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https://kwj2104.github.io/2018/cppngan-2/
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## Generative Art with CPPN-GANs Part II
In the previous blog post I attempted to train a vanilla GAN with a CPPN-architecture and failed to find convergence, and in this post I reattempt generation using instead, a Wasserstein GAN, on a few different datasets. Code for the new CPPN-WGAN-GP can be found here.
### Wasserstein GANs
GANs are hard to train, for a variety of reasons. First, in the world of personifying neural nets, to repeat the oft-used analogy, the art forger always seems to have a harder job than the art critic; GAN balancing naturally becomes this black art of preventing the discriminator from overpowering the generator. Behind this simple, intuitive picture, lies a host of problems in the mathematical theory behind the architectures, which further explains exploding/vanishing gradients, mode collapse, and other not-so-fun issues.
Inherently, the problem of generative modelling is one of comparing, and mimicking probability distributions. How do we artificially create a function with neural nets, that emits the same densities as the true distribution of the training data? Backpropagation dictates then, that our loss function reflect our objective - this thus has led to the prevalence of metrics such as the Kullbeck-Leibler Divergence (e.g. in vanilla VAEs), Jenson-Shannon Divergence (e.g. indirectly, in vanilla GANs), and Total Variation Distance. Likewise, the WGAN proposes the Wasserstein metric, or often known as the “Earth-Mover’s distance”.
I won’t go into the mathematical details (you can find a great blog post here), but in summary the paper explains that the Wasserstein metric (and using the metric as a loss function for a GAN) is important because:
• There are many distributions out there that will converge under EM, but not under KL, reverse KL, TV, and JS divergences.
• Compared to other divergences, and under assumptions reasonable in a GAN model, only the EM is continuous everywhere, and differentiable almost everywhere, which is extremely important for backpropagation.
• Under assumptions reasonable in a GAN model, every distribution that converges in the other divergences will also converge with EM.
• In vanilla GANs you need to alternate training the discriminator and generator, but in WGANs on each iteration you can train the discriminator to convergence (which gives you the accurate distance metric and loss function $W(P_r, P_{\theta})$, and then train the generator.
• Does a bunch of mathematical manipulations to make this loss function tractable, and usable for training.
• Unlike the Vanilla GAN which has no quantitative metric for evaluating results, the Wasserstein distance is a direct (or at least well correlated) metric of sample quality!
Shortly after the WGAN paper was released, a few researchers tweaked the algorithm to improve WGAN training, referred to as WGAN with Gradient Penalty (WGAN-GP). In this project I apply the WGAN-GP variation to three datasets: MNIST, CIFAR10, and CASIA.
### MNIST
MNIST converged, unlike the vanilla GAN architecture, after a few hours of training, and the Wasserstein distance does seem to indeed correspond with image quality.
Enlarged images (500x500):
Interpolation:
However, blowing the images up yielding very different results than that of the Vanilla-GAN. The images had a much smokier and blurrier quality than the sharp pictures from the GAN. Its unclear to me why this is the result from using the Wasserstein objective as compared to the JS Divergence. This result is similar to David Ha’s CPPN-GAN-VAE, which he hypothesizes that the blurriness is caused by using the Maximum Likelihood objective in the VAE, yet this model does not.
### CIFAR10
I thought the natural extension would be an attempt to train this model on colored images. Unfortunately, Cifar 10 was unable to converge into any meaningful resemblance of the original dataset, and after a few hours of training the Wasserstein distance was unable to continue decreasing. Nonetheless, the images generated were still interesting, colorful, and resembled the original CPPN color generations.
Enlarged images:
Interpolation:
### CASIA
I’ve always wanted to mess with the Casia Chinese Handwriting dataset. A couple of user notes for anyone who’s considering using this in the future. This dataset was developed by the Chinese Academy of Sciences, Institute of Automation, and comes in filesizes of at least a couple hundred MB. This is only a subset of their entire dataset, whereby the full data can only be obtained through a manual request form. Even so, their data takes exceedingly long to download due to the location/infrastructure of their servers, and comes in forms that require some preprocessing. @lucaskjaero’s pycasia library is a good starting point for managing this data.
I scaled this dataset to 64x64, and trained for around 16 hours, obtaining reasonable results. Unlike the MNIST, the large format images produce interesting textures beyond the actual contours of the characters.
Samples:
Enlarged:
Interpolation:
### Conclusion
It’s nice to finally have a CPPN model that trains properly, and for the most part avoids mode collapse, but the blurriness is not ideal for generating aesthetically pleasing images.
As GAN literature continues to expand at an astounding pace (see the Gan Zoo!), there’s still an amazing amount of potential to find new models, architectures, and designs that create pleasing and interesting art. Wasserstein distance is also only one metric out of many other new approaches, such as least squares (LS-GAN). There certainly will be more progress to be made, as researchers continue to refine this area of study.
References:
• https://arxiv.org/pdf/1701.04862.pdf
• https://www.alexirpan.com/2017/02/22/wasserstein-gan.html
• https://arxiv.org/pdf/1704.00028.pdf
• http://blog.otoro.net/2016/04/01/generating-large-images-from-latent-vectors/
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2019-04-25 19:48:21
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https://studydaddy.com/question/it-236-latest-version-set-week-7-dq-1
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QUESTION
In this file of IT 236 Latest Version Set Week 7 Discussion Question 1 you will find the next information:
Please post a 150-300-word response to the following discussion question by clicking on Reply.
Using the library, research an article that describes the importance of usability when designing a website. Provide a summary of the article and explain the qualities and importance of effective design.
• @
Tutor has posted answer for $5.19. See answer's preview$5.19
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2018-04-21 19:22:35
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http://ngdr.net/Manifold/PortableRailway/Overview/index.html
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Go Back To A Portable Railway
## A Portable Railway Overview
My portable railway came about mainly because of the confluence of two drivers: the loss of my Garden Railway, due to a house move; and the need for our small group of live-steamers to have a railway for public show use. Such a railway imposes a lot of logistical, hard-work, and operational, difficulties on its users; none of which has anything to do with running trains. These three difficulties are (non-exhaustively!) exemplified by: storage and transportation, setup and teardown, and track levelling and reliable point switching. Clearly, these issues had to be addressed by the design to the extent possible. Also, I wanted to produce a flexible railway if at all possible, one that could be assembled with different track configurations (think H0 train sets); so the railway module design had to support extensibility by being standardized in a simple way.
When I started thinking and designing I thought that it was more likely than not that I should find it infeasible actually to make something: the railway would be time and money expensive, storage and transportation always are problematic, finding a site for regular use might be difficult, and so on. This pessimism was somewhat liberating! I could take my time. Also, I could design something that might be idealistic, but fun to do. As it has worked-out, hardware now exists - I was wrong in my pessimism; although it remains to be seen if building the railway is sensible or not.
My thought processes during design are largely lost in the mists of time; but I followed the fun path mentioned above and did what appealed to me. The railway is not up and running, as can be seen in the image, which shows half a dozen modules put together arbitrarily (think H0 train sets). Clearly, I have not yet got to track-laying.
The initial construction project comprises twenty-four of these modules; six are shown above, but all of the twenty-four are at the same stage of completion. These modules form a very ordinary oval that fits in a rectangle 32ft[11m] by 44ft[14m], with 15ft[4.6m] radius curves, and an oval centreline length of 115ft[35m]. Junctions for steamup bays, not yet made, also can be seen in the diagram. What follows is a high level description of how I arrived at this point.
Abstract Design
Go Back to the Top
Engineering and software coding feature heavily in my work background; so producing track design software was an early and natural activity for me. Non-technical people may find it frightening, but things such as the abstraction of using three two-dimensional vectors to model a module, together with appropriate vector arithmetic, etc., has been highly successful. Apart from the track design aspect - see the diagram above - the mathematical and computer models have provided a cohesive foundation for everything else, including module structural frame design, full-size paper templates for module manufacture, and module hardware size verification.
Concepts that come out of the abstraction are not always conventional. For example, curves are defined by their number of modules per circle and their arc length, and not by their radius; the latter being an arithmetic result of the former two. Thus, in the track diagram, the main curves are 16mpc, i.e. sixteen modules per circle, so they each have a turn of $22\frac{1}{2}$ degrees. And the arc length is specified to minimize the expense of rail/track wastage. Track and rail is sold in the USA in lengths of 72 inches (six feet, [1.83m]). In the railway now under construction, the curved modules' arc length is set to 69 inches; resulting in the longest rail on a double track, 16mpc, gauge 3, module being 71.45 inches. The tracks centreline spacing is 10 inches [254mm]; and the outer track, module, and inner track radii are 15.06[4.59], 14.64[4.46], and 14.23[4.34] feet[m]
Part of the "fun thing to do" of the mathematical and computer modelling was to implement transition curves in the form of Euler spirals. The Euler spiral is the modern transition of choice for roller coasters, roads, and railways, and I wanted to understand it. Thus, in the diagram, the four modules adjacent to the two straights are transitions to and from constant curvature modules.
Interface
Go Back to the Top
The first hardware task to be considered was the interface system, i.e., how the modules are connected together. As an acknowledgment, some encouragement and inspiration for my solution ideas came from reading about the Connect and Module systems developed by the Gauge 3 Society; although, for various reasons, my design is quite different. Note that the Society writings about the systems mentioned are not publically viewable.
The basic interface plate comprises an aluminium angle with a locating pin and locating hole. Two interface plates are held together by commercial over-centre latches mounted on the bottom side of the angle. These two things, pins and latches, provide a precise, quick, and secure, connection. However, the design does produce a rolling torque about the bottom edge of the aluminium angle, generated by the latches, and which must be resisted by the module that hosts the plate.
An interface plate can be attached to any module, there is no requirement about how the plate is used. However, a normal expectation is that the centreline of a track be aligned with (midway between?) a locating pin/hole pair. Normally, a basic module has two plates, one at each end; similarly, a module which is a simple single track railway junction will have three interfaces. More complex interface plates come into being as the result of the subject railway design; for example, a two-track railway usually has two interfaces, one for each track, although probably they will be implemented in one piece of metal. The key thing to grasp is that it is the module design that specifies the number and placement of interfaces; but all the interfaces are identical. The modules that I have constructed are standardized straights, curves, and straightforward junctions, so that railway re-configuration enables straightforward module re-use (think H0 train sets).
This image shows the features described above. The modules are laying upside-down showing a single track module latched to a double track module. The hole in the centre of the double track interface is a "service hole", it has no function, I put it there in case a hole in the module interface plate centre ever is needed. The next hole is the second track locating hole, and on the other side of the unused latch is the second track locating pin. The hole between these locators is offset from the centre of the (single) track interface, and also is a service hole; it is used by instrumentation that measures a finished module.
Structural Ladder Frame
Go Back to the Top
I decided to to use a ladder frame with a single stressed skin for my modules. The sides, being key structural members, are aluminium bar. However, the ladder ribs are wood. Aluminium ribs would be better from some points of view, for example, if a module were destined to be installed outside in the weather. However, module manufacture would be more complex with aluminium, probably requiring welding or pop-rivets, etc., especially when leg connections (next section) are brought into the picture. The stressed skin is, of course, the track deck. This, too, is wood, being Baltic Birch plywood. Although, if I had decided on aluminium ribs, probably I should have chosen some form of aluminium sandwich board for the deck. An all-aluminium module is very appealing for permanent outside use.
Modules are glued together using high-quality epoxy. Mechanical fasteners and welding were contenders; but, once it was decided to use wood in the construction, using glue appeared to be the simplest course. I solder, but do not weld, so it would have been necessary to buy equipment and learn how to weld: too hard at present. Mechanical fasteners were judged to involve too much accurate drilling and possibly thread tapping, again, too hard.
A second skin on the bottom of the ribs would make a module very stiff. However, first indications are that a single stressed skin is perfect for the application. The current modules are strong, light, and dimensionally stable; but, also, are able to twist slightly. This characteristic allows a little track super-elevation simply by using shorter legs on the inside of a curve.
A module assembly jig was made during prototype development. In use, a full-size paper template is installed in the jig. The jig has interface plates that can be moved to accomodate module size and interface plate alignments as shown on the template. Component pieces are prepared using the template; for example, aluminium sides need to be pre-curved if the module requires it. A module is glued together in the jig, and can be removed after a twenty-four hour glue cure.
The image shows a two-track, curved module, ladder frame being constructed. The four separate interface plates are for subsequent modules; they show the wood blocks used to glue the plates to the sides.
Then the ladder frame is glued to the plywood deck.
Legs
Go Back to the Top
In the early designs the legs were the usual folding-table steel-tube-frame hinged-at-the-top arrangement. But after some time I decided to separate modules and legs entirely.
The current design puts sockets in the module frame wood ribs. The sockets are aluminium tubing with steel washers at the bottom to protect the wood. Support for a module is left to be done in whatever way is appropriate for the operation site, but presumably using the sockets. Also shown in this image is a keeper, which is the manufacturer's term for an anchor for an over-centre latch.
Usually, legs are required. The current legs are assemblies of aluminium telescoping tubing with a collet lock for length adjustment. Each leg has a rubber foot, with a steel washer insert to spread the load. At the top end is a magnet glued onto a non-magnetic support washer glued into the upper tube. Most modules have six sockets, and a leg can be put into any of these. The leg magnet binds to the steel washer in the socket, holding the leg in place. The magnet is there solely for setup purposes: the magnet is strong enough to hold the leg in place during module maneuvering, but weak enough to release comfortably when the leg is pulled out. In use, the weight of modules and trains is supported by the steel washers, the aluminium tubing, the collet lock, and the rubber foot. The legs allow a track deck height from 24 inches [0.6m], which is kid's viewing height, to 42 inches [1.1m], which is old geezer's train-running height. The adjustable legs also enable setup on rough or sloping ground. The legs are stored in boxes separately from the modules, which aids in storage and transportation. One of these boxes is just visible under the junction module in the image below.
Miscellaneous
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The description above gives an overview and the status of this portable railway project. The project is continuing on the two areas of work required for project completion: racks for storage and transportation, and track-laying. What follows are a few items that may be of interest.
• The little green bugs on the tops of the module decks are spirit levels that indicate both along and across the module. It is hoped that these will be a lot less obtrusive once the tracks and other railway stuff are in place.
• Assembly of the oval railway by two people takes about one minute per module, and there are twenty-four modules. However, this time does not include fetching, carrying, etc. Assembly can be done by one person, but two is vastly better. What is needed is a strong person and an agile person, agile at the end to be connected, strong at the free end of the new module. Agile latches the new module to the old assembly; then strong holds the module exactly level and becomes an unmovable rock, while agile plugs in the required legs and tightens the collet locks; they both let go and check the installation, maybe adjust it - blaming each other for the discrepancy; and then they move on to the next module.
• All modules are self-standing with three legs; however, the single track modules are a bit wobbly by themselves. This condition changes as the railway is assembled; the railway acquires rigidity and heft as each module is added. It remains to be seen if leg stiffening is required at key points.
• Each module weighs less than 15lbf [6.8kgf]. Track will add to this.
• Each leg weighs 1.25lbf [0.6kgf]
• Storage racks (underway) are 76inches[1.9m] by 24inches[0.6m], and allow 4inches[10cm] height for each module. All modules fit into this envelope with space to spare for track.
• All the module aluminium was primed with etching primer, this is the green paint on the interface plates; the wood was primed with exterior water-based primer. The brown is two coats of water-based porch paint.
Go Back To A Portable Railway
last-modification-date: 25 Jul 2017
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2018-11-16 10:23:27
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http://www.reference.com/browse/wiki/Pseudo-Goldstone_boson
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Definitions
# Pseudo-Goldstone boson
Pseudo-Goldstone bosons arise in a quantum field theory with an approximate symmetry such that if the symmetry were exact, then there would be spontaneous symmetry breaking (SSB) and the consequent formation of Goldstone bosons. The properties of these pseudo-Goldstone bosons can often be found by an expansion around the symmetric theory in terms of the explicit symmetry breaking parameter.
Quantum chromodynamics (QCD) provides the most well-known example: see the article on QCD vacuum for details. Experimentally it is seen that the masses of the octet of pseudoscalar mesons (such as the pion) are very much lighter than the next heaviest states, ie, the octet of vector mesons (such as the rho).
In QCD this is interpreted as the spontaneous symmetry breaking of a version of QCD with 3 flavours of massless quarks. Such a theory has global $SU\left(3\right) times SU\left(3\right)$ chiral flavour symmetry. Through SSB this is broken to the diagonal $SU\left(3\right)$, generating eight Goldstone bosons, which are the pseudoscalar mesons which lie in the octet representation of flavour $SU\left(3\right)$.
In real QCD, the quark masses break the chiral symmetry explicitly. The masses of the true pseudoscalar meson octet are found by an expansion in the quark masses which goes by the name of chiral perturbation theory. The internal consistency of this argument is further checked by lattice QCD computations allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark mass is as required by chiral perturbation theory.
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2013-06-19 21:45:26
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https://www.biostars.org/p/482952/#9462625
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Another infer_experiment.py interpretation question
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Entering edit mode
11 months ago
Hello there,
So I was given a bacterial RNA-Seq experiment to analyze and after generating alignments I wanted to know if the reads were stranded or not. So I ran infer_experiment.py on each BAM file which returned the following results:
This is PairEnd Data
Fraction of reads failed to determine: 0.0000
Fraction of reads explained by "1++,1--,2+-,2-+": 0.1704
Fraction of reads explained by "1+-,1-+,2++,2--": 0.8296
This is PairEnd Data
Fraction of reads failed to determine: 0.0030
Fraction of reads explained by "1++,1--,2+-,2-+": 0.3413
Fraction of reads explained by "1+-,1-+,2++,2--": 0.6556
Unknown Data type
This is PairEnd Data
Fraction of reads failed to determine: 0.0000
Fraction of reads explained by "1++,1--,2+-,2-+": 0.2744
Fraction of reads explained by "1+-,1-+,2++,2--": 0.7256
This is PairEnd Data
Fraction of reads failed to determine: 0.0000
Fraction of reads explained by "1++,1--,2+-,2-+": 0.2090
Fraction of reads explained by "1+-,1-+,2++,2--": 0.7910
Unknown Data type
This is PairEnd Data
Fraction of reads failed to determine: 0.0015
Fraction of reads explained by "1++,1--,2+-,2-+": 0.2698
Fraction of reads explained by "1+-,1-+,2++,2--": 0.7288
Unknown Data type
This is PairEnd Data
Fraction of reads failed to determine: 0.0000
Fraction of reads explained by "1++,1--,2+-,2-+": 0.2101
Fraction of reads explained by "1+-,1-+,2++,2--": 0.7899
Clearly, this does not answer my question. I've read the many posts on here and elsewhere about interpreting infer_experiment results, but this appears to be a unique case. Do you think aberrant antisense transcription played a part?
My main questions are, (1) what might have caused this, and (2) is the data usable for downstream analysis (differential expression)?
Many thanks for your help, it is much appreciated.
Lyn
UPDATE
It's supposed to be STRANDED. So apparently that's not completely the case. Any advice before moving forward?
RNA-Seq sequencing • 650 views
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Entering edit mode
Are you not able to find out which kit was used to generate these libraries? Have you visually examined the alignments in a genome browser? That can give you an idea as well.
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Entering edit mode
Thanks GenoMax. Unfortunately I don't have any information about the kit that was used (but I'm working on it). I've just looked at the alignments in IGV and it appears to be an unstranded library:
https://imgur.com/2lLcNqN
But does the infer_experiment output indicate a bias towards fr-firststrand - could this be a problem when counting features? Or should I just treat it as unstranded and move on?
Thanks!
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Entering edit mode
Are you sure you are using the correct reference? Double check that. Perhaps the strain you have has genomic rearrangements (and you may be using standard reference from NCBI?) Data does look stranded was opinion of a fellow moderator.
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Entering edit mode
Thanks again. I've tried now with an NCBI, Ensembl, and custom reference and infer_experiment gives the same results. I've actually got PacBio data from the same samples used for RNA-Seq and de novo assembly of each gives a single contig that is identical to the reference. So at this point do you think that it could be a poor annotation for this organism (GTF from both NCBI and Ensembl) or a failed library prep? And is this going to pose a problem? Thaaanks! :-)
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Entering edit mode
Poor annotation might be the reason, Do you know from where the annotation is coming?
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Entering edit mode
8 months ago
dare_devil ★ 1.5k
Ironically the output of the tool supposed to help you decide only adds to the confusion.
Here is a better way to do it, that does not require running tools, instead asks you to understand your own data.
Go to the documentation for GUESSmyLT you don't have to run the tool, instead study and understand what the possible orientations are as described in the docs:
https://github.com/NBISweden/GUESSmyLT
Now open up your BAM and transcript GTG files in IGV and select "view as pairs" options (right-click the panel). Now all you need is to look at your data relative to a gene and visually evaluate which case do you have. The patterns are usually absolutely evident at first glance.
courtesy
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Entering edit mode
When you quote someome verbatim, you need to make that clear by using quotation marks or a blockquote liks so:
this text is quoted
(see the quote icon above when you are in edit mode)
In addition you need to provide a reference to the original source. What you wrote is a copy-paste from:
Answer: Infering strand specificity of bam files using rseqc?
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2021-12-03 20:28:22
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https://www.gradesaver.com/textbooks/math/geometry/CLONE-68e52840-b25a-488c-a775-8f1d0bdf0669/chapter-10-review-exercises-page-486/35
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## Elementary Geometry for College Students (6th Edition)
We say that one corner of the triangle is at the origin, the base is of length a, and the two congruent sides are of length 2b. Thus: $a^2 + b^2 = (d_1)^2$ $a^2 + b^2 = (d_2)^2$ It follows: $(d_2)^2=(d_1)^2 \\ d_2=d_1$ Hence, the medians have equal lengths.
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2021-03-01 11:07:53
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http://www.stephenkinsella.net/tag/indifference-curve/
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Economics for Business Lecture 9: Production
The theory of production of commodities by means of commodities has a long history in economics. From the earliest recorded thinking about how households manage their inputs and outputs through to the modern day, the concern with how and why and what people produce in order to satisfy their needs and desires has been at the core of economics.
So it's important you have a look at the main approach to thinking about production.
We have a function, which is a rule which specifies, for each value of $x$, in the domain of $f$, prescribes a unique value of $y$ such that $y=f(x)$. We always want a function to be able to answer the question: 'Given a numerical value of $x$, what is the unique corresponding numerical value $y = f (x)$'?
The core idea of today's lecture is that the firm is just a machine which produces outputs from given inputs. Workers work the machines and receive a wage, and capitalists own the machines, and received profits (or losses) for their investment in the machines.
We represent the relationship between a firm's inputs and its outputs through a production function, which specifies how many units of output you'll get for a given combination of inputs.
What kinds of inputs are we talking about here? We have to consider hours of labour, $L$, spent on the production of the good, the amount of capital $K$ spent in getting the productive capacity ready to actually crank out units of the good, and amounts of raw materials $M$ used up in the productive process. We summarise the relationship between inputs and outputs through the production function, which takes the form
$q = f(K, L, M, ldots)$.
We leave an elipsis (those dots) at the end of the function to indicate other things might actually affect the output of the good. Most of the time, we just consider how much capital, $K$, and labour, $L$, are used in the production of the good, so our production function becomes
$q = f(K, L)$.
How are inputs to the productive process decided upon? We look at their mariginal product.
The marginal product of a production process asks the same type of question as marginal utility. The basic idea is to ask how much extra output of $q$ one might get for a one unit increase in the level of capital or labour used as inputs.
We have two definitions then: marginal product of labour, and marginal product of capital. It's worth writing these down somewhere, and checking the textbook for examples.
Marginal product of labor (MPL): the extra output obtained by employing one more unit of labor while holding the level of capital equipment constant.
Marginal product of capital (MPK): the extra output obtained by using one more machine while holding the number of workers constant.
We'll look at the relationship between total product, marginal and average product, and levels of output, using this Mathematica demonstration.
Isoquants
Once you have calculated the optimal level of output consistent with marginal products of labour and capital, one can think about trying to get the most output from differing combinations of those inputs. We'll study Leontieff-type production functions using this Mathematica demonstration.
Image via Wikipedia
We use the isoquant to show us outputs of the same level produced with different combinations of labour and capital. The isoquant map is infinite, meaning for any combination of capital and labour, one can find an isoquant at that combination. The isoquant is similar to the indifference curve we saw in earlier lectures, because each represents the contour of a surface, or the 'altitude' of production. The slope of the isoquant is given by the change in capital input divided by the change in labour input, a relationship we'll call the rate of technical substitution.
The Marginal Rate ot Technical Substitution (RTS) is the amount by which one input can be reduced when one more unit of another input is added while holding output constant (i.e. negative of the slope of an isoquant).
Mathematically we can solve for the RTS by calculating
$RTS_{L, K} = frac{MP_{K}}{MP_{L}}.$
Returns to Scale
A production function is said to exhibit constant returns to scale if a doubling of all inputs results in a precise doubling of output. If doubling all inputs results in more than a doubling of output, the production function exhibits increasing returns to scale. Although an intellectual storm has raged about the assumptions underlying returns to scale, in practice most mainstream economists assume CRTS for simplicity. Whether this is a good idea is the subject of a more advanced course.
Changes in Technology
Technology affects the production of goods, so we add another element to the production function to take account of the changes in technology. Call this element $A$, and our production function becomes
$y = Af(K, L)$.
This function says technology affects capital and labour in the same way. We measure the components of $A$ using total factor productivity.
Exercises to try:
7.2, 7.3, 7,7.
Next lecture: Costs, read chapter 8.
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2014-09-19 05:47:50
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https://www.physicsforums.com/threads/an-electron-moving-across-capacitor.41330/
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# An electron moving across capacitor
1. Aug 31, 2004
### spatel600
An electron is launched at a 45 angle and a speed of 5.0*10^6 from the positive plate of the parallel-plate capacitor shown in the figure. The electron lands 4.0 cm away.
What is the electric field strength inside the capacitor?
What is the minimum spacing between the plates?
So what I have put together so far is that a=(q*E)/m but what does E equal to. Then once I have acceleration it becomes a kinematics.
#### Attached Files:
• ###### knight.Figure.26.54.jpg
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2. Aug 31, 2004
### chroot
Staff Emeritus
The acceleration is found via the Lorentz force law and Newton's second law. Once you have the acceleration, as you say, it's just kinematics. It sounds like you have this one already figured out -- where exactly are you getting stuck?
- Warren
3. Sep 1, 2004
### spatel600
We haven't covered Lorentz force law.
Any more tips?
Thanks a bunch.
4. Sep 1, 2004
### chroot
Staff Emeritus
The Lorentz force law is:
$$\mathbf{F} = q \mathbf{E} + \mathbf{v} \times \mathbf{B}$$
When the magnetic field is zero, it reduces to just
$$\mathbf{F} = q \mathbf{E}$$
Given the electric field strength, all you need to do to find the force on a particle is multiply by the particle's charge.
- Warren
5. Sep 1, 2004
### spatel600
But what is the E field...its not given...
Any input?
Thanks.
6. Sep 1, 2004
### spatel600
This what i have so far:
(x)t = Vox * t
t=1.14*10^-7 seconds
I broke up the v0 into xy components v0x=3.5*10^5
v0y=3.5*10^5
The problem now is that
F=qE =
a=qE/m
I don't know the acceleration nor the E so this is where i am stuck at
THANKS SO MUCH
7. Sep 1, 2004
### chroot
Staff Emeritus
Do you not have a book? All of the things you're asking should be in it, probably even indexed.
The electron begins with a vertical (upward) velocity of $v_{0y} = (5 \cdot 10^6~\textrm{m/s}) / \sqrt{2} = 3.5 \cdot 10^6~\textrm{m/s}$, and a horizontal (rightward) velocity of the same magnitude, as you found.
The electric field is perpendicular to the plate, so it affects only the vertical motion, not the horizontal motion -- just like gravity.
The electron travelled 4.0 cm at its constant horizontal velocity. That took
$$t = \frac{0.04 \textrm m \cdot \sqrt{2}}{5 \cdot 10^6~\textrm{m/s}} = 11.31 \cdot 10^{-9}~\textrm{s}$$
During that 11.31 nanosecond period, the electron's altitude above the plane went from zero, up to some maximal value, and back to zero again. The standard kinematic equation you need is
$$s(t) = v_0 t + \frac{1}{2} a t^2$$
Plug in the initial vertical velocity and solve for the acceleration.
Now that you have the acceleration, you can use Newton's second law to get the force.
Now that you have the force, you can use the Lorentz force law ($\mathbf F = q \mathbf E$) to get the E field.
The second part of the question is just asking you to the find the maximum altitude the electron gains above the plate. This is another basic kinematic result:
$$y_\textrm{max} = \frac{v_{0y}}{2a}$$
- Warren
8. Sep 1, 2004
### robphy
Think about the kinematics problem first.
Hint: Can you find the acceleration somehow?
Once you have acceleration, you can (using F=ma) find E since you already know (or can look up) the charge and mass of the electron.
To answer the last part, you might want to look at the kinematics problem again.
9. Sep 1, 2004
### spatel600
Thanks so much guys, I really appreciate it!!
!!
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2018-02-18 07:19:59
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https://math.stackexchange.com/questions/1859320/multiplicity-of-cartier-divisor-on-locally-noetherian-scheme-is-only-non-zero-at
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# Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point
I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'.
My question concerns Definition 1.27:
Let $A$ be a Noetherian local ring of dimension 1. For any regular element $\ f \in A, \text{length}_A(A/fA)$ is finite. The map $f \mapsto \text{length}_A(A/fA)$ extends to a group homomorphism $\text{Frac}(A)^\times \rightarrow \mathbb{Z}$. Moreover, its kernel contains the invertible elements of $A$. Thus we obtain a group homomorphism $\text{mult}_A:\text{Frac}(A)^\times/A^\times \rightarrow \mathbb{Z}$.
This map will be used to define $\text{mult}_x(D)$ of a Cartier Divisor $D$ at a point $x \in X$:
Let $X$ be a locally Noetherian scheme. Let $D \in \text{Div}(X)$ be a Cartier divisor. For any point $x \in X$ of codimension 1, the stalk of $D$ at $x$ belongs to $$(\mathcal{K}_X^\times/\mathcal{O}_X^\times)_x = \text{Frac}(\mathcal{O}_{X,x})^\times/\mathcal{O}_{X,x}^\times,$$ thus we can define $$\text{mult}_x(D) := \text{mult}_{\mathcal{O}_{X,x}}(D).$$ Now let $U$ be an open everywhere dense (does this simply mean dense?) subset of $X$ such that $D_U = 0.$ Then: Any $x \in X$ of codimension 1 such that $\text{mult}_x(D) \neq 0$ is a generic point of $X-U$.
My Question: Why does the last statement hold?
The result seems to be very important and fundemanetal (hence the question), since following the statement, a Cartier divisor is determined on the generic points of $X$. Further, for assigning Weil divisors to Cartier divisors, we need that there are only finitely many prime divisors at which the Cartier divisor is non-zero - which is (with the above) a consequence if $X$ is assumed to be Noetherian.
I would be very grateful for any kind of help.
• I don't know the exact answer to your question right away, but I note one red flag: the highlighted text talks about generic points of $X-U$, but you talk about generic points of $X$. These are very different things! Jul 14 '16 at 14:51
Note that $D_U=0$ shows that multiplicity non-zero in particular implies $x \in X-U$, hence the closure of $x$ is contained in $X-U$ and has the same dimension by assumption. ($X-U$ having codimension at least $1$ is the everywhere dense condition)
Here is a rigorous proof using your definitions.
Let $x$ be a point with multiplicity non-zero. As I stated above, $D_U=0$ implies $x \in X-U$.
Let $y \in X-U$ with $x \in \overline {\{y\}}$. We have to show $x=y$, which is equivalent to $\overline {\{x\}}=\overline {\{y\}}$ (This equivalence essentially uses, that we work with a scheme. On arbitrary topological spaces, this is of course wrong).
Assume this does not hold, then we have $\overline {\{x\}} \subsetneq \overline {\{y\}}$. By the co-dimension $1$ assumption for $x$, we get that $\overline {\{y\}}$ is a maximal irreducible subset of $X$. Note that $\overline {\{y\}} \subset X-U$, since $y \in X-U$.
Now let $\mathcal Z$ be the set of maximal irreducible closed subsets of $X$. I claim that $S = \bigcup\limits_{Z \in \mathcal Z, Z \neq \overline {\{y\}}} Z$ is a closed subset. In particular it is a proper closed subset (it does not contain $y$), which contains $U$. This is a contradiction to the assumption that $U$ is dense.
To proof my claim, I will proof the following lemma:
Let $X$ be a locally noetherian scheme and $Z_i, i \in I$ a collection of maximal irreducible closed subsets. Then their union is a closed set. (Note that infinite unions are not closed in general)
Proof: Let $U$ be an open affine subset. By the locally noetherian assumption, $U$ is noetherian. For any $i$ $U \cap Z_i$ is an irreducible subset of $U$. By the noetherian property, the union $\bigcup_{i \in I} U \cap Z_i$ is actually a finite union, i.e. closed. Since we can cover $X$ be such $U$'s and we can test being closed on an open cover, the result follows.
You should note that the proof becomes essentially easier, when one assumes $X$ to be noetherian, because we have finitely many irreducible components then and thus can immediately derive that $\overline {\{x\}}$ is an irreducible component of $X-U$, which is equivalent to be a generic point.
• So, $U \subset X$ everywhere dense $\Leftrightarrow\ \text{codim}_X(X-U) \geq 1$? Jul 14 '16 at 16:05
• Only the forward direction is true (Think of the union of a curve and an isolated closed point. Take the open subset to be the curve). If all irreducible components have the same dimension, then the equivalence holds.
– MooS
Jul 14 '16 at 16:12
• Then, would you mind telling me what the definition of everywhere dense is? What is the difference to just dense? Because this would imply that for any $x \in X-U$ we have $\text{codim}_X \{x\}^- \geq 1$. This should further imply (how?) that if the codimension of such $x$ is exactly 1, that $\{x\}^- = X-U$, right? Jul 14 '16 at 16:15
• It is the same as dense. Meaning that $U$ does not leave out an irreducible component.
– MooS
Jul 14 '16 at 16:17
• Do we have to worry about the underlying space being not necessarily noetherian?
– Hoot
Jul 14 '16 at 16:55
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2021-12-07 00:24:23
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https://www.math.ias.edu/seminars/abstract?event=3085
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# Explicit Constructions of Bipartite Ramsey Graphs
COMPUTER SCIENCE/DISCRETE MATH SEMINAR, II Topic: Explicit Constructions of Bipartite Ramsey Graphs Speaker: Boaz Barak and Guy Kindler Affiliation: IAS Date: Tuesday, October 26 Time/Room: 10:30am - 12:30pm/S-101
The main goal of this talk will be to present a proof of the following theorem. Theorem 1: For every fixed \delta >0 here is a polynomial time (in n = log N) computable function(s) f:[N]x[N]-->{0,1}, for which the following hold. For every two sets A, B of [N], each of size at least K=N^{\delta}, we have f(AxB)={0,1}. If one thinks of f as a 2-coloring of the edges of the complete NxN bipartite graph, then the edges of no KxK subgraph are monochromatic (indeed we'll guarantee that each color will be represented in a constant fraction of all edges). No explicit construction was known for \delta < 1/2. Quite a few other new constructions (interesting in their own right) are needed on the way to construct f itself, and we will describe them too. They include a constant seed condenser, a constant seed 2-source extractor, a deterministic 3-source extractor, among others. Similar techniques allow us to achieve another explicit Ramsey construction: For every \delta we have an explicit 2-coloring of the n-dimensional cube GF(2)^n, such that no affine subspace of dimension (\delta)n is monochromatic. Again, nothing was known for \delta <1/2. If time permits we will sketch what is needed for this result as well. An essential ingredient in all constructions is the recent "multiple source" extractor of Barak, Impagliazzo and Wigderson, which in turn was based on the sum-product theorem for finite fields of Bourgain, Katz and Tao. The talk will be self contained. Joint work with Benny Sudakov, Ronen Shaltiel and Avi Wigderson.
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2020-07-02 20:21:57
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https://www.scijournal.org/articles/mathbb-command-in-latex
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How to write with the mathbb in LaTeX?
# How to write with the mathbb in LaTeX?
This post may contain affiliate links that allow us to earn a commission at no expense to you. Learn more
This article aims to show you the simplest and easiest way to write with the mathbb command in LaTeX.
Maybe you have heard or read about special commands for different fonts to use in LaTeX, today you will learn how to use it in your document
## Blackboard Bold
This font is usually used in mathematics to represent the set of real numbers R, denoted by white spaces between each line composing the letters, usually just uppercase letters are used with this font. Some people called it Double-Struck.
## Mathbb Command in LaTeX
For some reason LaTeX does not have the ability to write with this font by default, therefore we will need to use external packages. Remember to load the packages in the preamble of your document.
There are two main packages that you can use, the amssymb packages or the amsfonts package. Both of them work great for our purpose. To write this symbol or sign, you use the command \mathbb{}, inside the brackets {} goes the argument you will want to write with the font.
For example
\documentclass{article}
\usepackage{amssymb} %we added the package to the document
\begin{document}
The set of real numbers is denoted by $\mathbb{R}$ %the command is used here
\end{document}
Output for \mathbb command
It is important to notice that the new command is used inside the math mode, so pay attention where you place this command or LaTeX will hit you with an error message. Another example could be
\documentclass{article}
\usepackage{amsfonts} %we added the package to the document
1) Other set of numbers can be $\mathbb{N}$ or $\mathbb{Q}$.\\
\vspace{5pt}
2) $\backslash$mathbb\{\} does not work with\\
Greek letters $\mathbb{\alpha}$ is the same as $\alpha$
\end{document}
One key aspect regarding the command is that it only works with letters, it does not work with greek letters or symbols or numbers. So better do your little research if you’re going to write with this font or command in other languages.
I hope this tutorial was helpful, and as always keep writing in LaTeX
All the images were created in LaTeX by the author.
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2022-08-16 06:32:54
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https://ask.sagemath.org/answers/46079/revisions/
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Note that the mapping is just taking the binary digits of the integer and using them as coefficients w.r.t. the basis $1,x,x^2,\ldots$
You can use the built-in integer_representation() method, or do the job manually:
sage: a = F.fetch_int(3)
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2021-03-02 16:51:17
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http://www.ck12.org/book/CK-12-Chemistry---Second-Edition/r13/section/22.3/
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Chemistry - Second Edition Go to the latest version.
# 22.3: Spontaneous Processes
Difficulty Level: At Grade Created by: CK-12
## Lesson Objectives
The student will:
• define a spontaneous and non-spontaneous reaction.
• identify processes as either spontaneous or non-spontaneous.
• describe how endothermic and exothermic reactions can be spontaneous or non-spontaneous.
• explain the lack of correlation between spontaneity and speed of reaction.
## Vocabulary
• non-spontaneous event
• spontaneous event
## Introduction
Some events or reactions occur without any outside forces. For example, if you drop a spoonful of sugar into a cup of water, it automatically dissolves. The sugar is said to spontaneously dissolve in water. Rusting of iron can be spontaneous under the right conditions, but to undo this process would not be spontaneous. In this lesson, we will consider spontaneous and non-spontaneous reactions in light of what we just learned with respect to enthalpy.
## Change That Occurs Without Outside Assistance
A spontaneous event (or reaction) is a change that occurs under a specific set of conditions. A ball rolling down a hill, the water falling down a waterfall, and the dispersion of the smell of a perfume sprayed in a room (expansion of a gas) are all examples of spontaneous events. In comparison, a non-spontaneous event (or reaction) would be a change that will not occur under a specific set of conditions. Can you picture a cold cup of hot chocolate on your desk becoming warmer as you sit and listen to your chemistry teacher? Probably not, because an external source of heat would be required to warm up the hot chocolate. For non-spontaneous events, something else outside of the reaction must be done in order to get the event (or reaction) to occur. This might be applying a force to make a ball roll up a hill or to ski up the mountain. Other factors that could drive a reaction could be the addition of heat, addition of a catalyst, or an increase in pressure.
Example:
Which of the following would be considered spontaneous? Which would be considered non-spontaneous? Explain.
1. cooling a cup of hot coffee at room temperature
2. ice melting at room temperature
3. compression of gas to fill a tire
4. water flowing downhill
Solution:
1. Cooling a cup of hot coffee is spontaneous because heat flows spontaneously from a hotter substance to a cooler one.
2. Ice melting is spontaneous above $0^\circ C$ because above this temperature, water is normally at the liquid state.
3. Compression of gas to fill a tire is non-spontaneous because a pressure has to be applied to a gas in order to compress it.
4. Water flowing downhill is spontaneous because water will always flow down.
## Exothermic or Endothermic Can Be Spontaneous
Spontaneous and non-spontaneous reactions can be either endothermic or exothermic. Consider the the endothermic and the exothermic potential energy diagrams shown below.
Endothermic Reaction
Exothermic Reaction
Often but not always, a spontaneous process will be one that will result in a decrease in the energy of a system. Therefore, water will spontaneously flow down a waterfall, a ball will roll down a hill, and gas will expand to fill a container. All of these changes will occur spontaneously, leaving the products (or product state) with less energy than the reactants (or reactant state). Furthermore, if a reaction is spontaneous in one direction, it is non-spontaneous in the opposite direction. For example, a ball rolling down a hill would be a spontaneous event, but a ball rolling up a hill would be non-spontaneous.
Being spontaneous doesn’t necessarily mean, however, that the reaction is exothermic. Highly exothermic reactions tend to be spontaneous, but weakly exothermic or endothermic reactions can be spontaneous under the right conditions. In other words, endothermic reactions can be spontaneous just like exothermic reactions can be non-spontaneous. Consider the equation below. This equation represents the phase change of solid water (ice) to liquid water at $25^\circ\mathrm{C}$.
$\mathrm{H}_2\mathrm{O}_{(s)} \rightarrow \mathrm{H}_2\mathrm{O}_{(l)} \ \ \ \ \ \triangle H = 6.01 \ \text{kJ/mol}$
We know that ice will spontaneously melt above $0^\circ\mathrm{C}$, and the equation above also indicates the phase change is spontaneous. Now consider the equation for combustion below. Combustion is an example of a sponataneous, exothermic reaction.
$\mathrm{C}_3\mathrm{H}_\mathrm{8(g)} + 5 \ \mathrm{O}_{2(g)} \rightarrow 3 \ \mathrm{CO}_{2(g)} + 4 \ \mathrm{H}_2\mathrm{O}_{(l)} \ \ \ \ \ \triangle H = -2219.9 \ \text{kJ/mol}$
Therefore, spontaneity does not dictate whether a reaction is endothermic or exothermic. A spontaneous reaction is more likely to be exothermic but can be endothermic. Non-spontaneous reactions are more likely to be endothermic but can be exothermic. The deciding factor for these systems is the temperature.
## Thermodynamics and Kinetics
A spontaneous process is all about the initial and final states. Reactions are considered spontaneous if, given the necessary activation energy, reactants form the products without any external forces. Therefore, an ice cube will melt, an iron nail will rust in the present of oxygen dissolved water, and a sparkler will burn. Some of these reactions are fast, and some are slow. The oxidation of iron, for example, is slow. In comparison, after the sparkler is lit, the reaction from start to finish is quite fast. Regardless of the speed of the reaction, they are both still spontaneous reactions. The rate of these reactions is the study of chemical kinetics; the spontaneity of a reaction is the study of thermodynamics. Whether the reaction occurs quickly or slowly has little to do with the reaction being spontaneous. A spontaneous reaction only means that it occurs without any continuous outside support.
## Lesson Summary
• A spontaneous event (or reaction) is a change that occurs under a specific set of conditions and without any continuous external support.
• A non-spontaneous event (or reaction) would then be a change that will not occur under a specific set of conditions.
• Being spontaneous does not indicate how fast a reaction occurs or if the reaction is exothermic or endothermic.
The learner.org website allows users to view streaming videos of the Annenberg series of chemistry videos. You are required to register before you can watch the videos but there is no charge. The website has one video that relates to this lesson called “The Driving Forces.”
## Review Questions
1. Distinguish between spontaneous and non-spontaneous reactions.
2. Why are spontaneous reactions usually exothermic (but still can be endothermic)?
3. Which of the following processes would be spontaneous?
1. dissolving table salt
2. climbing Mt. Everest
3. separating helium from nitrogen in a mixture of gases
4. none of these are spontaneous
4. Which of the following processes would be non-spontaneous?
1. iron rusting in air
2. ice melting at $10^\circ\mathrm{C}$
3. a wild fire
4. the reaction of $\mathrm{CO}_2$ and $\mathrm{H}_2\mathrm{O}$
5. Which of the following reactions are spontaneous?
1. I and II
2. I and III
3. II and IV
4. Not enough information is given
6. If a reaction is spontaneous and fast, draw a likely potential energy diagram.
Feb 23, 2012
Mar 26, 2015
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2015-11-30 05:08:23
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http://math.stackexchange.com/questions/462566/prove-that-n-geq-2n-1-for-n-geq1/462575
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# Prove that $n! \geq 2^{n-1}$ for $n\geq1$ [duplicate]
Mathematical Induction:-Prove that $n! \geq 2^{(n-1)}$ for $n\geq 1$.
I tried mathematical induction but could not
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## marked as duplicate by Andrey Rekalo, Start wearing purple, Davide Giraudo, MathOverview, Git GudAug 8 '13 at 11:27
If you search this site a little, you will find several very similar questions, for example, Proof the inequality $n! \geq 2^n$ by induction and Is this induction procedure correct? ($2^n<n!$). – Martin Sleziak Aug 8 '13 at 8:22
Base case: If n = 1, then $1 = 1! \geq 1 = 2^{1 - 1}$.
Induction step: Suppose that $n! \geq 2^{n - 1}$ for some $n \geq 1$; we must show that this holds if we replace $n$ by $n + 1$. Now we have $n + 1 \geq 2$, so
$$(n + 1)! = (n + 1) n! \geq 2 (n!) \geq 2(2^{n - 1}) = 2^{n} = 2^{(n + 1) - 1}$$
as desired. Note that the second inequality is where we use the inductive hypothesis.
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It's not that hard. First we show that if $n=1$ the claim holds: $$n! =1! =1 = 2^0 =2^{1-1}$$ Now suppose that there exist $n \in \mathbb{N}$, so that your statement holds. Then $$(n+1)!=(n+1)\cdot n! \geq (n+1) \cdot 2^{n-1}$$ The inequality above follows by your induction hypothesis. Now $$(n+1)! \geq n\cdot 2^{n-1} + 2^{n-1} \geq 1\cdot 2^{n-1} + 2^{n-1} =2\cdot 2^{n-1}=2^n$$ And this completes your proof. Is this clear?
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If $n=1$, clearly equality holds.
If $n \geq 2$, $n!=n(n-1)(n-2) \dots(2)(1)$ has the first $n-1$ factors $\geq2$, and the last factor $1$.
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2016-06-30 18:06:40
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https://www.physicsforums.com/threads/quick-limit-question.402759/
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# Quick limit question
1. May 12, 2010
### srfriggen
how do you calculate lim x-> inf 4^x/7^x+4
It seems obvious that the bottom is growing at a faster rate than the top, so the limit would be zero, but how would you show that algebraically? I tried using l'hopital and just wind up with:
4^x ln4 / 7^x ln7
and it doesn't seem that using l'hopital again and again would get me anywhere further.
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. May 12, 2010
### System
Personally, I would devide top and bottom by 7^x.
3. May 12, 2010
### srfriggen
ok so that leaves lim x->inf (4/7)^x
Is that enough proof to show it goes to zero?
4. May 12, 2010
### System
if |a|<1 then lim(x-->infinity) for a^x = 0
If you see the graph for any function a^x where |a|<1, you see this is true
You can prove that by using the convergence's tests for the infinite series ..
5. May 12, 2010
### srfriggen
ah ok,
so that would be a geometric series with l r l < 1 (which we know converges), or you can prove this converges by using the root test, 4/7 < 1 , so since the series converges the limit of the sequence or partial sums goes to zero?
do I have that all correct?
6. May 12, 2010
### srfriggen
sorry, I meant the limit of the nth partial sum goes to zero as n goes to infinity
7. Jun 3, 2010
### System
No.
The limit of the nth term is zero.
Not the limit of the sequence of the partial sums!
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2017-08-21 02:21:41
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http://strata.opengamma.io/apidocs/com/opengamma/strata/product/bond/ResolvedFixedCouponBond.html
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## Class ResolvedFixedCouponBond
• java.lang.Object
• com.opengamma.strata.product.bond.ResolvedFixedCouponBond
• All Implemented Interfaces:
ResolvedProduct, java.io.Serializable, Bean, ImmutableBean
public final class ResolvedFixedCouponBond
extends java.lang.Object
implements ResolvedProduct, ImmutableBean, java.io.Serializable
A fixed coupon bond, resolved for pricing.
This is the resolved form of FixedCouponBond and is an input to the pricers. Applications will typically create a ResolvedFixedCouponBond from a FixedCouponBond using FixedCouponBond.resolve(ReferenceData).
The list of FixedCouponBondPaymentPeriod represents the periodic coupon payments, whereas the nominal payment is defined by Payment.
The legal entity of this fixed coupon bond is identified by StandardId. The enum, FixedCouponBondYieldConvention, specifies the yield computation convention.
A ResolvedFixedCouponBond is bound to data that changes over time, such as holiday calendars. If the data changes, such as the addition of a new holiday, the resolved form will not be updated. Care must be taken when placing the resolved form in a cache or persistence layer.
#### Price
Strata uses decimal prices for bonds in the trade model, pricers and market data. For example, a price of 99.32% is represented in Strata by 0.9932.
Serialized Form
• ### Nested Class Summary
Nested Classes
Modifier and Type Class Description
static class ResolvedFixedCouponBond.Builder
The bean-builder for ResolvedFixedCouponBond.
static class ResolvedFixedCouponBond.Meta
The meta-bean for ResolvedFixedCouponBond.
• ### Method Summary
All Methods
Modifier and Type Method Description
static ResolvedFixedCouponBond.Builder builder()
Returns a builder used to create an instance of the bean.
boolean equals(java.lang.Object obj)
java.util.Optional<FixedCouponBondPaymentPeriod> findPeriod(java.time.LocalDate date)
Finds the period that contains the specified date.
Currency getCurrency()
Gets the currency of the product.
DayCount getDayCount()
Gets the day count convention applicable.
java.time.LocalDate getEndDate()
Gets the end date of the product.
double getFixedRate()
Gets the fixed coupon rate.
Frequency getFrequency()
Gets the frequency of the bond payments.
LegalEntityId getLegalEntityId()
Gets the legal entity identifier.
Payment getNominalPayment()
Gets the nominal payment of the product.
double getNotional()
Gets the notional amount, must be positive.
com.google.common.collect.ImmutableList<FixedCouponBondPaymentPeriod> getPeriodicPayments()
Gets the periodic payments of the product.
RollConvention getRollConvention()
Gets the roll convention of the bond payments.
SecurityId getSecurityId()
Gets the security identifier.
DaysAdjustment getSettlementDateOffset()
Gets the number of days between valuation date and settlement date.
java.time.LocalDate getStartDate()
Gets the start date of the product.
java.time.LocalDate getUnadjustedEndDate()
java.time.LocalDate getUnadjustedStartDate()
FixedCouponBondYieldConvention getYieldConvention()
Gets yield convention.
boolean hasExCouponPeriod()
Checks if there is an ex-coupon period.
int hashCode()
static ResolvedFixedCouponBond.Meta meta()
The meta-bean for ResolvedFixedCouponBond.
ResolvedFixedCouponBond.Meta metaBean()
ResolvedFixedCouponBond.Builder toBuilder()
Returns a builder that allows this bean to be mutated.
java.lang.String toString()
double yearFraction(java.time.LocalDate startDate, java.time.LocalDate endDate)
Calculates the year fraction within the specified period.
• ### Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait
• ### Methods inherited from interface org.joda.beans.Bean
property, propertyNames
• ### Method Detail
• #### getStartDate
public java.time.LocalDate getStartDate()
Gets the start date of the product.
This is the first coupon period date of the bond, often known as the effective date. This date has been adjusted to be a valid business day.
Returns:
the start date
• #### getEndDate
public java.time.LocalDate getEndDate()
Gets the end date of the product.
This is the last coupon period date of the bond, often known as the maturity date. This date has been adjusted to be a valid business day.
Returns:
the end date
public java.time.LocalDate getUnadjustedStartDate()
This is the unadjusted first coupon period date of the bond.
Returns:
public java.time.LocalDate getUnadjustedEndDate()
This is the unadjusted last coupon period date of the bond.
Returns:
• #### getCurrency
public Currency getCurrency()
Gets the currency of the product.
All payments in the bond will have this currency.
Returns:
the currency
• #### getNotional
public double getNotional()
Gets the notional amount, must be positive.
The notional expressed here must be positive. The currency of the notional is specified by getCurrency().
Returns:
the notional amount
• #### hasExCouponPeriod
public boolean hasExCouponPeriod()
Checks if there is an ex-coupon period.
Returns:
true if has an ex-coupon period
• #### findPeriod
public java.util.Optional<FixedCouponBondPaymentPeriod> findPeriod(java.time.LocalDate date)
Finds the period that contains the specified date.
The search is performed using unadjusted dates.
Parameters:
date - the date to find the period for
Returns:
Throws:
java.lang.IllegalArgumentException - if more than one period matches
• #### yearFraction
public double yearFraction(java.time.LocalDate startDate,
java.time.LocalDate endDate)
Calculates the year fraction within the specified period.
Year fractions on bonds are calculated on unadjusted dates.
Parameters:
startDate - the start date
endDate - the end date
Returns:
the year fraction
Throws:
java.lang.IllegalArgumentException - if the dates are outside the range of the bond or start is after end
• #### meta
public static ResolvedFixedCouponBond.Meta meta()
The meta-bean for ResolvedFixedCouponBond.
Returns:
the meta-bean, not null
• #### builder
public static ResolvedFixedCouponBond.Builder builder()
Returns a builder used to create an instance of the bean.
Returns:
the builder, not null
• #### metaBean
public ResolvedFixedCouponBond.Meta metaBean()
Specified by:
metaBean in interface Bean
• #### getSecurityId
public SecurityId getSecurityId()
Gets the security identifier.
This identifier uniquely identifies the security within the system.
Returns:
the value of the property, not null
• #### getNominalPayment
public Payment getNominalPayment()
Gets the nominal payment of the product.
The payment date of the nominal payment agrees with the final coupon payment date of the periodic payments.
Returns:
the value of the property, not null
• #### getPeriodicPayments
public com.google.common.collect.ImmutableList<FixedCouponBondPaymentPeriod> getPeriodicPayments()
Gets the periodic payments of the product.
Each payment period represents part of the life-time of the product. The start date and end date of the leg are determined from the first and last period. As such, the periods should be sorted.
Returns:
the value of the property, not null
• #### getFrequency
public Frequency getFrequency()
Gets the frequency of the bond payments.
This must match the frequency used to generate the payment schedule.
Returns:
the value of the property, not null
• #### getRollConvention
public RollConvention getRollConvention()
Gets the roll convention of the bond payments.
This must match the convention used to generate the payment schedule.
Returns:
the value of the property, not null
• #### getFixedRate
public double getFixedRate()
Gets the fixed coupon rate.
The periodic payments are based on this fixed coupon rate.
Returns:
the value of the property
• #### getDayCount
public DayCount getDayCount()
Gets the day count convention applicable.
The conversion from dates to a numerical value is made based on this day count. For the fixed bond, the day count convention is used to compute accrued interest.
Note that the year fraction of a coupon payment is computed based on the unadjusted dates in the schedule.
Returns:
the value of the property, not null
• #### getYieldConvention
public FixedCouponBondYieldConvention getYieldConvention()
Gets yield convention.
The convention defines how to convert from yield to price and inversely.
Returns:
the value of the property, not null
• #### getLegalEntityId
public LegalEntityId getLegalEntityId()
Gets the legal entity identifier.
This identifier is used for the legal entity that issues the bond.
Returns:
the value of the property, not null
• #### getSettlementDateOffset
public DaysAdjustment getSettlementDateOffset()
Gets the number of days between valuation date and settlement date.
This is used to compute clean price. The clean price is the relative price to be paid at the standard settlement date in exchange for the bond.
It is usually one business day for US treasuries and UK Gilts and three days for Euroland government bonds.
Returns:
the value of the property, not null
• #### toBuilder
public ResolvedFixedCouponBond.Builder toBuilder()
Returns a builder that allows this bean to be mutated.
