – working paper, Cornell University, and Bond pricing and the term structure of interest rates: a new methodology (1989) – working paper (revised ed.), Cornell University. It has its critics, however, with Paul Wilmott describing it as "...actually just a big rug for [mistakes] to be swept under".[2] ## Framework The key to these techniques is the recognition that the drifts of the no-arbitrage evolution of certain variables can be expressed as functions of their volatilities and the correlations among themselves. In other words, no drift estimation is needed. Models developed according to the HJM framework are different from the so called short-rate models in the sense that HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve (the short rate). However, models developed according to the general HJM framework are often non-Markovian and can even have infinite dimensions. A number of researchers have made great contributions to tackle this problem. They show that if the volatility structure of the forward rates satisfy certain conditions, then an HJM model can be expressed entirely by a finite state Markovian system, making it computationally feasible. Examples include a one-factor, two state model (O. Cheyette, "Term Structure Dynamics and Mortgage Valuation", Journal of Fixed Income, 1, 1992; P. Ritchken and L. Sankarasubramanian in "Volatility Structures of Forward Rates and the Dynamics of Term Structure", Mathematical Finance, 5, No. 1, Jan 1995), and later multi-factor versions. ## Mathematical formulation The class of models developed by Heath, Jarrow and Morton (1992) is based on modeling the forward rates, yet it does not capture all of the complexities of an evolving term structure. The instantaneous forward rate $f \left(t,T\right),t\leq T$ is the continuous compounding rate available at time $\, T$ as seen from time $\, t$. It is defined by: $f\left(t,T\right)=-\frac{1}{P\left(t,T\right)}\frac{\partial}{\partial T}P\left(t,T\right)=-\frac{\partial \textrm{log} P\left(t,T\right)}{\partial T},$ (1) The basic relation between the rates and the bond prices is given by: $P\left(t,T\right)=e^{-\int_t^T f\left(t,s\right)\,ds}.$ (2) Consequently, the bank account $\beta\left(t\right)$ grows according to: $\beta\left(t\right)=e^{\int_0^t f\left(s,s\right)\,ds}$ (3) since the spot rate at time $t$ is $r(t)=f(t,t)$. The assumption of the HJM model is that the forward rates $f\left(t,T\right)$ satisfy for any $\, T$: $df\left(t,T\right)=\mu\left(t,T\right)dt+\xi\left(t,T\right)dW\left(t\right)$ (4) where the processes $\mu\left(t,T\right),\xi\left(t,T\right)$ are continuous and adapted. For this assumption to be compatible with the assumption of the existence of martingale measures we need the following relation to hold: $\frac{dP\left(t,T\right)}{P\left(t,T\right)}=\left[r\left(t\right)-\alpha\left(t,T\right)\theta\left(t\right)\right]dt +\alpha\left(t,T\right)dW\left( t\right).$ (5) We find the return on the bond in the HJM model and compare it (5) to obtain models that do not allow for arbitrage. Let $X\left(t\right)=\textrm{log} P\left(t,T\right).$ (6) Then $X\left(t\right)=-\int_t^T f\left(t,s\right)\,ds.$ (7) Using Leibniz's rule for differentiating under the integral sign we have that: $dX=-d\left(\int_t^T f\left(t,s\right) d s\right)=-A\left(t,T\right)dt-\tau\left(t,T\right)dW\left(t\right),$ (8) where $A\left(t,T\right)=-r\left(t\right)+\int_t^T\mu\left(t,s\right)\,ds~~~\textrm{and}~~\tau\left(t,T\right)=\int_t^T\xi\left(t,s\right)\,ds.$ By Itō's lemma, $\frac{dP\left(t,T\right)}{P\left(t,T\right)}=dX+\frac{1}{2}(dX)^2.$ (9) It follows from (5) and (9), we must have that $\alpha(t,T)=-\int_t^T\xi\left(t,s\right)\,ds,$ (10) $\alpha(t,T)\cdot\theta(t)=\int_t^T\mu(t,s)\,ds-\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2.$ (11) Rearranging the terms we get that $\int_t^T\mu(t,s) \,ds=\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2-\theta(t)\, \int_t^T\xi\left(t,s\right)\,ds.$ (12) Differentiating both sides with respect to $\, T$, we have that $\mu(t,T)=\xi\left(t,T\right)\left(\int_t^T\xi\left(t,s\right)\,ds-\theta(t)\right).$ (13) Equation (13) is known as the no-arbitrage condition in the HJM model. Under the martingale probability measure $\ , \theta=0$ and the equation for the forward rates becomes: $df(t,T)=\xi\left(t,T\right)\left(\int_t^T\xi\left(t,s\right)\,ds\right)dt+\xi\left(t,T\right)d\tilde{W}.$ (14) This equation is used in pricing of bonds and its derivatives. ## External links and references Notes 1. M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling. 2nd ed. New York : Springer-Verlag, 2004. Print. Primary references • Heath, D., Jarrow, R. and Morton, A. (1990). Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation. Journal of Financial and Quantitative Analysis, 25:419-440. • Heath, D., Jarrow, R. and Morton, A. (1991). Contingent Claims Valuation with a Random Evolution of Interest Rates. Review of Futures Markets, 9:54-76. • Heath, D., Jarrow, R. and Morton, A. (1992). Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica, 60(1):77-105. doi:10.2307/2951677 • Robert Jarrow (2002). Modelling Fixed Income Securities and Interest Rate Options (2nd ed.). Stanford Economics and Finance. ISBN 0-8047-4438-6 Articles • Non-Bushy Trees For Gaussian HJM And Lognormal Forward Models, Prof Alan Brace, University of Technology Sydney • The Heath-Jarrow-Morton Term Structure Model, Prof. Don Chance E. J. Ourso College of Business, Louisiana State University • Recombining Trees for One-Dimensional Forward Rate Models, Dariusz Gatarek, Wyższa Szkoła Biznesu – National-Louis University, and Jaroslaw Kolakowski • Heath–Jarrow–Morton model and its application, Vladimir I Pozdynyakov, University of Pennsylvania • An Empirical Study of the Convergence Properties of the Non-recombining HJM Forward Rate Tree in Pricing Interest Rate Derivatives, A.R. Radhakrishnan New York University • Modeling Interest Rates with Heath, Jarrow and Morton. Dr Donald van Deventer, Kamakura Corporation:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 26, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8749467730522156, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/63177/for-any-natural-n-how-could-we-prove-that-sum-limits-i-1-n-i23i1-i	# For any natural $n$, how could we prove that $\sum\limits_{i=1} ^n (i^2+3i+1) i!= (n+3)(n+1)! - 3$ How could we prove this ? $$\sum_{i=1} ^n (i^2+3i+1)\times i!= (n+3) \times (n+1)!-3$$ I did with induction, what I want to know is about other ways to prove this. - I am rolling back @Arturo Magidin's edit,seems descent to me. – Quixotic Sep 9 '11 at 19:39 @Grigory M:I have seen solutions of these kinds of problems using combinatorics,calculus and even elementary number theory.However I don't have any problem if you remove it :-) – Quixotic Sep 9 '11 at 19:42 @ Srivatsan Narayanan:Fixed,my apologies,I somehow forget about that tag. – Quixotic Sep 9 '11 at 19:47 On proofs like this I almost always just start doing induction unless there is a compelling reason not to. – Tim Sep 10 '11 at 13:16 ## 4 Answers We first look at the simpler expression $$\sum_{k=1}^n k\cdot k!.$$ This is $$1\cdot 1!+2\cdot 2! +3\cdot 3!+\cdots +n \cdot n!.$$ In general, $k \cdot k!=(k+1)\cdot k!-k!=(k+1)!-k!$. It follows that the sum above is equal to $$(1!-0!)+(2!-1!)+(3!-2!) +(4!-3!)+\cdots +((n+1)!-n!).$$ Add up. There is a whole lot of cancellation, and we get $(n+1)!-1$. Now we turn to our problem, which can be rewritten as $$\sum_{k=1}^n(k+1)^2\cdot k! +\sum_{k=1}^n k \cdot k!,$$ since $k^2+3k+1=(k+1)^2+k$. We have already computed the second sum. The first sum is $$\sum_{k=1}^n (k+1)\cdot (k+1)!.$$ The same collapsing argument then shows that this sum is $$(n+2)! -2.$$ Or else we can recycle the previous result, noting that we are dealing with $\sum_{j=2}^{n+1}j\cdot j!$, which is $1$ less than $\sum_{j=1}^{n+1}j\cdot j!$. Finally, add up. We get $$[(n+2)(n+1)!-2] +[(n+1)!-1].$$ This is $(n+3)(n+1)!-3$. Comment: But the above solution actually does not answer the question! The OP asked that induction not be used. However, induction was used, albeit in a subtle hidden way. We saw the systematic cancellation, it was obvious. But a "proper" complete proof would use the cancellation up to the $k$-th term to prove cancellation up to the $(k+1)$-th term. Much of the time when one sees ellipses ($\dots$) in a mathematical expression, induction is, technically speaking, needed to fill in the full formal details. Not that this should make any practical difference in our mathematical behaviour: Obvious is still obvious. - (+1)The obvious proof is obvious! – The Chaz 2.0 Sep 9 '11 at 22:07 I would use $$\sum_{i=1} ^n (i+2)! - i! = (n+2)!+(n+1)! - 2! - 1 !$$ - Note that `(i^2+3i+1) = (i+2)(i+1)-1` – Foo Bah Sep 10 '11 at 0:08 @Foo Bah: Indeed. Similarly $(n+2)!+(n+1)! = (n+3)\times(n+1)!$ – Henry Sep 10 '11 at 0:16 As André Nicolas notes, the essence of the identity is showing that $$\sum_{k=1}^n k \cdot k! = (n+1)! - 1.$$ There's a nice combinatorial proof of this. (See Benjamin and Quinn, Proofs that Really Count, Identity 181 on p. 92.) I'll give it in its $\sum_{k=1}^{n-1} k \cdot k! = n! - 1$ form. Both sides count the number of permutations of $1, 2, \ldots, n$ that exclude the identity permutation. The right side is straightforward. For the left side, how many permutations have $n-k$ as the first number that does not get mapped to itself? There are $k$ choices ($n-k+1, n-k+2, \ldots, n$) for the number that appears in position $n-k$, and then there are $k!$ ways to choose the remaining $k$ numbers to complete the permutation. Adding up over all possible values of $k$ yields the left-hand side. - – Mike Spivey Sep 15 '11 at 3:52 If you observe that $i^2 + 3i + 1 = (i + 2)(i+1) - 1$, then your left-hand sum is $$\sum_{i=1}^n ((i+2)(i +1) - 1)i!$$ $$= \sum_{i=1}^n ((i + 2)! - i!)$$ As Henry noted, this sum telescopes into $$(n + 2)! + (n + 1)! - 2! - 1!$$ $$= (n+2)(n+1)! + (n+1)! - 3$$ $$= (n+3)(n+1)! - 3$$ -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 23, "mathjax_display_tex": 15, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9381065368652344, "perplexity_flag": "middle"}
http://physics.stackexchange.com/questions/16732/why-negative-energy-states-are-bad?answertab=oldest	# Why Negative Energy States are Bad The argument is often given that the early attempts of constructing a relativistic theory of quantum mechanics must not have gotten everything right because they led to the necessity of negative energy states. What's so wrong with that? Why can't we have negative energy states? As I understand it, we know now that these "negative energy states" correspond to antiparticles. So then, what's the difference between a particle with negative energy and an antiparticle with positive energy? It seems to me that there really is no difference, and that the viewpoint you take is simply a matter of taste. Am I missing something here? - ## 5 Answers The problem is that interacting systems, like particles, tend to transition into states of lower energy. (Technically, the universe transitions into states of higher entropy, but in the context of a particle that usually means lower energy.) So in order for particles to be stable, the energy spectrum has to have a lower bound. Otherwise, a particle could just keep dropping to lower and lower energy states, emitting photons at every step. Now, there is a sense in which a positive energy antiparticle state can just as well be considered a negative energy particle state. The solution to the Dirac equation looks the same in either case. In the early days of relativistic QM, it never occurred to anyone that there was any interpretation of these solutions other than being negative-energy states, which led to the invention of ideas like the Dirac sea, and the identification of holes in the sea with antiparticles. But by the time quantum field theory came along, people realized that it just made more sense to include antiparticles as proper objects in the theory, rather than trying to explain them as holes, because then there was no need to bother with negative energy states at all. - 1 You wrote: "Otherwise, particle could just keep dropping to lower and lower energy states, emitting photons at every step." But the negative energy in question is about energy spectrum of a free particle, no interaction with photons is involved. If the particle is in interaction with the photon filed, it is not a free and cannot have certain energy. Besides, how about 4-momentum conservation law for emission/absorption of photons by a free particle? – Vladimir Kalitvianski Nov 9 '11 at 11:00 1 the fact that we see systems to transition to lower energy states is based on our experience from positive energy systems: more fundamentally, what happens is that systems tend to split the energy evenly across degrees of freedom. It makes complete sense for negative energy systems to thermodinamically evolve into states with higher energy (i.e: toward zero energy per degree of freedom) – lurscher Nov 9 '11 at 15:18 @lurscher: that thought did occur to me, but I have heard from other sources that particles in negative-energy states still tend to lose energy. I wanted to do an explicit calculation of the entropy to resolve this one way or the other but I didn't have time. – David Zaslavsky♦ Nov 9 '11 at 17:32 @lurscher:You have said in your answer, that the energy of the particles should tend to go to the zero energy states, to achieve 'minimum' entropy. I can't understand, how is this consistent with the fact as david as outlined, that the ' the universe transitions into states of higher entropy'. Also, I don't really understand how you can use statistical concepts such as entropy for a single particle. – ramanujan_dirac Dec 30 '12 at 4:57 A negative kinetic energy is not physical. It is supposed to be observable, as well as the particle velocity and mass. So it is just a non physical solution. On the other hand, for completeness of Fourier transformation, those negative frequencies must be present in the solution. They were made present as "antiparticle" solutions in a multi-particle construction. It means, the Dirac's equation solutions came in handy in QED and are not really physical in one-particle relativistic QM. - For very simple cases only, free quantum fields, we can certainly map negative frequencies (not energies, but the two things are conflated by most authors) to positive frequencies and vice versa in various ways. The details of this for the Klein-Gordon field are published as EPL 87 (2009) 31002, http://arxiv.org/abs/0905.1263v2; for the electromagnetic field, there is http://arxiv.org/abs/0908.2439v2 (which I recently almost completely rewrote). Up to a point, these papers put Vladimir Kalitvianski comment into one mathematical form (but other mathematical forms for his comment are certainly possible). FWIW, the presence of measurement incompatibility is tied in with whether one allows negative frequency modes. HOWEVER, I have no idea what the construction in those papers looks like if one uses similar mathematical transformations for the whole of the standard model of particle physics. In fact, over a number of years I have failed to get such an approach to work. It's necessary to get it right for the whole of a system that comes close to reproducing the phenomenology of the standard model (or something slightly different in an experimentally useful way or in a way that is useful for engineering) before many Physicists are likely to take the idea very seriously. The stability argument given by David Zaslavsky is completely right by conventional wisdom, but it assumes, for starters, that energy and action are viable concepts in a QFT context. In the algebraic context I currently work in, energy and action are not viable concepts. There is also no "axiom of stability" in quantum field theory, so there is no proof of a no-go theorem that there is no way to ensure stability except by having only positive frequencies; there is, instead, an "axiom of positive frequency" in the Wightman axioms. Note that a well-formulated axiom of stability would be far less theoretical and more natural than an axiom of positive frequency. - 1 as i commented on David's answer, if one wants to extrapolate thermodynamics arguments about stability to negative energy systems, one might want to consider the fact that positive energy systems go to lower energy states because that is what happens if you split their energy evenly across degrees of freedom. With negative energy systems the same principle would make then decay into higher energy levels (toward zero-energy states) – lurscher Nov 9 '11 at 15:32 @lurscher Interesting comment, but I'm not yet sure what to make of it in detail. You've clearly thought about this, so are there any references that you think are relevant? – Peter Morgan Nov 9 '11 at 16:15 sorry, i don't have any concrete reference that i'm aware. However i think the main problematic assumption here is that the Boltzmann factor $e^{ - \beta E_{i} }$ is unchanged when a system allows negative energy states. In this case, there is a certain $E_0$ for which entropy is exactly 0 (the Lorentz-invariant vacuum), so the Boltzmann factor needs to change near the vacuum energy (or any entropy local minima, for what its worth) – lurscher Nov 9 '11 at 17:36 The usual argument is that negative energy states are inherently unstable; if energy states are not bounded from below, a negative energy state can always become more negative, emitting positive energy radiation continuously. It turns out, this is more or less what it is believed that happened in the inflationary era: 1) an accelerated expanding cosmos 2) all the positive energy matter we see today. So negative energy states are only "bad" (or let say just wildly inconvenient) in our currently asymptotically flat space-time, but they probably existed at the very beginning in vast quantities. They probably marginally exist still today in the form of dark energy. However, i am confused why people extrapolate the idea that states will always try to decay to lower energy states (even if already negative): What happens at a more fundamental level is that systems try to achieve equilibrium by spreading energy evenly across degrees of freedom of all fields. Entropy is nothing but a logarithm in the number of available states reachable for a degree of freedom at a given, well defined energy. This entropy has a minima at zero energy, not at $- \infty$, as would be implied by the common lore. So it is not unreasonable to expect that, negative energy systems would decay to higher energy states, toward the zero energy states that we associate with the vacuum. - While I recognize that you might have the right idea about entropy, you're contradicting yourself here: the first part of your answer says that negative energy states will decay to states of more negative energy, but the second part says that they don't. Which is it? – David Zaslavsky♦ Nov 9 '11 at 17:35 yes, the first part is trying to answer according to the current lore. The last part is just me trying to stir up the notions as we believe to understand them. In any case, it might be well possible that there are systems of both kinds (unstable negative energy and stable negative energy states) or even that the stable systems are just adiabatic approximations of the unstable ones (small time scales) – lurscher Nov 9 '11 at 17:41 The point of energy is that it is distributed with an additive constraint--- the total is fixed. If you remove the additive constraint, it isn't that the energy runs to -infinity, it runs to -infinity while dumping more energy in modes that simultaneously run to +infinity. So the distribution of energies broadens until it is arbitrarily broad--- there is no additive conserved quantity anymore limiting growth, and you have instability. This is not conjecture, you can see it explicitly in phi^3 model as the field runs away to infinity, producing positive energy domain wall to balance energy. – Ron Maimon Jan 25 '12 at 5:41 I complete analogy with classical mechanics: We define the proper velocity: $$\eta ^\mu :=\frac{dx^\mu}{d\tau},$$ where $\tau$ is proper time. We likewise define (relativistic) momentum: $$p^\mu :=m\eta ^\mu .$$ And finally we define the (relativistic) energy (up to multiples of $c$) as the time-component of $p^\mu$. This happens to be $$\frac{mc^2}{\sqrt{1-(v/c)^2}},$$ which obviously must be positive. Thus, in order to be consistent with our relativistic definition of energy, we can't have particles with negative energy. This almost makes it tautological, but it is straightforward and precise. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9505674242973328, "perplexity_flag": "head"}
http://en.wikipedia.org/wiki/Two-point_tensor	Two-point tensor "Double vector" redirects here. For dual vectors, see dual space. For bivectors, see bivector. Two-point tensors, or double vectors, are tensor-like quantities which transform as vectors with respect to each of their indices and are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates. Examples include the first Piola-Kirchhoff stress tensor. As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates. Thus, a two-point tensor will have one capital and one lower-case index; for example, AjM. Continuum mechanics A conventional tensor can be viewed as a transformation of vectors in one coordinate system to other vectors in the same coordinate system. In contrast, a two-point tensor transforms vectors from one coordinate system to another. That is, a conventional tensor, $\mathbf{Q} = Q_{pq}(\mathbf{e}_p\otimes \mathbf{e}_q)$, actively transforms a vector u to a vector v such that $\mathbf{v}=\mathbf{Q}\mathbf{u}$ where v and u are measured in the same space and their coordinates representation is with respect to the same basis (denoted by the "e"). In contrast, a two-point tensor, G will be written as $\mathbf{G} = G_{pq}(\mathbf{e}_p\otimes \mathbf{E}_q)$ and will transform a vector, U, in E system to a vector, v, in the e system as $\mathbf{v}=\mathbf{GU}$. The transformation law for two-point tensor Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as $v'_p=Q_{pq}v_q$. For tensors suppose we then have $T_{pq}(e_p \otimes e_q)$. A tensor in the system $e_i$. In another system, let the same tensor be given by $T'_{pq}(e'_p \otimes e'_q)$. We can say $T'_{ij}=Q_{ip} Q_{jr} T_{pr}$. Then $T'=QTQ^T$ is the routine tensor transformation. But a two-point tensor between these systems is just $F_{pq}(e'_p \otimes e_q)$ which transforms as $F'=QF$. The most mundane example of a two-point tensor The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that $v'_p=Q_{pq}u_q$. Now, writing out in full, $u=u_q e_q$ and also $v=v'_p e_p$. This then requires Q to be of the form $Q_{pq}(e'_p \otimes e_q)$. By definition of tensor product, $(e'_p\otimes e_q)e_q=(e_q.e_q) e'_p = e'_p\qquad(1)$ So we can write $u_p e_p = (Q_{pq}(e'_p \otimes e_q))(v_q e_q)$ Thus $u_p e_p = Q_{pq} v_q(e'_p \otimes e_q) e_q$ Incorporating (1), we have $u_p e_p = Q_{pq} v_q e_p$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 20, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8847382664680481, "perplexity_flag": "middle"}
http://rjlipton.wordpress.com/2012/11/08/the-power-of-guessing/	a personal view of the theory of computation by Some results and ideas on guessing and nondeterminism David Doty and Michael Wehar are theorists who work in different areas of computational complexity, and are at different stages of their careers. Doty is at the California Institute of Technology as a fellow in the DNA and Natural Algorithms Group, while Wehar just finished his BS and MS from Carnegie Mellon University at age nineteen. Doty is one of the experts on self-assembly systems, while Wehar is focused—at least for now—on complexity theory: his honors thesis is titled “Intersection Emptiness for Finite Automata.” Today Ken and I wish to discuss the power of guessing, the power of nondeterminism, and what we might learn about the question ${\mathsf{P} =? \mathsf{NP}}$ from simulations and from contexts in which ${\mathsf{NP}}$ can be replaced by ${\mathsf{P}}$ or at least sub-exponential time. While DNA self-assembly systems and complexity theory seem far apart, they are tied by a common thread: what makes a system universal? This thread began for Doty in structural complexity theory and Jack Lutz’s program of resource-bounded theories of measure and dimensionality, and as with Lutz himself moved into biological computing models. This involved it in questions dear to me—self-assembly is one area covered by a conference that I help start, with Eric Baum, almost twenty years ago. The 19th DNA conference is scheduled for next fall—see here for details. It is exciting to see that while the conference has “mutated” some, it is still going strong. Besides proving nice results we will mention shortly and working with top people at CMU including Klaus Sutner, Manuel Blum, and Richard Statman, Wehar is noted for work on Rubik’s Cube. We quote him: Back in 2009 when I was 16 years old, I solved the 20x20x20 cube in 20 hours 37 minutes and 7.42 seconds. It was quite an accomplishment even though it was a rather slow solve. I think I could solve it in somewhere between 8 and 14 hours if I were to give a second stab at it. I just wanted to share this with all of you. The “7.42 seconds” part is what I like best—as complexity theorists we might have been satisfied to say “order-of 20 hours” or even “polynomial in 20 hours.” Efficiency and Guessing This degree of time-precision brings to mind the main point of a paper at the just-held FOCS conference co-authored by Doty with Lutz and Matthew Patitz, Robert Schweller, Scott Summers, and Damien Woods. This is on a tight notion of universality for self-assembling tile systems, analogous to a kind of linear-time universality in cellular automata. The tile assembly process itself is nondeterministic and asynchronous. When a cellular automaton model ${{\cal C}}$ is said to be universal, for itself or for some other model ${{\cal M}}$ such as Turing machines, it can mean several things. It can mean there is an easily computed mapping ${f}$ from computations in ${{\cal M}}$ to computations in ${{\cal C}}$ such that the results by a machine ${M \in {\cal M}}$ on an input ${x}$ can be quickly inferred from notionally-equivalent results of the automaton ${C = f(M)}$ running on the same (or lightly encoded) ${x}$. Or, stronger, it can mean ${f}$ is a mapping from configurations ${I,J}$ of an ${M}$ to corresponding configurations ${I',J'}$ of a ${C}$, such that ${I}$ goes to ${J}$ in one step if and only if ${I'}$ goes to ${J'}$ in some number of steps. The latter number of steps and sizes of ${I',J'}$ could still be huge. This was so with Matthew Cook’s seminal exponential-time simulation of arbitrary Turing machines by the cellular automaton Rule 110. Woods and his student Turlough Neary got this down to a polynomial overhead in their 2006 ICALP paper. The new result is that a standard tile-assembly model is universal for itself, under an even more special mapping ${f}$. Here, an individual tile ${T}$ of a system ${M}$ being simulated is represented as a “super-tile” ${T'}$ of the fixed universal system ${U}$. In fact, owing to the nondeterministic and asynchronous nature of the simulator, there may be many super-tiles that encode an individual tile ${T}$, so ${f}$ is a many-one function from super-tiles of ${U}$ to tiles of ${M}$. The super-tile is a legal combination of ${m^2}$ tiles from the finite set that constitutes the simulator ${U}$, where the number ${m}$ depends only on the simulated system ${M}$. Assemblies of tiles ${T}$ belonging to ${M}$ are directly mimicked by assemblies of the super-tiles ${T'}$, as are the dynamics of the assembly process. Note that there is an ${m^2}$ blowup from ${T}$ to ${T'}$, but this re-scaling depends only on the fixed ${M}$. So this is a linear-scaled simulation with a known, small constant scale factor, and gives rise to a notion called intrinsic universality (IU). One point of IU is that the requirements on ${f}$ and efficiency are so tight that communication complexity and Kolmogorov complexity techniques can be used to prove that certain other kinds of celullar automata are not IU. In other words, such weak celullar automata are provably not universal under this strict notion of linear-time simulation. As usual see the full paper for details. § On the power of guessing in general, I wish there were some great breakthrough to report, but there are a couple of results and an idea that I would like to share with you. All are related in one way or another to the power of guessing. Perhaps the key question in all of complexity theory is, what is the power of guessing, of nondeterminism? Of course that is the core of the ${\mathsf{P} = \mathsf{NP}}$ question, but it extends to many other parts of theory. The three questions we will discuss are: • Can we show that nondeterminism can be simulated better than by brute-force search? • Can we simulate the product of finite automata better than the “obvious” method? • Can we trade off guessing for correctness? Let’s look at each in turn. An Oracle Result I believe that while ${\mathsf{P}\neq\mathsf{NP}}$ may be true, nondeterministic machines can still be simulated better than by brute-force. Color me a strong disbeliever in the Exponential Time Hypothesis. Ken believes ETH is off only by a log factor in the exponent, in the sense of a ${2^{O(n/\log n)}}$ upper bound. We will see. In a paper a while ago with Subrahmanyam Kalyanasundaram (Subruk), Ken, and Farbod Shokrieh, we worked on a modest improvement to the best known deterministic simulation of a Nondeterministic Turing Machine (NTM). Here is the abstract from our paper—I quote it to save me some writing. The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size ${S}$ of this graph. This paper presents a new simulation method that runs in time ${O(\sqrt S)}$. The search savings exploit the one- dimensional nature of Turing machine tapes. In addition, we remove most of the time-dependence on nondeterministic choice of states and tape head movements. Doty proved a pretty oracle result that shows that any great improvement to what we did would have to be non-relativizing. Again I will quote from his paper: Hartmanis used Kolmogorov complexity to provide an alternate proof of the classical result of Baker, Gill, and Solovay that there is an oracle relative to which P is not NP. We refine the technique to strengthen the result, constructing an oracle relative to which a conjecture of Lipton is false. Is it cool to be mentioned in an abstract even if it shows that you were wrong? I guess it’s okay, since the proof is only for oracles. I still stand by the conjecture, which was that ${\mathsf{NTIME}(n)}$ is contain in ${\mathsf{DTIME}(c^{n})}$ for some constant ${c<2}$. But the result of Doty does show that any proof will have to be non-relativizing. An Automata Result One of the great mysteries to me is how can the “obvious” algorithm for the intersection of finite automata be the best possible? The usual intersection algorithm forms the direct product of the two machines and then uses the emptiness test on the product machine. For ${k}$ machines of ${n}$ states each, this leads to an ${n^{k}}$ order algorithm. This beautiful result—due to Michael Rabin and Dana Scott—cannot be the best. Or can it? We have found better methods for so many other problems that I find it hard to believe that by using randomness, or some new data structure, or some other trick, we cannot reduce the running time below ${n^{k}}$ But it seems that doing much better is unlikely because a strong improvement will lead to very surprising results in complexity theory. Years ago working with George Karakostas and Anastasios Viglas, we wrote a paper “On the complexity of intersecting finite state automata” where we proved many consequences of the assumption that the product emptiness problem could be solved in ${n^{o(k)}}$ time for fixed ${k}$. As we discussed in greater detail here, one was that $\displaystyle \mathsf{NL} \neq \mathsf{NP}.$ Wehar in his thesis has proved much more. In particular, he shows that the same assumption yields that if the intersection problem is in ${\mathsf{DTIME}(n^{o(k)})}$, then ${\mathsf{NL} \neq \mathsf{P}}$. This of course greatly improves our result. See also his overview. A New Idea? Define ${\mathsf{NTIME}(T(n),G(n))}$ to be those languages accepted by a nondeterministic Turing Machine (NTM) that runs in time ${T(n)}$ and uses at most ${G(n)}$ guesses. Note, if ${G(n) = T(n)}$, then this is just the usual definition of ${\mathsf{NTIME}(T(n))}$. Here is the idea that I have been playing with recently. Suppose that we consider a set ${S}$ in ${\mathsf{NTIME}({\text{poly}(n)},\delta n)}$ where ${\delta < 1}$. Then, there is a polynomial size circuit that solves membership for ${S}$ correctly for at least ${2^{(1-\delta)n}}$ inputs of length ${n}$. The proof is trivial: each input of length ${n}$ uses some guessing string of length ${\delta n}$, so by the pigeonhole principle there is a guess that works for the required number of inputs. I find this simple observation to be quite intriguing. For example, suppose that ${\mathsf{SAT}}$ could be solved by a NTM that runs in polynomial time and makes only ${\delta n}$ guesses. Then again there would be an exponentially large fraction of ${\mathsf{SAT}}$ problems that would be easy for some circuit. Two remarks about this idea” Since ${\mathsf{SAT}}$ can be checked we should be able to arrange that when the algorithm says “yes” it is always correct. Also we can avoid a trivialization of this observation that relates to the encoding of the problems. In the “usual” encoding many strings will not correspond to a ${\mathsf{SAT}}$ problem. We can fix this by using a more clever encoding. I will stop now and come back to this idea later with more details. Open Problems The gap between what we believe about the power of guessing and what we can actually prove remains huge. As we have discussed many times before, it remains open even to prove that ${\mathsf{NTIME}(n)}$ is not contained in ${\mathsf{DTIME}(n\log n)}$. Also if you are interested in the last idea, we can try and work out the details together. Update 11/10: Michael Wehar’s papers are now online, added links above as well. Like this: from → All Posts, Ideas, P=NP, Results 9 Comments leave one → 1. B. November 9, 2012 4:52 am Another way of “improving” the \$n^k\$ bound for the intersection of finite automata would be an FPT algorithm running in time \$f(k) n^c\$ for some function \$f\$ and constant \$c\$. Do you know of any result in this direction? Could this have strong consequences like P ≠ NL? • Magnus November 10, 2012 12:12 pm Yes, such an FPT algorithm would imply the result. One can use the self-reducibility of the question, to reduce a question of k machines with at most n states each to a question with, say, k/log k machines of at most n^(log k) states each, and then apply the FPT algorithm. 2. November 10, 2012 3:21 pm Yes! If I understand you correctly, I proved a more general statement. http://www.andrew.cmu.edu/user/mwehar/papers.html If you want, you can read “Hardness Results for Intersection Emptiness”. Please give me feedback. • November 10, 2012 3:56 pm Thanks! We have added links to the papers in the post itself. As an aside, the Neary-Woods result for Rule 110 is the most economical demonstration I (Ken) know of universal complexity emerging feasibly from a simple rule. A similar P-completeness result for John Horton Conway’s “Game of Life” underscores the last chapter of Hawking and Mlodinow’s popular book The Grand Design. The P-completeness text by Greenlaw, Hoover, and Ruzzo is a stem reference for the latter result; on p211 (p225 of the PDF file) they gave as open whether a one-dimensional cellular automaton can simulate TM’s, which Matthew Cook showed for exponential time and Neary-Woods answered fully. • November 10, 2012 4:09 pm Thanks! 3. Russell Impagliazzo November 11, 2012 1:09 pm I want to make just a few clarifying comments. First, the Exponential Time Hypothesis is a question, not a conjecture. We were not asking anyone to believe it, just pointing out that it would be interesting to know it is true, and interesting to know that it is false. However, I think strong belief or disbelief is premature at this point. Second, the statements you suggest as being contrary to ETH are not necessarily in contradiction to it. You need to be very careful about what n is in your simulation. In some contexts, its the input size, but it could also mean the solution size or the total time of the non-deterministic algorithm. In ETH, n refers to the solution size, the number of guessed bits, and ETH refers to a specific problem, 3-SAT. It is shown to be equivalent to sub-exponential time algorithms in the solution size for a wide variety of NP-complete problems. But the instance size for even a sparse formula on n Boolean variables is O(n \log n) bits long. So if n is the instance size or time of a non-deterministic algorithm, a 2^{O(n/\log n) algorithm does not contradict ETH. In the reverse direction, the best reductions from TM’s to SAT blow up the instance and solution by a logarithmic factor, so the negation of ETH would not imply a sub-exponential simulation of arbitrary TMs. Similarly, I don’t see that your conjecture that there is a c^n simulation of linear time NTMs with c< 2 contradicts even strong ETH, the conjecture that the constants for k-SAT go to 2 as k increases. • November 12, 2012 9:24 pm Thanks, Russ. I think we’ve studiously avoided calling ETH a “conjecture”. You are right that there are two parallel statements, one about SAT and one about NTM’s, with no proven relation. I’ve wondered whether Etienne Grandjean’s 1988 SICOMP paper and this followup could be tweaked to fuse them, but it’s never made my front burner. Dick led an effort to try to show time c^n for all c > 1 (I prefer to think 2^{\epsilon n} for all \epsilon > 0) two years ago, but it didn’t hit any interesting obstacles. • November 25, 2012 10:14 pm Ah no, it was called a conjecture here, wherein I also missed your similar comment then (Sep. 2010). I found it because I’ve been riffling through the comments in 2010 for the Sunday 11/25 post… 4. abhinav November 12, 2012 9:26 am yaa its nice concept can give in a great detail %d bloggers like this:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 80, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9415279626846313, "perplexity_flag": "middle"}
http://mathoverflow.net/revisions/9756/list	## Return to Question 3 additional info I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group. I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not. Edit (12/27) I added a bounty today and here is some additional information that may be useful: It should come out that the matrices representing the $y$-morphism $\mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$ are polynomials in the dimension $n$. When increasing $n$, we change the partition $\pi$ by adding to the first (largest) element and leaving the rows below fixed. Since a simple object in $Rep(S_t)$ for $t$ not an integer can be represented as a normal Young diagram (of size, say, $N$) with a "very long line" of "size" $t-N$ at the top, this kind of a polynomial interpolation will allow for an interpolation of the $M^{\pi}_c$ to complex rank. My claim that it should come out as a polynomial was based upon studying induction directly on these categories and finding it was interpolable. However, I don't know how to compute the $y$'s directly as matrices via the simple object decomposition.My hope was that there is a clean not-too-computationally-intensive way to do this, but unfortunately I'm not yet sufficiently comfortable with the theory of the symmetric group to have any ideas as to how to proceed. I am also interested in the degenerate affine Hecke algebra of type A, where these kinds of standard induced modules can be defined similarly. Their simple quotients form a complete collection of irreducible modules for the Hecke algebra according to a theorem of Zelevinsky, and I know that these, too, can be interpolated by reasoning on the definitions in Deligne's paper (so one gets objects in the interpolated category $Rep(H_t)$, which is defined in Etingof's talk). But I am curious here too how it is possible to compute the $y$-morphisms as matrices using the decomposition into irreducibles in $Rep(S_t)$. 2 Fix a display I've gotten stuck in a project I have been working on, essentially on the following question. One can obtain a 1-dimensional representation $M^n_c$ of the algebra $T_n := S_n \rtimes \mathbb{C}[y_1, \dots, y_n]$ by letting each $y_i$ act by $c$ and $S_n$ act trivially. Given a partition $\pi$ of $n =n_1 + \dots + n_k$ and $c_1, \dots, c_k \in \mathbb{C}$, we can consider the standard induced module [ ```$$ M^{\pi}_c = \mathrm{Ind} ( M^{n_1}_{c_1} \otimes \dots \otimes M^{n_k}_{c_k} ] M^{n_k}_{c_k}) $$``` where the induction is from the subalgebra $T_{n_1} \otimes \dots \otimes T_{n_k} \subset T_n$ to $T_n$. As far as I know, the simple quotients of these standard modules form a complete class for the simple representations of $T_n$. (I think this is a general fact about semidirect products of a finite group with a commutative algebra, together with the fact that representations of the symmetric group $S_n$ can be obtained by taking quotients induction of the trivial representation of a Young subgroup $S_{n_1} \times \dots \times S_{n_k}$.) My question is how to represent this explicitly in terms of the simple objects in the semisimple category $Rep(S_n)$. First of all, the $S_n$-module $M^{\pi}_c$ can be described in a combinatorial manner in terms of the simple objects in $Rep(S_n)$ using the Kostka numbers. The action of the $y_i$'s on $M^{\pi}_c$ can be given in terms of a suitable morphism $y: \mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$, where $\mathfrak{h} \in Rep(S_n)$ is the regular representation. The Pieri rule allows one to compute the decomposition in irreducibles (as parametrized by Young diagrams, of course) of $\mathfrak{h} \otimes M^{\pi}_c$, so we can view $y$ as a bunch of matrices based upon this decomposition (matrices w.r.t. the simple objects in $Rep(S_n)$, not as vector spaces). Is there an approach to compute these matrices? I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not. 1 # Explicit computation of induced modules of semidirect products with the symmetric group I've gotten stuck in a project I have been working on, essentially on the following question. One can obtain a 1-dimensional representation $M^n_c$ of the algebra $T_n := S_n \rtimes \mathbb{C}[y_1, \dots, y_n]$ by letting each $y_i$ act by $c$ and $S_n$ act trivially. Given a partition $\pi$ of $n =n_1 + \dots + n_k$ and $c_1, \dots, c_k \in \mathbb{C}$, we can consider the standard induced module [ M^{\pi}_c = \mathrm{Ind} ( M^{n_1}_{c_1} \otimes \dots \otimes M^{n_k}_{c_k} ] where the induction is from the subalgebra $T_{n_1} \otimes \dots \otimes T_{n_k} \subset T_n$ to $T_n$. As far as I know, the simple quotients of these standard modules form a complete class for the simple representations of $T_n$. (I think this is a general fact about semidirect products of a finite group with a commutative algebra, together with the fact that representations of the symmetric group $S_n$ can be obtained by taking quotients induction of the trivial representation of a Young subgroup $S_{n_1} \times \dots \times S_{n_k}$.) My question is how to represent this explicitly in terms of the simple objects in the semisimple category $Rep(S_n)$. First of all, the $S_n$-module $M^{\pi}_c$ can be described in a combinatorial manner in terms of the simple objects in $Rep(S_n)$ using the Kostka numbers. The action of the $y_i$'s on $M^{\pi}_c$ can be given in terms of a suitable morphism $y: \mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$, where $\mathfrak{h} \in Rep(S_n)$ is the regular representation. The Pieri rule allows one to compute the decomposition in irreducibles (as parametrized by Young diagrams, of course) of $\mathfrak{h} \otimes M^{\pi}_c$, so we can view $y$ as a bunch of matrices based upon this decomposition (matrices w.r.t. the simple objects in $Rep(S_n)$, not as vector spaces). Is there an approach to compute these matrices? I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 71, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9182260632514954, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/132199/non-disjoint-partition-of-a-graph-into-cliques-of-bounded-size	# Non-disjoint partition of a graph into cliques of bounded size I am looking for a method to list cliques in a graph such that: • All vertices of the graph are included in at-least one clique • The size of every clique is no greater than a bound K The problem seems to be similar to the Clique Cover Problem. However, my problem places a bound on the size of cliques rather than on the number of cliques listed in the solution. Also in my problem, a vertex may appear in more than one clique. In other words, I am not looking for disjoint partitions into cliques. I would prefer to have a small number cliques in the output, but a minimum solution is not necessary. I don't have too much experience with graph-theory or graph-algorithms, so I would appreciate any pointers towards a relevant method for this situation. Thanks! EDIT Adding more context.. What I started with originally is a graph-representation of my problem (say graph G). There are certain nodes (and groups of nodes) in G which are independent. I want to add more edges to this graph with the aim of making G more 'connected' to optimize my original problem. (I have a bunch of sets of edges over G, and the objective is to select a minimum number of sets to include all nodes) So I construct the complement graph of G ($G_C$), and try to find cliques in $G_C$, to help me find groups of nodes to attempt to connect in G. The computation of new edges is expensive, so I want to do it as few times as possible, and also keep a bound on the number of nodes in consideration at a time. I realize that I am probably being too vague.. I tried to be general enough while avoiding telling a long story :). The bottomline objective is to find cliques of moderate size in $G_C$ that I can attempt to break-up. So I am trying to think of ways to list cliques to cover all nodes, while keeping as less overlap between cliques as possible. - ## 1 Answer There are a number of trivial solutions: Take each node as its own clique, or take each edge as a $K_2$ (2 node clique). Do you have more parameters or context for this problem? If you don't want a minimum solution, what is the most desired output, as there are many ways this can be done? Many clique-related problems are NP complete, so are you looking for a poly-time algorithm to find said "most desired output"? - I am looking for heuristics or greedy solutions at this point, so "most desired output" is not a constraint. That said, I want to avoid singular cliques; and triangles and cliques of size K are preferred over cliques of size 2. – Dhrumeel Bakshi Apr 15 '12 at 20:19 Degree-based clique growing could be an interesting attempt. Start by computing the degree of each node (note that each node has a set of neighbors [edges]). Start with your highest degree node. Pick the neighbor with highest degree and intersect the neighbor sets [denote this set $S$]. Pick the node $n$ from $S$ with highest degree and intersect the neighbors of $n$ with $S$ and rename that $S$. Repeat. Once $S$ is empty, pick a new node by taking the highest degree node that is not yet in a clique and repeat. This has lots of flaws, but it's hard to optimize without a precise goal. – Justin Apr 15 '12 at 20:52 Thanks @Justin this sounds like a good place to start! I'm gonna try this and maybe it will be sufficient for what I need. I'll also maybe edit the question to add more context. – Dhrumeel Bakshi Apr 17 '12 at 2:00 Reading the context you added - if you just want to make sure the graph is connected, computing the connected components of a graph is quite easy. From there it is a simple matter of adding edges between the components. I'm not sure how the "sets" you've described work (where do new edges go?), but there are a number of metrics which are easy to compute which may assist you in determining edges to add. Edge-connectivity, min-cuts, diameter of a graph, and max distance from a vertex to any other vertex are all useful and can be ways to measure "connectivity". – Justin Apr 18 '12 at 2:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9568375945091248, "perplexity_flag": "head"}
http://mathoverflow.net/questions/72736?sort=newest	## ZFC, set membership and FOL ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Hi, Is set membership defined in the signature of ZFC, or is it *specified" in the signature of ZFC? The wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory says that the signature has set membership, but what does it mean by has? If I understand correctly, the axioms specify the properties of set membership. Given set membership, how come ZFC can be foramlised in FOL when some axioms, e.g., axiom of infinity, require quantification of sets? Aren't sets unary predicates since S(a) = True iff a is a member of S? Quantification over unary predicates is a feature of second-order though. I must be missing something... Thanks - Internal sets in ZFC are just elements in a model of ZFC. They're not subsets of the model (although they can contain all of the elements in a subset of the model). – Qiaochu Yuan Aug 12 2011 at 3:25 1 If you look at the article you'll see that ZFC is a one sorted theory in FOL, where sets are the individuals in the universe. Hence variables range only over sets, and set membership is a binary relation in the signature. Axiom of infinity and all axioms quantifying over sets are thus expressed in FOL. – godelian Aug 12 2011 at 3:38 This question should be community wiki or asked at mathstockexchange. – SNd Aug 12 2011 at 15:29 ## 3 Answers A specification is a prescription of how things are supposed to be. A model (or implementation in computer-sciency talk) is an instance of something that satisfies a specification. In first-order logic and model theory a specification is called a theory, and it consists of two parts: • a signature which prescribes the non-logical part of the language (constants, function symbols, and relation symbols) • axioms, which are first-order statements written in the language that includes the parts provided by the signature The division of a theory into two parts is convenient, but is not always possible outside first-order logic. For example, if we wanted to include Russell's definite operator $\iota x . \phi(x)$ ("the $x$ such that $\phi(x)$") then our syntax would get intertwined with logic because in order to even form the term $\iota x . \phi(x)$ one first has to prove something. Anyhow, the theory ZFC specifies that there is one relation symbol $\in$, which we usually read as "element of". A model of ZFC interprets the symbol $\in$ as a binary relation, so in this sense it defines its meaning. In a context where theory and its intended semantics are bunched together, such as the Wikipedia page seems to be, the distinction between the specification and implementation will be obsucred. In fact, most mathematicians seem to be a bit unclear about this distinction, as they never really have to distinguish the language they use from its meaning. - I think most mathematicians are clear about the distinction up to algebraic theories: they know the difference between a group and the theory of groups. It's when you add quantifiers/binders and implication/negation that they get fuzzy, since for some reason we are rarely taught the deductive systems we actually use. (I remember when I finally realized that dependent records gave a syntax for indexed sets -- embarrassingly, this was a couple of years after I learned about dependent types!) – Neel Krishnaswami Aug 12 2011 at 9:55 Just to get my terminology right: If I understand correctly, a theory specifies a symbol in its signature and defines the symbol's properties in its axioms -- is that right? – kate.r Aug 12 2011 at 12:21 @kate: Essentially, yes, but note that axioms can describe combined properties of symbols, such as distributivity of $\cdot$ over $+$, and sometimes symbols are not mentioned in the axioms at all (for example, uninterpreted constants). – Andrej Bauer Aug 13 2011 at 11:46 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Properly speaking, the signature of ZFC includes a binary relation symbol rather than a binary relation. In set theory this symbol is usually denoted $\in$ but it could be denoted equally well as $R$ or $\prec$. In an arbitrary model of set theory, the "sets" might actually be any objects: cats, books, chairs, etc. But if we're are only interested in the elements qua elements of that model, we would likely call them "sets" anyway, and call the relation "membership", in the context of that model. It is very common, when talking about a first-order theory, to conflate the symbols in the theory with their intended interpretations. For example, when we define Peano arithmetic in the signature of ordered rings, we might say that the signature has a single binary addition function $+$. Of course we already know what the "addition" function is on natural numbers, but the interpretation of the $+$ function in an arbitrary model of PA may have very little to do with addition on natural numbers. Nevertheless we usually call the elements of an arbitrary model of PA the "numbers" of the model, and we call the interpretation of the $+$ symbol the "addition" on those numbers. It's simply too cumbersome to say "The objects in the model which are intended to be numbers" or "the function in the model which is intended to be addition". Similarly, even though the elements of an arbitrary model of ZFC might not "really" be sets, or the interpretation of the $\in$ symbol may not really be set membership, we often speak as if they are. The key observation is that, if someone "lived inside" the model, and only had access to the $\in$ relation, that person would have no way to tell that the things they see are not sets. One way of making this observation precise is the following lemma, which is proved from "outside" a model $(X, R)$ of set theory. Mostowski Collapsing Lemma. Suppose that $R$ is a binary relation in an arbitrary class $X$ (of arbitrary objects) such that: • For each $y \in X$, the collection $\{ x \in X : xRy\}$ is a set • The model $(X,R)$ is well founded – every subset of $X$ has an $R$-minimal element • The model $(X,R)$ satisfies the axiom of extensionality Then there is a transitive class $C$ (of sets) such that the structure $(C, \in)$ is isomorphic to $(X, R)$, and both $C$ and the isomorphism are uniquely determined by $X$ and $R$. This lemma says that if we look from the outside at a model that looks even vaguely like a (well-founded) model of ZFC, we can replace it with an isomorphic model whose elements are actually sets and whose binary relation is actually set membership. This doesn't work formally for non-well-founded models, because the actual set membership relation is well founded. But "from the inside" we wont be able to tell that any model of ZFC is not well founded. - Mostowski, right? Gerhard "Ask Me About System Design" Paseman, 2011.08.12 – Gerhard Paseman Aug 12 2011 at 19:27 @Gerhard: Thank you. – Carl Mummert Aug 13 2011 at 0:22 Two small comments. 1.- It would moreover be considered improper to use + for a noncommutative operation - this is another place where the syntax 'implies' some semantics. 2.- An interesting case in point is that structures of the form $(A,\prec)$ where $\prec$ is a linear ordering of $A$ tend to satisfy an insanely large fragment of ZFC. – François G. Dorais♦ Aug 13 2011 at 0:37 The axioms of ZFC describe the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate. In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates. The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC. Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$. I hope this helps. - Very clearly, it is specified in the signature, at least where I come from. – Andrej Bauer Aug 12 2011 at 7:20 In the signature it is specified that there is a binary relation, which we happen to call set membership. But nothing about its properties is said. This is what the axioms are for. But you are right, of course, this is just a matter of terminology. – Stefan Geschke Aug 12 2011 at 7:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 59, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9511730670928955, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/275556/find-the-marginal-density-f-yy-of-a-random-vector-x-y	# Find the marginal density $f_Y(y)$ of a random vector $(X,Y)$. Given a (continuous) bivariate random vector $(X,Y)$ with a probability density: $$f_{XY}(x,y)=\left\{ \begin{array}{l}\tfrac{1}{y}e^{-(y+\tfrac{x}{y})} & \text{if }x>0 \text{ and } y>0 \\ 0 & \text{otherwise}\end{array} \right.$$ Find the marginal density $f_X(x)$ and $f_Y(y)$ and show whether $X$ and $Y$ are stochastically independent. In my attempt, I tried to calculate the marginal densities as follows: $$\begin{align}f_Y(y) &= \int_{-\infty}^{\infty}f_{XY}(x,y)\mathrm{d}x \\ &= \int_{0}^{\infty}\tfrac{1}{y}e^{-(y+\tfrac{x}{y})}\mathrm{d}x\\ &= -\lim_{N \to \infty} \left[ e^{-(y+\tfrac{x}{y})} \right]_0^N \\ &= -\lim_{N \to \infty} (e^{-(y+\tfrac{N}{y})}-e^{-(y+\tfrac{0}{y})})\\ &= e^{-y}-0 = \frac{1}{e^y} \end{align}$$ $\\$ $$\begin{align}f_X(x) &= \int_{-\infty}^{\infty}f_{XY}(x,y)\mathrm{d}y \\ &= \int_{0}^{\infty}\tfrac{1}{y}e^{-(y+\tfrac{x}{y})}\mathrm{d}y\\ \end{align}$$ But I get stuck at solving this last integrals. How should this be solved? Thanks in advance. - This looks a lot like homework, and if so, please add the `homework` tag. Also, could you please check to make sure that you have copied the problem correctly? If $X$ and $Y$ are stochastically independent, then you should be able to write $f_{X,Y}(x,y)$ as $g(x)h(y)$ and this does not seem obviously so. – Dilip Sarwate Jan 11 at 2:01 Actually, it's an exercise from last years exam for which I'm preparing. And I saw my mistake, it should have read "show whether X and Y are stochastically independent". – Jeroen Jan 11 at 2:12 ## 1 Answer The expression for $f_X(x)$ is not elementary. In fact, the integral is a defining integral for the Macdonald function: $$f_X(x) = \int_0^\infty \exp\left(-y - \frac{x}{y}\right)\frac{\mathrm{d}y}{y} = \left.\left( \underbrace{\left(\frac{z}{2}\right)^\nu \int_0^\infty \exp\left(-t-\frac{z^2}{4t} \right) \frac{\mathrm{d}t}{t^{\nu+1}}}_{2 K_\nu\left(z\right)}\right)\right\|_{\nu=0,z=2 \sqrt{x}} = 2 K_0\left(2 \sqrt{x}\right)$$ where $x$ was assumed positive. Anyhow, you readily conclude that $f_{X,Y}(x,y) \not= f_X(x) f_Y(y)$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9334946870803833, "perplexity_flag": "head"}
http://mathoverflow.net/questions/57904/why-not-define-triangulated-categories-using-a-mapping-cone-functor	## Why not define triangulated categories using a mapping cone functor? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called `$[1]$` in which certain diagrams, of the form `$X \to Y \to Z \to X[1]$` are called "exact", satisfying certain axioms. Two of these axioms are that (1) Given $X \to Y$, it can be extended to an exact triangle $X \to Y \to Z \to X[1]$ and (2) Given a commuting diagram `$$\begin{matrix} X & \to & Y \\ \downarrow & & \downarrow \\ X' & \to & Y' \end{matrix}$$` and exact triangles `$X \to Y \to Z \to X[1]$` and `$X' \to Y' \to Z' \to X'[1]$`, there is a map $Z \to Z'$ making the obvious diagram commute. And, as every source on triangulated categories points out, one of the standard problems with the theory is that there is no uniqueness statement in these axioms. So, why not make one? I envision a definition as follows: For any category $C$, let $Ar(C)$ be the category whose objects are diagrams $X \stackrel{f}{\longrightarrow} Y$ in $C$ and whose morphisms are commuting squares in $C$. Note that there are obvious functors $\mathrm{Source}$ and $\mathrm{Target}$ from $Ar(C) \to C$, and a natural transformation $\mathrm{Source} \to \mathrm{Target}$. Define a conical category to be an additive category $C$ with a self-equivalence $[1]$ and a functor $\mathrm{Cone} : Ar(C) \to C$, equipped with a natural transformations $\mathrm{Target} \to \mathrm{Cone} \to \mathrm{Source}[1]$, obeying certain axioms. I noticed one place you have to be careful. In a triangulated category, if $X \to Y \to Z \to X[1]$ is exact, then so is $Y \to Z \to X[1] \to Y[1]$ (with a certain sign flip). The most obvious analoguous thing in a conical category would be for $\mathrm{Cone}(Y \to \mathrm{Cone}(X \to Y))$ to equal $X[1]$; the right thing is to ask for a natural isomorphism instead. But everything else seems work out OK, at least in the case of the homotopy category of chain complexes. And it seems much more natural. What goes wrong if you try this? I know that this is the sort of subject where people tend to mention $\infty$-categories; please bear in mind that I don't understand those very well. Everything I've said above just used ordinary $1$-categories, as far as I can tell. - 1 You will end up needing more and more choices: apart from assigning cones to arrows, you need to make choices (at least!) for squares and cubes and so on. Bernhard Keller studied a similar situation, if I recall correctly, in his thesis. The idea converges to dérivateurs. – Mariano Suárez-Alvarez Mar 9 2011 at 2:45 9 The key point is that $Ho(Ar(C))\to Ar(Ho(C))$ is not an equivalence. It is the first category that has a cone functor, not the second. That's Neil's answer. But you can turn it around and keep track of $Ho(Ar(C))$ and similar categories--that is a derivateur. – Ben Wieland Mar 9 2011 at 3:42 ## 2 Answers Let's take $C$ to be the category of chain complexes of abelian groups, and homotopy classes of maps. What do you want the cone functor to be? An object of $Ar(C)$ consists of a pair of chain complexes $X$ and $Y$, together with a homotopy class of chain maps between them. To construct the mapping cone in the usual way, you need an actual chain map, not just a homotopy class. I don't think that this $C$ supports the structure that you describe. You could instead take $C$ to be the more rigid category of chain complexes and chain maps, and try to axiomatise the structure that allows you to form $Ho(C)$ and to prove that $Ho(C)$ is triangulated. One approach is to demand that $C$ is a model category in the sense of Quillen, with an extra stability axiom. This is discussed in Hovey's book on model categories. If you want something closer to the approach outlined in your question, you could look at the work of Baues. He has axioms for a category with a cylinder functor, and he develops a rich theory of unstable homotopy based on that, including Puppe sequences. I don't think he ever discusses the additional stability condition needed to give a triangulation, but it can't be too hard. A third approach would be to use stable infinity categories. That route involves a lot of statements and constructions that look very simple on the surface, but there tend to be a lot of elaborate technicalities hidden by the formalism that can jump out and bite you unexpectedly. - 1 Thank you, I was being dumb. I checked that a homotopy class of maps between arrows gave me a homotopy class of class of cones, but didn't notice that this meant I was going from (Arrows in chain complexes) to (Homotopy chain complexes). Yeah, that's going to get messy to fix... – David Speyer Mar 9 2011 at 2:34 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Verdier, Astérisque 239, Ch. II, Prop. 1.2.13 (p. 104) says that a triangulated category (with countable coproducts or products) equipped with a cone functor has to split. Ben Wieland is right - you get a cone functor when working with dérivateurs (or filtered derived categories), but that functor is no longer intrinsic to the base category, and you have to carry diagram categories along. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 33, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9475700259208679, "perplexity_flag": "head"}
http://mathoverflow.net/questions/115145?sort=oldest	## are immersions/submersions captured in generalised smooth spaces by some universal property? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Immersions & sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle. I realise that generalised smooth spaces do not have a canonical tangent bundle. But they have better categorical properties, but nastier objects. Is it possible to define what a submerssion/immersion is here by a universal property? - ## 2 Answers Diffeological spaces are examples of generalized smooth spaces, they give a complete and cocomplete category of spaces including smooth spaces as a full subcategory. A diffeological space is a set together with a diffeology: a collection of plots (maps from a numerical domain) which determines the smooth structure. For these diffeological spaces you have the concept of inductions and subductions. Let me focus on inductions they are morphisms $f:(X,D)\rightarrow (X',D')$ such that $f^*(D')=D$ ($D$ is the pull-back diffeology of $D'$ along $f$) and $f$ is injective. For smooth manifolds with their induced diffeology, every induction is an immersion but an immersion is not necessarily an induction even it is injective. Rather an immersion is a local induction. Thus at least for diffeological spaces if you want to generalize immersions you have to work locally (with a superset of every point). You also have local inductions, subductions. You can have a look at P. Iglesias-Zemmour's book "Diffeology" (published by the AMS). - Nice answer. Immersions are defined locally. What does it mean that an immersion is a local induction? That one must reduce its domain of definition further? For subductions, are they the dual of what you've stated about inductions? – Mozibur Ullah Dec 2 at 14:44 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. A surjective submersion can be characterised in the category of smooth manifolds as being an element in the largest pullback-stable class of regular epis. Also, submersions have local sections through every point in their codomain. Or they look locally like a projection out of a cartesian product with fibre a vector space. This latter is in fact how submersions are defined for infinite-dimensional (well, Fréchet) manifolds. The first definition works fine for smooth spaces, but proofs may not generalise if they need some of the other properties. The second one works if you are considering concrete smooth spaces, but is meaningless otherwise. The third definition implies the second, but restricts the sort of fibres. Edit: Another property of surjective submersions is that they are the largest class of morphisms which admit sections over open covers (which we can assume are made up of charts) of which all pullbacks exist. This means that surjective submersions are in some sense a 'saturation' of the singleton pretopology consisting of covers of the form $\coprod U_i \to X$. - Thanks, I wasn't aware of the first definition. Off-topic, it seems to me that the all three definitions are available for topological manifolds, whether any of them are provably equivalent is a different question. Are you aware as to whether they're useful distinctions to make in that category? – Mozibur Ullah Dec 2 at 14:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9438170194625854, "perplexity_flag": "head"}