Returns:
the mutable builder, not null
• #### equals
public boolean equals(java.lang.Object obj)
Overrides:
equals in class java.lang.Object
• #### hashCode
public int hashCode()
Overrides:
hashCode in class java.lang.Object
• #### toString
public java.lang.String toString()
Overrides:
toString in class java.lang.Object
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2019-06-26 12:43:50
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https://cforall.uwaterloo.ca/trac/changeset/a2ea829dc9b8ad1d478141b95be3b1e4b558d102
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# Changeset a2ea829
Ignore:
Timestamp:
Nov 6, 2017, 12:23:13 PM (4 years ago)
Branches:
aaron-thesis, arm-eh, cleanup-dtors, deferred_resn, demangler, jacob/cs343-translation, jenkins-sandbox, master, new-ast, new-ast-unique-expr, new-env, no_list, persistent-indexer, resolv-new, with_gc
Children:
b9d0fb6
Parents:
bbeb908
Message:
updated first 3 chapters
Location:
doc/proposals/concurrency
Files:
4 edited
Unmodified
Removed
• ## doc/proposals/concurrency/text/basics.tex
rbbeb908 The following is a quick introduction to the \CFA language, specifically tailored to the features needed to support concurrency. \CFA is a extension of ISO-C and therefore supports all of the same paradigms as C. It is a non-object oriented system language, meaning most of the major abstractions have either no runtime overhead or can be opt-out easily. Like C, the basics of \CFA revolve around structures and routines, which are thin abstractions over machine code. The vast majority of the code produced by the \CFA translator respects memory-layouts and calling-conventions laid out by C. Interestingly, while \CFA is not an object-oriented language, lacking the concept of a receiver (e.g., this), it does have some notion of objects\footnote{C defines the term objects as : [Where to I get the C11 reference manual?]}, most importantly construction and destruction of objects. Most of the following code examples can be found on the \CFA website \cite{www-cfa} \CFA is a extension of ISO-C and therefore supports all of the same paradigms as C. It is a non-object oriented system language, meaning most of the major abstractions have either no runtime overhead or can be opt-out easily. Like C, the basics of \CFA revolve around structures and routines, which are thin abstractions over machine code. The vast majority of the code produced by the \CFA translator respects memory-layouts and calling-conventions laid out by C. Interestingly, while \CFA is not an object-oriented language, lacking the concept of a receiver (e.g., this), it does have some notion of objects\footnote{C defines the term objects as : region of data storage in the execution environment, the contents of which can represent values''\cite[3.15]{C11}}, most importantly construction and destruction of objects. Most of the following code examples can be found on the \CFA website \cite{www-cfa} \section{References} Like \CC, \CFA introduces references as an alternative to pointers. In regards to concurrency, the semantics difference between pointers and references are not particularly relevant but since this document uses mostly references here is a quick overview of the semantics : Like \CC, \CFA introduces rebindable references providing multiple dereferecing as an alternative to pointers. In regards to concurrency, the semantic difference between pointers and references are not particularly relevant, but since this document uses mostly references, here is a quick overview of the semantics: \begin{cfacode} int x, *p1 = &x, **p2 = &p1, ***p3 = &p2, &r1 = x, &&r2 = r1, &&&r3 = r2; &r1 = x, &&r2 = r1, &&&r3 = r2; ***p3 = 3; //change x r3 = 3; //change x, ***r3 sizeof(&ar[1]) == sizeof(int *); //is true, i.e., the size of a reference \end{cfacode} The important thing to take away from this code snippet is that references offer a handle to an object much like pointers but which is automatically derefferenced when convinient. The important take away from this code example is that references offer a handle to an object, much like pointers, but which is automatically dereferenced for convinience. \section{Overloading} Another important feature of \CFA is function overloading as in Java and \CC, where routine with the same name are selected based on the numbers and type of the arguments. As well, \CFA uses the return type as part of the selection criteria, as in Ada\cite{Ada}. For routines with multiple parameters and returns, the selection is complex. Another important feature of \CFA is function overloading as in Java and \CC, where routines with the same name are selected based on the number and type of the arguments. As well, \CFA uses the return type as part of the selection criteria, as in Ada\cite{Ada}. For routines with multiple parameters and returns, the selection is complex. \begin{cfacode} //selection based on type and number of parameters double d = f(4); //select (2) \end{cfacode} This feature is particularly important for concurrency since the runtime system relies on creating different types to represent concurrency objects. Therefore, overloading is necessary to prevent the need for long prefixes and other naming conventions that prevent name clashes. As seen in chapter \ref{basics}, routines main is an example that benefits from overloading. This feature is particularly important for concurrency since the runtime system relies on creating different types to represent concurrency objects. Therefore, overloading is necessary to prevent the need for long prefixes and other naming conventions that prevent name clashes. As seen in chapter \ref{basics}, routine \code{main} is an example that benefits from overloading. \section{Operators} Overloading also extends to operators. The syntax for denoting operator-overloading is to name a routine with the symbol of the operator and question marks where the arguments of the operation would be, like so : Overloading also extends to operators. The syntax for denoting operator-overloading is to name a routine with the symbol of the operator and question marks where the arguments of the operation occur, e.g.: \begin{cfacode} int ++? (int op); //unary prefix increment \section{Parametric Polymorphism} Routines in \CFA can also be reused for multiple types. This is done using the \code{forall} clause which gives \CFA it's name. \code{forall} clauses allow seperatly compiled routines to support generic usage over multiple types. For example, the following sum function will work for any type which support construction from 0 and addition : Routines in \CFA can also be reused for multiple types. This capability is done using the \code{forall} clause which gives \CFA its name. \code{forall} clauses allow separately compiled routines to support generic usage over multiple types. For example, the following sum function works for any type that supports construction from 0 and addition : \begin{cfacode} //constraint type, 0 and + \end{cfacode} Since writing constraints on types can become cumbersome for more constrained functions, \CFA also has the concept of traits. Traits are named collection of constraints which can be used both instead and in addition to regular constraints: Since writing constraints on types can become cumbersome for more constrained functions, \CFA also has the concept of traits. Traits are named collection of constraints that can be used both instead and in addition to regular constraints: \begin{cfacode} trait sumable( otype T ) { \section{with Clause/Statement} Since \CFA lacks the concept of a receiver, certain functions end-up needing to repeat variable names often, to solve this \CFA offers the \code{with} statement which opens an aggregate scope making its fields directly accessible (like Pascal). Since \CFA lacks the concept of a receiver, certain functions end-up needing to repeat variable names often. To remove this inconvenience, \CFA provides the \code{with} statement, which opens an aggregate scope making its fields directly accessible (like Pascal). \begin{cfacode} struct S { int i, j; }; int mem(S & this) with this //with clause int mem(S & this) with (this) //with clause i = 1; //this->i j = 2; //this->j struct S1 { ... } s1; struct S2 { ... } s2; with s1 //with statement with (s1) //with statement { //access fields of s1 //without qualification with s2 //nesting with (s2) //nesting { //access fields of s1 and s2 } } with s1, s2 //scopes open in parallel with (s1, s2) //scopes open in parallel { //access fields of s1 and s2
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2021-09-27 04:12:23
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https://worldbuilding.stackexchange.com/questions/107827/why-would-be-preferable-use-crossbows-than-firearms
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# Why would be preferable use crossbows than firearms? [closed]
First of all, I am aware that its impossible for a planet with less gravity than the Earth, to naturally maintain an atmosphere with the same terrestrial gases, in an equal or greater proportion. Therefore, this proportion of gases is maintained artificially.
Second, although I have notions of what the ballistic coefficient implies, I have no idea how to calculate it or in what proportion its affected by the atmospheric density and gases of an atmosphere.
Following with this line of thought, we have the force of gravity of the planet X that is 8.04 m / s².
Let us assume then that the atmospheric pressure of this planet, whose atmosphere is being altered artificially, is approximately 2 atm.
Now with all the elements previously presented, I would like to know two things:
• 1) Would this atmosphere sustain intelligent life in a realistic way? It is necessary to know if a human civilization could develop there.
• 2) Would the effect of this atmosphere be enough to significantly affect the ballistic coefficient, to the point where its preferable to a civilization that has discovered firearms, have to continue using crossbows for more effective shots at a distance?
Note:Take into account side effects that may present apparently inert gases at high levels of partial pressure, such as nitrogen narcosis.
• I'm just going to tell now that this question will get cut down by the admins they don't like mutable questions change it to one question that's to do with the title don't add more. it sucks i know but that's the way this site operates – Creed Arcon Mar 25 '18 at 3:30
• I am aware that its impossible for a planet with less gravity than the Earth, to naturally maintain an atmosphere with the same terrestrial gases, in an equal or greater proportion depends on how much less gravity you have. I think that if the only difference is the gravity, with 8.04 m/s2 it would be doable to have much the same atmosphere. – Renan Mar 25 '18 at 4:17
• @CreedArcon, it isn't just this site. All Stack Exchange sites are one-question-one-best-answer format. The reality is that this is not a forum. JAMS, these are two very different questions. Together they are too broad. The question about life on a low-G-high-P world might, itself, be too broad (it might even be a duplicate). And for the record, the only practical reasons for preferring a crossbow over a gun is limited resources for guns. Nothing about the world would ever make crossbows preferrable due to the range + damage + weather benefits of firearms. – JBH Mar 25 '18 at 4:55
• (0) There is no need for artificial intervention. It is perfectly possible for your planet to sustain the required atmosphere naturally; see, for example, Venus. (1) At 2 atm total pressure the partial pressure of oxygen would be 0.42 atm, which is all right even for terrestrial humans. Yes, it can sustain intelligent life. Nitrogen narcosis is not an issue for life evolved on that planet, and even for humans it's borderline at only 2 atm. (3) Why would spring-powered weapons be preferable to firearms? Just like on Earth, crossbows would be preferable to firearms only in special situations. – AlexP Mar 25 '18 at 5:04
• @AlexP But in the case of venus it has an atmospheric pressure many times greater than the earth, due to the type of gases found in its atmosphere. – JAMS Mar 25 '18 at 5:19
an atmosphere with the same terrestrial gases, in an equal or greater proportion.
78% Nitrogen 21% Oxygen 01% Other stuff (mostly Argon)
the atmospheric pressure of this planet, whose atmosphere is being altered artificially, is approximately 2 atm.
Note that since the partial pressure of oxygen would effectively twice that of Earth, fires would burn faster.
Would this atmosphere sustain intelligent life in a realistic way?
Native life would/could have adapted to it, so there's no reason to say "no" to this hypothetical question. So... yes.
It is necessary to know if a human civilization could develop there.
If these are "humans which evolved on Earth, and then transported to this planet", then the answer is maybe. That's because nitrogen narcosis can begin even at relatively low depths when scuba diving, and 2 ATM of pressure is the same as diving to 10m.
Thus, enough people just might be loopy enough to die off within a few generations.
Would the effect of this atmosphere be enough to significantly affect the ballistic coefficient, to the point where its preferable to a civilization that has discovered firearms, have to continue using crossbows for more effective shots at a distance?
The question is flawed, because we didn't develop firearms because they shot better and farther. We developed them because they were easier.
Note that since there's so much more oxygen at 2 ATM, the gunpowder would burn a lot faster, generating a lot more pressure in the gun breaches. It's very possible that would have delayed the development of firearms (and cannons) until the development of better metallurgy.
• But in case firearms were developed with better metallurgy, a primitive harquebus would still have problems due to the shape of its projectiles. Wouldnt long and thin projectiles have an advantage due to air resistance? – JAMS Mar 25 '18 at 4:41
• We developed firearms because they are handy to breach walls. Try to do it with the crossbow. As of faster burnout of the gunpowder, the recipe is easily adjusted. – user58697 Mar 25 '18 at 4:49
• The single-user firearms follow the cannons, just because the cannon was a success in inflicting more casualties, on the greater distance, against better armed enemy. And the original fire__arm__ was a culverin, aka hand-held cannon. – user58697 Mar 25 '18 at 5:09
• Gunpowder contains its own oxidiser (saltpeter), it doesn’t use atmospheric oxygen. So the increased partial pressure of oxygen wouldn’t affect its rate of burning in a firearm. – Mike Scott Mar 25 '18 at 6:00
• @pojo-guy Crossbows are not silent. They are certainly far more quiet than a firearm, but you're not making a stealth-kill with it (plus bolts travel overwhelmingly slower than the speed of sound, where as the bullet can hit you before you've heard the shot). – pluckedkiwi Mar 26 '18 at 16:12
Would the effect of this atmosphere be enough to significantly affect the ballistic coefficient
No, and at a first glance neither would the higher oxygen pressure.
But you might imagine have large quantities of dry pollen floating in the denser atmosphere. This would require a great care in lighting fires (fireplaces would need something like a flame arrester mesh).
But in the open, firing a gun might be a deadly mistake - unless it had just stopped raining, or the weather conditions or season or geographical features otherwise allowed it. The combination of high oxygen pressure and suspended combustible particles would turn the volume around the shooter into a dust bomb, a weak form of fuel-air bomb. With the shooter at the center.
In Earth atmosphere, it takes some doing for a dust explosion to take place - but if memory serves, a bunch of probationary cooks engaging in a flour battle near cooking fires in a kitchen in Paris was enough to kill one and maim two others, and wreck the kitchen.
Combine some local plant like lycopodium, but more so, and a higher oxygen availability, and wild gun shooting loses a lot of its appeal.
Firearms were adopted because they were easier to train large numbers of people to use quickly than long or recurve bows. They replaced crossbows because a crossbow bolt, fired from a steel crossbow and drawn with a spanning mechanism could provide about 200J of energy at the target, while an arquebus imparted enough energy to the projectile to deliver 1000J of energy, an order of magnitude difference.
Spanning (or drawing) a steel crossbow. This gets you to 200J of energy
A Medieval soldier with a firearm. A 1000J of energy can deal with even heavy plate armour
Any crossbow which can deliver that much energy would be either far too large to carry or use in any practical manner, or require such a massive and powerful spanning mechanism that would also be impractical for use in battle outside of special situations.
So regardless of if you are fighting on Earth, the hypothetical planet, or even the vacuum of space (both a crossbow and a firearm will work in a vacuum), a firearm will always outperform a bow.
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2020-07-06 12:28:41
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https://www.vedantu.com/question-answer/a-nucleus-disintegrates-into-two-nuclear-parts-class-12-physics-cbse-5fd7c19dcd67a76506ea6469
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Question
# A nucleus disintegrates into two nuclear parts which have their velocities in the ratio $2:1$. The ratio of their nuclear size will be:a. ${2^{\dfrac{1}{3}}}:1$ b. $1:{3^{\dfrac{1}{2}}}$c. ${3^{\dfrac{1}{2}}}:1$d. ${2^{\dfrac{1}{3}}}:1$
Verified
91.8k+ views
Hint: To solve the given problem consider the concept of the momentum of the system. Remember that the momentum of the given system is conserved.
Formula used:
Total momentum of the system:
$\Rightarrow M = {m_1}{v_1} + {m_2}{v_2}$
Where, ${m_1},{v_1}$ is the mass and velocity of the first nuclear part and ${m_2},{v_2}$ is the mass and velocity of the second nuclear part.
In the question, it is given that the nucleus disintegrates into two nuclear parts which have their velocities in the ratio $2:1$.
Consider the diagram. A nucleus disintegrates into two nuclear parts. The first nuclear part has the momentum ${m_1}$ and moves with the velocity of ${v_1}$. The second nuclear part has the momentum ${m_2}$ and moves with the velocity of ${v_2}$.
The masses of the system are constant. Thus, it is conserved in the total momentum of the system. That is the initial and the final momentum is conserved. We have the formula to calculate the total momentum.
Total momentum of the system:
$\Rightarrow M = {m_1}{v_1} + {m_2}{v_2}$
Where, ${m_1},{v_1}$ is the mass and velocity of the first nuclear part and ${m_2},{v_2}$ is the mass and velocity of the second nuclear part.
The total momentum is conserved and hence the value of ${m_1}{v_1} + {m_2}{v_2}$ is zero. We can represent as,
$\Rightarrow {m_1}{v_1} + {m_2}{v_2} = 0$
We can bring the term ${m_2},{v_2}$ to the right hand side. The sign of the term will be changed into the opposite sign.
$\Rightarrow {m_1}{v_1} = - {m_2}{v_2}$
We can rearrange the common terms. That is, we can bring the mass to the right hand side and velocity terms into the left hand side. We get,
$\Rightarrow \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{ - {v_2}}}{{{v_1}}}$
We have the value for $\dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{ - {v_2}}}{{{v_1}}}$ as $\dfrac{1}{2}$ that is,
$\Rightarrow \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{ - {v_2}}}{{{v_1}}} = \dfrac{1}{2}$
As discussed before the masses of the nuclear particles are the same. So, the densities are also the same. We know the value for the mass, that is,
$\Rightarrow m = \dfrac{4}{3}\pi {r^3}\rho$
Where $\rho$ is the density, $m$ is the mass and $r$ is the radius
Consider,
$\Rightarrow {m_1} = \dfrac{4}{3}\pi {r_1}^3\rho$
$\Rightarrow {m_2} = \dfrac{4}{3}\pi {r_2}^3\rho$
We can divide the masses. We get,
$\Rightarrow \dfrac{{\dfrac{4}{3}\pi {r_1}^3\rho }}{{\dfrac{4}{3}\pi {r_2}^3\rho }}$
To simplify the given equation, we can cancel out the common terms we get,
$\Rightarrow \dfrac{{{r_1}^3}}{{{r_2}^3}}$
We can take the cube as the whole term cube. That is,
$\Rightarrow {\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^3}$
The value of ${\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^3}$ is equal to $\dfrac{1}{2}$ as the value of $\dfrac{{{m_1}}}{{{m_2}}}$ is equal to $\dfrac{1}{2}$.
$\Rightarrow {\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^3} = \dfrac{1}{2}$
We can take the cube root for the left-hand side part to remove the cube power in the right-hand side. That is,
$\Rightarrow \left( {\dfrac{{{r_1}}}{{{r_2}}}} \right) = {\left( {\dfrac{1}{2}} \right)^{\dfrac{1}{3}}}$
$\Rightarrow \left( {\dfrac{{{r_1}}}{{{r_2}}}} \right) = \left( {\dfrac{1}{{{2^{\dfrac{1}{3}}}}}} \right)$
The ratio is ${1 : 2^{\dfrac{1}{3}}}$.
Hence, the correct answer is option (D).
Note: Students need to remember that, when we bring the mass and velocity to the other hand side the sign of the terms will get changed. If the term gets a negative sign it means that the velocity of the nuclear particle travels in the opposite direction.
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2021-10-16 21:42:24
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http://mathhelpforum.com/calculus/104277-evaluation-partial-derivative-using-chain-rule.html
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# Math Help - Evaluation of a partial derivative using the chain rule
1. ## [SOLVED] Evaluation of a partial derivative using the chain rule
I was solving a practice test for the GRE math subject test and I came across a problem which I thought was unsolvable. The solution provided in the back makes no sense to me, so I hope someone can shed some light on this situation.
Let f, g, and h be functions of two variables that are differentiable everywhere such that $z=f(x,y)$ where x and y in turn are functions of u and v. We know that
$x=g(u,v)$, $y=h(u,v)$ and that $g(0,1)=2$ and that $h(0,1)=1$. Let $P=(0,1)$ and $Q=(2,1)$. We are also given that
$\frac{\partial f}{\partial x}\bigg|_Q = 11, \;\; \frac{\partial f}{\partial y}\bigg|_Q = -3, \;\; \frac{\partial g}{\partial u}\bigg|_P = 1, \;\; \frac{\partial h}{\partial u}\bigg|_P = -3, \;\; \frac{\partial g}{\partial v}\bigg|_P = \frac{\partial h}{\partial v}\bigg|_P = 2$
The object is to find $\frac{\partial z}{\partial v} \bigg|_P$.
Now note that the partial derivatives given are evaluated at different points, but the solution is not afraid to say that
$\frac{\partial z}{\partial v} \bigg|_P = \frac{\partial f}{\partial x} \bigg|_Q \frac{\partial x}{\partial v} \bigg|_P + \frac{\partial f}{\partial y} \bigg|_Q \frac{\partial y}{\partial v} \bigg|_P = (11)(2) + (-3)(2) = 16$
Now, it makes no sense to me that you can just pick and choose where you evaluate the partials involved in the chain rule... but that's what the solution does. Shouldn't they all be evaluated at P? Why is it allowed to evaluate some at Q and some at P? Does the solution somehow implicitly make use of $g(0,1)$ and $h(0,1)$ which are given?
Thank you for taking the time to read my question!
2. I answered my own question after some research...
When evaluating the partial derivatives the formula is
$\frac{\partial{z}}{\partial{v}} \bigg|_P = \frac{\partial{z}}{\partial{x}} \bigg|_{(x(P), y(P))} \frac{\partial{x}}{\partial{v}} \bigg|_P + \frac{\partial{z}}{\partial{y}} \bigg|_{(x(P),y(P))} \frac{\partial{y}}{\partial{v}} \bigg|_P$
so the solution did use the fact that $(x(P), y(P)) = Q$. Ugh.
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2016-07-29 13:21:35
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https://calendar.math.illinois.edu/?year=2013&month=10&day=03&interval=day
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Department of
# Mathematics
Seminar Calendar
for events the day of Thursday, October 3, 2013.
.
events for the
events containing
Questions regarding events or the calendar should be directed to Tori Corkery.
September 2013 October 2013 November 2013
Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa
1 2 3 4 5 6 7 1 2 3 4 5 1 2
8 9 10 11 12 13 14 6 7 8 9 10 11 12 3 4 5 6 7 8 9
15 16 17 18 19 20 21 13 14 15 16 17 18 19 10 11 12 13 14 15 16
22 23 24 25 26 27 28 20 21 22 23 24 25 26 17 18 19 20 21 22 23
29 30 27 28 29 30 31 24 25 26 27 28 29 30
Thursday, October 3, 2013
8:00 am in Second Floor, Levis Faculty Center,Thursday, October 3, 2013
#### Second International Meeting of the Association for the Philosophy of Mathematical Practice
Abstract: The Second International Meeting of the APMP will be held October 3-4, 2013. All talks will take place on the second floor of the Levis Faculty Center. See the conference website http://institucional.us.es/apmp/index_APMP2013.htm for more information.
11:00 am in 241 Altgeld Hall,Thursday, October 3, 2013
#### Colored Partition Identities Arising from Ramanujan's Formulas for Multipliers
###### Bruce Berndt (UIUC Math)
Abstract: Beginning with work of Farkas and Kra, a large number of partition identities have been established from theta function identities and modular equations in the past dozen years. In this lecture we examine a particular type of modular equation, namely, formulas for multipliers, and derive several interesting identities for colored partitions. This is joint work with Rui Zhou, who was a visiting graduate student from Dalian University during the past year.
12:30 pm in 243 Altgeld Hall,Thursday, October 3, 2013
#### Ribbon categories and anyons
###### Abishek Roy (Illinois Physics)
Abstract: An expository talk on ribbon categories and their application to physics. Using the graphical method of Reshetikhin-Turaev diagrams, we will prove modular invariance. I shall try to include several models of physical anyons - abelian, Ising, Fibonacci and (if time permits) Kitaev's quantum double model.
1:00 pm in 241 Altgeld Hall ,Thursday, October 3, 2013
#### Quasiconformal Mappings on Planar Surfaces
###### Colleen Ackermann (UIUC Math)
Abstract: The geometric definition of quasiconformality is a condition involving all quadrilaterals (Jordan domains with four boundary points identified). Past research has shown that one need only look at rectangles to determine quasiconformality. We will investigate whether it suffices to just consider squares. Next we will look at a method of studying quasiconformal mappings on the Grushin plane. I will give a quasisymmetry from the Grushin plane to the complex plane and use it to develop an analytic definition of quasisymmetry on the Grushin plane. We will also discuss several characterizations of comformal mappings on the Grushin plane.
1:00 pm in Altgeld Hall 347,Thursday, October 3, 2013
#### Factors of IID on Trees
###### Russel Lyons (Indiana University)
Abstract: Classical ergodic theory for integer group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We discuss a few known results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. No background will be assumed.
2:00 pm in 149 Henry Administration Building,Thursday, October 3, 2013
#### Revisiting Lehmers' Picturesque Exponential Sums (joint with Dan Schultz)
###### Michael DiPasquale (UIUC Math)
Abstract: Let $\zeta$ be a $k$th root of unity and $b_k(i)$ be the sum of the digits of $i$ when $i$ is written in base $k$. We consider a graphical representation $G(n,k)$ of the partial sums $S(n,k)=\sum_{i=0}^n \zeta^{b_k(i)}x^i$, where $x$ is a complex root of unity. Graphical representations of sums of this type were considered by D.H. and Emma Lehmer for the case $x=\zeta$. If $x$ is replaced by an arbitrary $m$th root of unity, the resulting graph exhibits fractal-like properties which can be explained by a formula for the partial sums $S(n,k)$ which extends a previously known formula discovered by Apostol. There will be pictures.
3:00 pm in 243 Altgeld Hall,Thursday, October 3, 2013
#### Local cohomology modules over polynomial rings of prime characteristic - Part IV
###### Yi Zhang (UIUC)
Abstract: Let $R=k[x_1,\cdots, x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0.$ If $I$ is an ideal of $R,$ we denote $H^i_I(R)$ the $i$-th local cohomology module of $R$ with support in $I.$ We discuss an adjointness theorem of Frobenius map. Then we use this property to study the dimension of the associated primes of $H^i_I(R),$ the grading on
4:00 pm in 245 Altgeld Hall,Thursday, October 3, 2013
#### Random orderings and unique ergodicity of automorphism groups
###### Russell Lyons (Indiana University)
Abstract: Is there a natural way to put a random total ordering on the vertices of a finite graph? Natural here means that all finite graphs get an isomorphism-invariant random ordering and induced subgraphs get the random ordering that is inherited from the larger graph. Thus, the uniformly random ordering is natural; are there any others? What if we restrict to certain kinds of graphs? What about finite hypergraphs or finite metric spaces? We discuss these questions and sketch how their answers give unique ergodicity of corresponding automorphism groups; for example, in the case of graphs, the group is the automorphism group of the infinite random graph. This is joint work with Omer Angel and Alexander Kechris.
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2021-08-05 06:53:58
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https://testbook.com/question-answer/in-a-4-pole-dynamo-the-fluxpole-is-15-mwb-if-th--5ebfd512f60d5d399ec57d83
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# In a 4-pole dynamo, the flux/pole is 15 mWb. If the armature is driven at 600 rpm, the average emf induced in one of the armature conductors will be:
This question was previously asked in
ESE Electronics 2020: Official Paper
View all UPSC IES Papers >
1. 0.3 V
2. 0.4 V
3. 0.5 V
4. 0.6 V
Option 4 : 0.6 V
Free
CT 3: Building Materials
3014
10 Questions 20 Marks 12 Mins
## Detailed Solution
Concept:
Induced emf in a dc machine is given by:
$$E = \frac{{NP\phi Z}}{{60A}}$$
Where N = speed in rpm
P = number of poles
ϕ = flux per pole
Z = number of armature conductors
A = number of parallel paths
Calculation:
Give N = speed = 600 rpm
P = number of poles = 4
ϕ = flux per pole = 15 mWb
The average emf induced in one of the armature conductors will be:
$$E = \frac{{NP\phi }}{{60}}$$
$$E = \frac{{600 \times 4 \times 15\times 10^{-3}}}{{60}}$$
E = 0.6 V
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2021-10-28 02:42:34
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http://forum.math.toronto.edu/index.php?action=printpage;topic=1045.0
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# Toronto Math Forum
## APM346-2018S => APM346--Lectures => Topic started by: Jingxuan Zhang on February 27, 2018, 10:12:26 AM
Title: Take a correct branch
Post by: Jingxuan Zhang on February 27, 2018, 10:12:26 AM
In chap 5.2 repeatedly I find "we need to take a correct branch". For instance, for the Schrodinger equation with $t\gtrless 0$, what in fact is the reason of taking $\sqrt{i}=\pm e^{i\pi/4}$?
Title: Re: Take a correct branch
Post by: Victor Ivrii on February 27, 2018, 04:09:03 PM
If we solve a crossbreed of Schr\"odinger and Heat
u_t =au_{xx}
\tag{1}
with $a\in \mathbb{C}$ with the positive real part (which is well-posed in the direction of positive $t$) we get a formula with kernel $\frac{1}{\sqrt {4\pi a t}} e^{-x^2/4at}$. This square root is positive for real $a>0$.
Let us look, what happens if $a=bi+\varepsilon$, $\varepsilon \to +0$ and $b>0$. We get
$\frac{1}{\sqrt {4\pi (bi+\varepsilon) t}} e^{-x^2/4(bi+\varepsilon)t}$. But if $\varepsilon \to +0$ then this equation (1) becomes Schr\"odinger equation
u_t =ibu_{xx}
\tag{2}
exponent factor tends to $e^{-x^2/4b i t}=e^{ix^2/4b t}$, but since $bi +\varepsilon \to bi$ from the right, then its argument tends to $\pi/2$ from below, and argument of the square root tends to ${\pi/4}$, so we get
\frac{1}{\sqrt {4\pi bt}} e^{-i\pi/4}e^{ix^2/4b t}
\tag{3}
Recall that $b>0$, $t>0$.
The same is true for $b<0, t<0$. Indeed, changing $t\mapsto -t$, $b\mapsto -b$ preserves equation (2).
So (3) holds for $bt>0$. But taking complex conjugation and $b\to -b$ also preserves (2). This operation with (3) brings
\frac{1}{\sqrt {4\pi |bt|}} e^{i\pi/4}e^{ix^2/4b t}.
\tag{4}
So, correct formula (I suspect I skipped on the lecture) for convolution kernel is
\frac{1}{\sqrt {4\pi |bt|}} e^{\mp i\pi/4}e^{ix^2/4b t}\qquad \text{for} \ \ bt \gtrless 0.
\tag{5}
Title: Re: Take a correct branch
Post by: Jingxuan Zhang on February 27, 2018, 08:14:49 PM
Indeed, changing $t\mapsto -t$, $b\mapsto -b$ preserves equation (2).
So (3) holds for $bt>0$. But taking complex conjugation and $b\mapsto -b$ also preserves (2). This operation with (3) brings
\frac{1}{\sqrt {4\pi |bt|}} e^{i\pi/4}e^{ix^2/4b t}.
\tag{4}
Is this because (heuristically) since $u$ is real-valued
$$u_t=ibu_{xx}=\overline{i(-b)u_{xx}} \implies u = \frac{1}{\sqrt {4\pi bt}} e^{i (-\pi/4+x^2/4b t)} = \overline{\frac{1}{\sqrt {4\pi (-b)t}} e^{i (-\pi/4+x^2/4(-b) t)}}=\frac{1}{\sqrt {4\pi (-b)t}} e^{i (\pi/4+x^2/4bt)}?$$
I hesitated a lot before posting. This $-b$ under square root is really poignant. How would you then justify your "taking complex conjugation and $b\mapsto -b$ also preserves (2)"?
Title: Re: Take a correct branch
Post by: Victor Ivrii on February 28, 2018, 01:57:26 AM
Just check equation (2)
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2020-11-30 17:52:17
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https://mikesmathpage.wordpress.com/tag/computers/
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## Sharing Matt Zucker’s Shadertoy program with kids
Saw an amazing tweet from Matt Zucker yesterday:
My older son is on a school trip this weekend, so this project is just with my younger son (in 6th grade). I thought he’d had a lot of fun playing around with the program, so I let him explore it (with no instruction or even explanation) for about 10 min and then asked him what he thought was neat:
At the end of the last video he was playing around with the different numbers. I didn’t want to go into what those numbers represented, but I did think it would be great to hear some of his ideas and conjectures.
He found some ideas that seemed to work and a few that didn’t – so that was great to hear. By the end we’d found a shape that we could make from our Zometool set.
To finish this morning’s project we built the shape – here’s are his thoughts about having the shape in front of him vs seeing it on a computer screen:
This was a super fun project. I think it might be a nice challenge to try to dive a little deeper into the general Wythoff constructions that the Matt Zucker’s program is designed to explore. For now though, even with any details, the program is really fantastic for kids to play with.
## Talking about “sums of divisors” with kids and also a pi surprise
I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.
But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and $\pi$.
The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.
The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:
Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.
At the end we were looking to see if we could find patterns in the powers of 3.
It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.
At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.
To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.
While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately $(\pi^2 / 12)*n$.
Number theory sure has some fun surprises 🙂
This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.
## Sharing basic machine learning ideas with kids
Two weeks ago I saw an interesting lecture from Gil Strang at MIT about the math behind machine learning. Sharing some of those ideas with kids has been on my mind ever since. Today I finally got around to it!
We’ve done a few previous projects that touched on ways to make machine learning accessible to kids. The Martin Gardner hexapawn project is incredibly fun and also is accessible to really young kids, the other project below uses the same Tensorflow website that we played with today:
Introduction to machine learning for kids via Martin Gardner’s article on Hexapawn
Sharing a fun neural network program with kids
Here’s a link to the Tensorflow website:
A Neural Network Playground
Today I began be asking the boys what they knew about machine learning and then I explained a bit about classification problems:
Next I moved on to drawing a clumsy picture of what a neural network might look like and then did a clumsy explanation of how a neural network might work. My older son asked a really great question that gets to the difference between the Hexapawn game and how modern neural networks work – so we chatted about that for a bit.
Then I talked about the so-called “relu” firing function for neurons.
Before moving on to the Tensorflow program, I wanted to spend a few minutes talking about an idea that Gil Strang mentioned in his lecture. That idea is the connection between folding and classification.
This idea, I think, helps make the classification problem accessible to kids.
Next up was playing with the Tensorflow program and exploring some basic classification examples:
Then I let the kids play with the program by themselves for about 15 min – here are a few of the ideas that they found interesting:
Machine learning is an incredibly popular and growing area of math and computer science right now – the Tensorflow website is a great way to share some of the ideas in machine learning with kids.
## Sharing hinged tiling quadrilaterals with kids
Saw a really neat tweet from John Carlos Baez last week:
Finally got a chance to share this site with my younger tonight. This site is fantastic to share with kids – my son enjoyed playing around with the tiling patterns, and it was also really interesting to hear him try to describe what he was seeing.
Here’s his initial look at the site:
Here’s his reaction and play with the part of the site that allows you to create and manipulate new quadrilaterals:
This is a wonderfully easy site and a really fun idea to play with. I think with older kids it would be nice to see them try to think through why the cyclic quadrilaterals have this hinged tiling property, but I thought that might be a little much for my younger son. We’ll do a follow up exploring those ideas soon, though.
## Sharing the ABRACADABRA problem with kids
Yesterday we did a fun project on Markov chains and sharing the “COVFEFE” problem with kids:
Sharing Markov chains and the “covfefe” problem with kids
For me math behind this problem was the most interesting math I learned in 2017:
The most interesting piece of math I learned in 2017 -> the “covfefe” problem
Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.
First, we reviewed the ideas from yesterday:
After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:
Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:
Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:
There are other approaches to these problems – the approach via Martingales, for example:
What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.
Anyway – super fun project, and an great piece of math to share with kids.
## Sharing Markov chains and the “COVFEFE” problem with kids
In 2017, the most interesting piece of math I learned can via Christopher Long and Nassim Taleb and related to the “COVFEFE” problem:
I wrote about the problem here:
The most interesting piece of math I learned in 2017 -> the “covfefe” problem
It turned out that we’ve looked at Markov chains before thanks to this great video from Kelsey Houston-Edwards:
Sharing Kelsey Houston-Edward’s Markov chain video with kids
I’d forgotten about that project, but when I mentioned to my younger son that we’d be looking at Markov chains today he told me he already knew about them!
So, I started today by having the boys watch the PBS Infinite Series video again. Here’s what they thought:
Next I introduced the “COVFEFE” problem. I was really happy how quickly the boys were able to pick up on how Markov chains could be used to solve this problem.
Next we looked at Nassim Taleb’s Mathematica code – that code is so clear that the problem becomes instantly accessible to kids, which is pretty amazing.
Finally, since things were going so well this morning, I introduced the word that we’ll study tomorrow -> ABRACADABRA. The kids were able to see why the transitions in this word were different. I’m excited to see how they think through the “ABRACADABRA” problem tomorrow!
The math behind this problem really was the most interesting math that I learned in 2017. It is really important math, too, and I’m excited that the Mathematica code makes some of the ideas accessible to kids. This was a fun one!
## An attempt to share some Katherine Johnson’s math ideas from Hidden Figures with my son
For the last few months I’ve been daydreaming about ways to share some of the math from the movie Hidden Figures with kids. As part of that prep work I found one of Katherine Johnson’s technical papers on NASA’s website:
NASA’s Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson
As you’d expect, there’s a lot of trig, calculus and spherical geometry. I like finding ways to share the work that mathematicians do with kids, but this work is pretty technical and I wasn’t getting any great ideas.
Then my son had a homework problem from his Precalculus book that made me think it was time to stop daydreaming and just try something. Here is that problem, which is a completely standard law of cosines problem:
The problem reminded me of one of the equations for an ellipse used in the Technical Note. One surprising thing is that the equation of an ellipse in polar coordinates is that is is a rational function in $\cos{\theta}$.
So, I drew an ellipse and showed my son that equation.
One of the neat things about the Technical Note is that the solution to some of the complicated trig equations were found by an iteration method. The specific ideas for solving those equations are too advanced for kids, so I decided to show my son a different (and really simple) iteration method that converges to a well known number:
After that introduction to iteration methods, I decided to jump to a second and slightly more complicated example -> solving x = 3*x*(1 – x).
The ideas in the iteration method we use here can be explored purely geometrically:
Next we went upstairs to the computer to see some of the ideas we just talked about. The first idea was the polar coordinate equation for an ellipse:
Now we played with the second dynamical system -> solving x = 3*x(1-x).
By the way, the ideas here are incredibly fun to explore (especially seeing when this method converges and when it doesn’t), but the details of this method wasn’t really the idea here. I just wanted to show him what an iterative method looks like.
Finally, I showed him the actual paper and pointed out some of the parts we explored. Sorry that this film didn’t come out as well as I’d hoped, but you can view the paper from the first link in this post:
This was a fun project – even if it wasn’t planned really well. Showing some of the math behind Hidden Figures I hope helps motivate some of the topics that my son is studying right now. It will be fun to return to a second Hidden Figures project when he is studying calculus.
## A fun math surprise with a 72 degree angle.
We’ve been talking a lot about 72 degree angles recently. Yesterday’s project was about a question our friend Paula Beardell Krieg asked:
Paula Beardell Krieg’s 72 degree question
In that project we learned that a right triangle with angles 72 and 18 (pictured below)
Is nearly the same as a right triangle with sides of 1, 3, and $\sqrt{10}$
Today I wanted to show the boys a neat surprise that I stumbled on almost by accident. The continued fraction expansion for the cosine of the two large (~72 degree angles) are remarkable similar and lead to the “discovery” of a 3rd nearly identical triangle.
We got started by reviewing a bit about 72 degree angles:
Now we did a quick review of continued fractions and the “split, flip, and rat” method that my high school teacher, Mr. Waterman, taught me. Then we looked at the continued fraction for $1 / \sqrt{10}$:
Now we looked at the reverse process -> given a continued fraction, how do we figure out what number it represents? Solving this problem for the infinite continued fraction we have here is a challenging problem for kids. One nice thing here was that my kids knew that they could do it if the continued fraction had finite length – that made it easier to show them how to deal with the infinitely long part.
Finally, we went to the computer to see the fun surprise:
Here’s that 3rd triangle:
I love the surprise that the continued fractions for the cosine of the (roughly) 72 degree angles that we were looking at are so similar. It is always really fun to be able to share neat math connections like this with kids.
## Exploring Newton’s method with kids
Yesterday we had about a 30 min drive and I had the boys open up to a random page in this book for a few short discussions in the car:
There were some fun topics that were accessible for kids, but then Newton’s method came up. Ha ha – not really drive time talk 🙂
It did seem like it could be a fun project, though, so I took a crack at it today. The goal was not computation, but mainly just the geometric ideas. Here’s how we got started:
Next I asked the boys if they could find situations in which Newton’s method wouldn’t work as nicely as it did in the first video. They were able to identify a few potential problems:
Now I had both kids draw their own picture to play out what would happen when you used Newton’s method to find roots. I think there’s a lot of ways to used the exercise here to help older kids understand ideas about tangent lines and function generally. I mostly let the kids play around here, though, and the results were actually pretty fun:
Finally, we went to Mathematica to see some situations in which Newton’s method produces some amazing pictures. Here we switch from real-valued functions to complex valued functions. Since I wasn’t going into the details of now Newton’s method works, rather than using some easier to understand code, I just borrowed some existing code from here:
The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica
The boys were amazed by the pictures. For example, (and this is one we looked at with the camera off) here’s a picture showing which root Newton’s method converges to depending on where you start for the function $f(z) = z^3 - 2z + z - 1$:
Definitely a fun project. Even if the computational details are a bit out of reach, it is fun to share ideas like this with kids every now and then.
## Sharing Tim Gowers’s nontransitive dice talk with kids
During the week I attending a neat talk at Harvard given by Tim Gowers. The talk was about a intransitive dice. Not all of the details in the talk are accessible to kids, but many of the ideas are. After the talk I wrote down some ideas to share and sort of a sketch of a project:
Thinking about how to share Tim Gowers’s talk on intransitive dice with kids
One of the Gowers’s blog posts about intransitive dice is here if you want to see some of the original discussion of the problem:
One of Tim Gowers’s blog posts on intransitive dice
We started the project today by reviewing some basic ideas about intransitive dice. After that I explaine some of the conditions that Gowers imposed on the dice to make the ideas about intransitive dice a little easier to study:
The next thing we talked about was 4-sided dice. There are five 4-sided dice meeting Gowers’s criteria. I thought that a good initial project for kids would be finding these 5 dice.
Now that we had the five 4-sided dice, I had the kids choose some of the dice and see which one would win against the other one. This was an accessible exercise, too. Slightly unluckily they chose dice that tied each other, but it was still good to go through the task.
Now we moved to the computer. I wrote some simple code to study 4-sided through 9-sided dice. Here we looked at the 4-sided dice. Although it took a moment for the kids to understand the output of the code, once they did they began to notice a few patterns and had some new ideas about what was going on.
Having understood more what was going on with 4-sided dice, we moved on to looking at 6-sided dice. Here we began to see that it is actually pretty hard to guess ahead of time which dice are going to perform well.
Finally we looked at the output of the program for the 9-sided dice. It is pretty neat to see the distribution of outcomes.
There are definitely ideas about nontransitive dice that are accessible to kids. I would love to spend more time thinking through some of the ideas here and find more ways for kids to explore them.
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2018-03-17 10:47:28
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https://brianbi.ca/etingof/2.15
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Brian Bi
## Section 2.15. Representations of $$\mathfrak{sl}(2)$$
Problem 2.15.1
1. Let $$v \in V, \lambda \in \mathbb{C}$$. Then, using the commutation relations for $$E$$ and $$H$$, \begin{equation*} (H - (\lambda + 2)I)(Ev) = E(H - (\lambda + 2)I)v + 2Ev = E(H - \lambda I)v \end{equation*} It follows that \begin{equation*} (H - (\lambda + 2)I)^n (Ev) = E(H - \lambda I)^n v \end{equation*} so if $$v$$ is a generalized eigenvector of $$H$$ with eigenvalue $$\lambda$$, then $$Ev$$ is either a generalized eigenvector of $$H$$ with eigenvalue $$\lambda + 2$$, or zero. If $$\lambda$$ is the eigenvalue with maximum real part, then $$Ev$$ must be zero.
2. Lemma 1: If $$k \geq 1$$ is an integer, then $$[H, F^k] = -2k F^k \label{eqn:2.15.1b1}$$
Proof: Use the formula $$[A, BC] = B[A, C] + [A, B]C$$ with $$A = H, B = F^{k-1}, C = F$$ together with induction on $$k$$.
Lemma 2: If $$k \geq 1$$ is an integer, then $$[E, F^k] = kF^{k-1}H - k(k-1) F^{k-1} \label{eqn:2.15.1b2}$$
Proof: By induction. The base case $$k = 1$$ is easily verified. For $$k \geq 2$$, \begin{align*} [E, F^k] &= F[E, F^{k-1}] + [E, F]F^{k-1} \\ &= F((k-1) F^{k-2} H - (k-1)(k-2)F^{k-2}) + HF^{k-1} \\ &= (k-1)F^{k-1}H - (k-1)(k-2) F^{k-1} + F^{k-1} H + [H, F^{k-1}] \\ &= k F^{k-1}H - (k-1)(k-2) F^{k-1} - 2(k-1) F^{k-1} \\ &= k F^{k-1}H - k(k-1) F^{k-1} \end{align*}
Lemma 3: If $$k \geq 0$$ is an integer, then \begin{equation*} EH^k w = 0 \end{equation*}
Proof: Use the fact that $$EH^k = (HE + [E,H])H^{k-1}$$ together with induction on $$k$$.
It immediately follows that $$EP(H)w = 0$$ where $$P$$ is any polynomial.
Now we are ready to prove the main result. Let $$P$$ be a polynomial. Then \begin{align*} E^k F^k P(H) w &= E^{k-1} (E F^k) P(H) w \\ &= E^{k-1} (F^k E + [E, F^k]) P(H) w \\ &= E^{k-1} (k F^{k-1}H - k(k-1) F^{k-1}) P(H) w \\ &= E^{k-1} F^{k-1} k(H - (k-1)) P(H) w \end{align*} Therefore \begin{equation*} E^k F^k w = k! H (H - 1) (H - 2) \ldots (H - (k-1)) w \end{equation*}
3. This is very similar to part (a), but $$F$$ acts to lower the real part of the eigenvalue by 2, in contrast to $$E$$ which raises it. Since $$V$$ is finite-dimensional, it can only have finitely many eigenvalues, so the sequence $$v, Fv, F^2v, \ldots$$ must end in zero. Indeed, we can make the stronger statement that $$F^{\dim V}v = 0$$.
4. Lemma: If $$v$$ is a generalized eigenvector of $$H$$ with eigenvalue $$\lambda$$, and $$a_1, \ldots a_n \in \mathbb{C}$$, and $$(H-a_1)(H-a_2) \ldots (H-a_n)v = 0$$, then there is some $$i$$ such that $$a_i = \lambda$$.
Proof: By induction. For $$n = 1$$, $$v$$ is just an ordinary eigenvector with eigenvalue $$a_1$$. If $$n \geq 2$$, then either $$(H - a_n)v = 0$$, in which case $$\lambda = a_n$$, or $$(H - a_n)v$$ is a generalized eigenvector with eigenvalue $$\lambda$$, and by the inductive hypothesis, there is some $$i \leq n - 1$$ with $$a_i = \lambda$$.
Now use the Hint from the problem. Take $$N = \dim V$$. From (b), we have $$\label{eqn:2.15.1.d} 0 = E^N F^N v = N! H(H-1)(H-2)\ldots(H-N+1) v$$ By the Lemma, $$\lambda$$ is an integer between 0 and $$N-1$$. Since polynomials of $$H$$ commute with each other, we can rewrite $$(\ref{eqn:2.15.1.d})$$ as \begin{equation*} 0 = \left(\prod_{0 \leq i < N, i \neq \lambda} H-i\right) (H-\lambda)v \end{equation*} Let $$v' = (H-\lambda)v$$. By the contrapositive of the Lemma, $$v'$$ cannot be a generalized eigenvector of $$H$$ with eigenvalue $$\lambda$$, since none of the $$i$$'s can equal $$\lambda$$. Therefore $$v' = 0$$, and $$v$$ is an ordinary eigenvector. Since this holds for all $$v \in \overline{V}(\lambda)$$, it follows that $$H$$ is diagonalizable on $$\overline{V}(\lambda)$$.
5. This follows immediately from the Lemma proven in (d).
6. Suppose $$V$$ is an irreducible finite-dimensional representation of $$\mathfrak{sl}(2)$$. Let $$\lambda$$ be as in (a) and $$v \in V$$ be an eigenvector of $$H$$ with eigenvalue $$\lambda$$. From (e), we have that $$F^{\lambda+1}v = 0$$ and $$F^\lambda v \neq 0$$. Therefore $$v, Fv, \ldots, F^\lambda v$$ are eigenvectors of $$H$$ with eigenvalues $$\lambda, \lambda-2, \ldots, 2, 0, -2, \ldots, -\lambda$$, respectively, which implies that they are all linearly independent. Furthermore, $$\span\{v, Fv, \ldots, F^\lambda v\}$$ is obviously invariant under $$F$$ and $$H$$, and is also invariant under $$E$$ since, by $$(\ref{eqn:2.15.1b2})$$, for any positive integer $$k$$, \begin{equation*} EF^k v = F^k Ev + kF^{k-1}Hv -k(k-1)F^{k-1}v = k(\lambda-k+1) F^{k-1}v \end{equation*} Since $$V$$ is irreducible, this span must equal $$V$$.
With respect to the basis $$\{v, Fv, \ldots, F^\lambda v\}$$, the action of $$H$$ on $$V'$$ takes the matrix form $$\diag(N-1, N-3, \ldots, -(N-3), -(N-1))$$, while $$F$$ obviously has ones on the subdiagonal and zeroes everywhere else. The raising property of $$E$$ implies that it is nonzero only along the superdiagonal, and the equation $$[E, F] = H$$ then fixes $$E$$ as well, so that the superdiagonal entries are easily seen to be $$1(N-1), 2(N-2), \ldots, (N-1)(N-(N-1)) = N-1$$. Call this irreducible representation $$V_\lambda$$. It has dimension $$N = \lambda + 1$$; thus, the value of $$\lambda$$ fixes the dimension of the irreducible representation and also fixes the representation itself up to isomorphism.
We have not proven that $$V_\lambda$$ is actually irreducible. Let us do so now. Suppose $$w \in V_\lambda$$ is nonzero. Write $$w = \sum_i c_i F^i v$$, where $$Hv = \lambda v$$ as above. Let $$m$$ be the smallest integer such that $$c_m$$ is nonzero. Then $$\frac{1}{c_m} F^{\lambda - m} w = F^\lambda v$$ and by applying $$E$$ to $$F^\lambda v$$ we can regenerate the basis. We conclude that there is exactly one irreducible representation of each positive finite dimension up to isomorphism, namely $$V_\lambda$$ where $$\lambda$$ is one less than the dimension.
7. Using the identity $$[A, BC] = B[A, C] + [A, B]C$$: \begin{align*} [E, C] &= [E, EF + FE + H^2/2] \\ &= [E, EF] + [E, FE] + [E, H^2/2] \\ &= [E, E]F + E[E, F] + F[E, E] + [E, F]E + \frac{1}{2}H[E, H] + \frac{1}{2}[E, H]H \\ &= EH + HE - HE - EH \\ &= 0 \\ [F, C] &= [F, EF + FE + H^2/2] \\ &= [F, EF] + [F, FE] + [F, H^2/2] \\ &= E[F, F] + [F, E]F + F[F, E] + [F, F]E + \frac{1}{2}H[F, H] + \frac{1}{2}[F, H]H \\ &= -HF -FH + HF + FH \\ &= 0 \\ [H, C] &= [H, EF + FE + H^2/2] \\ &= [H, EF] + [H, FE] + [H, H^2/2] \\ &= E[H, F] + [H, E]F + F[H, E] + [H, F]E \\ &= -2EF + 2EF + 2FE - 2FE \\ &= 0 \end{align*} Since $$C$$ commutes with all the generators, it is central, and by the result of Problem 2.3.16(a), it therefore acts as a scalar on $$V_\lambda$$. If we take $$v$$ such that $$Ev = 0$$, we can readily compute $$Cv = \frac{\lambda(\lambda+2)}{2}v$$ using the result of the previous part.
8. Since $$V$$ is assumed indecomposable, we can use the result of Problem 2.3.16(b) to conclude that $$C$$ must act in $$V$$ as an operator with only one eigenvalue, namely, the scalar by which it acts on some irreducible subrepresentation. If that irreducible subrepresentation is $$V_\lambda$$, then that single eigenvalue is $$\lambda(\lambda+2)/2$$ by the result of the previous part.
9. Since $$W$$ is reducible and finite-dimensional, it has to have some irreducible proper subrepresentation, and if $$C$$'s single eigenvalue is $$\lambda(\lambda+2)/2$$, then that subrepresentation is isomorphic to $$V_\lambda$$. The quotient representation $$V/W$$ is also a nonzero representation of $$\mathfrak{sl}(2)$$ but is of lower dimension than $$V$$. Either $$V/W$$ is irreducible, in which case it must also be isomorphic to $$V_\lambda$$ since $$C$$ has only one eigenvalue, so that $$n = 1$$; or else, if $$V/W$$ is reducible, then since $$V$$ was assumed to be the smallest reducible representation which is not a direct sum of irreps, then $$V/W$$ must be a direct sum of irreps, and again, since $$C$$ has only the single eigenvalue, each of those irreps must be isomorphic to $$V_\lambda$$. In this case $$n \ge 2$$.
10. If the eigenspace $$V(\lambda)$$ of $$H$$ is $$m$$-dimensional, then, since the eigenspace $$W(\lambda)$$ is one-dimensional, it follows that the eigenspace $$(V/W)(\lambda)$$ is $$(m-1)$$-dimensional. If $$V/W = nV_\lambda$$, then $$m - 1 = n$$, so $$V(\lambda)$$ is $$(n+1)$$-dimensional.
Let $$\{v_1, \ldots, v_{n+1}\}$$ be a basis for $$V(\lambda)$$. The result of part (e) guarantees that for each $$j$$ in $$\{0, \ldots, \lambda\}$$, the vectors in the set $$S_j = \{F^j v_1, \ldots, F^j v_{n+1}\}$$ are all nonzero. We also know that the elements of $$S_j$$ are all eigenvectors of $$H$$ with eigenvalue $$\lambda - 2j$$. The result of part (b) implies that each of $$E^j F^j v_i$$ is a nonzero scalar multiple of $$v_i$$, so that $$\span\{E^j S_j\}$$ is $$(n+1)$$-dimensional, which implies that $$\span S_j$$ is likewise $$(n+1)$$-dimensional; so $$S_j$$ is linearly independent. Since eigenvectors with different eigenvalues are always linearly independent, the union $$S = \cup_j S_j$$ is linearly independent. Since $$|S| = (n+1)(\lambda+1) = \dim V$$, $$S$$ is a basis of $$V$$.
11. As $$i$$ ranges from 1 to $$n+1$$ and $$j$$ ranges from 0 to $$\lambda$$, the $$F^j v_i$$ range through all distinct basis vectors of $$V$$ as found in the previous part. By partitioning these basis vectors into the subsets $$T_i = \{F^j v_i \mid 0 \leq j \leq \lambda\}$$ for each $$i$$, we obtain the subspaces $$W_i = \span T_i$$ whose direct sum equals $$V$$. By the argument given in the solution to part (f), each of these subspaces is invariant under $$E$$, $$F$$, and $$H$$, and is therefore a subrepresentation of $$V$$. So we have derived a contradiction with the assumption that $$V$$ is not the direct sum of subrepresentations.
12. The Jordan form of $$E$$ in the irrep $$V_\lambda$$, as found in (f), is a single Jordan block with eigenvalue zero (and size $$\lambda + 1$$, of course). Therefore, the Jordan form of $$E$$ in the general representation $$V = \large\oplus_i V_{\lambda_i}$$ is the direct sum of Jordan blocks with eigenvalue zero and sizes $$\lambda_i + 1$$; since all finite-dimensional representations of $$\mathfrak{sl}(2)$$ are of this form, we can conclude that for every direct sum of Jordan blocks with eigenvalue zero, there is exactly one representation of $$\mathfrak{sl}(2)$$ in which $$E$$ takes that Jordan normal form. Given a nilpotent operator $$A : V \to V$$, its Jordan normal form is a direct sum of Jordan blocks with eigenvalue zero, so there is exactly one representation, up to isomorphism, in which $$E = A$$.
13. Following the Hint, we define the character of a finite-dimensional representation $$V$$ as $$\chi_V(x) = \tr(e^{xH})$$. If $$V$$ and $$W$$ are two representations, then \begin{align*} \chi_{V \oplus W}(x) &= \tr \exp(x\rho_{V\oplus W}(H)) \\ &= \tr \exp(x\rho_V(H) \oplus x\rho_W(H)) \\ &= \tr(\exp(x\rho_V(H)) \oplus \exp(x\rho_W(H))) \\ &= \tr \exp(x\rho_V(H)) + \tr \exp(x\rho_W(H)) \\ &= \chi_V(x) + \chi_W(x) \end{align*} Also, \begin{align*} \chi_{V \otimes W}(x) &= \tr \exp(x(\rho_V(H) \otimes \Id_W + \Id_V \otimes \rho_W(H))) \\ &= \tr[\exp(x\rho_V(H) \otimes \Id_W) \exp(x\Id_V \otimes \rho_W(H))] \\ &= \tr[(\exp(x \rho_V(H)) \otimes \Id_W ) (\Id_V \otimes \exp(x \rho_W(H)))] \\ &= \tr[\exp(x \rho_V(H)) \otimes \exp(x \rho_W(H))] \\ &= \tr \exp(x \rho_V(H)) \tr \exp(x \rho_W(H)) \\ &= \chi_V(x) \chi_W(x) \end{align*} where between the first and second lines we have used the fact that $$\rho_V(H) \otimes \Id_W$$ and $$\Id_V \otimes \rho_W(H)$$ commute; and between the second and third lines we have used the identity that $$\exp(A \otimes \Id) = \exp(A) \otimes \Id$$, which follows from the power series expansion of the operator exponential.
The character of $$V_\lambda$$ is easily seen to be $$e^{\lambda x} + e^{(\lambda - 2)x} + \ldots + e^{-\lambda x}$$ from the result of part (f). Using the formula derived in the previous paragraph, the character of $$V_\lambda \otimes V_\mu$$ is $$\chi_{V_\lambda}\chi_{V_\mu}$$. If we assume without loss of generality that $$\lambda \geq \mu$$, then \begin{equation*} (e^{\lambda x} + \ldots + e^{-\lambda x})(e^{\mu x} + \ldots + e^{-\mu x}) = e^{(\lambda + \mu)x} + 2e^{(\lambda + \mu - 2)x} + \ldots + \mu e^{(\lambda - \mu)x} + \mu e^{(\lambda - \mu - 2)x} + \ldots + \mu e^{-(\lambda - \mu)x} + (\mu - 1) e^{-(\lambda - \mu + 2)x} + \ldots + e^{-(\lambda + \mu)x} \end{equation*} To write this as the sum of characters of irreps is easy because if $$V_k$$ is the highest-dimensional irrep that appears in the sum, then $$e^{kx}$$ will be the highest term that appears in the character; so we can successively peel off $$\chi_{V_k}(x)$$ from $$\chi_{V_\lambda \otimes V_\mu}(x)$$ where at each stage $$k$$ is taken from the highest exponential remaining. The result is \begin{equation*} \chi_{V_\lambda \otimes V_\mu}(x) = \chi_{V_\lambda + V_\mu} + \chi_{V_{\lambda + \mu - 2}} + \ldots + \chi_{V_{\lambda - \mu}} \end{equation*} so the desired decomposition is \begin{equation*} V_\lambda \otimes V_\mu \simeq V_{\lambda + \mu} \oplus V_{\lambda + \mu - 2} \oplus \ldots \oplus V_{\lambda - \mu} \end{equation*}
14. Using (l), there exists a representation in which $$E = A = J_M(0) \otimes \Id_N + \Id_M \otimes J_N(0)$$. But this is just the tensor product of representations in which $$E = J_M(0)$$ and $$E = J_N(0)$$, which are the irreps $$V_{M-1}$$ and $$V_{N-1}$$. This representation therefore decomposes as the direct sum $$V_{M+N-2} \oplus V_{M+N-4} \oplus \ldots \oplus V_{M-N}$$, where, without loss of generality, we have assumed that $$M \geq N$$, so $$E$$'s Jordan normal form, and hence that of $$A$$, must be $$J_{M+N-1}(0) \oplus J_{M+N-3}(0) \oplus \ldots \oplus J_{M-N+1}(0)$$.
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2022-08-13 03:23:44
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http://partiallyattended.com/2016/12/15/hello-sage/
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## Partially Attended
an irregularly updated blog by Ian Mulvany
# Hello SAGE!
I joined SAGE at the start of September. Hello SAGE!! Here I outline some of my initial impressions.
First up, I’ve been really delighted to meet so many great people at SAGE. I’ve received great support from everyone in the company. I generally find publishing folk to be very friendly. This is a friendly industry, working on the fabric of knowledge, knowing that your work can help to make a difference, trying to make the work of academics a bit easier. I believe that these are all things that can help to create a good environment for an industry to be situated in. All that aside, I’ve still been really impressed by how lovely everyone is. I think that comes from some initial interactions that I had way back in my first week, and it’s only continued through the weeks.
One obvious change at SAGE is the scale of the company. It’s a good bit bigger than eLife, and I’ve not worked in a company close to this size since 2010. At Mendeley, and later at eLife, I saw what happens as a company starts to grow out beyond the point where not everyone is able to know everything that is going on (that’s not a bad thing at all, just an inevitable part of the development of a company). Back when I was working in Springer and Nature my work mostly involved interacting with people in close proximity to my project. What I’m working on now is of interest across the company. Communicating across the natural silos of information that will emerge in a large organisation such as SAGE has required some new thinking. The main thing to note is the existence of structure that is contingent on the history of how that structure emerged, and the best thing I’ve found for understanding that quickly is just to tap into the collective wisdom that already exists within the organisation. Basically asking people who have been around for a lot longer than I have about how to do things, or who to talk to. That’s mostly been successful. The one time where it didn’t work so well was when asked someone a few things, only to discover pretty quickly that they only knew fractionally more than I did because they had only been here about a week longer than me!
I’d never known a huge amount about SAGE before starting to think seriously about coming on board. I’d known a few people for a few years, whom I held in fairly high regards. I didn’t know that the name SAGE comes from Sara and George, the founders of the company. Sara is still very much involved in the company, and chairs the board meetings, as well as continuing to take a keen interest the strategic direction of the company. Since joining I’ve had the pleasure of meeting her a couple of times, and I’ve been hugely impressed with how impassioned she is for the important role that social research can play in society. One moment in particular stands out. It was a few weeks ago, just a few days after the US election. Moods were a little deflated. She stood up at a small meeting and simply articulated the importance of what social science researchers are doing for societal outcomes. She talked about how organisations like SAGE are in a privileged position, and being in that position almost sets a demand on them to do what they can to help support that role of social science.
I think this connects well with another thing that I’ve learnt in the last few months. For the particular project that I’m working on I’m spending a lot of time talking directly to researchers. They uniformly have a positive attitude to SAGE, and the things that SAGE builds in this space. It’s clear that the values of company really seep into how they act in the market.
So what is it that I’m doing now? My job title here at SAGE is Head of Product Innovation. For the time being that title sits in front of one very specific project. My main responsibility over the next year is to support the emerging field of what we might loosely call computational social science. Specifically the team I am in are working on finding services that SAGE can partner on, or build. It’s a pure greenfield product development position.
Here I’m not going to get into the nitty gritty of what to call a data intensive way of doing social science (there are subtleties around whether we call it A or B, or some other label), but I’ll tell you what we currently believe, and what we have observed.
We believe that data at scale is transforming many aspects of how social science is done, and with that transformation is will come the opportunity to answer questions that were previously intractable, as well as making it easier to tackle currently hard questions. We believe we are seeing the emergence of a new methodology for how social science can be done (but we also believe that this does not remove the need for existing methodologies, rather it will enhance them). My favourite analogy here is to think of this as akin to the creation of a new kind of telescope or instrument. It opens up new ways of viewing and understanding the world that builds upon and broadens what we already know.
We see some groups out there already doing this kind of work, and we see many others who are interested but who face a variety of barriers to starting with these techniques. This is where the project I am working on comes in. Initially we are trying to understand these barriers, and design things that can help reduce them.
There are many many reasons why this move towards data intensive social science may be important. At a very basic level expanding the tools available to scholars is always a good thing. Being able to make the most of the implicit data that is now a by-product of the digital interfaces of our lives may move us from a position where we may be haunted by that data to a position where we have have the means to understand how to deal with it. I feel that most importantly it’s also about bringing some humanity to the systems that we are building today. These digital systems and the data that we as a society, and as individuals, are generating are determinative to many social outcomes. If the only driver for the creation of these systems is the market then those outcomes are probably not going to be wholly fantastic. (Cathy O’Neil writes about this clearly in Weapons of Math Destruction). To help balance this I think social scientists need a seat at the table when it comes to the design and engineering of those systems.
Over its fifty year history the publication of methods has been core to what SAGE does. From this perspective finding a way for SAGE to support the emergence of a new class of methodologies makes perfect sense for us. We are not working in isolation either, rather, we are contributing to a strong trend in a way of thinking about the systems that surround us. We want to help by being an active partner in initiatives that can help with the agenda I’ve outlined above, and where we have the opportunity to build things that help move that forward, we will try to do so.
Our thinking about the kinds of things that we can help to create is still very open ended at this point in time. It is also almost impossible to predict what are the things that you do that will have a real impact, and what are the things that you do that end up not making much difference. What is clear is that you don’t have a chance of finding out if you don’t try. We aim to try, and to learn, and hopefully we can iterate on what we learn to find a way to make a meaningful contribution.
I’ve been asked by a lot of people why I decided to move from eLife to SAGE. I’ve already outlined here a bit about the project that I’m working on. Overall when I was approached with this opportunity I decided to weigh up three factors, what impact might the project have, what impact might I have on making the project a success, and overall how would working on this help me support my family.
It was clear to me pretty quickly that this project has the potential to be impactful, and certainly that the motivations behind the instigation of the project were very aligned with my own personal beliefs and interests. I also felt that my background in working on digital tools for researchers over the last few years was a good fit for the needs of the project. This past summer my family grew by one, and now with two young children to juggle (sometimes very literally), the opportunity to work on stuff that matters, and to do so from quite close to home, was one that I had to look very closely at. I made the jump, and now three months in I can honestly say I’m just getting more and more excited by where we are going with the project.
If you have managed to get this far and you are still interested in what we are working on, then maybe you might like thinking about joining us? We currently have a very small team. We decided from the outset to pursue a lean approach to product discovery and development. The team is myself, half of the amazing Katie Metzler and some time from the also awesome Martha Sedgwick.
After a lot of learning in the last three months we have decided that we want to bring in another person full time (on a 12 month contract) to help with the development of the project. We are initially setting the position to be a 12 month fixed contract, as we just have a high degree of uncertainty around what the ideal shape of the team will be in a year from now.
Initially we we want help with the following kinds of things (at the heart of which is helping us to understand the needs of researchers, and helping us to follow up on the many amazing conversations and leads that we are having right now), but we also have a fair expectation that the role, and the entire project, will continue to evolve over the coming year:
* Assist with market segmentation and market sizing
• Conduct competitor analysis product positioning
• Recruit a pool of relevant users for testing of product concepts with
• Conduct solution interviews
• Participate in usability and product concept testing
* Participate in product ideation workshops
* Synthesise and capture feedback from interactions with researchers, and share and distribute that feedback amongst other team members
* Provide product development support during the build phase from concept to MVP
* Be a voice for the user through the evolution of our product ideas.
If you are interested and want to find out more please reach out to me!
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2021-05-09 01:15:54
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|
https://pypi.org/project/scrapy-new/
|
A package providing code generation command for scrapy CLI
# scrapy-command-new
A package providing code generation command for scrapy CLI.
The project is a WIP, so expect major changes and additions (latter, mostly). Master branch is to be considered as always ready to use, with major changes/features introduced in feature branches.
This is a part of a bigger project - Scrapy Boilerplate.
The command works with a specific scrapy project structure (not the default one). Rationale for this is described here.
## Usage
This is a scrapy command to generate class files and automatically add imports to respective module's __init__ files. It can be used as follows:
scrapy new spider SampleSpider
The first argument (spider) is a type of class file to be generated, and can be one of the following:
• command
• extension
• item
• middleware
• model
• pipeline
• spider
The second argument is class name.
Also for pipeline and spider class an option --rabbit can be used to add RabbitMQ connection code to generated source.
Option --item with value CLASSNAME is supported for generating pipelines, which adds an import and type-check for a provided item class to the resulting code.
Option --settings is also supported for pipelines, extension, middlewares and spider middlewares. It has an optional integer value PRIORITY that adds specified priority. If only -s is used, settings file will be settings.py.
(experimental) Option --file is used for specifying settings file name (or class). You can use spider file for adding newly generated class to spiders' custom_settings property. If you enumerate file names (or class names) using , (like -f SomeSpider,AnotherSpider) - script will add generated class to custom_settings of each file. If only -f is used, will be used default priority (300).
Option --terminal will output 'custom_settings' code to terminal.
Option --custom can be used for custom template folder path. Template names should be like {}.py.mako. Option will enable usage of TEMPLATES_MODULE setting from projects settings.py. If this setting is not defined, will cause exception.
## Installation
This command is included in the Scrapy Boilerplate out of the box. If you want to install it manually, you can get it from PyPi:
pip install scrapy-new
Please note that this package won't work with default Scrapy project structure, it requires a specific custom one, as described here.
## Project details
Uploaded source
Uploaded py3`
|
2022-12-07 10:40:47
|
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|
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=127&t=18030
|
## Work of Expansion [ENDORSED]
Marc Farah 3D
Posts: 22
Joined: Wed Sep 21, 2016 2:59 pm
### Work of Expansion
For Work of expansion, why must we measure the changes in energy for in very small steps? And therefore having to use an integral for the infinitesimal changes in volume? When do we use w=-pV as opposed to the integral equation for w?
104822659
Posts: 23
Joined: Wed Sep 21, 2016 2:57 pm
### Re: Work of Expansion [ENDORSED]
I'm not sure if this anywhere near what answer you were looking for, but ..
I think for reactions that happen quickly (at infinitesimal changes), we can use $w=-P_{ex}\cdot \bigtriangleup V$. For reactions that happen more slowly, we use the integral (which is the cumulative sum of all the $w=-P_{ex}\cdot \bigtriangleup V$ at each infinitesimal change/each stage of expansion). In the end, they both would give approximately the same answer.
In our problems, the book usually compares the amount of work between a irreversible path and the reversible and isothermal path.
We use $w=-P_{ex}\cdot \bigtriangleup V$ because the an irreversible path indicates that there is the gradual reduction of force in external pressure that changes the opposing force, making the potential of the system to do work less than what is maximally possible.
For the reversible and isothermal path, the pressure of gas falls as it expands, so, to be reversible, the external pressure must fall as the volume expands so that at every stage of expansion, the external pressure matches with the pressure of the gas (which makes the potential of work at every stage of expansion done maximally).
Christine_Mavilian_3E
Posts: 30
Joined: Fri Jul 22, 2016 3:00 am
### Re: Work of Expansion
If I correctly understand your question, based on the lecture today, there was an example with a diagram where the system (ideal gas) was expanding to twice its volume. The mass would slowly decrease and the system would expand slowly, doing more work. During such process it will lose energy but it's in thermal equilibrium so whatever energy lost will be replaced by heat going into the system from the surroundings at constant temperature. If the process is done slowly the energy change will be minimal and the temperature will remain constant (isothermal).
You would use w=-pv when the pressure is constant. You would use the integral which we learned today that after simplifying equals w=-nRTln (V2/V1) when pressure changes at the same time as the volume; in other words, when work done by the system is not equal.
I hope this helps!
|
2020-08-11 22:57:51
|
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|
http://digital.auraria.edu/AA00004222/00001
|
Citation
## Material Information
Title:
Monomial hyperovals in Desarguesian planes
Creator:
Vis, Timothy L.
Publication Date:
Language:
English
Physical Description:
v, 74 leaves : ; 30 cm
## Subjects
Subjects / Keywords:
Ovals ( lcsh )
Ovals ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )
## Notes
Bibliography:
Includes bibliographical references (leaves 71-74).
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
## Record Information
Source Institution:
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
221615550 ( OCLC )
ocn221615550
Classification:
516.5 ( ddc )
Full Text
MONOMIAL HYPEROVALS IN DESARGUESIAN PLANES
by
Timothy L. Vis
B.A., Dordt College, 2005
M.S., University of Colorado Denver, 2008
A thesis submitted to the
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2010
This thesis for the Doctor of Philosophy
degree by
Timothy L. Vis
has been approved
by
Timothy J. Penttila
Vis, Timothy L. (Ph.D., Applied Mathematics)
Monomial Hyperovals in Desarguesian Planes
Thesis directed by Professor William E. Cherowitzo
ABSTRACT
In a finite projective plane of order n, an oval is a set of n + 1 points such
that no three of the points are collinear. Each point of an oval lies on a unique
line tangent to the oval. When n is even the tangent lines to an oval meet in a
common point with which the oval can be extended to form a hyperoval. In the
plane PG(2,q) constructed from GF(q)3, irreducible conics provide examples
of ovals. Segre [32, 33] showed that when q is odd, these are the only examples.
The situation is quite different when q is even: numerous families and exam-
ples exist and classification seems intractable. The work of Glynn [11], however,
provides a powerful tool for classifying monomial hyperovals (those described
by monomials), suggesting that this may be a tractable problem.
Every known monomial hyperoval is equivalent to one described with an
exponent whose binary expansion has three or fewer nonzero bits. Work of
Segre [34] and of Cherowitzo and Storme [5] has fully classified the monomial
hyperovals described by two or fewer bits. We extend this work by classifying
the monomial hyperovals described by three bits.
Following certain reduction arguments, we use Glynns criterion [11] to an-
alyze possible exponents. The possibilities are restricted to eleven broad cases,
which are eliminated systematically through more narrowly focused application
of Glynns criterion.
With this classification complete, we discuss several avenues of research with
the goal of showing that any monomial hyperoval can be described with three or
fewer bits. We discuss alternate forms of monomial hyperovals, giving all forms
for the known monomial hyperovals and some of the alternate forms for arbitrary
one, two, and three bit exponents. We also discuss two means of transforming
the points of a projective plane that may be useful in simplifying monomial
hyperovals by viewing them in relation to other well-studied structures.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
ACKNOWLEDGMENT
I would like to acknowledge Warren Batemans generous sponsorship of the
Lynn Bateman Memorial Doctoral Fellowship which allowed me four semesters
of release from teaching duties to concentrate on the work found in this thesis.
CONTENTS
Figures ................................................................ ix
Tables.................................................................. x
Chapter
1. MDS Codes and Arcs.................................................. 1
1.1 Error-Correcting Codes............................................ 1
1.2 The Singleton Bound and MDS Codes................................. 4
1.3 n-arcs in Projective Spaces....................................... 6
2. Background........................................................... 7
2.1 Projective Planes................................................. 7
2.2 Desargues Theorem and PG (2,K)................................... 9
2.3 Structure of Projective Planes................................... 11
2.4 Arcs, Ovals, and Hyperovals in PG (2,q).......................... 14
2.5 Monomial Hyperovals.............................................. 22
2.6 Proof of Glynns Criterion....................................... 24
3. Three-Bit Monomial Hyperovals....................................... 28
3.1 Reduction........................................................ 28
3.2 Common Divisor Conditions......................................... 30
3.3 Monomials With Exponent of the Form a + a1 + a?................... 40
3.4 Reducible Monomial Hyperovals..................................... 52
4. Proof of Theorem 3.29............................................... 55
vi
4.1 Case 1............................................................ 55
4.2 Case 2............................................................ 65
4.3 Case 3............................................................ 88
4.4 Case 4............................................................ 97
4.5 Case 5........................................................... 104
4.6 Case 6........................................................... 105
4.7 Case 7........................................................... 110
4.8 Case 8........................................................... 110
4.9 Case 9........................................................... 115
4.10 Case 10.......................................................... 117
4.11 Case 11 ......................................................... 150
4.12 Conclusion....................................................... 151
5. Projectively Equivalent Monomial Sets............................. 152
5.1 Alternative Forms of Monomial Hyperovals.......................... 152
5.1.1 Translation Hyperovals Q> (a)................................. 153
5.1.2 Segre Hyperovals ^ (6).......................................... 154
5.1.3 Glynn Hyperovals Q) (a + 7)..................................... 156
5.1.4 Glynn Hyperovals \$) (3cr + 4)................................... 157
5.2 Some Equivalent Bit Patterns...................................... 158
5.2.1 One Bit Patterns................................................ 160
5.2.2 Two Bit Patterns................................................ 161
6. Relating Projectively Inequivalent Sets of Points................. 164
6.1 Approach from Cremona Transformations............................. 164
6.2 Generalized Oval Derivation....................................... 166
vii
6.2.1 Derivation...................................................... 167
6.2.2 Collineations Under Derivation.................................. 171
6.2.3 Deriving a Desarguesian Plane When P = Q..................... 175
6.2.4 Deriving a Desarguesian Plane When P ^ Q..................... 181
7. Future Research.................................................... 185
References............................................................ 187
viii
FIGURES
Figure
2.1 The Fano Plane...................................................... 8
2.2 A Coordinatized Fano Plane......................................... 10
2.3 Triangles in Perspective from the Point P.......................... 11
2.4 Triangles in Perspective from the Line l .......................... 12
2.5 The Desargues Configuration........................................ 13
2.6 An Oval and its Nucleus in the Fano Plane ......................... 16
IX
TABLES
Table
4.1 -Case 1 Classification Outline ............................................ 66
4.2 Case 2 Classification Outline ............................................ 88
4.3 Case 3 Classification Outline ............................................ 98
4.4 Case 4 Classification Outline ........................................... 105
4.5 Case 6 Classification Outline ........................................... 109
4.6 Case 8 Classification Outline ........................................... 115
4.7 Case 9 Classification Outline ........................................... 117
4.8 Case 10 Classification Outline.......................................... 148
x
1. MDS Codes and Arcs
1.1 Error-Correcting Codes
A crucial element of information transmission is the ability to send a message
and ensure, with high probability, that the same message will be received. When
noise or other factors cause errors in the transmitted message, the capability to
detect and correct those errors is critical to effective communication. For this
purpose, error-correcting codes are used. For a more thorough introduction to
the theory of error-correcting codes, see [29]. Here, we shall only consider codes
in which all codewords have the same length.
Example 1. One means of detecting and correcting errors would be to repeat
each letter three times. This is the three-repeat code. When a received word (of
three letters) does not consist of the same letter repeated three times, an error
is detected; however, if some letter appears twice, the word can be corrected to
that letter repeated three times.
Formally, we define a code in the following way.
Definition 1.1 Denote the space of all vectors of length n over an alphabet
of q elements as V (n, q). A code C over an alphabet of q elements and
having length n is simply a subset of M elements ofV (n,q). An element
of V (n, q) is a word, and an element of C is a codeword.
In the case of the three-repeat code of Example 1, codewords are of length
n = 3, and the code C consists of all words that contain the same letter repeated
three times.
1
Unless an error changes a transmitted codeword to another codeword, the
received word will not be a codeword, and the error will be detected. Further-
more, if there is a clear indication of what error was made, the received word
may be corrected back to the codeword that was originally transmitted. In order
to determine the correct codeword for a given received word, we define a metric
on V (n, q) as follows.
Definition 1.2 Given two words u and v, the Hamming distance d (u, v) is the
number of positions in which the two words differ. The minimum distance
d of a code C is the smallest distance between distinct codewords. A code
with length n, M codewords, and minimum distance d is an (n, M, d) code.
Again, referencing the three-repeat code of Example 1, the distance between
any two distinct words (and therefore the minimum distance) is three. Thus, in
an alphabet with q letters, the three-repeat code is a (3, q, 3) code.
The Hamming distance defines a metric on V (n,q). This metric may be
used to determine how many errors can be effectively detected or corrected with
the following theorem.
Theorem 1.3 In a code with minimum distance d, any d 1 or fewer errors
can be detected, and any t or fewer errors can be corrected.
Proof: At least d errors are required for a codeword other than that transmit-
ted to be received, so any d 1 or fewer errors can be detected. If t or fewer
errors occur, the distance to any codeword other than that transmitted is at
least t -f 1, and the transmitted codeword is the unique codeword closest to the
received word. Thus, any t or fewer errors can be corrected.
2
Applying Theorem 1.3 to the three-repeat code indicates that any two er-
rors may be detected and any one error corrected. Indeed, if the word aaa is
transmitted, two or fewer errors must leave at least one a and cannot yield an-
other codeword, while three errors might yield the codeword bbb, so that only
two errors may be detected. If only one error occurs, the received word will
look like baa, aba, or aab, and can unambiguously be corrected to aaa. If two
errors occur, the received word could look like abb or abc. In the first case, the
word would be improperly corrected to bbb; in the second the word cannot be
definitively corrected to any of aaa, bbb, or ccc. Thus only one error may be
properly corrected.
In general, an error-correcting code need not have any algebraic structure.
However, many of the most useful codes can be thought of algebraically and
the underlying algebraic structure can often be used to aid in efficient error-
correction. In particular, when q (the size of the alphabet) is a prime power we
may treat V (n, q) as the vector space GF (q)n and we may use some subspace
of GF (q)n as the code C. We shall restrict our attention to these linear codes,
which we now formally define.
Definition 1.4 A linear [n,k,d] code is a k-dimensional subspace of GF (q)n
with minimum distance d.
The three-repeat code of Example 1 over the alphabet GF (q) provides a
simple example of a linear code spanned by the vector (1,1,1). It is thus a
[3,1,3] code.
1.2 The Singleton Bound and MDS Codes
3
There is a natural tension between the parameters n, k, and d in terms of
constructing efficient codes that correct many errors. It is desirable that n k be
small, to minimize the amount of extra information transmitted. Put another
way, for a given n, it is preferable to maximize the number of codewords so as to
maximize the efficiency of transmission. At the same time, it is desirable that d
be large, to maximize the error-correcting capabilities of the code.
These two goals are naturally at odds: increasing the number of codewords
and increasing their mutual distance are incompatible goals. It is natural to ask
what the best possible situation could be. The Singleton Bound answers that
question, and the optimal codes, in terms of minimum distance, are those codes
that attain the Singleton Bound. The Singleton Bound can be used to bound
any of the parameters n, k, and d with the other two. For our purposes, we
express it as a bound on d.
Theorem 1.5 (Singleton Bound) d < n k + 1.
Proof: The Singleton Bound may also be written k < n d + 1. Consider
the first n d + 1 entries in the codewords of an [n, k, d]-code. Since only d 1
other entries exist, any two codewords must differ in at least one of these entries.
Thus, the dimension of the code, k, can be no more than n d + 1.
Codes which meet the Singleton Bound are those which attain the maximum
possible minimum distance for a given choice of n and k. We give these codes a
special name.
Definition 1.6 A code C for which d = nk+1 is called a Maximum Distance
Separable or MDS code.
4
The three-repeat code of Example 1 meets the Singleton Bound and is an
MDS code when the codewords are considered to have length three. However,
in a broader perspective, the three-repeat code becomes highly inefficient. If,
instead of thinking of encoding individual letters, we think of encoding words
of length n by repeating each letter three times, we obtain a [3n, n, 3] code, and
the minimum distance 3 is much less than the value 2n +1 required for the code
to be MDS. In general then, it is far more efficient to use other codes which
more closely approach the Singleton Bound, such as MDS codes if they exist.
MDS codes have a nice geometric interpretation [15, 36] and serve as a
significant motivator for the study of certain geometric structures. In order to
discuss this interpretation, however, we require a few more basic concepts related
to linear codes.
Definition 1.7 Given a linear code C, let G be a matrix with linearly inde-
pendent rows such that C = row (G) and let H be a matrix with linearly
independent rows such that C null (H). Then G is a generator matrix
for C and H is a parity-check matrix for C.
It is easy to see that for an [n, k, d]-code, a generator matrix is k x n and a
parity-check matrix is n k x n. In addition, a parity-check matrix for a code
can be used to determine its minimum distance as follows.
Theorem 1.8 A linear code with parity-check matrix H has minimum distance
d if every set of d 1 columns of H is linearly independent and some set of d
columns of H is linearly dependent.
5
Thus, any MDS code has aiJ-lxn parity-check matrix with every set of
d 1 columns linearly independent and any such matrix is a parity-check matrix
for an MDS code.
1.3 n-arcs in Projective Spaces
Consider then, the columns of a parity-check matrix for an MDS code over
GF (q). These columns form a set of n vectors in GF (q)dl in which every subset
of d 1 or fewer vectors is linearly independent.
These vectors may also be considered as coordinates of points in the pro-
jective space PG (d 2,q). Any set of m points with linearly independent co-
ordinates spans an m 1-dimensional projective space. Thus, these vectors
correspond to a set of n points with the property that any set of m < d 1
points spans anm-1 dimensional projective space, no m points lie in any m 2
dimensional projective space.
Definition 1.9 A set ofn points of PG (r, q) is an n- arc if for every m < r+1,
no set of m points lies in any r l-dimensional subspace of PG (r, q).
It should be clear from this definition that the columns of a parity-check
matrix for an MDS code coordinatize the points of an n-arc; the converse is also
true. As such, the study of arcs in projective spaces (of any dimension) is of
interest in determining optimal codes.
6
2. Background
2.1 Projective Planes
In order to discuss monomial hyperovals in PG (2,2h), we must first discuss
what PG (2, K) is. We begin by defining a projective plane and discussing
certain important properties of projective planes.
Definition 2.1 A projective plane is a point-line incidence geometry satisfying
the following three axioms:
1. Any two distinct points are incident with a unique common line.
2. Any two distinct lines are incident with a unique common point.
3. There exists a set of four points such that no three are incident with
a common line.
The last axiom is a non-degeneracy axiom that excludes certain undesirable
and uninteresting cases.
Example 2. The smallest projective plane, pictured in Figure 2.1 has seven
points and seven lines. Note that the circle in the figure represents a line in this
plane. Each point is incident with three lines, and each line is incident with
three points.
Before we proceed further, we introduce several concepts.
Definition 2.2 A set of points in a projective plane is collinear if there exists
a line incident with each of the points. A set of lines in a projective plane
is concurrent if there exists a point incident with each of the lines.
7
Figure 2.1: The Fano Plane
Definition 2.3 A skewfield consists of a set S and two operations + and -,
such that (S, +) is an Abelian group with identity 0, (S \ {0} ) is a group,
and a -(b + c) = a- b + a- c for all a, b, and c (not necessarily distinct) in
S. In other words, a skewfield satisfies all of the axioms of a field except
possibly commutativity of multiplication.
Higher dimensional projective spaces can all be constructed from vector
spaces over skewfields; the same cannot be said for projective planes. In fact,
the classification of all projective planes remains perhaps one of the most out-
standing and significant open problems in combinatorics. Our interest, however,
shall be solely in those projective planes constructed from vector spaces over
skewfields: the planes PG (2, K).
2.2 Desargues Theorem and PG (2, K)
8
Definition 2.4 The geometry PG (2, K) is the point-line incidence structure
whose points are the one-dimensional vector subspaces of K3, lines are
the two-dimensional vector subspaces of K3, and incidence is the natural
subspace relation. When K is the finite field GF (q), we simply write
PG (2, q).
The geometry PG (2, K) is, as alluded to earlier, a projective plane. Proving
this is a simple exercise in linear algebra.
Theorem 2.5 The geometry PG (2, K) is a projective plane.
PG (2, K) has the desirable property that it can be easily coordinatized
with both points and lines receiving coordinates. We will represent points of
PG (2, K) as non-zero vectors in the corresponding space with parentheses and
lines of PG (2, K) as non-zero vectors in the orthogonal space with brackets. In
each case, two sets of coordinates that are scalar multiples of one another refer
to the same point or line. In Figure 2.2, we present such a coordinatization of
the Fano plane using GF (2).
In addition to the algebraic structure that defines PG (2, K), there is also a
geometric property that distinguishes these planes: they are precisely the planes
that satisfy Desargues Theorem. In order to describe Desargues Theorem, we
require the following definitions.
Definition 2.6 A triangle is a set of three non-collinear points. Triangles
QRS and Q'R!S' are said to be perspective from a point or central if
the lines QQ', RR', and SS1 are concurrent (see Figure 2.3). Triangles
9
Figure 2.2: A Coordinatized Fano Plane
are said to be perspective from a line or axial if the points QR D Q'R',
QS fl Q'S', and RS fl R'S' are collinear (see Figure 2.4).
Definition 2.7 A projective plane satisfies Desargues Theorem and is called
Desarguesian if and only if every pair of triangles that is perspective from
a point is perspective from a line. Figure 2.5 shows triangles QRS and
Q'R'S' that are perspective from the point P and from the line l in the
Desargues configuration.
The Fano plane is somewhat exceptional in that it does not contain any
pairs of disjoint triangles. In fact, the Desargues configuration (pictured in
Figure 2.5) contains ten points, and the Fano plane only seven. If, however, we
10
Figure 2.3: Triangles in Perspective from the Point P
do not require that the triangles are disjoint, any triangles in the Fano plane
that are perspective from a point are perspective from a line. If we do require
that the triangles are disjoint, the statement that any pair of disjoint triangles
that are perspective from a point are also perspective from a line is vacuously
true in the Fano plane. As such, we consider the Fano plane to be Desarguesian.
2.3 Structure of Projective Planes
In order to further our study of projective planes, we now turn to an exami-
nation of the structure of a projective plane. The following result, which is true
for any projective plane, establishes a great deal of structure within a projective
plane.
11
Figure 2.4: Triangles in Perspective from the Line l
Theorem 2.8 In a projective plane, given points P and Q and lines l and m,
there are bijections between each of the following: lines through P, lines through
Q, points on l, and points on m. In particular, in a finite projective plane, there
is some number n such that every line contains n + 1 points, every point lies on
n + 1 lines, and such that the plane contains n2 + n + 1 points and the same
number of lines. m
Definition 2.9 The number n described in Theorem 2.8 is called the order of
a plane.
Theorem 2.8 indicates that from the simple angle of counting the number
of lines incident with a point (or points incident with a line), all points (or all
lines) are equivalent. It is natural to consider when points or lines are equivalent
from a more meaningful perspective. The next definition describes the analogue
to an isomorphism in incidence geometry: a map that preserves the incidence
relations.
12
Figure 2.5: The Desargues Configuration
Definition 2.10 A bijective map a on the points and lines of a projective plane
is called a collineation of the plane provided that whenever a point P is
incident with a line l, the point Pa is incident with the line la.
It is not difficult to see that the set of all collineations of a projective plane
forms a group, called the full collineation group of the projective plane. When
the projective plane is PG (2, K), the structure of this group is known. Two
families of collineations form the basis for this group.
Definition 2.11 A homography is a collineation of PG (2, K) induced by mul-
tiplication of each point by an invertible 3x3 matrix.
13
Definition 2.12 An automorphic collineation is a collineation induced by ap-
plying the same automorphism of K to each coordinate of each point.
Theorem 2.13 (Fundamental Theorem of Field Planes) If F is a field,
every collineation of PG (2, F) is the product of a homography and an automor-
phic collineation.
2.4 Arcs, Ovals, and Hyperovals in PG(2,q)
In 1947, Bose [3] studied the application of finite geometries to the theory of
confounding in factorial designs. One of the particular questions he addressed
was the construction of symmetric factorial designs in which interactions of a
certain degree (t) or lower remained unconfounded. He was able to construct
such designs using sets of points in the projective space PG (r, q) such that every
t of them span a subspace of dimension t 1.
Notice that when t = r, such a set of points is an n-arc. Because our interest
from this point forward will lie strictly within projective planes, we redefine an
arc within this setting.
Definition 2.14 A k-arc (or simply an axe) in a projective plane is a set of k
points, no three collinear.
Theorem 2.15 (Bose, [3]) The maximum size of an arc in PG (2, q) is q + 2
when q is even and q + 1 when q is odd.
Given an arc JF in an arbitrary projective plane 7r, the lines of 7r can be
separated into three classes by the size of their intersection with JF. Lines not
14
intersecting are external lines; lines intersecting in a single point are
tangent lines; lines intersecting Jtf in two points are secant lines. Qvist was
able to extend Boses result to arbitrary projective planes in the following way.
Theorem 2.16 ([30]) In a plane of order q, if q is even the tangent lines to a
q + 1 arc all meet in a common point (called the nucleus); if q is odd every point
on a tangent line to a q + 1 arc lies on some other tangent line.
Corollary 2.17 In a plane of order q, every q +1 arc can be extended to a q + 2
arc when q is even, and no q + 2 arcs can exist when q is odd.
Our interest is in these arcs of maximum size, for which we make the fol-
lowing definition.
Definition 2.18 A q + 1 -arc in a projective plane of order q is an oval; a
q + 2-arc is a hyperoval.
Example 3. Figure 2.6 illustrates an oval in the Fano plane. Lines k, l, and
m are, respectively, external, tangent, and secant to the oval, while the point N
is its nucleus.
In the more general case of PG(2,q), an example of an oval is given by
the points whose coordinates satisfy an irreducible quadratic equation. When
q is even, extending this set of points yields an example of a hyperoval. These
examples are named in the following definition.
Definition 2.19 The set of points in PG (2, q) whose coordinates satisfy an
irreducible quadratic equation is a conic or regular oval. If q is even, the
extension of this set of points is a hyperconic or regular hyperoval.
15
k
l \m
Figure 2.6: An Oval and its Nucleus in the Fano Plane
In 1955, Segre proved his celebrated theorem, cementing the connection
between the combinatorially described oval (set of q+l points, no three collinear)
and the algebraically described conic (set of points satisfying an irreducible
Theorem 2.20 (Segre, [32, 33]) Every oval in PG(2,q), q odd, is a conic.u
When q is even, the situation is quite different, and a complete classification
of ovals in PG(2,q), q even seems at present to be intractable. A number of
families have been constructed, which we discuss after a few more preliminary
definitions and results.
Definition 2.21 The points (0,0,1), (0,1,0), (1,0,0), and (1,1,1) are to-
16
A consequence of the Fundamental Theorem (Theorem 2.13) is that given
any two ordered quadrangles of PG(2,q), there exists a collineation mapping
one to the other in the same order (essentially a change of basis). This gives us
the following equivalence.
Theorem 2.22 Every hyperoval in PG (2, q) is projectively equivalent to one
We may assume, therefore, that a hyperoval contains the fundamental quad-
rangle and consider the implications for the remaining points. Notice that any
point of the form (0, x, y) lies on the line joining (0,1,0) and (0, 0,1), so that the
remaining points all have non-zero first coordinate. Without loss of generality,
the remaining points have coordinates (1, x, y). Now if any two such points have
the same second (or third) coordinate, the point (0, 0,1) (or (0,1,0)) lies on the
line joining those two points. Thus, any two such points have distinct second
and third coordinates. Since there are q such points, we may assume these points
are of the form (1, x, / (x)), as x ranges over the elements of GF (q), and where
/ is a permutation polynomial. We define, more generally, the following set of
points.
Definition 2.23 The set of points Q) (/) is the following set:
(f) = {(l,x,f(x))\xeGF (<,)} U {(0,1,0), (0,0,1)} .
When f (x) xk, we will simply write (k) for Td (/).
17
Following Cherowitzo [6], we call any function / with / (0) = 0 and / (1) = 1
for which 2 (/) is a hyperoval an o-polynomial.
Considering the lines through the point (1, £,/(£)) and any other point of
the hyperoval, we also notice that the values are distinct if and only
if the lines are distinct. Thus, we obtain the following criterion, due to Segre
[34, 35] for a polynomial to be an o-polynomial.
Theorem 2.24 ([34, 35]) The permutation f is an o-polynomial if and only if
the following three conditions hold:
1. /(0) = 0,
2. /(1) = 1,
3. Each ft (x) is a permutation polynomial, where
X + t
We should note that the set 2 (/) is a hyperoval whenever the third condi-
tion holdsthe requirement that / (0) = 0 and / (1) = 1 simply forces (/) to
contain the fundamental quadrangle. In this notation, conics can be described in
several ways, most notably as (2). With this notation, we are able to discuss
the other known examples of hyperovals.
Two years after proving his famed theorem, Segre [34] constructed the trans-
lation hyperovals using field automorphisms of maximum order. These hyper-
ovals have the form 2 (21), (i, h) = 1 and exist distinctly from regular hyperovals
18
in PG (2, 2h) for h = 5 and h > 7. In the same paper, Segre established that
all hyperovals in PG (2, 2h), h < 3 were regular (projectively equivalent to con-
ics) and asked whether any irregular (non-conic) hyperovals existed in the two
remaining cases.
For PG (2,24), the answer came quickly when Lunelli and See [19] con-
structed an irregular hyperoval with o-polynomial 7712x2 + r710x4-f773x8 + r/12a:10-|-
rfx12 + r]4xu, 77 6 GF (24) with rj4 77 + 1. For PG (2, 26) the answer was much
longer in coming and several other families and examples appeared in the inter-
vening time.
Following these hyperovals, many further examples and families were found.
In 1962, Segre announced the construction of the Segre hyperoval 2 (6) in
PG (2, 2h), h odd [35]. The proof, however, did not appear until 1971 [31].
A few years later, Eich, Payne, and Hirschfeld [10, 14] showed that 2 (20) was
a hyperoval in PG (2, 27) distinct from the translation and Segre hyperovals.
With the aid of a computer, Hall [13] showed in 1975 that the only hyperovals
in PG (2, 24) were the regular hyperoval and the Lunelli-Sce hyperoval; in 1991,
OKeefe and Penttila [21] provided a computer-free proof of the result.
Glynn [11] introduced two further infinite families in 1983: 2 (3cr + 4) and
2) (cr + 7) in PG (2, 2h), h odd, where cr2 = 2 = q4 mod 2h 1. Glynns hyper-
ovals included the earlier example of [10, 14]. At the same time, he introduced
a computationally advantageous criterion for determining when 2 (k) is a hy-
peroval. As this criterion is critical to our work, we prove it at the end of this
chapter.
Definition 2.25 Define the partial order < as follows. Write x = Y^ox^1
19
and y = YlrLoVi^ so that each %i G {0,1} and each {0,1}. Then
x < y if and only if Xj < yi for all i. In other words, x ^ y if and only if
in each position in the binary expansion of x containing a one, the binary
expansion of y also contains a one.
Example 4. Since 11 = 2 + 21 + 23, 12 = 22 + 23, and 13 = 2 + 22 + 23,
11 2< 13, but 12 X 13.
Theorem 2.26 (Glynns Criterion [11]) f (t) = tk is an o-polynomial over
PG (2, q) if and only if d H kd, for all 1 < d < q 2, where kd is reduced modulo
q 1 under the convention that zero is reduced to zero and any other multiple
of q 1 is reduced to q 1.
In 1985, Payne [24] discovered the first infinite family of non-monomial
hyperovals with o-polynomial xs +x5 -frrl in PG (2, 2^), h odd, while studying
generalized quadrangles. Three years later in 1988, Cherowitzo [6] discovered the
initial examples of the Cherowitzo hyperovals with h odd, having o-polynomial
xa + xa+2 + x3tT+4, where a2 = 2 mod 2h 1. In 1998 [7], he showed this to be
an infinite family.
A few years later, Glynn [12] extended his criterion to arbitrary polynomials
as follows:
Theorem 2.27 ([11]) If f (0) = 0 and /(1) = 1, @ (/) is a hyperoval if and
only if for all pairs of integers (b, c) with l the coefficient on xc in [f (x)]b is zero.
20
In 1992, OKeefe and Penttila, examining possible stabilizers of hyperovals
in PG (2,25), produced a new irregular (non-conic) hyperoval with o-polynomial
x4 + x16 + x28 + rjn
(X6 + X10 + Xl4 + X18 + X22 + X26)
W (x8 + x20)+v6 (x12 + x24).
Two years later, Penttila and Pinneri [25] produced the first irregular hyper-
oval in PG (2, 26). Subsequently, Penttila and Royle [26, 27] produced further
examples of irregular hyperovals in PG (2, 2h) with 6 < h < 8. They further
proved that the only hyperovals in PG (2, 25) are the known ones, and that any
unknown hyperoval in PG (2, 26) has a trivial stabilizer.
The 1990s saw the development of techniques using flocks of cones to con-
struct o-polynomials, and Cherowitzo, Penttila, Pinneri, and Royle used flocks
of the quadratic cone to construct the two families of Subiaco hyperovals in
PG (2,2h). These have o-polynomials given as follows: Let g be an element of
GF (2h) such that t/-1 has trace one and 1 4- 77 + rj2 ^ 0. Let
g(x) = 12 +
rfx4 + rf (1 + 77 + 772) x3 + rf (1 + 77 + rf) x2 + rfx
(x2 + gx + l)2
and let
, _ 772 1 774x4 + ?73 (1 + 77 + T}2)2 x3 + rf (1 + t?2) x
77 (1 + 77 + 772) 77 (1 + 77 + 77s) (x2 + 77X + l)2
Then the o-polynomials are g (x) + tf (x) + t^x^, t E GF (2fc) and / (x). This
infinite family includes the Lunelli-Sce hyperoval.
Further new examples were developed in 1997 by Payne, Penttila, and Royle
[23] in the planes PG (2, 2l) for i E {10,12,14,16}.
21
In [7], Cherowitzo developed the theory to use flocks of cones over trans-
lation ovals to construct hyperovals, allowing him to establish the Cherowitzo
hyperovals as an infinite family. Five years later, Cherowitzo, OKeefe, and
Penttila [8] constructed the Adelaide hyperovals in PG (2, 2fe), h even with this
technique. These hyperovals have o-polynomial
T(ym)(t+1) , r((>t + >2) ) ,
T(,) T(ri)(t+ T(rnt< I l)" '
where rj E GF (2h) with r] ^ 1 and rj1'2+l = 1, T (x) = x + x, and m =
Together, the Adelaide and Subiaco hyperovals include each of the hyper-
ovals of [23], [25], [26], and [27], thereby placing every known hyperoval ex-
cept the OKeefe-Penttila in an infinite family. The known hyperovals then
are the translation hyperovals (including the regular hyperovals), Segre hyper-
ovals, Glynn hyperovals (two families), Payne hyperovals, Cherowitzo hyper-
ovals, Subiaco hyperovals, Adelaide hyperovals, and the OKeefe-Penttila hy-
peroval.
2.5 Monomial Hyperovals
Of the known hyperovals, the first four families discovered (translation hy-
perovals, Segre hyperovals, and both families of Glynn hyperovals) have mono-
mial o-polynomials. These hyperovals have, in general, a much greater degree
of symmetry than their non-monomial counterparts and lead to further combi-
natorial structures, making them an interesting subject of study.
One of the fundamental properties of a hyperoval is its stabilizer group in the
full collineation group of the ambient projective plane. In 1978, Korchmaros [17]
22
classified those hyperovals having a stabilizer acting transitively on the points
of the hyperoval.
Theorem 2.28 ([17]) The only hyperovals having a transitive stabilizer are the
regular hyperovals in PG (2,2) and PG (2,22) and the Lunelli-Sce hyperoval in
PG (2,24).
In 1994, OKeefe and Penttila [22] extended these results, considering hy-
perovals with large point orbits under the full collineation group of the plane.
Theorem 2.29 ([22]) The only hyperovals containing a transitive q + l-arc are
the regular hyperovals.
Theorem 2.30 ([22]) The only hyperovals containing a transitive q-arc are the
translation hyperovals.
Theorem 2.31 ([22]) The only hyperovals containing a transitive q 1-arc are
the monomial hyperovals unless q 212m+4.
As the transitive hyperovals, ovals, and q-arcs are thus completely charac-
terized (since regular and translation hyperovals are completely characterized),
a logical next step appears to be a complete characterization or classification
of monomial hyperovals. Along with a characterization of the q = 212m+4 case,
this would provide a complete classification of arcs with large point orbits under
their stabilizer groups.
Monomial hyperovals are also of interest because of their connection to cyclic
difference sets. Maschietti [20] has shown that whenever @ (k) is a hyperoval in
PG (2, 2^), the set {xk + x\x G GF (2ft) } is a difference set in Z2h_i- Because
23
of the interest in difference sets, a classification of monomial hyperovals is again
desirable.
A natural method of classification suggests itself from Glynns Criterion
(Theorem 2.26). This method considers the number of ones in the binary ex-
pansion of k, and some progress has been made in this direction. Segres work
in [34] completely characterizes all hyperovals of the form (2l). More re-
cent work of Cherowitzo and Storme [5] characterizes the hyperovals of the form
@ (2* + 2J). Since every known monomial hyperoval can be represented in either
one, two, or three bits, a two stage classification program is suggested:
1. Classify hyperovals of the form 3> (2l + 2n + 212).
2. Show that every monomial hyperoval can be represented in at most three
bits.
With extensive use of Glynns Criterion, we are able to complete the first part
of this program; the second remains an ongoing area of research.
In addition, Glynns Criterion allows for computational ease in classifying
monomial hyperovals in small planes. In [12], Glynn classified the monomial
hyperovals in PG (2,2h), for h < 30 as the known examples. Private com-
munication indicates that unpublished work has extended this classification to
considerably higher values of h. This suggests that new monomial hyperovals
are unlikely to exist.
2.6 Proof of Glynns Criterion
Because of its fundamental importance to our work, we now proceed to
provide a proof of Glynns Criterion. We begin with necessary preliminary re-
24
suits, starting with a monomial adaptation of Segres condition on o-polynomials
(Theorem 2.24).
Lemma 2.32 ([31, 35]) @ (k) is a hyperoval if and only if the following three
conditions hold: (k,q l) = 1, (k l,q 1) = 1, and ^-!l! permutes
GF(q)*.
Proof: The last condition requires k ^ 0. Thus, 0fc = 0 and \k = 1. Now xk is
a permutation if and only if (k, q 1) = 1 so by Theorem 2.24, we have only to
show that ***** is a permutation for each t. When t = 0, ^ is a permutation if
and only if (A: l,q 1) = 1. When t 0, is a permutation if and only if
(x+t)k+tk
is a permutation if and only if t
fc(iiri;
is a permutation if and only
r (x+l)k + l
if
is a permutation, establishing the result.
We also require certain algebraic results regarding polynomials over finite
fields.
Lemma 2.33 ([9]) The polynomial of degree at most q 1 representing a
function f (x) on a field has coefficients given as follows a0 = /(0), =
- ZteGFfo)* / (*) f~r 1 Theorem 2.34 (Hermite-Dickson Criterion [18]) Let GF (q) be a field of
characteristic p. Then f G GF (q) is a permutation polynomial of GF (q) if and
only if the following two conditions hold:
1. [f (f)]r has degree at most q 2 modulo tg t for r with 1 < r < q 2,
and r ^ 0 mod p.
2. f has exactly one root in GF (q).
25
Lemma 2.35 The expansion of (1 + x)k modulo xq x over GF (q) is given by
x md X<1 ~ x> where 0 < c < q 1.
With these results, we are prepared to prove Glynns Criterion.
Theorem 2.36 (2.26[11]) f (t) = tk is an o-polynomial over PG (2,q) if and
only if d ^ kd, for all 1 < d < q 2, where kd is reduced modulo q 1 under the
convention that zero is reduced to zero and any other multiple of q 1 is reduced
to q 1.
Proof: Let k be given as desired. Then by Lemma 2.32, xk is an o-polynomial
if and only if fk (x) = ^x+1^ -1 is a permutation. But ^X+1J +1 = ^z+i)+i > w^ich
simplifies to ^2i=o (x + l)*- In particular, fk (0) is simply the sum of k ones
and is zero or one as k is even or odd. For x ^ 0, fk (x) = 0 if and only if
(x + l)fc +1 = 0 if and only if (x + l)fe = 1 if and only if x +1 = lfc_1 if and only
if x 0, a contradiction. So to be a permutation, k must be even (to obtain
a root) and [fk (x)]r modulo xg x must have zero coefficient on xq~l for all
l By Lemma 2.33, this coefficient is
- E
teGF(q)* V
for all 1 < r < q 2.
Now let g (x) = ^(x + l)fc + 1 j There is a unique polynomial / (x) repre-
senting this of degree at most q 1, and by Lemma 2.33, / (x) has coefficient on
26
xr of X^teGF(g)* ^t+1f 'fl j Thus, fk (x) is a permutation if and only if the
coefficient of xr in + l)k + 1 j is zero for all 1 < r < q 2.
Equivalently, the expansion Yhd giyen by Lemma 2.35 has zero
coefficient on xr for all 1 < r < q 2. But the terms with an xr are exactly
J2d^r Ylr is equivalent to the statement that \{d : d ^ r ^ kd}\ = 0 mod 2 for all r.
With this given, suppose that there exists some d' such that d! -< kd! and
such that d' is minimal. Then r = d' yields {d \ d < d' -< kd) {d1}, a contra-
diction, so that this occurs if and only if d ^ kd for all d.
Now d kd for all d implies (k, q 1) = (k 1, q 1) = 1 in the following
way. If (k, q 1) = l > 1, m = satisfies mk = q 1 mod q 1. Then
m < mk, a contradiction, so that (k,q 1) = 1. If (k 1, q 1) = n ^ 1,
p = so p (k 1) = 0 mod q 1, which implies that pk = p mod q 1
giving them the same binary expansions, again a contradiction. So d ^ kd for
all d if and only if the three conditions of Lemma 2.32 are all satisfied.
27
3. Three-Bit Monomial Hyperovals
3.1 Reduction
To classify monomial hyperovals in PG (2,2h) of the form Q) (2* + 211 + 212),
it is helpful to restrict our attention to a more limited class of exponents in an
attempt to classify these hyperovals.
Definition 3.1 Let k = ]T)"_02a*, with a, < h. Further, let h = be. If, for
every pair {i,j}, ai ^ aj mod b, we say that k is 6-reducible with respect
to h. The number k' = \$^_02ai, where a' is the reduction modulo b of aj
to lie in [0, 61], is called the 6-reduction of k with respect to h. If k is
not b-reducible, we say k is 6-irreducible with respect to h.
More generally, we make the following definition.
Definition 3.2 If there exists a 6 such that b\h and k is b-reducible, we say that
k is reducible with respect to h. If, on the other hand, k is b-irreducible
for every divisor 6 of h, we say that k is irreducible with respect to h.
When we are working in a finite field of order 2h, we say reducible and
irreducible for reducible with respect to h and irreducible with respect to h re-
spectively.
The idea of reducibility allows a significant lessening of the scope of the
necessary classification, as the following results will indicate. Our first result
shows that if a 6-reduction of k does not determine a monomial hyperoval, k
does not determine a monomial hyperoval.
28
Theorem 3.3 Let h = be and let k' be a b-reduction of k with respect to h. If
(k1) is not a monomial hyperoval in PG (2,2b), then 3> (k) is not a monomial
hyperoval in PG (2, 2h).
Proof: Suppose (kr) is not a monomial hyperoval in PG(2,26). Then
there exists some d' such that 1 < d' < 2b 2 and d' < k'd' mod 2b 1. Let
d = YfiZo 2bxd'. Now 1 < d < 2h 1, so it remains only to show that d z< kd
mod 2h 1.
The binary expansion of the value of d we constructed consists precisely of
the binary expansion of d! repeated c times. As such, 2bid = d mod 2h 1
for any integer i. More generally, 2bl+ad = 2ad mod 2h 1 for any choice of i
and ait is only the congruence class of a mod b that determines the product
2ad. Since the congruence classes of the bits in k coincide with the congruence
classes of the bits in k', then kd = k'd mod 2h 1. Thus, kd = YliZo 2bik'dl
mod 2h 1. Since d is simply d' repeated c times and kd is simply k'd' repeated
c times, we must have d < kd, so that & (k) is not a monomial hyperoval in
PG (2,2h). m
This leads naturally to the question of when a potential hyperoval exponent
is reducible, and motivates the classification of irreducible exponents and of
those hyperovals having irreducible exponents. It follows immediately from the
work above that any ^-reduction of an exponent yielding a monomial hyperoval
must itself give a monomial hyperoval. This suggests the following method of
classification:
1. Classify all monomial hyperovals having irreducible exponents.
29
2. Use the classification of hyperovals having irreducible exponents to classify
monomial hyperovals that have reducible exponents.
In the case that k = 2l + 211 + 212 we may easily restrict the occurrence of
irreducible exponents.
Theorem 3.4 If k = 210 + 2n + 212 andp, q, and r are distinct primes dividing
h, then k is reducible.
Proof: Suppose h = paq^s, where p \ s, q\ s, and s > 1. Further suppose that
k = 2l + 2n + 212 and that k is irreducible. Since k is irreducible, any divisor
of h must divide at least one of the differences between distinct bits in k or
a reduction with respect to that divisor would necessarily exist. In particular,
paqP must divide one 0f z2 ix, U z0, and ?'0 i2. Without loss of generality,
then, paq0\io ii-
Again, since k is irreducible, pas must divide one of the three differences.
However, ifpQs|i0 i2, it must be that h\i0 z2, a contradiction. Thus, pas\ii i0
or pas|i2 i\. Without loss of generality, then, pas\i\ Iq. But then, since
pa\i0 i2 and pa|ii i0, pa\i2 i\.
Finally, since k is irreducible, q@s must divide one of the three differences.
However, if q^s\ia since pa\ia h, h\ia % So q^s cannot divide any of the
three differences, a contradiction, so that k must be reducible.
3.2 Common Divisor Conditions
If k = 2l + 211 + 212 it is helpful if we can normalize with respect to one of
the exponents. If, in fact, one of the exponents (without loss of generality io)
is relatively prime to /i, a convenient normalization is possible, allowing us to
30
adapt Glynns criterion to an oary expansion, where a = 2l, since the a-ary
expansion is simply a permutation of the indices on the binary expansion. Our
main result is that for an irreducible exponent k = 2l + 2n + 2*2, this is always
possible. Our first few results, however, do not require that k be irreducible.
The results that follow through the remainder of this chapter and the next
make extensive use of Glynns criterion to show that under certain conditions
on k and h, (k) is not a monomial hyperoval in PG (2,2h). Since Glynns
criterion will be used in the same manner throughout, it will be helpful to explain
the procedure for using Glynns criterion.
In each case, we shall choose some value of d and use this value to compute
the product kd. Since k = 2l + 211 + 2*2, this is accomplished by distributing
the product as 210d + 2Hd + 212d. In order to show that d ^ kd, thereby showing
that @ (k) is not a monomial hyperoval in PG (2, 2h), two conditions must be
satisfied. Every bit of d must appear in one of the three summed terms, and
none of the three summed terms can coincide, resulting in carries. This is, of
course, the simplest situation. In many cases some bits of d will not appear in
any of the summed terms, and some of the summed terms will coincide. When
some bits of d do not appear, we will always show that carries from terms which
coincide will produce those bits of d. When other terms coincide, we will also
show that they do not produce any bits of d. In essence, we examine all of the
terms which coincide and show that after performing the necessary carries, and
under the stated conditions, d < kd.
This procedure is used in two primary ways. In the initial results, a gen-
eral value of d is chosen and examined to determine which values of k can be
31
excluded with the value of d. This procedure excludes wide swaths of values of
k and provides a narrow focus for the remaining work. The remaining work, in
later results, typically starts with a specific value or family of values of k and
determines a value of d which excludes that value of k.
Our work with Glynns criterion requires that everything be reduced
mod 2h 1. Throughout, we represent values that are congruent mod 2h 1
as equal, allowing us to eliminate the visual clutter of constantly writing
mod 2h 1 and allowing us to reserve the use of congruence for instances when
some other modulus is employed.
Lemma 3.5 Suppose m > 1, and m divides each ofi0) i\, i2; and h. Then if
k = 2l + 2n +212, Q) (k) is not a hyperoval in PG (2, 2.
Proof: Let d = 2J'TO. Then kd = 3 2jm = + 2jm+1).
Since jm + 1 ^ am for any a, d -< kd and 1 < d < 2h 2. Thus, (k) is not a
hyperoval in PG (2, 2h).
We further restrict the form of k with the following lemmas.
Lemma 3.6 7/(z0, h) > 1, (*i, h) > 1, (z2, h) > 1 and there exists no m dividing
each ofio, i\, ?2 and h, then for some ij, there exists an n dividing ij and h, but
not dividing ii for any l ^ j.
Proof: Since there is no m dividing each of the ij, the numbers (i0, h), (i\, h),
and (i2,h) cannot all be equal. Choose a = max {(i0, h), (*i, h), (i2, h)}. If a
occurs only once as (ij, h), then n = a divides ij and h, but not b for any l ^ j.
On the other hand, if a occurs twice, then let (ij,h) 7^ a, so that either
(ij, h) \ a or (ij, h) is a common divisor of each and h greater than 1. Since
32
this is excluded by the hypothesis, (ij, h) \ it for / ^ j, and n = (ij) satisfies the
conditions of the lemma.
We now assume, without loss of generality that a number m exists dividing
io and h but neither i\ nor i2.
Lemma 3.7 If ii and i2 are not both congruent to 1 mod m and k = 2l +
2n + 212, (k) is not a hyperoval in PG (2, 2ft).
Proof: Let d = Then kd = (2^m + 2-?m+ri + 2-?m+r'2), where
r\ and r2 are the reductions of i\ and i2 mod m to lie in [0, m 1], If ri and
r2 are not both m 1, 2jm+ri + 2^m+r2 ^ 2^+1^m, so that d ^ kd and @ (k) is
not a hyperoval in PG (2, 2h).
In the case that (i0, h), (i\,h), and (i2, h) are all distinct, we can strengthen
this condition considerably.
Lemma 3.8 Suppose (i0,h), (ii,h), and (i2,h) are distinct and each greater
than one. Then = i2 = 1 mod (i0,fi), i0 = i2 = 1 mod (ii,fi), and
i0 = ii = 1 mod (i2, fi) or ^ (fc) is not a hyperoval in PG (2,2h).
Proof: Lemma 3.7 guarantees that, without loss of generality, i\ = i2 = 1
mod (i0,fi). Without loss of generality, (ii,fi) does not divide (i2,h) and thus
does not divide i2. Further, since i\ = 1 mod (i0, h), (i\,h) does not divide
(i0, h) and thus does not divide i0. So, applying Lemma 3.7 again, i0 = i2 = 1
mod (ii, h). But then since i2 = 1 mod (i0, h), (i2, h) does not divide io, and
since i2 = 1 mod (ii, h), (i2, h) does not divide i\. So Lemma 3.7 now implies
that io = h = 1 mod (i2,fi).
33
We are left with two possibilities: either all three of (i0, h), (ii, h), and (i2, h)
are distinct, or (i\,h) = (i2,h) = 1 mod (io,h). We are able to rule out the
latter case with a series of results that mimic similar results of Cherowitzo and
Storme in their classification of the monomial hyperovals of the form (2* + 2l1)
[5],
Lemma 3.9 Let k = 2l + 2n + 212, where 2| (ii,h) and 2| (i2,h) and where
2 f (io, h). Then (k) is not a hyperoval in PG (2, 2h).
Proof: Notice that io is necessarily odd, since i\ is even and congruent to
-1 mod (i0,h). Now let d = eIo2^'1- Then kd = 2EL^2i~l + Ezio22'-
Combining these two sums yields kd = 2 E/=o 22t ~ Ei=o 22l+1 = Ei=o 22j_1i so
that d < kd, and Q) (k') is not a hyperoval in PG (2, 2/l).
When q > 2, a little more work is needed. Our treatment of this case
is adapted from similar arguments in Cherowitzo and Storme [5] dealing with
k = 2i + 2T
Lemma 3.10 Let k = 22 + 211 + 2*2 and let c\ (*i, h) and c\ (i2, h), where c > 1,
with io ^ 0 mod c. Now let i\ = i'2 ijf and h' Further, let k' =
2li + 2*2. If there exists a d! such that d! -< k'd' and such that if E d' = Ereh 2r
and 2l2 d' = E re/22r, h L\ I2 = \$ (the products never coincide), then S> (k) is
not a hyperoval in PG (2,2h).
Proof: Where d' = EreD 2"> let d = EreD 2Cr Now since ^ld! = Ereh 2">
then 2ild = Ereh ^ Similarly, if 2^d' = Ereh^> then 2*ad = Ereh2^-
Since Ji fl I2 0 (the products never coincide), all terms remain distinct, and
34
since each r G D is realized as T for some r G /j U /2, each cr with r G D is
realized as 2cr for some r G I\ U /2.
Finally, since d = J2t w^ere r ^ 0
mod c. Thus, d -< kd, and (k) is not a hyperoval in PG (2,2h).
We are immediately able to require that (zi, h) (z2, h) c.
Lemma 3.11 Suppose k = 2l + 2n + 2l2; and let c\ (zl5 h) and c| (z2, h), where
(zi, h) { (z2, h), and where c > 1. Further, let zo ^ 0 mod c. Then S> (k) is not
a hyperoval in PG (2, 2^)
Proof: Let i[ = d-, z2 = and h! Then necessarily (i[,h') = ^ \
and \ (i'2,h') = Then the proof of Lemma 3.7 in [5] provides the
h'
value d! = 2a(tl,h ) for which all exponents in k'd' (as defined in Lemma
3.10) are distinct. Thus, this value of d' satisfies the conditions of Lemma 3.10
and @ (k) is not a hyperoval in PG (2, 2h).
The next lemma is an extremely close adaptation of Lemma 3.9 in [5].
Lemma 3.12 Suppose k = 2l + 2l1 + 212, and let c = ((zi,/i), (i2,h)), where
c > 1, and io ^ 0 mod c. Let i\ = ci'ly *2 = ci2, and h = ch!. If St (k) is a
hyperoval in PG (2, 2h), either i2 = 2i[ mod h' or i'2 = mod h', with h'
odd.
Proof: If c {h,h) or c / (z2,h), Lemma 3.11 applies and Q) (k) is not
a hyperoval in PG (2,2hy Then we must have c = (ii,h) = {i2,h), so that
(z^, h') = 1. Thus, we may write a = 2d and a1 = 2*2, and we may further write
h' = mi + j where m > 1 and j < i 1. In this case, the proofs of Lemmas
35
3.4 and 3.5 in [5] provide values d! = J2t=o at + a'+U and (if 3 = i 1),
d! = o at + oc2l~2 + YT=2 atl~l satisfying the conditions of Lemma 3.10 that
eliminate every case except z'2 = 2i[ mod h' and i'2 = mod h! with b!
odd.
We now consider the remaining possibilities. Since (z2, h!) = 1 as well, we
must, in addition to having either z2 = 2i[ mod h! or i2 = mod h' with
h' odd, have either i[ = 2i2 mod h' or i[ = mod h' with h! odd. Notice
in particular, that if i2 = i[^j^- mod h!, 2z2 = i[ (h' + 1) = i[ mod h'. As
such, we have two possibilities: either i\ = 2z2 mod h' with h! odd, or i\ = 2z'2
and i2 = 2i[.