http://mathoverflow.net/questions/70716/characterization-of-locally-free-modules-via-exterior-powers	## Characterization of locally free modules via exterior powers ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Let $X$ be a scheme and $\mathcal{F}$ be quasi-coherent module on $X$. It is clear that if $\mathcal{F}$ is locally free of rank $n$, then $\det(\mathcal{F}) := \wedge^n \mathcal{F}$ is invertible, i.e. locally free of rank $1$. But what about the converse? Question. Assume $\wedge^n \mathcal{F}$ is invertible. Does it follow that $\mathcal{F}$ is locally free (necessarily of finite rank $n$)? Of course we may assume that $X$ is affine. Then it is enough to prove that $\mathcal{F}$ is flat and of finite presentation, but I don't know how to prove either one. Also it seems to be hard to find counterexamples. - ## 2 Answers Here's an argument for $n=2$. After further localisation if necessary, we may assume that $X=\text{spec}(k)$ and that we have an isomorphism $\alpha:\Lambda^2(F)\simeq k$. As this is surjective we see that $1$ can be written as a sum of terms $\alpha(u\wedge v)$; after yet more localisation we may assume that some such term is invertible, and then we can adjust the choice of $v$ to ensure that $\alpha(u\wedge v)=1$. Now define $\beta:F\to k^2$ by $\beta(x)=(\alpha(x\wedge v),\alpha(u\wedge x))$, so $\beta(u)=(1,0)$ and $\beta(v)=(0,1)$. It follows that $u$ and $v$ generate a free submodule of $F$ and that $F=ku\oplus kv\oplus F_0$ where $F_0=\ker(\beta)$. This means that we can define a split monomorphism $\gamma:F_0\to\Lambda^2(F)$ by $\gamma(x)=u\wedge x$ but $\alpha\gamma=0$ by the definition of $F_0$ and $\alpha$ is an isomorphism so we must have $F_0=0$. - Can you expand/explain how you reduce to the case $X=\mathrm{Spec}(k)$? – Baptiste Calmès Jul 19 2011 at 9:52 1 @Baptiste: We may assume that $X$ is affine and I think that Neil means $k$ just to be a ring (not a field, in which case nothing has to be shown). @Neil: Thanks! I'll try to generalize this method of proof. – Martin Brandenburg Jul 19 2011 at 10:30 1 Neil, your proof generalizes to arbitrary $n$: Let $\wedge^n F$ be free of rank $1$. Writing a generator as sums of pure wedge products and localizing, we may arrange that the generator is pure, say $v_1 \wedge ... \wedge v_n$. Define $\phi : F \to (\wedge^n F)^n \cong A^n$ by $x \mapsto (v_1 \wedge ... \wedge v_{i-1} \wedge x \wedge v_{i+1} \wedge ... \wedge v_n)_{1 \leq i \leq n}$. Then $\phi$ is linear and satisfies $\phi(v_i)=e_i$. – Martin Brandenburg Jul 19 2011 at 15:31 1 Thus $v_1,...,v_n$ span a free submodule $U$ of $F$ and for $V:=\ker(\phi)$ we have $F = U \oplus V$. Thus $\wedge^n F$ is the direct sum of $\wedge^n U, \wedge^{n-1} U \otimes V, ... , \wedge^n V$. However, by construction $\wedge^n U \to \wedge^n F$ is an isomorphism. Thus all the other summands are zero, in particular $\wedge^{n-1} U \otimes V = 0$. Since $\wedge^{n-1} U$ is a free module of rank $\binom{n}{n-1}=n>0$, it follows $V=0$. Thus $F=U$ is free. – Martin Brandenburg Jul 19 2011 at 15:44 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. I think that $\mathcal F$ is indeed locally free of rank $n$: Pick a point $x\in X$. It will be enough to show that there is a neighbourhood of $x$ on which $\mathcal F$ is free of rank $n$. Now, the exterior power commutes with pullbacks (aka scalar extensions) so that in particular the fibre (in the sense of pullback to $\mathrm{Spec}k(x)$) of $\Lambda^m\mathcal F$ at $x$ equals `$\Lambda^m\mathcal{F}_x$`. This shows that `$\mathcal{F}_x$` is an $n$-dimensional vector space. After possibly shrinking $X$ we may assume that there is a an `$\mathcal{O}_X$`-map `$f\colon \mathcal{O}_X^n\to \mathcal F$` which induces an isomorphism on fibres at $x$. Thus $\Lambda^nf$ is a map between locally free modules (of rank $1$) that gives an isomorphism on fibres at $x$ and hence is an isomorphism in a neighbourhood of $x$ so that we may assume that it is a global isomorphism. The wedge product induces pairings $\mathcal{F}\times\Lambda^{n-1}\mathcal{F}\to \Lambda^{n}\mathcal{F}$ and ```$\mathcal{O}_X^n\times\Lambda^{n-1}\mathcal{O}_X^n\to \Lambda^{n}\mathcal{O}_X^n$```, the latter being a perfect pairing. Composing the second with $\Lambda^nf$ gives a pairing `$\mathcal{O}_X^n\times\Lambda^{n-1}\mathcal{O}_X^n\to \Lambda^{n}\mathcal{F}$`. As $\Lambda^\ast f$ is multiplicative we get that the composite ```$$\mathcal{O}_X^n\xrightarrow{f}\mathcal{F}\to \mathrm{Hom}(\Lambda^{n-1}F,\Lambda^{n}\mathcal{F})\xrightarrow{\Lambda^{n-1}f^*}\mathrm{Hom}(\Lambda^{n-1}\mathcal{O}_X^n,\Lambda^{n}\mathcal{F})$$``` equals the map induced by the pairing for `$\mathcal{O}_X^n$`. This is an isomorphism (as `$\mathcal{O}_X^n$` is free of rank $n$ and $\Lambda^nf$ is an isomorphism) so we get that $f$ is split injective and we may write $\mathcal{F}$ as `\mathcal{O}_X^n\bigoplus \mathcal G$` for some quasi-coherent sheaf $\mathcal{G}$. Now, `$\Lambda^n(\mathcal{O}_X^n\bigoplus \mathcal{G})$` splits up as $$\bigoplus_{i+j=n}\Lambda^i\mathcal{O}_X^n\bigotimes \Lambda^j\mathcal{G}$$ and $\Lambda^nf$ is the inclusion into the $j=0$ factor. As that inclusion is an isomorphism, the other factors are zero but $\Lambda^{n-1}\mathcal{O}_X^n\bigotimes \Lambda^1\mathcal{G}$ has $\mathcal{G}$ as a direct factor and hence $\mathcal{G}=0$. - Great! This equals Nick's proof (generalized to arbitrary $n$ as I've done in the comments there). – Martin Brandenburg Jul 19 2011 at 15:40 Very nice, Torsten: this explains a lot! – Georges Elencwajg Jul 19 2011 at 16:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 88, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9365273118019104, "perplexity_flag": "head"}
http://unapologetic.wordpress.com/2008/01/18/the-heine-borel-theorem/?like=1&source=post_flair&_wpnonce=c327cf8c6a	# The Unapologetic Mathematician ## The Heine-Borel Theorem We’ve talked about compact subspaces, particularly of compact spaces and Hausdorff spaces (and, of course, compact Hausdorff spaces). So how can we use this to understand the space $\mathbb{R}$ of real numbers, or higher-dimensional versions like $\mathbb{R}^n$? First off, $\mathbb{R}$ is Hausdorff, which should be straightforward to prove. Unfortunately, it’s not compact. To see this, consider the open sets of the form $(-x,x)$ for all positive real numbers $x$. Given any real number $y$ we can find an $x$ with $\|y\|<x$, so $y\in(-x,x)$. Therefore the collection of these open intervals covers $\mathbb{R}$. But if we take any finite number of them, one will be the biggest, and so we must miss some real numbers. This open cover does not have a finite subcover, and $\mathbb{R}$ is not compact. We can similarly show that $\mathbb{R}^n$ is Hausdorff, but not compact. So, since $\mathbb{R}^n$ is Hausdorff, any compact subset of $\mathbb{R}^n$ must be closed. But not every closed subset is compact. What else does compactness imply? Well, we can take the proof that $\mathbb{R}^n$ isn’t compact and adapt it to any subset $A\subseteq\mathbb{R}^n$. We take the collection of all open “cubes” $(-x,x)^n$ consisting of $n$-tuples of real numbers, each of which is between $-x$ and $x$, and we form open subsets of $A$ by the intersections $U_x=(-x,x)^n\cap A$. Now the only way for there to be a finite subcover of this open cover of $A$ is for there to be some $x$ so that $U_x=A$. That is, every component of every point of $A$ has absolute value less than $x$, and so we say that $A$ is “bounded”. We see now that every compact subset of $\mathbb{R}^n$ is closed and bounded. It turns out that being closed and bounded is not only necessary for compactness, but they’re also sufficient! To see this, we’ll show that the closed cube $\left[-x,x\right]^n$ is compact. Then a bounded set $A$ is contained in some such cube, and a closed subset of a compact space is compact. This is the Heine-Borel theorem. In the $n=1$ case, we just need to see that the interval $\left[-x,x\right]$ is compact. Take an open cover $\{U_i\}$ of this interval, and define the set $S$ to consist of all $y\in\left[-x,x\right]$ so that a finite collection of the $U_i$ cover $\left[-x,y\right]$. Then define $t$ to be the least upper bound of $S$. Basically, $t$ is as far along the interval as we can get with a finite number of sets, and we’re hoping to show that $t=x$. Clearly it can’t go past $x$, since $S\subseteq\left[-x,x\right]$. But can it be less than $x$? In fact it can’t, because if it were, then we can find some open set $U$ from the cover that contains $t$. As an open neighborhood of $t$, the set $U$ contains some interval $(t-\epsilon,t+\epsilon)$. Then $t-\epsilon$ must be in $S$, and so there is some finite collection of the $U_i$ which covers $\left[-x,t-\epsilon\right]$. But then we can just add in $U$ to get a finite collection of the $U_i$ which covers $\left[-x,t+\frac{\epsilon}{2}\right]$, and this contradicts the fact that $t$ is the supremum of $S$. Thus $t=x$ and there is a finite subcover of $\left[-x,x\right]$, making this closed interval compact! Now Tychonoff’s Theorem tells us that products of closed intervals are also compact. In particular, the closed cube $\left[-x,x\right]^n\subseteq\mathbb{R}^n$ is compact. And since any closed and bounded set is contained in some such cube, it will be compact as a closed subspace of a compact space. Incidentally, since $n$ is finite, we don’t need to wave the Zorn talisman to get this invocation of the Tychonoff magic to work. As a special case, we can look back at the one-dimensional case to see that a compact, connected space must be a closed interval $\left[a,b\right]$. Then we know that the image of a connected space is connected, and that the image of a compact space is compact, so the image of a closed interval under a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ is another closed interval. The fact that this image is an interval gave us the intermediate value theorem. The fact that it’s closed now gives us the extreme value theorem: a continuous, real-valued function $f$ on a closed interval $\left[a,b\right]$ attains a maximum and a minimum. That is, there is some $c\in\left[a,b\right]$ so that $f(c)\geq f(x)$ for all $x\in\left[a,b\right]$, and similarly there is some $d\in\left[a,b\right]$ so that $f(d)\leq f(x)$ for all $x\in\left[a,b\right]$. About these ads Like Loading... ## 6 Comments » 1. [...] Okay, the Heine-Borel theorem tells us that a continuous real-valued function on a compact space takes a maximum and a [...] Pingback by \| January 21, 2008 \| Reply 2. [...] Maxima and Minima From Heine-Borel we know that a continuous function on a closed interval takes a global maximum and a minimum. [...] Pingback by \| January 25, 2008 \| Reply 3. [...] the Heine-Borel theorem says that this sphere, being both closed and bounded, is a compact subset of . We’ll define a [...] Pingback by \| November 13, 2009 \| Reply 4. [...] collection of points with one or two spheres. These spheres are closed and bounded, and thus compact. The collection must be finite or else it would have an accumulation point by Bolzano-Weierstrass, [...] Pingback by \| March 1, 2010 \| Reply 5. [...] corresponding , and so we find that is a closed interval covered by the open intervals . But the Heine-Borel theorem says that is compact, and so we can find a finite collection of the which cover . Renumbering the [...] Pingback by \| April 15, 2010 \| Reply 6. [...] linear operators — is continuous by our assumption, the function is continuous as well. The extreme value theorem tells us that since is compact this continuous function must attain a maximum, which we call [...] Pingback by \| May 4, 2011 \| Reply « Previous \| Next » ## About this weblog This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”). I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now. • ## RSS Feeds RSS - Posts RSS - Comments • ## Feedback Got something to say? Anonymous questions, comments, and suggestions at Formspring.me! %d bloggers like this:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 73, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9503511786460876, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/317874/calculate-the-angle-between-two-vectors	# Calculate the angle between two vectors I comes from Stack Overflow and I though my question was more related to this forum. The problem is i'm not a matematician, so please excuse me if I ask a dumb question. I'm trying to get the angle between two vectors. As numbers of posts says, here or here, I tried this solution. But my angle must be "oriented": If th angle between u⃗ and v⃗ is θ, the angle between v⃗ and u⃗ must be -θ. Is there a mathematical solution to this? Edit : Here's the formula I implemented for the points $a = (x_1, y_1)$ and $b = (x_2, y_2)$ representing the vectors: $$\mathrm{angle} = \arccos \left(\frac{x_1 \cdot x_2 + y_1 \cdot y_2}{\sqrt{x_1^2+y_1^2} \cdot \sqrt{x_2^2+y_2^2}} \right)$$ - vectors in 2D or 3D? – Emanuele Paolini Mar 1 at 14:05 Sorry, it's 2d vectors. – Martin Mar 1 at 14:14 What you did used the dot product, which is usually the standard way to find angles. However, because of your requirement for angles to be oriented, the dot product will not work. I'll elaborate a bit more in my answer where I can more freely LaTeX. – Draksis Mar 1 at 14:20 For some basic information about writing math at this site see e.g. here, here, here and here. – Américo Tavares Mar 1 at 14:25 This formula does not take orientation into account. – julien Mar 1 at 14:35 ## 2 Answers I assume $u$ and $v$ are both nonzero. Let $\theta\in (-\pi,\pi]$ modulo $2\pi$ be the oriented angle between $u$ and $v$. Using $$\cos\theta=\frac{(u,v)}{\\|u\\|\\|v\\|}$$ you can find the value of $\cos\theta$. Taking $\arccos$ of the latter will, you get $\theta_0$ in $[0,\pi]$ such that $$\theta=\theta_0 \quad\mbox{mod} \;2\pi\quad\mbox{or}\quad \theta=-\theta_0 \quad\mbox{mod} \;2\pi.$$ Now to determine the orientation of $(u,v)$, you must compute the $2\times 2$ determinant of the matrix whose first column is $u$, and second column is $v$. If this is $0$, this means $u$ and $v$ are parallel. Write $u=\lambda v$. If $\lambda >0$, then $\theta=0$ mod $2\pi$. If $\lambda<0$, then $\theta=\pi$ mod $2\pi$. If the determinant is positive, this means $\theta=\theta_0$ modulo $2\pi$. If the determinant is negative, you have $\theta=-\theta_0$ modulo $2\pi$. - Yes, there it is! Calculating the determinant was the solution. Thanks! – Martin Mar 1 at 14:47 Find the cross product of the two vectors, then divide by the magnitudes of each vector, and take the inverse sine. For example, say the vectors were $a = (4,3)$ and $b = (6,8)$. The cross product length is $a_1b_2-a_2b_1 = 4 \times 8 - 6 \times 3 = 14$. The length of the cross product is $\|a\|\|b\|\sin(\theta) = 14$, and since $\|a\| = 5$ and $\|b\| = 10$, $\sin(\theta) = 14/50 = 0.28$. Taking the inverse sine, one obtains the angle from $a$ to $b$ as being $16.26 ^\circ$. Notice that if we had reversed $a$ and $b$, the cross product vector length would have been $-14$, leading to an angle of $-16.26 ^\circ$. - The cross product is a vector. If you divide by magnitudes, it remains a vector. So it does not make sense to take the inverse sine. You simply forgot the word length, or magnitude. – julien Mar 1 at 14:19 Thanks. Editing... – Draksis Mar 1 at 14:20 The length of a vector is always nonnegative. So you will never find $-14$. – julien Mar 1 at 14:21 Thanks for the reply, but now θ is evoluating between `[-π/2, π/2]` instead of `[-π, π]`. – Martin Mar 1 at 14:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 45, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.927058756351471, "perplexity_flag": "head"}
http://mathhelpforum.com/advanced-applied-math/131266-help-me-ques.html	# Thread: 1. ## Help me with this ques. Hey i have a question i cant solve it but m very curious to know its solution Help me!! Here it is: Two rays are drawn through a point at an angle of 30.A point B is taken on one of them at a distance d from the point A.A perpendicular is drawn from the point B to other ray and another perpendicular is drawn from as foot to meet AB at another point from where similar process is repeated indefinitely. Calculate the length of the resulting infinite polygonal line. 2. Originally Posted by mrbains21 Hey i have a question i cant solve it but m very curious to know its solution Help me!! Here it is: Two rays are drawn through a point at an angle of 30.A point B is taken on one of them at a distance d from the point A.A perpendicular is drawn from the point B to other ray and another perpendicular is drawn from as foot to meet AB at another point from where similar process is repeated indefinitely. Calculate the length of the resulting infinite polygonal line. I suppose point A is the intersection point of both rays...then, after choosing point B on ray 1 and drawing from it a perpendicular to ray 2 you get a straight-angle triangle 30-60-90, sometimes aka golden triangle, with hipotenuse d, and thus the leg opposite to the 90 deg. angle is half the hipotenuse ==> the first part of the polygonal line is $\frac{1}{2}d$, and the other leg's length is, by Pythagoras, $\frac{\sqrt{3}}{2}d$. Repeat the process, again you get a 30-60-90 triangle but with hipotenuse $=\frac{\sqrt{3}}{2}d$, so this time the polygonal line's length is $\frac{\sqrt{3}}{4}d$ , and etc. The whole polygonal line's length is thus $\sum\limits_{k=0}^\infty \frac{1}{2}\left(\frac{\sqrt{3}}{2}\right)^k=\frac {\frac{1}{2}}{1-\frac{\sqrt{3}}{2}}$ $=\frac{1}{2-\sqrt{3}}=2+\sqrt{3}$ Tonio 3. Originally Posted by tonio I suppose point A is the intersection point of both rays...then, after choosing point B on ray 1 and drawing from it a perpendicular to ray 2 you get a straight-angle triangle 30-60-90, sometimes aka golden triangle, with hipotenuse d, and thus the leg opposite to the 90 deg. angle is half the hipotenuse ==> the first part of the polygonal line is $\frac{1}{2}d$, and the other leg's length is, by Pythagoras, $\frac{\sqrt{3}}{2}d$. Repeat the process, again you get a 30-60-90 triangle but with hipotenuse $=\frac{\sqrt{3}}{2}d$, so this time the polygonal line's length is $\frac{\sqrt{3}}{4}d$ , and etc. The whole polygonal line's length is thus $\sum\limits_{k=0}^\infty \frac{1}{2}\left(\frac{\sqrt{3}}{2}\right)^k=\frac {\frac{1}{2}}{1-\frac{\sqrt{3}}{2}}$ $=\frac{1}{2-\sqrt{3}}=2+\sqrt{3}$ Tonio when i solved it i got d(2+sqrt(3)) 4. Originally Posted by mrbains21 when i solved it i got d(2+sqrt(3)) Of course: I just forgot to multiply by d all along, but d appears in my post , too. Tonio	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9200615882873535, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/tagged/farey-sequences	# Tagged Questions The Farey sequence of order $n$ is the sequence of all lowest-terms fractions between 0 and 1 whose denominators do not exceed $n$, in increasing order. 1answer 59 views ### How to prove that construction of Farey sequence by mediant is coverage? Farey sequence of order $n+1$ ($F_{n+1}$) can be construct by adding mediant value (${a+c \over b+d}$) into $F_{n}$, where ${a \over b}$ and ${c \over d}$ are consecutive term in $F_{n}$, and \$b+d = ... 1answer 95 views ### Farey sequences for polynomials? Does a notion of Farey sequence (or something equivalent) exist for polynomials over finite fields? 3answers 158 views ### Constructing Farey sequences inductively Objective: I'd like to prove that $F_{n+1}$ (the Farey sequence of order $n+1$) is obtained form the Farey sequence $F_n$ of order $n$ by adding all fractions of the form $\frac{a+c}{b+d}$ when ... 1answer 103 views ### What is the sum of the squares of the differences of consecutive element of a Farey Sequence A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example \$F_6= ...	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9327451586723328, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/86259/mathbbq-sqrt1-sqrt2-is-galois-over-mathbbq?answertab=votes	# $\mathbb{Q}(\sqrt{1-\sqrt{2}})$ is Galois over $\mathbb{Q}$ I am trying to show that $E = \mathbb{Q}\left(\sqrt{1-\sqrt{2}}\right)$ is galois over $\mathbb{Q}$. The extension has the minimal polynomial $$\left(x-\sqrt{1-\sqrt{2}}\right)\left(x+\sqrt{1-\sqrt{2}}\right)\left(x-\sqrt{1+\sqrt{2}}\right)\left(x+\sqrt{1+\sqrt{2}}\right)$$ but I can't manage to show using elementary arithmetic that $\sqrt{1+\sqrt{2}}$ is in $E$ (i.e. multiplying, adding etc.) so that its the splitting field. The trick would be to use that $\sqrt{1+\sqrt{2}} = \frac{i}{\sqrt{1-\sqrt{2}}}$ but we only know that $i\sqrt{\sqrt{2}-1}$ is in the field, we dont know if we have $i$. Can someone give a hint on how to proceed, maybe there is a theorem I could use that I'm not thinking of. - 6 Maybe I am mistaken, but if $E$ was galois this would imply that $E = \mathbb{Q}\left(\sqrt{1+\sqrt{2}}\right)$, right? But $\mathbb{Q}\left(\sqrt{1+\sqrt{2}}\right) \subset \mathbb{R}$ which is not the case for $E$. – Matthias Klupsch Nov 28 '11 at 5:09 1 Perhaps the problem is flawed? It was made up by my teacher so this is a possibility.. – AnonymousCoward Nov 28 '11 at 5:27 3 I think there are issues with the problem, though I might be wrong. The minimal polynomial of $\sqrt{1 - \sqrt{2}}$ is $f(x) := x^{4} - 2x^{2} - 1$ which has 2 real roots and 2 imaginary roots. If $E/\mathbb{Q}$ was a Galois extension, it would be the splitting field of $f$, but $E \subset \mathbb{R}$ and hence $E$ can't be the splitting field of $f$, a contradiction. – Shayla Nov 28 '11 at 6:00 5 My uneducated guess is that your teacher meant to write down something like $\sqrt{2 - \sqrt{2}}$. – Dylan Moreland Nov 28 '11 at 6:11 3 @GottfriedLeibniz: Adjoining $\sqrt{1+\sqrt{2}}$ gives you an extension of degree 4 as does adjoining with $\sqrt{1 - \sqrt{2}}$. If $\mathbb{Q}(\sqrt{1-\sqrt{2}})$ were to contain all four roots you would have $\mathbb{Q}(\sqrt{1+\sqrt{2}}) \subset \mathbb{Q}(\sqrt{1-\sqrt{2}})$ and by equality of degrees you have equality of the extensions. – Matthias Klupsch Nov 28 '11 at 7:04 show 2 more comments ## 1 Answer The given field $E = \mathbb{Q}\left(\sqrt{1-\sqrt{2}}\right)$ is not a galois extension. If it were it would contain $\sqrt{1 + \sqrt{2}}$ as stated in the question and then $\mathbb{Q}\left(\sqrt{1-\sqrt{2}}\right)$ would contain $\mathbb{Q}\left(\sqrt{1+\sqrt{2}}\right)$ which is an extension of degree 4 as is $\mathbb{Q}\left(\sqrt{1-\sqrt{2}}\right)$ (the minimal polynomial is $X^4-2X^2-1$ for both numbers), thus this inclusion would be an equality which is not the case because $\mathbb{Q}\left(\sqrt{1+\sqrt{2}}\right) \subset \mathbb{R}$ and $\sqrt{1-\sqrt{2}} \in \mathbb{C}$ but $\sqrt{1-\sqrt{2}} \notin \mathbb{R}$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 34, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.977527379989624, "perplexity_flag": "head"}
http://mathhelpforum.com/pre-calculus/89400-half-life-decay.html	# Thread: 1. ## Half Life and Decay Hi guys, New to this forum so let me know If I am doing anything wrong. My question: Ive decided to take a calculus course after many years of not doing any math.. If you can give me hints in regards to this question it would be greatly appreciated. A 96-milligram sample of a radioactive substance decays according to the equation N=96 ·e-0.034 ·t where N is the number of milligrams present after t years. (a) Find the half-life of the substance to the nearest tenth of a year. TIA 2. Originally Posted by mvho Hi guys, New to this forum so let me know If I am doing anything wrong. My question: Ive decided to take a calculus course after many years of not doing any math.. If you can give me hints in regards to this question it would be greatly appreciated. A 96-milligram sample of a radioactive substance decays according to the equation N=96 ·e-0.034 ·t where N is the number of milligrams present after t years. (a) Find the half-life of the substance to the nearest tenth of a year. TIA The general form of exponential decay is $N(t)=N_0e^{-kt}$ where • $N(t)$ is the amount at a given time t • $N_0$ is the amount at t=0 • $k$ is a positive constant • $t$ is time From the definition of half life $\frac{N_0}{2} = N_0e^{-kt_{1/2}}$ and so $N_0$ will cancel $\frac{1}{2} = e^{-kt_{1/2}}$ $ln(\frac{1}{2}) = -kt_{1/2}$ Bear in mind that $ln(\frac{1}{2}) = ln(2^{-1}) = -ln(2)$ $-ln(2) = -kt_{1/2}$ $t_{1/2} = \frac{ln(2)}{k}$ From your equation k = 0.034 so the half life is $\frac{ln(2)}{0.034}$ 3. Thank you very much. Oh man, I need to review my logs To summarize, I am given K or some variable. Plug known variable in and solve for Time in this case. I understand looking at your steps, but I have a stupid question to ask: Why do we ln both sides again? Was it to bring down the exponent? Its been a really, really, long time.. 4. Yes; the only way to get the power down where you could work with it was to take logs. 5. Originally Posted by mvho Thank you very much. Oh man, I need to review my logs To summarize, I am given K or some variable. Plug known variable in and solve for Time in this case. I understand looking at your steps, but I have a stupid question to ask: Why do we ln both sides again? Was it to bring down the exponent? Its been a really, really, long time.. Yeah, $ln(x) = log_e(x)$ and logs and exponentials are inverse operations like adding and subtracting. In my post everything except the last two lines is deriving the general half life formula so if you don't need to know it's derivation you can stick with $<br /> t_{1/2} = \frac{ln(2)}{k}<br />$ k will always be the number next to t and will be a power along with t as well as k > 0. The units will be 1 over the units of k for any exponent must be dimensionless. If k is in per minute, t will be in minutes 6. Thanks everyone. Never even knew this site existed, will have to share with everyone. Math is funny; once you get going, it seems all your tools come back pretty quickly.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9480099678039551, "perplexity_flag": "middle"}
http://en.wikipedia.org/wiki/Ito_process	# Itō calculus (Redirected from Ito process) Itō integral of a Brownian motion with respect to itself. Some or all of the formulas presented in this article have missing or incomplete descriptions of their variables, symbols or constants which may create ambiguity or prevent full interpretation. Please assist in recruiting an expert or improve this article yourself. See the talk page for details. (November 2010) Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion (Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itō stochastic integral. This is a generalization of the ordinary concept of a Riemann–Stieltjes integral. The generalization is in two respects. Firstly, we are now dealing with random variables (more precisely, stochastic processes). Secondly, we are integrating with respect to a non-differentiable function (technically, stochastic process). The Itō integral allows one to integrate one stochastic process (the integrand) with respect to another stochastic process (the integrator). It is common for the integrator to be the Brownian motion (also see Wiener process). The result of the integration is another stochastic process. In particular, the integral from 0 to any particular t is a random variable. This random variable is defined as a limit of a certain sequence of random variables. (There are several equivalent ways to construct a definition.) Roughly speaking, we are choosing a sequence of partitions of the interval from 0 to t. Then we are constructing Riemann sums. However, it is important which point in each of the small intervals is used to compute the value of the function. Typically, the left end of the interval is used. (It is conceptualized in mathematical finance as that we are first deciding what to do, then observing the change in the prices. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment.) Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. The limit then is taken in probability as the mesh of the partition is going to zero. (Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions.) The usual notation for the Itō stochastic integral is: $Y_t=\int_0^t H_s\,dX_s$ where X is a Brownian motion or, more generally, a semimartingale and H is a locally square-integrable process adapted to the filtration generated by X (Revuz & Yor 1999, Chapter IV). The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. In particular, it is not differentiable at any point and has infinite variation over every time interval. As a result, the integral cannot be defined in the usual way (see Riemann–Stieltjes integral). The main insight is that the integral can be defined as long as the integrand H is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black–Scholes). Then, the Itō stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount Ht of the stock at time t. In this situation, the condition that H is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time. This prevents the possibility of unlimited gains through high frequency trading: buying the stock just before each uptick in the market and selling before each downtick. Similarly, the condition that H is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums (Revuz & Yor 1999, Chapter IV). Important results of Itō calculus include the integration by parts formula and Itō's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms. ## Notation The process Y defined as before as $Y_t = \int_0^t H\,dX\equiv\int_0^t H_s\,dX_s ,$ is itself a stochastic process with time parameter t, which is also sometimes written as Y = H · X (Rogers & Williams 2000). Alternatively, the integral is often written in differential form dY = H dX, which is equivalent to Y − Y0 = H · X. As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P}) .$ The sigma algebra Ft represents the information available up until time t, and a process X is adapted if Xt is Ft-measurable. A Brownian motion B is understood to be an Ft-Brownian motion, which is just a standard Brownian motion with the properties that Bt is Ft-measurable and that Bt+s − Bt is independent of Ft for all s,t ≥ 0 (Revuz & Yor 1999). ## Integration with respect to Brownian motion The Itō integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that B is a Wiener process (Brownian motion) and that H is a left-continuous, adapted and locally bounded process. If {πn} is a sequence of partitions of [0, t] with mesh going to zero, then the Itō integral of H with respect to B up to time t is a random variable $\int_0^t H \,d B =\lim_{n\rightarrow\infty} \sum_{[t_{i-1},t_i]\in\pi_n}H_{t_{i-1}}(B_{t_i}-B_{t_{i-1}}).$ It can be shown that this limit converges in probability. For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. If H is any predictable process such that ∫0t H2 ds < ∞ for every t ≥ 0 then the integral of H with respect to B can be defined, and H is said to be B-integrable. Any such process can be approximated by a sequence Hn of left-continuous, adapted and locally bounded processes, in the sense that $\int_0^t (H-H_n)^2\,ds\to 0$ in probability. Then, the Itō integral is $\int_0^t H\,dB = \lim_{n\to\infty}\int_0^t H_n\,dB$ where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itō isometry $\mathbb{E}\left[ \left(\int_0^t H_s \, dB_s\right)^2\right]=\mathbb{E} \left[ \int_0^t H_s^2\,ds\right ]$ which holds when H is bounded or, more generally, when the integral on the right hand side is finite. ## Itō processes An Itō process is defined to be an adapted stochastic process which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time, $X_t=X_0+\int_0^t\sigma_s\,dB_s + \int_0^t\mu_s\,ds.$ Here, B is a Brownian motion and it is required that σ is a predictable B-integrable process, and μ is predictable and (Lebesgue) integrable. That is, $\int_0^t(\sigma_s^2+\|\mu_s\|)\,ds<\infty$ for each t. The stochastic integral can be extended to such Itō processes, $\int_0^t H\,dX =\int_0^t H_s\sigma_s\,dB_s + \int_0^t H_s\mu_s\,ds.$ This is defined for all locally bounded and predictable integrands. More generally, it is required that Hσ be B-integrable and Hμ be Lebesgue integrable, so that $\int_0^t (H^2 \sigma^2 + \|H\mu\| )ds < \infty.$ Such predictable processes H are called X-integrable. An important result for the study of Itō processes is Itō's lemma. In its simplest form, for any twice continuously differentiable function f on the reals and Itō process X as described above, it states that f(X) is itself an Itō process satisfying $df(X_t)=f^\prime(X_t)\,dX_t + \frac{1}{2}f^{\prime\prime} (X_t) \sigma_t^2 \, dt.$ This is the stochastic calculus version of the change of variables formula and chain rule. It differs from the standard result due to the additional term involving the second derivative of f, which comes from the property that Brownian motion has non-zero quadratic variation. ## Semimartingales as integrators The Itō integral is defined with respect to a semimartingale X. These are processes which can be decomposed as X = M + A for a local martingale M and finite variation process A. Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process H the integral H · X exists, and can be calculated as a limit of Riemann sums. Let πn be a sequence of partitions of [0, t] with mesh going to zero, $\int_0^t H\,dX = \lim_{n\rightarrow\infty} \sum_{t_{i-1},t_i\in\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}}).$ This limit converges in probability. The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itō's Lemma, changes of measure via Girsanov's theorem, and for the study of stochastic differential equations. However, it is inadequate for other important topics such as martingale representation theorems and local times. The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem holds. That is, if Hn → ;H and \|Hn\| ≤ J for a locally bounded process J, then $\int_0^t H_n dX \to \int_0^t H dX,$ in probability. The uniqueness of the extension from left-continuous to predictable integrands is a result of the monotone class lemma. In general, the stochastic integral H · X can be defined even in cases where the predictable process H is not locally bounded. If K = 1 / (1 + \|H\|) then K and KH are bounded. Associativity of stochastic integration implies that H is X-integrable, with integral H · X = Y, if and only if Y0 = 0 and K · Y = (KH) · X. The set of X-integrable processes is denoted by L(X). ## Properties The following properties can be found for example in (Revuz & Yor 1999) and (Rogers & Williams 2000): • The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. • The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time t is Xt − Xt−, and is often denoted by ΔXt. With this notation, Δ(H · X) = H ΔX. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous. • Associativity. Let J, K be predictable processes, and K be X-integrable. Then, J is K · X integrable if and only if JK is X integrable, in which case $J\cdot (K\cdot X) = (JK)\cdot X$ • Dominated convergence. Suppose that Hn → H and \|Hn\| ≤ J, where J is an X-integrable process. then Hn · X → H · X. Convergence is in probability at each time t. In fact, it converges uniformly on compacts in probability. • The stochastic integral commutes with the operation of taking quadratic covariations. If X and Y are semimartingales then any X-integrable process will also be [X, Y]-integrable, and [H · X, Y] = H · [X, Y]. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process, $[H\cdot X]=H^2\cdot[X]$ ## Integration by parts As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itō integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itō calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If X and Y are semimartingales then $X_tY_t = X_0Y_0+\int_0^t X_{s-}\,dY_s + \int_0^t Y_{s-}\,dX_s + [X,Y]_t$ where [X, Y] is the quadratic covariation process. The result is similar to the integration by parts theorem for the Riemann–Stieltjes integral but has an additional quadratic variation term. ## Itō's lemma Main article: Itō's lemma Itō's lemma is the version of the chain rule or change of variables formula which applies to the Itō integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous d-dimensional semimartingale X = (X1,…,Xd) and twice continuously differentiable function f from Rd to R, it states that f(X) is a semimartingale and, $df(X_t)= \sum_{i=1}^d f_{i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^d f_{i,j}(X_{t})\,d[X^i,X^j]_t.$ This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xi,Xj ]. The formula can be generalized to non-continuous semimartingales by adding a pure jump term to ensure that the jumps of the left and right hand sides agree (see Itō's lemma). ## Martingale integrators ### Local martingales An important property of the Itō integral is that it preserves the local martingale property. If M is a local martingale and H is a locally bounded predictable process then H · M is also a local martingale. For integrands which are not locally bounded, there are examples where H · M is not a local martingale. However, this can only occur when M is not continuous. If M is a continuous local martingale then a predictable process H is M-integrable if and only if $\int_0^t H^2 d[M] <\infty,$ for each t, and H · M is always a local martingale. The most general statement for a discontinuous local martingale M is that if (H2 · [M])1/2 is locally integrable then H · M exists and is a local martingale. ### Square integrable martingales For bounded integrands, the Itō stochastic integral preserves the space of square integrable martingales, which is the set of càdlàg martingales M such that E[Mt2] is finite for all t. For any such square integrable martingale M, the quadratic variation process [M] is integrable, and the Itō isometry states that $\mathbb{E}\left [(H\cdot M_t)^2\right ]=\mathbb{E}\left [\int_0^t H^2\,d[M]\right ].$ This equality holds more generally for any martingale M such that H2 · [M]t is integrable. The Itō isometry is often used as an important step in the construction of the stochastic integral, by defining H · M to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes. ### p-Integrable martingales For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales. These are càdlàg martingales such that E(\|Mt\|p) is finite for all t. However, this is not always true in the case where p = 1. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales. The maximum process of a càdlàg process M is written as Mt = sups ≤t \|Ms\|. For any p ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales M such that E[(Mt)p] is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities. The Burkholder–Davis–Gundy inequalities state that, for any given p ≥ 1, there exist positive constants c, C that depend on p, but not M or on t such that $c\mathbb{E} \left [ [M]_t^{\frac{p}{2}} \right ] \le \mathbb{E}\left [(M^_t)^p \right ]\le C\mathbb{E}\left [ [M]_t^{\frac{p}{2}} \right ]$ for all càdlàg local martingales M. These are used to show that if (Mt)p is integrable and H is a bounded predictable process then $\mathbb{E}\left [ ((H\cdot M)_t^*)^p \right ] \le C\mathbb{E}\left [(H^2\cdot[M]_t)^{\frac{p}{2}} \right ]<\infty$ and, consequently, H · M is a p-integrable martingale. More generally, this statement is true whenever (H2 · [M])p/2 is integrable. ## Existence of the integral Proofs that the Itō integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such simple predictable processes are linear combinations of terms of the form Ht = A1{t > T} for stopping times T and FT-measurable random variables A, for which the integral is $H\cdot X_t\equiv \mathbf{1}_{\{t>T\}}A(X_t-X_T).$ This is extended to all simple predictable processes by the linearity of H · X in H. For a Brownian motion B, the property that it has independent increments with zero mean and variance Var(Bt) = t can be used to prove the Itō isometry for simple predictable integrands, $\mathbb{E} \left [ (H\cdot B_t)^2\right ] = \mathbb{E} \left [\int_0^tH_s^2\,ds\right ].$ By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying $\mathbb{E} \left[ \int_0^t H^2 ds \right ] < \infty,$ in such way that the Itō isometry still holds. It can then be extended to all B-integrable processes by localization. This method allows the integral to be defined with respect to any Itō process. For a general semimartingale X, the decomposition X = M + A for a local martingale M and finite variation process A can be used. Then, the integral can be shown to exist separately with respect to M and A and combined using linearity, H · X = H · M + H · A, to get the integral with respect to X. The standard Lebesgue–Stieltjes integral allows integration to be defined with respect to finite variation processes, so the existence of the Itō integral for semimartingales will follow from any construction for local martingales. For a càdlàg square integrable martingale M, a generalized form of the Itō isometry can be used. First, the Doob–Meyer decomposition theorem is used to show that a decomposition M2 = N + <M> exists, where N is a martingale and <M> is a right-continuous, increasing and predictable process starting at zero. This uniquely defines <M>, which is referred to as the predictable quadratic variation of M. The Itō isometry for square integrable martingales is then $\mathbb{E} \left [(H\cdot M_t)^2\right ]= \mathbb{E} \left [\int_0^tH^2_s\,d\langle M\rangle_s\right],$ which can be proved directly for simple predictable integrands. As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying E[H2 · <M>t] < ∞. This method can be extended to all local square integrable martingales by localization. Finally, the Doob–Meyer decomposition can be used to decompose any local martingale into the sum of a local square integrable martingale and a finite variation process, allowing the Itō integral to be constructed with respect to any semimartingale. Many other proofs exist which apply similar methods but which avoid the need to use the Doob–Meyer decomposition theorem, such as the use of the quadratic variation [M] in the Itō isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itō isometry. The latter applies directly to local martingales without having to first deal with the square integrable martingale case. Alternative proofs exist only making use of the fact that X is càdlàg, adapted, and the set {H · Xt: \|H\| ≤ 1 is simple previsible} is bounded in probability for each time t, which is an alternative definition for X to be a semimartingale. A continuous linear extension can be used to construct the integral for all left-continuous and adapted integrands with right limits everywhere (caglad or L-processes). This is general enough to be able to apply techniques such as Itō's lemma (Protter 2004). Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands (Bichteler 2002). ## Differentiation in Itō calculus The Itō calculus is first and foremost defined as an integral calculus as outlined above. However, there are also different notions of "derivative" with respect to Brownian motion: ### Malliavin derivative Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula (Nualart 2006). ### Martingale representation The following result allows to express martingales as Itô integrals: if M is a square-integrable martingale on a time interval [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted square integrable process α on [0, T] such that $M_{t} = M_{0} + \int_{0}^{t} \alpha_{s} \, \mathrm{d} B_{s}$ almost surely, and for all t ∈ [0, T] (Rogers & Williams 2000, Theorem 36.5). This representation theorem can be interpreted formally as saying that α is the “time derivative” of M with respect to Brownian motion B, since α is precisely the process that must be integrated up to time t to obtain Mt − M0, as in deterministic calculus. ## Itō calculus for physicists In physics, usually stochastic differential equations, also called Langevin equations, are used, rather than general stochastic integrals. A physicist would formulate an Itō stochastic differential equation (SDE) as $\dot{x}_k=h_k+g_{kl} \xi_l,$ where $\xi_j$ is Gaussian white noise with $\langle\xi_k(t_1)\,\xi_l(t_2)\rangle=\delta_{kl}\delta(t_1-t_2)$ and Einstein's summation convention is used. If $y=y(x_k)$ is a function of the xk, then Itō's lemma has to be used: $\dot{y}=\frac{\partial y}{\partial x_j}\dot{x}_j+\tfrac{1}{2}\frac{\partial^2 y}{\partial x_k \, \partial x_l} g_{km}g_{ml}.$ An Itō SDE as above also corresponds to a Stratonovich SDE which reads $\dot{x}_k = h_k + g_{kl} \xi_l - \frac{1}{2} \frac{\partial g_{kl}}{\partial {x_m}} g_{ml}.$ SDEs frequently occur in physics in Stratonovich form, as limits of stochastic differential equations driven by colored noise if the correlation time of the noise term approaches zero. For a recent treatment of different interpretations of stochastic differential equations see for example (Lau & Lubensky 2007). ## References • Bichteler, Klaus (2002), Stochastic Integration With Jumps (1st ed.), Cambridge University Press, ISBN 0-521-81129-5 • Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore); Paperback ISBN 981-238-107-4. Fifth edition available online: PDF-files, with generalizations of Itō's lemma for non-Gaussian processes. • He, Sheng-wu; Wang, Jia-gang; Yan, Jia-an (1992), Semimartingale Theory and Stochastic Calculus, Science Press, CRC Press Inc., ISBN 7-03-003066-4,0-8493-7715-3 Check \|isbn= value (help) • Karatzas, Ioannis; Shreve, Steven (1991), Brownian Motion and Stochastic Calculus (2nd ed.), Springer, ISBN 0-387-97655-8 • Lau, Andy; Lubensky, Tom (2007), "State-dependent diffusion", Phys. Rev. E 76 (1): 011123, doi:10.1103/PhysRevE.76.011123 • Nualart, David (2006), The Malliavin calculus and related topics, Springer, ISBN 3-540-28328-5 • Øksendal, Bernt K. (2003), Stochastic Differential Equations: An Introduction with Applications, Berlin: Springer, ISBN 3-540-04758-1 • Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4 • Revuz, Daniel; Yor, Marc (1999), Continuous martingales and Brownian motion, Berlin: Springer, ISBN 3-540-57622-3 • Rogers, Chris; Williams, David (2000), Diffusions, Markov processes and martingales - Volume 2: Itô calculus, Cambridge: Cambridge University Press, ISBN 0-521-77593-0 • Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 33, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.885794460773468, "perplexity_flag": "head"}
http://www.amamanualofstyle.com/view/10.1093/jama/9780195176339.001.0001/med-9780195176339-div1-215?rskey=qHBfrc&result=1&q=	• or Create Advanced Search ## AMA Manual of Style Committee Publisher: Oxford University Press Print Publication Date: Mar 2007 Print ISBN-13: 9780195176339 Published to AMA Manual of Style: Apr 2009 DOI: 10.1093/jama/9780195176339.001.0001 • Print • Save • Cite • Email • Share • Text size: A • A # Glossary of Statistical Terms Chapter: Study Design and Statistics Author(s): ### Stephen J. Lurie UPDATE: Although our style manual recommends (Section 20.9, page 888 in the print) that "[expressing] P to more than 3 significant digits does not add useful information to P<.001," in certain types of studies (particularly GWAS [genome-wide association studies] and other studies in which there are adjustments for multiple comparisons, such as Bonferroni correction, and the definition of level of significance is substantially less than P<.05) it may be important to express P values to more significant digits. For example, if the threshold of significance is P<.0004, then by definition the P value must be expressed to at least 4 digits to indicate whether a result is statistically significant. GWAS express P values to very small numbers, using scientific notation. If a manuscript you are editing defines statistical significance as a P value substantially less than .05, possibly even using scientific notation to express P values to very small numbers, it is best to retain the values as the author presents them. This change was made August 16, 2011. In the glossary that follows, terms defined elsewhere in the glossary are printed in this font. An arrowhead (→) indicates points to consider in addition to the definition. For detailed discussion of these terms, the referenced texts and the resource list at the end of the chapter are useful sources. Eponymous names for statistical procedures often differ from one text to another (eg, the Newman-Keuls and Student-Newman-Keuls test). The names provided in this glossary follow the Dictionary of Statistical Terms38 published for the International Statistical Institute. Although statistical texts use the possessive form for most eponyms, the possessive form for eponyms is not used in JAMA and the Archives Journals (see 16.0, Eponyms). Most statistical tests are applicable only under specific circumstances, which are generally dictated by the scale properties of both the independent variable and the dependent variable. Table 3 presents a guide to selection of commonly used statistical techniques. This table is not meant to be exhaustive but rather to indicate the appropriate applications of commonly used statistical techniques. Table 3. Selection of Commonly Used Statistical Techniquesa Scale of Measurement Intervalb Ordinal Nominalc 2 Treatment groups Unpaired t test Mann-Whitney rank sum test χ2 Analysis-of-contingency table; Fisher exact test if ≤6 in any cell ≥3 Treatment groups Analysis of variance Kruskal-Wallis statistic χ2 Analysis-of-contingency table; Fisher exact test if ≤6 in any cell Before and after 1 treatment in same individual Paired t test Wilcoxon signed rank test McNemar test Multiple treatments in same individual Repeated-measures analysis of variance Friedman statistic Cochran Q Association between 2 variables Linear regression and Pearson product moment correlation Spearman rank correlation Contingency coefficients a Adapted with permission from Glantz, Primer of Biostatistics.39 © The McGraw-Hill Companies, Inc. b Assumes normally distributed data. If data are not normally distributed, then rank the observations and use the methods for data measured on an ordinal scale. c For a nominal dependent variable that is time dependent (such as mortality over time), use life-table analysis for nominal independent variables and Cox regression for continuous and/or nominal independent variables. • abscissa: horizontal or x-axis of a graph. • absolute risk: probability of an event occurring during a specified period. The absolute risk equals the relative risk times the average probability of the event during the same time, if the risk factor is absent.40(p327) See absolute risk reduction. • absolute risk reduction: proportion in the control group experiencing an event minus the proportion in the intervention group experiencing an event. The inverse of the absolute risk reduction is the number needed to treat. See absolute risk. • accuracy: ability of a test to produce results that are close to the true measure of the phenomenon.40(p327) Generally, assessing accuracy of a test requires that there be a criterion standard with which to compare the test results. Accuracy encompasses a number of measures including reliability, validity, and lack of bias. • actuarial life-table method: see life table, Cutler-Ederer method. • adjustment: statistical techniques used after the collection of data to adjust for the effect of known or potential confounding variables.40(p327) A typical example is adjusting a result for the independent effect of age of the participants (age is the independent variable). • aggregate data: data accumulated from disparate sources. • agreement: statistical test performed to determine the equivalence of the results obtained by 2 tests when one test is compared with another (one of which is usually but not always a criterion standard). → Agreement should not be confused with correlation. Correlation is used to test the degree to which changes in a variable are related to changes in another, whereas agreement tests whether 2 variables are equivalent. For example, an investigator compares results obtained by 2 methods of measuring hematocrit. Method A gives a result that is always exactly twice that of method B. The correlation between A and B is perfect since A is always twice B, but the agreement is very poor; method A is not equivalent to method B (written communication, George W. Brown, MD, September 1993). One appropriate way to assess agreement has been described by Bland and Altman.41 • algorithm: systematic process carried out in an ordered, typically branching sequence of steps; each step depends on the outcome of the previous step.42(p6) An algorithm may be used clinically to guide treatment decisions for an individual patient on the basis of the patient’s clinical outcome or result. • α (alpha), α level: size of the likelihood acceptable to the investigators that a relationship observed between 2 variables is due to chance (the probability of a type I error); usually α = .05. If α = .05, P < .05 will be considered significant. • analysis: process of mathematically summarizing and comparing data to confirm or refute a hypothesis. Analysis serves 3 functions: (1) to test hypotheses regarding differences in large populations based on samples of the populations, (2) to control for confounding variables, and (3) to measure the size of differences between groups or the strength of the relationship between variables in the study.40(p25) • analysis of covariance (ANCOVA): statistical test used to examine data that include both continuous and nominal independent variables and a continuous dependent variable. It is basically a hybrid of multiple regression (used for continuous independent variables) and analysis of variance (used for nominal independent variables).40(p299) • analysis of residuals: see linear regression. • analysis of variance (ANOVA): statistical method used to compare a continuous dependent variable and more than 1 nominal independent variable. The null hypothesis in ANOVA is tested by means of the F test. In 1-way ANOVA there is a single nominal independent variable with 2 or more levels (eg, age categorized into strata of 20 to 39 years, 40 to 59 years, and 60 years and older). When there are only 2 mutually exclusive categories for the nominal independent variable (eg, male or female), the 1-way ANOVA is equivalent to the t test. A 2-way ANOVA is used if there are 2 independent variables (eg, age strata and sex), a 3-way ANOVA if there are 3 independent variables, etc. If more than 1 nonexclusive independent variable is analyzed, the process is called factorial ANOVA, which assesses the main effects of the independent variables as well as their interactions. An analysis of main effects in the 2-way ANOVA above would assess the independent effects of age group or sex; an association between female sex and systolic blood pressure that exists in one age group but not another would mean that an interaction between age and sex exists. In a factorial 3-way ANOVA with independent variables A, B, and C, there is one 3-way interaction term (A × B × C), 3 different 2-way interaction terms (A × B, A × C, and B × C), and 3 main effect terms (A, B, and C). A separate F test must be computed for each different main effect and interaction term. If repeated measures are made on an individual (such as measuring blood pressure over time) so that a matched form of analysis is appropriate, but potentially confounding factors (such as age) are to be controlled for simultaneously, repeated-measures ANOVA is used. Randomized-block ANOVA is used if treatments are assigned by means of block randomization.40(pp291-295) → An ANOVA can establish only whether a significant difference exists among groups, not which groups are significantly different from each other. To determine which groups differ significantly, a pairwise analysis of a continuous dependent variable and more than 1 nominal variable is performed by a procedure such as the Newman-Keuls test or Tukey test, as well as many others. These multiple comparison procedures avoid the potential of a type I error that might occur if the t test were applied at this stage. Such comparisons may also be computed through the use of orthogonal contrasts. → The F ratio is the statistical result of ANOVA and is a number between 0 and infinity. The F ratio is compared with tables of the F distribution, taking into account the α level and degrees of freedom (df) for the numerator and denominator, to determine the P value. Example: The difference was found to be significant by 1-way ANOVA (F2,63=61.07; P < .001).43 The dfs are provided along with the F statistic. The first subscript (2) is the df for the numerator; the second subscript (63) is the df for the denominator. The P value can be obtained from an F statistic table that provides the P value that corresponds to a given F and df. In practice, however, the P value is generally calculated by a computerized algorithm. Because ANOVA does not determine which groups are significantly different from each other, this example would normally be accompanied by the results of the multiple comparisons procedure.43 Other models such as Latin square may also be used. • ANCOVA: see analysis of covariance. • ANOVA: see analysis of variance. • Ansari-Bradley dispersion test: rank test to determine whether 2 distributions known to be of identical shape (but not necessarily of normal distribution) have equal parameters of scale.35(p6) • area under the curve (AUC): technique used to measure the performance of a test plotted on a receiver operating characteristic (ROC) curve or to measure drug clearance in pharmacokinetic studies.42(p12) When measuring test performance, the larger the AUC, the better the test performance. When measuring drug clearance, the AUC assesses the total exposure of the individual, as measured by levels of the drug in blood or urine, to a drug over time. The curve of drug clearance used to calculate the AUC is also used to calculate the drug half-life. → The method used to determine the AUC should be specified (eg, the trapezoidal rule). • artifact: difference or change in measure of occurrence of a condition that results from the way the disease or condition is measured, sought, or defined.40(p327) Example: An artifactual increase in the incidence of AIDS was expected because the definition of AIDS was changed to include a larger number of AIDS-defining illnesses. • assessment: in the statistical sense, evaluating the outcome(s) of the study and control groups. • assignment: process of distributing individuals to study and control groups. See also randomization. • association: statistically significant relationship between 2 variables in which one does not necessarily cause the other. When 2 variables are measured simultaneously, association rather than causation generally is all that can be assessed. Example: After confounding factors were controlled for by means of multivariate regression, a significant association remained between age and disease prevalence. • attributable risk: disease that can be attributed to a given risk factor; conversely, if the risk factor were eliminated entirely, the amount of the disease that could be eliminated.40(pp327-328) Attributable risk assumes a causal relationship (ie, the factor to be eliminated is a cause of the disease and not merely associated with the disease). See attributable risk percentage and attributable risk reduction. • attributable risk percentage: the percentage of risk associated with a given factor among those with the risk factor.40(pp327-328) For example, risk of stroke in an older person who smokes and has hypertension and no other risk factors can be divided among the risks attributable to smoking, hypertension, and age. Attributable risk percentage is often determined for a population and is the percentage of the disease related to the risk factor. See population attributable risk percentage. • attributable risk reduction: the number of events that can be prevented by eliminating a particular risk factor from the population. Attributable risk reduction is a function of 2 factors: the strength of the association between the risk factor and the disease (ie, how often the risk factor causes the disease) and the frequency of the risk factor in the population (ie, a common risk factor may have a lower attributable risk in an individual than a less common risk factor, but could have a higher attributable risk reduction because of the risk factor’s high prevalence in the population). Attributable risk reduction is a useful concept for public health decisions. See also attributable risk. • average: the central tendency of a number of measurements. This is often used synonymously with mean, but can also imply the median, mode, or some other statistic. Thus, the word should generally be avoided in favor of a more precise term. • Bayesian analysis: theory of statistics involving the concept of prior probability, conditional probability or likelihood, and posterior probability.38(p16) For interpreting studies, the prior probability is based on previous studies and may be informative, or, if no studies exist or those that exist are not useful, one may assume a uniform prior. The study results are then incorporated with the prior probability to obtain a posterior probability. Bayesian analysis can be used to interpret how likely it is that a positive result indicates presence of a disease, by incorporating the prevalence of the disease in the population under study and the sensitivity and specificity of the test in the calculation. → Bayesian analysis has been criticized because the weight that a particular study is given when prior probability is calculated can be a subjective decision. Nonetheless, the process most closely approximates how studies are considered when they are incorporated into clinical practice. When Bayesian analysis is used to assess posterior probability for an individual patient in a clinic population, the process may be less subjective than usual practice because the prior probability, equal to the prevalence of the disease in the clinic population, is more accurate than if the prevalence for the population at large were used.32 • β (beta), β level: probability of showing no significant difference when a true difference exists; a false acceptance of the null hypothesis.42(p57) One minus β is the statistical power of the test to detect a true difference; the smaller the β, the greater the power. A value of .20 for β is equal to .80 or 80% power. A β of .1 or .2 is most frequently used in power calculations. The β error is synonymous with type II error.43 • bias: a systematic situation or condition that causes a result to depart from the true value in a consistent direction. Bias refers to defects in study design (often selection bias) or measurement.40(p328) One method to reduce measurement bias is to ensure that the investigator measuring outcomes for a participant is unaware of the group to which the participant belongs (ie, blinded assessment). • bimodal distribution: nonnormal distribution with 2 peaks, or modes. The mean and median may be equivalent, but neither will describe the data accurately. A population composed entirely of schoolchildren and their grandparents might have a mean age of 35 years, although everyone in the population would in fact be either much younger or much older. • binary variable: variable that has 2 mutually exclusive subgroups, such as male/female or pregnant/not pregnant; synonym for dichotomous variable.44(p75) • binomial distribution: probability with 2 possible mutually exclusive outcomes; used for modeling cumulative incidence and prevalence rates42(p17) (for example, the probability of a person having a stroke in a given population over a given period; the outcome must be stroke or no stroke). In a binomial sample with a probability p of the event and n number of participants, the predicted mean is p × n and the predicted variance is p(p −1). • biological plausibility: evidence that an independent variable can be expected to exert a biological effect on a dependent variable with which it is associated. For example, studies in animals were used to establish the biological plausibility of adverse effects of passive smoking. • bivariable analysis: see bivariate analysis. • bivariate analysis: used when 1 dependent and 1 independent variable are to be assessed.40(p263) Common examples include the t test for 1 continuous variable and 1 binary variable and the χ2 test for 2 binary variables. Bivariate analyses can be used for hypothesis testing in which only 1 independent variable is taken into account, to compare baseline characteristics of 2 groups, or to develop a model for multivariate regression. See also univariate and multivariate analysis. → Bivariate analysis is the simplest form of hypothesis testing but is often used incorrectly, either because it is used too frequently, resulting in an increased likelihood of a type I error, or because tests that assume a normal distribution (eg, the t test) are applied to nonnormally distributed data. • Bland-Altman plot: a method to assess agreement (eg, between 2 tests) developed by Bland and Altman.41 • blinded (masked) assessment: evaluation or categorization of an outcome in which the person assessing the outcome is unaware of the treatment assignment. Masked assessment is the term preferred by some investigators and journals, particularly those in ophthalmology. → Blinded assessment is important to prevent bias on the part of the investigator performing the assessment, who may be influenced by the study question and consciously or unconsciously expect a certain test result. • blinded (masked) assignment: assignment of individuals participating in a prospective study (usually random) to a study group and a control group without the investigator or the participants being aware of the group to which they are assigned. Studies may be single-blind, in which either the participant or the person administering the intervention does not know the treatment assignment, or double-blind, in which neither knows the treatment assignment. The term triple-blinded is sometimes used to indicate that the persons who analyze or interpret the data are similarly unaware of treatment assignment. Authors should indicate who exactly was blinded. The term masked assignment is preferred by some investigators and journals, particularly those in ophthalmology. • block randomization: type of randomization in which the unit of randomization is not the individual but a larger group, sometimes stratified on particular variables such as age or severity of illness to ensure even distribution of the variable between randomized groups. • Bonferroni adjustment: one of several statistical adjustments to the P value that may be applied when multiple comparisons are made. The α level (usually .05) is divided by the number of comparisons to determine the α level that will be considered statistically significant. Thus, if 10 comparisons are made, an α of .05 would become α = .005 for the study. Alternatively, the P value may be multiplied by the number of comparisons, while retaining the α of .05.44(pp31-32) Alternatively, the P value may be multiplied by the number of comparisons, while retaining the α of .05. For example, a P value of .02 obtained for 1 of 10 comparisons would be multiplied by 10 to get the final result of P = .20, a nonsignificant result. → The Bonferroni test is a conservative adjustment for large numbers of comparisons (ie, less likely than other methods to give a significant result) but is simple and used frequently. • bootstrap method: statistical method for validating a new diagnostic parameter in the same group from which the parameter was derived. Thus, the validation of the method is based on a simulated sample, rather than a new sample. The parameter is first derived from the entire group, then applied sequentially to subsegments of the group to see whether the parameter performs as well for the subgroups as it does for the entire group (derived from “pulling oneself up by one’s own bootstraps”).42(p32) For example, a number of prognostic indicators are measured in a cohort of hospitalized patients to predict mortality. To determine whether the model using the indicators is equally predictive of mortality for subsegments of the group, the bootstrap method is applied to the subsegments and confidence intervals are calculated to determine the predictive ability of the model. The jackknife dispersion test also uses the same sample for both derivation and validation. → Although the preferable means for validating a model is to apply the model to a new sample (eg, a new cohort of hospitalized patients in the previous example), the bootstrap method can be used to reduce the time, effort, and expense necessary to complete the study. However, the bootstrap method provides less assurance than validation in a new sample