The next two lemmas, closely adapted from Lemmas 3.11 and 3.12 in [5]
rule out these cases.
Lemma 3.13 Suppose k = 2* + 2n + 22n, where (ii, h) = (2*i, h) = c > and
(z0, c) = 1, with h! = ^ odd. Then £> (k) is not a hyperoval in PG (2, 2h).
Proof:
Let d = Yt=o 22tl1 Then the following is obtained for kd:
h'-l
2
h' + l
2
h'-l
2
l+*0
kd = E 22(t+l)ii _j_ E 22til + 22til
t=0 Z=1 t=0
h'-3 h'-l
2 2 _____________________
= 2h + ^2 22(m)il + 1 + 22til + 2h + 22til
t=l t=i
ft'-i
= 1 + 2il+1 + ^ 2th + ]P 22til+io
h'-l
2
t=0
+20
Z=2
t=0
Since Zi + 1 = 1 mod c, while each ti\ = 0 mod c, and since 2ti\ + z0 ^ 0
mod c the only terms which may possibly not be distinct are 2ll+1 and 22Ul+t.
But if c > 2, even a possible 2 2n+l =2 mod c, so that the resulting terms are
36
distinct and d < kd. On the other hand, if c = 2, Lemma 3.9 yields the desired
result, so that (k) is not a hyperoval in PG (2,2h).
Lemma 3.14 Let k = 2l+2n+2t2, with (i1; h) = (i2, h) = c > 1 and (i0, c) = 1.
Let h' = and i'2 = and suppose that i2 = 2i\ mod h! and that
= 2i2 mod h'. Then @ (k) is not a hyperoval in PG (2, 2h).
Proof: Since i2 = 2i[ mod h' and i[ = 2i2 mod h1, then z2 = 2ix mod h and
i\ = 2i2 mod h, so that {*i,*2} = {|, ^}. But then h' = 3, which is odd, and
Lemma 3.13 applies, so that (k) is not a hyperoval in PG (2, 2ft).
Theorem 3.15 Let k = 2l + 2n + 212, with (io,h) > 1 and(i\,h) = (z2,/i) > 1,
where i0 ^ 0 mod (zl7 h). Then (k) is not a hyperoval in PG (2, 2h).
Proof: The conditions stated satisfy the hypotheses of Lemmas 3.9, 3.10, 3.12,
3.13, and 3.14, which leave no possibilities for k to describe a hyperoval.
We note that, although the previous restrictions allowed us to assume that
(*i,h) = (i2,h), this was not assumed for the initial lemmas and was, in fact,
ruled out in Lemma 3.11, so that this series of lemmas in fact ruled out the
existence of any m dividing i\, *2, and h, but not i0.
With the added condition that k be irreducible, we can, in fact enforce a
further restriction.
Lemma 3.16 Suppose k = 2l + 2n +2*2 is irreducible and (i0, h) > 1, (i\,h) >
1, and (*2, h) > 1. Then |{(*0, h), (ii,h), (*2, h)}\ = 2 or @ (k) is not a hyperoval
in PG (2,2h).
37
Proof: Lemma 3.5 guarantees that |{(z0, h), {i\,h), (z2, h)}| ^ 1, so suppose
that |{(zo, h), (ii, h), (z2, h)}| = 3. Then Lemma 3.8 guarantees that z0 = i\ =
1 mod (z2,/z), i0 = i2 = 1 mod (z^/z), and zi = z2 = 1 mod (z0,h).
These conditions ensure that any prime dividing (z0, h) does not divide i\ or z2,
any prime dividing (z!, h) does not divide z0 or z2, and any prime dividing (z2, h)
does not divide z0 or i\. Thus, there are at least three distinct primes dividing h,
so that k is reducible, contradicting Theorem 3.4. So |{(z0, h), (zi, h), (z2, h)}\ =
2, as desired.
With these preliminary steps in place, we are now prepared to prove the
main theorem.
Theorem 3.17 Suppose k 2l + 2n + 212 is irreducible with (io,h) > 1,
(zi, h) > 1, and (z2, h) > 1. Then (k) is not a hyperoval in PG (2, 2h).
Proof: Lemma 3.16 ensures that under these conditions, exactly one of the
greatest common divisors of z0, ii, and z2 with h is distinct. Without loss of
generality, assume (z0, h) ^ (zi, h) (z2, h). Further, by Theorem 3.4, h = paqP.
Lemma 3.7 guarantees that i\ = z2 = 1 mod (io,h), so that (io,h) has no
common factors with i\ or z2. Thus, without loss of generality, (io,h) = pa and
(ii,h) = qb, where 1 < a < a and 1 < b < (5. Now let {z/,zm,zn} = {zo,ii,z2}
in some order. Since k is irreducible, potqli~l divides one of the differences,
without loss of generality in im. Further, pa~lq@ divides one of the differences,
which cannot be in im lest h divide in im. Without loss of generality, then
pa-iq0\^m ih So pQ_1^_1 divides both in im and im ii, and therefore also
divides Z/ in. Thus, pa~lqP~1 divides each of zq z2, i\ z0, and z2 i\.
38
Now define o' = min {a 1, o} and b' = min {(3 1, b}. So pa' divides each
of i0, *o *2, and i\ i0, implying that pa' also divides i\, i2, {ii, h), and (i2, h).
Since (fy h) = q13, a' = 0, so that a a = 1. Similarly, qb> divides each of i\, and
i\ i0, implying that qv also divides i0 and thus (iQ, h). Since (i0, h) = pa, b' = 0,
so that b = j3 1. Thus, h = pq, q divides both ii and i2, and i\ = i2 = 1
mod p. Since the p multiples of q have distinct congruences modulo p, ii = i2,
a contradiction. So (k) is not a hyperoval in PG (2, 2h). m
Thus, any monomial hyperoval in PG (2,2h) with irreducible exponent k =
2l + 211 + 2t2 must satisfy (fy h) = 1 for some j.
Note, however, that the sequence |(21>)I| is merely a permutation of the
sequence {2*}^. If then we let a = 2fy we can express each integer in Z2h_!
in the form n = q ai(xl, where Oj {0,1} and the summation is reduced
modulo 2h 1 to lie in [0, 2h l] with the convention that 0 is reduced to 0
and any nonzero multiple of 2h 1 to 2h 1. Further, the a* are simply a
permuatation of the bits in the binary expression of n, and this permuation
is invariant under the choice of n. Defining the same partial ordering for this
expansion as in Definition 2.25, it is immediately evident that a -< b in the
binary expansion if and only if a ^ b in the expansion with respect to a. It
follows then, that Theorem 2.26 applies equally well to this expansion. We may
thereby restrict our attention to those exponents of the form a + a1 + cF, and,
in so doing, reduce the number of parameters to be considered by one.
Although Glynns criterion applies to this expansion, there is an added dif-
ficulty when terms coincide. In the binary expansion, the term 2 2a is easily
simplified to 2a+1. In the a-ary expansion, the simplification depends completely
39
on the choice of a. In a sense, then, we have not eliminated a parameter, but
have simply changed its nature. Our work with the a-ary expansion will often
require the parameter r, where ar = 2, to simplify any terms which coincide as
2 a = aa+r.
3.3 Monomials With Exponent of the Form a + a1 + ai
It is convenient to consider the relationships between h, j, and i. Without
loss of generality, we will assume j > i. We assume throughout that a = 2lh
where {ij,h) = 1. Using the division algorithm twice, we may express h as
h = mj + ni + l, where ni + l < j and l < i. We are immediately able to
exclude all but a few monomials from the set of monomial hyperovals. Although
reducibility of the exponent guarantees that the exponent have this form to be
a hyperoval exponent, we do not here assume that the chosen values of k be
reducible, and the results that follow are true for any value of k that can be
expressed as a + a1 + a].
Lemma 3.18 Let k = ot+a%+a?, where a = 2l for (i0, h) = 1 and suppose that
for some s > 1, s|i, s\j, and s|h. Then @ (k) is not a hyperoval in PG (2, 2h).
Proof: Without loss of generality, suppose that a1 = 2n and ad = 212. Then
s|ii and s|2, but s \ *o, and the results of Lemma 3.11 and Theorem 3.15
preclude Q) (k) from being a hyperoval in PG (2,2h).
Our next result shows that, in fact, once k can be expressed as a + a1 + af
one of i and j must be relatively prime to h, and therefore k can also be expressed
as (3 + Px + /3y, where fdx = a and (3 = a1 if (z, h) = 1 or (3 a.i if (j, h) = 1.
40
Theorem 3.19 Let k = a + a1 + a3, where a = 2J and aT = 2. Suppose that
(i, h) = a and (j, h) = b for a > 1 and b > 1. Then & (k) is not a hyperoval in
PG( 2,2h).
Proof: By Lemma 3.18, we may assume that (a, b) = 1, so that ab\h. If
i^l mod b, let d = ^2b=o a<:bi so that kd = J2c=o (c6+1 + a.cb+1 + acb+3).
Since b\j, Yhc=o Oicb+3 = Y2c= o aCb = d. No terms coincide as i ^ 0 mod b
(since b \ z)and z ^ 1 mod b. On the other hand, if i = 1 mod b and r ^
1 mod 6, kd \aCb + otcb+r+1) and no terms coincide, so that d ^ kd
__^ h_2
in either case. Similarly, if j ^ 1 mod a, let d = X^c=o aCa> so that kd =
X^c=o + ca+1 + a004"-7)- Again, no terms coincide as j ^ 0 mod a (since
a f j) and j ^ 1 mod 6. Once again, if j = 1 mod a and r ^ 1 mod a,
fe _2
kd = Yhc=o (QCa + aca+r+1) and no terms coincide, so that d < kd in either case.
Thus, the only possibility is that i = 1 mod b, j = 1 mod a, and r = 1
mod ab. Notice then that i+j = 1 mod ab, since i+j = 1 mod a and i+j = 1
mod b. Notice further that j ^ 2i, as a \ j, and that i > 2 (and consequently
j > 3), as b > 1.
If now j 2i 1, we choose d = (CQb + acab+l_1), so that kd =
Ecto1 (ac6 + ca6+1 + 2acaf,+i + acab+2i-1+aca6+J). With r = -1 mod ab,
Z£01 2acab+t = ]P=o1 acab+t_1 and no other terms coincide, hence d + kd.
On the other hand, if j 2i, let d = q1 (ctcab + acab+3~1), so that kd =
o1 (acab + acab+1 + acab+1 + acab+3~l + 2o;cab+-?). Again, r = 1 mod ab, so
that J^c=o 2aC(lb+] = f2c=o oaib+3'"1 and no other terms coincide, so again
d + kd.
Thus, having covered all possible cases, if (k) is a hyperoval in PG (2, 271),
41
either (i,h) 1 or (j, h) = 1.
We now use five related values of d to place severe restrictions on the param-
eters of a potential hyperoval. After each, we consider the restrictions on the
parameters resulting from combining the restrictions from all of the values of d
used up to that point. This results in a new set of restrictions on the parameters
that is carried forward and combines the information in all of the values of d
used to that point.
Lemma 3.20 Let k = a + al -\-ai, and let h = mj + ni + l, where ni + l < j and
l < i. If *3) (k) is a hyperoval in PG (2,2h}, then at least one of the following
conditions holds:
1. i-l = l
2. j i 3. j 1 = ni + l.
Proof: Suppose that none of the above conditions hold, and let d = Y^!a=o a<1 +
Eteo'a^' + KTo1 acj+m+i Thjg ieacis to the following value for kd.
I-1 711
(oa+1 + aa+i + aa+j
a=0 6=0
m 1
_j_ ^acj+ni+l+1 acj+ni+l+i _|_ Qcj+ni+l+j
c=0
Under the conditions set out, d is defined. Further, all terms of d are present and
no terms coincide, so that d < kd and & (k) is not a hyperoval in PG (2, 2h).
) + >>
+ a(
bi+l+i
+ '
bi+l+j'j
42
Lemma 3.21 Let k = a + al + aj, and let h mj + ni + l, where ni + l < j and
l < i. If (k) is a hyperoval in PG (2,2h), then at least one of the following
conditions holds:
1. n < 1
2. i = 2
3. rn = 1 and j = ni + l + 1
4- m = 1 and j = ni + l + i 1
5. m = 1 and j = ni + l + i
6. 1 = 0.
Proof: Suppose that none of the above conditions hold, and let d = Y?a=o a<1 +
oP+i-i + ai+m+l~l + ac^+TU+b We then obtain the following
expression for kd.
i-i
n1
+ aj+ni+l~i+1 -p (yi+ni+l a2j+ni+l-i
m 1
+
Again, under the conditions imposed, d is defined, all terms of d are present,
and no duplicate terms appear. Thus, d < kd and (k) is not a hyperoval in
Corollary 3.22 Let k = a + a1 + aJ, and let h = mj + ni + 1, where ni + l < j
and l < i. If (k) is a hyperoval in PG (2,2hthen at least one of the following
conditions holds:
1. n = 1 and i 1 = l
2. n = 0 and i 1 = l
3. i 2 and l = 1
4 n = 1 and j i 5. n = 0 and j i 6. m = 1 and j = ni + Z + i 1
7. m = 1 and j = ni + l + i
8. j = i + 1+ 1 and n = 1
P. m = 1 and j = ni + l + 1
.70. j = m + 1.
Proof: We simply check that anything satisfying each set of conditions from
Lemma 3.21 must also satisfy one of the conditions above. Notice that if S (k)
is a hyperoval some set of conditions from Lemma3.20 must also be satisfied.
If n > 1 and i 1 = l, either condition 1 or condition 2 holds. If n > 1 and
j i 1 and
j l ni + l, then n = 1 and condition 10 holds.
44
If % 2 and i l = l, condition 3 holds. If i = 2 and j i < ni + l < j i + l,
then either l = 1 and condition 3 again holds, or l = 0 and thus j = ni+i so that
2| (i, h) and 2| (j, h), a contradiction of Theorem 3.19. If i = 2 and j 1 = ni + l,
either l = 1 and condition 3 holds, or l = 0 and condition 10 holds.
Conditions 3, 4, and 5 of Lemma 3.21 are identical to conditions 9, 6, and
7 respectively.
If l = 0, % 1 ^ l as i > 1. If / = 0 and j i and i divides each of i, j, and h, a contradiction of Theorem 3.19. Finally, if
l = 0 and j 1 = ni + l, condition 10 holds.
Lemma 3.23 Let k = a + a1 + cF, and let h = mj + ni + l, where ni + l < j and
l < i. If (k) is a hyperoval in PG (2, 2h), then at least one of the following
conditions holds:
1. n 0
2. i = 2
3. j = ni + l + 1
4. j i 5. j 2 i
6. 1 = 0.
Proof: Suppose that none of the above conditions hold and let d = a 0,1+i-i _|_ abl+l + Y1T=() Q'CJ+m+/. We then obtain the following expression
45
for kd.
i-1
kd = J2 (Q+1 + a+i + a+j) + i+l + 2i+'_1 +
a=0
n1
+ ^ (<*W+,+1 + abi+l+l + aj+bi+l)
6=1
m1
^cj+m+z+l acj+m+i+; _|_ Qcj+j+ni+l^
c=0
Again, under the conditions imposed, d is defined, all terms of d appear, and
no duplicate terms appear. Thus, d A kd and (k) is not a hyperoval in
PG( 2,2h). u
Corollary 3.24 Let k = a + a* + ckj, and let h mj + ni + 1, where ni + l < j
and l < i. If & (k) is a hyperoval in PG (2, 2h), then at least one of the following
conditions holds:
1. n = 1, i \ l, and j = 2i
2. n = 0 and i 1 = l
3. i = 2 and l = 1
4- n = 1 and j i 5. m = 1 and j = ni + l + i 1
6. m = n = l = 1 and j = 2i
7. m = 1, j = ni + i 1 and l = 0
5. m 1 and j = ni + l + i
46
9. n = 1 and j = i + l + 1
10. m = 1 and j = ni + l + 1
j = ni + 1 and l = 0
12. n = 0 and j < i + l
Proof: Conditions 2, 3, 4, 5, 6, 7, 8, and 9 of Corollary 3.22 are identical to
conditions 2, 3, 4, 8, 5, 12, 9, and 10 respectively.
Condition 1 of Corollary 3.22 contradicts conditions 1 and 6 of Lemma 3.23,
satisfies condition 3 whenever it satisfies condition 3 of Lemma 3.23, satisfies
condition 1 whenever it satisfies either of conditions 3 or 5 of Lemma 3.23, and
satisfies condition 4 whenever it satisfies condition 4 of Lemma 3.23.
Condition 10 of Corollary 3.22 contradicts conditions 1, 4, and 5 of Lemma
3.23, satisfies condition 3 or 11 whenever it satisfies condition 2 of Lemma 3.23,
and satisfies condition 11 whenever it satisfies either of conditions 3 of 6 of
Lemma 3.23.
Lemma 3.25 Let k a + cd + ed, and let h = mj + ni + l, where ni + l < j and
l < i. If Q) (k) is a hyperoval in PG (2,2^), then at least one of the following
conditions holds:
1. l = i- 1
2. j = 2i
3. j = i + 1
4- m = 1 and j = ni + l + 1
47
5. m = 1 and j = ni + i + l
6. n = 0
7. m = 1, n = 1 and 2i < j < 2i + l
Proof: Suppose that none of the above conditions hold and let d = X^L=o a +
Eto1 ^+ + QI+"-'+' + E27 acj+m+i' yye obtain the following expression for
kd:
i-1
711
*<*=£(
oa+1 + aa+i
+ aa+j) + Y (aW+m + a+ aj+bi+l)
a=0
6=0
_|_ aj+ni-i+l+1 _|_ aj+ni+l _|_ ^2j+ni-i+l
m 1
+ (acj+ni+l+1 +
a
cj+ni+i+l _|_ acj+j+ni+l
')
c= 1
Again, under the conditions, d is defined, d appears in the expansion of kd, and
no duplicate terms appear in the expansion of kd so that d ^ kd and @ (k) is
not a hyperoval in PG (2,2h).
Corollary 3.26 Let k = alpha + cd + a?, and let h = mj + ni + l, where
ni + l < j and l < i. If £> (k) is a hyperoval in PG (2, 2h), then at least one of
the following conditions holds:
1. n = 0 and i 1 = l
2. i 2 and l = 1
3. n = 1, i 1 = l, and 2i < j < 3i 1
4- m = 1, j = ni + 2i 2, and i 1 = l
48
5. n = 1 and j = 2i
6. n 1, l = 0, and j = i + 1
7. m = 1 and j = ni + l + 1
5. m = 1 and j = ni + i + l
9. m = 1, n = 0; and j = z + l 1
m = n 1 and 2i < j <2i + l
11. j < i + l and n = 0.
Proof: Conditions 1, 2, 6, 8, and 10 of Corollary 3.24 and conditions 4, 5, and
7 of Lemma 3.25 imply conditions 3, 1, 10, 8, 7, 7, 8, and 10 respectively.
Condition 1 of Lemma 3.25 satisfies condition 3 whenever it satisfies either
of conditions 4 or 9 of Corollary 3.24, satisfies condition 4 whenever it satisfies
condition 5 of Corollary 3.24, and contradicts conditions 7 and 11 of Corollary
3.24.
Condition 2 of Lemma 3.25 satisfies condition 5 whenever it satisfies either
of conditions 4 and 9 of Corollary 3.24 and contradicts conditions 5, 7, and 11
of Corollary 3.24.
Condition 3 of Lemma 3.25 satisfies condition 6 whenever it satisfies any of
conditions 7, 9, and 10 oc Corollary 3.24, and contradicts conditions 4 and 5 of
Corollary 3.24.
Condition 6 of Lemma 3.25 satisfies condition 9 whenever it satisfies condi-
tion 5 of Corollary 5 and contradicts conditions 4, 7, 9, and 11 of Corollary 5.
49
Lemma 3.27 Let k = a + al + aj, and let h = mj + ni + l, where ni + l < j and
l < i. If 31 (k) is a hyperoval in PG (2, 2h), then at least one of the following
conditions holds:
l.i = 2
2. j = i + 1
3. ni + i < j < ni + i + l 1
4- m = 1
5. 1 = 0.
Proof: Suppose, as usual, that none of the above conditions hold, and let
d = aa+Sb=i of1*11+aj+ni+l~1^ aci+ni+l. We obtain the following
expression for kd:
i-1
kd
=E(
att+1 + aa+i + aa+j
')+£(
abi+l + a6i+i+i-l
+ a'
bi+j+l-
)
a=0
6=1
_|_ cyj+ni+l _|_ aj+ni+i+l-1 a2j+ni+l-l
m1
acj+ni+l+1 acj+ni+i+l _|_ cj+j+ni+l^
C=1
As before, d is defined, the terms of d appear, and no duplicate terms appear,
so that d -< kd and 3! (k) is not a hyperoval in PG (2, 2h).
Our final combination of conditions results in eleven cases. Because this
provides the structure for much of the work which follows, we state this as a
separate theorem rather than a corollary.
50
Theorem 3.28 Let k = a + a1 + aj, where a = 2io and (io,h) = 1. Further
let h = mj + ni + l, where ni + l < j and l < i. If St (k) is a hyperoval in
PG (2, 2h), at least one of the following sets of conditions holds:
1. n = 1, m = 1, and 2i + l 2. i = 2 and l = 1
3. m = 1, j = i + l 1, and n 0
4- i 1 = l, m = 1, and j ni + 2i 2
5. j = ni + l + 1 and m = 1.
6. n = 1, j = 2i, and l ^ 0
7. m = 1 and j = ni + i + l
8. n = 1, j = i + 1, and l = 0
9. i 1 = l, n = 1, and 2i < j < 3i 2
10. n 0 and j < i + l
11. i 1 = l, n = 0, and m = 1
Proof: Conditions 2, 4, 5, 6, 7, 8, 9, 10, and 11 of Corollary 3.26 fall into
cases 2, 4, 1 (since if l 0, i divides each of i, j, and h, contradicting Theorem
3.19), 8, 5, 7, 3, 1, and 10, leaving only conditions 1 and 3 of Corollary 3.26 to
be considered.
Condition 1 of Corollary 3.26 falls into case 2 whenever it satisfies condition
1 of Lemma 3.27, falls into case 10 whenever it satisfies conditions 2 or 3 of
51
Lemma 3.27, falls into case 11 whenever it satisfies condition 4 of Lemma 3.27,
and contradicts condition 5 of Lemma 3.27.
Finally, condition 3 falls into case 2 whenever it satisfies condition 1 of
Lemma 3.27, contradicts conditions 2 and 5 of Lemma 3.27, falls into case 9
whenever it satisfies condition 3 of Lemma 3.27, and falls into case 1 whenever
it satisfies condition 4 of Lemma 3.27.
We examine each of these cases in turn as the major portion of this classi-
fication, obtaining the following theorem. As the proof consists of a very long
case argument that follows the pattern of the preceding lemmas and is not par-
ticularly instructive, it is left to the next chapter.
Theorem 3.29 Let k = a + a1 + a?, where a = 2l and (z0, h) = 0. Let
h = mj + ni + l, where ni + l < j and l < i. If & (k) is a hyperoval in PG (2,2h)
it is a translation hyperoval, Segre hyperoval, or Glynn hyperoval.
3.4 Reducible Monomial Hyperovals
Having completed the classification of monomial hyperovals with irreducible
exponents and those of the form k = a + a1 + af where a = 2l with (i0, h) =
1, we now turn our attention to the remaining case: those having reducible
exponents which cannot be expressed asa + a' + a^ under the restrictions on a.
We thus consider k = 2l + 2n + 212. The results of Lemmas 3.6 and 3.7
force (z0, h), (A, h), and (z2, h) to be distinct, and thus i\=ii = 1 mod (z0, h),
iQ = z2 = 1 mod (ii, h) and z0 = Zi = 1 mod (z2, h). With the appropriate
reduction and a reference to the classification result obtained for those hyper-
ovals of the form St {a + a1 + cF) with a = 210 and (zq, h) = 1, we are able to
52
rule out this final case as well.
Theorem 3.30 If (i0,h), (i\,h), and (i2,h) are distinct and each greater than
one, £> (k) is not a hyperoval m PG (2,2h).
Proof: By Lemma 3.8, i\ = i2 = 1 mod (z0,fi), i0 = i2 = 1 mod (i\,h),
and i0 = i\ = 1 mod (i2,h). It follows that (i0,h), and (i2,h) are
pairwise relatively prime. So (i0,h) (i\,h) divides h.
Now, since (io,h) (i\,h) is the least common multiple of (i0, h) and
*2 = 1 mod (i0, h) (*!, h). Further, i0 is congruent mod (*0, h) (ij, h) to
a multiple of (i0,h) and is congruent mod (i0,h)(ii,h) to a multiple of
(i\,h). Since, again, the product (z0,/i)(ii,fi) is the least common multi-
ple of the two factors, neither of which is one, 0, *i, and i2 have distinct
residues mod (i0, h) (ii, h). In particular, a ^-reduction of k exists with
k' = 2a<'l0,h) -f 2b(ll'h'> + 2(o.h)(*iA)-1, where 1 < a < (ii, h) and 1 < h < (io, h).
In such a reduction, neither a(i0,h) nor b(i\,h) can be relatively prime to
(io,hfc>r However> since 1 necessarily is relatively prime to {ioA^h),
this reduction describes a monomial hyperoval if and only if it is one of the
monomial hyperovals determined in Theorem 3.29. Since each such monomial
hyperoval of the form 2l + 2-y + 2h~1 has at least one of i or j relatively prime
to h, the reduction is not a hyperoval, and thus, by Theorem 3.3, 3 (k) is not a
hyperoval in PG (2, 2h).
Theorem 3.31 Let k = 2l + 2n + 212, with (i0, h), (ix, h), and (i2, h) all greater
than one. Then @ (k) is not a hyperoval in PG (2, 2h).
Proof: This is an immediate consequence of Lemma 3.5, Lemma 3.6, Lemma
53
3.8, Theorem 3.30, and Theorem 3.15.
We finally state the major theorem resulting from all of this work.
Theorem 3.32 If Qt (2l + 2n + 212) is a hyperoval in PG (2, 2h), it is a trans-
lation hyperoval, Segre hyperoval, or Glynn hyperoval.
Proof: This follows immediately from Theorem 3.29 which classifies those with
(ix, h) = 1 for some x {0,1,2} as the translation hyperovals, Segre hyperovals,
and Glynn hyperovals, and from Theorem 3.31 which rules out the possibility
of such a hyperoval with (ix, h) > 1 for all r G {0,1,2}.
54
4. Proof of Theorem 3.29
In this chapter, we exhibit a complete proof of Theorem 3.29.
Theorem 3.29 Let k = a + a1 + a?, where a 2l and (io,h) = 0. Let
h = mj + ni + l, where ni + l < j and l < i. If (k) is a hyperoval in PG (2,2^)
it is a translation hyperoval, Segre hyperoval, or Glynn hyperoval.
Theorem 3.28 places restrictions on the relationships among the parameters,
dividing these parameters into eleven distinct cases. We examine each of these
cases in turn, establishing for each a lemma requiring that any hyperoval satis-
fying the assumptions of that case be a translation hyperoval, Segre hyperoval,
or Glynn hyperoval.
4.1 Case 1
We turn now to case 1, in which n 1, m = 1, and 2i + 1 < j < 2i + l. For
ease of notation, let j 2i + x, where 1 < x < l. Further, let r be the unique
power such that ar = 2, with 1 < r < h 1. We use the family {da} of values
for d, where
la i+l
da = 'Y2ab + ab
6=0 b=i+la
and 0 < a < l.
In this case we obtain the following product kda:
la+l i+l+1 2 i+l 2 i+x+la Si+x+l1
kda ^ ab + ^ ab + ^ ac + ^ ac + ^ ab.
6=0 b=i c=2i+la c=2i+x b=3i+x+la
Notice that within this product, the only terms which may possibly coincide
are those indexed by c, unless l = i 1 and a = 0 or a = l. Thus, the
55
powers of a that appear multiple times are precisely the powers ac, where
max {2i + x, 2i + l a} < c < min {2i + l + x a, 2i + /}. Notice also that
all terms of da appear in kda. The use of {da} severely restricts values which
may be taken by r.
Proposition 4.1 Let k = a + a1 + a2l+x, where a = 2l with aT = 2 and
1 < r < h l. Further, let h = 3i + l + x, where l < i and 1 < x < l. If
min {x,l x} + l 3i + l 1, orifl^i 1 and r E {i + l + 2, 2i + l 1,2i + l + x + 2, i + x 1},
then (k) is not a hyperoval in PG (2, 2h).
Proof: Given a value of a, we determine the values for r for which otc+r does
not appear for any max {2i + x,2i + l a} < c < min {2i + l + x a,2i + 1}
and thus da X kda. We consider each of the four possible combinations for
the maximum and minimum terms separately. If 2i + x > 2i + l a and
2i +1 + x a > 2i +1, the acceptable values of r are i + 2l a + 2 < r < 2i + x 1,
and a satisfies l x < a < x. If2i + x>2i + l a and 2i + l > 2i + l + x a, the
acceptable values for r are i+21a+2 and lx+1 < r < i1, and a satisfies max {x,l x} < a < l. If 2i+la > 2i+x
and 2i + l + x a > 2i + l, the acceptable values for r are i + l + x + 2 > 2i + x l,
2i + l + x + a + 2 < r < 3i + 2x 1, and x + 1 < r < i + x a 1, and a satisfies
a < min {x, l x}. Finally, if 2i + l a > 2i + x and 2i + l > 2i + l + x a, the
acceptable values of r are i+l+x+2 and a + l Considering at once all possible values of a in a given situation, the values
i + l + 3 56
x-f3 min {x,l x} + 1 < r < i + x 2 (min {x, l x}-\-l each have some choice of a with 1 < a < Z 1 (0 < a < Z if l < i 1) such that
da zf: kda, yielding the desired result.
Proposition 4.2 Let k = a + a1 + a2l+x, where a = 210, ar = 2, and h =
3i + l + x, where l < i and 1 < x < l. If r < min {x, l x} or r > 3z + l, (k)
is not a hyperoval in PG (2, 2h) or it is a translation hyperoval, Segre hyperoval,
or Glynn hyperoval.
Proof: Consider the value kdj. Notice that kdi has only the duplicate terms
2a2i+i-i an(j 2a2l+l and no terms ac with 2i + x + l a + 1 appear. Since this amounts to i consecutive terms and i > min (x, l x}, there
exist q, q such that a2l+l~1+qT and a2l+l+q'r do not otherwise appear in kd\,
while for s < q, s' < q', a2l+l~l+sr and a2l+t+s'r never appear in d\, so that
d\ < kd\.
In similar fashion, notice that r > Si + Z is equivalent to h r < x. No
terms ac with i + l + 2 i + x l 2>x consecutive terms, except when Z = i 1 and x > i 2. Thus,
unless Z = i 1 and x > i 2, using the same reasoning as above, cZi < kd\.
Now if Z = i 1 and x = i 2, the only duplicate terms are 2o?l~2 and 2a3l_1,
while if Z = x = z 1, the only duplicate term is 2a3,_1, and neither aM~2+r nor
q3ji+r appears otherwise for r > 3% + 1 + 1. Thus, we have only to consider
r = Si + Z in these two situations.
57
If Z = i l=x+l = h r 1, assume i > 3 (if i = 2, Lemma 4.21 yields
the result), and let d = 1 + cd_1 + o?l~x + a3*-2, so that kd 1 + a + ed-1 +
2cd 4- 2l_1 + a21 + a3l~2 + 2a3*-1. As 2cd = a2 and 2a31'1 = o:2l+1, neither of
which terms appear otherwise, d < kd.
If Z = z = d r = i 1, let d = 1 + Y^c=i\i aC> so that = 1 + a + cd +
Sc=^+2 a + 2o;3l_1 + Y^c=Ai a- Since 2a3*-1 = a2t and 2a21 = cd+1, d < kd, so
that the result follows.
The preceding propositions now restrict the values taken by r in this case
toi + r i + x 1 < r < i +1 + 2 and 2i + l 1 examine these remaining cases, beginning with 2i +1 1 < r < 2i +Z + x + 2. In
this case we are immediately able to show that r is restricted to the boundaries
of this range.
Proposition 4.3 Let k = a + cd + a2l+x, where a = 2l with cd = 2 and
1 < r < h l. Further, let h = 3z + l + x, where l < i and 1 < x < l. If
2i + l Proof: Choose d = YX=o6 + a3l+l+x~r~1. Since r < 2i + x + l, all of these
terms are distinct. Then kd = Y2l=.v ab + a>> + a3l+l+x~r + a4l+l+x~r~l +
a5i+i+2x-r-i' gjnce r > 2i + l > i + l + x + 1, Qpt+t+x-r-1 appears in kd,
and a3l+l+x~r appears exactly twice unless r = 2i + l and l x = i 1.
But 2a3l+l+x~r = 1, and if no further powers of a coincide, d < kd. Since
r > 2i + l > 2i + x, 5i + l + 2x r 1 < 3i + l + x, and a5l+l+2x~1-1 does not
coincide with any other terms. Finally, since r > 2i + l, 4i + l + x r 1 < 2i + x,
58
so that a4l+l+x~r~1 does not coincide with any other terms. Thus, d ^ kd, and
(k) is not a hyperoval.
In the case r = 2i + l and l x = i 1, let d = Ec=o a + so that
kd = I^cIoac+2cd+X^=i+i ac+2a3*'~1 + E^='3i ftC- Now 2cd+2a3_1 = cd+a:4i,
and d < kd again.
This leaves r = 2i + l 1 (when l = i 1), r = 2? + / + x + 1, and
r = 2i + l + x + 2 (when l = i 1) among the higher set of remaining values for
r.
Proposition 4.4 Let k = a + cd + a2l+x, where a = 2l with 0i2l+l+x+l = 2.
Further, let h = 3i + l + x, where l < i and 1 < x < l. Then ^ (k) is not a
hyperoval in PG (2,2h).
Proof: If l < i 2, let d = E^o0^ + cd+* (previously denoted by do).
Then kd = ES + ElHi b + EjS+iab + ai+l+1 + <*2l+l + a3i+Z+X- Note
that a3l+l+x = 1 and that the only power of a that appears twice is a2l+l.
But 2a2l+l = a1+l+1. Since this already appears, we realize that 2o1+/+1 = al+2,
which cannot appear twice in the product kd. Thus, since all powers of a present
in d are present in kd, d < kd and L2) (k) is not a hyperoval in PG (2, 2h).
On the other hand, if l > i 2, let d = Eb^oab + ct2l_3- Then kd =
ab + 2a2t_3 + Eb^h+x ab + ol3i~3 + ait+x~3. It may happen that a 2o3t~3
term is present. But this is equal to cc2l~2 and we have either a 2a2*-2 or 3a2l~2
term present. In either case, this results in a second cd_1 term, which simplifies
to a 1. Thus, d < kd.
Proposition 4.5 Let k = a + cd + a2l+x, where a = 2l with a2l+l+x+2 = 2.
59
Further, let h = 3i + l + x, where l = i 1 and 1 < x < l. Then Q) (k) is not a
hyperoval in PG (2,2h).
Proof: Let d = 1 + Yl[=2+ al+l Then kd = a + a1 + a2l+x + ^6=3a>) +
ab + Y^b=2i+l+2ab + a21 + a2t+l + a3l+l+x. It may be that a 2a2l+l term
is present. But this is simply a2l+1, which does not otherwise appear in the
product, unless x = 1, in which case 2a2l+1 = cd+1, which does not appear.
In addition, a 2al term is present, which simplifies to a2, which is not
otherwise present. Thus, all powers of a present in d are also present in kd, so
that ^ (k) is not a hyperoval in PG (2, 2/l).
Proposition 4.6 Let k a + a1 + a2l+x, where a = 2l with a2lJrl~l = 2.
Further, let h = 3i + l + x, where l i 1 and 1 < x < l. Then @ (k) is not a
hyperoval in PG (2, 2h) unless it is a Segre hyperoval.
Proof: Consider di = 1 + ab as defied earlier. Then kdi = ]T^=0 ab +
J2b=i ab + Oi2l+x + a + q6- The only powers that appear more
than once are 2a21 and 2a2l+x. Now 2a21 = a4l+i_1 = a3t+l+l = al~x, and
so long as l x > 2, no further terms coincide. On the other hand, 2a2l+x =
a4i+i+x-1 which does not already appear. Since all terms of d are present
in kd, d When l x = 1, k = 2 + 2h~3 + 2h~3. For > 3, let d = YLc=oa<:> so that
kd = &c + &h~5 + ah~4 + 2ah~3 + ah~2 + a^1. As 2ah~3 + ah~2 + a^1 = 1,
d A kd. If % = 2, k = 2 + 22 + 24 and h = 7, so that ^ = 6 and (k) is a
Segre hyperoval.
60
Finally, when l = x, notice that if i is even, 2| (3i 2, 5i 2). Hence i is
odd. We consider i = 1 mod 4 and i = 3 mod 4.
If i = 4s + 1, it is clear that h = 20s + 3. Further, it is easy to verify that
a = 215s+1, a1 = 210s+1, and a3l_1 = 215s+2. In this case, set d = l + 25s + 210s+1,
so that kd = 25'-1 + 25s + 210s+1 + 2 215s+1 + 215s+2 + 220s+1 + 2 220s+2, which
simplifies to 1 + 25s_1 + 25s + 210s+1 + 215s+3 + 220s+1, so that d ^ kd.
If i = 4s + 3, it is clear that h = 20s + 13. Further, it is easy to verify that
a = 25s+2, a1 = 210s+6, and a3l~l = 25s+3. In this case, set d = l + 25s+3 + 210s+7,
so that kd = 1 + 25s+2 + 25s+3 + 210s+5 + 2 210s+6 + 2 215s+9 + 215m+1, which
simplifies to 1 + 25s+2 + 25s+3 + 210s+7 + 215s+u, so that d ^ kd. m
We now address the situation in which i + x 1 < r < i + l + 2. We are again
able to immediately show that r is restricted to the boundaries of this range.
Proposition 4.7 Let k = a + a1 + a2l+x, where a 2l with ar = 2. Further,
let h = 3i + l + x, where l < i and 1 < x < l. Ifi + x not a hyperoval in PG (2,2h).
Proof: Consider d = J2b^oX~r ab + a2l+(+x_r. Then kd = ^^1+x_r+1 ab +
ES<+Ir+i ab + ab + a3i+l+x~r + a4i+l+2x~r. Notice that under the
restrictions, 4i + l + 2x r < 3i + l + x and the only power of a that appears
twice is ocil+l+x-r. But 2a3t+l+x~r = a3l+l+x = 1, which does not otherwise
occur. Thus, d - This leaves r = i + x, r = i + l + 1, r = i + l + 2 (when l = i 1), and
r = i + x 1 (when l = i 1).
Proposition 4.8 Let k = a = a1 + a2l+x, where a = 210 with at+x 2. If
61
h = 3i + l + x, where l < i and 1 < x < l, then 2> (k) is not a hyperoval in
PG(2, 2h).
Proof: Let d = 0^ + cd+/_x_1. Then kd = Y^b=i a& + &l+l~x + ot2l+l~x +
Y^b=2%+x ab + Oi3l+l. Now 2a'l+l~x = a2l+l appears, yielding 2a2l+l = 1. It is
possible that 2a2l+l~x = a3l+l appears. In this case, 2a3l+l = a\ 2al = a2l+x,
and 2oi2i+x = a3t+2x, which does not otherwise appear unless l = x, in which
case, a second a2l+x no longer appears. Thus, d X kd. m
Proposition 4.9 Let k = a + a1 + a2l+x, where a = 2l with al+x_1 = 2.
Further, let h = 3i + l + x, where l < i and 1 < x < l. Then 2 (k) is not a
hyperoval in PG (2,2h).
Proof: Setd=£'lx+V + a2l+'. Then kd = <*b + ^ +
Yll^i+x ab + 2a2l+,+1 + a3l+l. We notice that the only doubled powers of a are
2a1 (or 3a1 when x 1) and 2a2l+l+1. Now 2a1 = a2l+x~l, which does not occur
elsewhere in the product. Further, 2a2l+l+1 = o:3l+i+x=1, So that d -< kd and
2 (k) is not a hyperoval in PG (2, 2^).
Proposition 4.10 Let k = a + cd + Oi2l+x, where a = 210 with cd+*+1 = 2.
Further, let h = 3i + l + x, where l < i 2 and 1 < x < l. Then 2 (k) is not a
hyperoval in PG (2, 2h).
Proof: Consider di = 1 + ab Then kd = a + a1 + a2l+x + E&ti+i ab +
Y^b=2iab + Eiltl+xSince a3l+l+x = 1, all terms of d are present in the
product. In addition, a 2oi2i+x appears. But 2q;2j+x = a, and 2a = a1+l+2.
Since l < i 2, cd+i+2 does not otherwise appear in the product, and thus
62
d -< kd.
Proposition 4.11 Let k = a + a1 + a2t+x, where a = 2l with cP+i+1 = 2.
Further, let h = 3i + l + x, where l = i 2 and 1 < x < l. Then Lfl (k) is not a
hyperoval in PG (2, 2h).
Proof: Set d = Et=o ocb+ al+x~l+ a2l+2x~2. Then kd = EEi o^ + Ejii""1 ab +
ESS1 ab + ai+x + + a3i+2x~1 + a2i+2x^ + a3i+2x~2 + a4i+3x~2. Now
a4t+3x2 a2x jt js possible that a 22x term exists, however, no 2x can be
present in d. If a 2a:2x term exists, it simplifies (since l = i 2) to a2t+2x~1.
As such, either a 3a2l+2x_1 term or a 2a2l+2x_1 term exists. In either case, an
additional ax term is produced. But 2ctx = a2l+x~l, and the resulting 2a2l+x_1
term reduces to 1. All powers of a present in d being retained, d < kd and 2 (k)
is not a hyperoval in PG (2,2h).
Where l = i 1 rather than i 2, a slight adjustment to the value of d
eliminates most of the remaining possibilities.
Proposition 4.12 Let k = a + a1 + a2l+x, where a = 2l with al+l+1 = 2.
Further, let h = 3i + l + x, where l = i 1 and 1 < x < l. Then & (k) is not a
hyperoval in PG (2, 2h).
Proof: We consider three cases on the values of x. If x > 3, we consider
d = EIZl + ES-J + a2l+2*-3- In this case, kd = ££ + ElS +
a21-2 + ESS-2 ab + 2o2i+2l_2 + ESS-3 ab- is possible that a 2q;2x-2 =
q,2z+2x-2 ^erm exists, so either a 3a2l+2x~2 term or a 2a2t+2x~2 term exists. In
either case, an additional ax_1 term is produced. But 2ax_1 = a2l+x~4, and the
63
resulting 2a2l+x 1 term reduces to 1. All powers of a present in d being present
in kd, with no further terms coinciding, d A kd.
If x = 2, set d = ELo a6*. Then kd = EL0 aM+1 + ELi W + Ej=26i+2-
Now a4l+2 = ct, so that a 2a appears. But 2a = a2l+l, and 2a2l+1 = 1, so that
d < kd in this case.
Finally, if x = 1, h = 4i and r = 2i. This case cannot occur, since (h, r) > 1.
Thus, (k) is not a hyperoval in PG (2, 2h).
All that remains now is r = i + l + 2 with l = i 1.
Proposition 4.13 Let k = a + a1 + a2l+x, where a = 2l with al+l+2 = 2.
Further, let h = 3i + l + x, where l = i 1 and 1 < x < l. Then (k) is not a
hyperoval in PG (2, 2ft).
Proof: We consider five cases on the values of x. If x > 5, let d = Y,t:oV +
Et+:
b
i+x1
i+x4
a
+ a2l+2x~6. Then kd = a* + £12 a" + a2-5 + ESSE +
2x5 Q,2i+2x4
a2t+2x-5 _|_ q,3j+2x-6 _|_ Efe^3i+2x-4 It Is possible that a 2a
term appears, in which case 2a2l+2x~4 = ax~2, which cannot otherwise appear.
Necessarily, however, a 2a2l+2x~b = ax~z appears, and 2ax_3 = a2l+x~2 and
2a2l+x~2 = 1 so that d ^ kd.
If now x = 4 consider d = Eb=ofti" + a2l+1- Then kd = EfLo0^1 +
ELi W+Et2 abi+4+a2i+2+a3i+1+a4i+5. Now a4l+4 = a, so that a 2a = a2l+2
appears. Now 2a2l+2 = 1, so that d kd in this case.
When x = 3, h = M + 2 and r = 2i + 1, so that (h,r) > 1, and this case
does not occur.
64
When x = 2, consider di-i. Now 2a2l+2 + 2a2l+3 appears in kdi_x; however,
2a2l+2 _|_ 2a2l+3 + a2 = a2l+3 + a3, neither of which terms otherwise appear, so
that di^i -< kdi-i.
Finally, when x = 1, let d ab + a2j_1. Then kd = JJb=o ab + 2a +
Ylll=i+i ab + 2a3*-1 -f a3'. Now 2a3-1 + 2al = a1 + aZl+l so that d < kd.
Thus, S (k) is not a hyperoval in PG (2, 2h).
Lemma 4.14 Let k = a+o:l + Q!2l+x, where a = 2l with (io, h) = 1 and ar = 2.
Further, let h = 3i + l + x, where l < i and 1 < x < l. Then (k) is not a
hyperoval in PG (2, 2h).
Proof: Table 4.1 illustrates how Propositions 4.1-4.13 together classify this
case.
With all possibilities covered, the result follows.
4.2 Case 2
In our examination of case 2, we are very quickly able to use values of d
similar to those used above to severely restrict the parameters m, n, and j. Our
first proposition states that m and n may not simultaneously be greater than
one.
Proposition 4.15 Let k = a + a2 + aj, where ar 2. Further, let h =
mj + 2n + 1, where 2n + 1 < j. If m > 2 and n > 2, then (k) is not a
hyperoval in PG (2, 2h).
Proof: Define the family {datb} of values for d in the following way. If a < n b,
hand, a > n-b, da, =
65
Table 4.1: Case 1 Classification Outline Conditions Proposition
r < min {x, l x} 4.2
min {x, l x} + 1 < r < i + x 2 4.1
r i + x 1 4.9
i+x t = i + Z-t-1, 2 4.10
t = i 1 1, l % 2 4.11
r i + l + 1, l = i 1 4.12
r = i + l + 2, l < i 1 4.1
r = i + l + 2, l = i 1 4.13
i + l + 3 r = 2i + l 1, l < i 1 4.1
r = 2i + l 1, l = i 1 4.6
2i-\rl r 2i + l + x + l 4.4
t = 2i 1 x 2, l i 1 4.1
r = 2i + l + x + 2, l = i 1 4.5
2i + l + x + 3 3 i + l < r 4.2
Note that dn_b,b has been defined in two different ways. When we wish to use
the value defined under the a < n b assumption, we will denote it by d~_bb,
and when we wish to use the value defined under the a > n b assumption, we
66
will denote it by d^_b b to avoid confusion.
We now determine the value kd^b for each of the definitions of daIf
a < n b, we find the following value for kd^b.
2a+l
2(n6)+3
nb 1
kda,b = Y olc + 2a2a+2 + Y aC + J2 ^ + J2 ^+2C+1
c=0
j+2(n+l)+l
c=2a+3
n1
c=0
+ E c+ E
OL
2j+2c+\
c=j+2(nb)+l c=nb
m1
+ Y (acj+2n+1 + acj+2n+2 + acj+2n+3) .
c= 2
The exponents which appear are distinct whenever 1 < b < n 1. Every power
of a present in dab is present in kdatb, and the only duplicate term which appears
is 2a2a+2.
In a similar fashion, we compute the value kda,b fr the case in which a >
n b. In this case, we obtain the following:
2(n6+1) n-6-1 j+2a+l j+2a+3
kda^ a+ aj+2c _j_ E ' + 2++2+2+ £
c=0 c=0 c=j+2(n-6) c=j+2a+3
a
n1
m1
+ E
a
2j+2c
E
a
2j+2c+J
+ E(
acj+2n+l _j_ acj+2n+2 ^ acj+2n+3^ _
c=nb ca c= 2
Once again, all of the exponents are distinct (so long as 1 < b < n 1), all
powers of a that appear in da,b also appear in kdaib, and the only duplicate term
which appears is 2oP+2a+2.
We can now determine the values for r that are not eliminated by each of
do,6) dn-b,b' dn-b,bi anCl dn,6-
67
For d0)b the values that remain are
1 < r <2(n 6) 1
r = J 2
r = j + 2c 1 : 0 < c < n 6 1
j + 2 (n 6) 1 < r < j + 2n + 1
r = 2j + 2c 1 : n b < c < n 1
r = cj + 2n 1 : 2 < c < m 1
r = cj + 2n: 2 < c < m 1
r = cj + 2n + l : 2 < c < m 1
mj + 2n 1 < r < mj + 2n.
For d~_b b the values that remain are
r = 1
r = j 2c : 1 < c < n 6+1
j 1 < r < j -f 2b + 1
r = 2 j + 2c 1 : 0 < c < b 1
r = cj + 2b 1 : 2 < c < m 1
r = cj + 2b : 2 < c < m 1
r = cj + 26 + 1 : 2 < c < m 1
mj + 26 1 < r < mj + 2n.
68
For b the values that remain are
1 < r < 2b + 1
r = j ~ 2
r = j + 2c 1 : 0 < c < b 1
r = cj + 261: 1 < c < m 2
r = cj + 2b : 1 < c < m 2
r = cj + 26 + 1 : 1 < c < m 2
(m 1) j + 2b 1 < r < (m 1) j + 2n + 1
r = mj + 2c1: b < c < n 1
77V? + 2n 1 < r < mj + 2n
For dn b the values that remain are
r = 1
r = j 2c : 1 < c < b + 1
r = cj 1 : 1 < c < m 2
r = cj : 1 < c < m 2
r = cj + 1 : 1 < c < m 2
(m 1) j 1 < r < (m 1) j + 2 (n b) + 1
r = mj + 2c 1 : 0 mj + 2(n b) l Under the assumption that 1 < b < n1, examining the remaining values for
these values of da^ reveals that only the following values for r are not eliminated
69
by at least one of the da^.
r = 1
r = j ~ 2
r = j- 1
r = j + 1
r = 2 j + 1
r (m 1) j + 26+ 1
r = mj + 2n 1
r = mj + 2 n.
We eliminate these values for r in turn. For r = 1, the only duplicate term in
kdnis 2oJ+2,l+2 = cF+2n+3, and 2aJ+2"+3 = ct7+2n+4, which does not otherwise
appear when j ^ 2n + 2. When j = 2n + 2, 2cF+2n+4 = aj+2n+5, which does not
appear, so that dn If r = j 2 and j 7^ 2n + 3, the only duplicate term in kd~0 is 2a2n+2
or 3a2n+2, and 2a2n+2 = a2-74-2"-2, which does not otherwise appear, so that
d~0 ^ kd~ o- When j = 2n + 3, let d = X^c=o a2 + Qfc-7+2n+1. Then kd =
£c=o V+2c*2n+3+a2"+4 + £c=i cF+2c+2a7+2"+2+7+2"+3 + £^=21 (aC7+2n+1 +
Q,cj+2ri+2 _|_ acj+2n+3^ Now 22ri+3 = Qf7"1"2"4"1, so all terms of d appear in kd, and
2od+2n+2 = a2-74"2", which does not otherwise appear so that d ^ kd.
Now if r = j 1 and j > 2n + 3, the only duplicate term in kd0^0 is
2a2 = a-74-1 term. Now 2a-7+1 = a2-7, which cannot otherwise appear in kdo^,
so that do,o ^ kdo Q. If j = 2n + 3, let d = £^o a2c + 1l!c=2 ac^+2n+l. Then
kd = £c2:t2 c + 2a2n+3 + a2n+A + £"£' cF+2c + (aC7+2n+2 + ac-*+2n+3) +
££3 a0-74"24-1. As 2a2n+3 = o;t+2n+2 anc[ 2o;t+2n+2 Q,2j+2n+i^ ^ ^ Finally,
70
if j 2n+2, let d = EEo <*2c+E / otCJ+2n+l, so that kd = Ec^o1 c+2a2n+2+
EEi aj+2c + aj+2n+2 + ctj+2n+3 + J2Z~2 (uCJ+2n+1 + acj+2n+2 + acj+2n+3). Since
2a2n+2 = ot?+2n+1, d X kd.
If r = j + 1, consider the value d0,i- Since 2a2 = od+3 and 2aJ+3 = a2j+4,
which cannot otherwise appear, d0,i ^ A;d0,i.
When r = 2j + 1, take d = Ec=o ^ + a2"-1 + a-7+2n_1 + EEi^ aCJ+2n+1, so
that fcd = EcE* c + 2a2" + E=o qJ+2c + aJ+2"_1 + aJ+2" + aj+2n+1 + 0i2i+2n-1 +
E^1 (cj+2n+2 + o;ct+2n+3) + Er=3^+2n+1. Since 2a2n = a2i+2n+l, d r< fcd.
For r = (m 1) j + 2n 1, let d = 1 + a + Q-(m_1W+2Tl+1. Then kd =
l+a+2a2+a3+aJ+aJ+1+a(m~lb'+2"+2Wm_1)j+2"+3. But 2a2 = c*(-W+2n+i
so that d ^ fcd.
If r = mj + 2n 1, take d 1 + a. Then kd = a + 2a2 + a3 + aJ + aJ+1,
and 2a2 = 1, so that again d ^ fcd.
Finally, when r = mj + 2n, if j > 2n + 4 set d = E"=o q2c + EEi* &cj+2n+1-
So kd = E2o3c + Ec=oj+2c + 2a-7+2n+2 + a7+2"+3 + EE21 (acj+2n+1 +
acj+2n+2 _|_ 0,^4-271+3^ now 2ai+2n+2 a-?+2"+1, so that d < kd. On the other
hand, if j < 2n + 4, set d = 1 + a2n+2 + YZ?=i acj+2n+1 so that kd = 1 + a + a2 +
a2n+3 + a2n+4 + aj + 2aj+2n+2 + aj+2n+3 + Y^ (aci+2n+l + acj+2n+2 + ^+277+3)
Now a-7 G {a2n+2, a2n+3, a2n+4}. As 2a2"+4 = a2n+3 and 2a2n+3 = a2"+2, and
0,271+2 (p)es no^ appear apart from a?, a2n+2 appears in each case. Further,
2a.i+2n+2 = ctJ+2n+1, so that d A kd.
Thus, for every choice of r, some value of d was found for which d < kd, so
that, if m > 2 and n> 2, @ (k) is not a hyperoval in PG (2, 2h).
We next consider the situation in which m = 1 and j > 2n + 3.
71
Proposition 4.16 Let k = a + a2 + a3, where a = 2l, ar = 2, j > 2n + 3, and
h = j + 2n + 1. Then if @ (k) is a hyperoval in PG (2,2h~), it is a translation
hyperoval.
Proof: As defined in the proof of Proposition 4.15, consider da0. Note now
that kdafi can be simplified to a + 2a2a+2 + a + Sc=o aj+2 +
aJ+2c+1. Notice also that only the following values for r are not ruled out
for a particular a:
l r j 2c : 1 < c < a + 1
r = j + 2c 1 : 0 j + 2 (n a) 1 < c < j + 2n.
For o = 0, this equates to the following values for r:
1 < r < 2n + 1
r = j -2
r = j + 2c 1 : 0 < c < n 1
j + 2n 1 < r < j + 2n
while for a = n, this leaves the following values for r:
r = 1
r = j 2c : l j 1 < r < j + 2n.
72
Combining these two lists, only the following values for r remain (noting that
the lists are identical if n = 0):
r = 1
r = j 2c : 2 [?r2~1j < c r = j 2
r = j + 2c 1 : 0 < c < n 1
j + 2n 1 < r < j + 2n.
We again rule out each of the remaining cases in turn. When r = 1, consider d~0.
Since 2a2n+2 a2n+3 and 2a2n+:i a2n+4 (which may not otherwise appear in
the product) and 2a2n+4 = 2n+5, where a2n+5 does not appear in the product
kd~o otherwise, d~0 < kdnt0. This fails only if n = 0 and j 4, in which case,
k = 2 + 22 + 24, and 1 j^ = 22, so that 2> (k) is a translation hyperoval in
PG (2,4).
For r = j 2c, we consider two cases depending on the parity of j. If j
is even, consider d0i0. In the product kd0t0 a 2a2 appears. Now with j even,
r is also even. But every power of a greater than 2 in d is odd. Furthermore,
h is odd, and the powers a^+2n~2c for 0 < c < n 1 are all absent from the
product kdofi- Since this r value is necessarily less than 2n + 1, if new powers
of a continue to arise from the 2ct2 through continued carrying, one of these
powers must be of the form a^+2n~2c with 0 This again fails for n = 0, however, n = 0 implies that j 1 < 2, so that j = 3,
h = 4, and | = 2, so that Q> (k) is a hyperconic.
If j is odd, and j 2c < 2n + 1, we use d = 2 a29 + o?^n+c~Lr )+1 +
a2n+l. Then kd = a9 + 2a2n+2c~j+1 + a^n+2c-j+2 + a2n+2 + a2n+3 +
73
X!g=o 2 +29 + cr'+2(n+c~i2_)+1, with all powers distinct. But 2a2n+2c~i+1 =
an+l, so that d ^ kd.
On the other hand, if j 2c 2n + 1, we divide into two further
cases. If j > 4n + 4, kdoto has a 2a2 term. Now 2a2 = a2n+3 and
2a2n+3
= a4n+4. Since 4n + 4 < j, this does not otherwise appear, so
that d0,0 Z. kd0io- If j < 6n + 3, consider d = ^1 + a^ X]g=o 1 0Z9
Here, kd = ^1 + a^ 2"a;9 + 1 aj~%+29^J. Notice that with
j < 6n + 3, all of the exponents in the second factor are less than |. Now j |
is equal to either 2 [|J 2n 1 or 2 [|J 2n, and in either case, 2a= a%,
so that all terms of d arise in kd.
Now, if r = j 2 we consider four cases. For j > 2n + 6, we consider
d^0 In the product kdn (h 2a2n+2 = aJ+2n, and 2a3+2n = a-7-3 which does not
appear, so that d+0 < kd0. If j = 2n + 6, let d = 1 == J2c=oa2c+3 so that
kd = 1 + a + 2a2 + ^=4o:c + YZUZo ad+2c+3. Now 2a2 = cd and 2aJ = a3,
so that d < kd. This fails when n = 0, in which case ^ = 6 and @ (k) is
a Segre hyperoval. When j = 2n + 5, h = 4n + 6 and r = 2n + 3 = |, so
that (r, h) ^ 1, a contradiction. Finally, if j 2n + 4 let d = a2c- Then
kd = 2a + JZiZ3 qC + 2ad + X]c=i a2c+2- But 2a? = a, 3a = a + a-7"-1, and
2aj~1 1, so that d ^ kd.
When r j + 2c 1 with 1 < c < n 1, we use d = Z2gZo ft2c + a2(n~c^+l,
so that kd = ZZfZZ^1 a9 + 2a2(n~c'l+2 + a2^n_^+3 + ZZI^Zq o?+2a + ad+2(n_c)+i_
Now 2a2(n~c)+2 = 1, so that d ^ /cd.