that the model is generalizable to another population. • Brown-Mood procedure: test used with a regression model that does not assume a normal distribution or common variance of the errors.38(p26) It is an extension of the median test. • C statistic: a measure of the area under a receiver operating characteristic curve. • case: in a study, an individual with the outcome or disease of interest. • case-control study: retrospective study in which individuals with the disease (cases) are compared with those who do not have the disease (controls). Cases and controls are identified without knowledge of exposure to the risk factors under study. Cases and controls are matched on certain important variables, such as age, sex, and year in which the individual was treated or identified. A case-control study on individuals already enrolled in a cohort study is referred to as a nested case-control study.42(p111) This type of case-control study may be an especially strong study design if characteristics of the cohort have been carefully ascertained. See also 20.3.2, Observational Studies, Case-Control Studies. → Cases and controls should be selected from the same population to minimize confounding by factors other than those under study. Matching cases and controls on too many characteristics may obscure the association of interest, because if cases and controls are too similar, their exposures may be too similar to detect a difference (see overmatching). • case-fatality rate: probability of death among people diagnosed as having a disease. The rate is calculated as the number of deaths during a specific period divided by the number of persons with the disease at the beginning of the period.44(p38) • case series: retrospective descriptive study in which clinical experience with a number of patients is described. See 20.3.3, Observational Studies, Case Series. • categorical data: counts of members of a category or class; for the analysis each member or item should fit into only 1 category or class38(p29) (eg, sex or race/ethnicity). The categories have no numerical significance. Categorical data are summarized by proportions, percentages, fractions, or simple counts. Categorical data is synonymous with nominal data. • cause, causation: something that brings about an effect or result; to be distinguished from association, especially in cohort studies. To establish something as a cause it must be known to precede the effect. The concept of causation includes the contributory cause, the direct cause, and the indirect cause. • censored data: censoring has 2 different statistical connotations: (1) data in which extreme values are reassigned to some predefined, more moderate value; (2) data in which values have been assigned to individuals for whom the actual value is not known, such as in survival analyses for individuals who have not experienced the outcome (usually death) at the time the data collection was terminated. The term left-censored data means that data were censored from the low end or left of the distribution; right-censored data come from the high end or right of the distribution42(p26) (eg, in survival analyses). For example, if data for falls are categorized as individuals who have 0, 1, or 2 or more falls, falls exceeding 2 have been right-censored. • central limit theorem: theorem that states that the mean of a number of samples with variances that are not large relative to the entire sample will increasingly approximate a normal distribution as the sample size increases. This is the basis for the importance of the normal distribution in statistical testing.38(p30) • central tendency: property of the distribution of data, usually measured by mean, median, or mode.42(p41) • χ2 test (chi-square test): a test of significance based on the χ2 statistic, usually used for categorical data. The observed values are compared with the expected values under the assumption of no association. The χ2 goodness-of-fit test compares the observed with expected frequencies. The χ2 test can also compare an observed variance with hypothetical variance in normally distributed samples.38(p33) In the case of a continuous independent variable and a nominal dependent variable, the χ2 test for trend can be used to determine whether a linear relationship exists (for example, the relationship between systolic blood pressure and stroke).40(pp284-285) → The P value is determined from χ2 tables with the use of the specified α level and the df calculated from the number of cells in the χ2 table. The χ2 statistic should be reported to no more than 1 decimal place; if the Yates correction was used, that should be specified. See also contingency table. Example: The exercise intervention group was least likely to have experienced a fall in the previous month (χ23 = 17.7, P = .02). Note that the df for χ23 is specified using a subscript 3; it is derived from the number of cells in the χ2 table (for this example, 4 cells in a 2 × 2 table). The value 17.7 is the χ2 value. The P value is determined from the χ2 value and df. Results of the χ2 test may be biased if there are too few observations (generally 5 or fewer) per cell. In this case, the Fisher exact test is preferred. • choropleth map: map of a region or country that uses shading to display quantitative data.42(p28) See also 4.2.3, Visual Presentation of Data, Figures, Maps. • chunk sample: subset of a population selected for convenience without regard to whether the sample is random or representative of the population.38(p32) A synonym is convenience sample. • Cochran Q test: method used to compare percentage results in matched samples (see matching), often used to test whether the observations made by 2 observers vary in a systematic manner. The analysis results in a Q statistic, which, with the df, determines the P value; if significant, the variation between the 2 observers cannot be explained by chance alone.38(p25) See also interobserver bias. • coefficient of determination: square of the correlation coefficient, used in linear or multiple regression analysis. This statistic indicates the proportion of the variation of the dependent variable that can be predicted from the independent variable.40(p328) If the analysis is bivariate, the correlation coefficient is indicated as r and the coefficient of determination is r2. If the correlation coefficient is derived from multivariate analysis, the correlation coefficient is indicated as R and the coefficient of determination is R2. See also correlation coefficient. Example: The sum of the R2 values for age and body mass index was 0.23. [Twenty-three percent of the variance could be explained by those 2 variables.] → When R2 values of the same dependent variable total more than 1.0 or 100%, then the independent variables have an interactive effect on the dependent variable. • coefficient of variation: ratio of the standard deviation (SD) to the mean. The coefficient of variation is expressed as a percentage and is used to compare dispersions of different samples. The smaller the coefficient of variation, the greater the precision.43 The coefficient of variation is also used when the SD is dependent on the mean (eg, the increase in height with age is accompanied by an increasing SD of height in the population). • cohort: a group of individuals who share a common exposure, experience, or characteristic, or a group of individuals followed up or traced over time in a cohort study.38(p31) • cohort effect: change in rates that can be explained by the common experience or characteristic of a group or cohort of individuals. A cohort effect implies that a current pattern of variables may not be generalizable to a different cohort.38(p328) Example: The decline in socioeconomic status with age was a cohort effect explained by fewer years of education among the older individuals. • cohort study: study of a group of individuals, some of whom are exposed to a variable of interest (eg, a drug treatment or environmental exposure), in which participants are followed up over time to determine who develops the outcome of interest and whether the outcome is associated with the exposure. Cohort studies may be concurrent (prospective) or nonconcurrent (retrospective).40(pp328-329) See also 20.3.1, Observational Studies, Cohort Studies. → Whenever possible, a participant’s outcome should be assessed by individuals who do not know whether the participant was exposed (see blinded assessment). • concordant pair: pair in which both individuals have the same trait or outcome (as opposed to discordant pair). Used frequently in twin studies.42(p35) • conditional probability: probability that an event E will occur given the occurrence of F, called the conditional probability of E given F. The reciprocal is not necessarily true: the probability of E given F may not be equal to the probability of F given E.44(p55) • confidence interval (CI): range of numerical expressions within which one can be confident (usually 95% confident, to correspond to an α level of .05) that the population value the study is intended to estimate lies.40(p329) The CI is an indication of the precision of an estimated population value. → Confidence intervals used to estimate a population value usually are symmetric or nearly symmetric around a value, but CIs used for relative risks and odds ratios may not be. Confidence intervals are preferable to P values because they convey information about precision as well as statistical significance of point estimates. → Confidence intervals are expressed with a hyphen separating the 2 values. To avoid confusion, the word to replaces hyphens if one of the values is a negative number. Units that are closed up with the numeral are repeated for each CI; those not closed up are repeated only with the last numeral. See also 20.8, Significant Digits and Rounding Numbers, and 19.4, Numbers and Percentages, Use of Digit Spans and Hyphens. Example: The odds ratio was 3.1 (95% CI, 2.2-4.8). The prevalence of disease in the population was 1.2% (95% CI, 0.8%-1.6%). • confidence limits (CLs): upper and lower boundaries of the confidence interval, expressed with a comma separating the 2 values.42(p35) Example: The mean (95% confidence limits) was 30% (28%, 32%). • confounding: (1) a situation in which the apparent effect of an exposure on risk is caused by an association with other factors that can influence the outcome; (2) a situation in which the effects of 2 or more causal factors as shown by a set of data cannot be separated to identify the unique effects of any of them; (3) a situation in which the measure of the effect of an exposure on risk is distorted because of the association of exposure with another factor(s) that influences the outcome under study.42(p35) See also confounding variable. • confounding variable: variable that can cause or prevent the outcome of interest, is not an intermediate variable, and is associated with the factor under investigation. Unless it is possible to adjust for confounding variables, their effects cannot be distinguished from those of the factors being studied. Bias can occur when adjustment is made for any factor that is caused in part by the exposure and also is correlated with the outcome.25(p35) Multivariate analysis is used to control the effects of confounding variables that have been measured. • contingency coefficient: the coefficient C (Note: not to be confused with the C statistic), used to measure the strength of association between 2 characteristics in a contingency table.44(pp56-57) • contingency table: table created when categorical variables are used to calculate expected frequencies in an analysis and to present data, especially for a χ2 test (2-dimensional data) or log-linear models (data with at least 3 dimensions). A 2 × 3 contingency table has 2 rows and 3 columns. The df are calculated as (number of rows − 1)(number of columns −1). Thus, a 2 x 3 contingency table has 6 cells and 2 df. • continuous data: data with an unlimited number of equally spaced values.40(p329) There are 2 kinds of continuous data: ratio data and interval data. Ratio-level data have a true 0, and thus numbers can meaningfully be divided by one another (eg, weight, systolic blood pressure, cholesterol level). For instance, 75 kg is half as heavy as 150 kg. Interval data may be measured with a similar precision but lack a true 0 point. Thus, 328C is not half as warm as 648C, although temperature may be measured on a precise continuous scale. Continuous data include more information than categorical, nominal, or dichotomous data. Use of parametric statistics requires that continuous data have a normal distribution, or that the data can be transformed to a normal distribution (eg, by computing logarithms of the data). • contributory cause: independent variable (cause) that is thought to contribute to the occurrence of the dependent variable (effect). That a cause is contributory should not be assumed unless all of the following have been established: (1) an association exists between the putative cause and effect, (2) the cause precedes the effect in time, and (3) altering the cause alters the probability of occurrence of the effect.40(p329) Other factors that may contribute to establishing a contributory cause include the concept of biological plausibility, the existence of a dose-response relationship, and consistency of the relationship when evaluated in different settings. • control: in a case-control study, the designation for an individual without the disease or outcome of interest; in a cohort study, the individuals not exposed to the independent variable of interest; in a randomized controlled trial, the group receiving a placebo or standard treatment rather than the intervention under study. • controlled clinical trial: study in which a group receiving an experimental treatment is compared with a control group receiving a placebo or an active treatment. See also 20.2.1, Randomized Controlled Trials, Parallel-Design Double-blind Trials. • convenience sample: sample of participants selected because they were available for the researchers to study, not because they are necessarily representative of a particular population. → Use of a convenience sample limits generalizability and can confound the analysis depending on the source of the sample. For instance, in a study comparing cardiac auscultation, echocardiography, and cardiac catheterization, the patients studied, simply by virtue of their having undergone cardiac catheterization and echocardiography, likely are not comparable to an unselected population. • correlation: description of the strength of an association among 2 or more variables, each of which has been sampled by means of a representative or naturalistic method from a population of interest.40(p329) The strength of the association is described by the correlation coefficient. See also agreement. There are many reasons why 2 variables may be correlated, and thus correlation alone does not prove causation. → The Kendall τ rank correlation test is used when testing 2 ordinal variables, the Pearson product moment correlation is used when testing 2 normally distributed continuous variables, and the Spearman rank correlation is used when testing 2 non-normally distributed continuous variables.43 → Correlation is often depicted graphically by means of a scatterplot of the data (see Example F4 in 4.2.1, Visual Presentation of Data, Figures, Statistical Graphs). The more circular a scatterplot, the smaller the correlation; the more linear a scatterplot, the greater the correlation. • correlation coefficient: measure of the association between 2 variables. The coefficient falls between 1 and 1; the sign indicates the direction of the relationship and the number the magnitude of the relationship. A positive sign indicates that the 2 variables increase or decrease together; a negative sign indicates that increases in one are associated with decreases in the other. A value of 1 or 1 indicates that the sample values fall in a straight line, while a value of 0 indicates no relationship. The correlation coefficient should be followed by a measure of the significance of the correlation, and the statistical test used to measure correlation should be specified. Example: Body mass index increased with age (Pearson r = 0.61; P < .001); years of education decreased with age (Pearson r = -0.48; P = .01). → When 2 variables are compared, the correlation coefficient is expressed by r; when more than 2 variables are compared by multivariate analysis, the correlation coefficient is expressed by R. The symbol r2 or R2 is termed the coefficient of determination and indicates the amount of variation in the dependent variable that can be explained by knowledge of the independent variable. • cost-benefit analysis: economic analysis that compares the costs accruing to an individual for some treatment, process, or procedure and the ensuing medical consequences, with the benefits of reduced loss of earnings resulting from prevention of death or premature disability. The cost-benefit ratio is the ratio of marginal benefit (financial benefit of preventing 1 case) to marginal cost (cost of preventing 1 case).42(p38) See also 20.5, Cost-effectiveness Analysis, Cost-Benefit Analysis. • cost-effectiveness analysis: comparison of strategies to determine which provides the most clinical value for the cost.43 A preferred intervention is the one that will cost the least for a given result or be the most effective for a given cost.30(pp38-39) Outcomes are expressed by the cost-effectiveness ratio, such as cost per year of life saved. See also 20.5, Cost-effectiveness Analysis, Cost-Benefit Analysis. • cost-utility analysis: form of economic evaluation in which the outcomes of alternative procedures are expressed in terms of a single utility-based measurement, most often the quality-adjusted life-year (QALY).42(p39) • covariates: variables that may mediate or confound the relationship between the independent and dependent variables. Because patterns of covariates may differ systematically between groups in a trial or observational study, their effect should be accounted for during the analysis. This can be accomplished in a number of ways, including analysis of covariance, multiple regression, stratification, or propensity matching. • Cox-Mantel test: method for comparing 2 survival curves that does not assume a particular distribution of data,44(p63) similar to the log-rank test.45(p113) • Cox proportional hazards regression model (Cox proportional hazards model): in survival analysis, a procedure used to determine relationships between survival time and treatment and prognostic independent variables such as age.37(p290) The hazard function is modeled on the set of independent variables and assumes that the hazard function is independent of time. Estimates depend only on the order in which events occur, not on the times they occur.44(p64) Thus, authors should generally indicate that they have tested the proportionality assumption of the Cox model, which assumes that the ratio of the hazards between groups is similar at all points in time. The proportionality assumption would not be met, for instance, if one group experienced an early surge in mortality while the other group did not. In this case, the ratio of the hazards would be different early vs late during the time of follow-up. • criterion standard: test considered to be the diagnostic standard for a particular disease or condition, used as a basis of comparison for other (usually noninvasive) tests. Ideally, the sensitivity and specificity of the criterion standard for the disease should be 100%. (A commonly used synonym, gold standard, is considered jargon by some.42(p70)) See also diagnostic discrimination. • Cronbach α: index of the internal consistency of a test,44(p65) which assesses the correlation between the total score across a series of items and the comparable score that would have been obtained had a different series of items been used.42(p39) The Cronbach α is often used for psychological tests. • cross-design synthesis: method for evaluating outcomes of medical interventions, developed by the US General Accounting Office, which pools results from databases of randomized controlled trials and other study designs. It is a form of meta-analysis (see 20.4, Meta-analysis).42(p39) • crossover design: method of comparing 2 or more treatments or interventions. Individuals initially are randomized to one treatment or the other; after completing the first treatment they are crossed over to 1 or more other randomization groups and undergo other courses of treatment being tested in the experiment. Advantages are that a smaller sample size is needed to detect a difference between treatments, since a paired analysis is used to compare the treatments in each individual, but the disadvantage is that an adequate washout period is needed after the initial course of treatment to avoid carryover effect from the first to the second treatment. Order of treatments should be randomized to avoid potential bias.44(pp65-66) See 20.2.2, Randomized Controlled Trials, Crossover Trials. • cross-sectional study: study that identifies participants with and without the condition or disease under study and the characteristic or exposure of interest at the same point in time.40(p329) → Causality is difficult to establish in a cross-sectional study because the outcome of interest and associated factors are assessed simultaneously. • crude death rate: total deaths during a year divided by the midyear population. Deaths are usually expressed per 100 000 persons.44(p66) • cumulative incidence: number of people who experience onset of a disease or outcome of interest during a specified period; may also be expressed as a rate or ratio.42(p40) • Cutler-Ederer method: form of life-table analysis that uses actuarial techniques. The method assumes that the times at which follow-up ended (because of death or the outcome of interest) are uniformly distributed during the time period, as opposed to the Kaplan-Meier method, which assumes that termination of follow-up occurs at the end of the time block. Therefore, Cutler-Ederer estimates of risk tend to be slightly higher than Kaplan-Meier estimates.40(p308) Often an intervention and control group are depicted on 1 graph and the curves are compared by means of a log-rank test. This is also known as the actuarial life-table method. • cut point: in testing, the arbitrary level at which “normal” values are separated from “abnormal” values, often selected at the point 2 SDs from the mean. See also receiver operating characteristic curve.42(p40) • DALY: see disability-adjusted life-years. • data: collection of items of information.42(p42) (Datum, the singular form of this word, is rarely used.) • data dredging (aka “fishing expedition”): jargon meaning post hoc analysis, with no a priori hypothesis, of several variables collected in a study to identify variables that have a statistically significant association for purposes of publication. → Although post hoc analyses occasionally can be useful to generate hypotheses, data dredging increases the likelihood of a type I error and should be avoided. If post hoc analyses are performed, they should be declared as such and the number of post hoc comparisons performed specified. • decision analysis: process of identifying all possible choices and outcomes for a particular set of decisions to be made regarding patient care. Decision analysis generally uses preexisting data to estimate the likelihood of occurrence of each outcome. The process is displayed as a decision tree, with each node depicting a branch point representing a decision in treatment or intervention to be made (usually represented by a square at the branch point), or possible outcomes or chance events (usually represented by a circle at the branch point). The relative worth of each outcome may be expressed as a utility, such as the quality-adjusted life-year.42(p44) See Figure 2. • degrees of freedom (df): see df. • dependent variable: outcome variable of interest in any study; the outcome that one intends to explain or estimate40(p329) (eg, death, myocardial infarction, or reduction in blood pressure). Multivariate analysis controls for independent variables or covariates that might modify the occurrence of the dependent variable (eg, age, sex, and other medical diseases or risk factors). • descriptive statistics: method used to summarize or describe data with the use of the mean, median, SD, SE, or range, or to convey in graphic form (eg, by using a histogram, shown in Example F5 in 4.2.1, Visual Presentation of Data, Figures, Statistical Graphs) for purposes of data presentation and analysis.44(p73) • df (degrees of freedom) (df is not expanded at first mention): the number of arithmetically independent comparisons that can be made among members of a sample. In a contingency table, df is calculated as (number of rows − 1)(number of columns − 1). → The df should be reported as a subscript after the related statistic, such as the t test, analysis of variance, and χ2 test (eg, χ23 = 17.7, P = .02; in this example, the subscript 3 is the number of df). • diagnostic discrimination: statistical assessment of how the performance of a clinical diagnostic test compares with the criterion standard. To assess a test’s ability to distinguish an individual with a particular condition from one without the condition, the researcher must (1) determine the variability of the test, (2) define a population free of the disease or condition and determine the normal range of values for that population for the test (usually the central 95% of values, but in tests that are quantitative rather than qualitative, a receiver operating characteristic curve may be created to determine the optimal cut point for defining normal and abnormal), and (3) determine the criterion standard for a disease (by definition, the criterion standard should have 100% sensitivity and specificity for the disease) with which to compare the test. Diagnostic discrimination is reported with the performance measures sensitivity, specificity, positive predictive value, and negative predictive value; false-positive rate; and the likelihood ratio.40(pp151-163) See Table 4. → Because the values used to report diagnostic discrimination are ratios, they can be expressed either as the ratio, using the decimal form, or as the percentage, by multiplying the ratio by 100. Example: The test had a sensitivity of 0.80 and a specificity of 0.95; the false-positive rate was 0.05. Or: The test had a sensitivity of 80% and a specificity of 95%; the false-positive rate was 5%. → When the diagnostic discrimination of a test is defined, the individuals tested should represent the full spectrum of the disease and reflect the population on whom the test will be used. For example, if a test is proposed as a screening tool, it should be assessed in the general population. • dichotomous variable: a variable with only 2 possible categories (eg, male/female, alive/dead); synonym for binary variable.44(p75) → A variable may have a continuous distribution during data collection but is made dichotomous for purposes of analysis (eg, age <65 years/age 65 years). This is done most often for nonnormally distributed data. Note that the use of a cut point generally converts a continuous variable to a dichotomous one (eg, normal vs abnormal). • direct cause: contributory cause that is believed to be the most immediate cause of a disease. The direct cause is dependent on the current state of knowledge and may change as more immediate mechanisms are discovered.40(p330) Example: Although several other causes were suggested when the disease was first described, the human immunodeficiency virus is the direct cause of AIDS. • disability-adjusted life-years (DALY): A quantitative indicator of burden of disease that reflects the years lost due to premature mortality and years lived with disability, adjusted for severity.45 • discordant pair: pair in which the individuals have different outcomes. In twin studies, only the discordant pairs are informative about the association between exposure and disease.42(pp47-48) Antonym is concordant pair. • discrete variable: variable that is counted as an integer; no fractions are possible.44(p77) Examples are counts of pregnancies or surgical procedures, or responses to a Likert scale. • discriminant analysis: analytic technique used to classify participants according to their characteristics (eg, the independent variables, signs, symptoms, and diagnostic test results) to the appropriate outcome or dependent variable,44(pp77-78) also referred to as discriminatory analysis.37(pp59-60) This analysis tests the ability of the independent variable model to correctly classify an individual in terms of outcome. Conceptually, this may be thought of as the opposite of analysis of variance, in that the predictor variables are continuous, while the dependent variables are categorical. • dispersion: degree of scatter shown by observations; may be measured by SD, various percentiles (eg, tertiles, quantiles, quintiles), or range.38(p60) • distribution: group of ordered values; the frequencies or relative frequencies of all possible values of a characteristic.40(p330) Distributions may have a normal distribution (bell-shaped curve) or a nonnormal distribution (eg, binomial or Poisson distribution). • dose-response relationship: relationship in which changes in levels of exposure are associated with changes in the frequency of an outcome in a consistent direction. This supports the idea that the agent of exposure (most often a drug) is responsible for the effect seen.40(p330) May be tested statistically by using a χ2 test for trend. • Duncan multiple range test: modified form of the Newman-Keuls test for multiple comparisons.44(p82) • Dunnett test: multiple comparisons procedure intended for comparing each of a number of treatments with a single control.44(p82) • Dunn test: multiple comparisons procedure based on the Bonferroni adjustment.44(p84) • Durbin-Watson test: test to determine whether the residuals from linear regression or multiple regression are independent or, alternatively, are serially correlated.44(p84) • ecological fallacy: error that occurs when the existence of a group association is used to imply, incorrectly, the existence of a relationship at the individual level.40(p330) • effectiveness: extent to which an intervention is beneficial when implemented under the usual conditions of clinical care for a group of patients,40(p330) as distinguished from efficacy (the degree of beneficial effect seen in a clinical trial) and efficiency (the intervention effect achieved relative to the effort expended in time, money, and resources). • effect of observation: bias that results when the process of observation alters the outcome of the study.40(p330) See also Hawthorne effect. • effect size: observed or expected change in outcome as a result of an intervention. Expected effect size is used during the process of estimating the sample size necessary to achieve a given power. Given a similar amount of variability between individuals, a large effect size will require a smaller sample size to detect a difference than will a smaller effect size. • efficacy: degree to which an intervention produces a beneficial result under the ideal conditions of an investigation,40(p330) usually in a randomized controlled trial; it is usually greater than the intervention’s effectiveness. • efficiency: effects achieved in relation to the effort expended in money, time, and resources. Statistically, the precision with which a study design will estimate a parameter of interest.42(pp52-53) • effort-to-yield measures: amount of resources needed to produce a unit change in outcome, such as number needed to treat43; used in cost-effectiveness and cost-benefit analyses. See 20.5, Cost-effectiveness Analysis, Cost-Benefit Analysis. • error: difference between a measured or estimated value and the true value. Three types are seen in scientific research: a false or mistaken result obtained in a study; measurement error, a random form of error; and systematic error that skews results in a particular direction.42(pp56-57) • estimate: value or values calculated from sample observations that are used to approximate the corresponding value for the population.40(p330) • event: end point or outcome of a study; usually the dependent variable. The event should be defined before the study is conducted and assessed by an individual blinded to the intervention or exposure category of the study participant. • exclusion criteria: characteristics of potential study participants or other data that will exclude them from the study sample (such as being younger than 65 years, history of cardiovascular disease, expected to move within 6 months of the beginning of the study). Like inclusion criteria, exclusion criteria should be defined before any individuals are enrolled. • explanatory variable: synonymous with independent variable, but preferred by some because “independent” in this context does not refer to statistical independence.38(p98) • extrapolation: conclusions drawn about the meaning of a study for a target population that includes types of individuals or data not represented in the study sample.40(p330) • factor analysis: procedure used to group related variables to reduce the number of variables needed to represent the data. This analysis reduces complex correlations between a large number of variables to a smaller number of independent theoretical factors. The researcher must then interpret the factors by looking at the pattern of “loadings” of the various variables on each factor.43 In theory, there can be as many factors as there are variables, and thus the authors should explain how they decided on the number of factors in their solution. The decision about the number of factors is a compromise between the need to simplify the data and the need to explain as much of the variability as possible. There is no single criterion on which to make this decision, and thus authors may consider a number of indexes of goodness of fit. There are a number of algorithms for rotation of the factors, which may make them more straightforward to interpret. Factor analysis is commonly used for developing scoring systems for rating scales and questionnaires. • false negative: negative test result in an individual who has the disease or condition as determined by the criterion standard.40(p330) See also diagnostic discrimination. • false-negative rate: proportion of test results found or expected to yield a false-negative result; equal to 1 − sensitivity.40 See also diagnostic discrimination. • false positive: positive test result in an individual who does not have the disease or condition as determined by the criterion standard.40(p330) See also diagnostic discrimination. • false-positive rate: proportion of tests found to or expected to yield a false-positive result; equal to 1 − specificity.40 See also diagnostic discrimination. • F distribution: ratio of the distribution of 2 normally distributed independent variables; synonymous with variance ratio distribution.42(p61) • Fisher exact test: assesses the independence of 2 variables by means of a 2 × 2 contingency table, used when the frequency in at least 1 cell is small44(p96) (usually <6). This test is also known as the Fisher-Yates test and the Fisher-Irwin test.38(p77) • fixed-effects model: model used in meta-analysis that assumes that differences in treatment effect in each study all estimate the same true difference. This is not often the case, but the model assumes that it is close enough to the truth that the results will not be misleading.46(p349) Antonym is random-effects model. • Friedman test: a nonparametric test for a design with 2 factors that uses the ranks rather than the values of the observations.38(p80) Nonparametric analog to analysis of variance. • F test (score): alternative name for the variance ratio test (or F ratio),42(p74) which results in the F score. Often encountered in analysis of variance.44(p101) Example: There were differences by academic status in perceptions of the quality of both primary care training (F1,682 = 6.71, P = .01) and specialty training (F1,682 = 6.71, P = .01). [The numbers set as subscripts for the F test are the df for the numerator and denominator, respectively.] • funnel plot: in meta-analysis, a graph of the sample size or standard error of each study plotted against its effect size. Estimates of effect size from small studies should have more variability than estimates from larger studies, thus producing a funnel-shaped plot. Departures from a funnel pattern suggest publication bias. • gaussian distribution: see normal distribution. • gold standard: see criterion standard. • goodness of fit: agreement between an observed set of values and a second set that is derived wholly or partly on a hypothetical basis.38(p86) The Kolmogorov-Smirnov test is one example. • group association: situation in which a characteristic and a disease both occur more frequently in one group of individuals than another. The association does not mean that all individuals with the characteristic necessarily have the disease.40(p331) • group matching: process of matching during assignment in a study to ensure that the groups have a nearly equal distribution of particular variables; also known as frequency matching.40(p331) • Hartley test: test for the equality of variances of a number of populations that are normally distributed, based on the ratio between the largest and smallest sample variations.38(p90) • Hawthorne effect: effect produced in a study because of the participants' awareness that they are participating in a study. The term usually refers to an effect on the control group that changes the group in the direction of the outcome, resulting in a smaller effect size.44(p115) A related concept is effect of observation. The Hawthorne effect is different than the placebo effect, which relates to participants' expectations that an intervention will have specific effects. • hazard rate, hazard function: theoretical measure of the likelihood that an individual will experience an event within a given period.42(p73) A number of hazard rates for specific intervals of time can be combined to create a hazard function. • hazard ratio: the ratio of the hazard rate in one group to the hazard rate in another. It is calculated from the Cox proportional hazards model. The interpretation of the hazard ratio is similar to that of the relative risk. • heterogeneity: inequality of a quantity of interest (such as variance) in a number of groups or populations. Antonym is homogeneity. • histogram: graphical representation of data in which the frequency (quantity) within each class or category is represented by the area of a rectangle centered on the class interval. The heights of the rectangles are proportional to the observed frequencies. See also Example F5 in 4.2.1, Visual Presentation of Data, Figures, Statistical Graphs. • Hoeffding independence test: bivariate test of nonnormally distributed continuous data to determine whether the elements of the 2 groups are independent of each other.42(p93) • Hollander parallelism test: determines whether 2 regression lines for 2 independent variables plotted against a dependent variable are parallel. The test does not require a normal distribution, but there must be an equal and even number of observations corresponding to each line. If the lines are parallel, then both independent variables predict the dependent variable equally well. The Hollander parallelism test is a special case of the signed rank test.38(p94) • homogeneity: equality of a quantity of interest (such as variance) specifically in a number of groups or populations.38(p94) Antonym is heterogeneity. • homoscedasticity: statistical determination that the variance of the different variables under study is equal.42(p78) See also heterogeneity. • Hosmer-Lemeshow goodness-of-fit test: a series of statistical steps used to assess goodness of fit; approximates the χ2 statistic.47 • Hotelling T statistic: generalization of the t test for use with multivariate data; results in a T statistic. Significance can be tested with the variance ratio distribution.38(p94) • hypothesis: supposition that leads to a prediction that can be tested to be either supported or refuted.42(p80) The null hypothesis is generally that there is no difference between groups or relationships among variables and that any such difference or relationship, if found, would occur strictly by chance. Hypothesis testing includes (1) generating the study hypothesis and defining the null hypothesis, (2) determining the level below which results are considered statistically significant, or α level (usually α = .05), and (3) identifying and applying the appropriate statistical test to accept or reject the null hypothesis. • imputation: a group of techniques for replacing missing data with values that would have been likely to have been observed. Among the simplest methods of imputation is last-observation-carried-forward, in which missing values are replaced by the last observed value. This provides a conservative estimate in cases in which the condition is expected to improve on its own, but may be overly optimistic in conditions that are known to worsen over time. Missing values may also be imputed based on the patterns of other variables. In multiple imputation, repeated random samples are simulated, each of which produces a set of values to replace the missing values. This provides not only an estimate of the missing values but also an estimate of the uncertainty with which they can be predicted. • incidence: number of new cases of disease among persons at risk that occur over time,42(p82) as contrasted with prevalence, which is the total number of persons with the disease at any given time. Incidence is usually expressed as a percentage of individuals affected during an interval (eg, year) or as a rate calculated as the number of individuals who develop the disease during a period divided by the number of person-years at risk. Example: The incidence rate for the disease was 1.2 cases per 100 000 per year. • inclusion criteria: characteristics a study participant must possess to be included in the study population (such as age 65 years or older at the time of study enrollment and willing and able to provide informed consent). Like exclusion criteria, inclusion criteria should be defined before any participants are enrolled. • independence, assumption of: assumption that the occurrence of one event is in no way linked to another event. Many statistical tests depend on the assumption that each outcome is independent.42(p83) This may not be a valid assumption if repeated tests are performed on the same individuals (eg, blood pressure is measured sequentially over time), if more than 1 outcome is measured for a given individual (eg, myocardial infarction and death or all hospital admissions), or if more than 1 intervention is made on the same individual (eg, blood pressure is measured during 3 different drug treatments). Tests for repeated measures may be used in those circumstances. • independent variable: variable postulated to influence the dependent variable within the defined area of relationships under study.42(p83) The term does not refer to statistical independence, so some use the term explanatory variable instead.38(p98) Example: Age, sex, systolic blood pressure, and cholesterol level were the independent variables entered into the multiple logistic regression. • indirect cause: contributory cause that acts through the biological mechanism that is the direct cause.40(p331) Example: Overcrowding in the cities facilitated transmission of the tubercle bacillus and precipitated the tuberculosis epidemic. [Overcrowding is an indirect cause; the tubercle bacillus is the direct cause.] • inference: process of passing from observations to generalizations, usually with calculated degrees of uncertainty.42(p85) Example: Intake of a high-fat diet was significantly associated with cardiovascular mortality; therefore, we infer that eating a high-fat diet increases the risk of cardiovascular death. • instrument error: error introduced in a study when the testing instrument is not appropriate for the conditions of the study or is not accurate enough to measure the study outcome40(p331) (may be due to deficiencies in such factors as calibration, accuracy, and precision). • intention-to-treat analysis, intent-to-treat analysis: analysis of outcomes for individuals based on the treatment group to which they were randomized, rather than on which treatment they actually received and whether they completed the study. The intention-to-treat analysis generally avoids biases associated with the reasons that participants may not complete the study and should be the main analysis of a randomized trial.44(p125) See 20.2, Randomized Controlled Trials. → Although other analyses, such as evaluable patient analysis or per-protocol analyses, are often performed to evaluate outcomes based on treatment actually received, the intention-to-treat analysis should be presented regardless of other analyses because the intervention may influence whether treatment was changed and whether participants dropped out. Intention-to-treat analyses may bias the results of equivalence and noninferiority trials; for those trials, additional analyses should be presented. See 20.2.3, Randomized Controlled Trials, Equivalence and Noninferiority Trials. • interaction: see interactive effect. • interaction term: variable used in analysis of variance or analysis of covariance in which 2 independent variables interact with each other (eg, when assessing the effect of energy expenditure on cardiac output, the increase in cardiac output per unit increase in energy expenditure might differ between men and women; the interaction term would enable the analysis to take this difference into account).40(p301) • interactive effect: effect of 2 or more independent variables on a dependent variable in which the effect of an independent variable is influenced by the presence of another.38(p101) The interactive effect may be additive (ie, equal to the sum of the 2 effects present separately), synergistic (ie, the 2 effects together have a greater effect than the sum of the effects present separately), or antagonistic (ie, the 2 effects together have a smaller effect than the sum of the effects present separately). • interim analysis: data analysis carried out during a clinical trial to monitor treatment effects. Interim analysis should be determined as part of the study protocol prior to patient enrollment and specify the stopping rules if a particular treatment effect is reached.7(p130) • interobserver bias: likelihood that one observer is more likely to give a particular response than another observer because of factors unique to the observer or instrument. For example, one physician may be more likely than another to identify a particular set of signs and symptoms as indicative of religious preoccupation on the basis of his or her beliefs, or a physician may be less likely than another physician to diagnose alcoholism in a patient because of the physician’s expectations.44(p25) The Cochran Q test is used to assess interobserver bias.44(p25) • interobserver reliability: test used to measure agreement among observers about a particular measure or outcome. → Although the proportion of times that 2 observers agree can be reported, this does not take into account the number of times they would have agreed by chance alone. For example, if 2 observers must decide whether a factor is present or absent, they should agree 50% of the time according to chance. The κ statistic assesses agreement while taking chance into account and is described by the equation [(observed agreement) (agreement expected by chance)]/(1 agreement expected by chance). The value of κ may range from 0 (poor agreement) to 1 (perfect agreement) and may be classified by various descriptive terms, such as slight (0-0.20), fair (0.21-0.40), moderate (0.41-0.60), substantial (0.61-0.80), and near perfect (0.81-0.99).48(pp27-29) → In cases in which disagreement may have especially grave consequences, such as one pathologist rating a slide “negative”’ and another rating a slide “invasive carcinoma,” a weighted κ may be used to grade disagreement according to the severity of the consequences.48(p29) See also Pearson product moment correlation. • interobserver variation: see interobserver reliability. • interquartile range: the distance between the 25th and 75th percentiles, which is used to describe the dispersion of values. Like other quantiles (eg, tertiles, quintiles), such a range more accurately describes nonnormally distributed data than does the SD. The interquartile range describes the inner 50% of values; the interquintile range describes the inner 60% of values; the interdecile range describes the inner 80% of values.38(pp102-103) • interrater reliability: reproducibility among raters or observers; synonymous with interobserver reliability. • interval estimate: see confidence interval.40(p331) • intraobserver reliability (or variation): reliability (or, conversely, variation) in measurements by the same person at different times.40(p331) Similar to interobserver reliability, intraobserver reliability is the agreement between measurements by one individual beyond that expected by chance and can be measured by means of the κ statistic or the Pearson product moment correlation. • intrarater reliability: synonym for intraobserver reliability. • jackknife dispersion test: technique for estimating the variance and bias of an estimator, applied to a predictive model derived from a study sample to determine whether the model fits subsamples from the model equally well. The estimator or model is applied to subsamples of the whole, and the differences in the results obtained from the subsample compared with the whole are analyzed as a jackknife estimate of variance. This method uses a single data set to derive and validate the model.48(p131) → Although validating a model in a new sample is preferable, investigators often use techniques such as jackknife dispersion or the bootstrap method to validate a model to save the time and expense of obtaining an entirely new sample for purposes of validation. • Kaplan-Meier method: nonparametric method of compiling life tables. Unlike the Cutler-Ederer method, the Kaplan-Meier method assumes that termination of follow-up occurs at the end of the time block. Therefore, Kaplan-Meier estimates of risk tend to be slightly lower than Cutler-Ederer estimates.40(p308) Often an intervention and control group are depicted on one graph and the groups are compared by a log-rank test. Because the method is nonparametric, there is no attempt to fit the data to a theoretical curve. Thus, Kaplan-Meier plots have a jagged appearance, with discrete drops at the end of each time interval in which an event occurs. This method is also known as the product-limit method. • κ (kappa) statistic: statistic used to measure nonrandom agreement between observers or measurements.42(p94) See interobserver and intraobserver reliability. • Kendall τ (tau) rank correlation: rank correlation coefficient for ordinal data.48(p134) • Kolmogorov-Smirnov test: comparison of 2 independent samples of continuous data without requiring that the data be normally distributed44(p136); may be used to test goodness of fit.43 • Kruskal-Wallis test: comparison of 3 or more groups of nonnormally distributed data to determine whether they differ significantly.44(p137) The Kruskal-Wallis test is a nonparametric analog of analysis of variance and generalizes the 2-sample Wilcoxon rank sum test to the multiple-sample case.38(p111) • kurtosis: the way in which a unimodal curve deviates from a normal distribution; may be more peaked (leptokurtic) or more flat (platykurtic) than a normal distribution.44(p137) • Latin square: form of complete treatment crossover design used for crossover drug trials that eliminates the effect of treatment order. Each patient receives each drug, but each drug is followed by another drug only once in the array. For example, in the following 4 × 4 array, letters A through D correspond to each of 4 drugs, each row corresponds to a patient, and each column corresponds to the order in which the drugs are given.8(p142) First Drug Second Drug Third Drug Fourth Drug Patient 1 C D A B Patient 2 A C B D Patient 3 D B C A Patient 4 B A D C See also 20.2.2, Randomized Controlled Trials, Crossover Trials. • lead-time bias: artifactual increase in survival time that results from earlier detection of a disease, usually cancer, during a time when the disease is asymptomatic. Lead-time bias produces longer survival from the time of diagnosis but not longer survival from the time of onset of the disease.40(p331) See also length-time bias. → Lead-time bias may give the appearance of a survival benefit from screening, when in fact the increased survival is only artifactual. Lead-time bias is used more generally to indicate a systematic error arising when follow-up of groups does not begin at comparable stages in the natural course of the condition. • least significant difference test: test for comparing mean values arising in analysis of variance. An extension of the t test.40(p115) • least squares method: method of estimation, particularly in regression analysis, that minimizes the sum of the differences between the observed responses and the values predicted by the model.44(p140) The regression line is created so that the sum of the squares of the residuals is as small as possible. • left-censored data: see censored data. • length-time bias: bias that arises when a sampling scheme is based on patient visits, because patients with more frequent clinic visits are more likely to be selected than those with less frequent visits. In a screening study of cancer, for example, screening patients with frequent visits is more likely to detect slow-growing tumors than would sampling patients who visit a physician only when symptoms arise.44(p140) See also lead-time bias. • life table: method of organizing data that allows examination of the experience of 1 or more groups of individuals over time with varying periods of follow-up. For each increment of the follow-up period, the number entering, the number leaving, and the number dying of disease or developing disease can be calculated. An assumption of the life-table method is that an individual not completing follow-up is exposed for half the incremental follow-up period.44(p143) (The Kaplan-Meier method and the Cutler-Ederer method are also forms of life-table analysis but make different assumptions about the length of exposure.) See Figure 3. → The clinical life table describes the outcomes of a cohort of individuals classified according to their exposure or treatment history. The cohort life table is used for a cohort of individuals born at approximately the same time and followed up until death. The current life table is a summary of mortality of the population over a brief (1- to 3-year) period, classified by age, often used to estimate life expectancy for the population at a given age.42(p97) • likelihood ratio: probability of getting a certain test result if the patient has the condition relative to the probability of getting the result if the patient does not have the condition. For dichotomous variables, this is calculated as sensitivity/(1 − specificity). The greater the likelihood ratio, the more likely that a positive test result will occur in a patient who has the disease. A ratio of 2 means a person with the disease is twice as likely to have a positive test result as a person without the disease.43 The likelihood ratio test is based on the ratio of 2 likelihood functions.38(p118) See also diagnostic discrimination. • Likert scale: scale often used to assess opinion or attitude, ranked by attaching a number to each response such as 1, strongly agree; 2, agree; 3, undecided or neutral; 4, disagree; 5, strongly disagree. The score is a sum of the numerical responses to each question.`44(p144) • Lilliefors test: test of normality (using the Kolmogorov-Smirnov test statistic) in which mean and variance are estimated from the data.38(p118) • linear regression: statistical method used to compare continuous dependent and independent variables. When the data are depicted on a graph as a regression line, the independent variable is plotted on the x-axis and the dependent variable on the y-axis. The residual is the vertical distance from the data point to the regression line43(p110); analysis of residuals is a commonly used procedure for linear regression. (See Example F4 in 4.2.1, Visual Presentation of Data, Figures, Statistical Graphs.) This method is frequently performed using least squares regression.37(pp202-203) → The description of a linear regression model should include the equation of the fitted line with the slope and 95% confidence interval if possible, the fraction of variation in y explained by the x variables (correlation), and the variances of the fitted coefficients a and b (and their SDs).37(p227) Example: The regression model identified a significant positive relationship between the dependent variable weight and height (slope = 0.25; 95% CI, 0.19-0.31; y = 12.6 Þ 0.25x; t451 = 8.3, P < .001; r2 = 0.67).43 (In this example, the slope is positive, indicating that as one variable increases the other increases; thet test with 451 df is significant; the regression line is described by the equation and includes the slope 0.25 and the constant 12.6. The coefficient of determination r2 demonstrates that 67% of the variance in weight is explained by height.)43 → Four important assumptions are made when linear regression is conducted: the dependent variable is sampled randomly from the population; the spread or dispersion of the dependent variable is the same regardless of the value of the independent variable (this equality is referred to as homogeneity of variances or homoscedasticity); the relationship between the 2 variables is linear; and the independent variable is measured with complete precision.40(pp273-274) • location: central tendency of a normal distribution, as distinguished from dispersion. The location of 2 curves may be identical (means are the same), but the kurtosis may vary (one may be peaked and the other flat, producing small and large SDs, respectively).49(p28) • logistic regression: type of regression model used to analyze the relationship between a binary dependent variable (expressed as a natural log after a logit transformation) and 1 or more independent variables. Often used to determine the independent effect of each of several explanatory variables by controlling for several factors simultaneously in a multiple logistic regression analysis. Results are usually expressed by odds ratios or relative risks and 95% confidence intervals.40(pp311-312) (The multiple logistic regression equation may also be provided but, because these involve exponents, they are substantially more complicated than linear regression equations. Therefore, in JAMA and the Archives Journals, the equation is generally not published but can be made available on request from authors. Alternatively, it may be placed on the Web.) → To be valid, a multiple regression model must have an adequate sample size for the number of variables examined. A rough rule of thumb is to have at least 25 individuals in the study for each explanatory variable examined. • log-linear model: linear models used in the analysis of categorical data.38(p122) • log-rank test: method of using the relative death rates in subgroups to compare overall differences between survival curves for different treatments; same as the Mantel-Haenszel test.38(pp122,124) • main effect: estimate of the independent effect of an explanatory (independent) variable on a dependent variable in analysis of variance or analysis of covariance.44(p153) • Mann-Whitney test: nonparametric equivalent of the t test, used to compare ordinal dependent variables with either nominal independent variables or continuous independent variables converted to an ordinal scale.42(p100) Similar to the Wilcoxon rank sum test. • MANOVA: multivariate analysis of variance. This involves examining the overall significance of all dependent variables considered simultaneously and thus has less risk of type I error than would a series of univariate analysis of variance procedures on several dependent variables. • Mantel-Haenszel test: another name for the log-rank test. • Markov process: process of modeling possible events or conditions over time that assumes that the probability that a given state or condition will be present depends only on the state or condition immediately preceding it and that no additional information about previous states or conditions would create a more accurate estimate.44(p155) • masked assessment: synonymous with blinded assessment, preferred by some investigators and journals to the term blinded, especially in ophthalmology. • masked assignment: synonymous with blinded assignment, preferred by some investigators and journals to the term blinded, especially in ophthalmology. • matching: process of making study and control groups comparable with respect to factors other than the factors under study, generally as part of a case-control study. Matching can be done in several ways, including frequency matching (matching on frequency distributions of the matched variable[s]), category (matching in broad groups such as young and old), individual (matching on individual rather than group characteristics), and pair matching (matching each study individual with a control individual).42(p101) • McNemar test: form of the χ2 test for binary responses in comparisons of matched pairs.42(p103) The ratio of discordant to concordant pairs is determined; the greater the number of discordant pairs with the better outcome being associated with the treatment intervention, the greater the effect of the intervention.44(p158) • mean: sum of values measured for a given variable divided by the number of values; a measure of central tendency appropriate for normally distributed data.49(p29) → If the data are not normally distributed, the median is preferred. See also average. • measurement error: estimate of the variability of a measurement. Variability of a given parameter (eg, weight) is the sum of the true variability of what is measured (eg, day-to-day weight fluctuations) plus the variability of the instrument or observer measurement, or variability caused by measurement error (error variability, eg, the scale used for weighing). The intraclass correlation coefficient R measures the relationship of these 2 types of variability: as the error variability declines with respect to true variability, R increases, up to 1 when error variance is 0. If all variability is a result of error variability, then R = 0.46(p30) • median: midpoint of a distribution chosen so that half the values for a given variable appear above and half occur below.40(p332) For data that do not have a normal distribution, the median provides a better measure of central tendency than does the mean, since it is less influenced by outliers.47(p29) • median test: nonparametric rank-order test for 2 groups.38(p128) • meta-analysis: See 20.4, Meta-analysis. • missing data: incomplete information on individuals resulting from any of a number of causes, including loss to follow-up, refusal to participate, and inability to complete the study. Although the simplest approach would be to remove such participants from the analysis, this would violate the intention-to-treat principle. Furthermore, certain health conditions may be systematically associated with the risk of having missing data, and thus removal of these individuals could bias the analysis. It is generally better to attempt imputation of these missing values, which are then included in the analysis. • mode: in a series of values of a given variable, the number that occurs most frequently; used most often when a distribution has 2 peaks (bimodal distribution).49(p29) This is also appropriate as a measure of central tendency for categorical data. • Monte Carlo simulation: a family of techniques for modeling complex systems for which it would otherwise be difficult to obtain sufficient data. In general, Monte Carlo simulations use a computer algorithm to generate a large number of random “observations.” The patterns of these numbers are then assessed for underlying regularities. • mortality rate: death rate described by the following equation: [(number of deaths during period) × (period of observation)]/(number of individuals observed). For values such as the crude mortality rate, the denominator is the number of individuals observed at the midpoint of observation. See also crude death rate.44(p66) → Mortality rate is often expressed in terms of a standard ratio, such as deaths per 100 000 persons per year. • Moses ranklike dispersion test: rank test of the equality of scale of 2 identically shaped populations, applicable when the population medians are not known.38(p134) • multiple analyses problem: problem that occurs when several statistical tests are performed on one group of data because of the potential to introduce a type I error. The problem is particularly an issue when the analyses were not specified as primary outcome measures. Multiple analyses can be appropriately adjusted for by means of a Bonferroni adjustment or any of several multiple comparisons procedures. • multiple comparisons procedures: any of several tests used to determine which groups differ significantly after another more general test has identified that a significant difference exists but not between which groups. These tests are intended to avoid the problem of a type I error caused by sequentially applying tests, such as the t test, not intended for repeated use. Authors should specify whether these tests were planned a priori, or whether the decision to perform them was post hoc. → Some tests result in more conservative estimates (less likely to be significant) than others. More conservative tests include the Tukey test and the Bonferroni adjustment; the Duncan multiple range test is less conservative. Other tests include the Scheffé test, the Newman-Keuls test, and the Gabriel test,38(p137)as well as many others. There is ongoing debate among statisticians about when it is appropriate to use these tests. • multiple regression: general term for analysis procedures used to estimate values of the dependent variable for all measured independent variables that are found to be associated. The procedure used depends on whether the variables are continuous or nominal. When all variables are continuous variables, multiple linear regression is used and the mean of the dependent variable is expressed using the equation Y = α + β1χ1 + β2χ2 + ··· + βkχk, where Y is the dependent variable and k is the total number of independent variables. When independent variables may be either nominal or continuous and the dependent variable is continuous, analysis of covariance is used. (Analysis of covariance often requires an interaction term to account for differences in the relationship between the independent and dependent variables.) When all variables are nominal and the dependent variable is time-dependent, life-table methods are used. When the independent variables may be either continuous or nominal and the dependent variable is nominal and time-dependent (such as incidence of death), the Cox proportional hazards model may be used. Nominal dependent variables that are not time-dependent are analyzed by means of logistic regression or discriminant analysis.37(pp296-312) • multivariable analysis: another name for multivariate analysis. • multivariate analysis: any statistical test that deals with 1 dependent variable and at least 2 independent variables. It may include nominal or continuous variables, but ordinal data must be converted to a nominal scale for analysis. The multivariate approach has 3 advantages over bivariate analysis: (1) it allows for investigation of the relationship between the dependent and independent variables while controlling for the