If r = j 1, we consider several cases. If j > 2n + 6, consider d_i;o (note
that with 0 74
a 2a2n appears. Now 2a2n cP+2n~1 and 2aj+2n~l = aj~3. Since j > 2n + 6,
j 3 > 2n + 3, so that d_1;0 A ^dn-i,o- Now, if j = 2n + 6 and n > 1, set
d = X^c=o a2c + a2_3 + o:2"+1, so that kd = X^c=o3 c + 2a2"-2 + a2n-1 + a2n+2 +
a2"1-3 + ^02 a3+2c + a-7-1"2-3. Now 2a2n_2 = aj+2n~3 and 2cr7+2n~3 = a2n+1,
so that d < kd. If j 2n + 6 and n = 1, set d = 1 + a2 + a6, so that kd =
a + Q;2 + 2a!3 + a:4 + a:7 + 2ai8 + a10. But 2a3 = a10 and 2a10 = a6, while 2a8 = a4
and 2a4 = 1, so that d ^ kd. If j 2n+5, then h = 4n+6 and r = 2n+4, so that
(r, h) ^ 1, a contradiction. Finally, if j = 2n + 4, consider d = 1 + ]>Zc=i a2c+1,
so that kd = 1 + a + 2a2 + c + 22n+4 + a2n+5 + aj+2c+1. Now
2ct2"+4 = a2, 3a2 = ct2+5 + a2, and 2a2+5 = a3, so that d -< kd.
We next consider r = j + 2n l. As in the proof of Proposition 4.15, we use
d 1 + a, so that kd = a + 2a2 + a3 + a? + cP+1. So long as n ^ 0, Of-7+1 ^ 1,
in which case 2a2 = 1 and d < kd. If n = 0, consider d = 1 + a2 + a3. Then
kd = 2a + 2ct2 + a3 + 2a4 + a5 + a?. Now 2a4 = a2, 3a2 = a2 + 1] 2a = cr7, and
2cP = a-7-2, which no longer otherwise appears unless j G {3, 4, 5, 7}. If j = 3,
1 k = 2 and @ (k) is a hyperconic; if j = 4, 1 k = 6 and @ (k) is a Segre
hyperoval; if j = 5 or j = 7, (r, h) 2, a contradiction.
Finally, if r = j + 2n, the values of d used in the proof of Proposition 4.15
work equally well with m = 1 for all values of n.
Thus, in all cases, either a value of d was presented with d ^ kd, or @ (k)
was shown to be a translation hyperoval or a Segre hyperoval.
We next deal with the situation in which j = 2n + 3. In such a situation,
we have the following result.
Proposition 4.17 Let k = a + a2 + a^, where a 2*, ar = 2, j = 2n + 3, and
75
h = j + 2n + 1. Then if (k) is a hyperoval in PG (2, 2h) it is a hyperconic.
Proof: We split into cases for different values for r. Note that h = A(n + 1),
so we need only consider odd values for r. If r = 1, then consider d~0. A
2a2n+2 term appears, creating a 3a2n+3 term, which in turn creates an ot2n+i
term, which does not otherwise appear unless n 0, in which case j = 3, and
1 (a + a2 + a3) = a. But a = 2, so that @ (a + a2 + a3) is a hyperconic in
PG (2,24). Thus, for n > 0, d~0 ^ kd~0.
Now, if 3 < r < 2n 1, consider d = 1 + a + V'T! .r+i a2c+1. Then kd =
C~ 2
l+a + 2o2+a3 + Ec=r+3c + 2^2rl+3 + 2n+4 + EcJ4i J+2c+1. But 2a2 = ar+2,
while 2o2+3 = a2n+r+3, which does not otherwise appear, so d < kd.
When r = 2n+l, we set d = 1+a, so that kd = a + 2Q;2 + Q!3 + a2Tl+3 + Q;2n+4,
and 2a2 = o2"+3, giving rise to a 2o2+3 = 1, so that d z< kd.
On the other hand, when r = 2n + 3, we use d0i0. Then 2a2n+3 = a2, and
3a2 = a2 + a2n+5, which does not otherwise appear unless n = 0, in which case,
k = a + a2 + a3 and h = 4, so that k = 2 + 22 + 23, and 1 k = 2, making @ (k)
a hyperconic in PG (2,24). For n > 0, though, d0,o ^ kd0p-
If 2n + 5 < r < An + 1, define d 1 + V r-2n-i o?c + ct2n+1, so that
2
kd l+a + a2 + E2^2nc + 2a2n+2 + 2a2n+3 + E"=^i j+2c- But 22"+2 =
olt~2"-2, while 2a2"+3 = ar~2n~x, neither of which terms otherwise appear, so
that d -< kd.
Finally, if r = 4n + 3 = j + 2n, the same value of d as used in the proofs of
Propositions 4.15 and 4.16 still works.
Thus, in all cases in which (k) was not a hyperconic, a value of d with
d -< kd was exhibited.
We obtain a similar result when j = 2n + 2.
Proposition 4.18 Let k = a + a^ + cP, where a = 2l, ar = 2, j 2n + 2, and
h = j + 2n + 1. Then if 2 (k) is a hyperoval in PG (2, 2h), it is either a Segre
hyperoval or a Glynn hyperoval.
Proof: As usual, we divide into cases based on the different possible values for
r. Notice that n > 1, as n = 0 implies that j = 2, a contradiction.
For r = 1, the value d~0 as defined in the proof of Proposition 4.15 gives
a product with the only repeated term being a 3a2n+2. But 3o2n+2 = o-2n+2 +
a2"1'3, 2ai2ri+3 = a2"+4, and 2a2n+4 = a2n+5 which does not otherwise appear so
long as n > 2. On the other hand, when n = 1, we obtain k = 2 + 22 + 24. But
then 1 £ = 6, so that 2 (k) is equivalent to the Segre hyperoval in PG (2,27).
If 2 < r < 2n 1, define d = 1 + a + ^"=j- 2 ^ ar+2c + a2n+1. Then kd
l+a + 2a2 + a3 + ac+{2+tf [=JJ) a2n+2 + 22"+3 + aJ+2c+r.
Now, since [2 [|JJ is zero or one as r is odd or even, the coefficient on a2n+2 is
either 2 or 3. But then a 2n+2+r term arises, which does not otherwise arise, and
the 2a2n+3 likewise gives rise to an a2n+3+r term which also does not otherwise
arise. Finally, 2a2 = ar+2 so that d < kd for these values for r.
Now if r = 2n, we examine d~0, and notice that a 3a2n+2 term arises. But
3a2n+2 = a2n+2 + ai4n+2, 2a4n+2 = a2n_1, and 2a2n_1 = a4"-1, which does
not otherwise arise, so long as n > 3. Now, if n = 1, k = 24 + 2 + 22 since
h = 7, so that 2 (k) was earlier shown to be equivalent to the Segre hyperoval
in PG (2,27). If, on the other hand, n = 2, k = 23 + 26 + 27 with h = 11. Then
1 | = 23 + 26, where (23)4 = (26)2 = 2, so that 2 (k) is equivalent to the
77
Glynn hyperoval (a + 7) in PG (2,211).
For r = 2n + 1, d = E"=o q2c giyes a product of kd = EcHt* a<: + 2a2n+2,
and 2a2"+2 = 1, so that d ^ fcd.
When r = 2n + 2, consider d = E"=o a2c + ce2n_3 + a2"-1 + a2", so that
kd = ESc/V + 2a2"-2 + a2"1 + a2n + 2a2"+1 + a2n+2 + Ec=o <*2n+2c+2 +
0,471-1 _|_ a4n+1 4- ain+2. Now 2a2"-2 = a4", which does not appear elsewhere in
the product. Furthermore, 2a2"+1 = 1, so that d < kd, as desired. This fails
precisely when n < 3. But when n = 1, we again have k = 2 + 22 + 24 and
h = 7, giving a Segre hyperoval, and when n = 2, h = 11, and k = 2 + 22 + 24,
where ^ = 22 + 26 + 27, or 3a + 4, a Glynn hyperoval.
If r = 2n + 2c + 1 with 1 < c < n, set d = YZ^Zo0-29 + a2^"-c)+1. Then
kd = E2LV)+1 a + 2a2("-c)+2 + a2("-c)+3 + ZgZo a2n+2+2 + a4"-2c+3. But
2a2(n-c)+2 = so that d ^ kd.
If r = 2n + 2c + 2 with 1 < c < n, set d = 1 + J2g=c 0(29+1 > so that
kd = 1 + a + a2 + Eg=2c+2 a9 + 2a2"+2 + a2n+3 + YZgZl a2"+2ff+3. But since
2a2"+2 = a2c+1, d < kd.
Having covered all possibilities for r, it follows that in this case any hyperoval
is a Glynn hyperoval or a Segre hyperoval.
We finally address the case in which m >2 and n < 2 in the two propositions
that follow.
Proposition 4.19 Let k = a + a2 + a-7, where a = 2l, ar = 2, and h = mj +1.
Then if @ (k) is a hyperoval in PG (2, 2h), it is a translation hyperoval or a
Glynn hyperoval.
78
Proof: We first consider which values are ruled out by g^0, noting that kd,Q 0 =
l + a + 2a2 + a3 + a-7 + (a^+1 + + oc-7+3)- Subtracting all powers
of a in the product, we are left with r G {1, j 2, cj 1, cj, cj + 1, mj 1, mj}
where 1 < c < m 1. But further for j > 4, if r = 1, 2a2 = a3, and 2a3 = a4
which does not otherwise appear.
Now, if j > 6, and r = j 2, notice that 2a2 = aj and 2(P = a2j~2, which
does not otherwise appear in the product.
If j > 5, and r = cj + 1, for 1 < c < m 1, 2ct2 = a o2c-?+4. If 2cj + 4 < h, this term does not otherwise appear, while if 2cj + 4 > h,
2cj + 4 gj + 3 for some value of g. Now there exists some first multiple of
r (say q) such that 2 + q ^ 3 mod j. But then, necessarily, 2 + r (q 1) = 3
mod j, so that 2 + qr = 4 mod j. Furthermore, a2+9m appears for all m < q,
but does not then appear for m = q. Thus, d -< kd.
This analysis of 0 fails for j = 3, when an additional doubled term appears;
however, the values remaining comprise all values between 1 and h 1 when
j = 3, so that the following analysis is in fact complete.
We now analyze the remaining values for r.
If r = 1 and j = 3, let d = J2T=o ct3c+a3^m^+1. Then kd = ac+
2a3(m-2)+2+2a3(m-l)+2a3(m-l) + l+a3(m-l)+2+a3m_ Now 2a3(m-2)+2+2a3(m-1) +
2a3(m-1)+1 + a3m = a3('m~1'> + a3(m-1)+i + 1, so that d ^ kd.
If, on the other hand, j = 4, set d = 1 + a2 + o;4c+1, so that kd
^3=0 ac + 2a4 + 2ct6 + a7 + Y^=2 (a4c + 4c+1 + ct4c+2). Now 2a4 = a5, while
2a6 = a7 and 2a7 = a8, which does not otherwise occur in the product so that
d -< kd.
79
Now if r = j 2 and 4 < j < 5 (since 3 2 = 1 we need not consider
j = 3 again), consider d = E^2^ + c^m-2^+1 + a^m_2^+3, in which case
kd = ZZo2 (aCJ+1 + c^'+2) + ZZl1 + u{m~2)j+2 + a^-2^ + c^-W+i +
a(m-2);+4 _|_ Q,(m-2)j+5 _j_ a(m-l)j+3_
If j 4, the multiple powers that appear are precisely 2a^rn~2^+2 = ,
2cdm-2^+4 2with 3a^m_1^ = + Q/i771-1)^2 which does not
otherwise appear, and 2a^m~1^+1 = a^m_1h+35 which already appears, so that
2a{m-\)j+z j an(j d < kd.
If j = 5, the multiple powers that appear are precisely 2cdm-2h+2 = a(m~i)j
and 2a^m~1^, so that the resulting 3a^m_1^ = + a(m_1h+3. But
2Q,(m-i)j+3 so that d -< kd.
If now r = j 1 and j = 3, we use d = ZT=o ^C + a3m_4, so that
kd = Ec3=r4c+23m-3+3m-2+3"1-1. But 2a3m~3 = a31 and 2a3771-1 1,
so that d < kd.
If j = 4, we note that if m = 2, h = 9 and j 1 = 3, so that this case
cannot occur as (3,9) ^ 1. We use d = EEo3q;4c + a4^m_3^+2 + Q,4(m_2)+2, so
that fed = £r=o3 (a4c+1 + a4c+2) + E? 4c + a4^-3)+3 + a4(m:2) + a4^+2 +
Q,4(m-2)+3 _|_ _|_ 0,4777-2 now a 2a4(77i-2) ^4(771-2)4-3 term appears, but
2Q;4(77l-2)+3 -- a4m_2; and 2o;4m2 = 1, so that d X A;d.
Finally, if j > 4, consider d E^o* ^ + a(m-2^+2, in which case kd =
Z7=o (<*cj+1 + acj+2) + Ei <*cj + (m_2)i+3 + aim~2)j+4 + a^m^j+2. Here, a
2a{m-i)j+2 appears, but this simplifies to 1, so that d -< kd.
When r = cj 1 for 2 < c < m 1 and j = 3, we set d =
Er=oC_1 39 + a3(m"c)-1 + Q!3(m_c)+1. Then kd = EsLTcM a + 2a3(m~c) +
80
a3(m-c) + l + 2a3(m-c)+2 + a3(m~c)+3 + a3(m-c)+4_ Nqw 2aHm-c) = a3m-l which
does not otherwise appear, and 2a3(-m~c'l+2 = 1, so that d < kd.
On the other hand, if j ^ 3, set d = Y^=o 0/99 + a;(m~ch'+1) so that kd =
EJTcT1 (9j+1 + a"+2) + Er=T+1 a9j + a(m_cb+1 + 2a(m-cW+2 + cdm-c)J+3 +
o;(m_c+ib'+i_ gince 2cdm-ch'+2 = 1, d ^ fcd.
If r = j and j ^ 3, d = 1 + a2 + EEV 0/09+1 gives the product kd =
1 + a + a2 + a3 + a4 + a? -f 2aJ+2 + aJ+3 + EE^ (Q;C:+1 + + aC9+3). Now
2aJ+2 + E^1 olC9+2 = a, and 2a = aJ+1 so that d -< kd. (If j 4, 2a4 = a2j,
which does not otherwise appear.) When j = 3, k = 2 + 2m+1 + 22m+1. Notice
then that | = Ec=m 2C, so that 1 £ = 2m, where (m, 3m + 1) = 1, and ^ (fc)
is then a translation hyperoval.
Now if r = cj with 2 < c < m 1, for j 3, consider d = 0/99 +
a(m.ci)j+i, where kd = Y^=o~2 (aS'7+1 + a9^+2) + "Y^=\+l 0/99 + cdm~c~:Lh+1 +
2a(m_c_1h+2 + Q!(1-c-1)f+3 _p 2a^m_c^+1 + a(rn_ch'+2_ Now 2a^m_c_1^+2
Q,(mi)j+2 ^jch (joes not otherwise appear in the product, and 2a^m_c_1^+1 = 1,
so that d -< kd.
When j = 3 and c > 2, consider d = Y^=o~2 0/39 + a3(m~c_2)+1 +
a3(m~c~i)+i-f a3(m-c), so that kd = E^Ti~C~2^+1 ct9 + 2a3(m-c_2)+2 + 2a3(m~c~1) +
a3(m-c-l) + l _|_ a3(m-c-l)+2 _|_ a3(m-c) 2a3(m_c)+1 _p Q,3(m-c)+2 a3(m-c+l)_ Now
2a3(m-c-2)+2 = a3(m-2)+2^ 2a3(m~c~1') = a3^m-1\ and 2a3^m_c^+1 = 1, so that
d ^ kd.
Finally, if j = 3, r = 6, and m > 6, consider d = ESu)6 a3c + a3^m_6^+2 +
a3(m_5)+1 + o;3(m_4) + Q,3(m3)+2^ jn this case; kd = X)^_6')+2 ac + 2a3^m_5^ +
a3(m-5)+l _p 2o;3(m-5)+2 -p Q,3(m-4) -p 2ct3(m_4) + 1 + a3(m-4)+2 _|_ ^3(771-3) a3(m-2) _|_
81
a3(m~2)+i + a3(m-2)+2 j\jow 2c*3h"~5) = o;3(m-'3) and 2c*3h"-3) c*^"1-1) which
does not otherwise appear. Further, 2c*3h"-5)+2 = ct3^-3^2, which does not
otherwise appear in kd but does appear in d. Finally, 2c*3hn-4)+1 = a3(m-2)+1,
and 2a3(m~2)+1 = 1, so that d ^ kd.
In the case that r = cj, this leaves only the possibility that j = 3, r = 6,
and m < 6. Now if m is odd, h is even and (r, h) > 1, so that we need only
consider m = 4 and m = 2. If m = 4, notice that k 27 + 29 + 2U, with h = 13.
Then | = ]Cc=2 2C + ]Cc=9 2C, so that 1 | = 22 + 27 + 28 = 3cr + 4, making
@ (k) a Glynn hyperoval. When m = 2, notice that k = 24 + 25 + 26 and h = 7,
so that 1 k = 24 and ^ (fc) is a translation hyperoval.
We next consider r = cj + 1 when j < 4. Here let d YgYtf1 ft9J +
a(m-c)j-2j so kd = ^rgn~o~l (9j+1 + a9j+2)+Er=7~X a^+(m-c)^1+2a(m-cb +
^(m-c+ip-2 gut then cj j 2 6 {(m c 1) j + 1, (m c 1) j + 2},
2c*h"-cb-1 = c*^ which does not otherwise appear, and 2cdm_ch = 1, so that
d -< kd.
When r = mj 1, as in the proof of Proposition 4.15, d = 1 + a suffices
so long as j 3. If j = 3 and m > 2, let d = 1 + a3 + c*5, so that kd
Ec=i Q^ + Sa;6 + c*7 + c*8. Now 2c*6 = c*4, 2c*4 = a2, and 2c*2 = 1, which does not
otherwise appear, so that d z< kd. If, on the other hand, m = 2, A: = 22 + 23 + 26,
and 1 k = 22 + 24 + 25 = 3cr + 4, so that @ (k) is a Glynn hyperoval.
Finally, when r mj and j > 4, choose d = 1 + aj~l + YZ2 aC9+\ so that
kd = 1 + c* + c*2 + 2c*J + c*2j_1 + Y=2 (&cj+1 + c*c:,+2 + c*CJ+3). Now 2c*J = cP-1,
so that d z< kd. When j = 4, let d = EEo1 ^C + c*4m_5 + c*4m_3, so that
kd = YZo1 (4c + c*4c+1 + c*4c+2) + c*4-4 + c*4m~3 + c*4m~2 + 2c*4"1'1 + c*4m.
82
Now ^Zc=4m-4 ac + 2a4m~1 = o4m_5, which does not otherwise appear, and thus
d kd. On the other hand, whenj = 3, k = 2h~z+2h~2-\-2h~l, and 1 k = 2h~z.
Since 3 does not divide h, (k) is a translation hyperoval.
Thus, having presented either a d with d kd, or an equivalence to a known
hyperoval in all cases, the result follows.
We finally classify the hyperovals with n = 1 and m >2.
Proposition 4.20 Let k = a + a2 + od, where a = 2l, ar = 2, j > 3, and
h = mj + 3. Then if (k) is a hyperoval in PG (2,2h), it is a Glynn hyperoval.
Proof: We first examine the four relevant da b values with the same analysis
as in the proof of Proposition 4.15. If j > 5, do,o leaves the following values for
r:
1 < r < 3
j-2 cj + l mj + 1 < r < mj + 2
If j 7^ 5, df o leaves
r = 1
r = j 4
r = j -2
cj l raj 1 < r < mj + 2
83
If j > 4, d~t0 leaves
r 1
r = j 2
cj l mj j + 2 < r < mj j + 3
r = raj 1
mj + 1 < r < mj + 2
Finally, if j >4, leaves
r = 1
j 2 < r < j
cj + 1 2j l = r
mj + 1 < r < mj + 2
84
Putting this together, at most the following values remain to be classified:
r = 1
j 2 < r < j 1
r = j = 5
r = 2j 1 = 9
r = cj 1 2 < c < m 1 j = 4
r cj 1 < c < m 1 J = 4
r = cj + l 1 < c < ra + 1
mj 3 < r < 2 j = 5
raj 1 < r < mj j 4
r mj + 1
r = mj + 2
We classify these in turn.
If r = 1, consider d0,o- In this product, we have 2a:2 + a3 + aA + a5 = a6,
which does not appear unless j = 5 or j = 6. If j = 5, 2a6 = a7, which does
not appear, and if j = 6, 2a6 + a7 = a8, which does not appear, so d z< kd.
For j / 5 and r j 2, consider d^0. We obtain a 2a4 = a?+2 (or
3a4 = a4 + ai+2 if j = 4), and 2a4+2 = a2-7 which does not otherwise appear
unless j = 4. Now if j = 4, 2a2-7 = a2-74"2 which does not otherwise appear so
that d ^ kd.
When j = 5, consider d = 5c + a5(m_P+1, in which case the product
kd = £r=o2 (<*5c+1 + a5c+2) + Er=" q5c + a5(m_1)+1 + 2a5(-1)+2 + ^(m-D+3 +
a5m + a5m+1. Now 2a5(m_1)+2 = a5m and 2a5m = 1, so that d < kd. For
r j 1 and j > 4, consider d = a>+ a^m_2^+2 + a^m_2^+4 + a^m_1b+2,
85
so that kd = ES2 (acS+I + <*+2) + EE',' o'3 + ESm'+s c + a<"">W+2 +
a(m-i)j+3 _|_ 2a(m-l)j+4 + aT?y'+2. Now 2a^m_1^+4 = 1, and, if a 2ahn~1b term
appears, 2cdm_1) = a-7-1, which cannot otherwise appear with j > 4, so that
d -< kd as desired.
On the other hand, if j 4. consider d = Y^T=o a4c + a4m_2, so that
fcd = E^o1 (4c+1 + 4c+2) + E^Ti1 a4c + a4m"1 + 2a4m + a4m+2. Now 2a4m = 1,
and d ^ kd.
Now when r j and j = 5, consider the value d0,o- In the product fcdo.o, we
obtain a 2a5 term and a 2ct2 term. Now 2a5 + E^=2 ^c = (where the sum is
assured to appear in the product), and now 3a2 a2 + a7, where a7 does not
otherwise appear, so that do,o ^ kd00.
If now r 2j 1 and j = 5, let d = Ecl^2 a5c + a5^m_2^+2 + a5^m_2^+3. Then
fed = Er02 (5c+1 + 5c+2) + EEf ft5c + a5(m-2)+3 + 2a5(m"2)+4 + 2a5(m_1) +
Q,5(m-i)+2_j_Q,5(m-i)+3 ]\jow 2a5(m-2)+4 j. 20,5(771-1) = a, 2a = a10 which does
not otherwise appear if m = 2, 2a10 = a19, which does not otherwise appear.
Thus, d < kd.
For the case in which r = cj 1, for j 4 and with no restric-
tions on the value of c (thus encompassing also r = mj 1), consider
d = E^a4* + a4(c)+2. Then kd = E^o" (49+1 + <*49+2) + EJLTa4 +
a4(m-c)+3 + 2a4(m-c+l) + a4(m-C+l)+2 Nqw 2a^m-c+l) = l SQ that d kd.
When r = cj, for j = 4 and c < m, consider d = a4c +
Q,4(m-c-l)+2 _|_ Q,4(rrec1)+3 _|_ Qf4(m-c)+2^ go that fcj ^^c-1 (a4ff+1 + a4ff+2) +
^m-c-1 ^4^ ^4(mc1)+3 _|_ gQ,4(m-c) _|_ ^(m-cj+l _|_ a4(m-c)+2 _|_ 2Q,4(m-c)+3
a4(m-of 1) + a4(m-C+l)+2_ Her6) 3a4(m-c) = a4(7n-c) + whjch doeg not oth-
86
erwise appear in the product unless c = 1, (where a4m + Y^=o a4ff+1 = a4m+1,
which does not otherwise appear), while 2a4^m_c^+3 = 1, so that d < kd.
In the case in which r = cj + 1 with no restrictions on either c or j (thus
encompassing r = mj + 1 as well), we use d Y=o + o^m~c^ + a.^m~c^+1,
so that kd = (9j+1 + 9j+2 + a(s+1)j) + a{m~c)j+1 + 2a(m_c>J'+2 +
Q(m-c)j+3 + a(m-c+l)j + a(m-C+l)j + l Since 2a(m-c)j+2 = ^ -< kd.
For r = mj 3 with j = 5, consider d = 1 + a + a5, where kd = a + 2a2 +
a3 + a5 + 2a6 + a7 + a10. Since 2a2 which cannot appear elsewhere,
and since 2a6 = 1, d < kd.
For r = mj 2 with j = 5, consider in similar fashion to the preceding case
d = 1 + a + ct3, where kd = a + 2a2 + a3 + a4 + 2a5 + a6 + a8. Here, 2a2 = amj,
which does not otherwise appear, and because 2a5 = 1, d ^ kd.
Now when j = 4 and r = mj, if m > 3, consider d 1 + a + a5 + a7, so
that kd = a + 2c? + ^c=3aC + 2a9 + a11. Now 2a2 = ami+2, which appears
nowhere else, while 2a9 = a6, 2a6 = a3, and 2a3 = 1, so that d -< kd. On the
other hand, when m = 2, k 23 + 26 + 27, so that = 23 + 26 = a + 7, so
that 3 (k) is a Glynn hyperoval in PG (2, 211).
Finally, when r mj + 2, set d = Y^=o a+ cdm-2^+2 + ajm-1b'+2 +
a(m~ 1b+4. In this case, kd = 2a + a2 + ]G(!!r(2 (aCJ + aCJ+1 + aC9+2) + o;(rn-2b+3 +
Q,(m-2p+4 _|_ a(m-i)j _|_ ac + a7"-74-2. Now 2a = 1, and if j = 4,
2Q;4m+2_|_^4m+i^2 _ 0,401-3^ does not otherwise appear, so that d < kd.
Thus, in all cases, either a d with d < kd or an equivalence to a Glynn
hyperoval was presented.
We finally summarize the results of the preceding propositions into a clas-
sification lemma.
Lemma 4.21 Ifk = a+oP+ai, where a 2J and 2 = ar, with h mj+2n+l,
then if @ (k) is a hyperoval, it is a translation hyperoval, a Segre hyperoval, or
a Glynn hyperoval.
Proof: Table 4.2 illustrates how Propositions 4.15-4.20 together classify this
case.
Table 4.2: Case 2 Classification Outline
Conditions Proposition
m > 2, n > 2 4.15
m = 1, j > 2n + 3 4.16
m = 1, j 2n + 3 4.17
m = 1, j = 2n + 2 4.18
n 0 4.19
n = 1 4.20
With all possibilities covered, the result follows.
4.3 Case 3
In case 3, where j = i + l 1 and h = j +1, we find it useful to deviate from
our usual technique of decomposing h as mj + ni + l and to instead decompose h
as ni + l'. The following proposition uses this idea to rule out many possibilities.
88
Proposition 4.22 Let k = a + cd + a?, where a = 2l, 2 = ar, h = j + l,
l + 2 < < 21 1, and j = i + l 1. Then (k) is not a hyperoval in
PG (2, 2h).
Proof: Consider d = X^=o_1 a+(^2l~1- Then kd = Y?l=l a +
a2i+i-i _j_ oca + a3l+l~2. Note that a3,+I_1 = al~l. We make several
observations regarding this product. Notice first of all that all terms of d are
present in kd, as 21 + i 1 = h and thus q;2/+*-1 = 1. Further notice that
21 i < l 1, since l + 1 < i; 21 < i + l 1, since l < i 1; 31 2 < h\ and
21 i < l 1, since again l < i 1. As such, no terms can possibly coincide,
and d ^ kd. m
We may further, with a little work, eliminate the possibility that l = i 1,
as the following proposition establishes.
Proposition 4.23 Let k = ot + a* + aj, where a 2l, 2 = ar, h = j + i 1
and j = 2i 2. Then Q) (k) is not a hyperoval in PG (2, 2h).
Proof: Let l = 2 1. We consider the congruence classes of l and r modulo
3 together with the value d = ^c=oa3c- Notice that h = 3i 3, so that
r ^ 0 mod 3, lest 3| (r, h). If l = 1 mod 3, kd = X^c=o (a3c+1 + 2a3c+2),
which reduces to either X^c=oa3c (when r = 2 mod 3) or (a3c + <*3c+1)
(when r = 1 mod 3), so that d < kd. If l = 0 mod 3 or l = 2 mod 3,
kd = X^c=o (a3c + 2a3c+1). Then, if r = 1 mod 3, kd = ]Cc=o (a3c + a3c+2) and
d ^ kd, so we need only consider r = 2 mod 3.
Similarly, however, consider d = ad. Here kd = (;d + 2ac,+ 1),
and unless r = l 1 mod l, d < kd. Thus, it remains only to consider the case
89
r = 2 mod 3, with l = 0 mod 3 or l = 2 mod 3, and r = l 1 mod l. It
follows immediately then that if l = 2 mod 3, the only remaining value for r is
r = 31 1, while if Z = 0 mod 3, we must also consider r = l 1 and r = 21 1.
Now if r = 3/ 1, let d = 1 + a1, so that Zed = 1 + a + 2al+1 + a2i + a2l+l.
As 2cd+1 = a1, d < kd.
If r = 21 1 and l > 6, consider d = X^c=o aC + a/+3 so that Zed = a +
a2 -f 2a3 + ^=]+i a + Y^2i a + 2/+4- Now 2a3 = a2?+2, 2a2l+2 = a/+1, and
2a/+1 = 1, so that d X Zed. If, on the other hand, i 3, set d = 1 + a + a5, so
that Zed = 1 + a + 2a2 + a4 + a5 + 2a6 + a7. Here, 2ct6 + 2a2 + a7 = a2 + a3,
so that d ^ Zed.
Finally, if r = l 1, consider d = X)c=o aC + cd+1 so that Zed = 2a + a2 + a3 +
al+l+2al+2+a/+3+a21+a2l+1+2a2/+2. Now 2a2l+2 = a, and 3a = a+al, which
does not otherwise appear, while 2ai+2 = a2/+1, and 2a2/+1 = 1, so that d < kd,
as desired, unless 1 = 3. If l = 3, we obtain additionally a 2a21 = a3*-1 = a2i+2
term, and we obtain a 2a; = ai+2 term. However, we still have d < kd.
Thus, a value of d with d ^ Zed was exhibited in all cases.
Our next series of results places restrictions on the values which r may
assume, restricting the value of l as little as possible.
Proposition 4.24 Let k = a + a1 + a-7, where a = 2l, 2 = aT, l ^ i 1,
j = i + l 1, and h = j + 1. If l + 2 < r <21 then @ (Ze) is not a hyperoval
in PG (2,2'*).
Proof: Consider the value d = l + ^Cc=r-z ftC- Then kd = 1 + a + ]Cc=t-m aC +
a1 + Ec=!+r-/ aC + 2aj + cd+1 + Y^cZ]+r-iaC- But 2qJ = aT~l-> so that d^kd.m
90
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2018-04-26 15:23:21
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{"extraction_info": {"found_math": false, "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": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8439754247665405, "perplexity": 2958.7918148409963}, "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}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948285.62/warc/CC-MAIN-20180426144615-20180426164615-00327.warc.gz"}
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http://fossil.include-once.org/upgradephp/info/a33d1695b1c130aa
|
PHP userland backwards compatibility layer that emulates PHP 5.5+ core functions.
## Check-in [a33d1695b1]
Overview
Comment: Variant of var_export that utilizes php 5.4 array [] syntax. family | ancestors | descendants | both | trunk files | file ages | folders a33d1695b1c130aa4239e6898a01d086416cd3b5 mario 2014-08-14 07:22:40
Context
2015-05-07 03:41 Fix runkit json_decode mapping, as reported by Lifeboat Fndt. check-in: 87087bb336 user: mario tags: trunk 2014-08-14 07:22 Variant of var_export that utilizes php 5.4 array [] syntax. check-in: a33d1695b1 user: mario tags: trunk 2014-04-27 17:08 Fix json_encode to always output object keys as strings. check-in: 6c8e7d4e1f user: mario tags: trunk
Changes
> > > > > > > > > > > > > > > > > > > > > > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 $value) {$r[] = "$indent " . ($indexed ? "" : var_export54($key) . " => ") . var_export54($value, "$indent "); } return "[\n" . implode(",\n",$r) . "\n" . $indent . "]"; case "boolean": return$var ? "TRUE" : "FALSE"; default: return var_export($var, TRUE); } } print var_export54(["B"=>1.2,"C"=>TRUE,"D"=>3,"E"=>[4,"\\text \"\$here\r\n\t",6,[7,8]]]);
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2018-09-20 09:18:14
|
{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5756933093070984, "perplexity": 4027.6371022885637}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156423.25/warc/CC-MAIN-20180920081624-20180920101624-00094.warc.gz"}
|
http://tex.stackexchange.com/questions/80338/wrapfigures-and-quote-environment-not-working-together?answertab=active
|
# wrapfigures and quote environment not working together
I want to have a figure floating next to a block quote. So I used the wrapfig package and the quote environment.
If you typeset the following example, you will see that the figure is printed on the next page instead of being next to the block quote. It works perfectly with normal text. Is there any solution to that?
\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{wrapfig}
\begin{document}
\blindtext[3]
\begin{wrapfigure}{R}{0.5\textwidth}
\centering
\rule{5cm}{5cm}
\caption{Foo bar foobar baz baq}
\end{wrapfigure}
\begin{quote}
\blindtext
\end{quote}
\blindtext
\end{document}
-
It seems to work OK using r instead of R (lowercase specifiers don't allow the object to float); I also added some value for the hanging indentation:
\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{wrapfig}
\begin{document}
\blindtext[3]
\begin{wrapfigure}{r}[-10pt]{0.5\textwidth}
\vspace*{\topsep}
\centering
\rule{5cm}{5cm}
\caption{Foo bar foobar baz baq}
\end{wrapfigure}
\begin{quote}
\blindtext
\end{quote}
\blindtext
\end{document}
Notice the additional \vspace*{\topsep} inside wrapfigure; in this case, this is necessary to have the top of the figure aligned with the top of the first line of the quote environment (thanks to barbara beeton for suggesting this improvement).
-
Thanks, but it only works with the indentation! Otherwise I get this. – Ps0ke Nov 4 '12 at 3:07
@Ps0ke I don't understand your comment. My code produces the effect illustrated in the image I attached; if you are still experiencing undesired results, please edit your question and add some complete and minimal document illustrating the problem. – Gonzalo Medina Nov 4 '12 at 3:09
I'm sorry, I must've made a mistake. Your code works perfectly--with or with out the indentaion. – Ps0ke Nov 4 '12 at 3:16
@Ps0ke ah, OK then. – Gonzalo Medina Nov 4 '12 at 3:24
@GonzaloMedina -- is there any way to have the top of the figure aligned with the top of the first line of the quote? it would look nicer. – barbara beeton Nov 4 '12 at 14:10
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2014-07-23 14:34:05
|
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|
https://math.stackexchange.com/questions/437234/gp-1-4-4-an-extension-of-partial-converse-of-preimage-theorem
|
# GP 1.4.4 An extension of partial converse of preimage theorem.
This is exercise 1.4.4 on Guillemin and Pollack's Differential Topology
Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Then there exist independent functions $g_1, \dots, g_l$, on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W = \{y \in W : g_1(y) = 0, \dots, g_l(y) = 0\},$$ $$X \cap W = \{y \in W : g_i(y) = 0, \dots , g_m(y) = 0\},$$ where $l-m$ is the codimension of $Z$ in $X$.
I tried to set up the proof as following:
Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Let $Z$ and $X$ have codimensions $l$ and $m$ in $Y$, $Z$ has codimension $l-m$ in $X$. From the partial converse to the preimage theorem, there exist independent functions $f_1, \dots f_m$ on a neighborhood $U$ of $z$ in $Y$ such that $X \cap U$ is the common vanishing set of the $f_i$.
We also know that there exist independent functions $h_{m+1}, \dots, h_l$ on a neighborhood $V$ of $z$ in $X$ such that $Z \cap V$ is the common vanishing set of the $h_i$.
And then I don't know why $h_i$s are smooth, and how should I continue.
Any ideas? Thank you.
• why you assumed that $Z$&$X$ have codimensions $l$ and $m$ in Y? – Idonotknow Oct 9 '18 at 14:23
• In the question in the second intersection it is $g_{1}$ not $g_{i}$ – hopefully Oct 17 '18 at 13:05
tl;dr: straighten the neighborhood in $Y$ so that $Z$ looks flat, straighten the neighborhood in $Z$ so that $X$ looks flat, and extend the latter straightening some way into $Y$.
I don't know the theorem names off the top of my head, but here's how I'd approach this:
WLOG $z=0$ and $Y = \mathbb{R}^n$ for some $n$. Since $Z$ is a smooth submanifold of codimension $l$, there is some smooth $f\times g\colon U\cong V\times W$ where $U\subseteq\mathbb{R}^n$, $V\subseteq\mathbb{R}^l$, $W\subseteq\mathbb{R}^{n-l}$, and for $x\in U$, $x\in Z$ iff $f(x)=0$. (I want to say this is the domain-straightening theorem?) The neighborhood of $x$ in $Z$ looks like $W$, so we'll identify the two for convenience.
Now $X$ is a smooth submanifold of $Z$ (codimension $l$), so by the same theorem, after possibly shrinking $W$ (and $U$ to match), there is some smooth $h\times k\colon W\cong S\times T$ where $S\subseteq\mathbb{R}^{l-m}$, $T\subseteq\mathbb{R}^{n-m}$, and for $x\in W$, $x\in X$ iff $h(x)=0$.
Of the $g_i$, the first $l$ are the components of $f$, and the remaining $m-l$ are the components of $h\circ g$. Apologies for the notational weirdness.
• Could you provide more details please? – Idonotknow Oct 9 '18 at 17:40
• what do you mean by "so we'll identify the two for convenience."? sorry my mother tongue is not English. – hopefully Oct 17 '18 at 10:19
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2019-04-20 17:06:40
|
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|
http://www.koreascience.or.kr/article/JAKO200922335514059.page
|
VERIFICATION OF A PAILLIER BASED SHUFFLE USING REPRESENTATIONS OF THE SYMMETRIC GROUP
• Cho, Soo-Jin (DEPARTMENT OF MATHEMATICS AJOU UNIVERSITY) ;
• Hong, Man-Pyo (DEPARTMENT OF INFORMATION AND COMPUTER ENGINEERING AJOU UNIVERSITY)
• Published : 2009.07.31
Abstract
We use an idea of linear representations of the symmetric group to reduce the number of communication rounds in the verification protocol, proposed in Crypto 2005 by Peng et al., of a shuffling. We assume Paillier encryption scheme with which we can apply some known zero-knowledge proofs following the same line of approaches of Peng et al. Incidence matrices of 1-subsets and 2-subsets of a finite set is intensively used for the implementation, and the idea of $\lambda$-designs is employed for the improvement of the computational complexity.
References
1. M. Abe, Mix-networks on permutation networks, Advances in cryptology-ASIACRYPT '99 (Singapore), 258–273, Lecture Notes in Comput. Sci., 1716, Springer, Berlin, 1999
2. P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, London Mathematical Society Student Texts, 22. Cambridge University Press, Cambridge, 1991
3. D. Chaum, Untraceable electronic mail, return addresses, and digital pseudonyms, Commun. ACM 24 (1981), no. 2, 84–88 https://doi.org/10.1145/358549.358563
4. S. Cho and M. Hong, Proving a shuffle using representations of the symmetric group, ICISC 2008 (P. J. Lee and J. H. Cheon, eds.), 354–367, Lecture Notes in Computer Science, vol. 5461, Springer, 2009 https://doi.org/10.1007/978-3-642-00730-9_22
5. G. Danezis, Mix-networks with restricted routes, Privacy Enhancing Technologies (Roger Dingledine, ed.), 1–17, Lecture Notes in Computer Science, vol. 2760, Springer, 2003 https://doi.org/10.1007/b94512
6. G. Danezis, R. Dingledine, and N. Mathewson, Mixminion: Design of a type iii anonymous remailer protocol, IEEE Symposium on Security and Privacy, 2–15, IEEE Computer Society, 2003
7. Y. Desmedt and K. Kurosawa, How to break a practical mix and design a new one, EUROCRYPT, 557–572, 2000 https://doi.org/10.1007/3-540-45539-6_39
8. C. Diaz, S. Seys, J. Claessens, and B. Preneel, Towards measuring anonymity, in Dingledine and Syverson [11], pp. 54–68
9. R. Dingledine, M. J. Freedman, D. Hopwood, and D. Molnar, A reputation system to increase mix-net reliability, Information Hiding (Ira S. Moskowitz, ed.), 126–141, Lecture Notes in Computer Science, vol. 2137, Springer, 2001 https://doi.org/10.1007/3-540-45496-9_10
10. R. Dingledine, N. Mathewson, and P. F. Syverson, Tor: The second-generation onion router, USENIX Security Symposium, 303–320, USENIX, 2004
11. R. Dingledine and P. F. Syverson (eds.), Privacy enhancing technologies, Second international workshop, pet 2002, san francisco, ca, usa, april 14-15, 2002, revised papers, Lecture Notes in Computer Science, vol. 2482, Springer, 2003 https://doi.org/10.1007/3-540-36467-6
12. P. Frankl, Intersection theorems and mod p rank of inclusion matrices, J. Combin. Theory Ser. A 54 (1990), no. 1, 85–94 https://doi.org/10.1016/0097-3165(90)90007-J
13. W. Fulton and J. Harris, Representation Theory, A First Course, Graduate Texts in Mathematics 129, Springer 1991
14. J. Furukawa and K. Sako, An efficient scheme for proving a shuffle, Advances in cryptology-CRYPTO 2001 (Santa Barbara, CA), 368–387, Lecture Notes in Comput. Sci., 2139, Springer, Berlin, 2001 https://doi.org/10.1007/3-540-44647-8_22
15. E.-J. Goh, Encryption schemes from bilinear maps, Ph. D. thesis, Department of Computer Science, Stanford University, Sep. 2007
16. P. Golle, M. Jakobsson, A. Juels, and P. F. Syverson, Universal re-encryption for mixnets, Topics in cryptology-CT-RSA 2004, 163–178, Lecture Notes in Comput. Sci., 2964, Springer, Berlin, 2004
17. P. Golle, S. Zhong, D. Boneh, M. Jakobsson, and A. Juels, Optimistic mixing for exitpolls, Advances in cryptology-ASIACRYPT 2002, 451–465, Lecture Notes in Comput. Sci., 2501, Springer, Berlin, 2002 https://doi.org/10.1007/3-540-36178-2_28
18. J. Groth, A verifiable secret shuffle of homomorphic encryptions, Public key cryptography-PKC 2003, 145–160, Lecture Notes in Comput. Sci., 2567, Springer, Berlin, 2002 https://doi.org/10.1007/3-540-36288-6_11
19. J. Groth and S. Lu, Verifiable shuffle of large size ciphertexts, Public key cryptography-PKC 2007, 377–392, Lecture Notes in Comput. Sci., 4450, Springer, Berlin, 2007 https://doi.org/10.1007/978-3-540-71677-8_25
20. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992
21. M. Mitomo and K. Kurosawa, Attack for flash MIX, Advances in cryptology-ASIACRYPT 2000 (Kyoto), 192–204, Lecture Notes in Comput. Sci., 1976, Springer, Berlin, 2000 https://doi.org/10.1007/3-540-44448-3_15
22. C. A. Neff, A verifiable secret shuffle and its application to e-voting, ACM Conference on Computer and Communications Security, 116–125, 2001 https://doi.org/10.1145/501983.502000
23. L. Nguyen, R. Safavi-Naini, and K. Kurosawa, Verifiable shuffles: A formal model and a Paillier-based efficient construction with provable security, ACNS (Markus Jakobsson, Moti Yung, and Jianying Zhou, eds.), 61–75, Lecture Notes in Computer Science, vol. 3089, Springer, 2004
24. W Ogata, K Kurosawa, K Sako, and K Takatani, Fault tolerant anonymous channel, Proc. ICICS '97, 440–444, Lecture Notes in Comput. Sci., 1334, Springer-Verlag, 1997 https://doi.org/10.1007/BFb0028500
25. P. Paillier, Public-key cryptosystems based on composite degree residuosity classes, Advances in cryptology-EUROCRYPT '99 (Prague), 223–238, Lecture Notes in Comput. Sci., 1592, Springer, Berlin, 1999 https://doi.org/10.1007/3-540-48910-X_16
26. C. Park, K. Itoh, and K. Kurosawa, Efficient anonymous channel and all/nothing election scheme, Advances in cryptology-EUROCRYPT '93 (Lofthus, 1993), 248–259, Lecture Notes in Comput. Sci., 765, Springer, Berlin, 1994 https://doi.org/10.1007/3-540-48285-7_21
27. K. Peng, C. Boyd, and E. Dawson, Simple and efficient shuffling with provable correctness and ZK privacy, Advances in cryptology-CRYPTO 2005, 188–204, Lecture Notes in Comput. Sci., 3621, Springer, Berlin, 2005 https://doi.org/10.1007/11535218_12
28. K. Peng, C. Boyd, E. Dawson, and K. Viswanathan, A correct, private, and efficient mix network, Public key cryptography-PKC 2004, 439–454, Lecture Notes in Comput. Sci., 2947, Springer, Berlin, 2004
29. B. Pfitzmann and A. Pfitzmann, How to break the direct RSA-implementation of mixes, EUROCRYPT, 373–381, 1989 https://doi.org/10.1007/3-540-46885-4_37
30. B. Pfitzmann, M. Schunter, and M. Waidner, How to break another provably secure payment system, EUROCRYPT, 121–132, 1995
31. B. E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991
32. A. Serjantov and G. Danezis, Towards an information theoretic metric for anonymity, in Dingledine and Syverson [11], pp. 41–53
33. D. Wikstrom, A sender verifiable mix-net and a new proof of a shuffle, Advances in cryptology-ASIACRYPT 2005, 273–292, Lecture Notes in Comput. Sci., 3788, Springer, Berlin, 2005 https://doi.org/10.1007/11593447_15
34. R. M. Wilson, A diagonal form for the incidence matrices of t-subsets vs. k-subsets, European J. Combin. 11 (1990), no. 6, 609–615
|
2021-09-21 19:23:12
|
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|
https://www.physicsforums.com/threads/algae-carbon-capture.1004876/
|
Algae carbon capture
• Chemical/Paint
CCatalyst
TL;DR Summary
I want to save the planet.
I'm just devastated by the news of climate change as of late, and I knew it was worrying but I've never been more worried about it in my life than I am right now. Have we truly passed the point of no return? Are we all doomed?
Or do we still have time? Is algae carbon capture the way out? And also how do I turn into biofuel?
So I want to make my own algae tank and use it to absorb carbon dioxide. Any advise and suggestions? And how do I get started?
2milehi and Ivan Seeking
Summary:: I want to save the planet.
I want to make my own algae tank and use it to absorb carbon dioxide.
At individual level, there is very little chance that this will yield negative carbon footprint.
I suggest you plant some trees instead.
hutchphd, Astronuc and Ivan Seeking
Staff Emeritus
Gold Member
Summary:: I want to save the planet.
I'm just devastated by the news of climate change as of late, and I knew it was worrying but I've never been more worried about it in my life than I am right now. Have we truly passed the point of no return? Are we all doomed?
Or do we still have time? Is algae carbon capture the way out? And also how do I turn into biofuel?
So I want to make my own algae tank and use it to absorb carbon dioxide. Any advise and suggestions? And how do I get started?
Firstly, there is always hope. The choices are now more about how to best manage problems and through what means. Now we are likely looking at geoengineering solutions and not just reduced emissions.
I agree with Rive. In fact I actually started a company with an impressive team, with the intent to start producing fuel from algae at a competitive price, back around 2008. But there was still so much costly research needed it just wasn't practical for a small company to make a go of it. And one thing that becomes abundantly clear as one tries to design practical applications for this, is the scale of the problem. In my own model, I had to go to a 50,000 acre site before I could expect profitability. It is not an easy problem. And algae farming is not for amateurs.
Think globally. Act locally. Figure out how you can best contribute to the solution as part of the big picture.
hutchphd and Spinnor
Gold Member
More than ten years ago I visited an algae growing aquaculture facility in an arid area in California somewhere (don't remember where).
Its function was to make Spirulina algae for food production (humans and other things).
Besides the water maintaining equipment, it was mostly an oval shaped raceway of circulating water, maybe 20-50 feet wide but only a a foot or two deep. (The color was very green and the light probably wouldn't penetrate very far.)
The water circulation (due to paddle-wheel) keeps the algae in suspension. Otherwise it would create production problems.
This looks like a cement lined pond, but it might be cheaper to just dig a depression and use a pond liner.
This would be far more efficient than a set-up that does not use natural light.
It was a big set-up for the amount production, but was economically feasible due to the prices of the product, which would be hard to get in large pure quantities by other means.
I can't see something like this having much impact on global CO2 in the air.
When it comes to making fuel, it might be economically feasible, depending on the fuel produced.
One advantage of algae is that it can be fairly easily engineered genetically (being single cells). This could lead to production benefits.
Staff Emeritus
Gold Member
More than ten years ago I visited an algae growing aquaculture facility in an arid area in California somewhere (don't remember where).
Its function was to make Spirulina algae for food production (humans and other things).
Besides the water maintaining equipment, it was mostly an oval shaped raceway of circulating water, maybe 20-50 feet wide but only a a foot or two deep. (The color was very green and the light probably wouldn't penetrate very far.)
The water circulation (due to paddle-wheel) keeps the algae in suspension. Otherwise it would create production problems.
View attachment 285599
This looks like a cement lined pond, but it might be cheaper to just dig a depression and use a pond liner.
This would be far more efficient than a set-up that does not use natural light.
It was a big set-up for the amount production, but was economically feasible due to the prices of the product, which would be hard to get in large pure quantities by other means.
I can't see something like this having much impact on global CO2 in the air.
When it comes to making fuel, it might be economically feasible, depending on the fuel produced.
One advantage of algae is that it can be fairly easily engineered genetically (being single cells). This could lead to production benefits.
That is a classic raceway. The problem with using Spriulina and other nutritional supplments as a comparison to algae for fuel is that the end product for Spirulina sells for around $15 for 500 mg [$30 a gram]. Algae fuel needs to sell for [today] $4 for 7.2 pounds [one gallon]. It is a very different problem by two orders of magnitude. I am still convinced that algae is key to a renewable energy future. However, we need to grow about as much algae [by area] as we do corn. That is in the neighborhood of 90 million acres. Some companies were trying to start giant algae plumes in the ocean to offset the CO2 they produce in industry. I don't know the state of this. Given that ideally you can produce as much as 5000 gallons of fuel per acre-year of algae growth, that also suggests an upper limit for the CO2 one acre of algae can absorb in one year. Most of the carbon absorbed goes to either fatty acids [used for biodiesel] or sugar [used for ethanol]. Last edited: BillTre Staff Emeritus Science Advisor Gold Member Staff Emeritus Science Advisor Gold Member Spirulina sells for around$15 for 500 mg [$30 a gram]. Algae fuel needs to sell for [today]$4 for 7.2 pounds [one gallon]. It is a very different problem by two orders of magnitude.
Ack! Memory error plus a typo. Spirulina now sells for as little at $10 a pound but the point is the same. The markup in some "health foods" is crazy. BillTre Science Advisor Gold Member Thought that price was a bit high. Spirulina is a pretty pure product as far as I have seen. Maybe this is not such an issue in fuel production? Algae blooms can be made in some conditions just by adding iron to the water (which is pretty cheap I've been told). There are two arguments going on. One concerns doing things that would have a positive effect on climate change. The other is non-extractive fuel production. I guess that's because on the bipartite nature of this thread. I can see how this would remove CO2 from the atmosphere, and perhaps eventually sequestered to the bottom of the oceans, as dead things that have eaten the algae. However, if the carbon then just burned as fuel, I'm not sure it does anything for carbon balance. How you could harvest fuel from the algae you release into the ocean? Other than plankton netting or net penning the whole area where the algae are growing. That would not be popular with a lot of people. Done in tanks on land, these problems would not be problems. Staff Emeritus Science Advisor Gold Member Thought that price was a bit high. Spirulina is a pretty pure product as far as I have seen. Maybe this is not such an issue in fuel production? Algae blooms can be made in some conditions just by adding iron to the water (which is pretty cheap I've been told). There are two arguments going on. One concerns doing things that would have a positive effect on climate change. The other is non-extractive fuel production. I guess that's because on the bipartite nature of this thread. I can see how this would remove CO2 from the atmosphere, and perhaps eventually sequestered to the bottom of the oceans, as dead things that have eaten the algae. However, if the carbon then just burned as fuel, I'm not sure it does anything for carbon balance. How you could harvest fuel from the algae you release into the ocean? Other than plankton netting or net penning the whole area where the algae are growing. That would not be popular with a lot of people. Done in tanks on land, these problems would not be problems. Purity is a HUGE issue in fuel production. Firstly, most strains are not sufficient for producing fuel. And strains are being hybridized to maximize the yields. Strains tend to mutate from good producers to poor producers. So maintaining a high-yield strain is challenging. Also, algae is highly vulnerable to invasive algae species, bacteria, viruses, and parasites. These are but a few of the challenges in producing fuel from algae. Wild strains left to grow naturally will never produce enough fuel per acre-year to be competitive at the pump. For CO2 sequestration, the idea is to do what nature has done for millions of years. Algae in the ocean absorb CO2 to grow. They eventually die and sink in the deep ocean where it is preserved by the high pressure and low temperatures until eventually turned over and buried by tectonic motion. It turns out that a significant percentage of petroleum actually has it's origins in ancient algae. When used as a fuel, algae fuels can be carbon neutral. They can only release as much carbon as was absorbed to grow. Due to the vast amounts of water required to grow algae for fuel production, [note the number of hydrogen atoms per molecule of biodiesel] the only viable option I see is to grow the algae in closed bioreactors [essentially giant baggies] in the ocean and possibly large lakes. Abandoned oil platforms in the Gulf of Mexico would make excellent hubs for oceanic algae farms. And the infrastructure for fuel production is already in place. Ocean farming also helps to reduce the problem of excess heat, which is a challenge for land-based algae farming. Open raceways like that used for Spirulina are not acceptable for fuel production. They will certainly be closed batch systems. The purity of the strain could never be maintained for long in an open. continuous production system. And even running a paddle wheel in a configuration like that could eat up most of your energy gains from the algae produced. The key to starting an algae bloom is to dump vast amounts of nitrogen into the water. In fact nitrogen accounts of a significant percentage of the cost of farming algae. Runoff from farms and other sources rich in nitrogen such as sewage, often spawn wild algae blooms. Also, algae can be used to remediate nitrogen pollution in a controlled manner, rather than allowing wild blooms to choke off the oxygen supply for wildlife. When I first started looking into algae for fuel, perhaps as far back in 2006, biodiesel from algae cost up to$50 a gallon to produce. I have read recently that some producers are claiming to have broken the $5 per gallon barrier. Last edited: BillTre CCatalyst With all due respect, shouldn't removing carbon dioxide be of a higher priority than creating fuel? Sure it may be expensive but wouldn't it be worth it if it saves the whole planet? Also, couldn't we use algae and some other renewable in tandem? We would use solar and/or wind to circulate the algae and run the pumps to collect carbon dioxide and remove the oxygen. So couldn't that solve the problem? Also I think the sources (or at least the sources I've read) state that algae is a better if not the best at absorbing carbon dioxide. Or how well does it compare to other carbon capture methods? I know this may take some R&D but the fossil fuel industry had 50 years to do so. If they shared our concern they would have solved it by now. Instead they let their greed blind them and now we are looking down the barrel of the next mass extinction. We cannot let this happen. Ivan Seeking Science Advisor With all due respect, shouldn't removing carbon dioxide be of a higher priority than creating fuel? No. The first priority right now is to reduce carbon footprint. Any 'remove carbon' is just empty word till actual usage depends on fossil carbon and overall footprint is vastly positive. CCatalyst No. The first priority right now is to reduce carbon footprint. Any 'remove carbon' is just empty word till actual usage depends on fossil carbon and overall footprint is vastly positive. But won't removing carbon dioxide be an example of reducing our carbon footprint? And doesn't algae absorb carbon dioxide faster than trees? Answer this for me, how well does algae absorb carbon dioxide in comparison to oh say, magnesium oxide? Staff Emeritus Answer this for me, how well does algae absorb carbon dioxide in comparison to oh say, magnesium oxide? How about in comparison to a tree? Last edited: Rive and berkeman CCatalyst Mentor Note -- As a reminder, this thread is going okay so far, but please keep in mind the current PF rules about Climate Change discussions: https://www.physicsforums.com/threads/climate-change-global-warming-policy.757267/ Thanks. Fair enough, I think we can all accept that climate change IS real and it IS the result of human activity. After all we cannot fix the problem if we do not think it is happening. THAT debate should be over, we should be discussing how to fix it. How about in comparison to a tree? You tell me, that's why I'm here. But from a quick Google search, I was able to determine that algae can absorb 1 to 3 times its weight in carbon dioxide. Besides, don't trees have a very low metabolism? And aren't they burning causing less of them to be available to recycle the carbon dioxide? I'm not saying we should not be planting any trees, please do. I just don't think it will be enough. Can we plant trees faster than they are being burned down? Good luck with that. Is it not true that if we were to bring our net carbon dioxide output to zero, we would still be past the point of no return? The carbon dioxide in the atmosphere is still causing elevated temperatures, resulting in more forest fires, resulting in more carbon dioxide, resulting in more elevated temperatures, resulting in more forest fires and so on? Mentor and it IS the result of human activity. That is not what the PF rules say. we should be discussing how to fix it. Sure. Science Advisor we should be discussing how to fix it. But this particular topic seem to be about something different: Summary:: I want to save the planet. So I want to make my own algae tank and use it to absorb carbon dioxide. You were provided examples, hints and references about the difficulties regarding usage of algae. At home, there is no way you can make this even just comparable to trees. But from a quick Google search, I was able to determine that algae can absorb 1 to 3 times its weight in carbon dioxide. The end product of algae 'gardening' is a stinky, wet and 'dirty' goo. Especially if you do this at home. It requires quite amount of processing till it becomes either some kind of alcohol and/or some kind of oil/fat, ready for storage. Without processing (and circulating the waste), it quickly depletes the available nutrients, what you provided with quite a carbon footprint. Compared to the straightforward carbohydrate production of trees, it's just far more troublesome. Especially if you do this at home. If done big, that's a slightly different matter but still: the end product is 'dirty'. It's best to use it in an industry where this 'dirt' is called 'nutrients' instead and highly valued - food. But then it is no longer about carbon capture. Cool down and instead of pressing this dead end just listen to others. Last edited: BillTre and berkeman Science Advisor Gold Member Hi Ivan, I remember a long thread you had about this. Must have been on PF version 1. Maybe @Greg Bernhardt can find it. Ivan Seeking CCatalyst If done big, that's a slightly different matter but still: the end product is 'dirty'. It's best to use it in an industry where this 'dirt' is called 'nutrients' instead and highly valued - food. But then it is no longer about carbon capture. Cool down and instead of pressing this dead end just listen to others. Well fine but take a look at this. Algae might be a secret weapon to combating climate change It says that algae can be 400 times more effective than trees at absorbing carbon dioxide. Now everyone keeps saying it is difficult to scale up. But why is that exactly? Plus some people are making their own biodiesel from algae. Staff Emeritus Science Advisor Gold Member Hi Ivan, I remember a long thread you had about this. Must have been on PF version 1. Maybe @Greg Bernhardt can find it. Not that long ago. :) https://www.physicsforums.com/threads/algae-to-the-rescue.211274/ Near the end I spill my guts about some things we learned while researching this and in countless hours of discussions with experts. I had my company for about two years before we decided the cost of development was still just too high for a small company. The scale of the problem just keeps growing and growing, forgive the pun. Even ignoring much of the development cost, the first profitable point I could find using my model was a 50,000 acre farm. Note also that in addition to an algae biologist and a noted land use expert, I managed to recruit the chemist who was the nation's leading expert on biodiesel at the time. We had quite a team! This was a serious effort. I was most proud my solution for nitrogen. In short, nitrogen is a significant cost of farming algae. It turns out that diesel engines used to power the farm could be used to produce free nitrogen. I did the calculations and it appears that as much as 100% of the nitrogen might be produced onsite that way. Diesel engines are mass producers of oxides of nitrogen, which make acid rain- also known as nitrogen fertilizer. This also allow for much higher compression ratios in the engines, and therefore higher efficiencies in running the farm. Note that compression ratios are artificially limited to REDUCE the production of oxides of nitrogen. But we want to crank that up as high as we can and then use the nitrogen. There also are other tricks like increasing the size of the fuel droplets, and advancing the timing. In fact when you work through the practical aspects of farming algae, it becomes clear that the diesel engines can satisfy several critical roles and end up as the heart of the operation in a number of ways. As I said, I discuss all of this near the end of the thread linked. Last edited: dlgoff and berkeman Science Advisor Gold Member Not that long ago. :) https://www.physicsforums.com/threads/algae-to-the-rescue.211274/ Near the end I spill my guts about some things we learned while researching this and in countless hours of discussions with experts. I had my company for about two years before we decided the cost of development was still just too high for a small company. The scale of the problem just keeps growing and growing, forgive the pun. Even ignoring much of the development cost, the first profitable point I could find using my model was a 50,000 acre farm. I was most proud my solution for nitrogen. In short, nitrogen is a significant cost of farming algae. It turns out that diesel engines used to power the farm could be used to produce free nitrogen. I did the calculations and it appears that as much as 100% of the nitrogen might be produced onsite that way. Diesel engines are mass producers of oxides of nitrogen, which make acid rain- also known as nitrogen fertilizer. This also allow for much higher compression ratios and higher efficiencies in running the farm. Note that compression ratios are artificially limited to REDUCE the production of oxides of nitrogen. But we want to crank that up as high as we can and then use the nitrogen. In fact when you work through the practical aspects of farming algae, it becomes clear that the diesel engines can satisfy several critical roles and end up as the heart of the operation in a number of ways. As I said, I discuss all of this near the end of the thread linked. Thanks for the link Ivan. I've now got it bookmarked. That was back in 2008; a long time IMO. Ivan Seeking Staff Emeritus Science Advisor Gold Member Well fine but take a look at this. Algae might be a secret weapon to combating climate change It says that algae can be 400 times more effective than trees at absorbing carbon dioxide. Now everyone keeps saying it is difficult to scale up. But why is that exactly? Plus some people are making their own biodiesel from algae. Making fuel from algae is fairly easy. Making fuel from algae at a price that can compete with petroleum is extremely difficult. Most of these DIYers are probably paying$20-$30 a gallon for their fuel. When we can compete with the price of petroleum at the pump, we can easily convert to carbon-neutral fuels from algae. It is ALL about price. dlgoff Staff Emeritus Science Advisor Gold Member If done big, that's a slightly different matter but still: the end product is 'dirty'. The idea of starting huge blooms in the ocean is interesting. But the drawback is that a highly productive bloom, meaning one active enough to absorb vast quantities of carbon, also depletes the O2 from the water and creates a dead zone. I don't know if moderated algae growth is a reasonable option or not. I've never looked at the math for a sustainable bloom. We ruled out continuous production and opted for a high-yield batch process early on. One of the biggest mistakes DIYers make is they allow the algae to go anaerobic. That is when you get a big stinky mess. My algae all smelled like the fertilizer or it was mostly odorless. But you have to provide a great deal of aeration. Spinnor and BillTre Science Advisor Making fuel from algae is fairly easy. Making fuel from algae at a price that can compete with petroleum is extremely difficult. Well, I'll believe those youtubers only if/when they can take out their family car for a spin running only on that. I mean: repeatedly and without consequences. Till that it's just some lamp oil or so. And then, the carbon footprint. One would think that the lesson of other biofuels (many of them with bigger carbon footprint than classic oil) got learned by now. The idea of starting huge blooms in the ocean is interesting. The idea really looked like a big life saver back then. It's just when the few nutrients on short supply got boosted, the sinking algae/plankton from the bloom carried everything to the bottom, creating a short supply of all nutrients. So at the end you need to resupply all nutrients to keep the bloom go on... I still think the idea could be utilized somehow, but the recirculation of nutrients must be solved first, and we should somehow limit the harvest to the good old CHON(PS)... Feels so easy to talk about it Ivan Seeking and BillTre Staff Emeritus Science Advisor Gold Member Well, I'll believe those youtubers only if/when they can take out their family car for a spin running only on that. I mean: repeatedly and without consequences. Till that it's just some lamp oil or so. haha true. But it is fairly easy to make some oil... at$30 a gallon
And then, the carbon footprint. One would think that the lesson of other biofuels (many of them with bigger carbon footprint than classic oil) got learned by now.
This was one of the critical variables I found in researching all of this. Almost everyone I studied did not account for the energy required to grow the algae. I found that to be an incredibly difficult problem to beat. We need algae that produces 25-30% or more fuel by weight [either oil or sugar] for processing efficiency. But growing high-yield algae takes energy. Even running a paddle wheel as is typically done for raceways could kill most of the energy benefit of a batch. So design of the photobioreactors as well as the design of the farm are energy critical. Aeration is another big one; especially when you consider that you really need HEPA filters to avoid contamination. Even the energy required to process the nutrients such as nitrogen fertilizer, become important. That was another reason my solution for nitrogen was significant.
I was finally able to produce a model that was energy net positive but it was painstaking. In the end I was convinced that net 2000-2500 gallons per acre year was doable.
Rive and BillTre
But it is fairly easy to make some oil... at \$30 a gallon
Yeah, if somebody has some talent for chemistry then it can be done even from chickens
I was finally able to produce a model that was energy net positive but it was painstaking.
Quite a feat, I would say
Really rare to care about such 'small' details.
Ivan Seeking
Staff Emeritus
Gold Member
And then, the carbon footprint. One would think that the lesson of other biofuels (many of them with bigger carbon footprint than classic oil) got learned by now.
I sort of derailed this thought with my comments about energy. But they are essentially the same problem. If the system is truly energy net positive then it is likely carbon neutral. By closing the system and only powering the farm using fuel produced onsite, the idea of stealing energy and leaving a carbon footprint is mostly moot. But all inputs to the system have to be considered. And nitrogen was a big one!
And as it turns out, in my own efforts the generator engines become critical components of the system in several ways. For example, in addition to the nitrogen supply for fertilizer from the NOxs produced, we can capture additional CO2 needed to accelerate algae growth. The engine's high temps and pressures also act as an air purifier and kills any potential biological contamination. Everything coming out of the exhaust can be captured and recycled for the next batch of algae. This all helps to eliminate or reduce energy losses.
Keith_McClary and Rive
Staff Emeritus
Gold Member
Note also that the generator exhaust is significant. In all likelihood, half of the fuel produced is needed just to run the farm. So on the average, half of everything from the last crop is being burned in the generator engines and fed to the current crop as nutrients.
It may be reasonable to use the biomass [fiber] from the processed algae as another source of nutrients and energy through combustion. But that is also a high-quality feed for cattle. Either way an algae farm at scale will produce vast quantities of biomass that can be used or sold.
It may be reasonable to use the biomass [fiber] from the processed algae as another source of nutrients and energy through combustion. But that is also a high-quality feed for cattle. Either way an algae farm at scale will produce vast quantities of biomass that can be used or sold.
Difficulties at every corner. If you remove biomass from the loop, you need to replace the relevant nutrients (at a cost of increased footprint).
If you don't utilize the surplus biomass, then you run on deficit.
It may look good on first sight to incorporate the cattle into the loop, but bringing back the nutrients through manure into the loop is difficult since you have to avoid the biological contamination of your pools.
Compost heaps and the heterogeneous nature of soil life can cover up so many trouble in classical agriculture
Last edited:
Staff Emeritus
Gold Member
Difficulties at every corner. If you remove biomass from the loop, you need to replace the relevant nutrients (at a cost of increased footprint).
That is mostly carbon, oxygen, and hydrogen. So again the biggest concern is a water supply.
Staff Emeritus
Gold Member
It may look good on first sight to incorporate the cattle into the loop, but bringing back the nutrients through manure into the loop is difficult since you have to avoid the biological contamination of your pools.
Compost heaps and the heterogeneous nature of soil life can cover up so many trouble in classical agriculture
Yes, the use of something like animal fertilizer is impractical. The bacteria would run amok and contaminate the entire system. If the fertilizer is sterilized then you have probably just killed your energy budget.
Algae can be used to remediate contamination but that likely isn't going to be a high-yield strain.
That is mostly carbon, oxygen, and hydrogen.
Cattle just burn carbohydrates, but does not really live on it: the nutrients (missing from the pool) makes the difference.
If you can have (only) carbohydrates (you can sufficiently separate them), then maybe you can aim for alcohols instead, and keep the (still mostly sterile) nutrients within the loop.
Staff Emeritus
Gold Member
Cattle just burn carbohydrates, but does not really live on it: the nutrients (missing from the pool) makes the difference.
If you can have (only) carbohydrates (you can sufficiently separate them), then maybe you can aim for alcohols instead, and keep the (still mostly sterile) nutrients within the loop.
On the fatty acid side of things, some groups were having luck migrating the oil out of the algae without having to kill the algae. I believe they were using ultrasound. I have not heard of anything along those lines for the sugar. Also, I don't know how that affected later yields.
The biomass remaining after removing either sugar or oil is a source of protein.
Seaweed and microalgae are considered a viable source of protein. Some species of seaweed and microalgae are known to contain protein levels similar to those of traditional protein sources, such as meat, egg, soybean, and milk [3,4]. Algae use for protein production has several benefits over traditional high-protein crop use in terms of productivity and nutritional value. Seaweed and microalgae have higher protein yield per unit area (2.5–7.5 tons/Ha/year and 4–15 tons/Ha/year, respectively) compared to terrestrial crops, such as soybean, pulse legumes, and wheat (0.6–1.2 tons/Ha/year, 1–2 tons/Ha/year, and 1.1 tons/Ha/year, respectively) [5]. Terrestrial agriculture already requires approximately 75% of the total global freshwater with animal protein in particular requiring 100 times more water than if the equivalent amount of protein was produced from plant sources [6,7].
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5447909/
Staff Emeritus
Gold Member
Is it not true that if we were to bring our net carbon dioxide output to zero, we would still be past the point of no return? The carbon dioxide in the atmosphere is still causing elevated temperatures, resulting in more forest fires, resulting in more carbon dioxide, resulting in more elevated temperatures, resulting in more forest fires and so on?
I strongly suspect that the first, or one of the first attempts at geoengineering to reduce warming, will be to release substances into the upper atmosphere that will reflect some percentage of sunlight away before it can cause any warming of the planet. I remember that Alcoa Aluminum had a patent on a bright-white aluminum oxide powder that can be added to jet fuel. The jets fly at 30,000+ feet and release the powder in the engine exhaust. At that point the powder is light enough to stay aloft for up to two years or so. I did a quick search to find a report on that and didn't spot it yet.
What follows is from a recent effort to explore this technology. Environmental concerns have temporarily stopped the testing planned but this is just one example. This idea has been debated and planned for probably 30 years now.
Scopex is intended to better understand one form of solar geoengineering: injecting substances into the air to reflect some of the sun’s rays back to space and thus reduce global warming relatively quickly.
Solar geoengineering has long been a subject of intense debate among scientists and policymakers, often seen as a desperate, potentially dangerous measure that could have unintended consequences
https://www.nytimes.com/2021/04/02/climate/solar-geoengineering-block-sunlight.html
We may soon decide that the risk of not reducing warming is greater than the risk of spreading aluminium oxide, or other compounds, all over the planet.
https://www.space.com/global-warming-aerosol-reflector-block-sunlight
Last edited:
dlgoff
Staff Emeritus
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2022-12-04 18:39:06
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https://superuser.com/questions/259353/7-zip-windows-7-make-extract-to-folder-default-on-double-click
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# 7-zip & Windows 7: Make “Extract to <folder>” default on double-click
I'm trying to find a way to make the action you can perform from the context menu, "Extract to <folder_same_as_file_name>" the default action when double-clicking the file instead of simply launching 7-zip. Is there a simple way to do this?
In the alternative, I gather I could try passing parameters into the following:
7z x <filename> -o<filename>
But I'm not sure how to set this up (how to pass the filename parameter, and can I do this directly or will I have to write a batch file instead and pass the filename to it? The latter I find irritatingly unelegant, but whatever works.
Unfortunately, afrazier's batch program method won't work; Windows doesn't handle opening multiple files like that. When you try to open multiple files with a program, Windows doesn't open a single instance of the program and pass the files as multiple arguments to that one instance. Instead, Windows opens many instances of the program (as many instances as there are files), passing one file to each instance. It would be nice if you could just use %* and pass a bunch of files to a single .bat, and have that .bat run a loop processing each file one at a time, but unfortunately you can only use %1 when setting these kinds of actions in the registry.
Someone with some time on their hands could write a program that uses a mutex object to check if there is another instance already running, and if there is, to pass it's file to that instance and then close, whereon the original instance will put that file in a queue and get to it once it's done processing its own file. a batch could do the trick using tasklist and find, too, but that's not as good of a solution as mutex.
Anyway, try this for your extract command registry value to get the right folder name:
"\path\to\7z.exe" x "%1" -o* -aou
This will create a new folder in the same directory as the source archive with the same name as the source archive (sans the file extension).
Also, I added the -aou switch to automatically avoid filename conflicts (7z will append a number to the end of a file instead prompting you whether you want to overwrite or whatever).
• Is there a way to do as -o* does, but only if there's more than one file in the archive? As in, can I make it extract to the current directory if there's only one file or folder in the archive? – mjohnson Jun 26 '16 at 0:18
• Default in Windows 10 is "C:\Program Files\7-Zip\7zFM.exe" x "%1" -o* -aou so change the 7zFM to just 7z "C:\Program Files\7-Zip\7z.exe" x "%1" -o* -aou – jsherk Dec 2 '17 at 17:36
This thread has become a bit confusing because of contradicting answers (it took me quite some time to figure out which was the right solution) so I thought it might be a good idea to summarize the results from afrazier's and Justin Roettger's posts combinded with my own experiences:
1. Start regedit as administrator
2. Open HKEY_CLASSES_ROOT\7-Zip.7z
3. Under that key, expand the Shell sub-key
4. Set the (Default) value to the string extract
5. Create a new sub-key named extract
6. Set the (Default) value for the extract key to Extract to Folder
7. Create a new sub-key under extract named command
8. Set the (Default) value of the command key to:
C:\Program Files\7-Zip\7zG.exe x "%1" -o*
(you might have to adjust this to match the path of your 7-Zip installation)
Instead of 7z with -aou like Justin Roettger suggested I ended up using 7zG, because this way you can choose to overwrite if you like just like extracting with the normal context menu.
That's it! 7z files are now extracted to a folder with their own name by double click. For other extensions like .rar and .zip you need to repeat these steps for the according keys. (i.e. HKEY_CLASSES_ROOT\7-Zip.rar and HKEY_CLASSES_ROOT\7-Zip.zip and so on)
Oh and to clarify: It does work with multiple files selected as well. No batch file need.
• I don't see that registry path. I'm on Windows 8 w/ 64-bit 7zip. – phillipwei Jun 18 '14 at 18:40
• Lots has changed since this post, but it still works great ... for me, it was under 7z_auto_file, not 7-Zip.7z. All I did was change the existing "command" to C:\Program Files\7-Zip\7zG.exe x "%1" -o* and voila. – neokio Dec 21 '15 at 7:35
• On Windows 10, I also didn't see the path until I associated at least one file with 7-Zip in the 7-Zip File Manager (via Tools -> Options... menu). – Chris Nolet May 4 '17 at 20:50
• Anyone know why this no longer works on Windows 10? The paths seem to be the same, but I just get a black console window that closes immediately, and nothing is extracted. Tried all the methods in this thread. – Ryan Weiss May 29 '19 at 16:34
### The easy way
Install ExtractNow. You can configure it to do exactly what you want.
### The hard way
Manual registry modification as follows...
• Start regedit as administrator
• Open HKCR\.7z and look at the (Default) value. Take note of what that is (in my case, as a PowerArchiver user, it's PASZIP)
• Go to the registry key in HKCR named that. (in my case HKCR\PASZIP)
• Under that key, expand the Shell sub-key
• Set the (Default) value to the string extract
• Create a new sub-key named extract
• Set the (Default) value for the extract key to Extract to Folder
• Create a new sub-key under extract named command
• Set the (Default) value of the command key to
C:\Program Files\7-Zip\7zG.exe x "%1" -o* -aou
(you might have to adjust the path)
Thanks to Justin Roettger for pointing out the correct name variable needed for this.
That should be it. Now 7z files are extracted to a folder with their own name by double click. For other extensions like .rar and .zip you need to repeat this steps for the according keys.
If you only want to make the changes on your user account instead of system-wide, modify HKCU\Software\Classes instead of HKCR. HKCR is a virtual key that's a union of HKLM\Software\Classes and HKCU\Software\Classes where the data in your account (HKCU) overrides the system-wide data (HKLM). Normally running regedit as an Administrator means that modifying HKCR alters system-wide data in HKLM.
### Extracting multiple files
Of course, this won't work if you have multiple files selected. If you want that to work, you need to create the following batch file:
@echo off
:top
if "%1"=="" goto :EOF
7z.exe x "%1" -o"%~dpn1"
shift
goto top
Now, follow the instructions above. In the very last step, set the (Default) value of the command key to C:\Path\To\File.bat %*
All of the registry modifications are untested from memory, but should be correct.
• Does %~dpn1 work in file associations? (It doesn't in Windows XP. Also, you forgot x for extension.) – user1686 Mar 18 '11 at 20:46
• @grawity: You don't want the x in the output folder name. As for it working... That's a good question. I'll edit to address... – afrazier Mar 18 '11 at 20:48
• Hm, good point. – user1686 Mar 18 '11 at 21:12
• So close, but not quite there yet. After playing with the code both afrazier and grawity provided, I have the default key value set as "C:\Program Files\7-Zip\7z.exe" x "%1" -o"%~dpn1" I wasn't getting any benefit from running cmd.exe that I could tell, and 7-zip's default open didn't bother with it. However, this is just unzipping to %~dpn1 in the same folder as the zip file. The \"%1.d\" gives a \filename.zip.d\ subdirectory in the same folder as the zip file, so it's closer, but not stripping off the extension. I've tried hybrids of the two, no luck. Any other suggestions? – schodge Mar 25 '11 at 1:43
• You'll probably have to use a batch file. – afrazier Mar 25 '11 at 3:51
Here's PowerShell script I wrote based on @haiggoh's answer. Before you run it, you need to open 7-zip, go to Tools->Options and associate 7-zip with wanted file extensions. After that, run the following PowerShell script (with admin rights):
$7zInstallationFolder = 'C:\Program Files\7-Zip'$reg = [Microsoft.Win32.RegistryKey]::OpenBaseKey([Microsoft.Win32.RegistryHive]::ClassesRoot, [Microsoft.Win32.RegistryView]::Default)
$subKeys =$reg.GetSubKeyNames() | where { $_ -match '7-Zip.' } foreach ($keyName in $subKeys) {$key = $reg.OpenSubKey($keyName + '\shell\open\command', $true)$key.SetValue('', '"' + $7zInstallationFolder + '\7zG.exe" x "%1" -o*') } Of course, make sure that $7zInstallationFolder variable contains correct path to your 7-zip installation.
• How to revert it? I uninstalling and reinstalling didn't work – Lombas May 9 '17 at 12:44
• @Lombas try associating 7-zip again with those extensions, if it doesn't work remove registry keys and try it again. – xx77aBs May 10 '17 at 10:58
• associating 7-zip again with the extensions worked. Thank you very much! – Lombas May 12 '17 at 10:49
Pass parameters like this:
7z x "%1" -o"%1.d"
I recall hearing complaints about Windows 7 not allowing to directly edit file actions. I don't know if this is true or not... but if it is, save the following as a *.reg file and import it.
Windows Registry Editor Version 5.00
[HKEY_CURRENT_USER\Software\Classes\7-Zip.7z\shell\extract]
@="Extract to folder"
[HKEY_CURRENT_USER\Software\Classes\7-Zip.7z\shell\extract\command]
@="7z.exe x \"%1\" -o\"%1.d\""
Here is a .reg file that configures Extract to folder as the default behavior when double-clicking files with one of the following extensions - 7z/CAB/GZ/GZIP/RAR/TAR/ZIP. You could do it for other file extensions using the same approach. I prefer the Windows default behavior for ISO/VHD mounting, so I didn't change that, and 7-Zip supports many other file types that I don't commonly encounter so I didn't change it for those.
You can revert this by going into 7-Zip File Manager, Tools, Options, and change the file associations as desired.
I tested on Windows 10 x64/7-Zip 15.12 x64. Because it uses C:\Program Files for the path to 7-Zip, you definitely need x64 Windows + x64 7-Zip, but I only tested on Windows 10.
You will have the same right-click options as before, this only changes what happens when you double-click, and only for those seven file types (again, you could do this for other file types using the same approach).
If there is already a folder of that name, it will give you the same 7-Zip prompt you get normally, with options for Yes/Yes to All/Auto Rename/No/No to All.
Windows Registry Editor Version 5.00
[HKEY_CURRENT_USER\SOFTWARE\Classes.7z]
@="7-Zip.7z"
[HKEY_CURRENT_USER\SOFTWARE\Classes.cab]
@="7-Zip.cab"
[HKEY_CURRENT_USER\SOFTWARE\Classes.gz]
@="7-Zip.gz"
[HKEY_CURRENT_USER\SOFTWARE\Classes.gzip]
@="7-Zip.gzip"
[HKEY_CURRENT_USER\SOFTWARE\Classes.rar]
@="7-Zip.rar"
[HKEY_CURRENT_USER\SOFTWARE\Classes.tar]
@="7-Zip.tar"
[HKEY_CURRENT_USER\SOFTWARE\Classes.zip]
@="7-Zip.zip"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.7z]
@="7z Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.7z\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,0"
[HKEY_CLASSES_ROOT\7-Zip.7z\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.7z\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.7z\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.7z\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.7z\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.cab]
@="cab Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.cab\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,7"
[HKEY_CLASSES_ROOT\7-Zip.cab\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.cab\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.cab\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.cab\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.cab\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gz]
@="gz Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gz\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,14"
[HKEY_CLASSES_ROOT\7-Zip.gz\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.gz\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.gz\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gz\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gz\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gzip]
@="gzip Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gzip\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,14"
[HKEY_CLASSES_ROOT\7-Zip.gzip\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.gzip\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.gzip\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gzip\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.gzip\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.rar]
@="rar Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.rar\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,3"
[HKEY_CLASSES_ROOT\7-Zip.rar\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.rar\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.rar\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.rar\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.rar\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.tar]
@="tar Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.tar\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,13"
[HKEY_CLASSES_ROOT\7-Zip.tar\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.tar\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.tar\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.tar\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.tar\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.zip]
@="zip Archive"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.zip\DefaultIcon]
@="C:\Program Files\7-Zip\7z.dll,1"
[HKEY_CLASSES_ROOT\7-Zip.zip\shell]
@="extract"
[HKEY_CLASSES_ROOT\7-Zip.zip\shell\extract]
@="Extract to Folder"
[HKEY_CLASSES_ROOT\7-Zip.zip\shell\extract\command]
@="\"C:\Program Files\7-Zip\7zG.exe\" x \"%1\" -o*"
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.zip\shell\open]
@=""
[HKEY_CURRENT_USER\SOFTWARE\Classes\7-Zip.zip\shell\open\command]
@="\"C:\Program Files\7-Zip\7zFM.exe\" \"%1\""
|
2021-05-14 14:46:56
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{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44935867190361023, "perplexity": 4745.985354065115}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00548.warc.gz"}
|
https://www.physicsforums.com/threads/serway-9th-ed-sliding-friction-and-problem.737423/
|
# Serway 9th ed - sliding friction and problem
Gold Member
## Homework Statement
I am absolutely stumped on a problem from a trig based college physics text by Serway et. al. 8th edition. This is exercise 4.14 on page 107. The example attached to it is a 10.0 kg block M, on top of which rests a 5.0 kg block m. The coefficient of friction between the two blocks is 0.350 and the coefficient on the ground is zero (the 10.0 kg block has no friction on the ground surface). The example calculates the maximum force (tension T) that can be applied to a cable attached to the 10 kg block that will not cause the 5 kg one on top to slip. The answer is 51.5 N, and I got that part.
The exercise that follows has the same set-up, but now the cable is attached to the 5 kg block on top (see attached image), and it asks for the maximum force that can be applied to the top block without causing it to slip. The answer in the text is 27.715 N. I reasoned that the force cannot exceed the force of friction (17.5 N) between the blocks and this would pull the two blocks across the frictionless ground surface.
## Homework Equations
$F_s = μ_s mg$
## The Attempt at a Solution
Some reverse engineering got me
$\frac{M+m}{M}F_s = 27.715 N$
This is consistent with the answer in the text, but is it in fact the answer? How do I get there? If it is not the answer what is it? I don't even know where to start.
Regards
David C.
#### Attachments
• ex-4.14-p-107-03.jpg
12.8 KB · Views: 531
tiny-tim
Homework Helper
Hi David!
i] If the tension is T, what is the acceleration, a?
ii] If the acceleration is a, what is the force on the lower block?
Gold Member
Hello tiny-tim,
I tried, the tension on m is $F_s = T = 17.15 N$. So
$a = \frac{17.15 N}{M}$
The acceleration of M, yes? So the force on M+m is
$(M+m) a = 15.0 * 1.715 \frac{kg \ast m}{s^2} = 25.72 N$
Still stumped. Thnx for the help btw. I seem to have no insight into the physics of the problem yet.
Last edited:
tiny-tim
Homework Helper
Hello GreyNoise!
I don't really understand what you're doing.
Assume the two blocks are moving together (without slipping).
Call the tension "T".
i] In terms of T, what is the formula for the acceleration, a, of both blocks?
ii] Using that formula for a, what is the formula for the force on the lower block?
1 person
haruspex
Homework Helper
Gold Member
2020 Award
I reasoned that the force cannot exceed the force of friction (17.5 N) between the blocks
Just to explain why that's wrong...
That argument would work if the blocks were not accelerating. Think about the horizontal forces on the top block and apply ∑F = ma. It gives Tension - Friction = Mass * Acceleration, so the tension can exceed the frictional force without slipping.
1 person
Gold Member
Ahh yes, I believe that the problems is seeking a formula for acceleration of both blocks in terms of the tension T whilke the cable is attached to the top block.
Gold Member
The problem continues to vex me. The blocks are supposed to move with the same acceleration, no slipping. I calculated the acceleration on the light block $m_1$from
$F_s = \mu N = \mu m_1g = 17.5 N$
$a = \mu g = 0.35*9.8 = 3.43 \frac{m}{s^2}$
But I could just as easily calculate the acceleration from the heavy block $m_2$ by
$F_s = \mu N = \mu m_1g = 17.15 N$, same as above
$a = \frac{\mu g }{m_2} = \frac{17.15N}{10.0kg} = 1.715 \frac{m}{s^2}$, divide by $m_2$ which happens to equal $2m_1$
Followed by $(m_1+m_2) * a = 15.0 (kg) * 1.715 (m/s^2) = 25.73 N$ which is the correct answer. I am missing some physical insight here. Should I generally calculate the acceleration from the heavier weight in these sorts of problems? If so, how should I have known that? I am unsure of what I am missing here.
Last edited:
haruspex
Homework Helper
Gold Member
2020 Award
The problem continues to vex me. The blocks are supposed to move with the same acceleration, no slipping. I calculated the acceleration on the light block $m_1$from
$F_s = \mu N = \mu m_1g = 17.5 N$
17.15
$a = \mu g = 0.35*9.8 = 3.43 \frac{m}{s^2}$
What is this acceleration? Static friction opposes relative motion.
I am missing some physical insight here. Should I generally calculate the acceleration from the heavier weight in these sorts of problems? If so, how should I have known that? I am unsure of what I am missing here.
The safe way, pretty much always, is to treat the blocks separately. Introduce unknowns for the forces and accelerations coupling them. Sometimes you can take a short cut by taking some blocks as a unit.
Your mistake above is in the way you calculated the acceleration of the top block. What are the forces on it?
Gold Member
solved
Ok haruspex, how 'bout this?
(1) From the diagram, the forces acting on $m_2$ alone are
$F_s = \mu N_1 = \mu m_1 g = 0.350*5.0*9.8 N = 17.15 N$
(2) Under this force $m_2$ will accelerate
$m_2 a = 17.15 N$
$a = \frac{17.15 N}{m_2} = \frac{17.15 N}{10 kg} = 1.715(\frac{m}{s^2})$
(3) The blocks do NOT slip, so they accelerate together and
$(m_1 + m_2)a = (5.0+10.0)*1.715 N = 25.73 N$
$T = 25.73 N$
If my reasoning is correct then I should be able to get the same answer by picking on the smaller block $m_1$, and that is below
(1) From the diagram, the forces acting on $m_1$ alone are
$T - F_s = T - \mu N_1 = T - \mu m_1 g$
(2) Under this force $m_1$ will accelerate
$m_1 a = T - \mu m_1 g$
$a = \frac{T - \mu m_1 g}{m_1}(\frac{m}{s^2})$
(3) The blocks do NOT slip, so they accelerate together and
$(m_1 + m_2)a = (m_1 + m_2)*\frac{T - \mu m_1 g}{m_1}$
$T = (m_1 + m_2)*\frac{T - \mu m_1 g}{m_1}$
$T*[\frac{m_1 + m_2}{m_1} - 1] = (m_1 + m_2)*\mu g$
and then substituting known values, I get
$T*[\frac{5 + 10}{5} - 1] = (5 + 10)*0.350*9.8 N$
$T*[3 - 1] = 51.45 N$
$T = 51.45/2 = 25.73 N$
$T = 25.73 N$
So obtaining the same answer gives me confidence that I am doing this correctly now. A quick check on $a = (T - \mu m_1 g)/m_1$ returned the same acceleration as well. Any mistakes in the above?
#### Attachments
• ex-4.14-p-107-04.jpg
13.6 KB · Views: 472
haruspex
|
2021-12-01 00:59:55
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|
https://www.aimsciences.org/article/doi/10.3934/mbe.2018046?viewType=html
|
American Institute of Mathematical Sciences
August 2018, 15(4): 1033-1054. doi: 10.3934/mbe.2018046
Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model
1 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK 2 Institute for Mathematical Sciences, Renmin University of China, Beijing, China 3 Department of Mathematics, College of William and Mary, Williamsburg, VA, USA 4 Key Laboratory of Eco-environments in Three Gorges Reservoir Region, and School of Mathematics and Statistics, Southwest University, Chongqing, China 5 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA 6 Department of Applied Mathematics, University of Western Ontario, London, ON, Canada
* Corresponding author: Xingfu Zou
Received November 15, 2017 Accepted November 15, 2017 Published March 2018
Lyme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value $c^*$ which is at least a lower bound for the wave speed in the sense that, if $c < c^*$, there is no traveling wave front of speed $c$ connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that $c^*$ is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with $c^*$. We also explore the dependence of $c^*$ on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.
Citation: Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao, Xingfu Zou. Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 1033-1054. doi: 10.3934/mbe.2018046
References:
show all references
References:
The life-stage components of the model: questing larvae ($L$) find a host, feed and moult into questing nymphs ($N$), which then find a new host, feed and moult into questing adults ($A_q$). Adult females that find a deer host ($A_f$) feed, drop to the forest floor, lay $2000$ eggs and then die. Hatching eggs create the next generation of questing larvae. The $r$ parameters are the per-capita transition rates between each compartment
$H_1(\lambda, c)$ and $H_2(\lambda, c)$ for different $c$. (a) $c = 0.55$; (b) $c = c^* = 0.6176844021$, ($\lambda = 0.7081234538$); (c) $c = 0.7$. Here, the model parameters are taken as $b = 3000$, $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$, $h = 100$ and $D = 1$
Dependence of $c^*$ on $b$ and $D$ respectively: (a) with $D = 1$; (b) with $b = 3000$. Other parameters are taken as: $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$ and $h = 100$
There is no biologically relevant traveling wave front solution with speed $c = 0.1<c^* = 0.24$: $\phi_1$ may take negative values
There is a non-negative traveling wave front solution with speed $c = 0.4>c^* = 0.24$
(a): time evolution of $L(x, t)$; (b): time evolution of $N(x, t)$; (c): contours of (a) with region where $L(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $N(x, t)>0.1$ shown in grey
(a): time evolution of $A_q(x, t)$; (b): time evolution of $A_f(x, t)$; (c): contours of (a) with region where $A_q(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $A_f(x, t)>0.1$ shown in grey
Explanation of parameters
Parameters Meaning Value $b$ Birth rate of tick $3000$ $1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$ $1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$ $1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$ $r_4$ Proportion of fed adults that can lay eggs 0.03 $d_1$ per-capita death rate of larvae 0.3 $d_2$ per-capita death rate of nymphs 0.3 $d_3$ per-capita death rate of questing adults 0.1 $d_4$ per-capita death rate of fed adults 0.1 $\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days $\tau_2$ average time tick is attached to a deer $10$ days
Parameters Meaning Value $b$ Birth rate of tick $3000$ $1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$ $1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$ $1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$ $r_4$ Proportion of fed adults that can lay eggs 0.03 $d_1$ per-capita death rate of larvae 0.3 $d_2$ per-capita death rate of nymphs 0.3 $d_3$ per-capita death rate of questing adults 0.1 $d_4$ per-capita death rate of fed adults 0.1 $\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days $\tau_2$ average time tick is attached to a deer $10$ days
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2018 Impact Factor: 1.313
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2021-01-20 01:06:39
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https://eprint.iacr.org/2020/286
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### Shorter Non-Interactive Zero-Knowledge Arguments and ZAPs for Algebraic Languages
Geoffroy Couteau and Dominik Hartmann
##### Abstract
We put forth a new framework for building pairing-based non-interactive zero- knowledge (NIZK) arguments for a wide class of algebraic languages, which are an extension of linear languages, containing disjunctions of linear languages and more. Our approach differs from the Groth-Sahai methodology, in that we rely on pairings to compile a $\Sigma$-protocol into a NIZK. Our framework enjoys a number of interesting features: – conceptual simplicity, parameters derive from the $\Sigma$-protocol; – proofs as short as resulting from the Fiat-Shamir heuristic applied to the underlying $\Sigma$-protocol; – fully adaptive soundness and perfect zero-knowledge in the common random string model with a single random group element as CRS; – yields simple and efficient two-round, public coin, publicly-verifiable perfect witness-indistinguishable (WI) arguments (ZAPs) in the plain model. To our knowledge, this is the first construction of two-rounds statistical witness-indistinguishable arguments from pairing assumptions. Our proof system relies on a new (static, falsifiable) assumption over pairing groups which generalizes the standard kernel Diffie-Hellman assumption in a natural way and holds in the generic group model (GGM) and in the algebraic group model (AGM). Replacing Groth-Sahai NIZKs with our new proof system allows to improve several important cryptographic primitives. In particular, we obtain the shortest tightly-secure structure-preserving signature scheme (which are a core component in anonymous credentials), the shortest tightly-secure quasi-adaptive NIZK with unbounded simulation soundness (which in turns implies the shortest tightly-mCCA-secure cryptosystem), and shorter ring signatures.
Available format(s)
Category
Public-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
zero-knowledge argumentsnon-interactive zero-knowledge argumentssatistical witness-indistinguishabilitypairing-based cryptographytight securitystructure-preserving signatures.
Contact author(s)
couteau @ irif fr
Dominik Hartmann @ rub de
History
2020-03-06: revised
See all versions
Short URL
https://ia.cr/2020/286
CC BY
BibTeX
@misc{cryptoeprint:2020/286,
author = {Geoffroy Couteau and Dominik Hartmann},
title = {Shorter Non-Interactive Zero-Knowledge Arguments and ZAPs for Algebraic Languages},
howpublished = {Cryptology ePrint Archive, Paper 2020/286},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/286}},
url = {https://eprint.iacr.org/2020/286}
}
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2022-08-17 22:21:54
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http://mathgl.sourceforge.net/doc_ru/PDE-solving-hints.html
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Next: , Previous: , Up: Hints [Contents][Index]
#### 3.9.14 PDE solving hints
Solving of Partial Differential Equations (PDE, including beam tracing) and ray tracing (or finding particle trajectory) are more or less common task. So, MathGL have several functions for that. There are ray for ray tracing, pde for PDE solving, qo2d for beam tracing in 2D case (see Global functions). Note, that these functions take “Hamiltonian” or equations as string values. And I don’t plan now to allow one to use user-defined functions. There are 2 reasons: the complexity of corresponding interface; and the basic nature of used methods which are good for samples but may not good for serious scientific calculations.
The ray tracing can be done by ray function. Really ray tracing equation is Hamiltonian equation for 3D space. So, the function can be also used for finding a particle trajectory (i.e. solve Hamiltonian ODE) for 1D, 2D or 3D cases. The function have a set of arguments. First of all, it is Hamiltonian which defined the media (or the equation) you are planning to use. The Hamiltonian is defined by string which may depend on coordinates ‘x’, ‘y’, ‘z’, time ‘t’ (for particle dynamics) and momentums ‘p’=p_x, ‘q’=p_y, ‘v’=p_z. Next, you have to define the initial conditions for coordinates and momentums at ‘t’=0 and set the integrations step (default is 0.1) and its duration (default is 10). The Runge-Kutta method of 4-th order is used for integration.
const char *ham = "p^2+q^2-x-1+i*0.5*(y+x)*(y>-x)";
mglData r = mglRay(ham, mglPoint(-0.7, -1), mglPoint(0, 0.5), 0.02, 2);
This example calculate the reflection from linear layer (media with Hamiltonian ‘p^2+q^2-x-1’=p_x^2+p_y^2-x-1). This is parabolic curve. The resulting array have 7 columns which contain data for {x,y,z,p,q,v,t}.
The solution of PDE is a bit more complicated. As previous you have to specify the equation as pseudo-differential operator \hat H(x, \nabla) which is called sometime as “Hamiltonian” (for example, in beam tracing). As previously, it is defined by string which may depend on coordinates ‘x’, ‘y’, ‘z’ (but not time!), momentums ‘p’=(d/dx)/i k_0, ‘q’=(d/dy)/i k_0 and field amplitude ‘u’=|u|. The evolutionary coordinate is ‘z’ in all cases. So that, the equation look like du/dz = ik_0 H(x,y,\hat p, \hat q, |u|)[u]. Dependence on field amplitude ‘u’=|u| allows one to solve nonlinear problems too. For example, for nonlinear Shrodinger equation you may set ham="p^2 + q^2 - u^2". Also you may specify imaginary part for wave absorption, like ham = "p^2 + i*x*(x>0)" or ham = "p^2 + i1*x*(x>0)".
Next step is specifying the initial conditions at ‘z’ equal to minimal z-axis value. The function need 2 arrays for real and for imaginary part. Note, that coordinates x,y,z are supposed to be in specified axis range. So, the data arrays should have corresponding scales. Finally, you may set the integration step and parameter k0=k_0. Also keep in mind, that internally the 2 times large box is used (for suppressing numerical reflection from boundaries) and the equation should well defined even in this extended range.
Final comment is concerning the possible form of pseudo-differential operator H. At this moment, simplified form of operator H is supported – all “mixed” terms (like ‘x*p’->x*d/dx) are excluded. For example, in 2D case this operator is effectively H = f(p,z) + g(x,z,u). However commutable combinations (like ‘x*q’->x*d/dy) are allowed for 3D case.
So, for example let solve the equation for beam deflected from linear layer and absorbed later. The operator will have the form ‘"p^2+q^2-x-1+i*0.5*(z+x)*(z>-x)"’ that correspond to equation 1/ik_0 * du/dz + d^2 u/dx^2 + d^2 u/dy^2 + x * u + i (x+z)/2 * u = 0. This is typical equation for Electron Cyclotron (EC) absorption in magnetized plasmas. For initial conditions let me select the beam with plane phase front exp(-48*(x+0.7)^2). The corresponding code looks like this:
int sample(mglGraph *gr)
{
mglData a,re(128),im(128);
gr->Fill(re,"exp(-48*(x+0.7)^2)");
a = gr->PDE("p^2+q^2-x-1+i*0.5*(z+x)*(z>-x)", re, im, 0.01, 30);
a.Transpose("yxz");
gr->SubPlot(1,1,0,"<_"); gr->Title("PDE solver");
gr->SetRange('c',0,1); gr->Dens(a,"wyrRk");
gr->Axis(); gr->Label('x', "\\i x"); gr->Label('y', "\\i z");
gr->FPlot("-x", "k|");
gr->Puts(mglPoint(0, 0.85), "absorption: (x+z)/2 for x+z>0");
gr->Puts(mglPoint(0,1.1),"Equation: ik_0\\partial_zu + \\Delta u + x\\cdot u + i \\frac{x+z}{2}\\cdot u = 0");
return 0;
}
The last example is example of beam tracing. Beam tracing equation is special kind of PDE equation written in coordinates accompanied to a ray. Generally this is the same parameters and limitation as for PDE solving but the coordinates are defined by the ray and by parameter of grid width w in direction transverse the ray. So, you don’t need to specify the range of coordinates. BUT there is limitation. The accompanied coordinates are well defined only for smooth enough rays, i.e. then the ray curvature K (which is defined as 1/K^2 = (|r''|^2 |r'|^2 - (r'', r'')^2)/|r'|^6) is much large then the grid width: K>>w. So, you may receive incorrect results if this condition will be broken.
You may use following code for obtaining the same solution as in previous example:
int sample(mglGraph *gr)
{
mglData r, xx, yy, a, im(128), re(128);
const char *ham = "p^2+q^2-x-1+i*0.5*(y+x)*(y>-x)";
r = mglRay(ham, mglPoint(-0.7, -1), mglPoint(0, 0.5), 0.02, 2);
gr->SubPlot(1,1,0,"<_"); gr->Title("Beam and ray tracing");
gr->Plot(r.SubData(0), r.SubData(1), "k");
gr->Axis(); gr->Label('x', "\\i x"); gr->Label('y', "\\i z");
// now start beam tracing
gr->Fill(re,"exp(-48*x^2)");
a = mglQO2d(ham, re, im, r, xx, yy, 1, 30);
gr->SetRange('c',0, 1);
gr->Dens(xx, yy, a, "wyrRk");
gr->FPlot("-x", "k|");
gr->Puts(mglPoint(0, 0.85), "absorption: (x+y)/2 for x+y>0");
gr->Puts(mglPoint(0.7, -0.05), "central ray");
return 0;
}
Next: , Previous: , Up: Hints [Contents][Index]
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2017-03-23 02:17:31
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http://harvard.voxcharta.org/tag/core/
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Array ( [0] => tag/core/ ) core « Vox Charta
# Posts Tagged core
## Recent Postings from core
### Constraints on Core Collapse from the Black Hole Mass Function
We model the observed black hole mass function under the assumption that black hole formation is controlled by the compactness of the stellar core at the time of collapse. Low compactness stars are more likely to explode as supernovae and produce neutron stars, while high compactness stars are more likely to be failed supernovae that produce black holes with the mass of the helium core of the star. Using three sequences of stellar models and marginalizing over a model for the completeness of the black hole mass function, we find that the compactness xi(2.5) above which 50% of core collapses produce black holes is xi(2.5)=0.24 (0.15 < xi(2.5) < 0.37) at 90% confidence). While models with a sharp transition between successful and failed explosions are always the most likely, the width of the transition between the minimum compactness for black hole formation and the compactness above which all core collapses produce black holes is not well constrained. The models also predict that f=0.18 (0.09 < f < 0.39) of core collapses fail assuming a minimum mass for core collapse of 8Msun. We tested four other criteria for black hole formation based on xi(2.0) and xi(3.0), the compactnesses at enclosed masses of 2.0 or 3.0 rather than 2.5Msun, the mass of the iron core, and the mass inside the oxygen burning shell. We found that xi(2.0) works as well as xi(2.5), while the compactness xi(3.0) works significantly worse, as does using the iron core mass or the mass enclosed by the oxygen burning shell. As expected from the high compactness of 20-25Msun stars, black hole formation in this mass range provides a natural explanation of the red supergiant problem.
### A very deep Chandra observation of Abell 1795: The Cold Front and Cooling Wake
We present a new analysis of very deep \cha \ observations of the galaxy cluster Abell 1795. Utilizing nearly 750 ks of net ACIS imaging, we are able to resolve the thermodynamic structure of the Intracluster Medium (ICM) on length scales of $\sim 1 \kpc$ near the cool core. We find several previously unresolved structures, including a high pressure feature to the north of the BCG that appears to arise from the bulk motion of Abell 1795′s cool core. To the south of the cool core, we find low temperature ($\sim 3 \keV$), diffuse ICM gas extending for distances of $\sim 50 \kpc$ spatially coincident with previously identified filaments of H$\alpha$ emission. Gas at similar temperatures is also detected in adjacent regions without any H$\alpha$ emission. The X-ray gas coincident with the H$\alpha$ filament has been measured to be cooling spectroscopically at a rate of $\sim 1 \msolar \yr^{-1}$, consistent with measurements of the star formation rate in this region as inferred from UV observations, suggesting that the star formation in this filament as inferred by its H$\alpha$ and UV emission can trace its origin to the rapid cooling of dense, X-ray emitting gas. The H$\alpha$ filament is not a unique site of cooler ICM, however, as ICM at similar temperatures and even higher metallicities not cospatial with H$\alpha$ emission is observed just to the west of the H$\alpha$ filament, suggesting that it may have been uplifted by Abell 1795′s central active galaxy. Further simulations of cool core sloshing and AGN feedback operating in concert with one another will be necessary to understand how such a dynamic cool core region may have originated and why the H$\alpha$ emission is so localized with respect to the cool X-ray gas despite the evidence for a catastrophic cooling flow.
### A very deep Chandra observation of Abell 1795: The Cold Front and Cooling Wake [Replacement]
We present a new analysis of very deep Chandra observations of the galaxy cluster Abell 1795. Utilizing nearly 750 ks of net ACIS imaging, we are able to resolve the thermodynamic structure of the Intracluster Medium (ICM) on length scales of ~ 1 kpc near the cool core. We find several previously unresolved structures, including a high pressure feature to the north of the BCG that appears to arise from the bulk motion of Abell 1795′s cool core. To the south of the cool core, we find low temperature (~ 3 keV), diffuse ICM gas extending for distances of ~ 50 kpc spatially coincident with previously identified filaments of H-alpha emission. Gas at similar temperatures is also detected in adjacent regions without any H-alpha emission. The X-ray gas coincident with the H-alpha filament has been measured to be cooling spectroscopically at a rate of ~ 1 Solar Masses/ yr, consistent with measurements of the star formation rate in this region as inferred from UV observations, suggesting that the star formation in this filament as inferred by its H$\alpha$ and UV emission can trace its origin to the rapid cooling of dense, X-ray emitting gas. The H-alpha filament is not a unique site of cooler ICM, however, as ICM at similar temperatures and even higher metallicities not cospatial with H$\alpha$ emission is observed just to the west of the H-alpha filament, suggesting that it may have been uplifted by Abell 1795′s central active galaxy. Further simulations of cool core sloshing and AGN feedback operating in concert with one another will be necessary to understand how such a dynamic cool core region may have originated and why the H-alpha emission is so localized with respect to the cool X-ray gas despite the evidence for a catastrophic cooling flow.
### Thermal emission of neutron stars with internal heaters
Using 1D and 2D cooling codes we study thermal emission from neutron stars with steady state internal heaters of various intensities and geometries (blobs or spherical layers) located at different depths in the crust. The generated heat tends to propagate radially, from the heater down to the stellar core and up to the surface; it is also emitted by neutrinos. In local regions near the heater the results are well described with the 1D code. The heater’s region projects onto the stellar surface forming a hot spot. There are two heat propagation regimes. In the first, conduction outflow regime (realized at heat rates $H_0 \lesssim 10^{20}$ erg cm$^{-3}$ s$^{-1}$ or temperatures $T_\mathrm{h} \lesssim 10^9$ K in the heater) the thermal surface emission of the star depends on the heater’s power and neutrino emission in the stellar core. In the second, neutrino outflow regime ($H_0 \gtrsim 10^{20}$ erg cm$^{-3}$ s$^{-1}$ or $T_\mathrm{h} \gtrsim 10^9$ K) the surface thermal emission becomes independent of heater’s power and the physics of the core. The largest (a few per cent) fraction of heat power is carried to the surface if the heater is in the outer crust and the heat regime is intermediate. The results can be used for modeling young cooling neutron stars (prior to the end of internal thermal relaxation), neutron stars in X-ray transients, magnetars and high-$B$ pulsars, as well as merging neutron stars.
### Growth of Jupiter: Enhancement of Core Accretion by a Voluminous Low-Mass Envelope
We present calculations of the early stages of the formation of Jupiter via core nucleated accretion and gas capture. The core begins as a seed body of about 350 kilometers in radius and orbits in a swarm of planetesimals whose initial radii range from 15 meters to 50 kilometers. The evolution of the swarm accounts for growth and fragmentation, viscous and gravitational stirring, and for drag-assisted migration and velocity damping. During this evolution, less than 9% of the mass is in planetesimals smaller than 1 kilometer in radius; < ~25% is in planetesimals with radii between 1 and 10 kilometers; and < ~7% is in bodies with radii larger than 100 kilometers. Gas capture by the core substantially enhances the size-dependent cross-section of the planet for accretion of planetesimals. The calculation of dust opacity in the planet’s envelope accounts for coagulation and sedimentation of dust particles released as planetesimals are ablated. The calculation is carried out at an orbital semi-major axis of 5.2 AU and the initial solids’ surface density is 10 g/cm^2 at that distance. The results give a core mass of nearly 7.3 Earth masses (Mearth) and an envelope mass of approximately 0.15 Mearth after about 4e5 years, at which point the envelope growth rate surpasses that of the core. The same calculation without the envelope yields a core of only about 4.4 Mearth.
### Sterile neutrino oscillations in core-collapse supernova simulations
We have made core-collapse supernova simulations that allow oscillations between electron neutrinos (or their anti particles) with right-handed sterile neutrinos. We have considered a range of mixing angles and sterile neutrino masses including those consistent with sterile neutrinos as a dark matter candidate. We examine whether such oscillations can impact the core bounce and shock reheating in supernovae. We identify the optimum ranges of mixing angles and masses that can dramatically enhance the supernova explosion by efficiently transporting electron anti-neutrinos from the core to behind the shock where they provide additional heating leading to much larger explosion kinetic energies. We show that an interesting oscillation in the neutrino luminosity develops due to a cycle of depletion of the neutrino density by conversion to sterile neutrinos that shuts off the conversion, followed by a replenished neutrino density as neutrinos transport through the core.
### The hot core towards the intermediate mass protostar NGC7129 FIRS 2: Chemical similarities with Orion KL
NGC 7129 FIRS 2 (hereafter FIRS 2) is an intermediate-mass (2 to 8 Msun) protostar located at a distance of 1250 pc. High spatial resolution observations are required to resolve the hot core at its center. We present a molecular survey from 218200 MHz to 221800 MHz carried out with the IRAM Plateau de Bure Interferometer. These observations were complemented with a long integration single-dish spectrum taken with the IRAM 30m telescope. We used a Local Thermodynamic Equilibrium (LTE) single temperature code to model the whole dataset. The interferometric spectrum is crowded with a total of ~300 lines from which a few dozens remain unidentified yet. The spectrum has been modeled with a total of 20 species and their isomers, isotopologues and deuterated compounds. Complex molecules like methyl formate (CH3OCHO), ethanol (CH3CH2OH), glycolaldehyde (CH2OHCHO), acetone (CH3COCH3), dimethyl ether (CH3OCH3), ethyl cyanide (CH3CH2CN) and the aGg’ conformer of ethylene glycol (aGg’-(CH2OH)_2) are among the detected species. The detection of vibrationally excited lines of CH3CN, CH3OCHO, CH3OH, OCS, HC3N and CH3CHO proves the existence of gas and dust at high temperatures. In fact, the gas kinetic temperature estimated from the vibrational lines of CH3CN, ~405 K, is similar to that measured in massive hot cores. Our data allow an extensive comparison of the chemistry in FIRS~2 and the Orion hot core. We find a quite similar chemistry in FIRS 2 and Orion. Most of the studied fractional molecular abundances agree within a factor of 5. Larger differences are only found for the deuterated compounds D2CO and CH2DOH and a few molecules (CH3CH2CN, SO2, HNCO and CH3CHO). Since the physical conditions are similar in both hot cores, only different initial conditions (warmer pre-collapse phase in the case of Orion) and/or different crossing time of the gas in the hot core can explain this behavior.
### Asteroseismic measurement of surface-to-core rotation in a main sequence A star, KIC 11145123 [Replacement]
We have discovered rotationally split core g-mode triplets and surface p-mode triplets and quintuplets in a terminal age main sequence A star, KIC 11145123, that shows both $\delta$ Sct p-mode pulsations and $\gamma$ Dor g-mode pulsations. This gives the first robust determination of the rotation of the deep core and surface of a main sequence star, essentially model-independently. We find its rotation to be nearly uniform with a period near 100 d, but we show with high confidence that the surface rotates slightly faster than the core. A strong angular momentum transfer mechanism must be operating to produce the nearly rigid rotation, and a mechanism other than viscosity must be operating to produce a more rapidly rotating surface than core. Our asteroseismic result, along with previous asteroseismic constraints on internal rotation in some B stars, and measurements of internal rotation in some subgiant, giant and white dwarf stars, has made angular momentum transport in stars throughout their lifetimes an observational science.
### Mapping the particle acceleration in the cool core of the galaxy cluster RX J1720.1+2638
We present new deep, high-resolution radio images of the diffuse minihalo in the cool core of the galaxy cluster RX ,J1720.1+2638. The images have been obtained with the Giant Metrewave Radio Telescope at 317, 617 and 1280 MHz and with the Very Large Array at 1.5, 4.9 and 8.4 GHz, with angular resolutions ranging from 1" to 10". This represents the best radio spectral and imaging dataset for any minihalo. Most of the radio flux of the minihalo arises from a bright central component with a maximum radius of ~80 kpc. A fainter tail of emission extends out from the central component to form a spiral-shaped structure with a length of ~230 kpc, seen at frequencies 1.5 GHz and below. We observe steepening of the total radio spectrum of the minihalo at high frequencies. Furthermore, a spectral index image shows that the spectrum of the diffuse emission steepens with the increasing distance along the tail. A striking spatial correlation is observed between the minihalo emission and two cold fronts visible in the Chandra X-ray image of this cool core. These cold fronts confine the minihalo, as also seen in numerical simulations of minihalo formation by sloshing-induced turbulence. All these observations provide support to the hypothesis that the radio emitting electrons in cluster cool cores are produced by turbulent reacceleration.
### Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs
Giant planet formation in the core accretion (CA) paradigm is predicated by the formation of a core, assembled by the coagulation of grains and later by planetesimals within a protoplanetary disc. In contrast, in the disc instability paradigm, giant planet formation is believed to be independent of core formation: massive self-gravitating gas fragments cool radiatively and collapse as a whole. We show that giant planet formation in the disc instability model may be also enhanced by core formation for reasons physically very similar to the CA paradigm. In the model explored here, efficient grain sedimentation within an initial fragment (rather than the disc) leads to the formation of a core composed of heavy elements. We find that massive atmospheres form around cores and undergo collapse as a critical core mass is exceeded, analogous to CA theory. The critical mass of the core to initiate such a collapse depends on the fragment mass and metallicity, as well as core luminosity, but ranges from less than 1 to as much as $\sim80$ Earth masses. We therefore suggest that there are two channels for the collapse of a gaseous fragment to planetary scales within the disc instability model: (i) H$_2$ dissociative collapse of the entire gaseous clump, and (ii) core-assisted gas capture, as presented here. We suggest that the first of these two is favoured in metal-poor environments and for fragments more massive than $\sim 5-10$ Jupiter masses, whereas the second is favored in metal-rich environments and fragments of lower mass. [Abridged]
### Detecting scattered light from low-mass molecular cores at 3.6 $\mu$m - Impact of global effects on the observation of coreshine
Recently discovered scattered light at 3-5 $\mu$m from low-mass cores (so-called "coreshine") reveals the presence of grains around 1 $\mu$m, which is larger than the grains found in the low-density interstellar medium. But only about half of the 100+ cores investigated so far show the effect. This prompts further studies on the origin of this detection rate. From the 3D continuum radiative transfer equation, we derive the expected scattered light intensity from a core placed in an arbitrary direction seen from Earth. We use the approximation of single scattering, consider extinction up to 2nd-order Taylor approximation, and neglect spatial gradients in the dust size distribution. The impact of the directional characteristics of the scattering on the detection of scattered light from cores is calculated for a given grain size distribution, and local effects like additional radiation field components are discussed. The surface brightness profiles of a core with a 1D density profile are calculated for various Galactic locations, and the results are compared to the approximate detection limits. We find that for optically thin radiation and a constant size distribution, a simple limit for detecting scattered light from a low-mass core can be derived that holds for grains with sizes smaller than 0.5 $\mu$m. The extinction by the core prohibits detection in bright parts of the Galactic plane, especially near the Galactic center. For scattered light received from low-mass cores with grain sizes beyond 0.5 $\mu$m, the directional characteristics of the scattering favors the detection of scattered light above and below the Galactic center, and to some extent near the Galactic anti-center. We identify the local incident radiation field as the major unknown causing deviations from this simple scheme.
### Effect of core--mantle and tidal torques on Mercury's spin axis orientation
The rotational evolution of Mercury’s mantle and its core under conservative and dissipative torques is important for understanding the planet’s spin state. Dissipation results from tides and viscous, magnetic and topographic core–mantle interactions. The dissipative core–mantle torques take the system to an equilibrium state wherein both spins are fixed in the frame precessing with the orbit, and in which the mantle and core are differentially rotating. This equilibrium exhibits a mantle spin axis that is offset from the Cassini state by larger amounts for weaker core–mantle coupling for all three dissipative core–mantle coupling mechanisms, and the spin axis of the core is separated farther from that of the mantle, leading to larger differential rotation. The relatively strong core–mantle coupling necessary to bring the mantle spin axis to its observed position close to the Cassini state is not obtained by any of the three dissipative core–mantle coupling mechanisms. For a hydrostatic ellipsoidal core–mantle boundary, pressure coupling dominates the dissipative effects on the mantle and core positions, and dissipation together with pressure coupling brings the mantle spin solidly to the Cassini state. The core spin goes to a position displaced from that of the mantle by about 3.55 arcmin nearly in the plane containing the Cassini state. With the maximum viscosity considered of $\nu\sim 15.0\,{\rm cm^2/s}$ if the coupling is by the circulation through an Ekman boundary layer or $\nu\sim 8.75\times 10^5\,{\rm cm^2/s}$ for purely viscous coupling, the core spin lags the precessing Cassini plane by 23 arcsec, whereas the mantle spin lags by only 0.055 arcsec. Larger, non hydrostatic values of the CMB ellipticity also result in the mantle spin at the Cassini state, but the core spin is moved closer to the mantle spin.