effects of other independent variables; (2) it allows several comparisons to be made statistically without increasing the likelihood of a type I error; and (3) it can be used to compare how well several independent variables individually can estimate values of the dependent variable.40(pp289-291) Examples include analysis of variance, multiple (logistic or linear) regression, analysis of covariance, Kruskal-Wallis test, Friedman test, life table, and Cox proportional hazards model. • N: total number of units (eg, patients, households) in the sample under study. Example: We assessed the admission diagnoses of all patients admitted from the emergency department during a 1-month period (N = 127). • n: number of units in a subgroup of the sample under study. Example: Of the patients admitted from the emergency department (N = 127), the most frequent admission diagnosis was unstable angina (n = 38). • natural experiment: investigation in which a change in a risk factor or exposure occurs in one group of individuals but not in another. The distribution of individuals into a particular group is nonrandom and, as opposed to controlled clinical trials, the change is not brought about by the investigator.40(p332) The natural experiment is often used to study effects that cannot be studied in a controlled trial, such as the incidence of medical illness immediately after an earthquake. This is also referred to as a “found” experiment. • naturalistic sample: set of observations obtained from a sample of the population in such a way that the distribution of independent variables in the sample is representative of the distribution in the population.40(p332) • necessary cause: characteristic whose presence is required to bring about or cause the disease or outcome under study.50(p332) A necessary cause may not be a sufficient cause. • negative predictive value: the probability that an individual does not have the disease (as determined by the criterion standard) if the test result is negative.40(p334) This measure takes into account the prevalence of the condition or the disease. A more general term is posttest probability. See diagnostic discrimination. • nested case-control study: case-control study in which cases and controls are drawn from a cohort study. The advantages of a nested case-control study over a case-control study are that the controls are selected from participants at risk at the time of occurrence of each case that arises in a cohort, thus avoiding the confounding effect of time in the analysis, and that cases and controls are by definition drawn from the same population.40(p111) See also 20.3.1, Observational Studies, Cohort Studies, and 20.3.2, Observational Studies, Case-Control Studies. • Newman-Keuls test: a type of multiple comparisons procedure, used to compare more than 2 groups. It first compares the 2 groups that have the highest and lowest means, then sequentially compares the next most extreme groups, and stops when a comparison is not significant.39(p92) • n-of-1 trial: randomized controlled trial that uses a single patient and an outcome measure agreed on by the patient and physician. The n-of-1 trial may be used by clinicians to assess which of 2 or more possible treatment options is better for the individual patient.50 • nominal variable: also called categorical variable. There is no arithmetic relationship among the categories, and thus there is no intrinsic ranking or order between them (for example, sex, gene alleles, race, eye color). The nominal or discrete variable usually is assessed to determine its frequency within a population.40(p332) The variable can have either a binomial or Poisson distribution (if the nominal event is extremely rare, eg, a genetic mutation). • nomogram: a visual means of representing a mathematical equation. • nonconcurrent cohort study: cohort study in which an individual’s group assignment is determined by information that exists at the time a study begins. The extreme of a nonconcurrent cohort study is one in which the outcome is determined retrospectively from existing records.40(p332) • nonnormal distribution: data that do not have a normal (bell-shaped curve) distribution; includes binomial, Poisson, and exponential distributions, as well as many others. → Nonnormally distributed continuous data must be either transformed to a normal distribution to use parametric methods or, more commonly, analyzed by non-parametric methods. • nonparametric statistics: statistical procedures that do not assume that the data conform to any theoretical distribution. Nonparametric tests are most often used for ordinal or nominal data, or for nonnormally distributed continuous data converted to an ordinal scale40(p332) (for example, weight classified by tertile). • normal distribution: continuous data distributed in a symmetrical, bell-shaped curve with the mean value corresponding to the highest point of the curve. This distribution of data is assumed in many statistical procedures.40(p330) This is also called a gaussian distribution. → Descriptive statistics such as mean and SD can be used to accurately describe data only if the values are normally distributed or can be transformed into a normal distribution. • normal range: measure of the range of values on a particular test among those without the disease. Cut points for abnormal tests are arbitrary and are often defined as the central 95% of values, or the mean of values ± 2 SDs. • null hypothesis: the assertion that no true association or difference in the study outcome or comparison of interest between comparison groups exists in the larger population from which the study samples are obtained.40(p332) In general, statistical tests cannot be used to prove the null hypothesis. Rather, the results of statistical testing can reject the null hypothesis at the stated α likelihood of a type I error. • number needed to harm: computed similarly to number needed to treat, but number of patients who, after being treated for a specific period of time, would be expected to experience 1 bad outcome or not experience 1 good outcome. • number needed to treat (NNT): number of patients who must be treated with an intervention for a specific period to prevent 1 bad outcome or result in 1 good outcome.40(pp332-333) The NNT is the reciprocal of the absolute risk reduction, the difference between event rates in the intervention and placebo groups in a clinical trial. See also number needed to harm. → The study patients from whom the NNT is calculated should be representative of the population to whom the numbers will be applied. The NNT does not take into account adverse effects of the intervention. • odds ratio (OR): ratio of 2 odds. Odds ratio may have different definitions depending on the study and therefore should be defined. For example, it may be the odds of having the disease if a particular risk factor is present to the odds of not having the disease if the risk factor is not present, or the odds of having a risk factor present if the person has the disease to the odds of the risk factor being absent if the person does not have the disease. → The odds ratio typically is used for a case-control or cohort study. For a study of incident cases with an infrequent disease (for example, <2% incidence), the odds ratio approximates the relative risk.42(p118) When the incidence is relatively frequent the odds ratio may be arithmetically corrected to better approximate the relative risk.51 → The odds ratio is usually expressed by a point estimate and 95% confidence interval (CI). An odds ratio for which the CI includes 1 indicates no statistically significant effect on risk; if the point estimate and CI are both less than 1, there is a statistically significant reduction in risk; if the point estimate and CI are both greater than 1, there is a statistically significant increase in risk. • 1-tailed test: test of statistical significance in which deviations from the null hypothesis in only 1 direction are considered.40(p333) Most commonly used for the t test. → One-tailed tests are more likely to produce a statistically significant result than are 2-tailed tests. Since the use of a 1-tailed test implies that the intervention could have only 1 direction of effect, ie, beneficial or harmful, the use of a 1-tailed test must be justified. • ordinal data: type of data with a limited number of categories with an inherent ordering of the category from lowest to highest, but without fixed or equal spacing between increments.40(p333) Examples are Apgar scores, heart murmur rating, and cancer stage and grade. Ordinal data can be summarized by means of the median and quantiles or range. → Because increments between the numbers for ordinal data generally are not fixed (eg, the difference between a grade 1 and a grade 2 heart murmur is not quantitatively the same as the difference between a grade 3 and a grade 4 heart murmur), ordinal data should be analyzed by nonparametric statistics. • ordinate: vertical or y-axis of a graph. • outcome: dependent variable or end point of an investigation. In retrospective studies such as case-control studies, the outcomes have already occurred before the study is begun; in prospective studies such as cohort studies and controlled trials, the outcomes occur during the time of the study.40(p333) • outliers (outlying values): values at the extremes of a distribution. Because the median is far less sensitive to outliers than is the mean, it is preferable to use the median to describe the central tendency of data that have extreme outliers. → If outliers are excluded from an analysis, the rationale for their exclusion should be explained in the text. A number of tests are available to determine whether an outlier is so extreme that it should be excluded from the analysis. • overmatching: the phenomenon of obscuring by the matching process of a case-control study a true causal relationship between the independent and dependent variables because the variable used for matching is strongly related to the mechanism by which the independent variable exerts its effect.40(pp119-120) For example, matching cases and controls on residence within a certain area could obscure an environmental cause of a disease. Overmatching may also be used to refer to matching on variables that have no effect on the dependent variable, and therefore are unnecessary, or the use of so many variables for matching that no suitable controls can be found.42(p120) • oversampling: in survey research, a technique that selectively increases the likelihood of including certain groups or units that would otherwise produce too few responses to provide reliable estimates. • paired samples: form of matching that can include self-pairing, where each participant serves as his or her own control, or artificial pairing, where 2 participants are matched on prognostic variables.42(p186) Twins may be studied as pairs to attempt to separate the effects of environment and genetics. Paired analyses provide greater power to detect a difference for a given sample size than do nonpaired analyses, since interindividual differences are minimized or eliminated. Pairing may also be used to match participants in case-control or cohort studies. See Table 3. • paired t test: t test for paired data. • parameter: measurable characteristic of a population. One purpose of statistical analysis is to estimate population parameters from sample observations.40(p333) The statistic is the numerical characteristic of the sample; the parameter is the numerical characteristic of the population. Parameter is also used to refer to aspects of a model (eg, a regression model). • parametric statistics: tests used for continuous data and that require the assumption that the data being tested are normally distributed, either as collected initially or after transformation to the ln or log of the value or other mathematical conversion.40(p121) The t test is a parametric statistic. See Table 3. • Pearson product moment correlation: test of correlation between 2 groups of normally distributed data. See diagnostic discrimination. • percentile: see quantile. • placebo: a biologically inactive substance administered to some participants in a clinical trial. A placebo should ideally appear similar in every other way to the experimental treatment under investigation. Assignment, allocation, and assessment should be blinded. • placebo effect: refers to specific expectations that participants may have of the intervention. These can make the intervention appear more effective than it actually is. Comparison of a group receiving placebo vs those receiving the active intervention allows researchers to identify effects of the intervention itself, as the placebo effect should affect both groups equally. • point estimate: single value calculated from sample observations that is used as the estimate of the population value, or parameter40(p333); in most circumstances accompanied by an interval estimate (eg, 95% confidence interval). • Poisson distribution: distribution that occurs when a nominal event (often disease or death) occurs rarely.42(p125) The Poisson distribution is used instead of a binomial distribution when sample size is calculated for a study of events that occur rarely. • population: any finite or infinite collection of individuals from which a sample is drawn for a study to obtain estimates to approximate the values that would be obtained if the entire population were sampled.44(p197) A population may be defined narrowly (eg, all individuals exposed to a specific traumatic event) or widely (eg, all individuals at risk for coronary artery disease). • population attributable risk percentage: percentage of risk within a population that is associated with exposure to the risk factor. Population attributable risk takes into account the frequency with which a particular event occurs and the frequency with which a given risk factor occurs in the population. Population attributable risk does not necessarily imply a cause-and-effect relationship. It is also called attributable fraction, attributable proportion, and etiologic fraction.40(p333) • positive predictive value: proportion of those participants or individuals with a positive test result who have the condition or disease as measured by the criterion standard. This measure takes into account the prevalence of the condition or the disease. Clinically, it is the probability that an individual has the disease if the test result is positive.40(p334) See Table 4 and diagnostic discrimination. • posterior probability: in Bayesian analysis, the probability obtained after the prior probability is combined with the probability from the study of interest.42(p128) If one assumes a uniform prior (no useful information for estimating probability exists before the study), the posterior probability is the same as the probability from the study of interest alone. • post hoc analysis: analysis performed after completion of a study and not based on a hypothesis considered before the study. Such analyses should be performed without prior knowledge of the relationship between the dependent and independent variables. A potential hazard of post hoc analysis is the type I error. → While post hoc analyses may be used to explore intriguing results and generate new hypotheses for future testing, they should not be used to test hypotheses, because the comparison is not hypothesis-driven. See also data dredging. • posttest probability: the probability that an individual has the disease if the test result is positive (positive predictive value) or that the individual does not have the disease if the test result is negative (negative predictive value).40(p158) • power: ability to detect a significant difference with the use of a given sample size and variance; determined by frequency of the condition under study, magnitude of the effect, study design, and sample size.40(p128) Power should be calculated before a study is begun. If the sample is too small to have a reasonable chance (usually 80% or 90%) of rejecting the null hypothesis if a true difference exists, then a negative result may indicate a type II error rather than a true failure to reject the null hypothesis. → Power calculations should be performed as part of the study design. A statement providing the power of the study should be included in the “Methods” section of all randomized controlled trials (see Table 1) and is appropriate for many other types of studies. A power statement is especially important if the study results are negative, to demonstrate that a type II error was unlikely to have been the reason for the negative result. Performing a post hoc power analysis is controversial, especially if it is based on the study results. Nonetheless, if such calculations were performed, they should be described in the “Comment” section and their post hoc nature clearly stated. Example: We determined that a sample size of 800 patients would have 80% power to detect the clinically important difference of 10% at a ¼ .05. • precision: inverse of the variance in measurement (see measurement error)42(p129); the degree of reproducibility that an instrument produces when measuring the same event. Note that precision and accuracy are independent concepts; if a blood pressure cuff is poorly calibrated against a standard, it may produce measurements that are precise but inaccurate. • pretest probability: see prevalence. • prevalence: proportion of persons with a particular disease at a given point in time. Prevalence can also be interpreted to mean the likelihood that a person selected at random from the population will have the disease (synonym: pretest probability).40(p334) See also incidence. • principal components analysis: procedure used to group related variables to help describe data. The variables are grouped so that the original set of correlated variables is transformed into a smaller set of uncorrelated variables called the principal components.42(p131) Variables are not grouped according to dependent and independent variables, unlike many forms of statistical analysis. Principal components analysis is similar to factor analysis. • prior probability: in Bayesian analysis, the probability of an event based on previous information before the study of interest is considered. The prior probability may be informative, based on previous studies or clinical information, or not, in which case the analysis uses a uniform prior (no information is known before the study of interest). A reference prior is one with minimal information, a clinical prior is based on expert opinion, and a skeptical prior is used when large treatment differences are not expected.44(p201) When Bayesian analysis is used to determine the posterior probability of a disease after a patient has undergone a diagnostic test, the prior probability may be estimated as the prevalence of the disease in the population from which the patient is drawn (usually the clinic or hospital population). • probability: in clinical studies, the number of times an event occurs in a study group divided by the number of individuals being studied.40(p334) • product-limit method: see Kaplan-Meier method. • propensity analysis: in observational studies, a way of minimizing bias by selecting controls who have similar statistical likelihoods of having the outcome or intervention under investigation. In general, this involves examining a potentially large number of variables for their multivariate relationship with the outcome. The resulting model is then used to predict cases’ individual propensities to the outcome or intervention. Each case can then be matched to a control participant with a similar propensity. Propensity analysis is thus a way of correcting for underlying sources of bias when computing relative risk. • proportionate mortality ratio: number of individuals who die of a particular disease during a span of time, divided by the number of individuals who die of all diseases during the same period.40(p334) This ratio may also be expressed as a rate, ie, a ratio per unit of time (eg, cardiovascular deaths per total deaths per year). • prospective study: study in which participants with and without an exposure are identified and then followed up over time; the outcomes of interest have not occurred at the time the study commences.44(p205) Antonym is retrospective study. • pseudorandomization: assigning of individuals to groups in a nonrandom manner, eg, selecting every other individual for an intervention or assigning participants by Social Security number or birth date. • publication bias: tendency of articles reporting positive and/or “new” results to be submitted and published, and studies with negative or confirmatory results not to be submitted or published; especially important in meta-analysis, but also in other systematic reviews. Substantial publication bias has been demonstrated from the “file-drawer” problem.52 See funnel plot. • purposive sample: set of observations obtained from a population in such a way that the sample distribution of independent variable values is determined by the researcher and is not necessarily representative of distribution of the values in the population.40(p334) • P value: probability of obtaining the observed data (or data that are more extreme) if the null hypothesis were exactly true.44(p206) → While hypothesis testing often results in the P value, P values themselves can only provide information about whether the null hypothesis is rejected. Confidence intervals (CIs) are much more informative since they provide a plausible range of values for an unknown parameter, as well as some indication of the power of the study as indicated by the width of the CI.37(pp186-187) (For example, an odds ratio of 0.5 with a 95% CI of 0.05 to 4.5 indicates to the reader the [im]precision of the estimate, whereas P = .63 does not provide such information.) Confidence intervals are preferred whenever possible. Including both the CI and the P value provides more information than either alone.37(187) This is especially true if the CI is used to provide an interval estimate and the P value to provide the results of hypothesis testing. → When any P value is expressed, it should be clear to the reader what parameters and groups were compared, what statistical test was performed, and the degrees of freedom (df) and whether the test was 1-tailed or 2-tailed (if these distinctions are relevant for the statistical test). → For expressing P values in manuscripts and articles, the actual value for P should be expressed to 2 digits for P =.01, whether or not P is significant. (When rounding a P value expressed to 3 digits would make the P value nonsignificant, such as P ¼ .049 rounded to .05, the P value can be left as 3 digits.) If P < .01, it should be expressed to 3 digits. The actual P value should be expressed (P = .04), rather than expressing a statement of inequality (P < .05), unless P < .001. Expressing P to more than 3 significant digits does not add useful information to P < .001, since precise P values with extreme results are sensitive to biases or departures from the statistical model.37(p198) P values should not be listed simply as not significant or NS, since for meta-analysis the actual values are important and not providing exact P values is a form of incomplete reporting.37(p195) Because the P value represents the result of a statistical test and not the strength of the association or the clinical importance of the result, P values should be referred to simply as statistically significant or not significant; terms such as highly significant and very highly significant should be avoided. → JAMA and the Archives Journals do not use a zero to the left of the decimal point, since statistically it is not possible to prove or disprove the null hypothesis completely when only a sample of the population is tested (P cannot equal 0 or 1, except by rounding). If P < .00001, P should be expressed as P < .001 as discussed. If P > .999, P should be expressed as P > .99. • qualitative data: data that fit into discrete categories according to their attributes, such as nominal or ordinal data, as opposed to quantitative data.42(p136) • qualitative study: form of study based on observation and interview with individuals that uses inductive reasoning and a theoretical sampling model, with emphasis on validity rather than reliability of results. Qualitative research is used traditionally in sociology, psychology, and group theory but also occasionally in clinical medicine to explore beliefs and motivations of patients and physicians.53 • quality-adjusted life-year (QALY): method used in economic analyses to reflect the existence of chronic conditions that cause impairment, disability, and loss of independence. Numerical weights representing severity of residual disability are based on assessments of disability by study participants, parents, physicians, or other researchers made as part of utility analysis.42(p136) • quantile: method used for grouping and describing dispersion of data. Commonly used quantiles are the tertile (3 equal divisions of data into lower, middle, and upper ranges), quartile (4 equal divisions of data), quintile (5 divisions), and decile (10 divisions). Quantiles are also referred to as percentiles.38(p165) → Data may be expressed as median (quantile range), eg, length of stay was 7.5 days (interquartile range, 4.3-9.7 days). See also interquartile range. • quantitative data: data in numerical quantities such as continuous data or counts42(p137) (as opposed to qualitative data). Nominal and ordinal data may be treated either qualitatively or quantitatively. • quasi-experiment: experimental design in which variables are specified and participants assigned to groups, but interventions cannot be controlled by the experimenter. One type of quasi-experiment is the natural experiment.42(p137) • r: correlation coefficient for bivariate analysis. • R: correlation coefficient for multivariate analysis. • r2: coefficient of determination for bivariate analysis. See also correlation coefficient. • R2: coefficient of determination for multivariate analysis. See also correlation coefficient. • random-effects model: model used in meta-analysis that assumes that there is a universe of conditions and that the effects observed in the studies are only a sample, ideally a random sample, of the possible effects.34(p349) Antonym is fixed-effects model. • randomization: method of assignment in which all individuals have the same chances of being assigned to the conditions in a study. Individuals may be randomly assigned at a 2:1 or 3:1 frequency, in addition to the usual 1:1 frequency. Participants may or may not be representative of a larger population.37(p334) Simple methods of randomization include coin flip or use of a random numbers table. See also block randomization. • randomized controlled trial: see 20.2.1, Randomized Controlled Trials, Parallel-Design Double-blind Trials. • random sample: method of obtaining a sample that ensures that every individual in the population has a known (but not necessarily equal, for example, in weighted sampling techniques) chance of being selected for the sample.40(p335) • range: the highest and lowest values of a variable measured in a sample. Example: The mean age of the participants was 45.6 years (range, 20–64 years). • rank sum test: see Mann-Whitney test or Wilcoxon rank sum test. • rate: measure of the occurrence of a disease or outcome per unit of time, usually expressed as a decimal if the denominator is 100 (eg, the surgical mortality rate was 0.02). See also 19.7.3, Numbers and Percentages, Forms of Numbers, Reporting Proportions and Percentages. • ratio: fraction in which the numerator is not necessarily a subset of the denominator, unlike a proportion40(p335) (eg, the assignment ratio was 1:2:1 for each drug dose [twice as many individuals were assigned to the second group as to the first and third groups]). • recall bias: systematic error resulting from individuals in one group being more likely than individuals in the other group to remember past events.42(p141) → Recall bias is especially common in case-control studies that assess risk factors for serious illness in which individuals are asked about past exposures or behaviors, such as environmental exposure in an individual who has cancer.40(p335) • receiver operating characteristic curve (ROC curve): graphic means of assessing the extent to which a test can be used to discriminate between persons with and without disease,42(p142) and to select an appropriate cut point for defining normal vs abnormal results. The ROC curve is created by plotting sensitivity vs (1 − specificity). The area under the curve provides some measure of how well the test performs; the larger the area, the better the test. See Figure 4. The C statistic is a measure of the area under the ROC curve. → The appropriate cut point is a function of the test. A screening test would require high sensitivity, whereas a diagnostic or confirmatory test would require high specificity. See Table 4 and diagnostic discrimination. • reference group: group of presumably disease-free individuals from which a sample of individuals is drawn and tested to establish a range of normal values for a test.40(p335) • regression analysis: statistical techniques used to describe a dependent variable as a function of 1 or more independent variables; often used to control for confounding variables.40(p335) See also linear regression, logistic regression. • regression line: diagrammatic presentation of a linear regression equation, with the independent variable plotted on the x-axis and the dependent variable plotted on the y-axis. As many as 3 variables may be depicted on the same graph.42(p145) • regression to the mean: the principle that extreme values are unlikely to recur. If a test that produced an extreme value is repeated, it is likely that the second result will be closer to the mean. Thus, after repeated observations results tend to “regress to the mean.” A common example is blood pressure measurement; on repeated measurements, individuals who are initially hypertensive often will have a blood pressure reading closer to the population mean than the initial measurement was.40(p335) • relative risk (RR): probability of developing an outcome within a specified period if a risk factor is present, divided by the probability of developing the outcome in that same period if the risk factor is absent. The relative risk is applicable to randomized clinical trials and cohort studies40(p335); for case-control studies the odds ratio can be used to approximate the relative risk if the outcome is infrequent. → The relative risk should be accompanied by confidence intervals. Example: The individuals with untreated mild hypertension had a relative risk of 2.4 (95% confidence interval, 1.9-3.0) for stroke or transient ischemic attack. [In this example, individuals with untreated mild hypertension were 2.4 times more likely than were individuals in the comparison group to have a stroke or transient ischemic attack.] • relative risk reduction (RRR): proportion of the control group experiencing a given outcome minus the proportion of the treatment group experiencing the outcome, divided by the proportion of the control group experiencing the outcome. • reliability: ability of a test to replicate a result given the same measurement conditions, as distinguished from validity, which is the ability of a test to measure what it is intended to measure.42(p145) • repeated measures: analysis designed to take into account the lack of independence of events when measures are repeated in each participant over time (eg, blood pressure, weight, or test scores). This type of analysis emphasizes the change measured for a participant over time, rather than the differences between participants over time. • repeated-measures ANOVA: see analysis of variance. • reporting bias: a bias in assessment that can occur when individuals in one group are more likely than individuals in another group to report past events. Reporting bias is especially likely to occur when different groups have different reasons to report or not report information.40(pp335-336) For example, when examining behaviors, adolescent girls may be less likely than adolescent boys to report being sexually active. See also recall bias. • reproducibility: ability of a test to produce consistent results when repeated under the same conditions and interpreted without knowledge of the prior results obtained with the same test40(p336); same as reliability. • residual: measure of the discrepancy between observed and predicted values. The residual SD is a measure of the goodness of fit of the regression line to the data and gives the uncertainty of estimating a point y from a point x.38(p176) • residual confounding: in observational studies, the possibility that differences in outcome may be caused by unmeasured or unmeasurable factors. • response rate: number of complete interviews with reporting units divided by the number of eligible units in the sample.36 See 20.7, Survey Studies. • retrospective study: study performed after the outcomes of interest have already occurred42(p147); most commonly a case-control study, but also may be a retrospective cohort study or case series. Antonym is prospective study. • right-censored data: see censored data. • risk: probability that an event will occur during a specified period. Risk is equal to the number of individuals who develop the disease during the period divided by the number of disease-free persons at the beginning of the period.40(p336) • risk factor: characteristic or factor that is associated with an increased probability of developing a condition or disease. Also called a risk marker, a risk factor does not necessarily imply a causal relationship. A modifiable risk factor is one that can be modified through an intervention42(p148) (eg, stopping smoking or treating an elevated cholesterol level, as opposed to a genetically linked characteristic for which there is no effective treatment). • risk ratio: the ratio of 2 risks. See also relative risk. • robustness: term used to indicate that a statistical procedure’s assumptions (most commonly, normal distribution of data) can be violated without a substantial effect on its conclusions.42(p149) • root-mean-square: see standard deviation. • rule of 3: method used to estimate the number of observations required to have a 95% chance of observing at least 1 episode of a serious adverse effect. For example, to observe at least 1 case of penicillin anaphylaxis that occurs in about 1 in 10 000 cases treated, 30 000 treated cases must be observed. If an adverse event occurs 1 in 15 000 times, 45 000 cases need to be treated and observed.40(p114) • run-in period: a period at the start of a trial when no treatment is administered (although a placebo may be administered). This can help to ensure that patients are stable and will adhere to treatment. This period may also be used to allow patients to discontinue any previous treatments, and so is sometimes also called a washout period. • sample: subset of a larger population, selected for investigation to draw conclusions or make estimates about the larger population.52(p336) • sampling error: error introduced by chance differences between the estimate obtained from the sample and the true value in the population from which the sample was drawn. Sampling error is inherent in the use of sampling methods and is measured by the standard error.40(p336) • Scheffé test: see multiple comparisons procedures. • SD: see standard deviation. • SE: see standard error. • SEE: see standard error of the estimate. • selection bias: bias in assignment that occurs when the way the study and control groups are chosen causes them to differ from each other by at least 1 factor that affects the outcome of the study.40(p336) → A common type of selection bias occurs when individuals from the study group are drawn from one population (eg, patients seen in an emergency department or admitted to a hospital) and the control participants are drawn from another (eg, clinic patients). Regardless of the disease under study, the clinic patients will be healthier overall than the patients seen in the emergency department or hospital and will not be comparable controls. A similar example is the “healthy worker effect”: people who hold jobs are likely to have fewer health problems than those who do not, and thus comparisons between these groups may be biased. • SEM: see standard error of the mean. • sensitivity: proportion of individuals with the disease or condition as measured by the criterion standard who have a positive test result (true positives divided by all those with the disease).40(p336) See Table 4 and diagnostic discrimination. • sensitivity analysis: method to determine the robustness of an assessment by examining the extent to which results are changed by differences in methods, values of variables, or assumptions40(p154); applied in decision analysis to test the robustness of the conclusion to changes in the assumptions. • signed rank test: see Wilcoxon signed rank test. • significance: statistically, the testing of the null hypothesis of no difference between groups. A significant result rejects the null hypothesis. Statistical significance is highly dependent on sample size and provides no information about the clinical significance of the result. Clinical significance, on the other hand, involves a judgment as to whether the risk factor or intervention studied would affect a patient’s outcome enough to make a difference for the patient. The level of clinical significance considered important is sometimes defined prospectively (often by consensus of a group of physicians) as the minimal clinically important difference, but the cutoff is arbitrary. • sign test: a nonparametric test of significance that depends on the signs (positive or negative) of variables and not on their magnitude; used when combining the results of several studies, as in meta-analysis.42(p156) See also Cox-Stuart trend test. • skewness: the degree to which the data are asymmetric on either side of the central tendency. Data for a variable with a longer tail on the right of the distribution curve are referred to as positively skewed; data with a longer left tail are negatively skewed.44(pp238-239) • snowball sampling: a sampling method in which survey respondents are asked to recommend other respondents who might be eligible to participate in the survey. This may be used when the researcher is not entirely familiar with demographic or cultural patterns in the population under investigation. • Spearman rank correlation (ρ): statistical test used to determine the covariance between 2 nominal or ordinal variables.44(p243) The nonparametric equivalent to the Pearson product moment correlation, it can also be used to calculate the coefficient of determination. • specificity: proportion of those without the disease or condition as measured by the criterion standard who have negative results by the test being studied40(p326) (true negatives divided by all those without the disease). See Table 4 and diagnostic discrimination. • standard deviation (SD): commonly used descriptive measure of the spread or dispersion of data; the positive square root of the variance.40(p336)The mean 2 SDs represents the middle 95% of values obtained. → Describing data by means of SD implies that the data are normally distributed; if they are not, then the interquartile range or a similar measure involving quantiles is more appropriate to describe the data, particularly if the mean 2 SDs would be nonsensical (eg, mean [SD] length of stay = 9 [15] days, or mean [SD] age at evaluation = 4 [5.3] days). Note that the format mean (SD) should be used, rather than the construction. • standard error (SE): positive square root of the variance of the sampling distribution of the statistic.38(p195)Thus, the SE provides an estimate of the precision with which a parameter can be estimated. There are several types of SE; the type intended should be clear. In text and tables that provide descriptive statistics, SD rather than SE is usually appropriate; by contrast, parameter estimates (eg, regression coefficients) should be accompanied by SEs. In figures where error bars are used, the 95% confidence interval is preferred54 (see Example F10 in 4.2.1, Visual Presentation of Data, Figures, Statistical Graphs). • standard error of the difference: measure of the dispersion of the differences between samples of 2 populations, usually the differences between the means of 2 samples; used in the t test. • standard error of the estimate: SD of the observed values about the regression line.38(p195) • standard error of the mean (SEM): An inferential statistic, which describes the certainty with which the mean computed from a random sample estimates the true mean of the population from which the sample was drawn.39(p21) If multiple samples of a population were taken, then 95% of the samples would have means would fall within2 SEMs of the mean of all the sample means. Larger sample sizes will be accompanied by smaller SEMs, because larger samples provide a more precise estimate of the population mean than do smaller samples. → The SEM is not interchangeable with SD. The SD generally describes the observed dispersion of data around the mean of a sample. By contrast, the SEM provides an estimate of the precision with which the true population mean can be inferred from the sample mean. The mean itself can thus be understood as either a descriptive or an inferential statistic; it is this intended interpretation that governs whether it should be accompanied by the SD or SEM. In the former case the mean simply describes the average value in the sample and should be accompanied by the SD, while in the latter it provides an estimate of the population mean and should be accompanied by the SEM. The interpretation of the mean is often clear from the text, but authors may need to be queried to discern their intent in presenting this statistic. • standard error of the proportion:SD of the population of all possible values of the proportion computed from samples of a given size.39(p109) • standardization (of a rate): adjustment of a rate to account for factors such as age or sex.40(pp336-350) • standardized mortality ratio: ratio in which the numerator contains the observed number of deaths and the denominator contains the number of deaths that would be expected in a comparison population. This ratio implies that confounding factors have been controlled for by means of indirect standardization. It is distinguished from proportionate mortality ratio, which is the mortality rate for a specific disease.40(p337) • standard normal distribution: a normal distribution in which the raw scores have been recomputed to have a mean of 0 and an SD of 1.44(p245) Such recomputed values are referred to as z scores or standard scores. The mean, median, and mode are all equal to zero. • standard score: see z score.38(p196) • statistic: value calculated from sample data that is used to estimate a value or parameter in the larger population from which the sample was obtained,40(p337) as distinguished from data, which refers to the actual values obtained via direct observation (eg, measurement, chart review, patient interview). • stochastic: type of measure that implies the presence of a random variable.38(p197) • stopping rule: rule, based on a test statistic or other function, specified as part of the design of the trial and established before patient enrollment, that specifies a limit for the observed treatment difference for the primary outcome measure, which, if exceeded, will lead to the termination of the trial or one of the study groups.7(p258) The stopping rules are designed to ensure that a study does not continue to enroll patients after a significant treatment difference has been demonstrated that would still exist regardless of the treatment results of subsequently enrolled patients. • stratification: division into groups. Stratification may be used to compare groups separated according to similar confounding characteristics. Stratified sampling may be used to increase the number of individuals sampled in rare categories of independent variables, or to obtain an adequate sample size to examine differences among individuals with certain characteristics of interest.29(p337) • Student-Newman-Keuls test: see Newman-Keuls test. • Student t test: see t test. W. S. Gossett, who originated the test, wrote under the name Student because his employment precluded individual publication.42(p166) Simply using the term t test is preferred. • study group: in a controlled clinical trial, the group of individuals who undergo an intervention; in a cohort study, the group of individuals with the exposure or characteristic of interest; and in a case-control study, the group of cases.40(p337) • sufficient cause: characteristic that will bring about or cause the disease.40(p337) • supportive criteria: substantiation of the existence of a contributory cause. Potential supportive criteria include the strength and consistency of the relationship, the presence of a dose-response relationship, and biological plausibility.40(p337) • surrogate end points: in a clinical trial, outcomes that are not of direct clinical importance but that are believed to be related to those that are. Such variables are often physiological measurements (eg, blood pressure) or biochemical (eg, cholesterol level). Such end points can usually be collected more quickly and economically than clinical end points, such as myocardial infarction or death, but their clinical relevance may be less certain. • survival analysis: statistical procedures for estimating the survival function and for making inferences about how it is affected by treatment and prognostic factors.42(p163) See life table. • target population: group of individuals to whom one wishes to apply or extrapolate the results of an investigation, not necessarily the population studied.40(p337) If the target population is different from the population studied, whether the study results can be extrapolated to the target population should be discussed. • τ(tau): see Kendall τ rank correlation. • trend, test for: see χ2 test. • trial: controlled experiment with an uncertain outcome38(p208); used most commonly to refer to a randomized study. • triangulation: in qualitative research, the simultaneous use of several different techniques to study the same phenomenon, thus revealing and avoiding biases that may occur if only a single method were used. • true negative: negative test result in an individual who does not have the disease or condition as determined by the criterion standard.40(p338) See also Table 4. • true-negative rate: number of individuals who have a negative test result and do not have the disease by the criterion standard divided by the total number of individuals who do not have the disease as determined by the criterion standard; usually expressed as a decimal (eg, the true-negative rate was 0.85). See also Table 4. • true positive: positive test result in an individual who has the disease or condition as determined by the criterion standard.40(p338) See also Table 4. • true-positive rate: number of individuals who have a positive test result and have the disease as determined by the criterion standard divided by the total number of individuals who have the disease as measured by the criterion standard; usually expressed as a decimal (eg, the true-positive rate was 0.92). See also Table 4. • t test: statistical test used when the independent variable is binary and the dependent variable is continuous. Use of the t test assumes that the dependent variable has a normal distribution; if not, nonparametric statistics must be used.40(p266) → Usually the t test is unpaired, unless the data have been measured in the same individual over time. A paired t test is appropriate to assess the change of the parameter in the individual from baseline to final measurement; in this case, the dependent variable is the change from one measurement to the next. These changes are usually compared against 0, on the null hypothesis that there is no change from time 1 to time 2. → Presentation of the t statistic should include the degrees of freedom (df), whether the t test was paired or unpaired, and whether a 1-tailed or 2-tailed test was used. Since a 1-tailed test assumes that the study effect can have only 1 possible direction (ie, only beneficial or only harmful), justification for use of the 1-tailed test must be provided. (The 1-tailed test at α = .05 is similar to testing at α = .10 for a 2-tailed test and therefore is more likely to give a significant result.) Example: The difference was significant by a 2-tailed test for paired samples (t15 = 2.78, P = .05). → The t test can also be used to compare different coefficients of variation. • Tukey test: a type of multiple comparisons procedure. • 2-tailed test: test of statistical significance in which deviations from the null hypothesis in either direction are considered.40(p338) For most outcomes, the 2-tailed test is appropriate unless there is a plausible reason why only 1 direction of effect is considered and a 1-tailed test is appropriate. Commonly used for the t test, but can also be used in other statistical tests. • 2-way analysis of variance: see analysis of variance. • type I error: a result in which the sample data lead to a rejection of the null hypothesis despite the fact that the null hypothesis is actually true in the population. The α level is the size of a type I error that will be permitted, usually .05. → A frequent cause of a type I error is performing multiple comparisons, which increase the likelihood that a significant result will be found by chance. To avoid a type I error, one of several multiple comparisons procedures can be used. • type II error: the situation where the sample data lead to a failure to reject the null hypothesis despite the fact that the null hypothesis is actually false in the population. → A frequent cause of a type II error is insufficient sample size. Therefore, a power calculation should be performed when a study is planned to determine the sample size needed to avoid a type II error. • uncensored data: continuous data reported as collected, without adjustment, as opposed to censored data. • uniform prior: assumption that no useful information regarding the outcome of interest is available prior to the study, and thus that all individuals have an equal prior probability of the outcome. See Bayesian analysis. • unity: synonymous with the number 1; a relative risk of 1 is a relative risk of unity, and a regression line with a slope of 1 is said to have a slope of unity. • univariable analysis: another name for univariate analysis. • univariate analysis: statistical tests involving only 1 dependent variable; uses measures of central tendency (mean or median) and location or dispersion. The term may also apply to an analysis in which there are no independent variables. In this case, the purpose of the analysis is to describe the sample, determine how the sample compares with the population, and determine whether chance has resulted in a skewed distribution of 1 or more of the variables in the study. If the characteristics of the sample do not reflect those of the population from which the sample was drawn, the results may not be generalizable to that population.40(pp245-246) • unpaired analysis: method that compares 2 treatment groups when the 2 treatments are not given to the same individual. Most case-control studies also use unpaired analysis. • unpaired t test: see t test. • U test: see Wilcoxon rank sum test. • utility: in decision theory and clinical decision analysis, a scale used to judge the preference of achieving a particular outcome (used in studies to quantify the value of an outcome vs the discomfort of the intervention to a patient) or the discomfort experienced by the patient with a disease.42(p170) Commonly used methods are the time trade-off and the standard gamble. The result is expressed as a single number along a continuum from death (0) to full health or absence of disease (1.0). This quality number can then be multiplied by the number of years a patient is in the health state produced by a particular treatment to obtain the quality-adjusted life-year. See also 20.5, Cost-effectiveness Analysis, Cost-Benefit Analysis. • validity (of a measurement): degree to which a measurement is appropriate for the question being addressed or measures what it is intended to measure. For example, a test may be highly consistent and reproducible over time, but unless it is compared with a criterion standard or other validation method, the test cannot be considered valid (see also diagnostic discrimination). Construct validity refers to the extent to which the measurement corresponds to theoretical concepts. Because there are no criterion standards for constructs, construct validity is generally established by comparing the results of one method of measurement with those of other methods. Content validity is the extent to which the measurement samples the entire domain under study (eg, a measurement to assess delirium must evaluate cognition). Criterion validity is the extent to which the measurement is correlated with some quantifiable external criterion (eg, a test that predicts reaction time). Validity can be concurrent (assessed simultaneously) or predictive (eg, ability of a standardized test to predict school performance).42(p171) → Validity of a test is sometimes mistakenly used as a synonym of reliability; the two are distinct statistical concepts and should not be used interchangeably. Validity is related to the idea of accuracy, while reliability is related to the idea of precision. • validity (of a study): internal validity means that the observed differences between the control and comparison groups may, apart from sampling error, be attributed to the effect under study; external validity or generalizability means that a study can produce unbiased inferences regarding the target population, beyond the participants in the study.42(p171) • Van der Waerden test: nonparametric test that is sensitive to differences in location for 2 samples from otherwise identical populations.38(p216) • variable: characteristic measured as part of a study. Variables may be dependent (usually the outcome of interest) or independent (characteristics of individuals that may affect the dependent variable). • variance: variation measured in a set of data for one variable, defined as the sum of the squared deviations of each data point from the mean of the variable, divided by the df (number of observations in the sample 1).44(p266) The SD is the square root of the variance. • variance components analysis: process of isolating the sources of variability in the outcome variable for the purpose of analysis. • variance ratio distribution: synonym for F distribution.42(p61) • visual analog scale: scale used to quantify subjective factors such as pain, satisfaction, or values that individuals attach to possible outcomes. Participants are asked to indicate where their current feelings fall by marking a straight line with 1 extreme, such as “worst pain ever experienced,” at one end of the scale and the other extreme, such as “pain-free,” at the other end. The feeling (eg, degree of pain) is quantified by measuring the distance from the mark on the scale to the end of the scale.42(p268) • washout period: see 20.2.2, Randomized Controlled Trials, Crossover Trials. • Wilcoxon rank sum test: a nonparametric test that ranks and sums observations from combined samples and compares the result with the sum of ranks from 1 sample.38(p220) U is the statistic that results from the test. Alternative name for the Mann-Whitney test. • Wilcoxon signed rank test: nonparametric test in which 2 treatments that have been evaluated by means of matched samples are compared. Each observation is ranked according to size and given the sign of the treatment difference (ie, positive if the treatment effect was positive and vice versa) and the ranks are summed.38(p220) • Wilks Λ (lambda): a test used in multivariate analysis of variance (MANOVA) that tests the effect size for all the dependent variables considered simultaneously. It thus adjusts significance levels for multiple comparisons. • x-axis: horizontal axis of a graph. By convention, the independent variable is plotted on the x-axis. Synonym is abscissa. • Yates correction: continuity correction used to bring a distribution based on discontinuous frequencies closer to the continuous χ2 distribution from which χ2 tables are derived.42(p176) • y-axis: vertical axis of a graph. By convention, the dependent variable is plotted on the y-axis. Synonym is ordinate. • z-axis: third axis of a 3-dimensional graph, generally placed so that it appears to project out toward the reader. The z-axis and x-axis are both used to plot independent variables and are often used to demonstrate that the 2 independent variables each contribute independently to the dependent variable. See x-axis and y-axis. • z score: score used to analyze continuous variables that represents the deviation of a value from the mean value, expressed as the number of SDs from the mean. The z score is frequently used to compare children’s height and weight measurements, as well as behavioral scores.42(p176) It is sometimes referred to as the standard score. Figure 2. Decision tree showing decision nodes (squares) and chance outcomes (circles). End branches are labeled with outcome states. The subtrees to which the decision tree refers are depicted in a separate figure for simplicity. Adapted from Mason JJ, Owens DK, Harris RA, Cooke JP, Hlatky MA. The role of coronary angiography and coronary revascularization before noncardiac vascular surgery. JAMA. 1995;273(24):1919–1925. Table 4. Diagnostic Discrimination Test Result Disease by Criterion Standard Positive a (true positives) b (false positives) Negative c (false negatives) d (true negatives) a + c = total number of persons with disease b + d = total number of persons without disease Sensitivity = $aa+c$ Specificity = $db+d$ Positive predictive value = $aa+b$ Negative predictive value = $dc+d$ Figure 3. Survival curve showing outcomes for 2 treatments groups with number at risk at each time point. While numbers at risk are not essential to include in a survival analysis figure, this presentation conveys more information than the curve alone would. Adapted from Rotman M, Pajak TF, Choi K, et al. Prophylactic extended-field irradiation of para-aortic lymph nodes in stages IIB and bulky IB and IIA cervical carcinomas: ten-year treatment results of RTOG 79–20. JAMA. 1995;274(5):387–393. Figure 4. Receiver operating characteristic curve. The 45° line represents the point at which the test is no better than chance. The area under the curve measures the performance of the test; the larger the area under the curve, the better the test performance. Adapted from Grover SA, Coupal L, Hu X-P. Identifying adults at increased risk of coronary disease: how well do the current cholesterol guidelines work? JAMA. 1995;274(10):801–806. • [54.224.75.101] • 54.224.75.101 Edit Annotation Character Limit: 500/500 Annotation Character Limit: 0/500	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9134960174560547, "perplexity_flag": "middle"}
http://physics.stackexchange.com/questions/14043/plotting-a-wave-function-that-represents-a-particle/14046	# Plotting a wave function that represents a particle The problem is this: A particle is represented by the wave function $\psi = e^{-(x-x_{0})^2/2\alpha}\sin kx$. Plot the wave function $\psi$ and the probability distribution $\|\psi(x)\|^2$. This the problem 2.1 in the book Fundamental University Physics Volume III by Marcelo Alonso and Edward Finn. The thing is I don't know what values $k, \alpha$ and $x_{0}$ should have. Probably I don't know what $\psi$ really represents in this case. - ## 2 Answers The real purpose of this exercise seems to be exactly finding out the meaning of $\alpha$, $k$ and $x_0$. So just try modifying them all one by one and see how this affects the wave function. For instance, you will see that a very small $\alpha$ will give you a narrow-peaked distribution, while a big $\alpha$ leads to a widely spread function. - Then plot first with $x_0=0$, $\alpha=1$, and $k=1$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 14, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9337447285652161, "perplexity_flag": "head"}