### Seismic constraints on the radial dependence of the internal rotation profiles of six Kepler subgiants and young red giants
Context : We still do not know which mechanisms are responsible for the transport of angular momentum inside stars. The recent detection of mixed modes that contain the signature of rotation in the spectra of Kepler subgiants and red giants gives us the opportunity to make progress on this issue. Aims: Our aim is to probe the radial dependance of the rotation profiles for a sample of Kepler targets. For this purpose, subgiants and early red giants are particularly interesting targets because their rotational splittings are more sensitive to the rotation outside the deeper core than is the case for their more evolved counterparts. Methods: We first extract the rotational splittings and frequencies of the modes for six young Kepler red giants. We then perform a seismic modeling of these stars using the evolutionary codes CESAM2k and ASTEC. By using the observed splittings and the rotational kernels of the optimal models, we perform inversions of the internal rotation profiles of the six stars. Results: We obtain estimates of the mean rotation rate in the core and in the convective envelope of these stars. We show that the rotation contrast between the core and the envelope increases during the subgiant branch. Our results also suggest that the core of subgiants spins up with time, contrary to the RGB stars whose core has been shown to spin down. For two of the stars, we show that a discontinuous rotation profile with a deep discontinuity reproduces the observed splittings significantly better than a smooth rotation profile. Interestingly, the depths that are found most probable for the discontinuities roughly coincide with the location of the H-burning shell, which separates the layers that contract from those that expand. These results will bring observational constraints to the scenarios of angular momentum transport in stars.
### Limits on core driven ILOT outbursts of asymptotic giant branch stars
We find that single-star mechanisms for Intermediate Luminosity Optical Transients (ILOTs; Red Transients; Red Novae) which are powered by energy release in the core of asymptotic giant branch (AGB) stars are likely to eject the entire envelope, and hence cannot explain ILOTs in AGB and similar stars. There are singe-star and binary models for the powering of ILOTs, which are eruptive stars with peak luminosities between those of novae and supernovae. In single-star models the ejection of gas at velocities of ~500-1000 km/s and a possible bright ionizing flash, require a shock to propagate from the core outward. Using a self similar solution to follow the propagation of the shock through the envelope of two evolved stellar models, 6Mo AGB star and 11Mo yellow supergiant (YSG) star, we find that the shock that is required to explain the observed mass loss also ejects most of the envelope. We also show that for the event to have a strong ionizing flash the required energy also removes most of the envelope. The removal of most of the envelope is in contradiction with observations. We conclude that single-star models for ILOTs of evolved giant stars encounter severe difficulties.
### Limits on core driven ILOT outbursts of asymptotic giant branch stars [Replacement]
We find that single-star mechanisms for Intermediate Luminosity Optical Transients (ILOTs; Red Transients; Red Novae) which are powered by energy release in the core of asymptotic giant branch (AGB) stars are likely to eject the entire envelope, and hence cannot explain ILOTs in AGB and similar stars. There are single-star and binary models for the powering of ILOTs, which are eruptive stars with peak luminosities between those of novae and supernovae. In single-star models the ejection of gas at velocities of ~500-1000 km/s and a possible bright ionizing flash, require a shock to propagate from the core outward. Using a self similar solution to follow the propagation of the shock through the envelope of two evolved stellar models, a 6Mo AGB star and an 11Mo yellow supergiant (YSG) star, we find that the shock that is required to explain the observed mass loss also ejects most of the envelope. We also show that for the event to have a strong ionizing flash the required energy expels most of the envelope. The removal of most of the envelope is in contradiction with observations. We conclude that single-star models for ILOTs of evolved giant stars encounter severe difficulties.
### Photo-Disintegration of Heavy Nuclei at the Core of Cen A [Replacement]
Fermi LAT has detected gamma ray emissions from the core of Cen A. More recently, a new component in the gamma ray spectrum from the core has been reported in the energy range of 4 GeV to tens of GeV. We show that the new component and the HESS detected spectrum of gamma rays from the core at higher energy have possibly a common origin in photo-disintegration of heavy nuclei. Assuming the cosmic rays are mostly Fe nuclei inside the core and their spectrum has a low energy cut-off at 52 TeV in the wind frame moving with a Doppler factor 0.25 with respect to the observer on earth, the cosmic ray luminosity required to explain the observed gamma ray flux above 1 GeV is found to be $1.5\times 10^{43}$ erg/sec.
### Photo-Disintegration of Heavy Nuclei at the Core of Cen A [Replacement]
Fermi LAT has detected gamma ray emissions from the core of Cen A. More recently, a new component in the gamma ray spectrum from the core has been reported in the energy range of 4 GeV to tens of GeV. We show that the new component and the HESS detected spectrum of gamma rays from the core at higher energy have possibly a common origin in photo-disintegration of heavy nuclei. Assuming the cosmic rays are mostly Fe nuclei inside the core and their spectrum has a low energy cut-off at 52 TeV in the wind frame moving with a Doppler factor 0.25 with respect to the observer on earth, the cosmic ray luminosity required to explain the observed gamma ray flux above 1 GeV is found to be $1.5\times 10^{43}$ erg/sec.
### Photo-Disintegration of Heavy Nuclei at the Core of Cen A
Fermi LAT has detected gamma ray emissions from the core of Cen A. More recently, a new component in the gamma ray spectrum from the core has been reported in the energy range of 4 GeV to tens of GeV. We show that the new component and the HESS detected spectrum of gamma rays from the core at higher energy have possibly a common origin in photo-disintegration of heavy nuclei. This gives an indirect evidence of ultrahigh energy cosmic ray composition at the core of Cen A.
### Constraining the Origin of Magnetar Flares [Replacement]
Sudden relaxation of the magnetic field in the core of a magnetar produces mechanical energy primarily in the form of shear waves which propagate to the surface and enter the magnetosphere as relativistic Alfv\’en waves. Due to a strong impedance mismatch, shear waves excited in the star suffer many reflections before exiting the star. If mechanical energy is deposited in the core and is converted {\em directly} to radiation upon propagation to the surface, the rise time of the emission is at least seconds to minutes, and probably minutes to hours for a realistic magnetic field geometry, at odds with observed rise times of $\lap 10$ ms for both small bursts and for giant flares. Mechanisms for both small and giant flares that rely on the sudden relaxation of the magnetic field of the core are rendered unviable by the impedance mismatch, requiring the energy that drives these events to be stored in the magnetosphere just before the flare. A corollary to this conclusion is that if the quasi-periodic oscillations (QPOs) seen in giant flares represent stellar oscillations, they must be excited {\em by the magnetosphere}, not by mechanical energy released inside the star. Excitation of stellar oscillations by relativistic Alfv\’en waves in the magnetosphere could be quick enough to excite stellar modes well before a giant flare ends, unless the waves are quickly damped.
### Constraining the Origin of Magnetar Flares [Replacement]
Sudden relaxation of the magnetic field in the core of a magnetar produces mechanical energy primarily in the form of shear waves which propagate to the surface and enter the magnetosphere as relativistic Alfv\’en waves. Due to a strong impedance mismatch, shear waves excited in the star suffer many reflections before exiting the star. If mechanical energy is deposited in the core and is converted {\em directly} to radiation upon propagation to the surface, the rise time of the emission is at least seconds to minutes, and probably minutes to hours for a realistic magnetic field geometry, at odds with observed rise times of $\lap 10$ ms for both and giant flares. Mechanisms for both small and giant flares that rely on the sudden relaxation of the magnetic field of the core are rendered unviable by the impedance mismatch, requiring the energy that drives these events to be stored in the magnetosphere just before the flare. ends, unless the waves are quickly damped.
### Prospects of Turbulence Studies in High-Energy Density Laser-Generated Plasma: Numerical Investigations in Two Dimensions [Cross-Listing]
We investigate the possibility of generating and studying turbulence in plasma by means of high-energy density laser-driven experiments. Our focus is to create supersonic, self-magnetized turbulence with characteristics that resemble those found in the interstellar medium (ISM). We consider a target made of a spherical core surrounded by a shell made of denser material. The shell is irradiated by a sequence of laser pulses sending inward-propagating shocks that convert the inner core into plasma and create turbulence. In the context of the evolution of the ISM, the shocks play the role of supernova remnant shocks and the core represents the ionized interstellar medium. We consider the effects of both pre-existing and self-generating magnetic fields and study the evolution of the system by means of two-dimensional numerical simulations. We find that the evolution of the turbulent core is generally, subsonic with rms-Mach number $M_t\approx 0.2$. We observe an isotropic, turbulent velocity field with an inertial range power spectra of $P(k)\propto k^{-2.3}$. We account for the effects of self-magnetization and find that the resulting magnetic field has characteristic strength $\approx 3\times 10^{4}$ G. The corresponding plasma beta is $\approx 1\times 10^{4}$–$1\times 10^{5}$, indicating that the magnetic field does not play an important role in the dynamical evolution of the system. The natural extension of this work is to study the system evolution in three-dimensions, with various laser drive configurations, and targets with shells and cores of different masses. The latter modification may help to increase the turbulent intensity and possibly create transonic turbulence. One of the key challenges is to obtain transonic turbulent conditions in a quasi-steady state environment.
### Measuring the Angular Momentum Distribution in Core-Collapse Supernova Progenitors with Gravitational Waves
The late collapse, core bounce, and the early postbounce phase of rotating core collapse leads to a characteristic gravitational wave (GW) signal. The precise shape of the signal is governed by the interplay of gravity, rotation, nuclear equation of state (EOS), and electron capture during collapse. We explore the dependence of the signal on total angular momentum and its distribution in the progenitor core by means of a large set of axisymmetric general-relativistic core collapse simulations in which we vary the initial angular momentum distribution in the core. Our simulations include a microphysical finite-temperature EOS, an approximate electron capture treatment during collapse, and a neutrino leakage scheme for the postbounce evolution. We find that the precise distribution of angular momentum is relevant only for very rapidly rotating cores with T/|W|>~8% at bounce. We construct a numerical template bank from our baseline set of simulations, and carry out additional simulations to generate trial waveforms for injection into simulated advanced LIGO noise at a fiducial galactic distance of 10 kpc. Using matched filtering, we show that for an optimally-oriented source and Gaussian noise, advanced Advanced LIGO could measure the total angular momentum to within ~20%, for rapidly rotating cores. For most waveforms, the nearest known degree of precollapse differential rotation is correctly inferred by both our matched filtering analysis and an alternative Bayesian model selection approach. We test our results for robustness against systematic uncertainties by injecting waveforms from simulations using a different EOS and and variations in the electron fraction in the inner core. The results of these tests show that these uncertainties significantly reduce the accuracy with which the total angular momentum and its precollapse distribution can be inferred from observations.
### Measuring the Angular Momentum Distribution in Core-Collapse Supernova Progenitors with Gravitational Waves [Cross-Listing]
The late collapse, core bounce, and the early postbounce phase of rotating core collapse leads to a characteristic gravitational wave (GW) signal. The precise shape of the signal is governed by the interplay of gravity, rotation, nuclear equation of state (EOS), and electron capture during collapse. We explore the dependence of the signal on total angular momentum and its distribution in the progenitor core by means of a large set of axisymmetric general-relativistic core collapse simulations in which we vary the initial angular momentum distribution in the core. Our simulations include a microphysical finite-temperature EOS, an approximate electron capture treatment during collapse, and a neutrino leakage scheme for the postbounce evolution. We find that the precise distribution of angular momentum is relevant only for very rapidly rotating cores with T/|W|>~8% at bounce. We construct a numerical template bank from our baseline set of simulations, and carry out additional simulations to generate trial waveforms for injection into simulated advanced LIGO noise at a fiducial galactic distance of 10 kpc. Using matched filtering, we show that for an optimally-oriented source and Gaussian noise, advanced Advanced LIGO could measure the total angular momentum to within ~20%, for rapidly rotating cores. For most waveforms, the nearest known degree of precollapse differential rotation is correctly inferred by both our matched filtering analysis and an alternative Bayesian model selection approach. We test our results for robustness against systematic uncertainties by injecting waveforms from simulations using a different EOS and and variations in the electron fraction in the inner core. The results of these tests show that these uncertainties significantly reduce the accuracy with which the total angular momentum and its precollapse distribution can be inferred from observations.
### Thermal conduction by dark matter with velocity and momentum-dependent cross-sections
We use the formalism of Gould and Raffelt [1] to compute the dimensionless thermal conduction coefficients for scattering of dark matter particles with standard model nucleons via cross-sections that depend on the relative velocity or momentum exchanged between particles. Motivated by models invoked to reconcile various recent results in direct detection, we explicitly compute the conduction coefficients $\alpha$ and $\kappa$ for cross-sections that go as $v_{\rm rel}^2$, $v_{\rm rel}^4$, $v_{\rm rel}^{-2}$, $q^2$, $q^4$ and $q^{-2}$, where $v_{\rm rel}$ is the relative DM-nucleus velocity and $q$ is the momentum transferred in the collision. We find that a $v_{\rm rel}^{-2}$ dependence can significantly enhance energy transport from the inner solar core to the outer core. The same can true for any $q$-dependent coupling, if the dark matter mass lies within some specific range for each coupling. This effect can complement direct searches for dark matter; combining these results with state-of-the-art Solar simulations should greatly increase sensitivity to certain DM models. It also seems possible that the so-called Solar Abundance Problem could be resolved by enhanced energy transport in the solar core due to such velocity- or momentum-dependent scatterings.
### Thermal conduction by dark matter with velocity and momentum-dependent cross-sections [Replacement]
We use the formalism of Gould and Raffelt to compute the dimensionless thermal conduction coefficients for scattering of dark matter particles with standard model nucleons via cross-sections that depend on the relative velocity or momentum exchanged between particles. Motivated by models invoked to reconcile various recent results in direct detection, we explicitly compute the conduction coefficients $\alpha$ and $\kappa$ for cross-sections that go as $v_{\rm rel}^2$, $v_{\rm rel}^4$, $v_{\rm rel}^{-2}$, $q^2$, $q^4$ and $q^{-2}$, where $v_{\rm rel}$ is the relative DM-nucleus velocity and $q$ is the momentum transferred in the collision. We find that a $v_{\rm rel}^{-2}$ dependence can significantly enhance energy transport from the inner solar core to the outer core. The same can true for any $q$-dependent coupling, if the dark matter mass lies within some specific range for each coupling. This effect can complement direct searches for dark matter; combining these results with state-of-the-art Solar simulations should greatly increase sensitivity to certain DM models. It also seems possible that the so-called Solar Abundance Problem could be resolved by enhanced energy transport in the solar core due to such velocity- or momentum-dependent scatterings.
### Thermal conduction by dark matter with velocity and momentum-dependent cross-sections [Replacement]
We use the formalism of Gould and Raffelt to compute the dimensionless thermal conduction coefficients for scattering of dark matter particles with standard model nucleons via cross-sections that depend on the relative velocity or momentum exchanged between particles. Motivated by models invoked to reconcile various recent results in direct detection, we explicitly compute the conduction coefficients $\alpha$ and $\kappa$ for cross-sections that go as $v_{\rm rel}^2$, $v_{\rm rel}^4$, $v_{\rm rel}^{-2}$, $q^2$, $q^4$ and $q^{-2}$, where $v_{\rm rel}$ is the relative DM-nucleus velocity and $q$ is the momentum transferred in the collision. We find that a $v_{\rm rel}^{-2}$ dependence can significantly enhance energy transport from the inner solar core to the outer core. The same can true for any $q$-dependent coupling, if the dark matter mass lies within some specific range for each coupling. This effect can complement direct searches for dark matter; combining these results with state-of-the-art Solar simulations should greatly increase sensitivity to certain DM models. It also seems possible that the so-called Solar Abundance Problem could be resolved by enhanced energy transport in the solar core due to such velocity- or momentum-dependent scatterings.
### The IRAM-30m line survey of the Horsehead PDR: IV. Comparative chemistry of H2CO and CH3OH
Aims. We investigate the dominant formation mechanism of H2CO and CH3OH in the Horsehead PDR and its associated dense core. Methods. We performed deep integrations of several H2CO and CH3OH lines at two positions in the Horsehead, namely the PDR and dense core, with the IRAM-30m telescope. In addition, we observed one H2CO higher frequency line with the CSO telescope at both positions. We determine the H2CO and CH3OH column densities and abundances from the single-dish observations complemented with IRAM-PdBI high-angular resolution maps (6") of both species. We compare the observed abundances with PDR models including either pure gas-phase chemistry or both gas-phase and grain surface chemistry. Results. We derive CH3OH abundances relative to total number of hydrogen atoms of ~1.2e-10 and ~2.3e-10 in the PDR and dense core positions, respectively. These abundances are similar to the inferred H2CO abundance in both positions (~2e-10). We find an abundance ratio H2CO/CH3OH of ~2 in the PDR and ~1 in the dense core. Pure gas-phase models cannot reproduce the observed abundances of either H2CO or CH3OH at the PDR position. Both species are therefore formed on the surface of dust grains and are subsequently photodesorbed into the gas-phase at this position. At the dense core, on the other hand, photodesorption of ices is needed to explain the observed abundance of CH3OH, while a pure gas-phase model can reproduce the observed H2CO abundance. The high-resolution observations show that CH3OH is depleted onto grains at the dense core. CH3OH is thus present in an envelope around this position, while H2CO is present in both the envelope and the dense core itself. Conclusions. Photodesorption is an efficient mechanism to release complex molecules in low FUV-illuminated PDRs, where thermal desorption of ice mantles is ineffective.
### Collapse of a molecular cloud core to stellar densities: stellar core and outflow formation in radiation magnetohydrodynamics simulations
We have performed smoothed particle radiation magnetohydrodynamics (SPRMHD) simulations of the collapse of rotating, magnetised molecular cloud cores to form protostars. The calculations follow the formation and evolution of the first hydrostatic core, the collapse to form a stellar core, the launching of outflows from both the first hydrostatic core and stellar cores, and the breakout of the stellar outflow from the remnant of the first core. We investigate the roles of magnetic fields and thermal feedback on the outflow launching process, finding that both magnetic and thermal forces contribute to the launching of the stellar outflow. We also follow the stellar cores until they grow to masses of up to 20 Jupiter-masses, and determine their properties. We find that at this early stage, before fusion begins, the stellar cores have radii of $\approx 3$ R$_\odot$ with radial entropy profiles that increase outward (i.e. are convectively stable) and minimum entropies per baryon of $s/k_{\rm B} \approx 14$ in their interiors. The structure of the stellar cores is found to be insensitive to variations in the initial magnetic field strength. With reasonably strong initial magnetic fields, accretion on to the stellar cores occurs through inspiralling magnetised pseudo-discs with negligible radiative losses, as opposed to first cores which effectively radiate away the energy liberated in the accretion shocks at their surfaces. We find that magnetic field strengths of >10 kG can be implanted in stellar cores at birth.
### Ejecting the envelope of red supergiant stars with jets launched by an inspiraling neutron star [Replacement]
We study the properties of the jets launched by a neutron star spiraling inside the envelope and core of a red supergiant (RSG) star, and find that Thorne-Zytkow objects (TZO) cannot be produced via a common envelope (CE) evolution. We use the jet-feedback mechanism, where energy deposited by the jets drives the ejection of the entire envelope and part of the core, and find a very strong interaction of the jets with the core material at late phases of the CE evolution. Following our results we speculate on two rare processes that might take place in the evolution of massive stars. (1) In recent studies it was claim that the peculiar abundances of the HV2112 RSG star can be explained if this star is a TZO. We instead speculate that the rich-calcium envelope comes from a supernova explosion of a stellar companion that was only slight more massive that HV2112, such that during its explosion HV2112 was already a giant that intercepted a relatively large fraction of the SN ejecta. (2) We raise the possibility that strong r-process nucleosynthesis, where elements with high atomic weight of A>130 are formed, occurs inside the jets that are launched by the NS inside the core of the RSG star.
### From the crust to the core of Neutron Stars on a microscopic basis [Replacement]
Within a microscopic approach the structure of Neutron Stars is usually studied by modelling the homogeneous nuclear matter of the core by a suitable Equation of State, based on a many-body theory, and the crust by a functional based on a more phenomenological approach. We present the first calculation of Neutron Star overall structure by adopting for the core an Equation of State derived from the Brueckner-Hartree-Fock theory and for the crust, including the pasta phase, an Energy Density Functional based on the same Equation of State, and which is able to describe accurately the binding energy of nuclei throughout the mass table. Comparison with other approaches is discussed. The relevance of the crust Equation of state for the Neutron Star radius is particularly emphasised.
### Disruption of a Red Giant Star by a Supermassive Black Hole and the Case of PS1-10jh [Replacement]
The development of a new generation of theoretical models for tidal disruptions is timely, as increasingly diverse events are being captured in surveys of the transient sky. Recently, Gezari et al. reported a discovery of a new class of tidal disruption events: the disruption of a helium-rich stellar core, thought to be a remnant of a red giant (RG) star. Motivated by this discovery and in anticipation of others, we consider tidal interaction of an RG star with a supermassive black hole (SMBH) which leads to the stripping of the stellar envelope and subsequent inspiral of the compact core toward the black hole. Once the stellar envelope is removed the inspiral of the core is driven by tidal heating as well as the emission of gravitational radiation until the core either falls into the SMBH or is tidally disrupted. In the case of tidal disruption candidate PS1-10jh we find that there is a set of orbital solutions at high eccentricities in which the tidally stripped hydrogen envelope is accreted by the SMBH before the helium core is disrupted. This places the RG core in a portion of parameter space where strong tidal heating can lift the degeneracy of the compact remnant and disrupt it before it reaches the tidal radius. We consider how this sequence of events explains the puzzling absence of the hydrogen emission lines from the spectrum of PS1-10jh and gives rise to its other observational features.
### Exploding Core-Collapse Supernovae by Jets-Driven Feedback Mechanism [Replacement]
We study the flow structure in the jittering-jets explosion model of core-collapse supernovae (CCSNe) using 2.5D hydrodynamical simulations and find that some basic requirements for explosion are met by the flow. In the jittering-jets model jets are launched by intermittent accretion disk around the newly born neutron star and in stochastic directions. They deposit their kinetic energy inside the collapsing core and induce explosion by ejecting the outer core. The accretion and launching of jets is operated by a feedback mechanism: when the jets manage to eject the core, the accretion stops. We find that even when the jets’ directions are varied around the symmetry axis they inflate hot bubbles that manage to expel gas in all directions. We also find that although most of the ambient core gas is ejected outward, sufficient mass to power the jets is accreted (0.1Mo), mainly from the equatorial plane direction. This is compatible with the jittering jets explosion mechanism being a feedback mechanism.
### Does a prestellar core always become protostellar? Tracing the evolution of cores from the prestellar to protostellar phase
Recently, a subset of starless cores whose thermal Jeans mass is apparently overwhelmed by the mass of the core has been identified, e.g., the core {\small L183}. In literature, massive cores such as this one are often referred to as "super-Jeans cores". As starless cores are perhaps on the cusp of forming stars, a study of their dynamics will improve our understanding of the transition from the prestellar to the protostellar phase. In the present work we use non-magnetic polytropes belonging originally to the family of the Isothermal sphere. For the purpose, perturbations were applied to individual polytropes, first by replacing the isothermal gas with a gas that was cold near the centre of the polytrope and relatively warm in the outer regions, and second, through a slight compression of the polytrope by raising the external confining pressure. Using this latter configuration we identify thermodynamic conditions under which a core is likely to remain starless. In fact, we also argue that the attribute "super-Jeans" is subjective and that these cores do not formally violate the Jeans stability criterion. On the basis of our test results we suggest that gas temperature in a star-forming cloud is crucial towards the formation and evolution of a core. Simulations in this work were performed using the particle-based Smoothed Particle Hydrodynamics algorithm. However, to establish numerical convergence of the results we suggest similar tests with a grid-scheme, such as the Adaptive mesh refinement.
### Unveiling a network of parallel filaments in the Infrared Dark Cloud G14.225-0.506
We present the results of combined NH3(1,1) and (2,2) line emission observed with the Very Large Array and the Effelsberg 100m telescope of the Infrared Dark Cloud G14.225-0.506. The NH3 emission reveals a network of filaments constituting two hub-filament systems. Hubs are associated with gas of rotational temperature Trot \sim 25 K, non-thermal velocity dispersion ~1.1 km/s, and exhibit signs of star formation, while filaments appear to be more quiescent (Trot \sim 11 K, non-thermal velocity dispersion ~0.6 km/s). Filaments are parallel in projection and distributed mainly along two directions, at PA \sim 10 deg and 60 deg, and appear to be coherent in velocity. The averaged projected separation between adjacent filaments is between 0.5 pc and 1pc, and the mean width of filaments is 0.12 pc. Cores within filaments are separated by ~0.33 pc, which is consistent with the predicted fragmentation of an isothermal gas cylinder due to the ‘sausage’-type instability. The network of parallel filaments observed in G14.225-0.506 is consistent with the gravitational instability of a thin gas layer threaded by magnetic fields. Overall, our data suggest that magnetic fields might play an important role in the alignment of filaments, and polarization measurements in the entire cloud would lend further support to this scenario.
### The Nature of the H2-Emitting Gas in the Crab Nebula
Understanding how molecules and dust might have formed within a rapidly expanding young supernova remnant is important because of the obvious application to vigorous supernova activity at very high redshift. In previous papers, we found that the H2 emission is often quite strong, correlates with optical low-ionization emission lines, and has a surprisingly high excitation temperature. Here we study Knot 51, a representative, bright example, for which we have available long slit optical and NIR spectra covering emission lines from ionized, neutral, and molecular gas, as well as HST visible and SOAR Telescope NIR narrow-band images. We present a series of CLOUDY simulations to probe the excitation mechanisms, formation processes and dust content in environments that can produce the observed H2 emission. We do not try for an exact match between model and observations given Knot 51′s ambiguous geometry. Rather, we aim to explain how the bright H2 emission lines can be formed from within the volume of Knot 51 that also produces the observed optical emission from ionized and neutral gas. Our models that are powered only by the Crab’s synchrotron radiation are ruled out because they cannot reproduce the strong, thermal H2 emission. The simulations that come closest to fitting the observations have the core of Knot 51 almost entirely atomic with the H2 emission coming from just a trace molecular component, and in which there is extra heating. In this unusual environment, H2 forms primarily by associative detachment rather than grain catalysis. In this picture, the 55 H2-emitting cores that we have previously catalogued in the Crab have a total mass of about 0.1 M_sun, which is about 5% of the total mass of the system of filaments. We also explore the effect of varying the dust abundance. We discuss possible future observations that could further elucidate the nature of these H2 knots.
### The Abundance, Ortho/Para Ratio, and Deuteration of Water in the High-Mass Star Forming Region NGC 6334 I
We present Herschel/HIFI observations of 30 transitions of water isotopologues toward the high-mass star forming region NGC 6334 I. The line profiles of H_2^{16}O, H_2^{17}O, H_2^{18}O, and HDO show a complex pattern of emission and absorption components associated with the embedded hot cores, a lower-density envelope, two outflow components, and several foreground clouds, some associated with the NGC 6334 complex, others seen in projection against the strong continuum background of the source. Our analysis reveals an H2O ortho/para ratio of 3 +/- 0.5 in the foreground clouds, as well as the outflow. The water abundance varies from ~10^{-8} in the foreground clouds and the outer envelope to ~10^{-6} in the hot core. The hot core abundance is two orders of magnitude below the chemical model predictions for dense, warm gas, but within the range of values found in other Herschel/HIFI studies of hot cores and hot corinos. This may be related to the relatively low gas and dust temperature (~100 K), or time dependent effects, resulting in a significant fraction of water molecules still locked up in dust grain mantles. The HDO/H_2O ratio in NGC 6334 I, ~2 10^{-4}, is also relatively low, but within the range found in other high-mass star forming regions.
### Quark-hybrid matter in the cores of massive neutron stars
Using a nonlocal extension of the SU(3) Nambu-Jona Lasinio model, which reproduces several of the key features of Quantum Chromodynamics, we show that mixed phases of deconfined quarks and confined hadrons (quark-hybrid matter) may exist in the cores of neutron stars as massive as around 2.1 M_Sun. The radii of these objects are found to be in the canonical range of $\sim 12-13$ km. According to our study, the transition to pure quark matter does not occur in stable neutron stars, but is shifted to neutron stars which are unstable against radial oscillations. The implications of our study for the recently discovered, massive neutron star PSR J1614-2230, whose gravitational mass is $1.97 \pm 0.04 M_Sun$, are that this neutron star may contain an extended region of quark-hybrid matter at it center, but no pure quark matter.
### The structure and kinematics of dense gas in NGC 2068
We have carried out a survey of the NGC 2068 region in the Orion B molecular cloud using HARP on the JCMT, in the 13CO and C18O (J = 3-2) and H13CO+ (J = 4-3) lines. We used 13CO to map the outflows in the region, and matched them with previously defined SCUBA cores. We decomposed the C18O and H13CO+ into Gaussian clumps, finding 26 and 8 clumps respectively. The average deconvolved radii of these clumps is 6200 +/- 2000 AU and 3600 +/- 900 AU for C18O and H13CO+ respectively. We have also calculated virial and gas masses for these clumps, and hence determined how bound they are. We find that the C18O clumps are more bound than the H13CO+ clumps (average gas mass to virial mass ratio of 4.9 compared to 1.4). We measure clump internal velocity dispersions of 0.28 +/- 0.02 kms-1 and 0.27 +/- 0.04 kms-1 for C18O and H13CO+ respectively, although the H13CO+ values are heavily weighted by a majority of the clumps being protostellar, and hence having intrinsically greater linewidths. We suggest that the starless clumps correspond to local turbulence minima, and we find that our clumps are consistent with formation by gravoturbulent fragmentation. We also calculate inter-clump velocity dispersions of 0.39 +/- 0.05 kms-1 and 0.28 +/- 0.08 kms-1 for C18O and H13CO+ respectively. The velocity dispersions (both internal and external) for our clumps match results from numerical simulations of decaying turbulence in a molecular cloud. However, there is still insufficient evidence to conclusively determine the type of turbulence and timescale of star formation, due to the small size of our sample.
### CosmoHammer: Cosmological parameter estimation with the MCMC Hammer
We study the benefits and limits of parallelised Markov chain Monte Carlo (MCMC) sampling in cosmology. MCMC methods are widely used for the estimation of cosmological parameters from a given set of observations and are typically based on the Metropolis-Hastings algorithm. Some of the required calculations, such as evaluating the likelihood, can however be computationally intensive, meaning that a single long chain can take several hours or days to calculate. In practice, this can be limiting, since the MCMC process needs to be performed many times to test the impact of possible systematics and to understand the robustness of the measurements being made. To achieve greater speed through parallelisation, algorithms need to have short auto-correlation times and minimal overheads caused by tuning and burn-in. In order to efficiently distribute the MCMC sampling over thousands of cores on modern cloud computing infrastructure, we developed a Python framework called CosmoHammer which embeds emcee, an implementation by Foreman-Mackey et al. (2012) of the affine invariant ensemble sampler by Goodman and Weare (2010). We test the performance of CosmoHammer for cosmological parameter estimation from cosmic microwave background data. While Metropolis-Hastings is constrained by overheads, CosmoHammer is able to accelerate the sampling process from a wall time of 30 hours on a single machine to 16 minutes by the efficient use of 2048 cores. Such short wall times for complex data sets opens possibilities for extensive model testing and control of systematics.
### The Correlation of Dust and Gas Emission in Star-Forming Environments
We present ammonia maps of portions of the W3 and Perseus molecular clouds in order to compare gas emission with continuum thermal emission. These are commonly expected to trace the same mass component in star-forming regions, often under the assumption of LTE. The star-forming regions are found to have different physical characteristics consistent with their identification as low-mass and high-mass respectively. Accounting for the distance of the W3 region does not fully reconcile these differences, suggesting that there is an underlying difference in the structure of the two regions. Peak positions of submillimetre and ammonia emission do not correlate strongly. Also, the extent of diffuse emission is only moderately matched between ammonia and thermal emission. Source sizes measured from our observations are consistent between regions, although there is a noticeable difference between the submillimeter source sizes in the two observed regions. Fractional abundance measurements of ammonia indicate a dip in abundance at the positions of peak submillimetre flux. Although, we find that depletion of ammonia in our sources is unlikely. Virial ratios are determined which show that sources in Perseus are generally not gravitationally bound and that sources in W3 are, although there is considerable scatter in both samples. We find that this that external pressure is necessary for cores at small scales to be bound while sources and clusters are gravitationally bound on larger scales. Our results indicate that assumptions of local thermal equilibrium and/or the coupling of the dust and gas phases in star-forming regions may not be as robust as commonly assumed. Alternatively, the assumption that ammonia and thermal emission trace the same mass component in these regions may need to be revisited, along with the degree to which the excitation conditions within a star-forming region vary.
### Velocity width measurements of the coolest X-ray emitting material in the cores of clusters, groups and elliptical galaxies
We examine the velocity width of cool X-ray emitting material using XMM-Newton Reflection Grating Spectrometer (RGS) spectra of a sample of clusters and group of galaxies and elliptical galaxies. Improving on our previous analyses, we apply a spectral model which accounts for broadening due to the spatial extent of the source. With both conventional and Markov Chain Monte Carlo approaches we obtain limits, or in a few cases measurements, of the velocity broadening of the coolest X-ray material. In our sample, we include new observations targeting objects with compact, bright, line-rich cores. One of these, MACSJ2229.7-2755, gives a velocity limit of 280 km/s at the 90 per cent confidence level. Other systems with limits close to 300 km/s include A1835, NGC4261 and NGC4472. For more than a third of the targets we find limits better than 500 km/s. HCG62, NGC1399 and A3112 show evidence for ~400 km/s velocity broadening. For a smaller sample of objects, we use continuum-subtracted emission line surface brightness profiles to account for the spatial broadening. Although there are significant systematic errors associated with the technique (~150 km/s), we find broadening at the level of 280 to 500 km/s in A3112, NGC1399 and NGC4636.
### Dust continuum and Polarization from Envelope to Cores in Star Formation: A Case Study in the W51 North region
We present the first high-angular resolution (up to 0.7", ~5000 AU) polarization and thermal dust continuum images toward the massive star-forming region W51 North. The observations were carried out with the Submillimeter Array (SMA) in both the subcompact (SMA-SubC) and extended (SMA-Ext) configurations at a wavelength of 870 micron. W51 North is resolved into four cores (SMA1 to SMA4) in the 870 micron continuum image. The associated dust polarization exhibits more complex structures than seen at lower angular resolution. We analyze the inferred morphologies of the plane-of-sky magnetic field (B_bot) in the SMA1 to SMA4 cores and in the envelope using the SMA-Ext and SMA-SubC data. These results are compared with the B_bot archive images obtained from the CSO and JCMT. A correlation between dust intensity gradient position angles (phi_{nabla I}) and magnetic field position angles (phi_B) is found in the CSO, JCMT and both SMA data sets. This correlation is further analyzed quantitatively. A systematically tighter correlation between phi_{nabla I} and phi_B is found in the cores, whereas the correlation decreases in outside-core regions. Magnetic field-to-gravity force ratio (Sigma_B) maps are derived using the newly developed polarization – intensity gradient method by Koch, Tang & Ho 2012. We find that the force ratios tend to be small (Sigma_B <= 0.5) in the cores in all 4 data sets. In regions outside of the cores, the ratios increase or the field is even dominating gravity (Sigma_B > 1). This possibly provides a physical explanation of the tightening correlation between phi_{nabla I} and phi_B in the cores: the more the B field lines are dragged and aligned by gravity, the tighter the correlation is. Finally, we propose a schematic scenario for the magnetic field in W51 North to interpret the four polarization observations at different physical scales.
### Misalignment of Magnetic Fields and Outflows in Protostellar Cores
Theoretical models of star formation generally assume that bipolar outflows are parallel to the mean magnetic-field direction in protostellar cores. Here we present results of \lambda1.3 mm dust polarization observations toward 16 nearby, low-mass protostars, mapped with ~2.5" resolution at CARMA. The results show that magnetic fields in protostellar cores on scales of ~1000 AU are not tightly aligned with outflows from the protostars. If one assumes that outflows emerge along the rotation axes of circumstellar disks, then our results imply that these disks are not aligned with the fields in the cores from which they formed.
### Nonlinear Gravitational Recoil from the Mergers of Precessing Black-Hole Binaries [Cross-Listing]
We present results from an extensive study of 83 precessing, equal-mass black-hole binaries with large spins, a/m=0.8, and use these data to model new nonlinear contributions to the gravitational recoil imparted to the merged black hole. We find a new effect, the "cross kick", that enhances the recoil for partially aligned binaries beyond the "hangup kick" effect. This has the consequence of increasing the probabilities (by nearly a factor two) of recoils larger than 2000 km/s, and, consequently, of black holes getting ejected from galaxies and globular clusters, as well as the observation of large differential redshifts/blueshifts in the cores of recently merged galaxies.
### Associated 21-cm absorption towards the cores of radio galaxies
We present the results of Giant Metrewave Radio Telescope (GMRT) observations to detect H{\sc i} in absorption towards the cores of a sample of radio galaxies. From observations of a sample of 16 sources, we detect H{\sc i} in absorption towards the core of only one source, the FR\,II radio galaxy 3C\,452 which has been reported earlier by Gupta & Saikia (2006a). In this paper we present the results for the remaining sources which have been observed to a similar optical depth as for a comparison sample of compact steep-spectrum (CSS) and giga-hertz peaked spectrum (GPS) sources. We also compile available information on H{\sc i} absorption towards the cores of extended radio sources observed with angular resolutions of a few arcsec or better. The fraction of extended sources with detection of H{\sc i} absorption towards their cores is significantly smaller (7/47) than the fraction of H{\sc i} detection towards CSS and GPS objects (28/49). For the cores of extended sources, there is no evidence of a significant correlation between H{\sc i} column density towards the cores and the largest linear size of the sources. The distribution of the relative velocity of the principal absorbing component towards the cores of extended sources is not significantly different from that of the CSS and GPS objects. However, a few of the CSS and GPS objects have blue-shifted components $\gapp$1000 km s$^{-1}$, possibly due to jet-cloud interactions. With the small number of detections towards cores, the difference in detection rate between FR\,I (4/32) and FR\,II (3/15) sources is within the statistical uncertainties.
### Implementation of Sink Particles in the Athena Code
We describe implementation and tests of sink particle algorithms in the Eulerian grid-based code Athena. Introduction of sink particles enables long-term evolution of systems in which localized collapse occurs, and it is impractical (or unnecessary) to resolve the accretion shocks at the centers of collapsing regions. We discuss similarities and differences of our methods compared to other implementations of sink particles. Our criteria for sink creation are motivated by the properties of the Larson-Penston collapse solution. We use standard particle-mesh methods to compute particle and gas gravity together. Accretion of mass and momenta onto sinks is computed using fluxes returned by the Riemann solver. A series of tests based on previous analytic and numerical collapse solutions is used to validate our method and implementation. We demonstrate use of our code for applications with a simulation of planar converging supersonic turbulent flow, in which multiple cores form and collapse to create sinks; these sinks continue to interact and accrete from their surroundings over several Myr.
### The faint source population at 15.7 GHz - I. The radio properties
We have studied a sample of 296 faint (> 0.5 mJy) radio sources selected from an area of the Tenth Cambridge (10C) survey at 15.7 GHz in the Lockman Hole. By matching this catalogue to several lower frequency surveys (e.g. including a deep GMRT survey at 610 MHz, a WSRT survey at 1.4 GHz, NVSS, FIRST and WENSS) we have investigated the radio spectral properties of the sources in this sample; all but 30 of the 10C sources are matched to one or more of these surveys. We have found a significant increase in the proportion of flat spectrum sources at flux densities below approximately 1 mJy – the median spectral index between 15.7 GHz and 610 MHz changes from 0.75 for flux densities greater than 1.5 mJy to 0.08 for flux densities less than 0.8 mJy. This suggests that a population of faint, flat spectrum sources is emerging at flux densities below 1 mJy. The spectral index distribution of this sample of sources selected at 15.7 GHz is compared to those of two samples selected at 1.4 GHz from FIRST and NVSS. We find that there is a significant flat spectrum population present in the 10C sample which is missing from the samples selected at 1.4 GHz. The 10C sample is compared to a sample of sources selected from the SKADS Simulated Sky by Wilman et al. and we find that this simulation fails to reproduce the observed spectral index distribution and significantly underpredicts the number of sources in the faintest flux density bin. It is likely that the observed faint, flat spectrum sources are a result of the cores of FRI sources becoming dominant at high frequencies. These results highlight the importance of studying this faint, high frequency population.
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2014-07-23 23:40:56
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http://codereview.stackexchange.com/questions/18346/optimizing-divisor-sieve
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Optimizing Divisor Sieve
I have two sieves that I wrote in python and would like help optimizing them if at all possible. The divisorSieve calculates the divisors of all numbers up to n. Each index of the list contains a list of its divisors. The numDivisorSieve just counts the number of divisors each index has but doesn't store the divisors themselves. These sieves work in a similar way as you would do a Sieve of Eratosthenes to calculate all prime numbers up to n.
Note: divs[i * j].append(i) changed from divs[i * j] += [i] with speed increase thanks to a member over at stackoverflow. I updated the table below with the new times for divisorSieve. It was suggested to use this board instead so I look forward to your input.
def divisorSieve(n):
divs = [[1] for x in xrange(0, n + 1)]
divs[0] = [0]
for i in xrange(2, n + 1):
for j in xrange(1, n / i + 1):
divs[i * j].append(i) #changed from += [i] with speed increase.
return divs
def numDivisorSieve(n):
divs = [1] * (n + 1)
divs[0] = 0
for i in xrange(2, n + 1):
for j in xrange(1, n / i + 1):
divs[i * j] += 1
return divs
#Timer test for function
if __name__=='__main__':
from timeit import Timer
n = ...
t1 = Timer(lambda: divisorSieve(n))
print n, t1.timeit(number=1)
Results:
-----n-----|--time(divSieve)--|--time(numDivSieve)--
100,000 | 0.333831560615 | 0.187762331281
200,000 | 0.71700566026 | 0.362314797537
300,000 | 1.1643773714 | 0.55124339118
400,000 | 1.63861821235 | 0.748340797412
500,000 | 2.06917832929 | 0.959312993718
600,000 | 2.52753840891 | 1.17777010636
700,000 | 3.01465945139 | 1.38268800149
800,000 | 3.49267338434 | 1.62560614543
900,000 | 3.98145114138 | 1.83002270324
1,000,000 | 4.4809342539 | 2.10247496423
2,000,000 | 9.80035361075 | 4.59150618897
3,000,000 | 15.465184114 | 7.24799900479
4,000,000 | 21.2197508864 | 10.1484527586
5,000,000 | 27.1910144928 | 12.7670585308
6,000,000 | 33.6597508864 | 15.4226118057
7,000,000 | 39.7509513591 | 18.2902677738
8,000,000 | 46.5065447534 | 21.1247001928
9,000,000 | 53.2574136966 | 23.8988925173
10,000,000 | 60.0628718044 | 26.8588813211
11,000,000 | 66.0121182435 | 29.4509693973
12,000,000 | MemoryError | 32.3228102258
20,000,000 | MemoryError | 56.2527237669
30,000,000 | MemoryError | 86.8917332214
40,000,000 | MemoryError | 118.457179822
50,000,000 | MemoryError | 149.526622815
60,000,000 | MemoryError | 181.627320396
70,000,000 | MemoryError | 214.17467749
80,000,000 | MemoryError | 246.23677614
90,000,000 | MemoryError | 279.53308422
100,000,000 | MemoryError | 314.813166014
Results are pretty good and I'm happy I was able to get it this far, but I'm looking to get it even faster. If at all possible, I'd like to get 100,000,000 at a reasonable speed with the divisorSieve. Although this also brings into the issue that anything over 12,000,000+ throws a MemoryError at divs = [[1] for x in xrange(0, n + 1)]) in divisorSieve. numDivisorSieve does allow the full 100,000,000 to run. If you could also help get past the memory error, that would be great.
I've tried replacing numDivisorSieve's divs = [1] * (n + 1) with both divs = array.array('i', [1] * (n + 1)) and divs = numpy.ones((n + 1), dtype='int') but both resulted in a loss of speed (slight difference for array, much larger difference for numpy). I expect that since numDivisorSieve had a loss in efficiency, then so would divisorSieve. Of course there's always the chance I'm using one or both of these incorrectly since I'm not used to either of them.
I would appreciate any help you can give me. I hope I have provided enough details. Thank you.
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What are you doing with the result? – Winston Ewert Nov 9 '12 at 2:04
If storing only prime factors counts as 'optimization', we can do ~3-4 times faster. – avip Nov 9 '12 at 2:31
Have you tested the application using Python 64bit? – cat_baxter Nov 9 '12 at 10:48
Thanks for the suggestion about Python 64bit. It's looking like it solves the memory issues. Will update once I've run all the tests – Jeremy K Nov 9 '12 at 22:29
You can use xrange's third param to do the stepping for you to shave off a little bit of time (not huge).
Changing:
for j in xrange(1, n / i + 1):
divs[i * j].append(i)
To:
for j in xrange(i, n + 1, i):
divs[j].append(i)
For n=100000, I go from 0.522774934769 to 0.47496509552. This difference is bigger when made to numDivisorSieve, but as I understand, you're looking for speedups in divisorSieve
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This was a great optimization! for n = 10,000,000 divisorSieve's time is down to **9.56704298066** and for n = 100,000,000 numDivisorSieve's time is down to **67.1441108416** which are both great optimizations. – Jeremy K Nov 9 '12 at 21:52
Wow... that's better than I expected! Glad I could help. – Adam Wagner Nov 9 '12 at 22:07
Well...apparently the reason it did so well is I had the range messed up. So while it's still an improvement, it's not quite as good as I thought. Doesn't make me appreciate your help any less, just makes me feel a little stupid. Guess that's what I get for not testing the output well enough. Will update the original post when I get a chance to recompute all the results – Jeremy K Nov 11 '12 at 4:14
@JeremyK That's fine. At least I know I'm not crazy now. :) – Adam Wagner Nov 11 '12 at 4:28
@JeremyK I was commenting on your OP about the erratic range before reading this. You should update the post - atleast change the code and say that the results are incorrect for now. – S Prasanth Dec 13 '12 at 7:03
EDIT: map(lambda s: s.append(i) , [divs[ind] for ind in xrange(i, n + 1, i)]) Seems to be ~0.2% faster ~2 times slower than Adam Wagner's (for n=1000000)
The infamous 'test the unit test' problem.
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Maybe I'm doing something incorrectly, but when I put this in there I get TypeError: list indices must be integers, not xrange – Jeremy K Nov 9 '12 at 21:52
~2 times slower Must be due to function call overhead for calling the lambda function. – S Prasanth Dec 13 '12 at 7:55
The following offers a very very small improvement to divisorSieve and a good improvement to numdivisorSieve. But the factors will not be sorted inside each list. For example the factors list of of 16 will be [4, 2, 8, 1, 16].
def divisorSieve(n):
divs = [[] for j in xrange(n + 1)]
nsqrt = int(sqrt(n))
for i in xrange(1, nsqrt + 1):
divs[i*i].append(i)
for j in xrange(i, i*i, i):
divs[j].append(j/i) #If j/i is replaced by i, a good improvement is seen. Of course, that would be wrong.
divs[j].append(i)
for i in xrange(i+1, n+1):
for j in xrange(i, n+1, i):
divs[j].append(j/i)
divs[j].append(i)
return divs
def numdivisorSieve(n):
divs = [1] * (n + 1)
divs[0] = 0
nsqrt = int(sqrt(n))
for i in xrange(2, nsqrt + 1):
divs[i*i] += 1
for j in xrange(i, i*i, i):
divs[j] += 2
for i in xrange(i+1, n+1):
for j in xrange(i, n+1, i):
divs[j] += 2
return divs
Unfortunately, modifying this definition to create two lists divsmall ([4,2,1]) and divlarge ([8,16]) and in the end doing divsmall[j].reverse(); divsmall[j].extend(divlarge[j]); return divsmall makes it slightly slower than the original.
Also, I think it makes more sense for divs[0] to be [] instead of [0]
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2015-05-03 10:54:26
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http://math.stackexchange.com/questions/61857/getting-the-name-of-combinatorial-problems/61864
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# Getting the name of combinatorial problems
I'll often find myself with some combinatorial problem that's obviously been studied before. For example, "Find the smallest set(s) of positive integers such that every integer from 1 to n is the sum of at most two elements of the set." Without becoming an expert on combinatorics, is there some way of finding out the name such a problem goes by in the literature, say in some sort of catalog? Googling and related tactics don't seem to be very helpful here, as most questions of this sort just consist of the words "set, smallest, such that, ..." repeatedly -- there's generally no unique word or phrase to latch on to. For instance, when I tried to Google the problem above, I got back the subset sum problem (given a set, determine whether some subset sums to zero) and the knapsack problem (given a set of objects with specific weights and values, find the most valuable subset under a given total weight), neither of which have anything to do with what I was actually looking for.
I'm not looking for the name of the problem above in particular (although it wouldn't hurt if anyone knows it), but rather some clean way of looking such things up for myself. Does such a catalog exist?
EDIT: My basic idea here is that a large number of combinatorial problems fall into some basic, MADLIBS-style patterns, for instance:
Choices of $m$ elements of the set ___, (with|without) repetition, (with|without) ordering, satisfying the additional constraint ____.
(Paths|circuits) through a (directed|undirected), (vertex|edge) weighted graph, which visit each (edge|vertex), such that the total weight is (maximal|minimal), and such that _.
An index that listed things in this manner would be helpful to non-experts.
-
You've tried checking the Handbook of Combinatorics? – Ｊ. Ｍ. Sep 4 '11 at 17:32
This isn't intended to sound snarky, but the best thing you can do is post such a question to SE. Until computers become really good at understanding English, the next best thing is to ask experts and see what they know. – Austin Mohr Sep 4 '11 at 17:50
The Handbook of Combinatorics does not appear to have any index of that sort -- it looks more like a survey text than a catalog. Am I missing something? – Daniel McLaury Sep 4 '11 at 20:43
There is the twelve-fold way, which is a classification of several of the most common combinatorial problems along the lines you are asking about in your edit. If you have $n$ balls to place into $x$ boxes, you can have the balls labeled or not, the boxes labeled or not, and whether the mapping of balls to boxes has no restrictions (choosing the boxes "with replacement," in some contexts), is injective (no more than one ball per box, or choosing the boxes "without replacement," in some contexts), or is surjective (at least one ball per box). Considering all these possibilities gives the twelve options in the name of the classification.
See also Richard Stanley's Enumerative Combinatorics, Vol. I, where it is discussed in detail.
Added (in response to OP's request for generalization): On p. 88 of Stanley's text (which you can download directly from his site using the link above) he says, "There are many possible generalizations of the Twelvefold Way and its individual entries." He then goes on to discuss one of them. In the notes (p. 107) on the chapter in which the Twelvefold Way appears Stanley also says, "An extension of the Twelvefold Way to a 'Thirtyfold Way' (and suggestion of even more entries) is due to R. Proctor." Tracking down the reference leads to Proctor's article "Let's Expand Rota's Twelvefold Way For Counting Partitions!", which I'm not familiar with but which certainly appears to be the kind of thing you're asking for.
-
This is exactly the sort of thing I was looking for. Do you know if anyone has extended this on a larger scale? – Daniel McLaury Sep 5 '11 at 3:22
@user3296: See my new edit. – Mike Spivey Sep 5 '11 at 4:39
One way is to calculate the terms for small $n$ (in your example the size of the smallest set with your desired property) and then look these up in the On-Line Encyclopedia of Integer Sequences
-
I try to use OEIS when possible, but I often find myself running into problems with the law of small numbers. Often a sequence will have some obstruction that only kicks in around, say, n = 15, at which point computing the terms is prohibitively expensive (say, because you're searching 2^15 different sets with an expensive function). – Daniel McLaury Sep 4 '11 at 20:43
@user3296: It depends on the size of your terms. It may be that if you calculate the first 10 (or whatever is reasonable) you only get a few dozen series. Then reading the notes to those you can pick out the one you want. True, if your terms are all single digits you will have too much to read through. But if you only have one term of order 10^5 you often can find the series you want. – Ross Millikan Sep 4 '11 at 21:17
Agreed. But, for instance, today I was looking at a problem which generated the sequence 2, 4, 6, 10, 14, 18, 22, 28, 34,..., where the terms after 22 became almost prohibitively expensive to calculate. The structure of this sequence is hard to discern: taking first differences, we have 2, 2, 4, 4, 4, 4, 6, 6, ..., but how many sixes should there be? (As it turns out, I was able to bound the sequence below and prove that it's not in the OEIS, but this illustrates the point.) – Daniel McLaury Sep 4 '11 at 23:12
The name that you are not looking for is "The postage stamp problem," C12 in Guy, Unsolved Problems In Number Theory. "A popular form of it concerns the design of a set of integer denominations of postage stamp, $A_k=\lbrace\,a_1,\dots,a_k\,\rbrace$ with $1=a_1\lt a_2\lt\cdots\lt a_k$ to be used on envelopes with room for at most $h$ stamps, so that all integer amounts of postage up to a given bound can be affixed. [...] At first the main interest was in the global problem: given $h$ and $k$, find an extremal basis $A_k'$ with largest possible $h$-range," that is, one which maximizes the smallest integer you can't get. Your problem is the case $h=2$. There is much information about this (and many, many other combinatorial number theory problems) in the book.
-
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2014-10-26 01:22:36
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https://www.gamedev.net/forums/topic/202185-windows-programone-big-class/
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Archived
This topic is now archived and is closed to further replies.
Windows program==one big class.
This topic is 5152 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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Is there anything wrong with making a windows program one big class? What I mean is having a class that has everything in it (includeing the main loop).
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Um, have you tried putting main inside the class? O___O
[edited by - nervo on January 17, 2004 12:11:20 AM]
Why, yes I have.
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Those are called God classes and usually aren''t well regarded.
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quote:
Original post by antareus
Those are called God classes and usually aren't well regarded.
Ok. Thank you. Can you tell me why though?
EDIT: Well, I did some research and it says that a god class is a class that has more than one purpose. The program I am writing is a simple map editor and everything done has to do with pretty much the same stuff. Is it still wrong to have a god class?
EDIT2: Also, this god class of mine has several other classes in it. Is it still considered a god class?
[edited by - tHiSiSbOb on January 17, 2004 12:27:20 AM]
[edited by - tHiSiSbOb on January 17, 2004 12:28:26 AM]
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Why even bother making it a class then. Just use namespaces and functions, cause it looks like to me you''re using classes just for the sake of using classes. If you want to do it the proper way, check out this tutorial.
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quote:
EDIT2: Also, this god class of mine has several other classes in it. Is it still considered a god class?
Yes. God classes are named that way because they generally do more than one thing and/or they 'know' too much. In software engineering terms, the God class is coupled to almost all of the other classes, inhibiting maintainability. Usually God classes end up mangling presentation of data with the actual logic that is done on the data. Although that sort of thing is fine for quick, small programs, (think VB) it can become harder to maintain.
Decouple the two using something like signals and slots. I have an entire app that is based only on signals and slots. They are very robust and scalable.
[edited by - antareus on January 18, 2004 1:12:25 AM]
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quote:
Original post by tHiSiSbOb
Is there anything wrong with making a windows program one big class? What I mean is having a class that has everything in it (includeing the main loop).
There isn''t anything wrong with it if you are using C# or Java. The C++ language tends to disagree with this practice, however.
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quote:
There isn''t anything wrong with it if you are using C# or Java.
o__O
"Everything is a class" != "it''s ok to use just one class". WTF.
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It''s already been done. MFC is an object oriented wrapper around win32.
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quote:
Original post by the_dannobot
It''s already been done. MFC is an object oriented wrapper around win32.
It is generally a bad idea because monolithic software is more difficult to maintain, and one main class would pretty much have to be monolithic.
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I plan to make my main() look like this:
int wWinMain( HINSTANCE, HINSTANCE, LPSTR, int ) { new Kernel(); Kernel::GetSingleton.Run(); Kernel::Destroy(); return 0;}
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quote:
Original post by the_dannobot
It''s already been done. MFC is an poorly designed and overly complex object oriented wrapper around win32.
There you go.
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quote:
Original post by Zahlman
quote:
There isn't anything wrong with it if you are using C# or Java.
o__O
"Everything is a class" != "it's ok to use just one class". WTF.
He did not specify the size of the program. Of course it's not very intelligent to always use a single class but for small programs in those languages a single class might be fine.
[edited by - Invader X on January 18, 2004 3:43:11 PM]
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One reason to try to split it up a little is because usually it''s better to make small parts of the working application that you can debug and make sure they''ll work.
If something goes wrong, it''s often easier to find the error this way.
You can also more easily work several people on the same project. If you''re not part of a team, this is no problem for you.
It''s also easier from a design point of view... searching through a large .cpp file is just time consuming, and using one(1) .h-file with several(>1) .cpp-files are just bad design and can confuse a lot IMO. Hope you understand what I mean by this.
But sure, for small->medium size applications I could imagine this being a good solution.
Albert
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2018-02-25 22:16:43
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http://nrich.maths.org/2286/index?nomenu=1
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We are first given that: $$x_1 = 2^2 + 3^2 + 6^2$$ $$x_2 = 3^2 + 4^2 + 12^2$$ $$x_3 = 4^2 + 5^2 + 20^2$$ Then show that $x_n$ is always a perfect square.
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2014-10-25 16:13:44
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https://hbfs.wordpress.com/
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## No post today
April 14, 2015
Except to say that there’sn’t a post today. Things have been busy lately, so no entry for this week. ¯\_(ツ)_/¯
## Beauxindex
April 7, 2015
Last week I presented a simple script that helped spot almost identical index entries. Since then, I’ve sought a better index style. While I found most book use a rather plain index style, I was looking for something that used space efficiently yet was easy to read.
The idxlayout package will help us customize the index somewhat. It will allow us to set the number of columns, set the distance between them, maybe with a line, set font size, and decide how we want topics arranged. In the image above, I have used the singlepar layout. It is invoked by adding the itemlayout=singlepar option as a package option:
\usepackage[itemlayout=singlepar]{idxlayout}
With an extra bit of configuration somewhere else in your LaTeX source:
\renewcommand{\see}[2]{\mdseries \emph{\seename} #1 }
\renewcommand{\indexsubsdelim}{~\raisebox{0.2ex}{\resizebox{1em}{0.5ex}{$\sim$}}
} % symbol(s) between subtopics
\setkeys{ila}{columns=3} % nb columns
\setkeys{ila}{rule=0pt} % width of line between cols (0=no line)
\setkeys{ila}{columnsep=1em} % spacing between columns
\setkeys{ila}{font=footnotesize} % base font size
\setkeys{ila}{indentunit=1em} % how much to indent after first line
\setkeys{ila}{justific=standard} % raggedright is a mess!
\setkeys{ila}{initsep=0.5em plus 0.5em minus 0.1em} % rubberband length between paragraphs
\printindex
The idxlayout package isn’t very well documented, so you might have to dig in the source code itself to find values that you’re interested in changing. It produces:
Which appears… flat.
These tweak make a much better-looking index than the makeindex default, but there are still a few things to tweak. The thing that bothered me most is that we do not really have a clear separation between the head topic and subtopics. To achieve good visual separation, we must redefine yet more variables. This time in makeindex. However, makeindex customization involves an “index style” file:
\makeindex[options=-s back-matter/index.ist]
The configurable elements are found in makeindex‘s man page. The necessary elements are set in the index.ist file, as follows:
item_0 "\n \\bfseries \\item "
item_1 "\n \\mdseries \\subitem "
item_01 "\n \\mdseries \\indexsubsdelim "
item_x1 "\n \\mdseries \\indexsubsdelim "
The item_0 variable defines the style with which a topic will be displayed. For now, in bold. The item_xy variables define the style change when you go from a level x item to a y level item. Since I want subtopic to appear “normal face”, \mdseries is inserted just before the subitem. The item_x1 seems to be also necessary to ensure that only the first topic in a paragraph is bold.
The changes produce the index as shown in the image at the beginning
## Checking a LaTeX Index
March 31, 2015
This week again, LaTeX. This time, the index. At the end of a document, you will usually find an index so that, if you don’t have a magical ctrl-f, you can find something fast in the book.