http://unapologetic.wordpress.com/2008/03/20/products-of-integrable-functions/?like=1&source=post_flair&_wpnonce=c327cf8c6a	The Unapologetic Mathematician Products of integrable functions From the linearity of the Riemann-Stieltjes integral in the integrand, we know that the collection of functions that are integrable with respect to a given integrator over a given interval form a real vector space. That is, we can add and subtract them and multiply by real number scalars. It turns out that if the integrator is of bounded variation, then they actually form a real algebra — we can multiply them too. First of all, let’s show that we can square a function. Specifically, if $\alpha$ is a function of bounded variation on $\left[a,b\right]$, and of $f$ is bounded and integrable with respect to $\alpha$ on this interval, then so is $f^2$. We know that we can specialize right away to an increasing integrator $\alpha$. This will work here (unlike for the order properties) because nothing in sight gets broken by subtraction. Okay, first off we notice that $f(x)^2$ is the same thing as $\|f(x)\|^2$, and so they have the same supremum in any subinterval of a partition Then the supremum of $\|f(x)\|^2$ is the square of the supremum of $\|f(x)\|$ because squaring is an increasing operation that preserves suprema (and, incidentally, infima). The upshot is that $M_i(f^2)=M_i(\|f\|)^2$. Similarly we can show that $m_i(f^2)=m_i(\|f\|)^2$. This lets us write $\displaystyle M_i(f^2)-m_i(f^2)=M_i(\|f\|)^2-m_i(\|f\|)^2=\left(M_i(\|f\|)+m_i(\|f\|)\right)\left(M_i(\|f\|)-m_i(\|f\|)\right)$ $\leq2M\left(M_i(\|f\|)-m_i(\|f\|)\right)$ where $M$ is an upper bound for $\|f\|$ on $\left[a,b\right]$. Riemann’s condition then tells us that $f^2$ is integrable. Now let’s take two bounded integrable functions $f$ and $g$. We’ll write $f(x)g(x)=\frac{1}{2}\left(\left(f(x)+g(x)\right)^2-f(x)^2-g(x)^2\right)$ and then invoke the previous result and the linearity of integration to show that the product $fg$ is integrable. Like this: Posted by John Armstrong \| Analysis, Calculus 8 Comments » 1. Typo: 2nd paragraph, 2nd line. I think you mean “…and if $f$ is bounded…” Comment by \| March 20, 2008 \| Reply 2. I came across your blog, I dont know how to sound less strange but then again I am a strange person. Well most of my friends say that I am intresting, howvever that isnt the point. I cant tell you how much I love math I dont know why but it intrests me. I could sit through math problems for 5 hours before puting them down. I am only 18 and a senoir in high school but my math teachers dont understand that I want to learn more. I ask in Hopes that you may have time and would teach me. I will understand if you wouldn’t want to. I just want to learn “feed my mind” kind of thing. Write me either way Yes or no. So I will know. thank you either way…. Comment by Kasey \| March 21, 2008 \| Reply 3. Kasey, I understand what you mean, and I’m sorry that your teachers aren’t in a better position to show you where to go next. If you don’t have easy access to a nearby college or university, you’re sort of stuck until you leave high school. But from this distance I’m not in much of a position to guide you directly either. The best I can suggest is to go through my archives. Start here, and anything you don’t understand either try to find links back, or search for it in the bar on the right. Alternatively, there are some great mathematics texts out there you could try to get ahold of from Amazon or something. Actually, I’ll make a post tomorrow about that sort of thing, but I’ve got to head off to something else right now… Comment by \| March 21, 2008 \| Reply 4. [...] math books Yesterday, someone left a comment, which I’m reinterpreting a bit as a call for help in what to do for self-directed study of [...] Pingback by \| March 22, 2008 \| Reply 5. Vishal, I did not become interested in mathematics until I turned 21 or so.. I did not go to college, my background is in computer science (been coding since i was 9 years old). I taught myself mathematics by buying books and learning them myself.. it has been very expensive.. I spend on average about \$2000 a year on mathematics/statistics books and I choose to learn from advanced research oriented material rather than the basics as I believe it is worth it in the long run. I suggest getting a copy of Maple and TexMacs for typesetting in a GUI very mathematical and professional looking papers. Comment by Stephen Crowley \| March 22, 2008 \| Reply 6. Stephen, I think you’re meaning that towards Kasey. Comment by \| March 22, 2008 \| Reply 7. My very first mathematics book was Measure Theory and Integration by MM Rao. Comment by Stephen Crowley \| March 22, 2008 \| Reply 8. John, you are right, my comment was directed towards Kasey. Comment by Stephen Crowley \| March 22, 2008 \| Reply « Previous \| Next » About this weblog This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”). I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 23, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9438633322715759, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/187958/number-theory-least-common-multiple-question	# Number Theory Least Common Multiple Question I don't know how I could find, WITH proof, the smallest possible least common multiple of three positive integers which sum to $2005$. If someone can provide a proof I would be very happy. - I suppose you want strictly positive integers, otherwise the answer is $0$. – Raskolnikov Aug 28 '12 at 16:11 2 $2005=5\cdot 401$ where both $5$ and $401$ are primes. Since $401$ is the larger one, it should be ideal if all summands contain that factor, that is, are of the form $k\cdot 401$. The three $k$-factors have to add to five, which for three positive integers is only possible for $1+2+2$, which gives $2005 = 401 + 802 + 802$ where $\operatorname{lcm}(401,802,802)=802$, therefore I'm pretty sure the correct answer is $802$. However I don't know how to prove it. – celtschk Aug 28 '12 at 16:12 4 @Raskolnikov: The way I learned it, "positive" means $>0$. – celtschk Aug 28 '12 at 16:14 Small lcm connote large gcd, which means a large prime factor shared in common, and $5$ and $401$ are the only prime factors of $2005$. – Michael Hardy Aug 28 '12 at 16:15 I got a rough lower bound that the LCM is greater than 668, and by getting the three numbers to divide each other, I got an LCM of 802. I see @celtschk went along the same lines. I think this is a pretty good candidate for a computer check :) – rschwieb Aug 28 '12 at 16:16 show 1 more comment ## 1 Answer $2005=4\cdot401$ with $401$ prime. Then $$2005=2\cdot401+2\cdot401+401 \,.$$ Thus we found three numbers with lcm 802. We claim that this is the smallest possible value. Suppose by contradiction that we can find $a+b+c =2005$ and $l=\operatorname{lcm}(a,b,c) < 802$. We can assume that $a \leq b \leq c$. Since $b, c \leq l$ it follows that $$2005=a+b+c \leq a+2l < a+2\cdot802$$ Thus $a > 401> \frac{l}{2}$. Since $a> \frac{l}{2}$ and $a\mid l$, it follows that $a=l$. Then $$l =a \leq b \leq c \leq l \Rightarrow a=b=c=l \Rightarrow 3l=2005$$ Contradiction. - You can write $5\cdot401$ or $5\times401$. There's no need for such crudidities as $5*401$. – Michael Hardy Aug 28 '12 at 16:18 Shouldn't it be "$b,c\le l$" in line 7 (and then of course also for the first inequality sign in the following line)? That of course doesn't invalidate the proof. – celtschk Aug 28 '12 at 16:25 Thank you very much! I was looking for a solution like this, and it is very elegant! – Nitesh Aug 28 '12 at 16:27 @celtschk Ty fixed. Nitesh You were welcome. The proof is pretty natural once you realize that the only way some numbers with fixed sum have low lmc is if they have a big common part. If 2005 would had have different prime factorization the problem could had been terrible.... – N. S. Aug 28 '12 at 16:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 30, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9576095342636108, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/2340/what-is-the-first-interesting-theorem-in-insert-subject-here/2351	## What is the first interesting theorem in (insert subject here)? [closed] ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack various definitions, i.e. they don't really say anything interesting about mathematics "in the wild." In various subjects, what would you consider to be the first theorem (say, in the usual presentation in a standard undergraduate textbook) with actual content? Some possible examples are below. Feel free to either add them or disagree, but as usual, keep your answers to one suggestion per post. • Number theory: the existence of primitive roots. • Set theory: the Cantor-Bernstein-Schroeder theorem. • Group theory: the Sylow theorems. • Real analysis: the Heine-Borel theorem. • Topology: Urysohn's lemma. Edit: I seem to have accidentally created the tag "soft-questions." Can we delete tags? Edit #2: In a comment, ilya asked "You want the first result after all the basic tools have been introduced?" That's more or less my question. I guess part of what I'm looking for is the first result that justifies the introduction of all the basic tools in the first place. - 3 This question seems to have long outgrown its usefulness (and some of the more recent additions have been, IMHO, lousy). Voting to close. – Todd Trimble Mar 1 2012 at 16:03 show 4 more comments ## 78 Answers Finite group theory: I would say Lagrange's theorem, that the order of a subgroup divides the order of the group. Certainly it's prior to the Sylow theorems, certainly it has content. - 4 I was about to post that. I should make the addendum though that while Lagrange stated his theorem in 1770, the first full proof occurred 30 years later at the same time as the first proof of the insolubility of the quintic. – Jason Dyer Oct 24 2009 at 20:42 2 LaGrange's theorem is just an application of the definition of an equivalence class. – Harry Gindi Dec 13 2009 at 15:26 22 Fermat's last theorem is just an application of the definition of a natural number. – Steven Gubkin Mar 5 2010 at 0:15 4 @Marcos : "I don't understand why he didn't prove it generally". Presumably because the definition of a group was only given in the mid-19th century. – Laurent Berger Mar 1 2012 at 14:03 show 1 more comment ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. The uncountability of the reals. This is such a classic fact that I'm not sure how to classify it. Set theory, perhaps. - Euler's theorem V-E+F=2, can be regarded as a first theorem in graph theory, or in the theory of convex polytopes, and probably 0th theorem in algebraic topology and other fields. - Algebraic Geometry: Bezout's Theorem. It's also good for selling what algebraic geometry is to people who've never heard of it before. - 1 Bezout's Theorem is a nice theorem but it is hardly surprising in its proper setting (algebraically closed field, taking into account multiplicities and points at infinity). – lhf Oct 30 2009 at 9:18 5 But it is a great motivator for schemes and cohomology: To put projective and affine space in the same framework, you need gluing. To get the right formulas for higher order contact, you need the scheme theoretic intersection of curves. When you approach the theorem cohomologically, it reduces to just intersecting lines (which is a conceptually beautiful way to approach the proof). So not only is it simple to understand, it can be used as motivation for very deep ideas. – Steven Gubkin Nov 12 2009 at 20:04 Graph theory: necessary and sufficient conditions for the existence of an Eulerian walk/cycle - Algebraic topology: Fundamental group of S^1. - 4 I'm not sure I like this one - although it can be hard to prove when you don't have much machinery, it's basically a tautology deriving from the definition of the fundamental group and not a very insightful result. I'd say the first nontrivial theorem of algebraic topology (at least on the homology side of things) is Poincaré duality - after all that was one of the things that launched the subject. Although there are easier results (Brouwer fixed point), they do not really pertain to algebraic topology itself. Considering homotopy, there should be something about homotopy groups of spheres. – Sam Derbyshire Oct 25 2009 at 2:47 1 This is an important first result because now we can prove the Brouwer fixed point theorem (usually the fundamental group comes before homology right? the other things you need for that result are much more straightforward). – Sean Tilson Mar 5 2010 at 21:05 3 This is the most important computation in algebraic topology, in my opinion. Everything else ultimately derives from it. – Jeff Strom Jul 23 2010 at 18:06 show 5 more comments Knot theory: sufficiency of the Reidemeister moves. - Quadratic reciprocity feels to me like a "first nontrivial statement" without an obvious branch of mathematics. Not number theory -- there are things like the infinitude of the primes, and "algebraic number theory" doesn't seem quite right either... - 2 The first nontrivial statement in class field theory? – James Cranch Mar 1 2012 at 15:35 Ring theory: If R is a UFD, then so is R[x]. - 6 For ring theory, I prefer R noetherian implies R[x] is. The Hilbert Basis Theorem. – Charles Siegel Oct 26 2009 at 20:56 1 Post this as a separate answer so that it can get voted independently. – lhf Oct 30 2009 at 9:20 Differential equations: Picard's theorem on existence and uniqueness. - Commutative Algebra: The Hilbert Basis Theorem: If R is noetherian, so is R[x]. - Category theory: The Yoneda lemma. - 2 I thought about that, but I think category theorists would consider the Yoneda lemma trivial. Not to say that it's easy to understand, but it does follow directly from the category axioms. – Qiaochu Yuan Oct 24 2009 at 20:36 19 Though Yoneda's lemma isn't non-trivial, I feel like understanding its significance is definitely a non-trivial step. Schur's lemma in representation theory and Nakayama's lemma in algebra have a similar feel to them. They're pretty trivial to prove, but can take a while to really grok them. – Anton Geraschenko♦ Oct 24 2009 at 20:56 10 Maybe it's more correct to say that Yoneda is the last trivial theorem? – Harrison Brown Oct 25 2009 at 5:52 show 3 more comments Number theory: Different undergraduate textbooks approach the subject differently, of course. But the irrationality of the square root of 2 and the infinitude of primes are contentful theorems that are certainly very early historically, and also very early in at least some textbook treatments of the subject. - 1 I have to admit when I say "number theory" I always think "in the sense of Gauss." The infinitude of the primes is a better candidate now that I think about it. – Qiaochu Yuan Oct 24 2009 at 20:39 4 Perhaps you could even argue that the irrationality of the square root of 2 and the infinitude of primes are the first nontrivial theorems in mathematics as a whole. The Pythagorean theorem seems to be another candidate in this direction. – Michael Lugo Oct 24 2009 at 21:59 3 Probably Pythagorean before "sqrt(2) is irrational". Without a^2 + b^2 = c^2 with a = b = 1, we have no reason to consider sqrt(2) in the first place. – Chad Groft Feb 22 2010 at 17:31 Linear algebra: rank-nullity theorem. - 2 Although of course, this is just a categorification of the corresponding statement for finite sets... – Scott Morrison♦ Oct 24 2009 at 21:42 2 You could say that the rank nullity theorem is a direct consequence of the fact that all modules over fields are free (and hence projective). This may be a warped way of looking at things but I have to admit, thats how I remember it now :) – Grétar Amazeen Oct 25 2009 at 2:08 2 I think that various theorems about bases (every two bases have the same cardinality, every linearly independent set can be extended to a basis, and so on) are nontrivial and certainly come earlier. Why non-trivial? Because they may fail for modules over other rings, even if the module is free (but the ring may be noncommutative) or for f.g. modules over commutative rings (maximal linearly independent systems may have different cardinality, mathoverflow.net/questions/30066/…) – Victor Protsak Jul 24 2010 at 5:33 show 2 more comments Probability theory: the Central Limit Theorem? - 12 I would consider the law of large numbers non-trivial, and that was covered in my probability courses long before the CLT. – Kevin P. Costello Oct 25 2009 at 0:19 Social choice theory: Arrow's Impossibility Theorem. - Functional Analysis: the Hahn-Banach Theorem. - Noncommutative ring theory: The Artin-Wedderburn Theorem. A ring R is semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. - Theory of Computation: The halting problem Set Theory: Cardinality of a set is strictly less than its powerset Not sure which field (one might learn it in an intro real analysis course): Reals are uncountable Complexity theory: The Time and Space Hierarchy theorems One might learn these results in different courses, but they're all the same beautiful idea: Cantor's diagonalization. - operator algebras: the Gelfand transform is an isomorphism for commutative $C^$-algebras. - Functional analysis: the open mapping theorem. - Finite group representation theory: the fact that the irreducible characters form a basis for the class functions. - Symplectic Geometry: Darboux's theorem - Computer science: sort requires `n log n` - 8 Yijie Han dx.doi.org/10.1016/j.jalgor.2003.09.001 has an algorithm for sorting natural numbers that takes time O(n log log n). So it really depends on the class of sorting algorithms. Any comparison sort needs at least $\log_2 n! > c n \log n$ comparisons. – Konrad Swanepoel Nov 11 2009 at 10:09 show 4 more comments Riemann Surfaces: Riemann-Roch Theorem - Although the Gauss-Bonnet theorem was cited for differential geometry of surfaces, I really think that the first striking result in this subject is Gauss's Theorema Egregium, which is not obvious from the definition of Gaussian curvature (which makes explicit reference to the ambient space). But the Gauss-Bonnet theorem is certainly the first really deep theorem one encounters in differential geometry. - 1 Theorema Egregium is certainly very deep! Without Theorema Egregium to back it up, the Gauss-Bonnet theorem isn't very meaningful, and of course, it comes later. – Victor Protsak May 23 2010 at 21:24 Combinatorics: the nth Catalan number is (2n choose n)/(n+1) - Algebraic topology: Poincaré Duality - In mathematical logic, Gödel's incompleteness theorem. - 9 I would think of Completeness or Compactness of first order logic as a more reasonable answer. Incompleteness is interesting, but it feels more like a side branch of the theorem tree than the main trunk (whatever that means). – Richard Dore Oct 25 2009 at 6:40 show 4 more comments Homotopy Theory: the Hopf Fibration? - 2 That's not a theorem. – André Henriques Jul 23 2010 at 17:44 2 There is a theorem there (more than one, in fact), e.g. "$\pi_3(S^2)$ is an infinite cyclic group generated by the class of the Hopf fibration". – Victor Protsak Jul 24 2010 at 5:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9226968884468079, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/44109/how-should-one-think-about-non-hausdorff-topologies/44113	## How should one think about non-Hausdorff topologies? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/nets have unique limits, compact sets are closed, etc.). Most topological spaces encountered in undergraduate studies are indeed Hausdorff, often even normed or metrizable. However, at some point one finds that non-Hausdorff spaces do come up in practice, e.g. the Zariski topology in algebraic geometry, the Fell topology in representation theory, the hull-kernel topology in the theory of C*-algebras, etc. My question is: how should one think about (and work with) these topologies? I find it very difficult to think of such topological spaces as geometric objects, due to the lack of the intuitive Hausdorff axiom (and its natural consequences). With Hausdorff spaces, I often have some clear, geometric picture in my head of what I'm trying to prove and this picture gives good intuition to the problem at hand. With non-Hausdorff spaces, this geometric picture is not always helpful and in fact relying on it may lead to false results. This makes it difficult (for me, at least) to work with such topologies. As this question is somewhat ambiguous, I guess I should make it a community wiki. EDIT: Thanks for the replies! I got many good answers. It is unfortunate that I can accept just one. - 2 Regarding the Zariski topology, I think about it as what's left if you forget "most" of the open sets that really "should" be there. This point of view makes it clear why it is not Hausdorff and it fits in both with the classical topology on complex varieties and the étale topology and its variations. – Dan Petersen Oct 29 2010 at 12:09 awesome question! – B. Bischof Oct 29 2010 at 20:57 2 Atleast on $\mathbb{A}^n$, I like to think of the open sets as the sets on which regular functions do not vanish. The fact that there are not enough regular functions to make the Zariski topology hausdorff just says to me that algebraic varieties are more like holomorphic manifolds than differential manifolds – Daniel Barter Oct 29 2010 at 23:39 Thank you! I've been thinking of asking essentially the same question for a while. – Darsh Ranjan Oct 30 2010 at 0:08 Cool question! How, for example, am I supposed to think about topologies (like the unitary dual of a discrete group $G$) where elements are in the closures of singletons comprised of other elements...I work with these, but am certainly not "used to" it! Is my "net" thinking off on this? Thanks for asking this! – Jon Bannon Oct 30 2010 at 14:40 ## 9 Answers For a variety of reasons, it's often useful to develop an intuition for finite topological spaces. Since the only Hausdorff finite spaces are discrete, one will have to deal with the non-Hausdorff case almost all the time. The fact of the matter is that the category of finite spaces is equivalent to the category of finite preorders, i.e., finite sets equipped with a reflexive transitive relation. In terms of a picture, draw an arrow $x \to y$ between points $x$ and $y$ whenever $x$ belongs to the closure of $y$ (or the closure of $x$ is contained in the closure of $y$). This defines a reflexive transitive relation. Two points $x$, $y$ have the same open neighborhoods if and only if $x \to y$ and $y \to x$. It follows that the topology is $T_0$ (the topology can distinguish points) if and only if the preorder is a poset, where antisymmetry of $\to$ is satisfied. The closure of a point $y$ is the down-set {$x: x \to y$}, and a set is closed iff it is downward closed in the preorder. In the finite case, I believe it is true that every closed irreducible set (one that isn't the union of two proper closed subsets) is the closure of a point = principal ideal; if the point is unique, the space is called sober. Sober spaces are the kinds of spaces that arise as underlying topological spaces of schemes, and it seems to be true that a finite space is sober iff it is $T_0$. - 5 I think this is a good intuitive model for e.g. the Zariski topology on Spec C[x_1, ... x_n]. One has closed points, then points corresponding to one-dimensional subvarieties whose closure contains the closed points on them, then points corresponding to two-dimensional subvarieties... – Qiaochu Yuan Oct 29 2010 at 14:25 6 There's even finite space models for the homotopy type of spheres :) – David Carchedi Oct 29 2010 at 17:44 6 One computer-sciency intuition for topology is that an open set corresponds to an observable property (i.e., a property which can be semidecided). This quickly leads to topological spaces which are also partial orders and whose open sets are upward closed. Thus Todd's answer is a good one: non-Hausdorff spaces are typically partial orders in which open sets are upward closed. We are in the land of order theory here. – Andrej Bauer Oct 29 2010 at 21:07 @Todd: it is for sure true that a finite space is sober iff it is $T_0$. Here is the guarantee: the Zariski spectrum of a commutative ring is sober (hence $T_0$), and the main theorem of Mel Hochster's thesis is that a topological space is homeomorphic to some Spec R iff it is an inverse limit of finite $T_0$ spaces. (As a proof, this is of course very complicated and very likely circular, but as a sanity check it works nicely. I presume it is no problem to give a direct proof...) – Pete L. Clark Feb 3 2011 at 13:07 @Pete: yes, thanks. A direct proof is not difficult: certainly sober spaces are $T_0$ because sobriety implies distinct points have distinct closures. On the other hand, since we have an ascending chain condition on closed sets in finite spaces, we can show that every closed irreducible is the closure of some point, and that point is unique under $T_0$, and therefore we have sobriety under $T_0$. – Todd Trimble Feb 3 2011 at 14:17 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. One can think of a topology on a space $X$ as abstracting all the "stable" information (or "physical measurements") one can say about a state $x$ in $X$ (i.e. the open neighbourhoods of $x$ in $X$). For instance, consider the real number $\pi$ in ${\bf R}$ (with the usual topology). We can't specify $\pi$ exactly in a stable manner, because we can perturb $\pi$ a little bit and it won't be $\pi$. (In other words, $\{\pi\}$ is not open.) But we can say, for instance, that $3.14 < \pi < 3.15$, and this is a stable piece of information (it is true even if we perturb $\pi$ a little bit). The Hausdorff nature of the real line then lets us demonstrate that two quantities are distinct even if we are only allowed to access them in a stable manner. For instance, $\pi$ and $e$ can be stably shown to be distinct, because we have a stable measurement $3 < \pi < 4$ of $\pi$ and a stable measurement $2 < e < 3$ that are disjoint from each other. Now we work instead with the Zariski topology. Here, we are not allowed to use the $<$ sign to make stable measurements (we are now in the algebraic world rather than the semi-algebraic world). The only way to make stable measurements, then, is to use the $\neq$ sign (in conjunction with the usual arithmetic operations). For instance, one can say that $\pi$ is not equal to $3$, that $\pi^2$ is not equal to $10$, and so forth. This is of course a much weaker topology. In particular, it is no longer possible to use stable measurements to stably separate $\pi$ from $e$. ($\pi$, of course, does obey the stable measurement $\pi \neq e$, and $e$ obeys the stable measurement $e \neq \pi$, but this does not help, because the stable (i.e. open) sets $\{ x: x \neq e \}$ and $\{ x: x \neq \pi\}$ are not disjoint, and so these stable measurements do not force distinctness. In more standard notation, the Zariski topology is $T_0$ but not Hausdorff.) [This has nothing to do with the transcendental nature of $\pi$ or $e$; one also fails to separate, say, $0$ and $1$, in the Zariski topology.] [One can also take a measurement-oriented perspective to other aspects of the Zariski topology. Thus, for instance, a set $E$ is Zariski-dense if there is no way to exclude an arbitrary point $x$ from lying in $E$ using only stable measurements of $x$. As the Zariski topology is so weak, this is a fairly weak property; there are a lot of Zariski-dense sets.] In general, non-Hausdorff topologies are usually extremely weak topologies, in which there are very few stable measurements available and so it is hard to stably separate distinct points from each other. The most extreme case is the trivial topology, in which no non-trivial measurements are available at all. - In a related direction, there is a connection between modal logic and topology, first developed by Tarski and McKinsey. To give a flavor of this: imagine that the points of the real line are continuously assigned colors in the spectrum, and you ask someone to identify which points are definitely green. There may be "grue" points on the boundary between green and blue points, but the 'definitely green' points tend to form an open set. This is related to the "necessarily true" operator used in modal logic. (This comment isn't directed at you particularly, Terry; it's more a FWIW comment.) – Todd Trimble Jun 10 2011 at 16:48 One way to get non-Hausdorff spaces from Hausdorff spaces is to take quotients under mildly bad equivalence relations. If your non-Hausdorff space comes from such a construction, then you can think of its points as being subsets of the bigger Hausdorff topological space of which it's a quotient. - Well, if your non-Hausdorff space comes from such a construction, not only you can think of its points in that way: they are subsets of the bigger Hausdorff topological space of which it's a quotient :) – Mariano Suárez-Alvarez Oct 29 2010 at 14:14 5 I like this because it makes it clear why you can't separate points- because they aren't points, they're subspaces of Hausdorff spaces and you definitely can't always separate subspaces! – Dylan Wilson Oct 29 2010 at 14:29 1 Let's make this very concrete. A topological group is Hausdorff iff the identity is a closed point. Therefore if N is a normal subgroup of a topological group G, G/N in the quotient topology is Hausdorff iff N is closed. Thus if you pick a non-closed normal subgroup N, G/N will not be Hausdorff. (Any open subgroup of a top. group is closed, so a non-closed subgroup of G will not be open either.) – KConrad Oct 29 2010 at 15:36 Also, every topological space $X$ has a minimal hausdorff quotient $X \to X/R$. Thus the equivalence classes may be separated in some sense. – Martin Brandenburg Oct 29 2010 at 16:01 2 In fact, ALL compactly generated spaces (with no separation axiom) arise as quotients of locally compact Hausdorff spaces. In fact, this is if and only if. – David Carchedi Oct 29 2010 at 17:47 show 4 more comments I don't know if this is relevant, but here is an easy and sometimes useful remark about spaces that have only finitely many points: The topology is determined by the relation between points "p is in the closure of q", and this may be any transitive and reflexive relation. This applies more generally to spaces in which the union of an arbitrary set of closed sets is closed - Arrgh, you beat me! I'll post anyway. – Todd Trimble Oct 29 2010 at 14:09 2 I just said it was easy. You took the trouble to give details. – Tom Goodwillie Oct 29 2010 at 14:20 If the relation is antisymmetric (i.e. the topology is $T_0$), your space is simply a finite poset in disguise. And it has the same weak homotopy type as the geometric realization of its nerve, which can be any finite simplicial complex. So these spaces look quite common, using homotopy glasses. – BS Oct 29 2010 at 14:22 See also this post and comments from the Secret Blogging Seminar: sbseminar.wordpress.com/2007/08/11/… – Todd Trimble Oct 29 2010 at 14:45 1 Just to drop a keyword for those who want to read more about such things: Alexandroff spaces. – Pete L. Clark Feb 3 2011 at 13:09 Some of the non-Hausdorff topologies that turn up are actually not that hard to get an intuition for. For example, you can think of the Zariski topology on a classical algebraic variety $V$ as just being a collection of information describing all the subvarieties of $V$ (e.g., the Zariski topology on $\mathbb{A}^3_k$ describes all the algebraic curves, surfaces, and points in 3-space). It might seem at first glance that the topologies involved get hard to understand when we move from varieties to schemes, but really the topology of a scheme is not hard to get a handle on either. The key to understanding the topologies of schemes is to understand the generic points and to understand these you just need to get some intuition about the concept of specialization and generalization. Given two points $x,y$ in a topological space $X$, we say that $x$ is a specialization of $y$ (or that $y$ is a generalization of $x$) if $x$ is contained in the closure of $y$. What this means is that $y$ is contained in every open neighbourhood of $x$. I like to think of this as meaning that $y$ is infinitesimally close to $x$. Similarly, given a subset $F\subseteq X$ we say that a point $x\in F$ is a generic point of $F$ if $F$ is the closure of $x$. Evidently a necessary condition for such an $F$ to possess a generic point is that $F$ be a (non-empty) irreducible closed subset of $X$. It is not hard to show that in a $T_0$-space every irreducible closed subset has at most one generic point. But in fact the topology of a scheme is nicer than this: the topology of a scheme has the nice property that every (non-empty) irreducible closed subset has a unique generic point. (Such as space is called a sober space). How should we think about this? Well, if $F$ is a closed irreducible subset of $X$ and $\xi$ is a generic point of $F$ then this means that every point of $F$ is a specialization of $\xi$; in other words $\xi$ is contained in every open neighbourhood of every point in $F$. So this generic point is infinitesimally close to all of the points in $F$. Now, in a sober space the map sending a point to its closure provides a bijection between the set of points of the space and the set of non-empty irreducible closed subsets of the space. So if you take any scheme $X$, the closed points are the points that you should think of as being the points forming a "geometric space", and all the other points are simply generic points of the various irreducible closed subsets of this space--each non-closed point describes a unique irreducible closed subset. For example, consider the scheme version of the affine plane: $\mathbb{A}^2_k=Spec(k[X,Y])$. The subspace of closed points (i.e., the maximal ideals) is homeomorphic to the usual variety affine plane with the Zariski topology; all the other points of the scheme are just generic points describing all the subvarieties of the affine plane. Some of this may be a bit vague or imprecise, but the point is that it isn't too hard to develop some intuition for the (non-Hausdorff) topologies arising in algebraic geometry. - 1 This is very nice! – David Carchedi Oct 29 2010 at 22:27 Since you referred to intuition, this is possibly not too off-topic. In a sense, the archetype of the topologic categories is, how very elementary beings perceive the world. If I was an amoeba, I'd possibly just understand space as places close, or less close to me, not otherwise structured. I'd have no particular metric idea of my own shape; I'd just feel more or less connected, &c. So, a possible answer to your question is: like a dull amoeba. To make an example possibly closer to us, think you're in a car in the urban traffic. Due to one-way streets, metric is not the best way to organize your perception of the space: actually, the proper topology to do that is possibly not Hausdorff (usually, you can't move to A without immediately finding yourself in B, and once you are in B, you are enormously far from A, even if you changed your mind about the opportunity of the movement.) - My question is: how should one think about (and work with) these topologies? I find it very difficult to think of such topological spaces as geometric objects, due to the lack of the intuitive Hausdorff axiom (and its natural consequences) A geometric object is not just a set together with a topology. It also consists (or sometimes, a priori only consists) of a set or rather sheaf of admissible or regular functions on it. I think that these are more important than the topology. Polynomial functions often cannot separate "points", whereas continuous functions in most application can. There you get the hausdorff property, it's already in the sheaf of regular functions. I never had any trouble concerning non-hausdorff spaces. For me, the cited little lemmas about hausdorff spaces are not natural at all. They are useful, of course, but what is so natural about the condition that all compact subsets of a topological space are closed? Every geometry has its characteristic models and methods. When you try a translation between two geometries, you have to make sure that all its 'partial translations' are compatible with each other. In the case of manifolds vs. varieties, the translation hausdorff <-> separated has been fruitful. - 1 Geometry may be hard to define, but I know it when I see it. – Tom Goodwillie Oct 29 2010 at 16:59 @Tom Goodwillie Awesome Kenneth Clarke reference!!! – B. Bischof Oct 29 2010 at 20:56 I thought it was Potter Stewart? – Todd Trimble Oct 29 2010 at 22:11 Todd, yes. And if I mean anything serious by this joke is that I would shy away from trying to insist that in order for topology to qualify as a kind of geometry it should conform to some general pattern that fits a number of other kinds of geometry. – Tom Goodwillie Oct 29 2010 at 23:53 Conversely, the fact that polynomial functions cannot separate points is one reason it took me a long time to accept that "algebraic geometry" is really a kind of geometry---so what's "intuitive" depends on where you're coming from. On the other hand, topology is arguably just the special case of geometry where functions take values in the poset $0\le 1$. – Mike Shulman Nov 1 2010 at 20:27 show 1 more comment I'll expand upon my comment in Andre's answer. In some sense (which I am about to make precise), non-Hausdorff spaces occur when trying to "naturally" close Hausdorff spaces under colimits. Let's say the only spaces you think are "real" are compact Hausdorff spaces (this is somewhat reasonable, from certain viewpoints). But then, you might want to consider an infinite disjoint union of such spaces as still being a space, so you arrive at having to consider locally compact Hausdorff spaces. In fact, EVERY compactly generated space (not assuming any separation axioms) is the quotient of a (possibly) infinite disjoint union of compact Hausdorff spaces. To see this: Any compactly generated space $X$ is a (possibly large) colimit of compact Hausdorff spaces. Consider the set $P(X)\O(X)$ of non-open subsets of $X$. Then for element $V$, there exists a map $p_V:T_V \to X$ from a compact Hausdorff space such that $p^{-1}\left(V\right)$ is not open. Now, the colimit of the diagram $\left(p_V:T_V \to X\right)$ is $X$. The colimit is ALMOST formed by taking a quotient of the disjoint union of each of these $T_V$s- this is true once we know that all points of $X$ are hit, but, we can fix this by adding in a bunch of constant maps to this family. The converse, that the quotient of a sum of compact Hausdorff spaces is compactly generated is clear. So, Andre's answer is the "total" answer, in that it includes all (compactly generated) spaces. So yes, (almost) every example of a non-Hausdorff space is really just considering points to actually be subsets of a particular Hausdorff one. - Very nice! This is quite intuitive. – Qiaochu Yuan Oct 29 2010 at 20:23 The fundamental category of Topological spaces (without the Hausdorff property) is the "Sober" spaces category, this is strictly related to Zariski topology, and to locales and Topos (growing in generalization). See EGA1 (Grothendieck, Dieudonne, Springer), or "Stone Spaces" of P. Johnstone. Of course Hausdorff Topological spaces are related (roughly) to a our usual way of see the geometrical spaces, in a non Hausdorff space points are related for other intrinsic (logical, geometrical, algebraic, orders) criteria, then is right that our usually intuitive point of view lack to represent them. But when we escape from Hausdorff propriety we are near to escape from "space as set of points" concept, see the concept of locales or frames ("Stone Spaces" Johnstone). -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 114, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9455922842025757, "perplexity_flag": "head"}
http://mathoverflow.net/revisions/9199/list	## Return to Answer 1 [made Community Wiki] I believe this is a problem given in J. Hirsch's Differential Topology. This may be much simpler than the ones posted here already. But for what it's worth, here it is. Show that given a collection of spheres the product manifold embeds into an Euclidean space of one dimension higher, viz., for instance $S^2 \times S^3$ embeds in $\mathbb{R}^6$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8912772536277771, "perplexity_flag": "middle"}
http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-525	New Titles \| FAQ \| Keep Informed \| Review Cart \| Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education Return to List Five Lectures in Complex Analysis Edited by: Manuel D. Contreras and Santiago Díaz-Madrigal, Universidad de Sevilla, Spain SEARCH THIS BOOK: Contemporary Mathematics 2010; 161 pp; softcover Volume: 525 ISBN-10: 0-8218-4809-7 ISBN-13: 978-0-8218-4809-8 List Price: US\$59 Member Price: US\$47.20 Order Code: CONM/525 This volume contains state-of-the-art survey papers in complex analysis based on lectures given at the Second Winter School on Complex Analysis and Operator Theory held in February 2008 at the University of Sevilla, Sevilla, Spain. Complex analysis is one of the most classical branches of mathematical analysis and is closely related to many other areas of mathematics, including operator theory, harmonic analysis, probability theory, functional analysis and dynamical systems. Undoubtedly, the interplay among all these branches gives rise to very beautiful and deep results in complex analysis and its neighboring fields. This interdisciplinary aspect of complex analysis is the central topic of this volume. This book collects the latest advances in five significant areas of rapid development in complex analysis. The papers are: Local holomorphic dynamics of diffeomorphisms in dimension one, by F. Bracci, Nonpositive curvature and complex analysis, by S. M. Buckley, Virasoro algebra and dynamics in the space of univalent functions, by I. Markina and A. Vasil'ev, Composition operators $$\heartsuit$$ Toeplitz operators, by J. H. Shapiro, and Two applications of the Bergman spaces techniques, by S. Shimorin. The papers are aimed, in particular, at graduate students with some experience in basic complex analysis. They might also serve as introductions for general researchers in mathematical analysis who may be interested in the specific areas addressed by the authors. Indeed, the contributions can be considered as up-to-the-minute reports on the current state of the fields, each of them including many recent results which may be difficult to find in the literature. This book is published in cooperation with Real Sociedad Matemática Española (RSME). Readership Graduate students and research mathematicians interested in complex analysis and operator theory. AMS Home \| Comments: webmaster@ams.org © Copyright 2012, American Mathematical Society Privacy Statement	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8673279881477356, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/205197/diagonalising-a-matrix/205209	# Diagonalising a matrix I've got a matrix $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}$ and wish to diagonalise it. I find the eigenvalues as below. $$\det(A - xI) = 0 = \det\begin{bmatrix}1-x&-1\\ 2&-1-x\end{bmatrix}$$ $$\det(A - xI) = (1-x)(-1-x) - (-2) = 1 + x^2$$ Then $x =i, -i$. So, now I need to find the eigenvectors, which is where I'm a little confused. In the case that $x = i$: $$E_1 = ker\begin{bmatrix}1-i&-1\\ 2&-1-i\end{bmatrix}$$ $$\begin{bmatrix}1-i&-1\\ 2&-1-i\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$$ At this point we usually row reduce and end up with a row of all zeros. However, in this case that isn't possible. So, I get $2a + (-1-i)b \implies a = (b+ i)/2$ from the second row. I want to write the kernel in terms of a spanning set, so get $E_1 = \{((1+i)/2, 1)\}$. Would this be a correct spanning set for $E_1$? Is the choice between which rows to use arbitrary? I'm just seeking clarification because all problems I've done before row reduce to only a single row. Thanks - ## 1 Answer Solving your linear system regarding $E_1$, $$\begin{bmatrix}1-i&-1\\2&-1-i\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix},$$ You can perform for operations as normal $R_2\leftarrow R_2-\frac{2}{1-i}R_1$, which will give the second row as all zeros (like 'normal'). Then you have $(1-i)a-b=0\Rightarrow a=\frac{1+i}{2}b$, as you had. Then, $$E_1=\operatorname{span}\left\{\begin{bmatrix}1+i\\2\end{bmatrix}\right\}.$$ The same process can then be taked for the other eigenpair. - Thanks! I don't know why I thought the matrix wouldn't work heh – user1520427 Oct 1 '12 at 3:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 9, "mathjax_display_tex": 6, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9448752999305725, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/33102/on-the-tightness-of-chernoff-bounds-for-sum-of-poisson-trials?answertab=votes	# On the tightness of Chernoff bounds for sum of Poisson trials For the sum $X$ of independent 0-1 random variables $X_i$ ($0 \le i \le n-1$) with $Pr(X_i)=p_i$, namely $X=\sum_{i=0}^{n-1}{X_i}$ the following Chernoff bound holds, $$Pr(X \ge (1+\delta)\mu) \le \left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^\mu$$ where $\mu=E[X]$ is the expected value of $X$. But how tight is this bound? Is there any lower bound for $Pr(X \ge (1+\delta)\mu)$. If exist, is it also exponential to $\mu$? - ## 1 Answer Any lower bound would have to be zero for $\delta$ large enough, namely $\delta > (n/\mu)−1$. And one can manage for $(n/\mu)−1$ to be as small as one wants. This suggests that universal lower bounds are unlikely. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9547678232192993, "perplexity_flag": "head"}
http://gauravtiwari.org/2011/07/13/famous-fox-rabbit-chase-problems-3/	MY DIGITAL NOTEBOOK A Personal Blog On Mathematical Sciences and Technology Home » Featured » Fox – Rabbit Chase Problems Fox – Rabbit Chase Problems Start here Part I: A fox chases a rabbit. Both run at the same speed $v$. At all times, the fox runs directly toward the instantaneous position of the rabbit , and the rabbit runs at an angle $\alpha$ relative to the direction directly away from the fox. The initial separation between the fox and the rabbit is $l$. When and where does the fox catch the rabbit (if it does)? If it never does, what is their eventual separation? Part II: Similarly think about the same situation, except now let the rabbit always move in the straight line of its initial direction in above part of the question. When and where does the fox catch the rabbit (if it does)? If it never does, what is their eventual separation? Solutions Part I The relative speed of the fox and the rabbit, along the line connecting them, is always $v_{\text{rel}}= v- v \cos \alpha$. Therefore, the total time needed to decrease their separation from $l$ to zero is $T=\dfrac{l}{v-v \cos \alpha} =\dfrac{l}{v(1-\cos \alpha)} \ \ldots (1)$ which is valid unless $\alpha=0$, in which case the fox never catches the rabbit. The location of their meeting is a little trickier to obtain. We have two methods to do : SLICK METHOD Imagine that the rabbit chases another rabbit, which chases another rabbit, etc. Each animal runs at an angle $\alpha$ relative to the direction directly away from the animal chasing it. The initial positions of all the animals lie on a circle, which is easily seen to have radius $R=\dfrac{l/2}{\sin (\alpha/2)} \ \ldots (2)$. The center of the circle is the point, O, which is the vertex of the isosceles triangle with vertex angle $\alpha$, and with the initial fox and rabbit positions as the othe two vertices. By symmetry, the positions of the animals at all times must lie on a circle with center O. Therefore, O, is the desired point where they meet. The animals simply spiral into O. Remark An equivalent solution is the the following: At all times, the rabbit’s velocity vector is obtained by rotating the fox’s velocity vector by angle $\alpha$. The meeting point O, is therefore the vertex of the above mentioned isosceles triangle, MESSIER METHOD The speed of the rabbit in the direction orthogonal to the line connecting the two animals in $v \sin \alpha$. Therefore, during a time $dt$, the direction of the fox’s motion changes by an angle $d\theta =\dfrac {v \sin \alpha}{l_t} dt$ , where $l_t$ is the separation at time $t$. Hence the change in the fox’s velocity has magnitude $\|d\overrightarrow{v}\|=v d\theta =v (v \sin \alpha dt/l_t)$. The vector $d\overrightarrow{v}$ is orthogonal to $\overrightarrow{v}$, therefore, to get the $x$-component of $d\overrightarrow{v}$, we need to multiply $\|d\overrightarrow{v}\|$ by $v_y/v$. Similar reasoning holds for $y$-component of $d\overrightarrow{v}$, so we arrive at the two equations $\dot{v_x}= \frac{vv_y \sin \alpha}{l_t} \ \ldots (3)$ $\dot{v_y}=- \frac{vv_x \sin \alpha}{l_t} \ \ldots (4)$ Now, we know that $l_t =\{ l-v(1-\cos \alpha) t \}$. Multiplying the above equations (3) and (4) by $l_t$ , and integrating from the initial to final times, yields $v_{x,0}l+v(1-\cos \alpha)X=v \sin \alpha \, Y \ \ldots (5)$ $v_{y,0}l+v(1-\cos \alpha)Y=-v \sin \alpha \, X \ \ldots (6)$ where (X,Y) is the total displacement vector and $(v_{x,0},v_{y,0})$ is the initial velocity vector. Putting all the X and Y terms on the right sides, and squaring and adding the equations, we get $l^2v^2=(X^2+Y^2)(v^2 \sin^2 \alpha +v^2{(1-\cos \alpha)}^2). \ \ldots (7)$ Therefore , the net displacement is $R=\sqrt{X^2+Y^2}=\dfrac{l}{\sqrt{2(1-\cos \alpha)}}=\dfrac{l/2}{\sin (\alpha/2)} \ \ldots (8)$ To find the exact location, we can, without loss of generality, set $v_{x,0} =0$, in which case we find $Y/X=(1-\cos \alpha)/\sin \alpha =\tan \alpha/2$. This agrees with the result of the first solution. $\Box$ Part II: SLICK METHOD Let $A(t)$ and $B(t)$ be the positions of the fox and the rabbit respectively. Let $C(t)$ be the foot of the perpendicular dropped from $A$ to the line of the rabbit’s path. Let $\alpha_t$ be the angle, dependent to the time, at which the rabbit moves relative to the direction directly away from the fox (so at $t=0, \ \alpha_0=\alpha$ and at $t=\infty , \ \alpha_{\infty}=0$). The speed at which the distance AB decreases is equal to $v-v \cos \alpha_t$. Therefore, the sum of the distances AB and CB doesn’t change. Initially, the sum is $l+l \cos \alpha$ and in the end , it is $2d$ where $d$ is desired eventual separation. Therefore, the desired eventual separation $d=\dfrac{l(1+\cos \alpha)}{2} \ \ldots (9)$ STRAIGHT FORWARD METHOD Let $\alpha_t$ be defined as in the first solution, and let $l_t$ be the separation at time $t$. The speed of the rabbit in the direction orthogonal to the line connecting the two animals is $v \sin \alpha_t$. The separation is $l_t$ , so the angle $\alpha_t$ changes at a rate $\dot{\alpha_{t}}= - \dfrac{v \sin \alpha_t}{l_t} \ \ldots (10)$. And $l_t$ changes at a rate $\dot{l_t}=-v(1-\cos \alpha_t) \ \ldots (11)$. Taking the quotient of the above two equations, separating variables, gives a differential equation $\dfrac{\dot{l_t}}{l_t} = \dfrac {\dot{\alpha_t}(1-\cos \alpha_t)}{\sin \alpha_t} \ \ldots (12)$ which on solving gives $\ln (l_t) = -\ln {(1+\cos \alpha_t)} + \ln (k) \ \ldots (13)$. Where $k$ is the constant of integration. Which gives $k=l_t (1+\cos \alpha_t) \ \ldots (14)$. Applying initial conditions $k_0 = l_0 (1+\cos \alpha_0)= l(1+\cos \alpha) \ \ldots (15)$. Therefore from (14), we get $l(1+\cos \alpha) = l_t(1+\cos \alpha_t)$, or $l_t= \dfrac{l(1+\cos \alpha)}{1+\cos \alpha_t}$. Setting $t= \infty$ and using $\alpha_{\infty} =0$, gives the final result $l_{\infty} = \dfrac{l(1+\cos \alpha)}{2}$. $\Box$. Remark: The solution of Part II is valid for all $\alpha$ except $\alpha= \pi$. If $\alpha = \pi$, the rabbit run directly towards the fox and they will meet halfway in time $l/2v$. 26.740278 83.888889 Like this: Tags: Dynamics, mathematical physics, Mechanics, Puzzles By in on . 2 Comments 1. Lance says: I am not sure what you mean by a “rabbit stop at angle alpha”. I’d like to solve the problem, but the only way I could get a hint is to read your solution. Frustrating. [Fixed - Thanks] • Gaurav Tiwari says: I’ve reddit this post to get improvements. . But I don’t know why the redditors usually don’t comment. This is my notebook and errors are possible as a student. • Smell everything with Google Nose! Just figured Google Nose (beta) in Google Search results, a great new feature which make you smell the things. Like how does a ‘wet dog’… • Life After Google Reader Seems like a false title, should have been “Life after the announcement of death of Google Reader” – whatever, Google Reader is reaching to its… • Happy Holi! : The Village Tour Holi, the festival of colors, was celebrated this year on 27th and 28th of March all over India. I decided to move to my own… • Euler’s (Prime to) Prime Generating Equation The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were… Cancel	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 73, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9043102264404297, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/282042/define-infinite-path-with-a-finite-relation-in-a-graph-with-least-fixed-point-lo?answertab=votes	# Define infinite path with a finite relation in a graph with Least Fixed Point logic Least Fixed Point(LFP) logic (p. 37ff) is an extension of first order logic which enables the usage of the least fixed point of FO-definable operators. For example consider a graph $G=(V,E)$ and binary operator $T(x,y)$ defined by $$x = y \lor \lor T(x,y) \lor \exists z (Exz \land Tzy).$$ The operator "updates" a binary relation $R \subseteq V \times V$ to $T(R)$. Now if we consider multiple application of $T$ onto a relation $R$ $$T(T(\dots T(R) \dots))$$ the operator reaches a fixed point such that $$T^n(R) = T^{n+1}(R).$$ Then $T^n$ is a fixed point. The fixed point which is minimal with respect to $\subseteq$ is the least fixed point. LFP allows FO-defined operators to define new relations. The least fixed point of $E$ and the operator above is the binary transtive closure relation $TC(E)$ of $E$. Given a directed (possibly not finite) graph $G = (V,E,P)$ with an unary relation symbol $P$. We want to define a LFP-formula $\varphi(v) \models G$ iff there is an infinite path starting in $v$ such that $P$ occurs only finitely often. If have several problems with this exercise. First of all I do not now how to define paths because it is possible to define relations on $V \times V$ but if an edge on infinite path would occur twice then a fixed point is imitatively reached even though the path is not finished. Second, I know that it is possible to define "finite" in a structure with a successor function $S$ with $LFP$ logic. But I don't think that such a function is definable. Is it a good approach to divide the properties or is it easier to use a more holistic approach? Any hints? - 1 I have only passing acquaintance with LFPL, but my thought would be to try to model the following statement: there is a node $u$ reachable from $v$ and there is an infinite path starting at $u$ with no occurrences of $P$. – Arthur Fischer Jan 19 at 16:14 Okay, the existence of a reachable node is similar to the transitive closure but how do I model an infinite path with a single formula without a set of formulas? – joachim Jan 19 at 17:12 1 If I had a better feel for the logic I would say more, but unfortunately I don't. I'll try to think about this more, however. – Arthur Fischer Jan 20 at 6:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 23, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.949445903301239, "perplexity_flag": "head"}
http://mathoverflow.net/questions/74338?sort=votes	## Taylor Series Remainder ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Suppose I have a $C^\infty$ smooth function $f$ defined on the reals. I can apply Taylor's formula and get the local expression $$f(x) = \sum_{i=0}^l\frac{f^{(i)}(0)}{l!}x^i+ f^{(l+1)}(\xi(x))x^{l+1}.$$ Question: Is the function $\xi$ smooth? The function $f$ can in principle be as nice as you want. - 2 This is a classical exercise in calculus classes. It is inappropriate to MO. – Denis Serre Sep 2 2011 at 13:14 Completely agree with @Denis – Igor Rivin Sep 2 2011 at 13:42 ## 1 Answer Note that the point $\xi$ in the expression of the remainder is not unique in general (as it is clear already for $l=0$). According to a common phenomenon, lack of unicity may cause a lack of continuity. For an example where there is no continuous $\xi$ (again for $l=0$) think of a smooth function $f$ which is positive and concave on $I:=(0,1)$; with $f(0)=f(1)=0$, with $f'(1) < 1$, and which is flat on an interval $J:=\{f'(x)=0\}\subset I$. Crossing the point $x_0=1$, the point $\xi(x)$ has to jump the interval $J$, causing a discontinuity at $x=1$. Note that in this example $f^{l+2}(\xi(x_0))=0$. On the other hand, going back to the general situation, if you have a point $\xi_0$ for the expression of the remainder corresponding to $x_0\neq0$, and if $f^{(l+2)}(\xi_0)\neq0$, then the implicit function theorem applies, giving a smooth function $\xi$ in a nbd of $x_0$. Finally note that any such function $\xi$ is certainly continuous at $x=0$, but it may have discontinuities in any nbd of $0$ even if $f$ is smooth (think of a proper version of the first example, with flat intervals accumulating at $0$). - 2 $f(x) = x (1-x)^2$ is a very simple example with $l=0$. $\lim_{x \to \infty} \frac{f(x)}{x} = \infty$ so you need $\xi(x) \to \infty$ as $x \to \infty$, but since $\frac{f(x)}{x} \ge 0$ for $x \ge 0$ it has to jump the interval $(1/3, 1)$ where $f' < 0$. – Robert Israel Sep 2 2011 at 19:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 37, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9290977120399475, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/9851/can-the-phase-of-a-function-be-extracted-from-only-its-absolute-value-and-its-fo/128373	# Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value? If for a function $f(x)$ only its absolute value $\|f(x)\|$ and the absolute value $\|\tilde f(k)\|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = \|f(x)\|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$) As Marek already stated, this is even not uniquely possible for $f(x)=c\in\mathbb C$, since the global phase cannot be re-determined. So please let me extend the question to Under what circumstances is the phase-retrieval (up to a global phase) uniquely possible, and what ambiguities could arise otherwise? - I think you mean $\tilde{f}(k)=N\int f(x)e^{-ikx}dx$. – Américo Tavares Nov 11 '10 at 13:30 @Américo Tavares: correct. although I could swap $x$ and $k$ as well :p – Tobias Kienzler Nov 11 '10 at 13:36 The sign in the argument of e is not important (or multiplying the argument by any real number for that matter -- e.g. 2\pi is a popular choice). It is just a convention. – Marek Nov 11 '10 at 13:50 @Marek: true. But since I use $x$ and $k$, the usual convention is $f(x)=\int \tilde f(k) e^{ikx} dx$ thus Américo's comment is fine with me – Tobias Kienzler Nov 11 '10 at 13:53 1 @Tobias Kienzler, @Marek, Thanks. Your comments explain why in different books the conventions are not always the same. – Américo Tavares Nov 11 '10 at 15:14 show 3 more comments ## 4 Answers Do the usual numerical phase retrieval algorithms not work? - – Tobias Kienzler Nov 18 '10 at 13:03 The short answer is no. Take constant function $f(x) \equiv C \in \mathbb{C}$. Disregarding normalization, we have $\hat{f} = C \delta$ (in the sense of distributions). Clearly, there is no way to recover the phase of $C$ once we take the absolute value on both sides. To make this a little more explicit, consider a lot easier version of the problem on the group $G = \mathbb{Z} / N\mathbb{Z}$. Its dual is $\hat{G} = G$. If you'll write out the Fourier transform equations (i.e. $\hat{f}(k) = \sum_{n=0}^{N-1} f(n) \exp(-{i k n \over 2 \pi})$), you'll obtain $2N$ real equations for $2N$ coefficients ($N$ Fourier phases and $N$ original phases). The properties of this system of equations are not clear to me, but the case $N=1$ (this is the same as in the first paragraph, but here we don't need to talk about distributions) already shows that the solutions need not be unique. I hope someone else can provide more information, I'd be also interested to see what conditions on $f$ one needs to assume to get a unique solution. Even for the case $G = \mathbb{Z} / N\mathbb{Z}$ this looks interesting enough. - +1 thanks, I should have thought of that example. I extended my question to include yours about conditions on uniqueness. – Tobias Kienzler Nov 11 '10 at 13:59 It is impossible to determine the phase uniquely, since a global phase could always be factored out that would be destroyed by taking absolute values. However the question is now, (when) is it possible to uniquely determine the relative phase, that is, the phase up to a global phase? – Tobias Kienzler Nov 12 '10 at 9:38 Yeah, I also realized that later. Also, solutions even need not exist. Intuitive reason: Fourier transform has to preserve certain absolute values, like \|\hat{f}(0)\| = \|f(0) + f(1) + ... \| so that you clearly can't impose arbitrary conditions on both sides (this is more obvious on finite groups). If I can make this a bit more formal later, I'll update my answer. – Marek Nov 12 '10 at 9:53 Just curious. How would one even define the phase of a constant function? – crasic Nov 12 '10 at 9:54 1 @crasic: "phase" is just a parlance for the phi(x) part of f(x) = R(x)exp(i phi(x)). For constant function phi doesn't depend on x. – Marek Nov 12 '10 at 9:57 The right way to ask the question is: given a function $f\in L²(\mathbb{R})$, can $f$ be determined from $\|f\|$ and $\|\widehat{f}\|$ up to a multiplicative constant $c$ of modulus $\|c\|=1$. This question dates back to Pauli and the answer is no. One can construct counter examples of the form $a\gamma(x-x_0)+b\gamma(x)+c\gamma(x+x_0)$ with $a,b,c$ properly chose ($\gamma(x)=e^{-\pi x²}$ the standard gaussian so that it is ots own Fourier transform). An other construction is as follows: take $\chi=\mathbf{1}_{[0,1/2]}$ $(a_j)_{j\in\mathbb{Z}}$ a sequence with finite support (to simplify) and $f(x)=\sum_j a_j\chi(x-j)$ so that $\hat f(\xi)=\sum_j a_je^{2i\pi j\xi}\hat\chi(\xi)$. Now we want to construct a sequence $(b_j)$ such that $\|a_j\|=\|b_j\|$ and $\left\|\sum_j a_je^{2i\pi j\xi}\right\|=\left\|\sum_j b_je^{2i\pi j\xi}\right\|$. This can be done via a Riesz product: take $\alpha_1,\ldots,\alpha_N$ a finite real sequence, $\varepsilon_1,\ldots,\varepsilon_N$ a finite sequence of $\pm1$ and consider $$\prod_{k=1}^N (1+i\alpha_j\varepsilon_j\sin 2\pi 3^j\xi)=\sum a_j^{(\varepsilon)}e^{2i\pi j\xi}.$$ Changing a $\varepsilon_j$ from $+1$ to $-1$ conjugates one of the factors on the left hand side, so it does not change the modulus. Now the same happens for the $a_j^{(\varepsilon)}$: each of them is either $0$ or a product of $i\alpha_j\varepsilon_j$ (up to a constant) -- the point is that it is not the sum of products of $i\alpha_j\varepsilon_j$'s, this is why we took the $3^j$ in the sine!. - I'm not sure if this applies, but for minimum-phase functions, the phase and magnitude of the Fourier Transform are related. See here for a brief overview. I've never actually used this relationship in practice, so can't really give you much more information. - sorry, I somehow missed the notification for your answer, so a very belated thanks for your answer. So does this simply state that the phase is the negate Hilbert transform of the absolute value's logarithm (plus some constant)? That almost sounds plausible, since the logarithm is analytical for positive arguments, but what about the essential singularity at 0? At the zeros, a phase is of course meaningless, so that might still be somehow okay. I wonder if there is some more rigorous work on this, Wikipedia doesn't seem to cite any source for that paragraph... – Tobias Kienzler Jan 12 '12 at 9:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 54, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9192747473716736, "perplexity_flag": "head"}
http://mathhelpforum.com/discrete-math/2177-recurrence-problem-print.html	# Recurrence Problem Printable View • March 11th 2006, 10:11 PM hotmail590 Recurrence Problem Suppose that the function f satisfies the recurrence relation 2 * f ( sqrt(n) ) + log(n) whenever n is a perfect square greater than 1 and f(2) = 1 For this problem I am asked to find f(16). If I pluged in n = 16 into the function above I will get 2 * 4 + log(16) which equals to aprox. 9.204 9.204 is not a perfect square so does that mean when n=16 there is no answer? I believe that we are supposed to plug the given n into the function and find its value, then if it is a perfect square, we plug that into the function again until that value is not a perfect square. Would this be correct? Thank you very much for your help! • March 12th 2006, 03:31 AM CaptainBlack Quote: Originally Posted by hotmail590 Suppose that the function f satisfies the recurrence relation 2 * f ( sqrt(n) ) + log(n) whenever n is a perfect square greater than 1 and f(2) = 1 As this is it is not a recurrence relation as it stands. It should look something like: $<br /> f(n)=2 f(\sqrt{n})+\log(n)<br />$ then as $f(2)=1$ we would have: $<br /> f(16)=2f(4) +\log(16)=2 (2f(2)+\log(4))+\log(16)<br />$ $<br /> =2(2+\log(4))+\log(16)=4+2\log(4)+\log(16)<br />$ RonL • March 12th 2006, 11:20 AM hotmail590 Thanks again CaptianBlack for the help! If I were to find a big O estimate for f(n) it would look like it is an O(log(n)) however I am not sure. Also there was a hint given in finding the big O estimate (Hint: Make the subsitution m = log n) What does that mean? If I replaced log(n) with m then f(n) would look something like the following $<br /> f(n)=2 f(\sqrt{n})+ m <br />$ Would m be just a constant and this f(n) will be an O(n^(1/2)) therefore $<br /> f(n)=2 f(\sqrt{n})+\log(n)<br />$ is O(n^(1/2)). Is this correct? Thanks again for the help!!! • March 15th 2006, 05:10 AM CaptainBlack Quote: Originally Posted by hotmail590 Thanks again CaptianBlack for the help! If I were to find a big O estimate for f(n) it would look like it is an O(log(n)) however I am not sure. Also there was a hint given in finding the big O estimate (Hint: Make the subsitution m = log n) What does that mean? If I replaced log(n) with m then f(n) would look something like the following $<br /> f(n)=2 f(\sqrt{n})+ m <br />$ Would m be just a constant and this f(n) will be an O(n^(1/2)) therefore $<br /> f(n)=2 f(\sqrt{n})+\log(n)<br />$ is O(n^(1/2)). Is this correct? Thanks again for the help!!! Is $n$ supposed to be equal to $2^{2^m}$ for some $m \in \mathbb{N_+}$ here? RonL All times are GMT -8. The time now is 11:33 AM.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9479964375495911, "perplexity_flag": "middle"}
http://physics.stackexchange.com/questions/tagged/torque	# Tagged Questions The torque tag has no wiki summary. 1answer 65 views ### How is torque equal to moment of inertia times angular acceleration divided by g? How is the following relation true $$\tau = \large\frac{I}{g} \times \alpha$$ where $\tau$ is torque, $I$ is moment of inertia, $g= 9.8ms^{-2}$, and $\alpha=$ angular acceleration. 2answers 63 views ### How do objects change their axis of rotation? If I hold a pencil at its end and spin it, throwing it upwards, it will spin about its end, but will soon start spinning around its center. How is this? I would draw the following torque diagram for ... 1answer 36 views ### Pendulum system: how is derived the output as Energy? Good day to everyone, I want to understand in which way the "Energy equation" is been implemented to this pendulum system. $x_1(t)$: The angular position of the mass $x_2(t)$: The angular velocity ... 0answers 67 views ### Torque, lever and mass The Force used in a catapult is exerted near its axis. If we double the length of the arm of the catapult, but still use the same Force at the same point as before near the same axis, does the ... 1answer 31 views ### Find the bending moment of a pole attached to a moving block I'm having trouble with the following problem. What I've done so far: x-y is the usual coordinate system. $a=\frac{F}{m}=\frac{800}{60}$ and the y component of this is $a_y=a\sin{60^\circ}$. To ... 1answer 58 views ### Direction of the torque In each one of the following figures there's a pole of length $1.2 \text{m}$ and there's a force $\vec F = 150 \text{N}$ acting on it. Determine the torque that is created by the force relative ... 0answers 104 views ### Deriving torque from Euler-Lagrange equation How could you derive an equation for the torque on a rotating (but not translating) rigid body from the Euler-Lagrange equation? As far as I know from my first class in Classical Mechanics, there is ... 1answer 112 views ### Calculating the acceleration of a car I'm trying to calculate the maximum acceleration a car can achieve with the current gear ratio. I am ignoring drag forces and friction to keep it simple. I'm doing this by: calculating the torque ... 2answers 69 views ### What fraction of peak horsepower do typical 4 door passenger vehicles use? I was surprised when I looked at the power rating of the engine used on a Humvee. It's only ~190 horsepower, which is exceeded by many sedan engines. So an obvious question is why doesn't my Camry SE ... 2answers 90 views ### Thrust center in space I have this dilemma: Suppose you have a space ship somewhere in deep space, where there is no drag force or substantial gravity. If the ship has a single engine situated in such a way that the center ... 1answer 180 views ### Is angular momentum always conserved in the absence of an external torque? Consider either the angular momentum of the earth around the sun or equivalently swinging a ball horizontally on a string. I know that with respect to the point of rotation of the swinging ball, ... 4answers 315 views ### Difference between torque and moment What is the difference between torque and moment? I would like to see mathematical definitions for both quantities. I also do not prefer definitions like "It is the tendancy..../It is a measure of ... 2answers 95 views ### comparing torque and 0-60 speed in cars Im looking a two cars, both are the same model but ones got a 2.0L turbo petrol engine and ones got a 3.0L turbo diesel engine. These are the full specs 2.0L Petrol 3.0L Diesel what im ... 0answers 37 views ### Truck driven from a small motor that can carry a heavy load, yet can travel fast. I am building a truck from trash as materials. I have one small motor and a a few small gears, but no other engineered materials are allowed. The truck must carry a load for a distance of 3m. The ... 1answer 734 views ### DC Motor Torque Constant I am very new to DC motors and to stackexchange. Please correct me if anything I said does not make sense. For DC motors, the equation looks like this: $P = \tau\dot{\theta}$ where $P$ is power, ... 1answer 105 views ### Repulsive Magnetic Hammering Experiment I have been thinking about something thing... I want to attach four or any number of magnets with arms to an axle.. Blue dots be magnets. Its is not a wheel configuration, all arms can move ... 0answers 83 views ### How much force is required to hold an umbrella? Assuming a small umbrella (e.g. http://www.amazon.com/ShedRain-WindPro-Umbrella-Close-Size/dp/B001DL5WN0). What are the factors that need to be considered? I assume where I hold the umbrella might ... 2answers 242 views ### Which is the axis of rotation? This should be simple, but it keeps bothering me. If a rigid body has no fixed axis, and a torque (defined relative to a point $A$) is applied, it will rotate around $A$. But often I can also ... 0answers 242 views ### Neglecting friction on a pulley? So, this is how the problem looks: http://www.aplusphysics.com/courses/honors/dynamics/images/Atwood%20Problem.png Plus, the pulley is suspended on a cord at its center and hanging from the ceiling. ... 1answer 194 views ### Calculation of torque for motor used in 4 wheel robot [duplicate] Possible Duplicate: Torque Required For a Motor to Move an Object on Wheels? I want to build a 4 wheel robot. the maximum weight of the robot is approximately 100 kg. and the radius of my ... 1answer 167 views ### Does weight distribution affect angular velocity? If an equal torque of equal radius and size is produced on two bodies of the same weight and same center of gravity, but with different weight distribution (say one has a 1kg mass 1 meter above the ... 1answer 47 views ### If a cart hits a wall, does the weight of it affect how it moves, when the center of gravity is constant? I have a model that represents a bicycle (a wood block with wheels), and I'm balancing the center of gravity so it's the same as a real bike. However, when the center of mass is kept constant, does ... 1answer 105 views ### Force and Torque Question on an isolated system If there's a rigid rod in space, and you give some external force perpendicular to the rod at one of the ends for a short time, what happens? Specifically: What dependence does the moment of inertia ... 0answers 67 views ### Physics Problem concerning Torque and nonuniform motion [closed] If the earth was spinning on its axis and it weighed 100 N with a radius of 42 m, and the earth spun 1 revolution every 24 hours, how many rockets would it take to counteract the earth spin? How do ... 1answer 89 views ### Does mass concentration affect the torque induced by a force? If you had two bodies with the same weight but one having mass concentrated more in the center, while the other had most mass concentrated on the outside, but both had the same center of mass and ... 1answer 141 views ### How can the torque a bicycle experiences be calculated based on the center of gravity, weight and a force? The position center of gravity of a bicycle and its rider is known, and the distance from it to the point of contact of the front wheel with the ground, in terms of horizontal and vertical distance (x ... 0answers 189 views ### Torque required to rotate Drum [closed] What Torque will require to rotate a Drum about horizontal axis XX. Drum is filled with industrial Oil. Weight of filled drum is $W=300 kg$. Length of Drum $(L)=900mm$ and Diameter of Drum is ... 0answers 134 views ### Torque required to rotate sand mixer [closed] the given data is- Mixer having one impeller(rotor) rotating in horizontal plane about vertical axis. capacity of mixer to design $C = 250kg$ rotor dia. $D = 45" = 1143mm$ friction coefficient ... 0answers 296 views ### How to find Rotational and Translational Equilibrium of Hanging Masses on a Bar? I am making a hanging mobile which needs to be done mathematically by calculating torque. The problem is, I can't seem to figure out how to solve for the distance of the two masses from the pivot ... 0answers 73 views ### Gear Ratio Required [closed] Assuming the following: -12V DC Motor, Free Speed: 5,310 RPM, Free Current: 2.7A, Max Power: 337W (at 2,655 RPM, 1.21 N-m @ 68A), Stall Torque: 2.42 N-m, Stall Current: 133A. -A 28"Wx28"Dx48"H box ... 3answers 557 views ### What determines the direction of precession of a gyroscope? I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the gyroscope. However, the direction, either clockwise or counterclockwise, of this ... 0answers 125 views ### How do you determine the torque caused by the mass of a lever? Suppose we have two objects sitting on two side of a lever, and the lever also has a mass, and those objects have masses. Then how we can balance $\sum τ$? This is what I have done: ... 1answer 79 views ### Stability of balanced masses with different surface areas Say I have this setup. The two round objects have equal mass and their centers of gravity at the same distance from the shaft. The objects only differ in that they have different surface areas ... 2answers 748 views ### Confused on how to properly use right hand rule I am having trouble using the right hand rule properly and often find myself putting my hand in awkward orientations. I know you point your hand in the direction of $r$ and then point your fingers in ... 0answers 129 views ### Torque required to spin a disk along its diameter How would I calculate (or simulate) this? I am only interested in the aerodynamic drag caused by the surface moving, not any other forces. As far as I know, the only variables needed are the drag ... 2answers 460 views ### Torque And Moment Of Inertia I am reading the two concepts mentioned in the title. According to the definition of torque and moment of inertia, it would appear that if I pushed on a door, with the axis of rotation centered about ... 3answers 229 views ### Meaning of the direction of the cross product I was doing calculations with torque and then I came across something very confusing: I understand that the magnitude of the torque is given by product of the displacement(from the center of ... 1answer 139 views ### Relationship between torques and centre of mass or centre of gravity [closed] 1) A wardrobe is $2$m high and $1.6$m wide. When empty, it has a mass of $110$kg and its centre of gravity is $0.8$m above the centre of its base. What is the minimum angle through which it must be ... 1answer 61 views ### How does the weight force distribute in roughly kite like kinematic to determine torques I have a kinematic model like the simplified following image. Assume that all points are fixed. I'm interested in how the force $\vec{m}$ distributes across $a_2$ and $b_2$ up to $a_1$ and $b_1$. ... 2answers 232 views ### Force applied off center on an object Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. ... 2answers 137 views ### Is it reasonable for a heavy door to “open by itself” (ie from differences in air pressure) if it had already been slightly ajar? If you consider a basically uniform massive door (say, 300 N) where there is some coefficient $\mu_{s,k}$ of static and kinetic friction between the thing on the inside of the door, and where the ... 2answers 857 views ### Calculating torque adjustment for offset of pivot point I'm trying to figure out how much to adjust torque, on a torque wrench when using an extension that offsets the pivot point. See this calculator for a diagram of what I'm talking about: The trouble ... 1answer 273 views ### Torque and equilibrium I am a little stuck on the concept involved in part b) of the following problem: A $5 m$ long road gate has a mass of $m = 15 kg$. The mass is evenly distributed in the gate. The gate is supported at ... 1answer 354 views ### Understanding moments as forces? I was watching this lecture on analysis of stress for mechanics of materials. At time 7:20, the lecturer says that in equilibrium, the sum of forces and "moments" in each direction (x,y,z) must be ... 0answers 25 views ### Determine the dilation temperature so as to double the speed There's a metallic rod of length $l_1$ which is spinning around a vertical ax which passes through its center. The ends of the rod are spinning with $\omega_1$ angular speed. Determine the temperature ... 1answer 326 views ### Calculating torque in a structure I posted this on math stack exchange but realize it is more a physics question. I have a structure which is set up as shown in the image. A weight hangs from point A with mass $m$. Joint B is free ... 0answers 258 views ### Torque required to rotate a cement mixer..? [closed] I need to design a motor to rotate a cement mixer which should mix one cubic meter. So, I calculated the required volume to be 1600 liters as it is an horizontal cylinder. Consider that the mixer ... 6answers 2k views ### Why is torque not measured in Joules? Recently, I was doing my homework and I found out that Torque can be calculated using $\tau = rF$. This means the units of torque are Newton meters. Energy is also measured in Newton meters which are ... 2answers 302 views ### Electric scooter acceleration I tested two electric scooters with almost same power (1500 W), but very different performances: A "Zem Star 45" (1500 W, 60 Nm) can reach 50 km/h in "a few seconds" (not measured actually), An ... 2answers 505 views ### Unit of torque with radians? Usually, the angular frequency $\omega$ is given in $\mathrm{1/s}$. I find it more consistent to give it in $\mathrm{rad/s}$. For the angular momentum $L$ is then given in \$\mathrm{rad \cdot kg \cdot ...	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 44, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9360954165458679, "perplexity_flag": "middle"}
http://stats.stackexchange.com/questions/33367/can-i-use-a-difference-score-as-my-outcome-variable-pre-post-change-in-a-moder	# Can I use a difference score as my outcome variable (pre-post change) in a moderated multiple regression equation? I am wanting to examine whether religiosity moderates intervention effects on stigmatized attitudes. Here are my variables: X = group (1 = experimental; 0 = control) Z = religiosity (14-item scale - using total score) Y = either post-test scores or pre-post change scores - Social Distance; Genderism/Transphobia I realize that I need to center my moderator variable and will also need to center my outcome variable (if I use post-test scores). My question is whether to use post-test scores or to use pre-post test change scores (and if I use the latter would I still center) - ## 1 Answer I see at least three reasonable options, although there is one I tend to do. • Compute the difference score, $D = Pre - Post$ and then predict that. The regression equation being something like: $$\hat{D}_{i} = b_0 + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}Genderism_{i}$$ One thing that is attractive about this is that it is straightforward to do. • Predict the post scores using your model, but also controlling for pre scores. This regression equation would look something like: $$\hat{Post}_i = b_0 + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}Genderism_{i} + b_5Pre_{i}$$ This tends to be my preferred approach. It does not require you to compute any new variables (which is not a big issue but can be slightly annoying). It also includes in the results an estimate of the strength of association between pre and post scores. It works because your treatment, $X$, predicts that portion of the post scores that are not explained by pre scores. The only drawback I see is that if you have a very small sample size, you lose an additional degree of freedom controlling for pre scores. That seems reasonable to me; however, because there is measurement error at both time points anyway, so it is not like computing the difference scores has zero error. • The last approach I see is to reshape the data from wide to long, and fit a mixed effects model. This would look something like: $$\hat{Outcome}_{ij} = b_0 + u_{0i} + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}Genderism_{i} + b_5time_{ij} + b_6time_{ij}X_{i}$$ The outcome is the score at the jth time point for the ith individual. The model includes a random intercept (captured by $u_{0i}$) for each individual. The time captures the change over time, and the interaction between time and the group variable is the "treatment" effect. Although kind of cool, I think this model is far too much work in the simple case where you only have pre and post scores (if you had 3+ time points, it would make sense). By the way, I do not particularly think that you need to center your variables prior to using them as moderators. I know many people teach that, but the models should work out the same. It can be slightly convenient as it makes the simple effects potentially more interpretable (0 = mean, therefore they are the effect of the variable at the mean of the other), but aside from that I see little gain. In more complex models, the reduction in collinearity between the variables and their product can also be helpful, but I have only ever see that matter in complicated random effects models or in some parallel latent growth models I fit once. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.955669105052948, "perplexity_flag": "middle"}
http://mathhelpforum.com/trigonometry/46692-trigonometry-triangles-question.html	# Thread: 1. ## A trigonometry in triangles question? In an isoceles triangle, if the altitudes intersect on the inscribed circle, then the cosine of the vertical angle A is? 2. The options are: A) 1/9 B) 1/3 C) 2/3 D) None How do I correlate the facts given in the question? 3. Originally Posted by fardeen_gen In an isoceles triangle, if the altitudes intersect on the inscribed circle, then the cosine of the vertical angle A is? were you given a diagram? because i am not sure where A is 4. No figure was given. But isn't the location of A obvious?(as it is the vertical angle of the isosceles triangle) 5. Originally Posted by fardeen_gen No figure was given. But isn't the location of A obvious?(as it is the vertical angle of the isosceles triangle) maybe it's just me, but i don't know what "vertical" angle of a triangle means. i'd look it up if i wasn't so lazy EDIT: ok, looked it up. it seems vertical angle has to be in reference to a given side, so i still don't know. are we talking the angle vertical to the side that is (perhaps) of a distinct length? that would make sense, maybe 6. Yes. The vertical angle is the angle between the two equal lengths. 7. Using cosine rule does not help. Is the distance between the orthocentre and side BC equal to 2r? 8. Here's a sketch. I think you just have to prove that O, I and A are aligned... looks like it's always the case. O represents the orthocenter (that is to say the intersection of the altitudes), center of the circumscribed circle c and I the center of the inscribed circle d. ------------------------------------------ The inscribed circle was drawn randomly, as well as the position of O on it. I then took a random point on circle c, named A. Then, I drew the tangents from A to the inscribed circle. One of this tangents intersects the circumscribed circle in B. Then I drew the circle e of center A and of radius AB, since ABC is an isosceles triangle. C must be on this circle e and also on the circumscribed circle c. AC must also be tangent to the inscribed circle d. And C is on these three elements iff A, O and I are aligned. This doesn't give the solution, but maybe it'll help... ? 9. Thanks for the figure. How is the angle A related to the altitude? I am still not able to figure that out. 10. My apologies to Jhevon for causing mindless confusion over "vertical angle". The question said "vertex angle" of the isosceles triangle. 11. Originally Posted by fardeen_gen My apologies to Jhevon for causing mindless confusion over "vertical angle". The question said "vertex angle" of the isosceles triangle. eh, don't worry about it. i should have probably known what it was anyway. 12. Ok, so geometry isn't my strong suit, but let me submit this anyway. maybe if someone could remind me how does the altitude of an isosceles triangle relate to the other sides. also, i have yet to use the fact that the triangle is inscribed in a circle. but i found that $\cos A = 2 - \frac 12 \bigg( \frac ab \bigg)^2$ and $\cos A = \frac {2d^2}{b^2} - 1$ here d is the height, b is the side opposite to angle B (one of the other angles) 13. Is there any theorem connecting altitudes of a triangle and the circles connected to them? #### Search Tags View Tag Cloud Copyright © 2005-2013 Math Help Forum. All rights reserved.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9499524831771851, "perplexity_flag": "middle"}
http://physics.stackexchange.com/questions/55391/heisenberg-uncertainty-principle-derivation-unexplained-factor-of-4-sigma-k?answertab=votes	Heisenberg uncertainty principle derivation - unexplained factor of $4 \sigma_k^2$ in Gaussian I did a Fourier transform of a gaussian function $\scriptsize \mathcal{G}(k) = A \exp\left[-\frac{(k-k_0)^2}{2 {\sigma_k}^2}\right]$ $$\scriptsize \begin{split} \mathcal{F}(x) &= \int\limits_{-\infty}^{\infty} \mathcal{G}(k) e^{ikx} \, \textrm{d} k = \int\limits_{-\infty}^{\infty} A \exp \left[-\frac{(k-k_0)^2}{2 {\sigma_k}^2}\right] e^{ikx}\, \textrm{d} k = A \int\limits_{-\infty}^{\infty} \exp \left[-\frac{(k-k_0)^2}{2 {\sigma_k}^2} \right] e^{ikx}\, \textrm{d} k =\\ &= A \int\limits_{-\infty}^{\infty} \exp \left[-\frac{m^2}{2 {\sigma_k}^2} \right] e^{i(m+k_0)x}\, \textrm{d} m = A \int\limits_{-\infty}^{\infty} \exp \left[-\frac{m^2}{2 {\sigma_k}^2} \right] e^{imx} e^{ik_0x}\, \textrm{d} m =\\ &= A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-\frac{m^2}{2 {\sigma_k}^2} \right] e^{imx}\, \textrm{d} m = A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-u^2 \right] e^{iu \sqrt{2} {\sigma_k} x} \sqrt{2} {\sigma_k} \textrm{d} u = \\ &=\sqrt{2} {\sigma_k} A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-u^2 \right] e^{iu \sqrt{2} {\sigma_k} x}\, \mathrm{d} u = \sqrt{2} {\sigma_k} A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-u^2 + i u \sqrt{2} {\sigma_k} x \right]\, \mathrm{d} u =\\ &= \sqrt{2} {\sigma_k} A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-\left(u + \frac{i {\sigma_k} x}{\sqrt{2}} \right)^2 - \frac{i^2 {\sigma_k}^2 x^2 }{2}\right]\, \mathrm{d} u =\\ &= \sqrt{2} {\sigma_k} A e^{ik_0x} \int\limits_{-\infty}^{\infty} \exp \left[-\left(u + \frac{i {\sigma_k} x}{\sqrt{2}} \right)^2 + \frac{{\sigma_k}^2 x^2 }{2}\right]\, \mathrm{d} u = \\ &= \sqrt{2} {\sigma_k} A e^{ik_0x} \int\limits_{-\infty}^{\infty} e^{-z^2} \exp \left[ \frac{{{\sigma_k}}^2 x^2 }{2} \right]\, \mathrm{d} z = \sqrt{2} {\sigma_k} A e^{ik_0x} \exp \left[ \frac{{{\sigma_k}}^2 x^2 }{2} \right] \underbrace{\int\limits_{-\infty}^{\infty} e^{-z^2} \, \mathrm{d} z}_{\text{Gauss integral}}=\\ &= \sqrt{2} {\sigma_k} A e^{ik_0x} \exp \left[ \frac{{{\sigma_k}}^2 x^2 }{2} \right] \sqrt{\pi}\\ \mathcal{F} (x)&= \sqrt{2\pi} {\sigma_k} A e^{ik_0x} \exp \left[ \frac{{{\sigma_k}}^2 x^2 }{2} \right]\\ \end{split}$$ It can be seen that Fourier transform equals $\scriptsize \mathcal{F} (x)= \sqrt{2\pi} {\sigma_k} A e^{ik_0x} \exp \left[ ({{\sigma_k}}^2 x^2) / 2\right]$. It is said on Wikipedia that the Gauss will be normalized only if $\scriptsize A=1 /(\sqrt{2 \pi} \sigma_k)$. I used this and got a result which corresponds with a result on Wikipedia - Fourier transform and characteristic function: $$\mathcal{F} (x)= e^{ik_0x} e^{\frac{{{\sigma_k}}^2 x^2 }{2}}\\$$ If i use a centralized Gauss whose mean value is $k_0=0$ i get: $$\mathcal{F} (x)= e^{\frac{{{\sigma_k}}^2 x^2 }{2}}\\$$ Which can be written as a: $$\mathcal{F} (x)= e^{\frac{x^2 }{2 \left(1/\sigma_k \right)^2}}\\$$ And i can see that $1/\sigma_k = \sigma_x$ BUT from this it follows that i get the Heisenberg uncertainty principle like this: $$\begin{split} \sigma_k \sigma_x &= 1\\ \Delta k \Delta x &= 1\\ \Delta p / \hbar \, \Delta x &= 1\\ \Delta p \Delta x &= \hbar\\ \end{split}$$ And this is a wrong result because i should get $\hbar/2$ in place of $\hbar$. Question: On our university professor derived this in a simmilar way but in the beginning in Gaussian he used $4{\sigma_k}^2$ instead of $2 {\sigma_k}^2$. This contributed to the right result $\hbar/2$ in the end. But i want to know why do we use factor $4$ instead of $2$? - 1 Just asking where you went wrong in your work isn't really appropriate for this site. I'd suggest editing this to focus more on your first question, namely why you don't get the HUP from your calculation. You don't need to show your complete work, but if outline what you did in just a little more detail (right here in the question, don't make people follow a link) it would be quite fine. – David Zaslavsky♦ Feb 28 at 3:00 1 I totaly rewrote the question and set some explicit questions which i cannot explain to myself. Today i was at university and my professor allso doesn't know why this is the case - i mean he couldn't explain to me where does facor 4 come from (please reread the question). – 71GA Feb 28 at 16:58 1 That's indeed a much better question, thanks! – David Zaslavsky♦ Feb 28 at 17:24 2 Answers You're treating this like a probability distribution instead of a wavefunction. Instead of assuming $2 \sigma_k^2$ vs. $4 \sigma_k^2$, I suggest setting the denominator equal to some constant and then finding the true variances in both position and wavenumber space directly--i.e. through the relation $$\sigma_a^2 = \langle \psi \| (\hat a - \bar a)^2 \| \psi \rangle$$ for any observable $\hat a$ with expectation $\langle \hat a \rangle = \bar a$. Break this into two integrals and see what you get for $\hat a = \hat k$ and $\hat a = \hat x$. - That's a long stream of equations to unravel, but it looks like you're trying to equate the standard deviation of the wavefunction in position space with $\Delta x$, which is not right. $\Delta x$ is defined by $$(\Delta x)^2 = \langle \psi \| X^2 \| \psi \rangle - (\langle \psi \| X\|\psi \rangle)^2$$ You have to use that form to calculate the uncertainty. For example, if your wavefunction is $\psi(x) = A e^{-x^2/\sigma^2}$, then $$(\Delta x)^2 = A^2 \int e^{-x^2/\sigma^2} x^2 e^{-x^2/\sigma^2}$$ - ,.Does this mean that my big integral is correct and i scew this up in my last steps? And what is that $X$ there. This notation $<\Psi\|X\|\Psi>$ is unknown to me. Is this maybee a average? Why is there $\Psi$ in notation? – 71GA Feb 28 at 20:57 – Mark Eichenlaub Feb 28 at 21:01 It is a bit long to explain in detail; your quantum mechanics textbook should explain it, though. I added a little detail on how to do this with a wavefunction. The answer to your question is essentially that the things you were call $\Delta x$ is not the $\Delta x$ in the uncertainty relationship. – Mark Eichenlaub Feb 28 at 21:05 I used $\psi$ because that is standard notation for a wavefunction – Mark Eichenlaub Feb 28 at 21:06 How did you know (i mean by what definition - please point me to this definition where i can read more) that: $(\Delta x)^2 = \int A e^{-x^2/\sigma^2} x^2 Ae^{-x^2/\sigma^2}$. So you think that my result $\sigma_k \sigma_x = 1$ is correct? Please confirm this. – 71GA Mar 1 at 18:04 show 2 more comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 30, "mathjax_display_tex": 8, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9429020285606384, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/88972/finding-an-orthogonal-basis-for-an-inner-product-space-mathbfp-2?answertab=oldest	# Finding an orthogonal basis for an inner product space $\mathbf{P}_2$ The problem reads as follows: Look at the inner product in $\mathbf{P}_2$ given by $\langle p,q \rangle = p(0)q(0) + p(\frac{1}{2})q(\frac{1}{2}) + p(1)q(1)$. Find an orthogonal basis $\mathbf{C}$ for $\mathbf{P}_2$. Now, I know I can use the standard basis for $\mathbf{P}_2$, i.e. $B=\{1,t,t^2 \}$ and apply Gram-Schmidt. So: $\mathbf{v}_1 = \mathbf{x}_1 = 1$ $\mathbf{v}_2 = \mathbf{x}_2 - proj_{\mathbf{v}_1} \mathbf{x}_2 = \mathbf{x}_2 - \frac{\langle \mathbf{x}_2, \mathbf{v}_1 \rangle}{\langle \mathbf{v}_1, \mathbf{v}_1 \rangle}\mathbf{v}_1$ and so on. My question is incredibly trivial: I just can't figure out how to calculate the inner products in the equation above using the one in the problem. From a solution of the problem I know that $\langle \mathbf{v}_1, \mathbf{v}_1 \rangle = 1$ and $\langle \mathbf{x}_2, \mathbf{v}_1 \rangle = \langle t,1\rangle= 3/2$. Could someone please show me the calculations, or how to insert these into the equation in the problem? - ## 1 Answer $\langle \mathbf{x}_2, \mathbf{v}_1\rangle = \mathbf{x}_2(0) \; \mathbf{v}_1(0) + \mathbf{x}_2(1/2) \;\mathbf{v}_1(1/2) + \mathbf{x}_2(1) \; \mathbf{v}_1(1)= 0 \cdot 1 + \frac{1}{2} \cdot 1 + 1 \cdot 1 = \frac{3}{2}$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9152309894561768, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/174674/if-n-balls-are-thrown-into-k-bins-what-is-the-probability-that-every-bin-gets-a	# If n balls are thrown into k bins, what is the probability that every bin gets at least one ball? If $n$ balls are thrown into $k$ bins (uniformly at random and independently), what is the probability that every bin gets at least one ball? i.e. If we write $X$ for the number of empty bins, what is $P(X=0)$? I was able to calculate the $E(X)$ and thus bound with Markov's inequality $P(X>=1) \le E(X)$ but I don't how to work out an exact answer. http://www.inference.phy.cam.ac.uk/mackay/itprnn/ps/588.596.pdf - ## 1 Answer What is the chance that all $k$ bins are occupied? For $1\leq i\leq k$, define $A_i$ to be the event that the $i$th bin stays empty. These are exchangeable events with $P(A_1\cdots A_j)=(1-{j\over k})^n$ and so by inclusion-exclusion, the probability that there are no empty bins is $$P(X=0)=\sum_{j=0}^k (-1)^j {k\choose j}\left(1-{j\over k}\right)^n.$$ Stirling numbers of the second kind can be used to give an alternative solution to the occupancy problem. We can fill all $k$ bins as follows: partition the balls $\{1,2,\dots, n\}$ into $k$ non-empty sets, then assign the bin values $1,2,\dots, k$ to these sets. There are ${n\brace k}$ partitions, and for each partition $k!$ ways to assign the bin values. Thus, $$P(X=0)={{n\brace k}\,k!\over k^n}.$$ - Thanks. I like it - the question is simple but the expression for the answer is disappointing complicated — we define a concise notation to make maths beautiful again. – Colonel Panic Aug 3 '12 at 15:47 – joriki Oct 11 '12 at 23:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9295879602432251, "perplexity_flag": "head"}
http://mathhelpforum.com/algebra/142189-relations-functions-ordered-pairs-print.html	# Relations, functions and ordered pairs. Printable View • April 29th 2010, 02:47 PM jamieirl Relations, functions and ordered pairs. Hello all and thanks in advance for all of the help you'll be giving me for the next couple of weeks. I'm just finishing a college algebra class and admittingly I haven't been the best student. The class is mostly online, that coupled with most of the in-class teachings consisting of calculator help (and the teacher stated at the start of the semester that she doesn't know how to use a Ti-89, which I have and I didn't want to buy another calculator after spending 130+ on mine) leaves me basically on my own. She (the professor) was nice enough to give us a review for the final exam (which I'll be asking plenty of questions about in the future) as well as a 3 question take home exam; which is where my first question is coming from. Anyway sorry for the long introduction, my first question from the take home exam: "1. One way to define a relation is "A set of ordered pairs". a. Give an example b. What is the difference between a function and a relation? Is your example in part a) a function? How can you tell?" I've scored fairly well on most of the assignments/test/quizzes thus far, but for some reason when I look at this take home exam it looks alien to me. Thanks for any replies. • April 29th 2010, 02:54 PM shenanigans87 A relation can be a set of ordered pairs, like you stated. i.e. {1,1} {3, 5} {4, 14} etc. A function is a formula from which you can derive ordered pairs. If the function is $y=x^2$ you can derive infinitely many relations by plugging in numbers. Let's get a relation for 0 < x < 3 {0, 0} {1, 1} {2, 4} {3, 9} • April 29th 2010, 03:07 PM jamieirl Quote: Originally Posted by shenanigans87 A relation can be a set of ordered pairs, like you stated. i.e. {1,1} {3, 5} {4, 14} etc. A function is a formula from which you can derive ordered pairs. If the function is $y=x^2$ you can derive infinitely many relations by plugging in numbers. Let's get a relation for 0 < x < 3 {0, 0} {1, 1} {2, 4} {3, 9} Thank you for the quick reply. I think I've gotten too used to the online-assignment format, when I have a written question in front of me I go blank. So for part a. I just give any random ordered pairs as an example, and for part b. I just state that no, part a. is not a function and "How can you tell?" I'd answer; " A function is a formula from which you can derive ordered pairs." and example a. (the ordered pairs example) is acquired from a function, not a function itself. Am I understanding this correctly? Thanks again. • April 29th 2010, 03:16 PM Tikoloshe I believe shenanigans87’s answer is misleading. A relation can be defined as any subset of the cartesian product of two sets (i.e., a set of ordered pairs). A function is a relation (a set of ordered pairs) in which each first coordinate has a unique second coordinate (i.e., if $(a,b)\in f$ and $(a,c)\in f$, then then b=c). These are standard definitions and have nothing to do with having a formula or infinitely many elements. • April 29th 2010, 03:45 PM shenanigans87 That's true. A function has one unique value of x for every y value. • April 29th 2010, 04:03 PM jamieirl Quote: Originally Posted by Tikoloshe I believe shenanigans87’s answer is misleading. A relation can be defined as any subset of the cartesian product of two sets (i.e., a set of ordered pairs). A function is a relation (a set of ordered pairs) in which each first coordinate has a unique second coordinate (i.e., if (a,b)=(a,c) then b=c). These are standard definitions and have nothing to do with having a formula or infinitely many elements. So based on what you're saying I could answer this question like this: 1. One way to define a relation is "A set of ordered pairs". a. Give an example: (0,3)(2,3) b. What is the difference between a function and a relation? Answer: A relation is any subset of the cartesian product of two sets. While a function is only one type of relation in which the first coordinate has a unique second coordinate. Is your example in part a) a function? Answer: No How can you tell? Answer: The relation does not meet the criteria required to be a function; a function must be a relation in which the first coordinate has a unique second coordinate (i.e., (a,b)=(a,c) then (b=c)). How does that look? Also, would this be an example of a function; (0,3),(2,2), since they have unique y's? I think I just confused myself. • April 29th 2010, 04:33 PM Tikoloshe Quote: Originally Posted by jamieirl Also, would this be an example of a function; (0,3),(2,2), since they have unique y's? Sure would. Now there is a small point that I think shenanigans87 initially tried to make. That is the definition of a function includes its domain. In some contexts of discrete mathematics there is a distinction between the domain and first set of your cartesian product. That is, let $f:A\to B$ be a function from $A$ to $B$. Then $f\subset A\times B$ such that $\forall a\in\mathrm{Dom}(f)$ there is a unique $b\in B$ such that $(a,b)\in f$. My point is the (possible) distinction between $A$ and $\mathrm{Dom}(f)$. Depending on what definition you choose to find useful, you might demand that the two sets are always equal for a function. In this case, $f=\{(0,3),(2,2)\}$ is not a function on $\mathbb{R}$, but is a function on the set $\{0,2\}$. • April 29th 2010, 04:40 PM shenanigans87 Not only that but consider the parabola $y^2=x$ You could pull infinitely many ordered pairs from that which don't repeat the same x value but that doesn't make it a function with respect to your definition. • April 29th 2010, 04:42 PM jamieirl Quote: Originally Posted by Tikoloshe Sure would. Now there is a small point that I think shenanigans87 initially tried to make. That is the definition of a function includes its domain. In some contexts of discrete mathematics there is a distinction between the domain and first set of your cartesian product. That is, let $f:A\to B$ be a function from $A$ to $B$. Then $f\subset A\times B$ such that $\forall a\in\mathrm{Dom}(f)$ there is a unique $b\in B$ such that $(a,b)\in f$. My point is the (possible) distinction between $A$ and $\mathrm{Dom}(f)$. Depending on what definition you choose to find useful, you might demand that the two sets are always equal for a function. In this case, $f=\{(0,3),(2,2)\}$ is not a function on $\mathbb{R}$, but is a function on the set $\{0,2\}$. I wish I could follow that better because it looks like very useful information, I understand about half of what you said there I think. Hehe. • April 29th 2010, 05:07 PM Tikoloshe It might help if I hadn’t made a typo (now fixed). I hope you reread my first post Here’s an example of what I mean: let $A=\{1,2,3,4,5\}$ and $B=\{6,7,8,9,10\}$. Suppose I define the following relations: $f=\{(1,6),(2,9),(3,10),(4,9),(5,7)\}$ $g=\{(2,7),(4,6),(5,7)\}$ $h=\{(1,10),(2,9),(3,8),(4,7),(5,6),(2,7)\}$ All standard definitions agree that h is not a function since $(2,9)\in h$ and $(2,7)\in h$, yet $7\neq9$. All standard definitions agree that $f:A\to B$ is a function. Some definitions will say that $g:A\to B$ is a function which is simply not defined at 1 or 3. Others will say that $g:A\to B$ is not a function, yet $g:\{2,4,5\}\to B$ is. • April 29th 2010, 05:09 PM Tikoloshe Note that no one says that a function must “hit” every point in $B$. The issue is whether it is defined for every point in $A$. • April 29th 2010, 07:42 PM jamieirl Quote: Originally Posted by Tikoloshe Note that no one says that a function must “hit” every point in $B$. The issue is whether it is defined for every point in $A$. Thanks for all of your replies, I'm going to let this simmer for a bit and see what happens :) All times are GMT -8. The time now is 12:59 PM.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 45, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9501574635505676, "perplexity_flag": "head"}
http://mathhelpforum.com/algebra/101674-fraction-hard.html	# Thread: 1. ## fraction hard! hi, i need to know what this is i know i need to change tham to a top heavey fraction but i don't know how to do it, please could you show me what to do with workings out!! 2. The question isn't very clear in my opinion. Can you make it neater? :O 3. sorry i just saw that, it was ok while i was writing it, sorry again 4. Yes, it would be wise to convert these "mixed fractions" to "top heavy fractions" or "proper fractions" as they're otherwise called! To do this, multiply the number in front of the fraction by the denominator and then add the numerator. eg: $4\frac{3}{2}$ Step 1: Multiply the number in front by the denominator (The number on the bottom of the fraction which happens to equal 2 in this example): $42=8$ Step 2: Add this number to the numerator of the fraction (The number on the top which happens to be 3 in this example): $8+3=11$ Step 3: Place this new number (11) over the original denominator (number at the bottom) which is 2: $\frac{11}{2}$ Voila! There is a mixed fraction converted to a "top heavy fraction" Try doing the question yourself following these steps! Otherwise, I'm happy to help further 5. Originally Posted by Finley Yes, it would be wise to convert these "mixed fractions" to "top heavy fractions" or "proper fractions" as they're otherwise called! To do this, multiply the number in front of the fraction by the denominator and then add the numerator. eg: $4\frac{3}{2}$ Step 1: Multiply the number in front by the denominator (The number on the bottom of the fraction which happens to equal 2 in this example): $42=8$ Step 2: Add this number to the numerator of the fraction (The number on the top which happens to be 3 in this example): $8+3=11$ Step 3: Place this new number (11) over the original denominator (number at the bottom) which is 2: $\frac{11}{2}$ Voila! There is a mixed fraction converted to a "top heavy fraction" Try doing the question yourself following these steps! Otherwise, I'm happy to help further thanks alot for that! so would mine be: $\frac{22}{7}$ - $\frac{11}{5}$ ? if so would the answer be: $\frac{11}{2}$ 6. My post was merely an example, the answer didn't apply to your question!! However, the first part of your answer is entirely correct! $\frac{22}{7} - \frac{11}{5}$ However, we need to go a few steps further to get the final answer!! Firstly, can you subtract two fractions that don't have a common denominator? 7. Originally Posted by Finley My post was merely an example, the answer didn't apply to your question!! However, the first part of your answer is entirely correct! $\frac{22}{7} - \frac{11}{5}$ However, we need to go a few steps further to get the final answer!! Firstly, can you subtract two fractions that don't have a common denominator? would it be $\frac{11}{2}$ ???? if not, how do i get the answer? 8. No, the answer is not $\frac{11}{2}$ To simplify we need to find the LOWEST COMMON DENOMINATOR of the two fractions. In other words, we need the two BOTTOM NUMBERS of the fractions to be THE SAME. The easiest way to find a COMMON DENOMINATOR is to multiply the two DENOMINATORS (bottom numbers) together: Aka. $75=CD$ Therefore Common Denominator = 35 Therefore, $\frac{110}{35}-\frac{77}{35}$ What I've done is place the original numerators (22 and 11) over the COMMON DENOMINATOR. What we do to the bottom, we must do to the top (this keeps the ratio even). In other words, to get from 7 (the denominator in the first fraction)) to 35 we needed to multiply it by 5! To get from 5 (the denominator in the second fraction) to 35 we needed to multiply it by 7! What we do to the bottom, we do to the top!! Keeping it short, 225 = 110 (New numerator for first fraction) 117 = 77 Now to simplify $\frac{110}{35}-\frac{77}{35}$! Deal with the numerators only: 110-77 = 33 Place the new numerator over the common denominator (35): 33/35 33/35 Can't be simplified any further, hence this is the answer! Answer: $\frac{33}{35}$ 9. i just looked in my book on how to do it, i know what the answer is now! it would be: $\frac{33}{35}$ 10. Originally Posted by Finley No, the answer is not $\frac{11}{2}$ To simplify we need to find the LOWEST COMMON DENOMINATOR of the two fractions. In other words, we need the two BOTTOM NUMBERS of the fractions to be THE SAME. The easiest way to find a COMMON DENOMINATOR is to multiply the two DENOMINATORS (bottom numbers) together: Aka. $75=CD$ Therefore Common Denominator = 35 Therefore, $\frac{110}{35}-\frac{77}{35}$ What I've done is place the original numerators (22 and 11) over the LOWEST COMMON DENOMINATOR. What we do to the bottom, we must do to the top (this keeps the ratio even). In other words, to get from 7 (the denominator in the first fraction)) to 35 we needed to multiply it by 5! To get from 5 (the denominator in the second fraction) to 35 we needed to multiply it by 7! What we do to the bottom, we do to the top!! Keeping it short, 225 = 110 (New numerator for first fraction) 117 = 77 Now to simplify $\frac{110}{35}-\frac{77}{35}$! Deal with the numerators only: 110-77 = 33 Place the new numerator over the new common denominator (35): 33/35 33/35 Can't be simplified any further, hence this is the answer! Answer: $\frac{33}{35}$ thanks, i just posted because i must have been writing it up when u posted, thanks alot for the help 11. No problem!! Hopefully it's beginning to make more sense Grappling the logic behind fractions will help enormously in later mathematics.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 25, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9221044182777405, "perplexity_flag": "middle"}
http://diracseashore.wordpress.com/2009/01/15/maldacenas-information-paradox/	# Shores of the Dirac Sea Feeds: Posts Comments ## Maldacena’s information paradox January 15, 2009 by Moshe Back in 2001, in a truly beautiful paper, Juan Maldacena formulated a version of Hawking’s information paradox, which has the added advantage that it could be discussed and analyzed in the context of a complete background independent theory of quantum gravity, namely that of the AdS/CFT correspondence. This variant is similar to the original paradox, formulated for black holes surrounded by flat space, in that it displays a sharp conflict between properties of black holes in classical General Relativity, and basic postulates of quantum mechanics. Alas, it is also different in many crucial ways from the original paradox. Despite that, Juan’s proposed resolution to his paradox seems to have led to Hawking’s arguments, who managed to convince himself (though I think it is fair to say not too many others, unless they were already convinced) that information is not lost after all in the process of black hole formation and evaporation. To present the paradox, first I’ll have to tell you just one fact about AdS spaces. AdS space has a center and a boundary, and a gravitational potential which makes it difficult to approach the boundary. Anything you throw out from the center towards the boundary of AdS space will eventually come back and hit you. It is as if you are surrounded by reflecting mirrors. This fact is a useful in many ways, in particular in stabilizing black holes. For certain black holes (the so-called large ones, which are the ones we’ll discuss) this means that the Hawking radiation emitted by the black hole is constantly reflected back towards the black hole. Soon enough the system will achieve an equilibrium state, where the black hole exists is in detailed balance with its Hawking radiation, it absorbs and emits the same amount of radiation. Which means that the black hole never disappears, unlike its cousin in asymptotically flat space, it is eternal. Therefore, the original information loss paradox (roughly, what happens to all the information stored in the BH after it completely evaporates?) cannot be formulated in this context (at least using the black holes I am describing). Tough luck. So, instead of looking at the complicated process of black hole formation and evaporation, let us start with one of those stable black hole and perturb it, add to it one bit of information, poke it just a little bit, you get the picture. Then we can try and follow the history of the perturbation and ask: after some time has passed, can we trace back the nature of the perturbation? can we retrieve the information we sent towards the black hole from the set of measurements we can do later in time? Turns out, this process is puzzling in the same way the original paradox was, but the advantage is that the new puzzle can be formulated in a complete microscopic theory of quantum gravity, namely in the gauge theory variables of the gauge-gravity duality. Let me describe the process and show where the problem lies. The original black hole, formulated in gauge theory variables, is simply a thermal state, it is a thermal equilibrium of quarks and gluons at a temperature which equals the black hole Hawking temperature. Now, imagine shooting a very energetic particle into that thermal gas. What will happen? it will start colliding with all the gas particles, thereby slowing down while speeding up the average speed of the other particles. Wait long enough and you will find a gas of particles in a new thermal equilibrium state, in a slightly higher temperature. This is exactly what happens in the gauge theory, and it has a simple interpretation in the gravity language: you send some bundle of energy towards the black hole, and it falls behind the horizon. The black hole “rings” for a little bit and then settles back to an approximate equilibrium state, which is a slightly larger black hole. The perturbation decays exponentially as function of time (the exponents are known as quasi-normal modes), after a little while you forget the black hole was perturbed, the information has disappeared behind the horizon. But now comes the paradoxical part: in quantum mechanics this description can only be an approximation to the complete situation, because information is never lost. In principle you can always perform precise enough measurements on that seemingly thermal state to retrieve the information about the original perturbation. This is the principle of quantum mechanical unitarity, evolution in quantum mechanics is reversible. The clearest way to see what is going on is looking at the behavior of the perturbation after a very long time. In the story we told, the effect of the perturbation dies off exponentially with time. In the complete quantum mechanical system that is what happens initially: for a very long time the effect dies off, but it cannot go all the way to zero, lest the information really be lost. Instead, after very long time the system will undergo Poincare recurrences: it will come back arbitrarily close to its initial configuration. So, looking at the behavior of the perturbation after extremely long times, of the order of the so-called Poincare recurrence time, is one clear way of deciding whether information is lost or not. This is very much “in theory” type of situation, the common (unfortunate) analogy seems to be trying to retrieve the contents of a burnt book from its ashes. Nevertheless, this is a question of principle: is there a set of measurements we can do on that perturbed black hole which can be used to retrieve the complete information in the original perturbation? Looking at the black hole solution of General relativity, if we take this seriously as the complete description of the situation, the precise answer is that the effect of perturbation dies of exponentially, for all times. Wait long enough and the system is arbitrarily close to an equilibrium state. In that case, you can never retrieve the information, it is lost. You can say words like “it continue to live behind the horizon”, but for you, the observer staying in the safety of the outer region, that information is lost. On the other hand, the system has a dual which is a more complete and microscopic description of the theory. When looking at the process in the complete theory you can see that information is never really lost. In fact, you can do more – you can also truncate the complete theory, taking into account classical gravity only, and observe that in fact with this truncation information really is lost. In other words, you can see the precise details of how the illusion of information loss comes about. So, what is the current state of affairs? in the dual language, it is clear that the system is unitary, but it also became clear that in the limit corresponding to classical gravity, there is an apparent information loss. In fact, this information loss looks like a generic phenomena of that (so-called large N) limit in a whole slew of models, even those not dual to any known gravity theories. So, it looks like this paradox is pretty much solved, at least no mystery as to whether information is really lost or not. What is missing at the moment is a phrasing of the resolution in the original, gravitational variables. Since we have a complete description of the story, it is known in detail that information is not lost, and why it seems to be lost as an artifact of the specific approximation we are making. What we are missing is the always elusive question of the “mecahnism” of retoring unitarity. What are the essential ingredients of the complete story that are needed in order to restore unitarity. Can we phrase the solution in terms of some coherent narrative phrased in the gravitational language? would be nice if we could, maybe we are not that far off, stay tuned. ### Like this: Posted in high energy physics, Quantum Gravity, string theory, thermodynamics \| 55 Comments ### 55 Responses 1. I don’t see where information would be lost, even if the perturbetions die off exponentially. thy are still there after any finite amount of time, or not? As a simple example, take a lightily damped classical harmonic oszillator. it’s amplitude falls of exponentially, but it never comes to a complete halt, if you are looking close enough. 2. The automatically generated possibly related posts make quite an interesting match. Question: Does that mean you can only trace the fate of information that was ‘thrown in’ from the boundary? What can you say about states that were not formed this way? Could you maybe comment on this paper?. 3. Matthias, damped system are exactly those where unitarity is violated, they are not time reversible. Wait long enough and all that distinguishes different states gets damped. Bee, I am only saying that we can form a clean puzzle for those set of states. We can also ask more general questions, and those would be more difficult to answer. The possibly related posts are, on this rare occasion, possibly related. At least some of them are. It’s a miracle! 4. on January 16, 2009 at 12:46 am Just Learning Incidently, there is an article in the January issue of Physics today by Klebanov and Maldacena called Solving quantum field theories via curved spacetimes 5. I ‘ve always been curious, where do gravitons come into the picture of AdS/CFT? One always hears about the dual picture; the gauge field and the gravitational field. Is the graviton a complication here? 6. Mark, precise details depend on context but very roughly speaking the graviton is a complicated state when written in terms of the gauge theory variables. It’s the no free lunch principle. 7. mark a. thomas: The gravitational field is a classical description of the dual to the boundary gauge quantum field theory. If you are in the large N limit, large lambda limit, then the first limit tells you that quantum corrections (handles in your stringy diagrams, if you wish) are suppressed, while the second limit tells you that while your gauge theory is strongly coupled, the dual bulk theory is weakly curved. As far as bulk calculations go, this is great, which is why people work in these limits, but the correspondence is conjectured to work away from this special point. Back to your question, gravitons are a quantum mechanical description of weak-field gravity, so unless you want quantum answers to quantum questions, you don’t need to consider gravitons. The big question is how is the information problem resolved? Is it because gravity is only a true description at the strictly infinite N limit, and somehow finite N resolves it (stringy black holes). In answering your question, I have managed to confuse myself, wondering if unitarity is what we’re really violating here (in the classical limit, presumably one needs a different word for unitarity). In any case, I think even the classical field theory should exhibit recurrence. 8. Lionel, I’d say that Liouville thm., about preservation of the phase space volume, is the classical counterpart of QM unitarity. But, classical in the bulk means infinite N on the bdy, the field theory is still QM (which has to do more with lambda, so classical bdy field theory is lambda=0). 9. Dear Lionel: Classical fields have infinite numbers of variables and don’t exhibit recurrence (except in integrable models). The main reason for this is that a ball of size one half in an infinite dimensional phase space can fit in a ball of size one an infinite number of times: hence the usual proof of Liouville’s theorem doesn’t do anything. In the quantum theory the high frequency modes get regulated away (they are too energetic because of Planck’s energy cost to excite a frequency $E=h \nu$ ). In classical fields one would expect that everything would dissipate into high frequency noise in the end. You can call this heat if you want to. 10. In this context though we are discussing field theory on a finite space (3-sphere), so I think both the classical and quantum field theory will have recurrences at finite N, and dissipation at infinite N. Unless of course I am missing something. 11. Great post. Thanks for taking the time to write it. Although I’ve personally always believed that the information is not destroyed, I can see why a true skeptic would not be moved by Maldacena’s argument. If we can’t actually tell an internally consistent story on the gravity side that explains how the outgoing radiation encodes the initial state, there will always be wriggle room. But hopefully, as you say, we are getting there! 12. Sean, I don’t believe it to be the case, but the logical possibility exists that the gravitational degrees of freedom are just the wrong set of variables to give you the correct answer. I find the case in AdS to be 100% fool proof, even if there is no story to tell in the (fundamentally incomplete) gravitational variables. But again, I do believe there is such a story, it is probably interesting and we might even learn something about the original information paradox. (Incidentally, Juan did attempt in that paper to resolve the paradox in the gravitational language, all the stuff with the sub-leading saddle points which Hawking took note of and tried to apply to his context, but that solution is incomplete for various reasons.) 13. on January 16, 2009 at 12:03 pm Pope Maledict XVI I agree that the case in AdS is “100% foolproof”. The problem is to turn this into an understanding of what happens to unitarity in the case of realistic Schwarzschild black holes, which are so very different. 14. on January 16, 2009 at 12:05 pm Just Learning “So, looking at the behavior of the perturbation after extremely long times, of the order of the so-called Poincare recurrence time, is one clear way of deciding whether information is lost or not.” Would it also be correct to assert that looking at a large number of similarly perturbed systems one would also be able to see if information is lost or not? Is the poincare recurrence time an expected value or not? Although I imagine it would be difficult to find a large collection of black holes. 15. Hi Moshe: The way I think of it is in terms of the UV catastrophe. The classical system will equipartition finite energy in an infinite number of modes, so all the energy ends up in the high frequency noise. There is an infinitude of frequencies there, and the Poincare recurrence time will become infinite. The UV catastrophe is what quantum mechanics cures. Infinite N is a different classical limit. There, the spectrum of the quantum system becomes continuous at high energies only in the strict infinite N value (it’s akin to going to an infinite box). At finite N the systems has a finite number of states below any energy. Dear Pope: I made a recent attempt with one of my students to explain how small black holes in AdS look like in the dual theory. These small black holes are going to be very similar to Schwarzschild. Maybe you might be interested. Here is the link arxiv:0809.0712. It turns out that quite a bit of the microscopic description is not all that different from the big black holes in AdS. The main difference is that the system is not in equilibirum, but the internal degrees of freedom of the black hole are in some type of dynamical thermos that makes it possible for them to have a different temperature than the ambient space. 16. Dear Moshe, I agree with David that the Poincare recurrence time is infinite in classical field theory. There are infinitely many Fourier modes and each of them can carry some, arbitrarily low, energy. So the energy will dissipate into them – mostly the infinite-frequency ones. In quantum field theory, one photon is a minimum, and that’s why you can only use the modes up to reasonable frequencies linked to the temperature. That’s why the finiteness of the Poincare recurrence time is only dictated by the IR physics, but this statement is only true at the quantum level because the hbar=0 limit is where all frequencies can participate (hf=0, too) and even a finite space contains infinitely many usable modes. Also, I agree with him that the spectrum of dimensions of operators in the N=4 is discrete for any finite N_colors because it is always a quantum theory on a compact S^3 times time. When one looks at the spectrum of operators and gives them no geometric interpretation, there is really no separate hbar because of all of them are just “some” operators. One might think that when one interprets the states geometrically, it is possible to be sending hbar to zero independently from N to infinity, effectively by considering large, slow, heavy objects and processes. But that’s an illusion because finite g, finite N really means that the volume and curvatures are close to the Planckian ones, and there are no large classical objects in a Planckian seed of space. One can be sending g and hbar to zero simultaneously to get the planar limit, too. I think it’s fair to say that it’s classical field theory, although equivalently described by the 1st quantized quantum theory of one string (the planar limit). David’s paper that I had missed is very provocative – approximately thermal states with respect to a dynamically generated subgroup of the gauge group – wow. It must clearly be possible because it’s about the localization on the sphere. For QCD, such a segregation of an SU(2) inside would probably not be possible, would it? Some comments about the comment #1 whether the information is getting lost if the perturbations “only” decrease exponentially: http://motls.blogspot.com/2009/01/killing-information-softly.html Best wishes Lubos 17. If a black hole is not to accrete faster than it radiates it must have a present temperature greater than 2.725 (+/-)0.0004 K cosmic background radiation. This requires it be less than ~4.7×10^22 kg (~0.79% of Earth’s mass). There are no obtainable circumstances in which a black hole can form in a universe cooler than its own temperature – with one exception: CERN’s LHC. GARDYLOO! 18. Sorry for the intrusion here from the “three dimensional world and time, into your higher abstract perspectives Again from a layman’s perspective it is extremely hard for the ordinary person to image what those strange worlds are, that mathematician and physics theorists are talking about. Now Imagine your going to turn my story of the Cave around? “Whereas Plato envisioned common perceptions as revealing a mere shadow of reality, the holographic principle concurs, but turns the metaphor on it’s head. The Shadows-then things that are flattened out and hence live on a lower-dimensional surface-are real, while what seemed to be more richly structured, higher dimensional entities (us; the world around us) are evanescent projections of the shadows.” Brian Greene-The Fabric of the Cosmos, pg. 482 He puts an asterisk beside that quote to show below on that support page, to provide the meaning of this. While this seems like a turning of the “thoughts about of Plato” I really don’t see it that way. For instance Greene goes on to write, “For instance, in Maldacena’s work, the bulk description and the boundary description are on an absolutely equal footing.” Pg 484. I think my illustration of the sun and the opening of the cave embeds the realization that in a “reductionistic view,” energy is well considered as we see the entropic values of the universe as descriptively as it is. Not to rewrite the work branes may have enforced from an “abstract and mathematical process” but to concur that the three dimensionality reality is “solidified” to the world around us, from that higher dimensional state. Geometry is a branch of mathematics that deals with points, lines, angles, surfaces and solids. One of Coxeter’s major contributions to geometry was in the area of dimensional analogy, the process of stretching geometrical shapes into higher dimensions. He is also famous for “Coxeter groups,” the inversive distance between two disjoint circles (or spheres). Now if one retains this understanding about Plato you might be moved to consider Coxeter views to those abstract objects. I do not believe that this is inconsistent with what is continued to be expressed in the Tomato soup analogy in recognition of Maldacena’s work in the five dimensional spacetime. Best, 19. David and Lubos, thanks, that makes perfect sense. I agree that David’s paper is interesting and provocative, blog post maybe? 20. on January 17, 2009 at 1:14 am Just Learning So the day’s discussion has made it clear: Information is not lost, only practically undecipherable. This is a similar statement to that of entropy, where energy is not lost, only made unavailable to do work. Utility is then maximized with the optimal pareto efficiency. Comments are closed. %d bloggers like this:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.935357928276062, "perplexity_flag": "middle"}
http://alanrendall.wordpress.com/2008/10/22/the-polarized-gowdy-equation/	# Hydrobates A mathematician thinks aloud ## The polarized Gowdy equation The polarized Gowdy solutions are the simplest solutions of the vacuum Einstein equations which are dynamical and not spatially homogeneous. Physically they represent a one-dimensional configuration of polarized gravitational waves propagating in a closed and otherwise empty universe. Here ‘closed’ means that space is compact. The simplest case is that in which space is a three-dimensional torus and here I will consider only that case. These solutions represent a simple model system for developing mathematical techniques for studying the Einstein equations. This system has accompanied me in my research for many years and now I want to take a minute to stand back and reflect on it. The central equation involved is $P_{tt}+t^{-1}P_t-P_{\theta\theta}=0$. This equation is also known under the name Euler-Poisson-Darboux equation but this does not seem to have helped much in the study of Gowdy solutions. This is perhaps due to the fact that the side conditions (boundary and initial conditions) are different from those of interest in other applications. In the Gowdy case $P$ is assumed to be periodic in the spatial variable $\theta$ and initial data can be prescribed on a hypersurface $t=t_0$ for some positive real number $t_0$. The data are of the form $(P_0,P_1)$ where $P_0$ and $P_1$ are the restrictions of $P$ and $P_t$ to the initial hypersurface $t=t_0$. This is a linear hyperbolic equation and it follows from standard results that there is a unique smooth (i.e. $C^\infty$) solution corresponding to any smooth initial data set. Thus all solutions can be parametrized by data on this initial hypersurface. The asymptotics of solutions in the limit $t\to 0$ are well understood. Any solution has an asymptotic expansion of the form $P(t,\theta)=-k(\theta)\log t+\omega(\theta)+R(t,\theta)$ where $R(t,\theta)=o(1)$ as $t\to 0$. Moreover $R$ has an asymptotic expansion of the form $R(t,\theta)\sim\sum_{j=1}^\infty (A_j(\theta)+B_j(\theta)\log t)t^j$. where all the coefficients $A_j$ and $B_j$ are determined uniquely in terms of $k$ and $\omega$. The asymptotic expansions may be differentiated term by term as often as desired. Conversely, given smooth functions $k$ and $\omega$ there is a solution $P$ for which the leading terms in the asymptotic expansion are exactly these functions. Thus $k$ and $\omega$ can be used to parametrize all solutions just as well as $P_0$ and $P_1$. After I wrote this I realized that published proofs of the statement about prescribing $k$ and $\omega$ apparently only cover the case where $k$ is everywhere positive. This restriction is not hard to remove using known techniques. What about the limit $t\to\infty$? It was proved by Thomas Jurke that any solution has an asymptotic expansion of the form $P(t,\theta)=A\log t+B+t^{-1/2}\nu(t,\theta)+R(t,\theta)$ where $A$ and $B$ are constants, $\nu$ satisfies the flat space wave equation $\nu_{tt}=\nu_{\theta\theta}$ and $R(t,\theta)=O(t^{-3/2})$ as $t\to\infty$. It was proved by Hans Ringström that given constants $A$ and $B$ and a solution $\nu$ of the flat-space wave equation there is a unique solution for which the leading terms in the asymptotic expansion are given by exactly these objects. In this way a third parametrization of the solutions is obtained. To mirror what is known for the limit $t\to 0$ it would be good to have an asymptotic expansion for the remainder $R$ arising in the limit $t\to\infty$. An expansion of this type has apparently never been derived. Supposing that a full expansion at late times had been obtained could it then be said that we knew essentially everything about solutions of the polarized Gowdy equations? I am not sure since I think there might still be some kind of intermediate asymptotics to be discovered. ### Like this: This entry was posted on October 22, 2008 at 6:30 am and is filed under general relativity. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. ### 2 Responses to “The polarized Gowdy equation” 1. Electrifying Gowdy « Hydrobates Says: January 27, 2009 at 9:02 am \| Reply [...] By hydrobates In a previous post I wrote about the polarized Gowdy equations. If the condition of polarization is dropped the full [...] 2. Cosmological perturbation theory, part 2 « Hydrobates Says: June 16, 2009 at 5:22 am \| Reply [...] equation in the limits and . It may be noted that this equation bears a certain resemblance to the polarized Gowdy equation. In fact we are able to import a number of techniques which have been used in the study of the [...] %d bloggers like this:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 46, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9547203779220581, "perplexity_flag": "head"}
http://www.physicsforums.com/showthread.php?p=4182849	Physics Forums Deriving the algebraic definition the dot product Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines? Recognitions: Gold Member Science Advisor Staff Emeritus What exactly do you mean by "the geometric definition"? The simplest "geometric" definition I know is that the dot product of vectors u and v is the product of the length of u and the length of the projection of v on u, but I suspect that you are thinking of "length of u times length of v times cosine of the angle between them". Since that has "cosine" already in it, if not using the "cosine law" per se, you certainly will need to use something that has a "cosine" in it. From the definition "length of u times the length of the projection of v on u", I would start by setting up a coordinate system in which the positive x-axis lies along vector u. Then u has components <a, 0>. Taking <b, c> to be the components of v, the "projection of v on u" is just <b, 0> so the dot product is ab= ab+ c0, the usual component formula for the dot product in this case. Get the general formula by rotating axes. Thanks HallsOfIvy, I'll try that. Recognitions: Gold Member Homework Help Science Advisor Deriving the algebraic definition the dot product Well, in the 2-D case, given vectors "A", "B", with $(x_{a},y_{a})$, $(x_{b},y_{b})$, lengths a and b, respectively you may think in terms of right-angled triangles, and set up: $$x_{a}=a\cos(\theta_{a})$$ $$y_{a}=a\sin(\theta_{a})$$ $$x_{b}=b\cos(\theta_{b})$$ $$x_{b}=b\sin(\theta_{b})$$ whereby we get: $$x_{a}x_{b}+y_{a}y_{b}=ab\cos(\theta_{a}-\theta_{b})$$ First define the dot product for A and B to be the product of their magnitudes and the cosine of the angle between them. We can see geometrically that A.(B + C) = A.B + A.C (think about the component of B and C along A), and therefore (A + B).(C + D) = A.C + A.D + B.C + B.D. Choosing perpendicular axis, every vector can be written in terms of components, so A = a_1i + a_2j and B = b_1i + b_2j. Therefore A.B = a_1b_1i.i + a_2b_2j.j + a_1b_2i.j + a_2b_1j.i. Because i and j are unit vectors and perpendicular, i.i = 1, j.j = 1, i.j = 0, j.i = 0. So we are left with A.B = a_1b_1 + a_2b_2. Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k. and of course here's more info on it from wikipedia: http://en.wikipedia.org/wiki/Vector_dot_product Recognitions: Gold Member Science Advisor Staff Emeritus Quote by jedishrfu Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k. and of course here's more info on it from wikipedia: http://en.wikipedia.org/wiki/Vector_dot_product Strictly speaking the term "dot product" is only used Euclidean space. In other vector spaces the term is "inner product". Of course, any n-dimensional vector space is isomorphic to Rn so the two work out to be "essentially" the same. And the cross product is only defined for R3, not general Euclidean spaces. Its also interesting to note the beauty of the inner and outer products. The inner product is a projection of one vector on another and the outer product is the non-projectable component of one vector on another. Since the projection interpretation is valid for either vector then the resultant vector must be perpendicular to both and thus it becomes normal to the plane containing the two vectors. Couldn't we just always work a coordinate system where the x-axis is parallel to the x-component of one vector? Or is that what you mean when you said 'get the general formula by rotating axes?'. And would the formula only work in two dimensions? If so, what about higher dimensions? EDIT: Woops, didn't see the new posts! SO for rotating the axes, would it much more complicated in 3D? Thanks arildno, when you times the multiply the components and add them up, that just comes from the definition of dot product that HallsOfIvy mentioned right? ('length of u times the length of the projection of v on u'). Recognitions: Science Advisor Quote by autodidude Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines? Project the vector A onto the line through B. Multiply the length of this projection by the length of B. By similar triangles this is the same as projecting the vector B onto the line through A and multiplying the length of the projection by the length of A. Call these products the dot product of A and B. Either product divided by AB gives you the same number which depends only on the lines and not on the particular vectors. Call this number the cosine of the angle between the vectors. P.S. In Euclidean geometry the law of similar triangles which is what is used here is logically equivalent to the Pythagorean Theorem so using the Law of Cosines is really no different than what is posted here. Thread Tools \| \| \| \| \|------------------------------------------------------------------------\|-------------------------------\|---------\| \| Similar Threads for: Deriving the algebraic definition the dot product \| \| \| \| Thread \| Forum \| Replies \| \| \| Introductory Physics Homework \| 4 \| \| \| Calculus & Beyond Homework \| 8 \| \| \| Linear & Abstract Algebra \| 3 \| \| \| Calculus & Beyond Homework \| 17 \| \| \| Introductory Physics Homework \| 2 \|	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 5, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9098858833312988, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/183045/can-one-solve-a-recurrence-that-contains-a-function	# Can one solve a recurrence that contains a function? I'd like to solve a recurrence, so I've been reading about solving recurrences, and all the ones I've seen solved involve only previous terms of the recurrence, and constants. My recurrence is $$t(n) = \frac{t(n-1)(1-\ln(n-1))}{\ln n}$$ and if I could get help solving it, I'd be much obliged; but I'd especially like to know if there are any general principles for solving recurrences of this nature (with functions in them). - ## 1 Answer When you have $t(n)=t(n-1)f(n)$ you can just write $t(n)=t(1)\prod_{i=2}^n f(i)$. In this case I started $i$ at $2$ because trying $n=1$ leads to division by zero. So $t(n)=t(1)\prod_{i=2}^n \frac {1-\ln (i-1)}{\ln i}$ - ... and it's quite unlikely that there is any simplification that can be usefully applied to that product. – Robert Israel Aug 16 '12 at 6:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9696860313415527, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/60143/maximum-area-with-a-given-perimeter	# Maximum area with a given perimeter An arbitrary pentagon (which is convex) is given to you, and it has a perimeter of $k$. Determine the i) maximum area, rigorously, in terms of $k$. ii) maximum INTEGRAL area, in terms of $k$. - 1 Is it that u want to know when the area can be maximum? if so its in the case of a regular pentagon and is given by {n(l^2)}/4tan(180/n) where n is number of sides and t is length of each side so nt=k – Bhargav Aug 27 '11 at 17:40 1 How rigorously? If we assume that there is a pentagon of perimeter $k$ which has maximum area, showing that this optimal pentagon is regular is not difficult. However, although the existence of an optimal pentagon is intuitively reasonable, proving it rigorously takes quite a bit of effort. – André Nicolas Aug 27 '11 at 19:52 2 Among all $n$-gons inscribed in a circle, which one has largest area and which one has largest perimeter? The regular polygon is the expected answer, and this turns out to be the case. Maxima and minima without calculus by Ivan Morton Niven Volume 6, 6.1 Introduction. – Américo Tavares Aug 27 '11 at 21:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9310542941093445, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/140352/surjective-graded-homomorphism-of-rings-also-an-isomorphism	# Surjective graded homomorphism of rings also an isomorphism? Suppose we are given two graded (commutative) rings $A$ and $B$ and a graded homomorphism $\psi:A\rightarrow{B}$ between them. Suppose moreover that $\psi$ is surjective in each degree i.e. that $\psi(A_n)=B_n$ for all $n$. If $\psi$ is injective in degrees $0$ and $1$, must it also be injective everywhere else? (all relations in $A$ come from these two degrees). If not in general, does it make a difference if we assume $B$ is an integral domain? (or even more specifically if it is a polynomial ring with coefficients in an integral domain). - ## 2 Answers The answer is indeed no, for both cases : take $A=R[X,X^2Y]$, graded by degree in $X$ and $B=R[X]$, also graded by degree in $X$, and $\psi:A\to B$ sending $Y$ to $0$. This is a graded morphism between two integral domain (as long as $R$ is one), that is surjective in each degree, injective in degrees 0 and 1 (since every polynomial containing $Y$s must contain twince as much $X$s) but not an isomoprphism. - Ok thanks for the answer! – Axiom May 9 '12 at 15:15 The answer is no. Consider $A = \mathbb{R}[t]$, and $B = \mathbb{R}[t]/(t^k)$, both graded by degree. The homorphism $\psi : A \to B$ is the natural projection, which is surjective in each degree. For any choice of $k \geq 1$, we have $\psi$ injective up to degree $k-1$, but not injective for degree $k$ and higher. About the follow-up question, if $B$ is an integral domain, let me think about that a little more. Hope this helps! - 2 This doesn't work : the natural projection homomorphism is not a graded morphism (e.g. $t^{k+1} \mapsto t$ lowers the degree by $k$. – Samuel T May 3 '12 at 12:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 31, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9683773517608643, "perplexity_flag": "head"}
http://mathoverflow.net/questions/48098/restriction-of-scalars-of-simple-algebraic-groups/112759	## Restriction of scalars of simple algebraic groups ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) I'm trying to understand the following basic property of the restriction of scalars: Given an absolutely simple algebraic groups $G$ defined over a number field $k$, are there at most finitely (up-to $k$-isomorphism) many absolutely simple algebraic groups $H$ defined over $k$ with $$Res_{k/\mathbb Q}G\cong_{\mathbb Q} Res_{k/\mathbb Q}H?$$ (i.e. their restrictions are isomorphic over $\mathbb Q$) I understand that it follows that $G$ and $H$ are $k$-forms of each other and good understanding of the possible forms over the completions of $k$ should lead to an answer. This is because there is a theorem by Borel and Serre that establish the finiteness of forms that agree at all but finitely many completions. Therefore my vague intuition is that the answer is yes, but I can't find a precise proof. Thinking about it also directed me to an even more basic question: Given $G$ and $H$, two forms of each other (i.e., $G$ and $H$ both defined over $k$, and are isomorphic over the algebraic closure $\bar k$.) Is $G$ and $H$ isomorphic over $k_v$ for all but finitely many valuations $v$ of $k$? I think that a positive answer to the latter should imply a positive answer to the former, but I think the latter might have a negative answer... I'll be a happy if someone can shed some light about it and\or guide me to the relevant results in the literature. Thanks! - For the latter question, take G=SL_3 and H=SU_3. They're isomorphic over the algebraic closure, but are only isomorphic at half of the completions, hence answer is no. – Peter McNamara Dec 2 2010 at 21:26 Indeed. But these are outer forms? (or maybe not? I'm quite confused). Is it also not true for inner forms? – Menny Dec 2 2010 at 21:44 [They are indeed outer forms] – Kevin Buzzard Dec 2 2010 at 21:45 1 Arithmetic & Galois actions on root data are irrelevant. First I give easy proof of something very general: for any field $K$, finite etale $K$-algebras $K'$ and $K''$, and ss gps $G'$ and $G''$ over $K'$ and $K''$ resp. with all fibers connd and abs. simple, every $K$-gp isom. R$_{K'/K}(G')\simeq$ R$_{K''/K}(G'')$ comes from a unique pair $(f,\alpha)$ consisting of a $K$-algebra isom $\alpha:K''\simeq K'$ and a gp isom $f:G'\simeq G''$ over Spec($\alpha$). Pf: By uniqueness and Galois descent, enough to check after base change to $K_s$. Then $K'$ and $K''$ are powers of $K$, so it's obvious! – BCnrd Dec 3 2010 at 6:38 1 OK, that being said, we now dispose of the original finiteness question using nothing subtle or arithmetic. Let $K'/K$ be a finite separable extension of fields and let $G'$ and $H'$ be two connected semisimple $K'$-groups that are abs. simple. Suppose the $K$-groups R$_{K'/K}(G')$ and R$_{K'/K}(H')$ are $K$-isomorphic. Previous comment shows $H'$ is an Aut($K'/K$)-twist of $G'$ (which is clearly "best possible"), and Aut($K'/K$) is a finite group. Hence, only finitely many possibilities for $H'$, and we've nailed down exactly which ones actually occur (e.g., if Aut($K'/K) = 1$ then $H'=G'$!) – BCnrd Dec 3 2010 at 6:43 show 4 more comments ## 2 Answers For the question in the comments, Kevin is right: every connected reductive group over k is an inner form of a unique quasi-split group, up to isomorphism. This follows from an argument using Galois cohomology and the splitting $$1\to\operatorname{Int}(G)\to\operatorname{Aut}(G)\to\operatorname{Aut}(\Psi_0(G))\to 1,$$ where $\Psi_0(G)$ is the based root datum of G. More details can be found on the first article of Corvallis, and surely in some other place (maybe Borel-Tits?). - ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. If I may be allowed, I will use a "high powered" theorem to deduce this is the case of number fields. It is indeed true that given an absolutely simple $k$-algebraic group $G$, the number of absolutely simple $k$-algebraic groups $H$ such that the restriction of scalars $R_{k/{\mathbb Q}}G$ and $R_{k/{\mathbb Q}}H$ are ${\mathbb Q}$-isomorphic, is finite. The hypothesis implies that as abstract groups $G(k)$ and $H(k)$ are isomorphic, since they are both ${\mathbb Q}$-rational points of the restriction of scalars group. Now by the Margulis superrigidity theorem (it was initially proved for ($S$)-arithmetic groups, but there is a version in his book for $k$-rational points which may be thought of as irreducible lattices in the adelic groups) such an abstract isomorphism $\theta$ arises from an isomorphism $\sigma:k \rightarrow k$ of the number field $k$, and a $k$-isomorphism $\phi: ^{\sigma }G \rightarrow H$ of $k$ algebraic groups. This means that $\theta (g)= (\phi (\sigma(g))$ for all $g\in G(k)$. $^{\sigma }G$ is the same group as $G$, twisted by the automorphism $\sigma$ on scalars. In particular, $H$ is isomorphic to $^{\sigma }G$ for some $\sigma$. Since the number of the $\sigma$'s is finite, it follows that the number of $H$ is finite. - Don't the comments to the original question provide a short elementary proof of the finiteness over any field using nothing beyond Galois descent (in particular, no input from number theory)? – nosr Nov 18 at 16:11 Indeed, but I find this answer very interesting also. Thanks, Askumadula! – Menny Nov 19 at 20:57 @Menny: OK, but doesn't Margulis' theorem (say in its adelic form) require an isotropicity hypothesis (or more)? It would be good to clarify this issue. – nosr Nov 20 at 4:14 No, Margulis' theorem does not require isotropy hypothesis; if $G$ is isotropic over $K$, then this result is a theorem of Borel and Tits (abstract homomorphisms paper). The Margulis theorem, on the other hand, does not use cocompact or non-cocompact (the proof in his book). – Aakumadula Nov 20 at 9:00 And thanks Menny for your remarks. – Aakumadula Nov 20 at 9:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 87, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9408405423164368, "perplexity_flag": "head"}
http://mathoverflow.net/questions/104731/instances-where-an-existence-result-precedes-the-constructive-version/104747	## Instances where an existence result precedes the constructive version ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. Here's the question: Are there good examples of instances in mathematics research where an existence-only result preceded - by some number of years - the corresponding constructive result? I am aware of one nice example from my own field, which I hope is illustrative of the type of answer that would be nice to collect here: Tucker's Lemma, which is a discrete version of the famous Borsuk-Ulam Theorem, was first proved in the following paper by contradiction: A. W. Tucker. Some topological properties of disk and sphere. In Proc. First Canadian Math. Congress, Montreal, 1945, pages 285–309. University of Toronto Press, Toronto, 1946. The first constructive proof (which is starkly different from the original), did not appear in the literature until R. M. Freund and M. J. Todd. A constructive proof of Tucker’s combinatorial lemma. J. Combin. Theory Ser. A, 30(3):321–325, 1981. I did try just searching google for "A constructive proof of" and similar search strings. This does provide some examples, but the results are not filtered by importance of the result in question, or the extent of difference between the old existence result and the newer constructive one. Only one example per answer, please! - 1 "In his seminal work of 1948, Shannon had characterized the highest rate (speed of transmission) at which one could reliably communicate over a discrete memoryless channel (a noise model); he called this limit the capacity of the channel. However, he used a probabilistic method in his proof and left open the problem of reaching this capacity with coding schemes of manageable complexities." From gilkalai.wordpress.com/2010/11/25/… – Steve Huntsman Aug 15 at 1:00 1 To clarify: I assume you want instances where there actually is a constructive proof. – Nate Eldredge Aug 15 at 3:01 2 I wonder if transcendental numbers qualify here. Does anybody know the history? – Yoav Kallus Aug 15 at 3:11 @Nate: yes. Is there another way to interpret the question? – Vidit Nanda Aug 15 at 3:23 1 There are existence proofs for which there is no constructive proof. The probabilistic method in combinatorics has produced many such examples, e.g., that $R(r,r)$ grows exponentially because if $n \gt c \sqrt2^r$ then the probability that a random coloring of $K_n$ has a monochromatic $K_r$ is less than $1$, or even less than $1/100$. No construction is known. – Douglas Zare Aug 15 at 6:09 show 1 more comment ## 12 Answers Steve Smale proved in 1958 that the 2-sphere in 3-space could be everted. Afterwards the first explicit model of an eversion was discovered in 1961 by Arnold Shapiro, but it took still a few more years before such models were explained to the larger mathematical community in Tony Phillips' 1966 article in Scientific American. - 3 1 Amazing film. It deserves a wide audience. – Felix Goldberg Aug 15 at 10:05 Yeah that's a great film. It is a continuation of the story of existence vs. construction. The construction depicted in that film was discovered by William Thurston, but in this case not as another isolated construction but as an implementation of general proof method for simple homotopy problems – Lee Mosher Aug 15 at 12:39 Oops, I meant to say "for regular homotopy problems". – Lee Mosher Aug 15 at 23:30 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Expander graphs were known to exist for many years before there was any provable example (essentially because it is not hard to show that a random sequence of graphs is expander with probability 1.) - 1 More generally, the probabilistic method (en.wikipedia.org/wiki/Probabilistic_method) as pioneered by Erdos comes to mind. – Michael Lugo Aug 15 at 3:04 1 It is pretty hard to construct a graph which is not expanding with some constant, and given a graph of a reasonable size, it is not so difficult to compute the expansion constant (the stupid algorithm is exponential, but for graphs up to around 30 vertices this should not be a problem). It is constructing an infinite family of such that is hard. – Igor Rivin Aug 15 at 4:06 4 Yes -- by "expander graph" I mean "infinite family of graphs with a spectral gap." Bad notation, I know. – JSE Aug 15 at 4:48 In his doctoral dissertation held in Koenigsberg in 1885, Minkowski proposed the conjecture that, unlike the quadratic case, nonnegative homogeneous polynomials of higher degree and more than two variables in general cannot be written as a sum of squares of real polynomials. The problem attracted the attention of Hilbert who in 1888 proved nonconstructively the existence of such polynomials. However, the first concrete example of a nonnegative polynomial which is not a sum of squares seems to have been given only in 1967 by T. S. Motzkin [The arithmetic-geometric inequality, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), pp. 205-224", Academic Press, New York, 1967]. - As I understand it, Hilbert's original solution to Gordan's Problem was nonconstructive, proving that every algebraic variety over a field had a finite generating set. (His result is now generally cited as "Hilbert's Basis Theorem", that polynomial rings over Noetherian rings are Noetherian.) In modern day algebraic geometry, Hilbert's nonconstructive argument is replaced by a very constructive process in which one generates a Groebner Basis for the algebraic set. - Didn't Hilbert himself produce a constructive proof a few years after his non-constructive one? – Arthur Fischer Aug 20 at 17:24 One example is indeed that of Transcendental numbers, as Yoav Kallus points out in the comments. Liouville showed in 1844 that numbers which do not satisfy a polynomial equation with integer coefficients exist, but he only gave an example in 1851, the famed Liouville constant, a celebrity among Transcendental numbers: $$\sum_{n=1}^\infty 10^{-n!}.$$ For more information, see here and here. You might need JSTOR access to read the first. Cantor however, whose proof of the existence of Transcendental numbers follows directly from the uncountability of the Reals, only came up with his proof in 1874. Whether his proof of the Uncountability of the Reals is constructive or not is something people are still debating, so I will not comment on that. For more information on this, see this Wikipedia Article - Your link entitled "Wikipedia Article" is broken. – Lee Mosher Aug 16 at 13:40 @LeeMosher Thanks! Fixed it. – LeBlanc Aug 17 at 3:08 2 The Cantor-style proof of the existence of transcendental numbers is certainly constructive. (I didn't see any coherent argument to the contrary on that Wikipedia talk page.) See for instance the comment of Joel David Hamkins here: mathoverflow.net/questions/10334/…, the Monthly article he cites, and my own answer here: mathoverflow.net/questions/20430/… – Tom Leinster Aug 20 at 18:42 One can give literally hundreds of examples from classical analysis. usually there are 4 stages: a) non-conscructive proof that some universal constant exists, usually by compactness arguments (in classical function theory this is called normal families argument). b) a proof which is constructive "in principle", and gives SOME numerical estimate. Sometimes this estimate is ridiculoisly large (or small), and the author does not even care to write it. c) Obtaining some reasonable estimate. On this stage, sometimes a competition starts for better and better estimates, until d) the exact constant is sometimes obtained. Which usually involves solving a variational problem and description of extremal configuration. Some stages can be skipped, of course. Some famous examples include "K\"obe constant", which turned out to be 1/4. Bloch's and Landau constants (they are almost century old, and currently they are on stage c)). Often the time between a) and b) is small. The largest time usually passes from c) to d), and many problems stop on stage c). Here are two examples from my own work: arXiv:math/0607743, the time lag between a) and b) was 65 years. arXiv:math/0510502, a problem still in stage a) The examples from analysis and number theory are so abundant that I don't think it is possible to catalog them. - Alexandre, this question is only about getting from a) to d). There are "literally hundreds of examples" because you have over-estimated the scope of the question. In particular, getting to stage c) is no guarantee whatsoever that stage d) will actually be reached and without that you are including all possible existence results for which no constructive version is known, rather than examples where the constructive version is known and came after many years. That being said, the K\'obe constant example sounds interesting and relevant. – Vidit Nanda Aug 15 at 17:20 1 Is these a misprint in your message: did you mean to write "from a) to b)" in the first line? This is how I understood the original question: from a) to b) from non-constructive to constructive. Yes, I widened the scope:-) But the two examples I gave from my work are indeed related to a) and b). – Alexandre Eremenko Aug 15 at 21:46 You're quite right, I had mis-read. Apologies! – Vidit Nanda Aug 17 at 17:09 The Lovasz Local Lemma establishes under certain assumptions that a collection of events can simultaneously fail to occur. The initial proofs from the 1970's were used the techniques of probabilistic techniques, hence were non-constructive. People worked on providing a constructive proof and had partial results based on substantially weaker assumptions about the relationship of the events. In 2008, Robin Moser greatly improved on these results, demonstrating an algorithm running in polynomial time that produces such an outcome for a standard case. With Gabor Tardos in 2009, he extended this result to recover every previously known application of the Local Lemma (though not quite the original statement). - Hironaka's original desingularization algorithm had constructive aspects but as I understand, also had aspects / computations which are not constructive and could not be done by hand or with a computer. Resolution algorithms now typically are based upon blowing up the strata of some invariant (where heuristically the singularities are at their worst) and so they are constructive if you can compute the invariant. But these invariants might be for all intents and purposes completely incomputable, for example based on infinite amounts of data. New algorithms have simplified many aspects on this, and indeed, algorithms have been implemented in computers, and it isn't so hard to write down all the various invariants as you follows modern algorithms. You can see http://mathoverflow.net/questions/4612/hironaka-desingularisation-theorem-new-proofs-in-literature for more discussion. - Gauss proved the Euclidean constructibality of the 257-gon (as well as the 17-gon and others) in 1796. In 1832, Friedrich Julius Richelot actually gave an explicit construction. - 6 Hm, I was under the impression a construction was extractable from the original proof, even if it wasn't actually extracted (making this not really an answer to the original question). Is that mistaken? – Harry Altman Aug 20 at 7:29 Indeed, we now have a construction for the regular $n$-gon for any Fermat prime $n$ - all we're lacking is the existence proof for Fermat primes exceeding 65537. – Gerry Myerson Sep 4 at 5:39 The Monster group is a finite simple group of order approx. $8\cdot 10^{53}$ whose existence was predicted by Fischer and Griess in about 1973. An explicit construction of a representation of dimension 196882 over $\mathbf Z_2$ was obtained by Robert Wilson with the aid of a computer (published in 1998); see Wikipedia article http://en.wikipedia.org/wiki/Monster_group. - Claudio, I believe this has already been covered in Lee Mosher's answer. – Vidit Nanda Aug 15 at 1:58 @Vel Nias: Sorry, I was beaten by Lee Mosher by 12 min or so and didn notice. I am replacing my answer by another now. – Claudio Gorodski Aug 15 at 2:18 There are certain "probabilistic constructions" in Banach space theory where one shows that something happens with positive probability, but where explicit examples still aren't known. (That said, I am not up to speed with the developments here, and I seem to recall reading that in several cases, via derandomization techniques one can turn the existence proofs into reasonable algorithms for producing examples.) The sort of thing I have in mind is the Johnson-Lindenstrauss lemma, but I can think of at least two regular MO users who would be better placed than me to comment on it. - 2 I think the question is asking for instances where existence < construction < now. Otherwise every nonconstructive result would be a candidate for an answer. – Nate Eldredge Aug 15 at 3:00 A striking example is the use of Buchberger's algorithm to construct Gröbner bases. The latter concept goes back to the end of the 19th century but it was only on the arrival of Buchberger's algorithm that it became the omnipresent computational tool that it is today. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9398354291915894, "perplexity_flag": "middle"}
http://mathhelpforum.com/differential-equations/190219-basic-laplace-transform-1st-order-differential-equation.html	# Thread: 1. ## Basic Laplace Transform of a 1st order Differential Equation Just wondering if someone could have a quick look over what I've done so far: Assuming zero initial conditions, solve the following DE: $\dot{x} + x = 2$ The Laplace Transform gives: $sX(s) - x_0 + \frac{1}{s^2} = \frac{2}{s}$ Rearranging and using the fact that $x_0 =0$ gives us $X(s) = \frac{2}{s^2} - \frac{1}{s^3}$ $X(s) = 2\frac{1!}{s^{1+1}} - \frac{1}{2} \frac{2!}{s^{2+1}}$ So $x(s) = 2t - \frac{t^2}{2}$? 2. ## Re: Basic Laplace Transform of a 1st order Differential Equation After taking the Laplace transform, where did the $\frac{1}{s^2}$ come from? 3. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by Danny After taking the Laplace transform, where did the $\frac{1}{s^2}$ come from? I took the Laplace transform of $x$, is this not right? 4. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by craig I took the Laplace transform of $x$, is this not right? $x$ is a dependent variable ( $x=x(t)$ ), so you'll obtain $s\mathcal{L}\{x\}-x_0+\mathcal{L}\{x\}=\frac{2}{s}\Rightarrow (s+1)\mathcal{L}\{x\}=x_0+\frac{2}{s}\Rightarrow \ldots$ 5. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by FernandoRevilla $x$ is a dependent variable ( $x=x(t)$ ), so you'll obtain $s\mathcal{L}\{x\}-x_0+\mathcal{L}\{x\}=\frac{2}{s}\Rightarrow (s+1)\mathcal{L}\{x\}=x_0+\frac{2}{s}\Rightarrow \ldots$ So given that $x_0 = 0$, does that mean that we just have $\mathcal{L}\{x\}=\frac{2}{s(s+1)}$. Which I presume we express in partial fractions, and then solve for $x(t)$? 6. ## Re: Basic Laplace Transform of a 1st order Differential Equation So for $\mathcal{L}\{x\}=\frac{2}{s(s+1)} = \frac{2}{s} - \frac{2}{s+1}$ we have: $\mathcal{L}\{x\}= 2\frac{1}{s} - 2\frac{1}{s-(-1)}$ So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ? Thanks again for the replies. 7. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by craig So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ? Right. 8. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by craig So for $\mathcal{L}\{x\}=\frac{2}{s(s+1)} = \frac{2}{s} - \frac{2}{s+1}$ we have: $\mathcal{L}\{x\}= 2\frac{1}{s} - 2\frac{1}{s-(-1)}$ So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ? Thanks again for the replies. Technically, since you're working with the one-sided LT (I assume), you have unit step functions multiplying everything. That is, $x(t)=2u(t)(1-e^{-t}),$ and the solution is valid for non-negative t's. 9. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by Ackbeet and the solution is valid for non-negative t's. Also considering the theorem about the necessary form of the solutions for the equation $x^{(n)}+a_{n-1}x^{(n-1)}+\ldots +a_1x'+a_0x=b(t)$ . 10. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by FernandoRevilla Also considering the theorem about the necessary form of the solutions for the equation $x^{(n)}+a_{n-1}x^{(n-1)}+\ldots +a_1x'+a_0x=b(t)$ . Sorry would you mind explaining what you meant by this? I understood the first part about the Unit-Step function, just not sure what you meant here? Thankyou again! 11. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by craig Sorry would you mind explaining what you meant by this? The Laplace transform of $x:[0,+\infty)\to\mathbb{R}$ is defined by $\mathcal{L}\{x(t)\}=\int_0^{+\infty}e^{-st}x(t)\;dt$ , so a priori we find solutions valid for $t\geq 0$. Your equation has (by a well known theorem) a determined form for its unique solution $x(t)=k+Ce^{at}$ valid for all $t\in\mathbb{R}$ so, necessarily is valid the solution found by LT for all $t\in\mathbb{R}$ . 12. ## Re: Basic Laplace Transform of a 1st order Differential Equation Originally Posted by FernandoRevilla The Laplace transform of $x:[0,+\infty)\to\mathbb{R}$ is defined by $\mathcal{L}\{x(t)\}=\int_0^{+\infty}e^{-st}x(t)\;dt$ , so a priori we find solutions valid for $t\geq 0$. Your equation has (by a well known theorem) a determined form for its unique solution $x(t)=k+Ce^{at}$ valid for all $t\in\mathbb{R}$ so, necessarily is valid the solution found by LT for all $t\in\mathbb{R}$ . Agreed. You could simply say that you extend the solution found by the LT method to the entire real line. The result of the inverse LT does have the unit step functions in it, and, in fact, does not satisfy the DE for negative t's. However, with the theorem you have invoked, you can extend the solution by eliminating the unit step function. 13. ## Re: Basic Laplace Transform of a 1st order Differential Equation Thankyou both! I don't think we've covered this in lectures yet but I think it makes sense.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 41, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9283609390258789, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/44528/a-simple-decomposition-for-fractional-brownian-motion-with-parameter-h1-2/101173	## Background Let `$X = \{X(t):t \geq 0\}$` be a (standard, real-valued) fractional Brownian motion (fBm) with parameter `$H \in (0,1)$`, i.e., a continuous centered Gaussian process with covariance function given, for `$0 \leq s \leq t$`, by $$C_X {(s,t)} := {\rm E}[X(s)X(t)] = \frac{1}{2}[t^{2H} + s^{2H} - (t - s)^{2H} ].$$ Writing `$C_X {(s,t)}$` as `$C_X {(s,t)} = \frac{1}{2}[t^{2H} - (t - s)^{2H} ] + \frac{1}{2}s^{2H}$`, gives rise to the decomposition of `$X$` as `$X = Y + Z$`, where `$Y$` is a centered Gaussian process with covariance function `$C_Y {(s,t)} = \frac{1}{2} [t^{2H} - (t - s)^{2H}]$`, independent of a time-changed Brownian motion `$Z$` (specifically, `$Z(t)=W(t^{2H}/2)$`, where `$W$` is a standard BM). However, in order for `$C_Y$` to be a valid covariance function it must be nonnegative definite. As indicated by numerical results (and can probably be easily proved), this is not the case for `$H>1/2$`. For `$H<1/2$`, on the other hand, `$C_Y$` is the covariance function of some interesting Gaussian process arising in the setting of Gaussian random fields. Since I plan to write a paper on this apparently new subject, I find it sensible not to give too much details here (maybe I'll add some details later on). Now to my questions. Have you encountered the aforementioned decomposition in the literature? (I haven't.) Does it correspond to some known (e.g., integral) representation of fBm? Can you think of some application of it? Finally, can you find a simple/useful representation for the process `$Y$` in that decomposition (simple/useful compared to the fBm case)? - ## 3 Answers Sorry I don't have time to write a better answer. I would be willing to bet Nualart has thought about this problem at least and his answer could very well be encompassed in this paper: (In particular your problem might be a special case described in section 3) P. Lei and D. Nualart: A decomposition of the bifractional Brownian motion and some applications. Statistics and Probability Letters 79, 619-624, 2009. http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.2227v1.pdf - That reference is certainly interesting in our context (and I might even cite it). However, the decomposition described there for fBm (cf. Proposition 1) is not similar to the one I indicated above, and is apparently much more complicated. – Shai Covo Nov 7 2010 at 11:55 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Interesting, I don't think I've seen that before. But there is a similar sort of decomposition in W. Li and W. Linde 1998, however I don't think it's quite the same. Cheridito 2003 (Mixed-FBM) tackles a tangential but not altogether unrelated question. One last comment-- you should probably put max(s,t) in your covariance function since you are implicitly assuming t>s. - It looks like this idea isn't new. There is something very similar in an article by Alos, Mazet, and Nualart (SPA 2000). -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9428973197937012, "perplexity_flag": "middle"}
http://mathhelpforum.com/calculus/50298-position-function.html	Thread: 1. Position Function? Hey, I am in MAT270 and I can't find how to do this problem anywhere in my book, can someone help me out? I am going to post a similar problem because I would like to figure out how to do this by the steps taken to find the answer to the similar problem. "Use the position function f(t) meters to find the velocity at time t=a seconds." √(t+16) [square root of (t+16)], (a) a=0; (b) a=2 2. Hello ! Originally Posted by Morpha Hey, I am in MAT270 and I can't find how to do this problem anywhere in my book, can someone help me out? I am going to post a similar problem because I would like to figure out how to do this by the steps taken to find the answer to the similar problem. "Use the position function f(t) meters to find the velocity at time t=a seconds." √(t+16) [square root of (t+16)], (a) a=0; (b) a=2 I'm not in MAT270, nor do I know what this stands for... but it doesn't matter But this is something commonly used in physics. If f(t) is the position function, then the speed function is $f'(t)$. And the velocity (which means speed at an instant) at time t=a is $f'(a)$ You can view it this way : - speed is a ratio distance/time - the derivative is a ratio "function displacement"/"units" A formal way to view the velocity is to see that velocity at a time t=a is the difference of distance that occurred instantaneously. More basically, it is the difference of distance that occurred within a very very very little interval of time. Let $\Delta t$ be this interval of time. $f(a+\Delta t)-f(a)$ denotes the difference of distance within this very little time (around t=a). Dividing by $\Delta t$ will make the speed. Because we want this interval to be very small, like null, we'll make $\Delta t \to 0$ So finally, we're looking for $\lim_{\Delta t \to 0} ~ \frac{f(a+\Delta t)-f(a)}{\Delta t}=f'(a)$, by definition. Is it clear ? NB : the acceleration function is $f''(t)$ 3. Given the position function $f(t) = \sqrt{t + 16}$, first calculate the derivative. The derivative is $f'(t) = \frac{1}{2\sqrt{t + 16}}$. Now that you have the derivative, you can evaluate f'(0) and f'(2).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.940822184085846, "perplexity_flag": "middle"}
http://www.scholarpedia.org/article/Zinn-Justin_equation	# Zinn-Justin equation From Scholarpedia Jean Zinn-Justin (2009), Scholarpedia, 4(1):7120. Curator and Contributors 1.00 - Jean Zinn-Justin Quantum field theories in a naive formulation lead to physical results plagued with infinities due to short distance singularities and require a regularization, operation by which their short-distance structure below a cut-off scale is modified in an unphysical way. For a class of quantum theories called, therefore, renormalizable, it is possible to construct a theory finite when the cut-off is removed by rendering the parameters of the initial Lagrangian cut-off dependent, a mathematical procedure called renormalization and whose deep meaning can only be understood in the framework of the renormalization group. Non-Abelian gauge theories are quantum theories at the basis of the Standard Model of particle physics (that describes fundamental interactions at the microscopic scale); their mathematical consistency requires their renormalizability and the preservation of some form of gauge invariance by the renormalization process. At the beginning of the 1970s, much effort was devoted to the proof of the perturbative renormalizability of non-Abelian gauge theories. Initial arguments based on Feynman diagrams ('t Hooft and Veltman 1972), and Lee-Zinn-Justin's proof (1972), based on Slavnov-Taylor identities (Slavnov 1972), (Taylor 1971), were simplified and generalized with the use of the BRST symmetry, discovered by Becchi, Rouet, Stora (1974,1975,1976) and by Tyutin (1975), that generalizes gauge invariance in this context. A general proof of renormalizability of non-Abelian gauge theories is based on the master equation, also called Zinn-Justin equation (Zinn-Justin, 1974). The Zinn-Justin equation (ZJ equation) is a quadratic equation satisfied both by the so-called one particle irreducible generating functional of Green's functions (or correlation functions) and by the quantized action. The ZJ equation can be shown to be perturbatively stable under renormalization. The general solution of this equation, taking into account locality, power counting and ghost number conservation, gives the general form of the renormalized action. In particular, the ZJ equation implies independence of physical observables from the gauge fixing procedure required to construct the quantum theory. ## Non-Abelian gauge theories: Classical field theory A classical non-Abelian gauge theory is a generalization of Maxwell's Electrodynamics, in which the gauge invariance is based on a non-Abelian gauge group $$G$$ in place of the Abelian $$U(1)$$ group underlying Maxwell's theory. We consider here only local field theories, that is, theories in which the action is the space-time integral of a function of fields and their derivatives (the Lagrangian density or simply Lagrangian). A classical gauge theory is a classical field theory whose action is invariant under gauge transformations. A gauge transformation is a space-time dependent representation of a matrix complex Lie group $$G\ ,$$ acting on the fields of the theory. If the group $$G$$ is non-Abelian, one speaks of non-Abelian gauge transformations and non-Abelian gauge theories. In what follows, in view of physics applications, we restrict the discussion to unitary groups (but orthogonal groups require trivial modifications) and 3+1 space-time dimensions. ### Gauge transformations and gauge fields We assume that matter fields $$\phi(x)$$ form complex vectors that, in a gauge transformation, transform like $\tag{1} \phi(x)\mapsto\phi^{\mathbf{g}}(x)= \mathbf{g}(x)\phi(x),\quad \mathbf{g}(x)\in G \quad\forall x\,,$ where $$\mathbf{g}(x)$$ smoothly maps the space time to the group $$G\ .$$ To construct a gauge theory, it is necessary to introduce a gauge field (or connection) $$\mathbf{A}_\mu(x)\ ,$$ also known as Yang-Mills field, which, for each space-time index $$\mu=0,1,2,3$$ takes values in the Lie algebra of $$G\ .$$ Gauge transformations act on the gauge field as $\tag{2} \mathbf{A}_{\mu}(x) \mapsto \mathbf{A}_{\mu}^{\mathbf g}(x) := \mathbf{g}(x) \mathbf{A}_{\mu}(x) \mathbf{g}^{-1}(x) + \mathbf{g}(x) \partial_{\mu} \mathbf{g}^{-1}(x).$ When $$\mathbf{g}(x)$$ is constant for all $$x\ ,$$ the gauge field $$\mathbf{A}_\mu(x)$$ transforms under the adjoint representation of the group $$G\ .$$ For generic $$\mathbf{g}(x)$$ the transformed field $$\mathbf{A}_{\mu}^{\mathbf g}(x)$$ is no longer linear in $$\mathbf{A}_\mu(x)$$ but affine. The form (2) ensures that $$\mathbf{A}_{\mu}^{\mathbf g}(x)$$ is still valued in the Lie algebra of $$G\ .$$ ### Covariant derivatives and curvature In a local, gauge invariant, field theory ordinary derivatives must be replaced by covariant derivatives that, because they transform linearly under gauge transformations, ensure the gauge invariance of the action. Covariant derivatives are constructed using the gauge connection $$\mathbf{A}_{\mu} \ .$$ Their explicit form depends on the representation under which fields are transforming. For example, for the matter fields that transform like in (1), the covariant derivative takes the form $$\mathbf{D}_{\mu}= \mathbf{1}\,\partial_{\mu} + \mathbf{A}_{\mu}$$ and transforms like $\tag{3} \mathbf{D}_{\mu} \mapsto \mathbf{D}_{\mu}^{\mathbf{g}} =\mathbf{1}\,\partial_{\mu} + \mathbf{A}_{\mu}^{\mathbf{g}} = \mathbf{g}(x) \mathbf{D}_{\mu} \mathbf{g}^{-1}(x) ,$ where the product has to be understood as a product of differential and multiplicative operators. Covariant derivatives of fields then transforms as $$\mathbf{D}_{\mu}^{\mathbf{g}}\, \mathbf{g}(x)\phi(x) =\mathbf{g}(x) \mathbf{D}_{\mu}\phi(x)\ .$$ As a consequence of the property (3), the commutator $\mathbf{F}_{\mu\nu}(x) = \left[ \mathbf{D}_{\mu},\mathbf{D}_{\nu}\right] = \partial_{\mu} \mathbf{A}_{\nu}(x) - \partial_{\nu} \mathbf{A}_{\mu}(x) + \left[ \mathbf{A}_{\mu}(x),\mathbf{A}_{\nu}(x)\right] ,$ which is no longer a differential operator, is also a tensor (the curvature of the connection) for gauge transformations: $\tag{4} \mathbf{F}_{\mu\nu}(x) \mapsto \mathbf{F}_{\mu\nu}^{\mathbf{g}}(x)= \mathbf{g}(x) \mathbf{F}_{\mu\nu}(x) \mathbf{g}^{-1}(x).$ It then follows from (4) that the local action for the gauge field $\tag{5} \mathcal{S}_{\mathrm{class.}}(\mathbf{ A})= {1\over 4 e^{2}} \int \mathrm{d}^4 x\, \mathrm{tr}\sum_{\mu,\nu} \mathbf{F} _{\mu \nu} (x ) \mathbf{F}^{\mu \nu} (x )\,,$ is gauge-invariant. ($$e$$ in (5) will characterize the strength of the interaction in the presence of matter fields.) More generally, it is important to realize that physical observables are related to gauge-invariant polynomials in the fields. When in the transformation (2), $$\mathbf{g}(x)$$ is close to the identity, that is, when $$\mathbf{g}(x)=\mathbf{1}+\omega(x) +O(\\|\omega\\|)^2\ ,$$ $$\omega(x)$$ being a `small' smooth map valued in the Lie algebra of $$G\ ,$$ the gauge transformation takes the form $\tag{6} \mathbf{A}_{\mu}^{\mathbf g}(x)-\mathbf{A}_{\mu}(x)=-\mathbf{D}_\mu \omega(x)+O(\\|\omega\\|)^2\quad \mathrm{with}\quad \mathbf{D}_\mu \omega(x)\equiv \partial_\mu \omega(x)+[\mathbf{A}_{\mu}(x),\omega(x)].$ Equation (6) gives the form of covariant derivative $$\mathbf{D}_\mu$$ when it is applied to fields valued in the Lie algebra of $$G\ ,$$ like $$\omega(x)\ .$$ ## Non-Abelian gauge theories: The quantized action Due to gauge invariance, in non-Abelian gauge theories like in Quantum Electrodynamics (QED), not all components of the gauge field are dynamical and a simple canonical quantization is impossible. However, in non-Abelian gauge theories the methods required for the construction of a quantized theory are more involved than in QED. The construction of local, relativistic-covariant quantum non-Abelian gauge theories, involves the so-called Faddeev-Popov determinant and the introduction of ghost fields and relies on manipulations of field integrals (Faddeev and Popov 1967). BRST symmetry emerges from this formalism. In what follows, as a slight simplification, we discuss only gauge theories without matter, the modifications due to the addition of matter fields being simple (even though, in the case of fermions and chiral gauge invariance it may lead to obstruction to quantization in the form of gauge anomalies). ### The quantized gauge action without matter Following Feynman (1948), one would naively expect the quantum evolution operator to be given by an integral over classical gauge fields of the form $\mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu]\exp\left({i\over \hbar}\mathcal{S}_{\mathrm{class.}}(\mathbf{A})\right),$ where $$\mathcal{S}_{\mathrm{class.}}(\mathbf{A})$$ is the classical action (5). However, as a consequence of gauge invariance, the integrand is constant along gauge orbits (the trajectories obtained by starting from one gauge field and acting on it with all gauge transformations) and thus the integral is not defined. The idea is then to introduce a surface (section) that cuts all gauge orbits once and to restrict the integral to this section with a measure that ensures that all choices of sections are equivalent. This surface is defined by a (Lie algebra valued) constraint of the form $$\mathbf{G}(\mathbf{A},x)=0\ ,$$ known as gauge fixing condition or simply gauge fixing. The appropriate integration measure has been identified by Faddeev and Popov and involves the determinant of a linear operator $$\mathbf{M}(\mathbf{A})$$ defined by $[\mathbf{M}(\mathbf{A})\omega](x) =\Delta_\omega \mathbf{G}(\mathbf{A},x),$ where $$\Delta_\omega \mathbf{G}$$ is the variation of $$\mathbf{G}$$ at first order in $$\omega$$ corresponding to the variation of $$\mathbf{A}_\mu$$ in (6): $\Delta_\omega \mathbf{G}(\mathbf{A},x)=\mathbf{G}(\mathbf{A}-\mathbf{D}\omega,x)-\mathbf{G}(\mathbf{A},x)+O(\\|\omega\\|)^2.$ In relativistic-covariant gauges $$\mathbf{M}(\mathbf{A})$$ is typically a differential operator. For example, in Landau's gauge, $\tag{7} \mathbf{G}(\mathbf{A},x)\equiv\sum_\mu \partial_\mu \mathbf{A}^\mu(x)=0 \ \Rightarrow \ [\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}](x)=-\sum_\mu\partial_\mu {\mathbf D}^\mu \bar{\mathbf{C}}(x)\,.$ It follows from the rules of integration in Grassmann algebras that determinants can be represented by integrals over generators of Grassmann algebras, which are anti-commuting variables (see the section #The origin of BRST symmetry and equation (30) in particular). In gauge theories, this corresponds to introducing two spinless fermion fields $$\mathbf{C}(x)$$ and $$\bar{\mathbf{C}}(x)\ ,$$ often called Faddeev-Popov ghosts, which are matrices belonging to the Lie algebra of $$G\ .$$ Such spinless fermions are unphysical because they violate the spin-statistics theorem. In addition, one introduces a scalar field $$\boldsymbol{\lambda}(x)\ ,$$ again belonging to the Lie algebra, which is used to enforce the gauge condition $$\mathbf{G}(\mathbf{A},x)=0$$ in the field integral (i.e., it acts as a Lagrange multiplier). The evolution of the quantized gauge theory is then described in terms of an integral over four types of fields, $$\mathbf{A}_\mu(x),\boldsymbol{\lambda}(x),\mathbf{C}(x),\bar{\mathbf{C}}(x),$$ of the form $\tag{8} \mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu][\mathrm{d}\boldsymbol{\lambda}][[\mathrm{d}\mathbf{C}\mathrm{d}\bar{\mathbf{C}}]\exp\left( {i\over \hbar}\mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})\right),$ where $$\mathcal{S}_{\mathrm{quant.}}$$ is a local action that reads $\tag{9} \mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\mathcal{S}_{\mathrm{class.}}(\mathbf{A})+\hbar\mathcal{S}_{\mathrm{gauge}} (\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})$ with $\tag{10} \mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x)\right].$ Finally, instead of using a constraint of the form $$\mathbf{G}(\mathbf{A},x)=0\ ,$$ it is often convenient to impose $$\mathbf{G}(\mathbf{A},x)=\mathbf{s}(x)$$ and to integrate over the field $$\mathbf{s}(x)$$ with a Gaussian distribution of width $$\xi\ .$$ This procedure corresponds to extending the field integral to a whole neighbourhood of the gauge section and, after the integration over $$\mathbf{s}(x)\ ,$$ it amounts to adding a term quadratic in $$\boldsymbol{\lambda}$$ to the gauge action, which becomes $\tag{11} \mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x) +{\xi\over 2} \boldsymbol{\lambda}^2(x)\right].$ In the limit $$\xi\rightarrow 0\ ,$$ Landau's gauge fixing and the correspondent gauge action (10) are recovered. ## BRST symmetry of the quantized action ### BRST symmetry Remarkably enough, the quantized action (9), (11), which is no longer gauge-invariant, is invariant, independently of the choice of the gauge condition $$\mathbf{G}(\mathbf{A},x)\ ,$$ under the following transformations: $\tag{12} \mathbf{A}_\mu(x)\mapsto \mathbf{A}_\mu(x)+\varepsilon \Delta \mathbf{A}_\mu(x),\quad \bar{\mathbf{C}}(x)\mapsto \bar{\mathbf{C}}(x)+ \varepsilon\Delta \bar{\mathbf{C}}(x),\quad \mathbf{C}(x)\mapsto \mathbf{C}(x)+ \varepsilon\Delta \mathbf{C}(x),\quad \boldsymbol{\lambda} (x)\mapsto \boldsymbol{\lambda} (x)+\varepsilon \Delta\boldsymbol{\lambda} (x)$ where $$\varepsilon$$ is a Grassmann constant (it thus anticommutes with $$\bar{\mathbf{C}}(x)$$ and $$\bar{\mathbf{C}}(x)\ ,$$ and $$\varepsilon^2=0$$) and $\Delta \mathbf{A}_\mu(x) =-\mathbf{D}_\mu \bar{\mathbf{C}}(x)\,,\quad \Delta \bar{\mathbf{C}}(x)=\bar{\mathbf{C}}^2(x),\quad \Delta\mathbf{C}(x) =\boldsymbol{\lambda} (x)\,,\quad \Delta\boldsymbol{\lambda} (x)=0\,.$ These transformations, in which bosons are transformed into fermions and conversely, are called BRST transformations and reflect the BRST symmetry of the action. Another way of expressing the symmetry is based on the introduction of a functional differential operator. Expanding all fields on a basis of (anti-hermitian) matrices $$\mathbf{t}^{a}$$ generating the Lie algebra of $$G\ ,$$ $\tag{13} \mathbf{A}_\mu(x)=\sum_{a} A^{a}_\mu(x)\mathbf{t}^{a},\quad \bar{\mathbf{C}}(x)=\sum_{a} \bar C^{a}(x)\mathbf{t}^{a}, \quad \mathbf{C}(x)=\sum_{a} C^{a}(x)\mathbf{t}^{a}, \quad \boldsymbol{\lambda} (x)=\sum_{a} \lambda^{a}(x)\mathbf{t}^{a},$ as well as $\Delta \mathbf{A}_\mu(x)=\sum_{a} \Delta A^{a}_\mu(x)\mathbf{t}^{a}, \quad \Delta \bar{\mathbf{C}}(x)= \sum_{a} \Delta\bar C^{a}(x)\mathbf{t}^{a},$ one can write it in the form ($$\delta$$ denotes functional differentiation) $\tag{14} \mathcal{D}=\int\mathrm{d}^4x\sum_{a}\left[\sum_\mu \Delta A^{a}_\mu(x) {\delta\over \delta A^{a}_\mu(x)}+ \Delta\bar C^{a}(x){ \delta\over\delta \bar C^{a}(x)}+\lambda^{a}(x) {\delta\over\delta C^{a}(x)}\right]\,.$ Then, the quantized action satisfies $\tag{15} \mathcal{D}\mathcal{S}_{\mathrm{quant.}}=0\,.$ In the field integral implementation of BRST symmetry, one must also verify the invariance of the integration measure in (8). This leads to the conditions $\tag{16} \sum_{ {a},\mu} {\delta \Delta A^{a}_\mu(x)\over \delta A^{a}_\mu(x)} =0\,,\quad \sum_{a} {\delta\Delta\bar C^{a}(x)\over\delta \bar C^{a}(x)} =0\,,$ that is, that the traces of the generators of the Lie algebra of the group $$G$$ in the adjoint representation (corresponding to the Lie algebra structure constants) must vanish, a property satisfied by compact Lie groups. Note, however, that in presence of matter fields, this condition may apply to generators in other representations and is then only satisfied for semi-simple Lie algebras. One verifies the very important property that the differential operator (14) satisfies $$\mathcal{D}^2=0$$ (nilpotency) so that $$\mathcal{D}$$ can be identified with a cohomology operator. In cohomological terminology, equation (15) states that the quantized action $$\mathcal{S}_{\mathrm{quant.}}$$ is BRST closed. Quantities of the form $$\mathcal{D}\Phi$$ are said to be BRST exact. The nilpotency of $$\mathcal{D}$$ implies that BRST exact quantities are BRST closed. One verifies that $\mathcal{S}_{\mathrm{gauge}}=\mathcal{D}\int\mathrm{d}^4 x\,\mathrm{tr}\,\mathbf{C}(x)\left(\mathbf{G}(\mathbf{A},x)+\textstyle{ {\xi\over2} }\boldsymbol{\lambda} (x)\right)\,,$ that is, that the gauge dependent part of the quantum action is BRST exact. This property is very important: it allows proving that gauge-invariant observables are insensitive to a modification of the gauge fixing function $$\mathbf{G}(\mathbf{A},x)\ ,$$ by using a simple integration by parts in the field integral. However, integration by parts implies acting with $$\mathcal{D}$$ on the left and, thus, commuting derivatives with coefficients in $$\mathcal{D}\ .$$ This commutation yields again the conditions (16), which we have discussed above. Gauge independence, in particular, is essential to prove that physical observables satisfy the unitarity requirement, a property that is not obvious for a quantized gauge theory. ### BRST invariant solutions Without entering into too many details, let us point out that one observation facilitates the construction of general BRST invariant polynomials in the fields. Setting, $\mathcal{D}=\mathcal{D}_+ + \mathcal{D}_-\quad\mathrm{with}\quad \mathcal{D}_+ =\int\mathrm{d}^4x\sum_{a}\left[ \Delta A^{a}_\mu(x) {\delta\over \delta A^{a}_\mu(x)}+ \Delta\bar C^{a}(x){ \delta\over\delta \bar C^{a}(x)}\right],\ \mathcal{D}_-= \int\mathrm{d}^4x\sum_{a}\,\lambda^{a}(x) {\delta\over\delta C^{a}(x)},$ one verifies that $\mathcal{D}_+ ^2=\mathcal{D}_-^2=0\,,\quad \mathcal{D}_+ \mathcal{D}_- + \mathcal{D}_- \mathcal{D}_+=0\,.$ One then uses the property that Grassmann algebras are graded algebras, that $$\mathcal{D}_+$$ increases the degree in $$\bar{\mathbf{C}}$$ while $$\mathcal{D}_-$$ leaves it unchanged, to expand an equation of the form $$\mathcal{D}\Phi=0$$ in powers of $$\bar{\mathbf{C}}\ .$$ ## Master (Zinn-Justin) equation ### Renormalization The initial field integral (8) of the quantized gauge theory is not defined, even in the sense of a perturbative expansion: the perturbative expansion is generated by keeping in the exponential the part of the classical action that is quadratic in the fields (free action) and expanding in a formal power series the exponential of the remaining part. This expansion is usefully described in terms of Feynman diagrams, each one representing a Feynman integral contribution to the perturbative series. In perturbation theory, the field integral (8) is not defined because, at space-time dimension $$d=4\ ,$$ divergent Feynman integrals (one also speaks of UltraViolet, UV divergences) arise at all orders, corresponding to short distance or large momentum singularities. A first necessary step is called regularization, in which one modifies in some unphysical way the classical action to render the expansion finite. The construction of quantum gauge theories and practical calculations are much simplified if the regularization preserves the BRST symmetry, even if, strictly speaking, this requirement is not mandatory. Different methods are available: one can modify the theory at short distance, in the continuum at a scale specified by a cut-off or by introducing a space-time lattice (lattice gauge theories). In theories without chiral fermion fields a BRST invariant regularization can also be achieved by using the so called dimensional regularization, which is based on the formal analytic continuation of Feynman integrals to arbitrary complex values of the space-time dimension $$d\ .$$ When $$d\rightarrow 4$$ poles appear in the dimensionally regularized Feynman integrals: these singularities are related to the initial short distance divergences. Renormalization consists into adding to the initial action so-called counter-terms, that is, space-time integrated monomials in the fields and their derivatives, multiplied by constant coefficients diverging when $$d\to4\ .$$ The coefficients are then fixed order by order in perturbative expansion in such a way that all singularities are cancelled. When the number of different field monomials, required to render finite the perturbative expansion, is bounded at all orders by some fixed number, one calls the quantum field theory (perturbatively) renormalizable by power counting. Gauge theories with properly chosen gauge sections satisfy this criterion, like, for example, Landau's gauge in (7). However, in gauge theories one still has to prove that renormalization can be achieved without spoiling the geometric structure that ensures that physical results do not depend on the choice of the gauge section. Initial proofs of gauge independence of the renormalized gauge theory were based on the Slavnov-Taylor identities, suitably extended to the situation of spontaneous symmetry breaking by Lee-Zinn-Justin. A more general and more transparent proof was then given by Zinn-Justin using BRST symmetry and the Zinn-Justin (ZJ) equation. Note that all these proofs apply to compact Lie groups with semi-simple Lie algebras; the simpler Abelian case has to be dealt with by different methods. Let us point out that if renormalization would preserve the BRST symmetry in the explicit form (12), gauge independence could be proved easily. However, this is not the case because counter-terms have the effect of rescaling fields, for example, $$\mathbf{A}_\mu \mapsto Z^{1/2}_A \mathbf{A}_\mu\ ,$$ where $$Z_A$$ is a divergent constant. Because the gauge transformation of $$\mathbf{A}_\mu$$ is affine, the form of the gauge transformation is modified for the rescaled fields. Other fields are similarly renormalized. Moreover, if the gauge fixing function $$\mathbf{G}(\mathbf{A},x)$$ is not linear in the fields, the gauge fixing equation is also modified and counter-terms quartic in the ghost fields are generated. ### ZJ equation In non-Abelian gauge theories, two BRST variations, $$\Delta\mathbf{A}_\mu$$ and $$\Delta\bar{\mathbf{C}}\ ,$$ are not linear in the dynamical fields. Local polynomials in the fields of degree larger than one are called composite operators. They generate new divergences and require new types of counter-terms. The renormalization of composite operators can be best discussed by introducing source fields that generate, by functional differentiation, their multiple insertions in correlation (or Green's) functions. We denote by $$\mathbf{K}^\mu$$ and $$\mathbf{L}$$ the sources for the $$\Delta\mathbf{A}_\mu$$ and $$\Delta\bar{\mathbf{C}}\ ,$$ respectively. Then, we consider $\mathcal{S}=\mathcal{S}_{\mathrm{quant.}}-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\left(\sum_\mu \mathbf{K}^\mu(x) \Delta\mathbf{A}_\mu(x) +\mathbf{L}(x)\Delta\bar{\mathbf{C}}(x)\right)\,,$ such that $\tag{17} {\delta {\mathcal S}\over \delta \mathbf{K}^\mu{} (x) }=-\hbar\Delta\mathbf{A}_\mu(x),\qquad{\delta {\mathcal S} \over \delta \mathbf{L}(x)}=-\hbar\Delta\bar{\mathbf{C}}(x)\,.$ The source $$\mathbf{K}^\mu(x)$$ is a Grassmann (anticommuting) field while $$\mathbf{L}(x)$$ is a complex field, and both are matrices that belong to the Lie algebra. Using $$\Delta\mathbf{A}_\mu=\mathcal{D}\mathbf{A}\ ,$$ $$\Delta\bar{\mathbf{C}}=\mathcal{D}\bar{\mathbf{C}}\ ,$$ $$\mathcal{D}^2=0$$ and the invariance of the sources under BRST transformation, it follows that $\tag{18} \mathcal{D}\mathcal{S}=0\ .$ We can expand $$\mathbf{K}^\mu(x)$$ and $$\mathbf{L}(x)$$ on the basis of generators of the Lie algebra as in (13). The ZJ equation then follows directly form BRST symmetry of $${\mathcal S}$$ and from relations (17), and can be written in the form $\tag{19} \int \mathrm{d}^4 x \sum_{a} \left( \sum_\mu {\delta {\mathcal S}\over \delta K^\mu{}^{a} (x) } {\delta {\mathcal S}\over \delta A_\mu^{a} (x) } +{\delta {\mathcal S} \over \delta L^{a}(x)}{\delta {\mathcal S}\over \delta \bar C^{a}(x)} -\lambda^{a}(x) {\delta {\mathcal S}\over \delta C^{a}(x)} \right)=0\,.$ To discuss in simpler terms equation (19), it is sometimes convenient to add to $$\mathcal{S}$$ the source for the $$\boldsymbol{\lambda}$$ field$\mathcal{S}\mapsto \mathcal{S}-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\boldsymbol{\lambda} (x)\boldsymbol{\mu}(x)\ .$ Equation (19) then takes the purely quadratic form $\tag{20} \int \mathrm{d}^4 x \sum_{a} \left( \sum_\mu {\delta {\mathcal S}\over \delta K^\mu{}^{a} (x) } {\delta {\mathcal S}\over \delta A_\mu^{a} (x)} +{\delta {\mathcal S} \over \delta L^{a}(x)} {\delta {\mathcal S}\over \delta \bar C^{a}(x)} +{\delta {\mathcal S}\over\delta \mu^{a}(x)} {\delta {\mathcal S}\over \delta C^{a}(x)} \right)=0\,.$ In contrast with equation (15) where BRST transformations (12) are explicit, equations (19), (20) can be proved to be stable under renormalization, that is, the renormalized action $${\mathcal S}_\mathrm{ren.}={\mathcal S}+{\mathcal S}_\mathrm{CT}\ ,$$ sum of the initial quantized action and properly chosen counter-terms, still satisfies equation (19). Its explicit solution, using locality (the action is a space-time integral over functions of fields and derivatives), power counting, which is a form of dimensional analysis, and ghost number conservation (if one assigns a ghost charge $$+1$$ to $$C$$ and $$-1$$ to $$\bar C\ ,$$ the action has total charge 0), yields the general form of the renormalized action. In particular, the renormalized action has still a BRST symmetry but with renormalized fields and parameters. In the example of gauge-fixing functions $$\mathbf{G}$$ linear in the fields, the $$\boldsymbol{\lambda}$$-quantum equation of motion yields additional relations between counterterms. In the simple example of the gauge (7) and in the absence of matter fields, the renormalized action $${\mathcal S}_\mathrm{ren.}$$ is then obtained from $${\mathcal S}$$ by the simple substitutions $e\mapsto Z_e^{1/2} e\,,\quad\mathbf{A}_\mu \mapsto Z^{1/2}_A \mathbf{A}_\mu\,,\quad \mathbf{C}\bar{\mathbf{C}} \mapsto Z_C \mathbf{C}\bar{\mathbf{C}},\quad \xi\mapsto Z_A \xi\,.$ (Since only the product $$\mathbf{C}\bar{\mathbf{C}}$$ always appears, only the renormalization of $$\mathbf{C}\bar{\mathbf{C}}$$ is defined.) By contrast, when the gauge-fixing function $$\mathbf{G}$$ is not linear in the fields, renormalization generates terms quartic in the ghost fields and, thus, the integration over $$\mathbf{C}$$ and $$\bar{\mathbf{C}}$$ no longer yields a simple determinant. Nevertheless, the ZJ equation still implies gauge independence (i.e., independence of the gauge section) of physical quantities and, thus, the field theory has the same physical properties. The ZJ equation can also be used to discuss the renormalization properties of gauge-invariant operators, which are related to physical observables. Finally, note that a (simpler) quadratic equation somewhat analogous to (19), (20) appears in the renormalization of the non-linear sigma model, a quantum field theory renormalizable in 1+1 space-time dimensions with an $$O(N)$$ orthogonal symmetry, in which the field belongs to a sphere $$S_{N-1}\ .$$ ## A few properties What distinguishes most the ZJ equation (20) (or equation (19)) from equation (18) is its quadratic structure, because the transformations of fields depend on the action itself. Thus, several properties of the equation can be understood by studying the more general equation $\tag{21} \int\mathrm{d}^4 x\,\mathrm{tr}\left({\delta \mathcal {S}\over\delta \mathbf{Q}(x)} {\delta \mathcal{S}\over\delta \mathbf{\Pi}(x)}\right)=0\,,$ where, with respect to the preceding section, the field $$\mathbf{Q}(x)$$ plays the role of the set of commuting fields $$\{\mathbf{A}_\mu,\mathbf{L},\boldsymbol{\mu}\}$$ and the field $$\mathbf{\Pi}(x)$$ the role of the set of anticommuting fields $$\{\mathbf{K}_\mu,\bar{\mathbf{C}},\mathbf{C}\}\ .$$ For notational simplicity, we replace equation (21) by a formally identical but simpler equation; the generalization to field theory is then straightforward. We assume that $$S$$ is a smooth function of $$N$$ real variables $$q_i$$ and $$N$$ generators $$\pi_i$$ of a Grassmann algebra$\pi_i\pi_j+\pi_j\pi_i=0\ ,$ which belongs to the commuting subalgebra and satisfies the equation $\tag{22} \sum_i{\partial S\over \partial q_i}{\partial S\over \partial \pi_i} =0\,.$ The index $$i$$ plays the role of space-time coordinate together with Lorentz and group indices in equation (21). Summation over $$i$$ replaces integration and summation. We also consider the differential operator $\tag{23} \mathcal{D} =\sum_i \left({\partial \mathcal{S} \over \partial \pi_i}{\partial \over \partial q_i}+{\partial {\mathcal S} \over \partial q_i}{\partial \over \partial \pi_i}\right)\,.$ One verifies that equation (22) is the necessary and sufficient condition for $$\mathcal{D}$$ to be a BRST cohomology operator, that is, for $$\mathcal{D}$$ to satisfy $$\mathcal{D}^2=0\ .$$ ### Special solutions It is possible to characterize all solutions of equation (22) of the special form $\mathcal{S}=\Sigma^{(0)}(q)+\sum_{i,j}\Sigma^{( 2)}_{ij}(q)\pi_i\pi_j \,,$ where $$\Sigma^{(0)}$$ and $$\Sigma^{( 2)} _{ij}=-\Sigma^{( 2)}_{ji}$$ are analytic functions of the $$q_i$$s. Equation (22) is equivalent to a system of two equations, corresponding to the vanishing of the terms of degree one and three in the generators $$\pi_i$$ in (22). The first equation, coming from the linear term, can be more easily expressed by introducing the differential operator $\mathrm{d}_i:=\sum_k \Sigma^{( 2)}_{ki}(q){\partial\over\partial q_k}\,.$ It then takes the form $\tag{24} \mathrm{d}_i \Sigma^{(0)}(q)=0\quad \forall\, i\,.$ The second equation takes the form $\tag{25} \sum_l \left\{ {\partial \Sigma^{( 2)}_{jk}\over\partial q_l}\Sigma^{( 2)}_{li}\right\}_{ijk}=0 \quad \forall \,i,j,k\,,$ where the notation means antisymmetrized over $$ijk\ .$$ From the latter equation, one derives the commutation relation $\tag{26} [\mathrm{d}_i,\mathrm{d}_j]=\sum_k {\partial\Sigma^{( 2)}_{ij} \over\partial q_k}\mathrm{d}_k\,.$ Equation (26) is the compatibility condition for the linear differential system (24). It also implies that the operators $$\mathrm{d}_i$$ are the generators of a Lie algebra in some non-linear representation. Finally, if $$\Sigma^{( 2)}_{ij}$$ is a first degree polynomial, $${\partial\Sigma^{( 2)}_{ij} \over\partial q_k}$$ are the structure constants of the Lie algebra and equation (Figure ) contains the corresponding Jacobi identity. ### Perturbative solutions We assume that we have found a solution $$\mathcal{S}^{(0)}$$ of equation (22), to which is associated a BRST operator $$\mathcal{D}_0$$ like in (23), and we look for solutions that can be expanded in terms of a real parameter $$\kappa$$ in the form $\mathcal{S}(\kappa)=\sum_{n \ge0}\kappa^n \mathcal{S}^{(n)} .$ Expanding equation (22) at order $$\kappa\ ,$$ one obtains the condition $$\mathcal{D}_0 \mathcal{S}_1=0\ .$$ Thus, one has to find $$\mathcal{D}_0$$ closed solutions. More generally, at order $$\kappa^n$$ the equation can be written as $\mathcal{D}_0\mathcal{S}^{(n)}=-\sum_{1\le m\le n-1}\sum_i\left({\partial\mathcal{S}^{(m)}\over\partial q_i}{\partial\mathcal{S}^{(n-m)}\over\partial \pi_i}\right).$ This reduces the problem of the recursive determination of the coefficients $$\mathcal{S}^{(n)}$$to an investigation of the properties of the $$\mathcal{D}_0$$ operator. ### Canonical invariance Equation (22) has properties reminiscent of those of the symplectic form $$\mathrm{d} p\wedge \mathrm{d q}$$ of classical mechanics; in particular it is invariant under some generalized canonical transformations. Indeed, after the change of variables $$(\pi ,q ) \mapsto (\pi ' ,q' )\ ,$$ $\tag{27} q_i = {\partial \varphi \over \partial \pi_i} (\pi,q' ), \quad \pi'_i = {\partial \varphi \over \partial q '_i} (\pi, q' ),$ in which $$\varphi (\pi, q' )$$ is a function belonging to the anticommuting part of the Grassmann algebra, one recovers equation (22) in the new variables: $\sum_i { \partial \mathcal{ S} \over \partial \pi '_i}{\partial \mathcal{ S} \over \partial q '_{i}}=0\,.$ The proof goes in two steps, which both involve the anticommutation of $$\pi_i$$ and $$\pi_j$$ or of the corresponding derivatives. One first goes from $$q_i$$ to $$q'_i$$ at $$\pi_i$$ fixed. One finds $\left.{\partial S\over\partial q'_i}\right\|_\pi=\sum_j {\partial q_j \over \partial q'_i}\left.{\partial S\over \partial q_j}\right\|_\pi=\sum_j {\partial^2 \varphi \over\partial q'_i \partial \pi_j} \left.{\partial S\over\partial q_j}\right\|_\pi\,,\quad \left.{\partial S\over \partial \pi_i}\right\|_{q'}=\left.{\partial S\over \partial \pi_i}\right\|_{q}\,.$ Then one changes from $$\pi_i$$ to $$\pi'_i\ :$$ $\left.{\partial S\over\partial \pi_i}\right\|_{q'}=\sum_j {\partial^2\varphi \over \partial \pi_i\partial q'_j}\left.{\partial S \over\partial\pi'_j}\right\|_{q'}\,,\quad \left.{\partial S\over \partial q'_i}\right\|_\pi =\left.{\partial S\over \partial q'_i}\right\|_{\pi'}+\sum_j {\partial^2 \varphi \over \partial \pi_i\partial q'_j}\left.{\partial S \over \partial \pi'_j}\right\|_{q'}\,.$ Collecting all terms, one verifies the property. ### Infinitesimal canonical transformations We now consider infinitesimal canonical transformations of type (27). The function $$\varphi(\pi ,q' ) =\sum_i\pi_iq'_i$$ corresponds to the identity. We then write the function $$\varphi$$ in terms of a real parameter $$\kappa$$ as $\varphi(\pi ,q' ) =\sum_i\pi_{i}q'_{i}+\kappa \psi (\pi ,q' ).$ The variation of $$S$$ at first order in $$\kappa$$ is $\mathcal {S} (\pi ' ,q' )-\mathcal {S} (\pi ,q )=\kappa\sum_i \left({ \partial \psi \over \partial q _i}{\partial \mathcal {S} \over \partial \pi_i}- { \partial \psi\over \partial \pi_i}{\partial \mathcal {S} \over \partial q_i}\right)+O\left(\kappa^2\right)=-\kappa\, \mathcal {D}\psi +O\left(\kappa^2\right)\, .$ One thus finds that an infinitesimal canonical transformation generates a BRST exact contribution and, conversely, any infinitesimal addition to $$\mathcal{S}$$ of a BRST exact term can be generated by a canonical transformation acting on $$\mathcal{S}\ .$$ One then verifies that, indeed, the additional contributions to the action due to the gauge-fixing procedure can also be generated by such a canonical transformation with $\kappa\, \psi (\pi ,q' )\mapsto \int\mathrm{d}^4x\,\mathrm{tr}\,\mathbf{C}(x)\left(\mathbf{G}(\mathbf{A},x)+\textstyle{ {1\over2} }\lambda(x)\right),$ acting on $\mathcal{S}_{\mathrm{class.}}(\mathbf{A}_\mu)-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\left(\sum_\mu \mathbf{K}^\mu(x) \Delta\mathbf{A}_\mu(x) +\mathbf{L}(x)\Delta\bar{\mathbf{C}}(x)+\boldsymbol{\mu}(x)\boldsymbol{\lambda} (x)\right).$ ## The origin of BRST symmetry One might be surprised that quantized gauge theories have this peculiar BRST symmetry. In fact, BRST symmetry is an automatic property of constraint systems handled in a specific way as we explain now. In particular, in gauge theories it is induced by the constraint of the gauge section (7) but its form is complicated by the choice of coordinates because the equation of the section applies to group elements $$\mathbf{g}(x)\ ,$$ the gauge transformations. Let $$\varphi^{a}$$ be a set of real quantities satisfying a system of real equations, $\tag{28} E_{a} (\varphi)=0\,,$ where the functions $$E_{a} (\varphi)$$ are smooth, and $$E_{a}=E_{a} (\varphi)$$ is a one-to-one map in some neighbourhood of $$E_{a}=0\ ,$$ which can be inverted in $$\varphi^{a}=\varphi^{a}(E).$$ In particular, this implies that equation (28) has a unique solution $$\varphi_{\rm s}^{a}\equiv \varphi^{a}(0)\ .$$ In the neighbourhood of $$\varphi_{\rm s}\ ,$$ the determinant $$\det\mathbf{E}$$ of the matrix $$\mathbf{ E}$$ with elements $E_{ {a}{b}}\equiv {\partial E_{a} \over\partial \varphi^{b} } \,,$ does not vanish and thus we choose $$E_{a}(\varphi)$$ such that it is positive. For any continuous function $$F(\varphi)$$ we now derive a formal expression for $$F (\varphi_{ {\mathrm s} } )$$ that does not involve solving equation (28) explicitly. We start from the simple identity $F(\varphi_{\mathrm{s}} )=\int\left\{ \prod_{ {a} } \mathrm{d} E_{a} \, \delta ( E_{a} ) \right\} F\bigl(\varphi(E)\bigr) ,$ where $$\delta(E)$$ is Dirac's $$\delta\hbox{-function}\ .$$ We then change variables $$E \mapsto \varphi\ .$$ This generates the Jacobian $$\mathcal{ J}(\varphi)=\det\mathbf{ E}>0\ .$$ Thus, $\tag{29} F(\varphi_{\mathrm{s}} )=\int\left\{ \prod_{ {a} } \mathrm{d} \varphi^{a} \, \delta \left[ E_{a}(\varphi) \right] \right\} \mathcal{ J}(\varphi)\, F(\varphi).$ We replace the $$\delta$$-function by its Fourier representation: $\prod_{ {a} }\delta \left[E_{a} (\varphi)\right] = \int \prod_{ {a} }{\mathrm{d}\bar\varphi^{a} \over 2i\pi} \exp\left(-\sum_{a}\bar\varphi^{ {a} } E_{a}(\varphi)\right),$ where the $$\bar\varphi$$ integration runs along the imaginary axis. Moreover, a determinant can be written as an integral over Grassmann variables (i.e., generators of a Grassmann or exterior algebra) $$\bar c^{a}$$ and $$c^{ {a} }$$ in the form $\tag{30} \det\mathbf{ E}= \int \prod_{ {a} } \left(\mathrm{d} \bar c^{a} \mathrm{d} c^{ {a} } \right)\exp\left(\sum_{ {a},{b} }c^{ {a} }E_{ {a}{b} } \bar c^{ {b} }\right).$ Expression (29) then becomes $\tag{31} F(\varphi_{\rm s}) =\mathcal{N} \int \prod_{ {a} } \left(\mathrm{d} \varphi^{a}\mathrm{d} \bar\varphi^{ {a} }\mathrm{d} \bar c^{a} \mathrm{d} c^{ {a} } \right)F(\varphi) \exp\left[-S (\varphi,\bar\varphi, c,\bar c)\right],$ in which $$\mathcal{N}$$ is a constant normalization factor and $\tag{32} S (\varphi ,\bar\varphi ,c ,\bar c)=\sum_{ {a} }\bar\varphi^{ {a} }E_{a} (\varphi)-\sum_{ {a},{b} } c^{ {a} }E_{ {a}{b} }(\varphi) \bar c^{b} \,,$ is a commuting element of the Grassmann algebra. Somewhat surprisingly, the function $$S$$ has a new type of symmetry, the BRST symmetry that we now describe. ### BRST symmetry The function $$S$$ defined by equation (32) is invariant under the BRST transformations $\tag{33} \varphi^{a} \mapsto \varphi^{a}+\delta_{\rm BRST}\varphi^{a} \ \mathrm{with}\ \delta_{\rm BRST}\varphi^{a}=\varepsilon \bar c^{a}, \quad \bar c^{a} \mapsto \bar c^{a} \,,$ and $c^{a}\mapsto c^{a} +\delta_{\rm BRST}c^a\ \mathrm{with}\ \delta_{\rm BRST}c^a= \varepsilon\bar\varphi^{a}\, ,\quad \bar\varphi^{a} \mapsto \bar\varphi^{a} ,$ where $$\varepsilon$$ is an anticommuting constant, an additional generator of the Grassmann algebra such that $$\varepsilon^{2}=0\, ,\quad \varepsilon \bar c^{a} + \bar c^{a} \varepsilon =0 \, ,\quad \varepsilon c^{ {a} }+ \varepsilon c^{ {a} }=0 \ .$$ Moreover, the integration measure in (31) is also invariant. The BRST transformation is clearly nilpotent (of vanishing square) since the variation of the variation always vanishes. Note that the transformations of $$c^{a}$$ and $$\bar\varphi^{a}$$ are identical to the transformations of the fields $$\mathbf{C}(x)$$ and $$\boldsymbol{\lambda}(x)$$ in (12). The BRST transformation can also be represented by a Grassmann differential operator $$\mathcal{D}$$ acting on functions of $$\{\varphi,\bar\varphi,c,\bar c\}\ :$$ $\mathcal{D} = \mathcal{D}_+ +\mathcal{D}_-\,,\quad \mathcal{ D}_+=\sum_{a} \bar c^{a} {\partial \over \partial \varphi^{a} }\,,\quad \mathcal{ D}_-=\sum_{a} \bar\varphi^{ {a} }{\partial \over \partial c^{ {a} } }\, .$ Then, $\mathcal{D}S=0\,.$ One verifies immediately that $\mathcal{D}_+ ^2=\mathcal{D}_-^2= \mathcal{D}_+ \mathcal{D}_- + \mathcal{D}_- \mathcal{D}_+ =0\,.$ The nilpotency of the BRST transformation follows: $\tag{34} \mathcal{ D}^2=0\,.$ The differential operator $$\mathcal{ D}$$ is a cohomology operator, generalization of the exterior differentiation of differential forms. In particular, the first term $$\mathcal{D}_+$$ in the BRST operator is identical to the exterior derivative of differential forms in a formalism in which the Grassmann variables $$\bar c^{a}$$ generate the corresponding exterior algebra. Equation (34) implies that all quantities of the form $$\mathcal{ D} Q(\varphi ,\bar\varphi ,c,\bar c)\ ,$$ (BRST exact), are BRST closed since $$\mathcal{ D} ( \mathcal{ D} Q(\varphi ,\bar\varphi ,c,\bar c))=0\ .$$ One verifies that the function $$S$$ (defined in equation (32)) is not only BRST closed but also BRST exact: $S=\mathcal{ D} \left(\sum_{a} c^{ {a} }E_{a} (\varphi) \right).$ The reciprocal property, the meaning and implications of the BRST symmetry follow from some simple arguments based on BRST cohomology. ### BRST symmetry and group elements We now assume that the variables $$\varphi^{a}$$ parametrize locally elements $$\mathbf{g}(\varphi)$$ of a group $$G$$ in some matrix representation. It is then natural to parametrize the BRST variation of $$\mathbf{g}$$ in terms of an element $$\bar{\mathbf{C}}$$ of the Lie algebra (being also a generator of a Grassmann algebra) in the form $\tag{35} \delta_{\rm BRST}\mathbf{g}=\varepsilon\,\bar{\mathbf{C}}\,\mathbf{g}\,.$ Calculating directly the variation of $$\mathbf{g}$$ from the variation (33) of $$\varphi^a\ ,$$ one obtains the relation $\bar{\mathbf{C}}\,\mathbf{g}=\sum_a {\partial\mathbf{g}\over\partial\varphi^a}\bar c^a\ \Rightarrow\ \bar{\mathbf{C}}=\sum_a {\partial\mathbf{g}\over\partial\varphi^a}\mathbf{g}^{-1}\bar c^a \,.$ The BRST variation of the matrix $$\bar{\mathbf{C}}$$ is $\delta_{\rm BRST}\bar{\mathbf{C}}=\varepsilon\sum_{a,b}\left( {\partial^2\mathbf{g}\over\partial\varphi^a\partial\varphi^b} \mathbf{g}^{-1} -{\partial\mathbf{g}\over\partial\varphi^a}\mathbf{g}^{-1}{\partial\mathbf{g}\over\partial\varphi^b}\mathbf{g}^{-1}\right)\bar{c}^b \bar{c}^a=\varepsilon\,\bar{\mathbf{C}}^2 ,$ where the anticommutation of $$\bar c^a$$ and $$\bar c^b$$ has been used. One recognizes the transformation of the ghost fields in non-Abelian gauge theories (second equation in (12)). Applying then the transformation (35) to the $$\mathbf{g}(x)$$ in (2), one can derive the first equation in (12). ## References • Barnich, G; Brandt, F and Henneaux, M (2000). Local BRST cohomology in gauge theories. Physics Reports 338: 439-569. • Barnich, G and Henneaux, M (1994). Renormalization of gauge invariant operators and anomalies in Yang-Mills theory. Physical Review Letters 72: 1588-1591. • Becchi, C; Rouet, A and Stora, R (1974). The Abelian Higgs-Kibble Model. Unitarity of the S Operator. Physics Letters B52: 344-346. • Becchi, C; Rouet, A and Stora, R (1975). Renormalization of the Abelian Higgs-Kibble Model. Communications in Mathematical Physics 42: 127-162. • Becchi, C; Rouet, A and Stora, R (1976). Renormalization of gauge theories. Annals of Physics 98: 287-321. • Brézin, E; Le Guillou, J C and Zinn-Justin, J (1976). Renormalization of the nonlinear $$\sigma$$ model in $$2+\epsilon$$ dimensions. Physical Review D14: 2615-2621. • Faddeev, L D and Popov, V N (1967). Feynman Diagrams for the Yang-Mills Field. Physics Letters B25: 29-30. • Feynman, R P (1948). Space-Time Approach to Non-Relativistic Quantum Mechanics. Review of Modern Physics 20(2): 367-387. • Kluberg-Stern, H and Zuber, J-B (1975). Ward Identities and Some Clues to the Renormalization of Gauge Invariant Operators. Physical Review D12: 467-481. • Lee, B W and Zinn-Justin, J (1972). Spontaneously Broken Gauge Symmetries I,II,III. Physical Review D5: 3121-3137, 3137-3155,3155-3160. • Lee, B W and Zinn-Justin, J (1973). Spontaneously Broken Gauge Symmetries IV. General Gauge Formulation. Physical Review D7: 1049-1056. • Slavnov, A A (1972). Ward identities in gauge theories. Theoretical and Mathematical Physics 10: 99-104. • Taylor, J C (1971). Ward identities and charge renormalization of the Yang-Mills field. Nuclear Physics B33: 436-444. • Tyutin, I (1975). Preprint of Lebedev Physical Institute, 39, 1975. • Weinberg, S (1996). The quantum theory of fields. Vol. 2: Modern Applications. Cambridge University Press, Cambridge. ISBN 0521550025 • Zinn-Justin, J (1975). Renormalization of gauge theories, Bonn lectures 1974, published in Trends in Elementary Particle Physics, Lecture Notes in Physics 37 pages 1-39, H. Rollnik and K. Dietz eds., Springer Verlag, Berlin. ISBN 978-3-540-07160-0 • Zinn-Justin J (1975) in Proc. of the 12th School of Theoretical Physics, Karpacz, Acta Universitatis Wratislaviensis 368. • Zinn-Justin, J (1984). Renormalization of gauge theories: Non-linear gauges. Nuclear Physics B246: 246-252. • Zinn-Justin, J (1999). Renormalization of gauge theories and master equation. Modern Physics Letters A14: 1227-1235. Internal references • Carlo Maria Becchi and Camillo Imbimbo (2008) Becchi-Rouet-Stora-Tyutin symmetry. Scholarpedia, 3(10):7135. • Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287. • Gerard ′t Hooft (2008) Gauge theories. Scholarpedia, 3(12):7443. • Andrei A. Slavnov (2008) Slavnov-Taylor identities. Scholarpedia, 3(10):7119. ## Further reading • Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. (2nd edition). Addison-Wesley Publishing Company, T. ISBN 0201524724. • Itzykson, C and Zuber, J B (2006). Quantum Field Theory. Dover Publications, New York. ISBN 0486445682 • Lai, C H ed. (1981). Gauge Theory of Weak and Electromagnetic Interactions. World Scientific Publishing, Singapore. ISBN 978-9971830236 • Weinberg, S (1996). The quantum theory of fields. Vol. 2: Modern Applications. Cambridge University Press, Cambridge. ISBN 0521550025 • Zinn-Justin, J (2002). Quantum Field Theory and Critical Phenomena (4th edition). Oxford University Press, Oxford. ISBN 0198509235 ## See also Gauge invariance, Gauge theories, Slavnov-Taylor identities, Becchi-Rouet-Stora-Tyutin symmetry	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 69, "mathjax_display_tex": 213, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8824594616889954, "perplexity_flag": "head"}
http://math.stackexchange.com/questions/231137/polynomial-orthogonal-complement/231170	# Polynomial Orthogonal Complement Let $V = \mathbb{P^4}$ denote the space of quartic polynomials, with the $L^2$ inner product $$\langle p,q \rangle = \int^1_{-1} p(x)q(x)dx.$$ Let $W = \mathbb{P^2}$ be the subspace of quadratic polynomials. Find a basis for and the dimension of $W^{\perp}$. The answer is $$t^3 - \frac{3}{5}t, t^4 - \frac{6}{7}t^2 + \frac{3}{35}; dim (W^{\perp}) =2$$ How did they get that? - "Space of quartic polynomials"? Do you mean the space of (real, complex, rational...) polynomials of degree $\,\leq 4\,$ and zero? – DonAntonio Nov 6 '12 at 2:57 @DonAntonio yes i do.. – diimension Nov 6 '12 at 3:01 Which one? Real? Complex? – EuYu Nov 6 '12 at 3:01 @EuYu it doesnt specify but im sure is real – diimension Nov 6 '12 at 3:06 1 It must be real or there would be a conjugate inside the inner product... – copper.hat Nov 6 '12 at 5:06 ## 4 Answers Well, for a finite dimensional vector space we have $\dim W + \dim W^\perp = \dim V$ so that covers the dimension part. For the basis of the orthogonal complement, we have $$\int_{-1}^1 ax^4 + bx^3 + cx^2 + dx + e\ dx = 0$$ $$\int_{-1}^1 x(ax^4 + bx^3 + cx^2 + dx + e)\ dx = 0$$ $$\int_{-1}^1x^2(ax^4 + bx^3 + cx^2 + dx + e)\ dx = 0$$ Because the standard basis of $\mathbb{P}^2$ must satisfy the orthogonality conditions. Therefore we get $$\frac{a}{5} + \frac{c}{3} + e = 0$$ $$\frac{b}{5} + \frac{d}{3} = 0$$ $$\frac{a}{7} + \frac{c}{5} + \frac{e}{3} = 0$$ Solving this system yields $$a = \frac{35}{3}e,\ \ b=-\frac{5}{3}d,\ \ c=-10e$$ with two parameters to vary. Your solutions follows by taking $(a=0,\ b=1)$ and $(a=1,\ b=0)$ respectively. - Thank you, but I am a bit confused I see how this is an orthogonal complement basis but how does it relate to $t^3 - \frac{3}{5}t, t^4 - \frac{6}{7}t^2 + \frac{3}{35}$?? – diimension Nov 6 '12 at 3:22 1 Take $a=0$ and $b=1$ and solve for $c,\ d,\ e$. See what you get. Remember $a,\ b,\ c,\ d,\ e$ are the coefficients of the standard basis. – EuYu Nov 6 '12 at 3:25 Well, for example choose, as written clearly in the above answer, $$a=1\,,\,b=0\Longrightarrow e=\frac{3}{35}\,,\,d=0\,,\,c=-10e=-\frac{6}{7}\Longrightarrow\, \text{one of the pol's is }\, x^4-\frac{6}{7}x^2+\frac{3}{35}$$ and etc. – DonAntonio Nov 6 '12 at 3:26 Yup, I understand now with you guys help! One last question, how did you know that a = 1, b = 0 , and vice versa? – diimension Nov 6 '12 at 3:31 Well, the coefficients kinda have to match. Both your polynomials are monic so those are the natural choices. – EuYu Nov 6 '12 at 3:34 show 1 more comment Well, one way would be to use Gram-Schmidt to produce an orthogonal basis, starting with the basis $1$, $t$, $t^2$, $t^3$, $t^4$. The result will be five polynomials $p_r(x)$ for $r=0,\ldots,4$ where $p_r$ has degree $r$. So $p_0$, $p_1$ and $p_2$ will span the space of quadratic polynomials and $p_3$ and $p_4$ will span a 2-dimensional space orthogonal to the quadratics. Since we know that the dimension of $W^\perp$ is two (because $\mathrm{dim}(W)+\mathrm{dim}(W^\perp)=5$), we see that $p_3$ and $p_4$ are a basis for $W^\perp$. - Thank you for explaining that. – diimension Nov 6 '12 at 3:23 Let $$p(x):=ax^4+bx^3+cx^2+dx+e\in W^\perp$$ Since $W:=\operatorname{Span}\{1,x,x^2\}\,$ , we get: $$(1)\;\;\;\;\;\;0=\langle\,p\,,\,1\,\rangle=\int_{-1}^1p(x)dx=\frac{2}{5}a+\frac{2}{3}c+2e$$ $$(2)\;\;\;\;\;\;\;\;\;\;\;0=\langle\,p\,,\,x\,\rangle=\int_{-1}^1xp(x)dx=\frac{2}{5}b+\frac{2}{3}d$$ $$(3)\;\;\;\;\;\;\;\;0=\langle\,p\,,\,x^2\,\rangle=\int_{-1}^1x^2p(x)dx=\frac{2}{7}a+\frac{2}{5}c+\frac{2}{3}e$$ The above relies on the easy results that the integral on a symmetric (above zero) interval of an even function is twice the value of its primitive on either of the two limits, whereas the same integral of an odd function is zero. Now solve the above linear system. - Thank you, but I am a bit confused I see how this is an orthogonal complement basis but how does it relate to $t^3 - \frac{3}{5}t, t^4 - \frac{6}{7}t^2 + \frac{3}{35}$?? – diimension Nov 6 '12 at 3:22 1 Read my comment below Euyu's answer: it applies exactly the same here. – DonAntonio Nov 6 '12 at 3:27 I understand now with guys help. Thank you very much , DonAntonio! – diimension Nov 6 '12 at 3:35 Let $v_k(x) = x^k$, $k=0,...,4$. Then $v_k$ is a basis for $\mathbb{P}^4$. (To see this note that any quartic can be written in terms of $v_k$, and if $\sum \alpha_k v_k = 0$, then by differentiating and evaluating at $x=0$ we can see that $\alpha_k = 0$, hence they are linearly independent.) By the same token, $v_k$, $k=0,1,2$ is a basis for $\mathbb{P}^2$. It follows that $\dim \mathbb{P}^2 = 3$, and since $\mathbb{P}^4 = \mathbb{P}^2 \oplus (\mathbb{P}^2)^\bot$, we see that $\dim (\mathbb{P}^2)^\bot = 2$. We can find a basis for $(\mathbb{P}^2)^\bot$ by projecting the $v_k$ onto $(\mathbb{P}^2)^\bot$. Clearly $v_k$, $k=0,1,2$ will project to zero. So the only remaining basis elements that need to be projected are $v_3,v_4$. Note in passing that $\langle v_j, v_k \rangle = \frac{1}{j+k+1}(1-(-1)^{j+k+1})$. To compute the projection of $x$ onto $(\mathbb{P}^2)^\bot$, we need to determine $\alpha \in \mathbb{R}^3$ such that $\langle x-\sum_{k=0}^2 \alpha_k v_k, v_j \rangle = 0$ for $j=0,1,2$. This is just the linear system $\langle x, v_j \rangle = \langle \sum_{k=0}^2 \alpha_k v_k, v_j \rangle$, or $$\begin{bmatrix} \langle v_0, v_0 \rangle & \langle v_1, v_0 \rangle & \langle v_2, v_0 \rangle \\ \langle v_0, v_1 \rangle & \langle v_1, v_1 \rangle & \langle v_2, v_1 \rangle \\ \langle v_0, v_2 \rangle & \langle v_1, v_2 \rangle & \langle v_2, v_2 \rangle \end{bmatrix} \alpha = \begin{bmatrix} \langle x, v_0 \rangle \\ \langle x, v_1 \rangle \\ \langle x, v_2 \rangle \end{bmatrix}$$ Grinding through the computations gives $\alpha = \frac{1}{5} (0,3,0)^T$ when $x=v_3$ and $\alpha = \frac{1}{35} (-3, 0, 30)^T$ when $x=v_4$. Hence a basis for $(\mathbb{P}^2)^\bot$ is $x \mapsto x^3-\frac{3}{5}x$, $x \mapsto x^4+\frac{3}{35}-\frac{6}{7}x^2$. - Wow, thank you very much! So there are many ways to do it, instead of just taking the $L^2$ inner product, i could have instead used gram matrix. Thank you for pointing that out , Copper.hat – diimension Nov 6 '12 at 4:00 1 You are welcome. Note that you are still taking the inner products, they just happen to be easy to compute for my particular choice of basis, $v_k$. – copper.hat Nov 6 '12 at 5:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 71, "mathjax_display_tex": 15, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9104325771331787, "perplexity_flag": "head"}
http://physics.stackexchange.com/questions/53999/industry-application-of-computational-quantum-mechanics/54000	# Industry application of computational quantum mechanics? I was wondering if anybody knew of an industry application of computational quantum mechanics. For example, the efficient placement of circuit elements on a PCB is in part motivated by classical FDTD simulations. I can make a long list of applications of classical methods and simulation, but I can't think of one where quantum mechanics is used. Additionally, I know of quantum mechanics simulators and their research applications, but I feel they are largely used to derive or very more succinct theories. What problems do people use computational quantum mechanics for? - What scale are you interested in? Single components or complete circuits? One of my professors often brings up tunneling junctions. – Michael Brown Feb 15 at 4:00 ## 1 Answer Computational quantum chemistry is one. Researchers in pharmaceutics use computational quantum chemistry programs to model the interactions of small molecules (drugs or fragments of drugs) with proteins/DNA and predict whether or not the designed drug may or may not be effective for its purpose. They can do that before having to spend time and money synthesizing and testing dozens of drugs which may not have a chance to work at all. A rather brief but nice explanation of how of computational quantum chemistry is applied in practice may be found in this website http://www.ccl.net/cca/documents/dyoung/topics-orig/contents.html. --EDIT-- (I didn't expected this to become an accepted answer, just a contribution. Since having just one industrial application for computational quantum mechanics looks a bit sad, I'll add some others from what I know.) The area I am working in is computational quantum chemistry so I know some more applications related to this. Computational quantum chemistry has become so widely used that developing software for it has become and industry by itself; examples are Schrodinger Inc. and Gaussian Inc. (just Google it). It is not only used in pharmaceutics, but also in the general chemistry industry. I know of big chemical companies that use similar simulations to the ones used in pharmaceutics, but for studying and developing catalysts rather than drugs. Computational quantum mechanics methods for periodic systems are also widely used in material science. Density Funtional Theory methods are among the most popular. In the chemical industry, this is also used for the study and development of catalysts (catalysts can also be extended systems) as well as materials which can adsorb toxic or greenhouse gases such as CO or CO$_2$. Closely related to computational quantum chemistry (or at least, the equations to solve and methods are similar) would be computational nuclear physics. I do not know to what extent do these methods are used in the nuclear energy industry, but I know the US Department of Energy funds this kind of research so I guess it must have some applications, although I am not qualified to develop more on this. On those lines, DoE also funds computational chemistry projects which can help develop alternative fuels. Unfortunately, I am not aware of the kind of applications in electronics that you are interested in. However, I consider myself an ignorant in this area and it is quite possible that there may be a few applications out there. - Sounds like there aren't many. – Mikhail Feb 22 at 0:35 @Mikhail Well, I intended my answer to be only a contribution and expected people would come up with many and compile them. Anyway, I edited my answer so that now it has some more applications I know of. – Goku Feb 22 at 4:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9503507614135742, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/60227/chern-character-of-the-index-bundle-for-a-family-of-dirac-operators/60309	## Chern character of the index bundle for a family of Dirac operators ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Suppose we have a family of compact oriented even dimensional spin manifolds ${Y_x}$ parameterized by a compact even dimensional manifold $X$. The $Y_x$'s are all diffeomorphic to some $Y$, of dimension $n$, and fit together to form a fiber bundle $\pi : Z \rightarrow X$ with fiber $Y_x=\pi ^{-1}(x)$. $TZ$ has the subbundle $V:=\text{ker }\pi_$ which is tangent to the fibers. There may be a family of coefficient bundles also and we obtain a family of twisted Dirac operators $D_x:\Gamma(S^+_x\otimes E_x)\rightarrow \Gamma (S^-_x\otimes E_x)$. The index of the family gives rise to an element $\text{ind} D \in K(X)$, which is the virtual vector bundle $[\text{ker } D_x]-[\text{coker }D_x]$ when the dimension of both spaces are constant. Finally, there is a map $\text{H} ^{}(Z,\mathbb{R})\rightarrow \text{H} ^{*-n}(X,\mathbb{R})$ known as the Gysin homomorphism or integration over the fibers map. We'll use the latter terminology writing the map $\int_Y$ and regarding cohomology classes as living in de Rham cohomology. The Atiyah-Singer index theorem gives $$\text{ch }(\text{ind } D)= \int _Y \hat A (V) \text{ch}(E)$$ What general results exist regarding the components of the Chern character of the index bundle, or equivalently the results of the integration over the fibers map, for twisted Dirac operators? To illustrate, an immediate answer is that the zero cohomology (virtual rank) is the index of the Dirac operator on $Y$. A more interesting answer is that in some cases that might be all one obtains: it is a result of Borel-Hirzebruch that the signature is strictly multiplicative in all bundles where $\pi_1$ of the base acts trivially on the rational cohomology of the fibers. The signature is the index of a certain twisted Dirac operator. If we have a family of these operators such that $Z\rightarrow X$ satisfies the condition involving the fundamental group, then the strict multiplicativity gives $\text{ch}(\text{ind }D)=\int_Y \hat A (V)ch(E)=\text{sign }(Y)$. A priori one could expect higher degree cohomology classes. It seems interesting that these vanish. If the question is too vague or broad, I would be happy knowing Are there any instances in which there are known relations between the Chern character of the index bundle and the Chern classes of $X$? - The answer to your second question is no. The smooth structure on the base space $X$ is completely irrelevant for index theory on fibre bundles on $X$. See Atiyah-Singer "Index of elliptic operators IV". – Johannes Ebert Apr 1 2011 at 9:04 That's good to know. Thank you for your response! I will read up on it some more. – charris Apr 1 2011 at 14:33 Pontryagin classes only depend on the topological structure of the manifold. Is it ever possible that there could be relations between the Chern character of the index bundle and the Pontryagin classes on $X$, or am I still missing something? – charris Apr 2 2011 at 1:15 Yes, you are missing something. The Atiyah-Singer family index theorem holds for any compact Hausdorff base $X$. The definition of a smooth bundle and a fibrewise elliptic operator needs to be adjusted a bit. Only the homotopy type of $X$ and isomorphism class of the smooth fibre bundle matters. Of course, if $X$ is a closed manifold, then so is $Z$. An elliptic operator $D$ on $Z$ restricts to an elliptic operator $D_{fib}$ on the fibres. Of course $ind(D)$ and the index bundle of $D_{fib}$ are related. – Johannes Ebert Apr 2 2011 at 9:19 ## 1 Answer Your question is not so clearly stated; so I take the opportunity to interpret it and write a little essay. As far as I understand your question, you found out that the multiplicativity result by "Borel-Hirzebruch" (I think it is actually due to Hirzebruch-Chern-Serre) implies that the higher Chern character of the index bundle of the signature operator are zero and now you want to see a more conceptual explanation. You should have a look into Atiyah's "The signature of fibre bundles". The following is definitly in the spirit of that paper, though I did not find the statement there. Theorem: "Let $Z \to X$ be an oriented smooth fibre bundle with fibre $Y$, closed, of dimension $4k$. Pick a fibrewise Riemann metric and let $D$ be the fibrewise signature operator. Assume that $\pi_1 (X)$ acts trivially on $H^{2k}(Y; \mathbb{R})$. Then the index bundle of the signature operator $ind(D)$ is trivial of virtual rank $sign(Y)$. Moreover, if the image of $\pi_1 (X)$ in $Gl(H^{2k} (Y); \mathbb{R})$ is finite, then the components of $ch(ind(D))$ in positive degree are zero." Proof: 1. A linear algebra fact. Given a finite-dimensional real vector space $V$ with a nondegenerate symmetric bilinear form $b$; of signature $(p,q)$. Consider the space $Q(b)$ of all pairs $(V_{+},V_{-})$, such that `$\pm b\|_{V_{\pm}} $` is positive definite and `$W_{-} \oplus W_{+}=V$`. $Q$ is a subspace of the product $Gr_p (V) \times Gr_q(V)$ of two Grassmannians. $Q(b)$ is diffeomorphic to the homogeneous space $O(p,q)/(O(p)\times O(q))$, and $O(p) \times O(q) \subset O(p,q)$ is a maximal compact subgroup. Therefore $Q$ is contractible (there should be a more direct proof, though). 2. Assume that the $\pi_1$-action on $V:=H^{2k} (Z_x; \mathbb{R})$ is trivial. There is the intersection form $b$ on $V$; pick a decomposition $V=W_{+} \oplus W_{-}$ as before. Now consider the bundle $E:=H^{2k} (Z/X) \to X$; the fibre over $x$ is the cohomology $H^{2k} (Y_x; \mathbb{R})$. Because the $\pi_1$-action is trivial, the bundle is trivial and the splitting above gives a splitting `$E=F_{+} \oplus F_{-}$` into two trivial subbundles. 3. Now pick a Riemann metric on the fibres. The Hodge theorem identifies $E$ with the bundle of harmonic $2k$-forms and the Hodge star $\ast$ is an involution and it splits $E= E_{+} \oplus E_{-}$ into the sum of eigenspaces. More or less by definition, $ind (D) = [E_{+}]-[E_{-}]$. Now I claim that $E_{\pm}\cong F_{\pm}$. This is because both decompositions can be viewed as sections to the bundle $X \times Q(b)$ and since $Q(b)$ is contractible, they are homotopic. 4. If the image of the monodromy is a finite group, say $\mu: \pi_1 (X) \to G$, you pass to the finite cover $\tilde{X} \to X$ corresponsing to the kernel of $\mu$. The pullback of $Z$ to $\tilde{X}$ has trivial monodromy and so the signature index bundle is trivial. But the induced homomorphism $H^{\ast} (X; \mathbb{Q}) \to H^{\ast}(\tilde{X}, \mathbb{Q})$ is injective, which allows to reduce the argument to the case of a trivial action. QED After I have written all this, I realize that the main point is that the bundle $E$ has structural group $O(p,q)$ and it is flat as such a bundle. It allows a reduction of the structural group to $O(p) \times O(q)$ (by the contractibility of $Q(b)$), but not as a flat bundle. If you find a flat reduction, then $ch(Ind(D))=0$ (in positive degrees) by Chern-Weil theory. The index theorem is not really relevant here; it identifies $ch(ind(D))$ with the fibre integral of the $L$-class of the vertical tangent bundle. - Thanks for the great response. Sorry my question wasn't completely clear. I was wondering if there were other situations like we're discussing where something "interesting" happens like the index bundle is actually trivial. But I was very interested in a conceptual understanding in this case, so your answer should really help me out a lot. I could be wrong, but I don't think the triviality of the index bundle (immediately) follow from the H-C-S result. Multiplicativity of the signature doesn't require strict multiplicativity (that the fiber integral ends up only in degree 0). – charris Apr 2 2011 at 0:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 83, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.933043360710144, "perplexity_flag": "head"}
http://physics.stackexchange.com/questions/5888/how-can-super-massive-black-holes-have-a-lower-density-than-water?answertab=votes	# How can super massive black holes have a lower density than water? I'm new here, so go easy! I heard on a podcast recently that the supermassive black holes at the centre of some galaxies could have densities less than water, so in theory, they could float on the substance they were gobbling up... can someone explain how something with such mass could float? http://www.universetoday.com/83204/podcast-supermassive-black-holes/ - Another way of sort of saying the same thing is that if you think of gravity as a flux, as would be suitable for a 1/r^2 gravity theory like Newton's, then there's no limit to the flux you can obtain no matter the lack of density of the mass you make it from; just make it big enough. – Carl Brannen Feb 26 '11 at 0:29 Floatation makes no sense in this context.. A supermassive black hole has more grav arrtn than the earth. So the blach hole will stay put at half the height of the fluid. – Manishearth♦ Feb 13 '12 at 15:45 ## 4 Answers Well, it can't (float), since a Black Hole is not a solid object that has any kind of surface. When someone says that a super massive black hole has less density than water, one probably means that since the density goes like $\frac{M}{R^3}$ where M is the mass and R is the typical size of the object, then for a black hole the typical size is the Schwarzschild radius which is $2M$, which gives for the density the result $$\rho\propto M^{-2}$$ You can see from that, that for very massive black holes you can get very small densities (all these are in units where the mass is also expressed in meters). But that doesn’t mean anything, since the Black Hole doesn’t have a surface at the Schwarzschild radius. It is just curved empty space. - 2 You should be clear that "it can't" refers to floating, not to having a low density. – dmckee♦ Feb 25 '11 at 17:23 1 As I’ve pointed out to your answer, since there is no surface for the BH and the matter is freefalling in the BH, there is no pressure. Thus there is no floating in any conventional way. The only thing that matters in that case is the total momentum transfer from the fluid, which will be pointing on the same way as the exterior field. It is a wrong, bad analogy. Sorry. – Vagelford Sep 3 '11 at 22:35 1 Free falling does not mean no pressure! It means that there is no pushing on the black hole itself. The pressure accelerates the infalling water, which enters with a greater kinetic energy (and up-momentum) at the bottom than at the top. The difference is exactly the bouyancy force in the fluid, as can be seen by analying the flow of up-momentum in a big sphere surrounding the black hole. – Ron Maimon Sep 3 '11 at 22:51 1 @vagelford: the whole construction is not oversimplified, it is not misleading, and it can be done in principle with a viscous enough liquid pool and a small enough black hole. The pool has to be in gravity or accelarated, to give it a pressure gradient. A black hole falling through such a pool (in gravity or accelerated) will absorb fluid in such a way that it feels a bouyancy force. The bouyancy force is the same as any other floating/sinking object, because it is determined by conservation laws. – Ron Maimon Sep 5 '11 at 14:28 1 @Vagelford: I did not make a mistake. – Ron Maimon Sep 6 '11 at 1:16 show 4 more comments The black hole would float in water, if you could make a large enough pool to submerge it, and with enough replenishment to replace the water that the black hole will sucks up. The black hole will remove water from its surroundings, but the water below will come into the horizon at higher pressure than the water above, so the velocity inward will not be uniform. If the black hole is denser than water, it will sink for a while, because the pressure difference is not enough to compensate for the pull of gravity. If the black hole has less density than water, it will float. It's like a balloon that sucks in water and expands, always maintaining a volume which is big enough to keep itself lighter than water. The problem is that when the black hole density is as that of water, a volume of water equal to the black hole's volume will not be stable to gravitational collapse, so it will be impossible to set up the pool. - Well, you are forgetting the effect of the self-gravity of the black hole which in the case of a balloon submerged in a fluid is negligible. In order to have the setting you are proposing the water and the BH would have to be embedded in an exterior, lets say uniform, gravitational field which would also define the difference between up and down. The first thing to point out is that in the free falling matter in the BH the important thing is the fluids kinetic energy and not the pressure. So, “under” the BH the gravitational pull towards the BH would be smaller relative to “over” the BH. – Vagelford Sep 3 '11 at 22:26 Thus the flow coming from the “up side” would have grater kinetic energy and thus it would excerpt a grater dynamic pressure (transfer more momentum to the BH). Whatever the case, it is a bad and misleading analogy the whole “floating” thing. – Vagelford Sep 3 '11 at 22:27 Ignoring the issue that the water would collapse under gravity, which invalidates the whole thing, the greater pressure below ensures that the water below will enter with a greater kinetic energy, and this is the reason for "floating". The water at the top will enter at a lower velocity, so there will be a total momentum transfer which is exactly as for any other submerged object. The problem with the density of water being unsustainable might be fixable by going to a more viscous fluid which will ooze into the black hole slowly, so you could have slow sucking up of material. – Ron Maimon Sep 3 '11 at 22:49 There is no hydrostatic pressure difference in freefall. – Vagelford Sep 4 '11 at 9:33 1 @Vagelford: it's not funny. The buoyancy force on a region is determined by conservation laws on a big sphere far away from the horizon, where the water is essentially stationary. On such a sphere, the total force is equal to the mass of displaced water, and the total mass is less than this because of the black hole's lower density. There is no way around it, because it is determined by conservation laws. – Ron Maimon Sep 7 '11 at 16:37 show 8 more comments The Schwarzschild radius scales with mass as $r~=~2GM/c^2$. What might be defined as a Schwarzschild volume would then be $V~=~4\pi r^3/3$ $=~(32/3)\pi(GM/c^2)^3$. So the density of matter defined by the horizon is $\rho~=~(3/32)(c^2/G)^3M^{-2}$. So density scales as the inverse square of the mass. A 10 billion solar mass black hole has a radius about $10^{10}$km, or a volume $V~\sim~10^{30}km^3$ $=~10^{39}m^3$. A solar mass is $10^{30}$kg and the density defined by the horizon is then $\rho~=~10^{-9}kg/m^3$. That is actually quite small. Of course if you fall into a black hole of any mass you encounter a region with enormous Weyl curvature and tidal forces. The source of this is in your future, and eventually you reach it --- it is inescapable. This region where curvature diverges is a spatial surface of infinite extent. - You should add some closing [/math] tags 'Sothedensityofmatterdefine...' yeah after reading that two times I could decipher it :) – datenwolf Feb 25 '11 at 17:31 I think it is actually misleading to make the claim that is puzzling you. "Density" suggests that the mass is distributed more or less uniformly within the black hole, and this is non-sense. The black hole is mostly empty, and all the mass is concentrated within a tiny region (clasically a point) in the center of the black hole. If you ignore this and pretend a black hole of mass $M$ and volume $V\propto r^3$ had a uniform density $\rho$ then you can calculate it, simply using $\rho=M/V$. Since for Schwarzschild black holes the radius of the black hole is proportional to its mass you obtain finally $\rho\propto 1/M^2$, so the heavier a black hole the smaller its density. But again, this provides a highly misleading picture of the mass distribution within the black hole. All its mass is in the center, so classically the density is infinite. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9413565397262573, "perplexity_flag": "middle"}
http://mathhelpforum.com/trigonometry/40921-i-need-help-trig.html	Thread: 1. I need help with trig! Hi i really really need to answer these last two questions by the next 30minutes!! (i have online math) can anyone help me? Use the formulas for functions of the difference and sum of two angles to find, without tables or calculator, the cosine, sine, and tangent of 15degrees(using 45degrees and 30degrees) tan 15degrees=? Using the given data, find the values of the sine, cosine, and tangent of ( AB) and of (A - B). Give the quadrant in which each of the angles (A + B) and (A - B) terminates: angle A in Quadrant II, tan A=-12/5 + angle B in Quadrant III, sin b=-3/5 sin a-b=? 2. Originally Posted by moneyca Hi i really really need to answer these last two questions by the next 30minutes!! (i have online math) can anyone help me? Use the formulas for functions of the difference and sum of two angles to find, without tables or calculator, the cosine, sine, and tangent of 15degrees(using 45degrees and 30degrees) tan 15degrees=? Using the given data, find the values of the sine, cosine, and tangent of ( AB) and of (A - B). Give the quadrant in which each of the angles (A + B) and (A - B) terminates: angle A in Quadrant II, tan A=-12/5 + angle B in Quadrant III, sin b=-3/5 sin a-b=? I'm not the trig guru on this forum, but this is what you need: $45\deg-30\deg=15\deg$ $\sin(s-t)=\sin s \cos t- \cos s \sin t$ $\cos(s-t)=\cos s cos t+\sin s sin t$ $\tan(s-t)=\frac{\tan s-\tan t}{1+\tan s \tan t}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8329956531524658, "perplexity_flag": "middle"}
http://physics.stackexchange.com/questions/43451/classical-mechanics-a-particle-move-in-one-dimension-under-the-influence-of-two	# Classical Mechanics: A particle move in one dimension under the influence of two springs [closed] A particle of mass $m$ can move in one dimension under the influence of two springs connected to fixed points a distance $a$ apart (see figure). The springs obey Hooke’s law and have zero unstretched lengths and force constants $k_1$ and $k_2$, respectively. a) Using the position of the particle from one fixed point as the generalized coordinate $q$, find the Lagrangian and the corresponding Hamiltonian. Is the energy conserved? Is the Hamiltonian conserved? b) Introduce a new coordinate $Q$ defined by $$Q= q-b \;\sin(\omega t) \\ b=\frac{k_2}{k_1+k_2} a$$ What is the Lagrangian in terms of $Q$? What is the corresponding Hamiltonian? Is the energy conserved? Is the Hamiltonian conserved? - 1 Is this one of those "do my homework for me" questions, or is there something you do not understand and need help with? – ja72 Nov 5 '12 at 0:14 ## closed as too localized by Martin Beckett, Ron Maimon, Mark Eichenlaub, Qmechanic♦, mbq♦Nov 5 '12 at 3:12 This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, see the FAQ.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.875719428062439, "perplexity_flag": "middle"}
http://mathhelpforum.com/differential-equations/189406-partial-differential-equation-method-characteristics-ode-solve.html	# Thread: 1. ## Partial Differential Equation -- method of characteristics (ode solve) $u_t + cos(\pi x) u_x = 0$ using the method of characteristics. This amounts to solving the following odes: $dt/ds = 1, t(0) = 0$ which implies $t = s$ and $dx/ds = cos(\pi x), x(0) = x_0$. This ode yields (if I'm right): $\|sec(\pi x) + tan(\pi x)\| = Ce^{\pi s}$. However, I cannot solve this explicitly for x, and therefore cannot solve the pde by the method of characteristics? Can anyone shed insight? Have I done something wrong? Is there a missing trig identity somewhere? Thanks 2. ## Re: Partial Differential Equation -- method of characteristics (ode solve) Originally Posted by davismj $u_t + cos(\pi x) u_x = 0$ using the method of characteristics. This amounts to solving the following odes: $dt/ds = 1, t(0) = 0$ which implies $t = s$ and $dx/ds = cos(\pi x), x(0) = x_0$. This ode yields (if I'm right): $\|sec(\pi x) + tan(\pi x)\| = Ce^{\pi s}$. However, I cannot solve this explicitly for x, and therefore cannot solve the pde by the method of characteristics? Can anyone shed insight? Have I done something wrong? Is there a missing trig identity somewhere? Thanks $u_{t}+ \cos (\pi x)\ u_{x}=0$ ... is 'nonhomogeneous' and equivalent to then system... $d t=\frac{d x}{\cos \pi x}$ $d u=0$ (2) ...the solution of which is... $c_{1}= v(x,t)= t-\frac{1}{\pi}\ \ln\ \|\tan (\frac{\pi}{2} x +\frac{\pi}{4})\|$ $c_{2}= u$ (3) ... so that the solution of (1) is... $u= \gamma\{v(x,t)\}$ (4) ... where $\gamma(*)$ is any continous function with continous derivative... Kind regards $\chi$ $\sigma$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9191806316375732, "perplexity_flag": "middle"}
http://mathoverflow.net/questions/120150/can-fpa-really-prove-its-consistency/120216	## Can FPA really prove its consistency? ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) I will ask the question first and then explain. QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency? FPA is a multi-sorted first-order theory, with lower-case or small letters (for numbers) and upper-case or big letters for relations of n-arity (n >= 1). (Practically, I think one can limit the theory to relations where n = 1, 2, or 3.) Full comprehension is assumed. FPA has a constant symbol 0, a 1-ary relationship N (natural number), and a 2-ary relationship symbol S (successoring). In this context the Peano Axioms can be written: (PA1) N0 (PA2) $\forall$n (Nn $\Rightarrow$ $\exists$m (Nm & Sn,m)) (PA3) $\forall$n$\forall$m$\forall$m' (Nn & Nm & Nm' & Sn,m & Sn,m' $\Rightarrow$ m = m') (PA4) $\forall$n$\forall$m$\forall$n' (Nn & Nm & Nn' & Sn,m & Sn',m $\Rightarrow$ n = n') (PA5) $\forall$n (Nn $\Rightarrow$ $\neg$ Sn,0) (PA6) $\forall$P (P0 & $\forall$n$\forall$m(Pn & Sn,m $\Rightarrow$ Pm) $\Rightarrow$ $\forall$n(Nn $\Rightarrow$ Pn)) FPA assumes all the Peano Axioms except (PA2), that is, everything except the totality of the successor relationship. It has as its standard models all the initial segments as well as the standard model of the natural numbers. {0} is a model. It is therefore agnostic as to whether the natural numbers go on and on. In FPA it is possible to define formula for addition, multiplication, less than, exponentiation, and, except obviously for totality and any related property, prove the usual properties. Intuitively, the existence of a natural number n implies that every number less than n exists. Predicates for numbers can be defined: one(n) if and only if S0,n & Nn, two(n) if and only if $\exists$x (one(x) & Sx,n) & Nn Obviously one cannot prove there exists any n such that one(n). But if such an n exists, then one can show it has all the usual properties of 1. Similarly for two, three, etc. FPA proves the Fundamental Theorem of Arithmetic. Recursion is available and it is possible to define formula expressing syntax in the Godel fashion. For instance, Term1(n) ("n is a lower-case term") might be defined as: seven(n) $\vee$ $\exists$y$\exists$z (+(y,y,n) & eleven(z) & z >= n). (Seven(n) expresses n is 0, and the lower-case variables are the even numbers >= eleven.) One can continue and define a formula GProof(n,x) which says that n is the Godel number of a proof in FPA of a wff whose Godel number is x. Letting $\mathcal{F}$ be "$\neg$ 0 = 0", then GCons(FPA) is the formula: $\neg$ $\exists$p (GProof(p,$\mathcal{F}$)) But FPA proves GCons(FPA). Intuitively the reasoning goes like this. Suppose $\neg$ GCons(FPA). Then there is a number p such that GProof(p,$\mathcal{F}$). But, because of the nature of Godelization, and its use of a single number to represent sequences of numbers via exponentiation, this is a very big number, easily bigger than what is required to define "true in {0}" for all propositions of length smaller than the propositions in the purported proof. FPA can show the axioms are true in {0}, that the deduction rules preserve truth in {0} for all steps in the proof, but of course $\mathcal{F}$ is not true in {0}. Contradiction, so FPA proves GCons(FPA). Well this does seem like a bit of a cheat, because it uses the fact that Godelization, by using numbers to represent sequences, needs very big numbers. Instead of lower-case numbers to code a sequence, one can use upper-case letters: R is a sequence if and only if dom(R) is {x : x <= n} for some natural number n. One can then redefine a formula RProof(R,S) which says that R is a sequence representing a proof of the proposition represented by the sequence S. And define a new formula RCons(FPA). The question is: does FPA prove RCons(FPA)? The proof for the Godelization formula doesn't go through because now only a small number (the length of the proof) is implied, and this is not enough, at least prime facie, to construct a model of true-in-{0} for the propositions in the proof. To show that FPA proves RCons(FPA), it would suffice to show that any proof of an inconsistency would have to be very, very long. To show that FPA doesn't prove RCons(FPA), maybe Godel's original proof would go through, but this makes me nervous, because of the problem with GCons(FPA). Sorry for the long question, but any help appreciated! - What do you mean one cannot prove there exists $n$ such that $one(n)$? (PA2) with $n=0$, together with (PA1), gives precisely $\exists m \mathrm{one}(m)$. – Amit Kumar Gupta Jan 29 at 9:11 @Amit Kumar Gupta: FPA omits PA2. – Ben Crowell Jan 29 at 15:12 ## 1 Answer In principle, the answer can depend on the proof system, but as long as you stick to some of the usual Hilbert-style or sequent proof systems, this shouldn’t matter. First, as explained in http://mathoverflow.net/questions/120106, the question is equivalent to provability of the consistency of FPA in $I\Delta_0+\Omega_1$. (The fact that you are using second order objects to encode proofs and formulas corresponds to using all numbers instead of just the logarithmically small ones in $I\Delta_0+\Omega_1$, hence you end up with the usual consistency statement.) Now, working in $I\Delta_0+\Omega_1$ (or equivalently, Buss’s $S_2$), the consistency of FPA is equivalent to the consistency of the second-order theory of the model with one-element first-order universe (in whatever finite language, it’s all equivalent), since the two theories are interpretable in each other. This in turn can be reduced to the quantified propositional calculus: since there is only one first-order element (and only one $n$-tuple of elements for every $n$), you can ignore first-order quantifiers and variables, and replace second-order variables with propositional variables both in second-order quantifiers and in atomic formulas. (Purely first-order atomic formulas such as $t=s$ can be replaced with the constant $\top$ for truth.) Thus, the question becomes whether $I\Delta_0+\Omega_1$ proves the consistency of the quantified propositional calculus ($G$). The answer is that this is one of the major open problems in the area, but it is conjectured to be false. There is a kind of correspondence of subsystems of bounded arithmetic to propositional proof systems; in particular, the fragments $T^i_2$ of $S_2$ (${}=I\Delta_0+\Omega_1$) correspond to the fragments $G_i$ of the quantified propositional calculus, obtained by restricting all formulas in the proof (or alternatively, all cut formulas in the sequent calculus formulation) to $\Sigma^q_i$ or $\Pi^q_i$ formulas (= formulas in prenex form with at most $i$ quantifier blocks). This means: • $T^i_2$ proves the consistency (and even some form of reflection principle) of $G_i$. • Conversely, $\mathrm{Con}_{G_i}$ implies over a weak base theory all $\forall\Delta^b_1$-consequences of $T^i_2$ (and more complex consequences of $T^i_2$ can be xiomatized by an appropriate reflection principle). A related fact is that if $T^i_2$ proves a $\forall\Sigma^b_i$ statement, one can translate it into a sequence of quantified propositional tautologies which will have polynomially bounded proofs in $T^i_2$. • If $P$ is any propositional proof system whose consistency is provable in $T^i_2$, then $G_i$ polynomially simulates $P$. $S_2$ is the union of its finitely axiomatizable fragments $T^i_2$. This means that $S_2$ proves the consistency of each fragment $G_i$, but on the other hand, if it proved the consistency of the full quantified propositional calculus $G$, it would imply that $G_i$ polynomially simulates $G$ for some $i$, and this is assumed to be false. To put it differently, the $\forall\Delta^b_1$-consequences of $S_2$ (as well as $S_2$ itself) are not assumed to be finitely axiomatizable. The correspondence of theories and propositional proof systems also extends to complexity classes. Sets definable by $\Sigma^b_i$ formulas in the standard model of arithmetic are exactly those computable in the $i$-level $\Sigma^P_i$ of the polynomial hierarchy. The theories $T^i_2$ have induction for $\Sigma^b_i$ formulas, and their provably total $\Sigma^b_{i+1}$-definable functions are $\mathrm{FP}^{\Sigma^P_i}$, so these theories correspond to levels of the polynomial hierarchy. On the propositional side, satisfiability of $\Sigma^q_i$ formulas is a $\Sigma^P_i$-complete problem. Taking the union, $S_2$ corresponds to the full polynomial hierarchy $\mathrm{PH}$. However, the complexity class corresponding to $G$ is $\mathrm{PSPACE}$, as satisfiability of unrestricted quantified propositional formulas is $\mathrm{PSPACE}$-complete. Thus, asking $S_2$ to prove the consistency of $G$ is in the same spirit as collapsing $\mathrm{PSPACE}$ to $\mathrm{PH}$ (and therefore to some its fixed level). (Don’t quote me on this. While the collapse of the $T^i_2$ hierarchy does imply the collapse of $\mathrm{PH}$, for propositional proof systems this becomes only a loose analogy.) In order to give also an upper bound on the consistency strength, the consistency of $G$, and therefore of FPA, is provable in theories corresponding to $\mathrm{PSPACE}$. The best known such theory is Buss’s theory $U^1_2$, which is a “second-order” extension of $S_2$ with comprehension for bounded sets defined by bounded formulas without second-order quantifiers, and length induction for bounded $\Sigma^1_1$-formulas. Notice that things get really messy here, as the first-order objects of $U^1_2$ correspond to second-order objects of FPA, and second-order objects of $U^1_2$ have no analogue in FPA. Alan Skelley formulated an equivalent (technically, RSUV-isomorphic) theory $W^1_1$. This is syntactically a third-order arithmetic, and it is more directly comparable to FPA (as numbers of one theory correspond to numbers of the other, and sets correspond to sets). $W^1_1$ proves the consistency of $G$, and thus of FPA. - Thank you very much, Emil. – abo Jan 29 at 19:15 And your reward is even more questions! – abo Jan 29 at 20:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 108, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9074258208274841, "perplexity_flag": "middle"}
http://mathoverflow.net/revisions/76832/list	## Return to Question 2 edited body There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that has all the following properties: • Minimal 6-connectedness: The discrete line is 6-connected. Also, assuming WLOG that the direction vector is in the first octant, for any integer N there is exactly one voxel $(x,y,z)$ of the line $(x,y,z)$ such that $x+y+z=N$ (the "minimal" part). • Well-behaved projections: The projections onto the $xy$, $xz$, and $yz$ planes are all 4-connected 2D discrete lines, such as would be produced by a modified Bresenham's algorithm. • Convexity: The voxels of the discrete line form a convex lattice set in $\mathbb Z^3$; that is, it is equal to the intersection of $\mathbb Z^3$ with its own convex hull. Is there a kind of discrete line that has all these properties? It's surprising and intersting to me that discrete lines are so easy to define in 2D, but so hard to pin down in 3D. I guess being codimension-1 makes it easy. 1 # Minimally 6-connected 3D discrete lines that are convex lattice sets There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that has all the following properties: • Minimal 6-connectedness: The discrete line is 6-connected. Also, assuming WLOG that the direction vector is in the first octant, for any integer N there is exactly one voxel of the line $(x,y,z)$ such that $x+y+z=N$ (the "minimal" part). • Well-behaved projections: The projections onto the $xy$, $xz$, and $yz$ planes are all 4-connected 2D discrete lines, such as would be produced by a modified Bresenham's algorithm. • Convexity: The voxels of the discrete line form a convex lattice set in $\mathbb Z^3$; that is, it is equal to the intersection of $\mathbb Z^3$ with its own convex hull. Is there a kind of discrete line that has all these properties? It's surprising and intersting to me that discrete lines are so easy to define in 2D, but so hard to pin down in 3D. I guess being codimension-1 makes it easy.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9291833639144897, "perplexity_flag": "middle"}
http://elidourado.com/blog/tag/copyright-term/	• Research • Code • Blog • Contact # Copyright Reform and the Incentive to Create Mercatus has a new book out on copyright, edited by Jerry Brito, called Copyright Unbalanced: From Incentive to Excess. I am pleased to be one of an otherwise-illustrious group of contributors. I expect that the book will create some controversy in policy circles. In this post, I want to address what is likely to be a knee-jerk response from our critics, that copyright reform will substantially decrease the incentive to produce creative works. Content creators anticipate that their products will generate some amount of revenue each year after they are released. The expectation is generally that the creative work will generate the highest revenue in the first year, and less revenue in each subsequent year. To model this revenue stream, I’m going to assume exponential decay. Exponential decay lets us pick a half-life, $h$, and assume that $h$ years after the work was released, it will generate revenue at half the initial rate. After $2h$ years, it will generate revenue at one-fourth the rate, and so on. In year $t$, the revenue that the content creator will receive if there is copyright is $e^{\frac{-t \ln2}{h}}$ times the initial revenue. Consequently, the total revenue that a copyright holder will receive over the life of a 95-year copyright term is $\sum\limits_{t=0}^{94} e^{\frac{-t \ln 2}{h}}$ times the initial revenue. However, content creators prefer revenue now to revenue 90 years from now. In order to calculate the present value of this revenue stream, we need to apply a discount rate $r$. The ex ante value of the revenue stream generated by the 95-year copyright term is therefore $\sum\limits_{t=0}^{94} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$ times the initial revenue. And of course, this calculation generalizes to different copyright terms. If we returned to a 28-year term, as Tom Bell advocates in his chapter of our book, the ex ante revenue stream would be valued at $\sum\limits_{t=0}^{27} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$ times the initial revenue. We’re now at a point where we can start to run some numerical calculations based on plausible values for $h$ and $r$. What is a reasonable ex ante expectation about the half-life of the revenue stream of a new creative work? I expect that for our book, the half-life will be something like 1 year or less; we will probably sell less than half as many books in the second year the book is out as in the first. But let’s not use $h=1$. Let’s estimate that $h=10$ to be extremely conservative and generous to our critics. What about $r$? Again, how about if we are conservative and give $r$ a low value, like $r=0.02$? Now we can run some calculations. Using the values above, the ex ante present value of a 95-year copyright is around 11.726 times the initial revenue. The ex ante present value of a 28-year copyright is around 10.761 times the initial revenue. Consequently, shortening the copyright term from 95 years to 28 years (less than 30% of the current term!) retains about 91.8 percent of the incentive effect of the current copyright term. It is unlikely that such a small decrease in the present-value of the revenue stream would reduce the amount of content production by much. To the extent that content producers cannot or do not substitute easily into other fields, they would simply take the 8.2 percent decline in compensation per project as a decrease in wages (not the end of the world), and there would be no decline in content production. To the extent that content producers can substitute into other fields, we would get less content, but we would also get more of other stuff—the welfare effects of less content are ambiguous, since there is a knowledge problem regarding the optimal amount of content. If you want to do the calculation with different half-lives and interest rates, be my guest. I am confident that for all plausible values of $h$ and $r$, you will find that shortening the copyright term will have at most a modest effect on the incentive to create. How about the value of the public domain? This is a little harder to model, because we care about the ex post value of works, not just the ex ante expectation that content creators have. In practice, there turn out to be works with much longer half-lives than others. This fact complicates any back-of-the-envelope calculation. We also don’t know exactly by how much content creation would fall. But let’s abstract from this and model the value of the public domain as the revenue stream for a given project that otherwise would have gone to copyright holders above. One difference for the public domain is that it no longer makes sense to discount the stream of value—future generations aren’t sitting around, waiting to be born so that they can watch Star Wars for the first time. Therefore, normalized to our original, first-year revenue stream, an estimate of the value of the public domain under a 95-year term is $\sum\limits_{t=95}^{\infty} e^{\frac{-t \ln 2}{h}}$. Under a 28-year term, the value is $\sum\limits_{t=28}^{\infty} e^{\frac{-t \ln 2}{h}}$. Plugging in the value we selected earlier for $h$, 10, the former expression yields around 0.021 and the latter about 2.144. In other words, the value of the public domain would be around 100 times higher per creative work if we shortened the term to 28 years. Again, this value is highly dependent on our selection of $h$, but the reason I am doing these calculations is so that my critics can repeat them with values they find more plausible, if they so choose. This analysis has been highly stylized, but it is also extremely conservative. The half-life of most creative works is probably much shorter than 10 years, and when valuing an uncertain revenue stream, most artists—and even content corporations—probably discount at a rate of higher than 2 percent. The value of the public domain has been understated in this analysis, because there are many works that turn out ex post to have longer half-lives (but it is still the ex ante estimate of value that matters for investment). I have also not factored in the gains from those derivative works that are impossible under the current regime due to transaction costs, or the savings in enforcement costs from having a shorter time during which enforcement is necessary, or indeed, many of the other issues discussed in our book. I would be interested in reading further analyses like the one above from anyone who supports the current copyright term or a longer one. How do you justify such a long term? You don’t have to use my assumptions, just make your own explicit so that people can see what they are and quarrel with them. How many fewer works do you really think would be created if we shortened the term from 95 years to 28 years? Would we really be worse off? Please show your work. ### Sharing buttons: This entry was posted by on December 3, 2012. Bell, Brito, copyright, copyright term, Copyright Unbalanced, exponential decay, incentives, intellectual property, Mercatus, present value, public domain	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 22, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9419177770614624, "perplexity_flag": "middle"}
http://mathforum.org/mathimages/index.php?title=First_Fundamental_Form&oldid=20631	# First Fundamental Form ### From Math Images Revision as of 10:10, 16 June 2011 by Kderosier (Talk \| contribs) (diff) ←Older revision \| Current revision (diff) \| Newer revision→ (diff) Let $M$ be a regular surface with $v_p,w_p$ points in the tangent space $M_p$ of $M$. Then the first fundamental form is the inner product of tangent vectors, $I(v_p,w_p)=v_p \cdot w_p$ The first fundamental form satisfies $I(ax_u+bx_v,ax_u+bx_v)=Ea^2+2Fab+Gb^2$ The first fundamental form (or line element) is given explicitly by the Riemannian metric $ds^2=Edu^2+2Fdudv+Gdv^2$ It determines the arc length of a curve on a surface. The coefficients are given by $E = \|\|x_u\|\|^2=\|\frac{\partial x}{\partial u}\|^2$ $F = x_u \cdot x_v= (\frac{\partial x}{\partial u}) * (\frac{\partial x}{\partial v})$ $G = \|\|x_v\|\|^2=\|\frac{\partial x}{\partial v}\|^2$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9207375049591064, "perplexity_flag": "middle"}
http://crypto.stackexchange.com/questions/2076/division-in-paillier-cryptosystem/2096	Division in paillier cryptosystem Is division possible in the Paillier Cryptosystem? i.e. given a the cipher-text $C$ of an integer $M$ the plain-text divisor $D$, and only the public key, can one compute the cipher-text of $M/D$ ? - 3 Answers In general, no. There are, however, specific conditions that, when met, make it possible. Working Mod $\phi(N^2)$ If given $D^{-1} \bmod{\phi(N^2)}$, then $\mathcal{E}(M/D)$ can be computed. So the question remains, when is computing $D^{-1} \bmod{\phi(N^2)}$ possible? It is only possible if $gcd(D, \phi(N^2))=1$. Note that $\phi(N^2)=N\cdot \phi(N)=N\cdot(p-1)\cdot(q-1)$. Since $N=p\cdot q$, $N$ is odd, but $p-1$ and $q-1$ are even. Thus, for any even $D$, $gcd(D, \phi(N^2))\neq 1$. So, $D$ can never be even. For odd $D$, it will work if the condition I specified holds. If $D\nmid M$ ($D$ does not divide $M$), even if the $gcd=1$, the result will not necessarily make sense. So, it will work under a few conditions, but in general it will not. Update, Working Mod $N$ As PulpSpy points out, one can also work Mod $N$. In that case, $gcd(D,N)=1$ for all $D\neq p,q$ (in most every practical case). So, the math works, but you still have issues in practice when $D\nmid M$. - Yes (and always). Given $\mathsf{Enc}(a)$ and $b$, you can compute $\mathsf{Enc}(a \cdot b^{-1} \bmod{n})$ by simply computing $\hat{b}=b^{-1} \bmod{n}$ and $Enc(a)^\hat{b} \bmod{n^2}$. Paillier encryption is built on the bijeective mapping from $(x,y)\in \mathbb{Z}_n \times \mathbb{Z}_n^$ to: $E_{g,n}(x,y)=g^x y^n \bmod{n^2}$. Generator $g$ is chosen to enforce that $x$ remains in $\mathbb{Z}_n$ (i.e., there exists a $y$ that makes $x$ unique in $\mathbb{Z}_n$). Specifically, it has order $\alpha n$ for a positive $\alpha < \mathrm{LCM}(p-1,q-1)$. A simple optimization is to set $\alpha=1$ by choosing $g=n+1$. Regardless of variant, in the encryption scheme, $x$ is the message and thus $\mathbb{Z}_n$ is the message space. Contrary to the other answers, this allows $b$ to be generally inverted $\bmod{n}$ (except when 0 or in the negligible case that it is either $p$ or $q$). Given the confusion, I did a sanity check in Mathematica to make sure (I picked small safe primes but you can set them using RandomPrime): ````p = 23; q = 83; n = pq; lam = LCM[p - 1, q - 1]; L[u_] := Quotient[u - 1, n]; Trap[y_] := L[PowerMod[y, lam, n^2]]; g = RandomInteger[{1, n}]; While[GCD[Trap[g], n] != 1, g = RandomInteger[{1, n}]] Enc[m_] := Module[{r, c}, r = RandomInteger[{1, n - 1}]; c = Mod[PowerMod[g, m, n^2]PowerMod[r, n, n^2], n^2] ] Dec[c_] := Mod[Trap[c]PowerMod[Trap[g], -1, n], n]; m = RandomInteger[{1, n - 1}]; s = RandomInteger[{1, n - 1}]; si = PowerMod[s, -1, n]; c = PowerMod[Enc[m], si, n^2]; Dec[c] == Mod[m*si, n] ```` This answer is the result of a crash course in Paillier. I typically work in the discrete logarithm setting. Feel free to edit if there are any errors (you probably do know more than me). - yes but like mikeazo pointed out, the answer does not make sense if $b\nmid a$ – user996522 Mar 13 '12 at 4:33 And the light came... Thanks for your patience. – fgrieu Mar 13 '12 at 8:38 1 @user996522 It depends on how you define "makes sense." If $a/b = c$, then $c \cdot b=a$. It is the same thing here. If you compute $a/b$ under encryption, you will get the encryption of the value $c$ such that $c \cdot b \bmod{n} = a$. You can think of it as "discrete division" akin to a discrete logarithm. – PulpSpy Mar 13 '12 at 14:50 One alternative that hasn't been mentioned, which may possibly be of interest, is doing plain old integer division in Paillier. The protocol is non-trivial, but, perhaps surprisingly, can be done somewhat efficiently. There was a paper at FC12: On Secure Two-party Integer Division [PDF] -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 56, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8705954551696777, "perplexity_flag": "middle"}
http://math.stackexchange.com/questions/320334/a-question-from-arhangelskii-buzyakova	# A question from Arhangel'skii-Buzyakova Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.) The fifth line from last. How could I see that for any $a\in H$ and $z\in Z\setminus H$, there exists an element $V$ of $\mathcal{U}$ such that $a\in V$ and $z\notin V$? Thanks very much. - 1 If $a \in Y$ it is easy: then we can use one of members of $\gamma_a$ that misses $z$. If $a \in H \setminus Y$, it is in one of the $K_\alpha$, but then $z$ could be in all the members of the corresponding $\gamma_\alpha$: all we would get it that would not be in $X$ in that case (so the case $z \in X$ is also taken care of). I don't quite see how to handle $a \in H \setminus Y$ and $z \in Z \setminus X, z \notin H$. – Henno Brandsma Mar 4 at 21:40 @HennoBrandsma: yes. Maybe Something is wrong in this paper. – Paul Mar 5 at 3:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9568443894386292, "perplexity_flag": "head"}
http://mathoverflow.net/questions/104212?sort=newest	## Adjoining a new isolated point without changing the space ### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points) Suppose $X$ is a $T_1$ space with an infinite set of isolated points. Show that if $X^\sharp = X \cup \lbrace \infty \rbrace$ is obtained by adding a single new isolated point, then $X$ and $X^\sharp$ are homeomorphic. I am almost embarrased to raise this, which seems obvious. The proof must be simple, but it eludes me for now. Maybe it is an exercise in some textbook. You can clearly establish a 1-1 equivalence between the isolated points of $X$ and those of $X^\sharp$. But it is not clear how this equivalence would extend to the closure of the isolated points. The theorem is easy when $X$ is compact $T_2$ and $cl(D) = \beta(D)$, where $D$ is the set of isolated points. - ## 2 Answers In my answer to http://mathoverflow.net/questions/26414 I descibed a somewhat simpler-looking example than Nik's, but proving that it works may be harder. Take two copies of $\beta\mathbb N$ and glue each non-isolated point of one copy to the corresponding point of the other copy. Any way of "absorbing" a new isolated point into the two copies of $\mathbb N$ forces a relative shift of those two copies, which forces corresponding shifts of the non-isolated points, which in turn conflicts with the gluing. The perhaps surprising thing about this example is that, if you add two isolated points, the result is (easily) homeomorphic to the original. - Yeah, that's cleaner. – Nik Weaver Aug 7 at 20:50 Two good answers. Thanks. – Fred Dashiell Aug 7 at 23:11 ### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Well, I think this is false. Start with a family of $2^{2^{\aleph_0}}$ mutually non-homeomorphic connected spaces, and attach them to the non-isolated points of $\beta {\bf N}$. (I.e., start with the disjoint union of $\beta {\bf N}$ and the other spaces, and factor out an equivalence relation which identifies each point of $\beta {\bf N} - {\bf N}$ with a point of one of the other spaces.) Any homeomorphism between $X$ and $X^\sharp$ has to take isolated points to isolated points; taking closures, it takes $\beta{\bf N}$ onto itself; and by connectedness it takes each of the extra spaces onto itself. So it has to fix each point of $\beta {\bf N} - {\bf N}$. Now the question is whether a bijection between ${\bf N}$ and ${\bf N}$ minus a point can fix $\beta {\bf N} - {\bf N}$ pointwise. The answer is no because iterating the map, starting on the missing point, yields a sequence within ${\bf N}$ that gets shifted by the map, and it is easy to see that this shift does not fix the ultrafilters supported on that sequence. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 25, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9536922574043274, "perplexity_flag": "head"}

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