In LaTeX, creating an index is really easy. You include the package makeindex, and plant \index{topic!subtopic} tags in the text (preferably just besides the word you want to index, the \index command doesn’t understand paragraphs). You add \printindex somewhere else in your document and you run pdflatex (or just latex) to get the index generated. That’s all fine except that it doesn’t provide checks. It adds to the index whatever you typed, and doesn’t give warnings if you have an entry “compression” and an entry “compresion” (because, you know, typos happen). Let’s see how we can somewhat fix that.
## A LaTeX Bibliography Hack
March 24, 2015
Yet another LaTeX hack! This time to insert text between the bibliography/reference header and the actual references. I’ve had, I admit, no really good reason to do this, as I might have added some text before the bibliography, but it made a lot more sense to include the paragraph there.
## A (simple) LaTeX Example Environment
March 17, 2015
This week again, a little LaTeX. As you know, I’m putting the finishing touches to a manuscript I’ve been working on for a little bit more than a year now. Those “finishing touches” always start simple, then end up asking for quite a bit more LaTeX programming that I’d like.
This week, I worked quite a bit to repair my “example” environment that include example number, extra margin, and a different colored box around the text. This helps locate and distinguish examples from normal text. But my first version had little problems. Like equations centered not on the example text but still relative to the whole page. Bummer. Let’s see how I fixed all that.
## The Tosser
March 10, 2015
Here’s a LEGO dice-tossing machine I built.
The two complicated parts are the reduction gear train and the flaps that re-center the dice so that the machine can pick it up correctly each time.
The gear train is composed of several gear reducers, which are composed of a small gear driving a larger gear (thus many turns of the small, driving gear, are needed for the big one to make a complete rotation). Each large gear share its spindle with a small gear that drives the next stage. The gear box yields a 243:1 reduction (which is $3^5$ to 1). Otherwise the motor spins too fast and just, well, eject the dice from the machine.
The flaps are used to funnel the dice back in the middle of the tray, where it can be picked up again.
*
* *
The next step would be to use OCR to read the dice value and see if, in the long run, the tosser is a strong number generator or if it is flawed in some way. Can’t really remember where I got the 16-sided dice, but there are some to be found online.
## X marks the spot
March 3, 2015
I’m presently finishing a large text typeset in LaTeX. Of course, in many places I just put a note to “fill later” and moved on with the rest of the text. But now, I must hunt them down and fill the holes.
Good for me, I had the idea of writing a LaTeX command that not only marked the spot in the rendered page, like this:
…but also, by using the command itself, marked the spot in the source code itself. Otherwise, I’d have to find all the “fill me later” “add more” and whatnots.
The command itself is really not that complicated:
\usepackage{pifont}
\usepackage[usenames,dvipsnames,svgnames,table]{xcolor}
\definecolor{bloodred}{HTML}{B00000}
\newcommand{\fillme}[1]{\textcolor{bloodred}{\smash{\ding{54}}}\message{Forgotten fillme on input line \the\inputlineno}} % 54 big fat X
The command takes an argument, which is ignored, but is quite convenient to leave the future you a message:
\fillme{blurb something about leibniz vs newton}
The command depends on the packages pifont for the dingbat and xcolor to define a nice dark red. The command itself does two things. One is to typeset a blood red X in the generated document so that it is conspicuous. The other is to output a message that can be parsed automagically by a script, making their eradication easier.
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2015-04-21 23:20:36
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https://inbrain.tech/machine-learning-methodologies-in-bci/1235/
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# Machine Learning Methodologies in BCI
Home Behind BCIs Machine Learning Methodologies in BCI
BCI tools and techniques such as signal acquisition, signal processing, feature extraction, Machine Learning algorithms, and classification techniques contribute to the development and improvement of BCI technology.
Almost all BCI systems contain a Machine Learning algorithm as a central part. It learns from training data and derives a function that can be used to discriminate different models of brain activity. It adapts the BCI system to the brain of a particular subject. This reduces the imposed learning load in this regard. For simplicity and practical reasons, Machine Learning algorithms are usually divided into two modules: extraction and classification of features.
The feature extraction module transforms brain signals into a representation that makes the classification simpler, usually data arrays. The goal of extraction is also to remove noise and other unnecessary information from input signals while maintaining important information to discriminate between different classes of signals.
Feature vectors are extracted from brain signals by signal processing methods. The neurophysiological knowledge of the subject in question can help to decide which characteristic of the brain signal is to be expected. This is fundamental for deciding the most discriminating information for the chosen paradigm. These features are translated into a control signal by Machine Learning algorithms.
Classification is a challenging problem due to the low signal-to-noise ratio of EEG signals, the variance of EEG signals for a given subject and the variance between different subjects. Pre-processing techniques serve to improve EEG signals and remove noise and extract relevant features.
## Data preparation
Before being able to apply any Machine Learning methodology, it is necessary to go through some steps that will allow us to transform the signals into tractable data samples.
After a pre-processing phase, the data is provided in the form of a signal. Now it is necessary to separate the signal provided through two sets, of training and testing, in single segments. This is achieved by extracting the data samples between 0-650 ms after the start of each intensification (during the pre-processing phase each significant signal is intensified). Knowing the time of the samples in question helps to extract these segments in the best possible way.
Filtering is a crucial step. It allows noise reduction as most artifacts occur at known frequencies. At this point, the filtered signals were decimated according to the highest cutoff frequency. This leads to the construction of a vector of useful segments which is then normalized: mean zero and variance equal to one.
## Machine Learning and classifiers
One of the main purposes of Machine Learning is classification, that is the problem of identifying the class of a new target on the basis of knowledge extracted from a training set. A system that classifies is called a classifier. The classifiers extract a model from the dataset which then they use to classify the new instances. If a single instance can be expressed as a vector in a numerical space R ^ n the problem of classification can be traced back to the search for the closed surfaces that delimit the classes.
### Linear Classifiers
There are pairs (x, y) of inputs x ∈ X and desired outputs y ∈ Y. A learning algorithm must choose, based on the training examples, a function f: X → Y such that new examples, not contained in the training set, they are correctly mapped to the corresponding output. For practical reasons, the functions f are usually indexed by a set of parameters θ, ie $y = f (x; \theta)$. Therefore, the task is to choose the function equivalent to the choice of parameters. In the binary case Y = {1, -1}, the linear classifier is represented by a single discriminating function given by the vector of the input characteristics ω and the bias b:
$$$$\ f(x) = (ω · x ) + b$$$$
The input vector x is assigned to the class y∈ {1, -1} as follows:
$$\left\{\begin{array} \ +1\,if\,(\omega \cdot x) + b \geq0 \\ -1\,if\,(\omega \cdot x) + b\leq0 \end{array}\right.$$
Different algorithms based on linear classifiers determine the vector ω and the bias b. These parameters obtained in the training phase are used in the test phase to predict the class to which each test example belongs. Algorithms based on linear classifiers can be distinguished based on their performance.
### Linear Support Vector Machines (SVM)
The signals provided are high in size with a low signal-to-noise ratio. There is also another problem: the signal response varies due to EEG signal components unrelated to the brain activity of a single subject. SVM is a powerful approach for pattern recognition and in particular for large problems, so it is frequently used in BCI searches.
SVM is one of the most used tools for pattern classification because instead of estimating the probability densities of the classes, it directly solves the problem of interest, or determine the decisional surfaces between the classes (classification boundaries).
Given two classes of linearly separable multidimensional patterns, of all the possible separation hyperplanes, SVM determines the one capable of separating the classes with the greatest possible margin. The margin is the minimum distance of points of two classes in training set from the identified hyperplane.
Margin maximization is related to generalization. If the training set patterns are classified by a wide margin, one can “hope” that even test patterns close to the border between classes are managed correctly.
### Bayesian classifier
Bayes’ theorem is a fundamental technique for experience-based pattern classification (training set). Through the Bayesian approach, it would be possible to construct an optimal classifier if one were perfectly aware of both the a priori probabilities $p(y_i)$, and the densities conditioned to the class $p(x|y_i)$. Normally this information is rarely available and the approach adopted is to build a classifier from a set of examples.
To model $p(x|y_i)$ a parametric approach is normally used and when possible, this distribution is made to coincide with that of a Gaussian or spline functions.
The most widely used techniques for estimating are the Maximum-Likelihood (ML) and the Bayesian Estimate which, although different in logic, lead to almost identical results. The Gaussian distribution is normally an appropriate model for most pattern recognition problems.
Sources
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Heidi Garciahttps://systemscue.it
Graduate in Computer Engineering at the University of Florence. Editor-in-chief of the Systems: Informatica e Tecnologia su CuE section in the Close-up Engineering network and executive member of EUROAVIA Pisa. Great passion for aerospace, astronomy and electronics.
Human enhancement
A beautiful mind
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Editorial
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2020-07-10 15:49:38
|
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https://tex.stackexchange.com/questions/374376/how-to-set-the-space-between-columns-provided-by-the-multcol-package?noredirect=1
|
# How to set the space between columns provided by the multcol package? [duplicate]
According to the multcol package's user manual (2017/4/15, p. 3)
The space between columns is controlled by the length parameter \columnsep.
Unfortunately, there is no example showing how to use this parameter.
How can the space between columns be set?
Attempted solution
I tried @Troy's suggestion below, as follows:
\documentclass{article}
\usepackage{multicol}
\usepackage{lipsum}
\begin{document}
\begin{multicols}{2}
\setlength{\columnsep}{0cm}
\lipsum[1-4]
\end{multicols}
\end{document}
but the resulting pdf file is the same whether or not the line \setlength{\columnsep}{0cm} is commented out, namely
## marked as duplicate by Troy, TeXnician, Heiko Oberdiek, Stefan Kottwitz♦Jun 11 '17 at 9:59
• \setlength{\columnsep}{1cm}? – Troy Jun 11 '17 at 8:46
• @Troy: \setlength{\columnsep}{1cm} has no effect whatsoever. – Evan Aad Jun 11 '17 at 8:55
• @EvanAad place the \setlength{...} before beginning the multicols envt. – Troy Jun 11 '17 at 9:01
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2019-11-14 23:26:17
|
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http://regex.info/blog/lightroom-goodies/run-any-command
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Jeffrey’s “Run Any Command” Lightroom Export Plugin
This plugin for Adobe Lightroom provides four separate (but sort of related) functions:
1) It provides an export filter that allows you to run a command of your choosing with each exported image copy, as part of the export while its going on.
2) In the export filter, you can also apply a command to all exported image copies as a group, after the last one has been rendered.
3) In the export filter, you can also have processed photos added to a normal collection and/or a publish collection, and can optionally invoke a Publish operation on a publish collection or a full publish service.
4) Via a new entry to the File > Plugin Extras menu, it lets you apply a command of your choosing on the master image files selected in Library.
This plugin works in Lightroom Classic, and older versions as far back as Lightroom 3, though some features depend on the version of Lightroom.
Export-Filter Features
This plugin includes an export filter that provides access to the rendered copy of an image being exported, during the export. It can be used with any export or publish service, whether built into to Lightroom or provided by a plugin.
The plugin adds a section to the Export/Publish dialog like this:
The top half is for the command to be executed during each image's render, while the bottom box is for the after-all-have-been-processed actions. You can use any or all parts that suit your workflow.
When Things Happen
It's important to understand when during the export/publish operation this plugin's actions are taken, especially in th face of errors downstream from this plugin.
The per-image command, if you have one, is executed during the export operation at the point where it appears in the dialog. In a complex export involving many filters (such as, for example, my Metadata Wrangler and/or LR/Mogrify), the ordering can have a material impact, depending on what you're doing in the command. For example, if you're using the command to send a backup copy somewhere, the copy you're backing up is the pre-Metadata-Wrangler version if this plugin appears in the export dialog above the Metadata Wrangler, and the post-Wrangler version if below.
It's also important to realize that an export can ultimately fail even if it passes this plugin successfully. If the prior example is all within, say, an Export to Facebook operation, the export will ultimately fail if there is no network connectivity. If the commands you had the plugin invoked did things like keeping backups or somehow marking a database, you've now done these things for images that ultimately did not finish exporting. It's an important consideration, depending on the nature of the commands.
In particular, the add-to-collection features happen after this plugin processes the final image of the export, and if the final steps of the export end up failing for some images, those images still remain in the added-to collections. If those collections were intended to be the full set of, say, images uploaded to ...., in the face of such errors you'll end up with images in the collection that shouldn't be there.
So, depending on your workflow, take care with these features when you have export errors.
Command Metasequences
Before a command is actually executed, special token sequences are recognized and replaced by the plugin. For example, the token {desktop} is replaced by the full path to the user's Desktop folder.
The set of tokens recognized depends on the situation.
The per-image Command to Execute command supports all the general tokens of my plugins' preset templates, so you can use image-specific metadata on your command line.
In addition to those tokens, the following are also recognized within this command:
SequenceReplaced By
{FILE}The filename of the exported image copy, with full path.
{file}The filename of the exported image copy, without the leading path.
{NAME}The filename of the exported image copy, with full path, but without the file extension.
{name}The filename of the exported image copy, without the leading path and without the file extension.
{folder} The folder part of {FILE}
{home} Full path of the user-dependent home folder.
{desktop} Full path of the user-dependent Desktop folder.
{pictures} Full path of the user-dependent picture folder (My Pictures or Pictures).
{documents} Full path of the user-dependent document folder (My Documents or Documents).
{temp} Full path of a system-dependent temporary folder.
The overall Command to execute upon completion command supports the following tokens:
SequenceReplaced By
{FILES} The full path of all images rendered for the export, each surrounded by double quotes, separated by spaces.
{qFILES} The full path of all images rendered for the export, each surrounded by single quotes, separated by spaces.
{xFILES} The full path of all images rendered for the export separated by spaces.
{COUNT} The count of images rendered for the export.
{MANIFEST} A temporary file that includes the full path of all images rendered for the export, one per line. (The temporary file is deleted by the plugin after the command has executed.)
{home} Full path of the user-dependent home folder.
{desktop} Full path of the user-dependent Desktop folder.
{pictures} Full path of the user-dependent picture folder (My Pictures or Pictures).
{documents} Full path of the user-dependent document folder (My Documents or Documents).
{temp} Full path of a system-dependent temporary folder.
Command Quoting
You must be very careful about where to quote items in the command lines you enter in the plugin: anything that has a space or other special shell variables must be quoted, but exactly what must be quoted, when, and with what kind of quotes is dependent on the operating system. It's fairly straightforward on a Mac: single quotes will almost always be okay, double quotes probably okay in most cases as well. On Windows, use double quotes and cross your fingers.
Never use smart quotes, like the ones in this sentence surrounding the phrase smart quotes. Only use the old-style ASCII single ( ' ) and double ( " ) quotes.
You'll probably end up using "{FILE}" (or, on a Mac, '{FILE}') in every command.
Here's a concrete example using Phil Harvey's most-excellent exiftool to insert a copyright notice into the jpeg image-comment field. Here it is for Windows:
"C:\path\to\exiftool.exe" -overwrite_original "-Comment=Copyright {yyyy} {Artist}" "{FILE}"
And on a Mac:
'/path/to/my/exiftool' -overwrite_original '-Comment=Copyright {yyyy} {Artist}' '{FILE}'
Each uses the {FILE} token with appropriate quotes, and also two preset-template tokens.
Logging
Details about each command executed and its results are kept in the plugin log, which is referenced in the upper-right section of the Plugin Manager.
Collections and Publishing
After the last photo has been processed, you can have all processed photos added to a regular collection and/or a publish collection, and can optionally have a Publish operation invoked on a publish collection.
Take care not to create a never-ending circular publish chain. If you add this plugin to two separate Publish Services, and in each configure this plugin to auto-publish the other publish service, you could destroy the known universe. Please don't do that.
The Plugin-Extras Extras
The plugin provides a way to execute a command with the master image file for currently-selected photos. Select the photos and invoke File > Plugin Extras... and you'll see four options for this plugin:
Initially, only the first has meaning, and it brings up a dialog where you can configure up to three custom commands:
Once a custom command is configured, you can then invoke it through the Plugins-Extra menu.
Availability
This plugin is distributed as “donationware”. I have chosen to make it available for free — everyone can use it forever, without cost of any kind — but unless registered, its functionality is somewhat reduced after six weeks.
Registration is done via PayPal, and if you choose to register, it costs the minimum 1-cent PayPal fee; any amount you'd like to add beyond PayPal's sliding fees as a gift to me is completely optional, and completely appreciated.
Note: a Lightroom major upgrade, such as from Lr6 to Lr7 (or the equivalent under the hood for the Lightroom Classic subscription) de-registers the plugin in the upgraded version, so if you want to maintain registration, a new ($0.01 if you like) registration code is needed in the upgraded version. It makes for a hassle every couple of years, I know. Sorry. See this note for details. For details on plugin registration and on how I came into this hobby of Lightroom plugin development, see my Plugin Registration page. Version History ( Update Log via RSS ) 2.01907e+07 Fixed a problem related to template tokens and photos without capture times. Added functions uc(), ucFirst(), lc(), and lcFirst() to the LUA token. 2.01903e+07 Added the PEOPLE variable to the LUA token. Fixed a problem with the SpeedKPH token. Added TempC and TempF to the template tokens that my plugins understand. Added the TempC and TempF tokens. Updated the keyword-related tokens to accept standard filters. Work around a bug that sometimes causes plugins to be disabled when starting Lightroom via clicking on a catalog file. Fix an "Unknown key: captureTime" crash. Added the GPSCoords token. 2.0181e+07 Updates for Lr8 (Lightroom Classic CC Version 8). Added hierarchical options to the Keywords token. Added the special PP() function to the {LUA} token. Try to work around a Lightroom bug related to photo timezones and how Lightroom handles accessing plugin data. 2.0181e+07 Added the 'nicknames' modifier to the {People} token. Added the SST1, SST2, and SS3 tokens to the template tokens that the plugin understands. 2.01809e+07 Try to avoid having unexpectedly-long error messages create too-big a dialog. 2.01805e+07 Allowed the custom commands (in File > Plugin Extras) to be configured to be invoked on all selected photos as a group, or individually one by one. Fixed a bug when 'Token Examples' invoked in certain situations. Clicking on the version number in the Plugin Manager now copies version info to the clipboard Updated the PublishCollectionName token to allow numeric arguments along the lines of the CollectionName token. Added the folowing template tokens: {home}, {desktop}, {temp}, {pictures}, {documents}, {IptcDateTaken} Added a bunch of token filters: F2D F2S F2X B2D B2S B2X S2X A2D A2S A2X Added the 'PCH' variable to the {LUA} tag. 2.01712e+07 Updates to the data templates that my plugins understand: updated the Keywords token, added CollectionNames and CollectionFullNames tokens, and added a bunch of stuff (KWf, CN, CFN, CNf, CFNf) to the {LUA} token. 2.01711e+07 Fixed some stuff with the custom-command feature. 2.0171e+07 Oops, more Lr7 stuff. 2.0171e+07 Updates for Lightroom 7 2.0171e+07 Better handle some character-encoding issues related to template tokens. Allow the "If Exists" feature of Templat Tokens to work with the PluginProperty token. Update registration support to handle a stupid bug at PayPal that PayPal refuses to fix )-: 2.01709e+07 Fixed broken error reporting with the upon-completion command preparation execution. 2.01708e+07 Fixed a bug introduced in previous update. 2.01708e+07 Added support for generic template tokens in the after-export command, with per-photo tokens referring to the first photo successfully exported. 2.01707e+07 Fixed a bug introuded the other day in template tokens, related to Windows filenames. 2.01706e+07 Added the Newline template token. Enhanced the FolderName token Added the "only if it has a value" feature to template tokens. 2.01704e+07 An error message confusingly had the name of another plugin in it. 2.01703e+07 Fixed a bug with the keyword tables in the LUA token. Added Weekday, Wday, weekday, and wday to the list of template tokens that my plugins understand. Switch the log-sending mechanism to https. Added "ISO8601Date" to the template tokens that my plugins understand. Added the following tokens to the template tokens that my plugins understand: Artworks, ArtworkTitle, ArtworkCopyright, ArtworkSource, ArtworkCreator, ArtworkDateCreated, ArtworkInventoryNum 2.01607e+07 Oops, new collection/publish stuff added in previous update didn't respect the "enabled" checkbox. 2.01607e+07 Added the ability to add processed photos to collections, and to publish a collection. Try to avoid yet another place where Lightroom gets hung because it can't handle certain kinds of dialogs at the same time. Added Russian-langauge support for the People-Support {People} tag. Added the following tokens to the templates that my plugins understand: FileModYYYY, FileModYY, FileModMM, FileModDD, FileModHH, FileModMIN, FileModSS, FileYYYY, FileYY, FileMM, FileDD, FileHH, FileMIN, FileSS, {FilenameNumber}. 2.01512e+07 Fixed how custom {People} formatting works with people keywords that have no birthday associated with them. 2.01512e+07 Added ChildOf and DescendantOf filters to the {Keywords} and {KeywordsAll} template tokens that my plugins understand. 2.0151e+07 Added {SpeedKPH} and {SpeedMPH} to the list of template tokens supported by my plugins. The {People} token wasn't working properly for some keywords without a registered birthday. 2.01505e+07 Fixed the "SpecPeople:259: attemt to index al nil value" error. 2.01502e+07 Command wasn't being applied to rendered video files. On OSX, allow text in the error-report dialog to be selected and copied. 2.01502e+07 In the POODLE-vunerability dialog, display a raw URL of a page on my site that discusses the issue, so that folks can be independently sure that the dialog is indeed from me and not malware. 2.01501e+07 Fix to the date_diff() function supported by the LUA template token. Updated the camera-name code to try to guess the actual camera model of Hasselblad H5D files, since in their infinite wisdom Hasselblad decided to encode three distinct models with the same internal code, making it impossible to know for sure what camera produced a given image file. 2.01412e+07 Registration was broken on Lr2 2.0141e+07 Windows Only: Add a one-time check for the POODLE security vulnerability, and alert the user if it exists. 2.01409e+07 Added the LrMD5, LrLocalization, LrSystemInfo, and LrMath packages to the {LUA} template token. 2.01409e+07 Added the {MANIFEST} token 2.01409e+07 Added a rudimentary "apply custom command to selected master image files" functionality. See the File > Plugin Extras > jf Run Any Command menu to configure and launch. 2.01409e+07 New build system 2.01408e+07 Made the {GPSAltitude}, {Altitude}, and {GPSCoordinates} tokens subject to the geo-privacy settings like the other geo-related tokens. 2.01407e+07 Registration fix for Lr5.6 2.01407e+07 Previous updates broke support on Lightroom 2 2.01407e+07 More Creative-Cloud support. 2.01407e+07 Fixed an issue with Creative-Cloud revalidation. 2.01407e+07 Lr5.5 and later Creative-Cloud installs can now revalidate themselves if needed. 2.01407e+07 Sigh, had a bug in the Creative-Cloud support. 2.01407e+07 Now supports Lr5.5+ Creative-Cloud Installs. 2.01407e+07 Sigh, introduced an error for some folks with the rebuild the other day. 2.01406e+07 Build-system update 2.01406e+07 Added date_diff() and raw_time_diff() functions to the special {LUA} token understood by the plugin. 2.01405e+07 Added new tokens to the template language the plugin understands: LrVersion, LrVersionMajor, LrVersionMinor, LrVersionRevision, LrVersionBuild, Location, CatalogName, CatalogPath, OperatingSystem, OS Added new token filters: NS and LO 2.01404e+07 Fix a location-related template-token bug introduced in a recent build. 2.01404e+07 Fixed a bug in the "smoother revalidation" stuff recently added. 2.01404e+07 The {Empty} template token wasn't working properly. Make the revalidation process smoother, especially for folks using Lr5.4 and later. 2.01309e+07 Oops, fix a bug introduced in the previous update 2.01309e+07 Added a bunch of tokens to the preset templates supported: ExportFormat, ExportColorSpace, ExportBitDepth, ExportQuality, ExportSharpeningLevel, ExportSharpeningMedia, IpernityUrl, GoogleDriveUrl, and TumblrUrl. The token-examples dialog had been broken. Also deprecated Folder and Path tokens in preference to FolderName and FolderPath tokens. Fixed the KW/KWE tables used by the LUA token; they had been broken when using load for the script. 2.01306e+07 Better support for plugin revalidation. 2.01306e+07 Yet another Lr5 update 2.01305e+07 Apparently, a recent change broke things on Lr2, which some folks apparently still use. 2.01305e+07 Update for Lr5 2.01304e+07 Build system update. 2.01303e+07 Fix for the registration system. 2.01302e+07 Added support for some new template tokens: FlagStatus (requires Lr4.1 or later), and for Lr3 and later, a bunch of IPTC extended metadata: AdditionalModelInfo, CodeOfOrgShown, DigImageGUID, Event, ImageSupplierImageId, MinorModelAge, ModelAge, ModelReleaseID, ModelReleaseStatus, NameOfOrgShown, PersonShown, PlusVersion, PropertyReleaseID, PropertyReleaseStatus, and SourceType. 2.01302e+07 More build-system maintenance 2.01302e+07 Tweak for my registration system 2.0121e+07 Enhance the {EMPTY} template token so that it interrupts the squelching of superfluous joining characters. 2.01208e+07 Updates to the environment in the {LUA} token (in the template tokens in my plugins) to include photoTime() and currentTime(), and other changes to match the updated docs at that link.. 2.01206e+07 Fix an "attempt to perform arithmetic on field" error. 2.01205e+07 Update to handle the Mac App Store version of Lightroom. Tweak for Lr4.1RC2. Added to the template tokens supported by the plugin: {FullMasterFolder}, {FullExportedFile}, and {FullExportedFolder}, and to match, renamed the recently-added {FullMasterPath} to {FullMasterFile}. 2.01204e+07 Enhanced the send-log dialog to hopefully make reports more meaningful to me, yielding, I hope, the ability to respond more sensibly to more reports. Added {FullMasterPath} to the list of template tokens supported by the plugin. 2.01203e+07 Update to handle 4.1RC. 2.01203e+07 Had broken registrations in Lr2; Update to the debug logging to better track down timing issues that might arise. 2.01203e+07 More on the march toward Lr4, including upheaval in the code to handle Lightroom APIs being discontinued in Lr4. Added the {AspectRatio} token to the token templates understood by the plugin, and added the Length=num filter. 2.01201e+07 More tweaks for Lr4b 2.01201e+07 Update for Lr4 beta: explain in the plugin manager that the plugin can't be registered in the beta. 2.01112e+07 Had issues with the registration button sometimes not showing. When doing a plugin upgrade, offer the ability to flush all the old copies of the plugin. Added a system-clock check and reports to the user if the system clock is more than a minute out of date. An incorrect system clock can cause problems with various kinds of communication and authentication with some of my plugins, so I've just gone ahead and added this to every plugin. 2.0111e+07 The {Path} token wasn't working properly. 2.01106e+07 In some cases where the command could not be run, the plugin crashed instead of reporting the error. 2.01101e+07 Added {CroppedWidth} and {CroppedHeight} to the template tokens used by my plugins. 2.01008e+07 Made the revalidation process much simpler, doing away with the silly need for a revalidation file. 2.01008e+07 Discovered a bug in my plugin build system that caused horribly difficult-to-track-down errors in one plugin, so am pushing out rebuilt versions of all plugins just in case. 2.01008e+07 Added code to allow plugin revalidation after having been locked due to a bad Lightroom serial number. 2.01007e+07 Figured out how to add the ability to run a command after all files have exported. 2.01006e+07 Yikes, shaking out some more build issues. 2.01006e+07 Discovered a nasty build bug; pushing a new version in case it affects this plugin. 2.01006e+07 This version can be registered in Lightroom 3. It can run in Lightroom 2 or Lightroom 3; it does not work in the Lr3 betas. It uses my new registration system when run on Lightroom 3, which avoids some of the silly issues of the old one. Please take care to note the details on the registration page: use of this version (or later) of the plugin in Lightroom 3 requires a new registration code, even if you had registered some older version of the plugin. 2.01005e+07 Update for the Lr3 beta. 2.01003e+07 Added new tokens to the templates that my plugins use, CameraName, IfGeoencoded, Keywords, IfKeyword, IfExportedKeyword, and LUA. See the templates page for details. 2.01003e+07 Fixed up some issues with the help dialog, and added a warning referring to it if the user command doesn't seem to refer to the exported image. Changed the semantics of the Places filter (in the tokens understood by the preset templates of my plugins) in two ways: if applied to a string value rather than a number, it works on the first number found in the string. Another is that you can now use something like Places=-1 to round to the 10s, Places=-2 to round to the 100s, etc. 2.01002e+07 Fixed the {GPSAltitude} template token so that it should now actually work. Completely changed how the one-click upgrade applies the newly-downloaded zip file, in the hopes that it'll work for more people. Rather than unzipping over the old copy, it now unzips to a temporary folder, then moves the old folder out of the way and the new folder into place. Prior versions' folders are now maintained (with the version number in the folder) in case you want to revert a version; you may want to clear them out from time to time. Of course, it won't take affect until you try to upgrade after having upgraded to or beyond this version. 2.01001e+07 Added two new template tokens, {DaysSince} and {PhotoDaysSince}. They're a bit tricky, but could be useful. 2.00912e+07 Added the ability to save command presets. 2.00912e+07 Minor internal debugging tweaks. 2.00911e+07 Added an {Altitude} item to the templates understood by the plugin. It's the numeric altitude in meters, as opposed to the {GPSAltitude} item which is a description of the altitude along the lines of “32.7 m”. Also updated the Places filter so that it can be used on fields that merely begin with a number. 2.0091e+07 Added a first draft of some rudimentary support for Lightroom 3 Beta. See this important note about plugin support in Lightroom 3 Beta and Lightroom 3, including future plans for features and my registration system. 2.00907e+07 Enhanced the one-click upgrade stuff quite a bit, now detecting ahead of time when it will fail because the plugin is installed where Lightroom can't write (if Lightroom can't write to it, it can't update itself). I also added a progress bar, and now download in smaller chunks to avoid 'out of memory' errors on the larger plugins. Do remember that this new functionality becomes available after you upgrade to or past this version, when you then upgrade with it. 2.00906e+07 Fixed a boo-boo in the help dialog, and added a note that command-line quotes should be ASCII, never "smart quotes". 2.00905e+07 Fixed a "loadstring" error some users got. 2.00905e+07 Added a link in the Plugin Manager to the plugin's update-log RSS feed. 2.00904e+07 Tweaked how the plugin tries to update itself during the one-click upgrade process, to hopefully get things working for those few Windows users that have never had it work. Crossing fingers. We'll see. 2.00903e+07 It seems that PayPal doesn't give everyone a "Unique Transaction ID" in the registration confirmation mail; some people get a "Receipt Number". So, the registration dialog now accepts that as well. 2.00902e+07 Fixed a bug that caused a plugin crash if my server couldn't be reached during registration. 2.00902e+07 NOTE: you may need to restart Lightroom after installing to this (or a later) version from the previous (or an earlier) version. Please try a restart if you get an error the first time you try to use the plugin. As per the ongoing discussion on my blog, with this version this plugin moves over to a "donationware" model, in which the plugin remains free, but registration eventually becomes required (and an eventual donation hoped for 🙂 ). For details, see Lightroom Plugin Development: Now With Added Encouragement. (For info about what drove this decision, see What To Do When a Hobby Becomes Work?) The plugin no longer expires, and correspondingly, I will not pay much attention to reports of bugs that have already been fixed, so please check your version and the version history before submitting bugs or feature requests. There was a lot of internal upheaval in the code, so I expect that some boo-boos my surface. If something breaks for you with this version, please let me know, but until I fix it, feel free to revert to the previous version. 2.00902e+07 Bumping expiration into the future. 2.00901e+07 Oops, forgot to include the unzipper in the plugin, so the automatic upgrade won't work on Windows until after you manually upgrade to this version. 2.00901e+07 Added an option to ignore errors generated by the command. 2.00901e+07 Small housekeeping update for the new locales supported by Lightroom 2.3. 2.00901e+07 Initial public release The 30 most-recent comments (out of 60; see all), most recent last... I have an answer to my own question but if anyone has any better suggestions let me know. I can use an ImageMagick command to make a unique filename based on the input names. My execute on completion command looks like this: convert -loop 0 -delay 50 {FILES} -set filename:f “output_%t” “{pictures}\TEST\%[filename:f].gif” Note the use of the -set to make a filename based n the input. It turns out that IM uses the 1st filename in {FILES} which is perfect for my needs. It is very cool how you can mix the Plugin template values and IM escape variables in the same command. Robin — comment by Robin Cottiss on February 9th, 2012 at 1:39am JST (7 years, 5 months ago) comment permalink For completeness here is what I do for the actual run command step to place a timestamp onto each image so I can see the time of each image in the resulting animation: mogrify -fill white -stroke black -font Arial -pointsize 36 -gravity SouthWest -annotate 0 “{file} – {T2}” “{FILE}” Robin — comment by Robin Cottiss on February 9th, 2012 at 1:42am JST (7 years, 5 months ago) comment permalink i would like to be able to digimarc photos on export using your direct to smugmug plugin (via temporary files). is it possible to do that using a photoshop droplet (app) with your “run any command” post process filter? thank you for your plugins! Lee Yes, seems reasonable. I’ve heard folks run Droplets from “Run Any Command” successfully. —Jeffrey — comment by Lee on September 13th, 2013 at 1:56am JST (5 years, 10 months ago) comment permalink i have tried to do this without success. if someone else has managed to do it on a Mac i would appreciate seeing how your command line was structured. my command line looks like this: ‘/Volumes/MacPro RS2/Documents_RS2/Photoshop Actions/Digi 800px.app’ ‘{FILE}’ the error result was this http://tinyurl.com/lfw7tfy Lee You probably want this: open -a /Volumes/MacPro RS2/Documents_RS2/Photoshop Actions/Digi 800px.app' '{FILE}' —Jeffrey — comment by Lee on September 13th, 2013 at 12:12pm JST (5 years, 10 months ago) comment permalink that worked! thank you. if i understand this “filter” correctly though, the process only happens after the upload to smugmug. is there a way to make it happen to the image before going to smugmug (only choice seems to be to export to disk first — adding the digimarc — then upload to smugmug from the disk copy. Lee If you use it with the SmugMug plugin, it’ll happen before the upload, but you’ve got to make sure you update the image in place. —Jeffrey — comment by Lee on September 13th, 2013 at 11:48pm JST (5 years, 10 months ago) comment permalink Would it be possible to add a feature that would let you run a command after the ENTIRE export is done? Let say : run a bash script on the folder that ZIP’s everything. Or run a script that would FTP the entire folder? That’s already there, both in the run-any-command plugin and in the vanilla Lightroom export. —Jeffrey — comment by Gerard Henninger on October 13th, 2013 at 8:55am JST (5 years, 9 months ago) comment permalink Does this plugin work with LR Publish Services (such as jf Flickr)? I’m attempting to do some exiftool post-processing on images prior to publishing them to Flickr (e.g. remove/update CreateDate, CreateTime, etc.), but the capture time for the image on Flickr is set to the original photo’s capture time. Yes, it gets control of the copy of the image that Lightroom creates for the export, before the overally controlling plugin (e.g. Flickr plugin) gets it. —Jeffrey — comment by Sol on December 12th, 2013 at 6:45am JST (5 years, 7 months ago) comment permalink From Portugal. Is there any chance of running a command immediately activated when a shoot if made on studio as the file enters on the LR workspace? A kind of listener ignition to run a command… I want to do 2 things: 1- on each image raw entered, read some of the xmp properties and compare to the ones approved by me (to detect camera wrong configuration); 2- do the same to small video files (first extract xml with exifTool using command and then reading properties and comparing with the approved ones; The approved values depend on the studio, and on me and are to loaded from an xml. At the end of all this, the photographer should be immediately warned if camera is not write configured, avoiding major mistakes that could not be corrected after. Lightroom offers no on-import hooks for plugins. The only thing I could suggest would be to use my folder-watch plugin along with its method to do automatic export, there having a run-any-command entry check the metadata as you want. Very kludgy, but possible. —Jeffrey — comment by Pedro Marques on January 18th, 2014 at 6:24pm JST (5 years, 6 months ago) comment permalink Jeffery, i use the command as shown below, to set the file modified date: touch -am -t ‘{YYYY}{MM}{DD}{HH}{MIN}.{SS}’ ‘{FILE}’ but this command only works for image file, video files do not take any effect. do you have any idea to make it works for video files? thank you. It seems to work for me. Perhaps send a plugin log for a sample export. —Jeffrey — comment by cupuno on January 27th, 2015 at 10:22pm JST (4 years, 6 months ago) comment permalink Can this be used to run a Python script on images after they are exported? Yes, it can execute anything you can execute from the command line. —Jeffrey — comment by Paul on January 30th, 2016 at 1:07am JST (3 years, 6 months ago) comment permalink Hi Jeffrey, Thanks for the previous answer about using with Python. I am using a Mac with Python 3 but I get ‘sh: python3: command not found’ when I try to run a Python script upon completion. Any idea why? I tried on two separate Macs but got the same result. However, I don’t get the error with Python 2.7, only with Python 3. Thanks, Paul Try using the full paths to the python binary and your script. —Jeffrey — comment by Paul on January 31st, 2016 at 12:08am JST (3 years, 6 months ago) comment permalink Is there a way to invoke exiftool through this to write a uuid into a metadata field? For example, select N number of files and make md5 hash from field1, field2, field3. ExifTool has a command “NewGUID”—would that be able to be invoked through this app? You could use the UUID that Lightroom generates for every photo, via the {UUID} template token the plugin supports, injecting it with ExifTool. I’m not familiar with NewGUID, but it sounds like that’d work out well too. —Jeffrey — comment by justin on April 19th, 2016 at 6:20am JST (3 years, 3 months ago) comment permalink I am registered user using Lightroom CC on Mac OSX El Capitan, I try to run a shell script .sh file with some tasks to run after my export. ——————————– My command to execute upon completion is: sh /Volumes/Macbook13\ 500GB/reportajes/list.sh (I run sh script from terminal and it is running well) ——————————— I don’t get any message and I don’t get anything in log or diagnostic messages, but it doesn’t works. This is my sh file: #Create csv.csv file with list of jpg exported files. rm csv.csv echo “Title,Taxonomy,Path,Price” >> csv.csv; for f in ls *.jpg; do echo$f”,”${PWD##*/}”,”${PWD##*/}”/”\$f”,100″ >> csv.csv;
done
—-
Thank you Jeffrey
You probably need to put something in there so that your script knows what directory it should work in. When run from Lightroom, you can’t assume you’ll know what directory is the script’s current working directory. —Jeffrey
— comment by David Crespo on June 10th, 2016 at 9:29am JST (3 years, 1 month ago) comment permalink
Hey Jeff, excellent software, I am a big fan.
Would this plugin be able to publish ANOTHER publish service, using the first as a trigger? I find myself in a situation where I want to publish to my hard drive as well as to Cloudspot, without my input. Can “Run Any Command” do this?
Thanks
Seems like a good idea… I’ve just added it to the version I pushed. I’ve not yet updated the documentation here, though. —Jeffrey
— comment by Aaron on July 12th, 2016 at 7:02am JST (3 years ago) comment permalink
Hi Jeffrey, all,
Nice plugin.
However, I am struggling on one point : is there any variable to access the original file ?
In my case, I am trying to run a command like “rm ORIGINAL_FILES” upon completion, basically to export to JPG and delete the NEF file.
Thanks !
Sure, click on the after-processing help link in the plugin and you’ll find it, but wow, it seems scary to delete the master this way. Unless you have copies saved elsewhere upstream in your workflow, considering moving the file rather than deleting, just so you have a chance to recover it if need be, at least until you next clear out the move destination. —Jeffrey
— comment by Flo on July 24th, 2016 at 8:05pm JST (3 years ago) comment permalink
Can I use this plugin to add a logo to a jpg picture and then save or overwrite the existing picture? If possible, How do I do it?
Thanks,
Manny
This plugin can execute an external command, so you’d need an external command that does your logo thing (or perhaps a Photoshop “Droplet”, I think it’s called). As for overlaying a logo, you might consider the Lr/Mogrify plugin —Jeffrey
— comment by Manny on May 5th, 2017 at 5:11am JST (2 years, 2 months ago) comment permalink
Jeffrey,
This plugin seems to be exactly what I need, but before I get lost trying to figure out HOW to do this, I’m wondering if you could tell me whether or not it’s even possible.
I publish photos to my WordPress website using WP/LR sync. This allows for the standard Lightroom publish settings such as image size and quality, but what I am really missing is some way to compress them appropriately for web, using a service such as jpegmini or some other… to be determined.
Will this plugin allow me to click publish on the WP/LR sync plugin, have this plugin intercept the created jpg, compress it, then continue the upload to my site?
Hopefully I haven’t misunderstood what your plugin is capable of…. thanks in advance!
Joe
Yes, but if you have jpegmini, you can just use their Lightroom plugin directly. In any case, you can use this plugin to do whatever you want to the image file…. so long as the file ends up in the same place when you’re done, you can do whatever you like to it. —Jeffrey
— comment by Joe on July 2nd, 2017 at 7:57pm JST (2 years ago) comment permalink
Hello Jeffrey!
It seems there’s no notification via Run Any Command when a file is ‘deleted’ from a publish collection, which causes the file to be deleted from the publish destination. There’s not even a notification at the end of the publish operation. My usage (in conjunction with your Folder Publisher) is to trigger a program to update a web gallery database (Piwigo) with newly published images. That’s working fine, but it would be nice to also remove photos from the gallery using the same mechanism. Did I miss something? Or perhaps this is not supported by LR?! It’s not something that I plan on doing often. But it leaves an unsatisfying gap in my otherwise elegant gallery syncing process.
Thanks for the great plugins! 🙂
Ferg — from Bellevue, WA
That one bit that lacks elegance can be really painful, like a little pebble in your shoe. Sadly, Lightroom doesn’t provide for any way to call a plugin like this during deletion. I might be able to build something into Folder Publisher… let me add that to the todo list… —Jeffrey
— comment by Scott Ferguson on August 30th, 2017 at 2:41pm JST (1 year, 11 months ago) comment permalink
Hi again Jeffrey!
After aborting a publish operation I continue getting invocations from Run Any Command for files not actually published. In this case, three files are marked to be published: DSC05015.JPG, MAH02617.MP4 and DSC02612. I aborted the Publish operation immediately after starting it. NO files were actually ever copied to the final destination. My program received invocations for the two JPGs, but not for the MP4.
Sometime earlier, I had aborted a Publish with a large number of files yet unpublished. I can’t say how many of those received invocations, but it was also a large number.
It seems that this is not the desired behavior. Maybe a problem/limitation with LR? Also likely I can work around it easily enough. Haven’t tried yet.
Logs have been sent for Run Any Command and Folder Publisher as well.
Unfortunately, I don’t think there’s anything I can do here. Lightroom invokes the plugin when it’s got the rendered copy ready for export, so the plugin is just responding to that. You’d imagine that Lightroom would be checking whether the export is canceled at regular milestones, such as just before invoking a plugin, but from what you report it sounds as if they don’t. )-: I’m not sure what to do other than report it to Adobe. —Jeffrey
— comment by Scott Ferguson on September 4th, 2017 at 2:35pm JST (1 year, 10 months ago) comment permalink
Hello,
I am wondering if you can help with if a concept is achievable using Run Any Command. I use a number of publish services (e.g. Envira, jfFacebook, jfFlickr, 500px, LR/Instagram) and I would like to be able to place a backup of the file(s) I am exporting into a local folder which is backed up to Google Sync and Backup.
At the moment I use the Run Any Command on a per image basis to create dropshadows for the images before they go through LR/Morgify 2, is there a way I can install another version of the plugin, or run a command so that at the very end, after dropshadow is created, the image is morgify’d and just before it is uploaded it is copied to the backup folder?
Using windows specifically, the reason I ask is that some plugins don’t specify the folder to store the image, others only specify a temp folder so I thought run any command might be a good way to do this?
Many thanks!
It’s built into Lightroom publishing services that you can put the exported copy wherever you like, and have it left there after the publish service is otherwise done with it. Publish services will default this to a temp folder that’s cleaned up when done, but they should let you set it as you like. (They can remove your ability to set it as you like, but they have to explicitly go to the trouble to do it; if one of your publish-service plugins is doing that, contact the author to ask whether they might not let you make your own decision.) —Jeffrey
— comment by Travis H on September 9th, 2017 at 2:16pm JST (1 year, 10 months ago) comment permalink
I just installed this to try to make animated GIFs as suggested above. When I ran Custom Command 1, I got this error:
Assert Failed run-any-command#93 U[1506302414]+5226.7:[x6080005612c8] @InitPlugin line 1044:
Assertion failed(!):
[x6080005612c8] @InitPlugin line 1044
[x6080005612c8] @CustomCommand line 146
[x6080005612c8] @Trap line 222
Followed by another dialog with “Assert @Debug:880”.
The command I was trying was:
convert -loop 0 -delay 50 {FILES} -set filename:f “output_%t” “{desktop}/animated/%[filename:f].gif”
But it seems to happen even with simpler commands.
From Berkeley, CA.
It’s hard to guess without the plugin log… please make sure you’re using the most recent version of the plugin, then if you encounter the error again, please send a log. Thanks. —Jeffrey
— comment by David Glasser on September 25th, 2017 at 10:21am JST (1 year, 10 months ago) comment permalink
How can I batch rename all assets in LR based on a certain string scheme AND most importantly the UUID of each LR asset?
Lightroom doesn’t allow for a plugin to rename items, so any solution will involve something outside of Lightroom. You can make a list of file/UUID pairs via my Data Explorer plugin. When you enable the “collect per-image data for spreadsheet export” and then export that data, Lightroom’s UUID of the photo is included in the spreadsheet. —Jeffrey
— comment by Matt on October 1st, 2017 at 2:28am JST (1 year, 10 months ago) comment permalink
Thank you, Jeffrey. I was not aware of this restriction. Alternatively, is there a way to put the UUID in one of the iptc or exif meta fields? how long would that take for a library of about 600,000 pictures? Would the LR assets/pictures need to be ‘online’ (all external drives attached) for this to work? Also – on export – would it be possible to rename the files to something like FirstName_LastName_UUID?
You can fill in a standard field with the UUID via my bag-o-goodies plugin. No idea how long it might take for 600,000 photos, but I imagine a while. It’s a database-only operation, so image files don’t need to be online. As for naming files on export, it’s trivial if you’ve stuffed the data info fields available to Lightroom filename templates. You might stuff “FirstName_LastName_UUID” into, for example, the Caption field, and use a filename template that refers to the caption. Also, some of my plugins (such as Collection Publisher) have enhanced-renaming support. —Jeffrey
— comment by Matt on October 2nd, 2017 at 4:19am JST (1 year, 10 months ago) comment permalink
One last thing: Is there anything we can do about syncing LR with a PHP web gallery? Similar to LR’s flickr plugin. (Syncing image edits, meta data edits and order of images in the online gallery)?
This kind of thing can certainly be built, either with a normal plugin or with a Web-Module plugin, and I know such things have been done for various web-gallery platforms, but I don’t know anything specific about “PHP web gallery”. —Jeffrey
— comment by Matt on October 2nd, 2017 at 4:21am JST (1 year, 10 months ago) comment permalink
howdy. After years of getting “the droplet couldn’t communicate with photoshop” found your plugin.
I have got it to work running with each image, but it produces an error: ‘the command attempted by the “run any command” filter failed’. Then lists the command path and file path (correctly). (But the next bit – path sent to OS has a 1 appended and another path at the end that I dont understand).
This is my command per image:
“C:\Users\me\AppData\Roaming\Adobe\Lightroom\Export Actions\3-2013 REAL STYLE other agents with blur group.exe” “{FILE}”
Its working – the image is exported and the action applied. I just get the error at the end.
also how would I get the droplet to run on all the images after export? Do I use {manifest} or {FILES}??
About the error you’re seeing, please send a log the next time you encounter it. As for how to run just once after the export, on all the exported files, that depends on how your target command works. If your exe command can be given a bunch of files on the command line and then do the right thing, you’d want {FILES}. —Jeffrey
— comment by Gethin Coles on December 11th, 2017 at 8:07pm JST (1 year, 7 months ago) comment permalink
Can I use this plugin to add a attribute/rating/colour to the files as they are exported or after export?
Regards Ian
This plugin allows you to set those things during export (set them for the Lightroom photo, not the exported copy). —Jeffrey
— comment by Ian Stuart on April 30th, 2018 at 11:42am JST (1 year, 3 months ago) comment permalink
Hello! I was referred to the “Run Any Command” plugin as a way to get around the Lightroom to Photoshop Droplet error of “droplet couldn’t communicate with Photoshop”. I can run a droplet with a handful of images, when I export a larger batch (20+ it begins to error out)… I’m told it’s something to do with the limit of characters being passed (too many) with it being a limitation of Windows. I am wanting to export via a droplet (in PS run some plugins & save) and after it saves, I need it to stack with the original image in Lightroom. What do I need to change so that I stop getting this error? Thanks for your time! ~Cathy
I’m afraid that I can’t offer any help… I don’t really know anything about “droplets” nor how they communicate with Photoshop. Just guessing from what you’ve described, though, I wonder whether it’s not a problem of asynchronicity, that is, running more than one at a time. Perhaps they have to be synchronous, but aren’t? —Jeffrey
— comment by Cathy on June 3rd, 2018 at 5:20am JST (1 year, 2 months ago) comment permalink
Hi Jeffrey,
is this plugin somehow usable for initiating a second export?
The use-case is that I need to deliver two different resolutions of the same photo, and I am executing two different exports with two different presets which is very cumbersome.
Thanks B
Yes, it’s cumbersome, and if there were an easy way around it, believe me, I’d have implemented it. I export two versions of every photo I put on my blog, and indeed have to launch two exports from two different presets. It’s a pain. )-: —Jeffrey
— comment by Bence on October 4th, 2018 at 5:50am JST (9 months, 14 days ago) comment permalink
Hi Jeffrey,
I’m trying to export images to a folder named after the Title metadata tag. I used mkdir ‘{Title}’ but I get a permission denied error. I’ve tried the same command in terminal and it worked fine in the same folder as the export.
Exemple of what I’m trying to achieve:
Images 1 to image 10 have title tag: 101-345
Images 11 to image 15 have title tag: 101-346
Images 16 to image 29 have title tag: 101-350
Result would be:
Folder name: 101-345
Content: images 1-10
Folder name: 101-346
Content: images 11-15
Folder name: 101-350
Content: images 16-29
Thanks
The command is not run from within the folder of the export… it’s run, I suppose, from your home directory (but even that you shouldn’t count on). You probably want to use a fully-qualified path, such as with {folder}. —Jeffrey
— comment by Fabrice on October 29th, 2018 at 11:05am JST (8 months, 20 days ago) comment permalink
{GPSCoordinates} outputs a string containing single and doublequotes which are super hard to escape
8°50’57.76″ S 115°9’42.509″ E
It was impossible to pass these values to exiftool directly, had to clean them up and reformat in a separate batch.
It would be super awesome if the plugin would split up this value beforehand and substitute ° and ‘ by space and delete ”
{GPSLatitude} = 8 50 57.76 S
{GPSLongitude} = 115 9 42.509 E
in this format it can be passed to exiftool directly -exif:gpslatitude=”42 30 0.00 S”
The tokens {Latitude} and {Longitude} give simple floating-point values, which ExifTool accepts. —Jeffrey
— comment by Ostap on March 13th, 2019 at 10:40pm JST (4 months, 5 days ago) comment permalink
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2019-07-18 02:30:45
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{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3081307113170624, "perplexity": 3273.3535414177854}, "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}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525483.64/warc/CC-MAIN-20190718022001-20190718044001-00003.warc.gz"}
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https://gamedev.stackexchange.com/questions/48315/game-state-and-input-handling-in-component-based-entity-systems
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# Game state and input handling in component-based entity systems
My question is:
How can I handle game states in my entity system, without resorting to keeping a stack of game state objects around?
So the design of my entity system means that when an entity needs to register for input events for instance, the input component calls the input system and says "register this entity for this input". This is all fine and well, however if you add into this the concept of game states (say a pause screen), it becomes a problem to work out if an entity is in the current state and should receive the input.
I could augment the input component/system so that it says, "register this entity for this input while in these game states", but this requires that every entity know which states it's going to be used in, and that may not be obvious. Also, keeping a list of game states around per registered input (and other systems that use callbacks) doesn't sound too efficient.
Another idea I had is since there will be an entity that represents the game state, mark that as being disabled, then when generating the input event check that the entity is not a descendant of a disabled game state entity. Seems expensive to work out the parent for every callback.
Another idea is to have all the systems store their data keyed against the current state, that way when generating the input, the target entity won't even be a candidate. However this really hurts the ability to allow communication between entities in different states (not so much a problem for pause screens, but think lock picking in Oblivion/Skyrim).
The only other idea I've had is to have all components handle a state change events and communicate with their relevant system to disable anything they have registered, and re-enable it when switching back to this state.
The second (mark an object as disabled) and forth (have each component deal with state changes) seem like the best of my ideas, but none of them jump out at me as being particularly great.
Does anyone else have any other ideas on how to do this?
edit While I talk about input specifically in this question, it can mean any system capable of sending messages/events to entities, such as collisions, timer events, etc...
• I do it like this: I have Screens, MenuScreen PauseScreen GameScreen, each screen can create its own World (container for Entities) and systems (like RenderingSystem) and then in GameScreen i create World, Entity with CameraComponent, and set CameraComponent.RenderTarget to screens background. This way i can add InventoryScreen that will have own entities and systems (like simplified renderer). Input can be passed from screen to world, so your userinterface will decide if it will pass input to screen (if its focused, visible etc) and that will pass input to world and entities – Kikaimaru Jan 29 '13 at 11:42
• – MichaelHouse Jan 29 '13 at 15:01
• @Byte56 Not really, only the first one has to do with gamestates (the other 2 are states within entities), and that doesn't really tackle the same problem that I'm having. When the game is in the paused state, something has to happen to the input system to stop it sending a movement messages to the player entity (for example), I just can't figure out a good way to do this. – elFarto Jan 29 '13 at 16:18
• OK, consider them related then. Good question. – MichaelHouse Jan 29 '13 at 16:27
• Something else to take into account that have been an annoyance to my component-based systems in the past: multi-layer UI. Dialog poping on top of world or multi-level screens. It has come up so far in every game I've made so I'd say to make sure to consider an approoach that can solve that problem. – ADB Feb 4 '13 at 20:05
What is often used is an intermediate Intent System which abstracts the input and keeps track of the context and relevant gamestates.
The Intent system will stop transmitting inputs when the simulation is paused for example. It also handles the mapping between controller events and intents (move in direction, run, shoot, reload...).
This way your other conponents are not dependent on specific gamepads/inputs (BUTTON_A, BUTTON_B vs BUTTON_X, BUTTON_O...) but they all react to the same intents (IntentRun, IntentReload...).
Another advantage is that the intent system can be aware of available controllers being added/removed, as it can send intents to any subscriber even outside the simulation you can handle intents like AddPlayer(controllerID).
How much information about the game state you provide to the system either through events/message or directly is up to you. But the time invested in the Intent system is usually worth it.
You can manage Intent Contexts which will generate intents when they are attached to the system.
The context can be prioritized, i.e.:
• SimulationAvailableContext sends intents to the simulation while it is available (but not running) for example move the camera, zoom in zoom out, add/remove player...
• SimulationRunningContext sends intents to the simulation while it is not paused move player, send unit to position, shoot...
This way you can add and remove the contexts which are currently relevant.
And one thing about the whole intent systems is that it should run while the simulation is paused.
One way which is often used to play/pause the game simulation without breaking non simulation related updates is to use a different sets of times. i.e. GenericSystem::onTime(Long time, Long deltaTime, Long simTime, Long simDeltaTime).
With this approach your engine can simply block the increments on the games's simTime which in turn will block updates on the relevant animation & physics engines which use simTime and simDeltaTime while allowing continuous updates of your camera spring effect if it has to move even during pause, the animation of the loading effect on a virtual in-game billboard while data is being downloaded...
• I like the fact that this doesn't have to call a bunch of "State Changed" functions on all the entities. You have to worry about the wrong intents being sent at the wrong time, but I think that is better than the alternative. – Thomas Marnell Jan 30 '13 at 22:20
• your entities can ignore intents like Jump while their state doesn't allow them to jump (i.e. not touching the ground). but they don't have to worry about receiving such intents while the game is paused. – Coyote Jan 31 '13 at 7:18
• I had already thought of letting the entity tell the input system what states to deliver messages in, but I hadn't thought of putting the states on the input itself, which is a good idea. Also splitting the time and simTime apart is nice too. – elFarto Jan 31 '13 at 7:54
• You should avoid bloating your simulation related state with non simulation related things. Move all UI, and player related code as far as possible from the simulation itself and in the simulation concentrate only on the intents. – Coyote Jan 31 '13 at 12:40
• Hey @Coyote, this system sounds very interesting. Could you maybe provide some more information by answering this question? Thanks! – pek May 10 '13 at 2:53
How about creating a global event system and then have an event listener component for your each entity? After an event "Game State Change" you could fiddle with components individually for each particular entity.
Let's say you have an input component. After the event listener component receives the game state change event, it changes very specific values for that particular input component, so it wouldn't receive any input calls or wouldn't make any movement or response calls to the system or it's owner.
This works for me as most of my components are scripted (via Lua). I.e. I have an input component, which is triggered once when a key is pressed and it fires of a movement + direction and then it is triggered when the key is released and it fires of a stop + direction. There is also an event listener component which contacts the input component (if the game is paused) to stop doing any function firing and halt if necessary. I could easily then add another entity with a different reaction to the same events and keypresses using another script. This way you would save the interaction between different entities in different states and even make it much more customizable. What is more, some entities might not even have the event listener component in them. There is one drawback though - since it requires the engine core to call in scripts every time something happens, it might affect performance, but I personally haven't got any problems with this particular solution.
What I just explained is basically a practical example of your fourth solution.
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2020-08-13 04:08:43
|
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|
http://icpc.njust.edu.cn/Problem/Hdu/5940/
|
# Hazy String
Time Limit: 6000/3000 MS (Java/Others)
Memory Limit: 65536/65536 K (Java/Others)
## Description
Archaeologists find an ancient string on a monument, but some characters have become hazy with the passage of time, and we only know the remain characters.
Now archaeologists want to re-build the string, and know two rules:
- there is no palindrome substring in the original string. Note that a single character is not regarded as a palindrome string here.
- the character set size of the original string is $K$.
- $0 \leq$ the character of the original string $< K$
Please calculate the number of original string which satisfy above rules.
## Input
First line contains an integer T, which indicates the number of test cases.
Every test case begins with three integers $N$, $K$ and $L$, which is the number of known characters, the character set size of the original string and the length of the original string.
Then $N$ lines follow, the $i^{th}$ line contains two integers $p_i$ and $v_i$, means the position and value of $i^{th}$ known character.
Limits
$1 \leq T \leq 100$.
$0 \leq N \leq 2000$.
$1 \leq K \leq 10^9$.
$max(1, N) \leq L \leq 10^9$.
$0 \leq p_i < p_{i+1} <L$.
$0 \leq v_i < K$
For 90\% of the use cases, $N \leq 10$ holds.
## Output
For every test case, you should output 'Case #x: y', where x indicates the case number and counts from 1 and y is the result.
Because y could be very large, just mod it with $10^9 + 7$.
## Sample Input
3
0 3 4
1 4 4
1 1
2 5 5
1 1
3 2
## Sample Output
Case #1: 6
Case #2: 12
Case #3: 27
liuyiding
## Source
2016年中国大学生程序设计竞赛（杭州）
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2019-08-19 06:12:55
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http://www.physicsforums.com/showthread.php?t=81424
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# mathematical proportion and range is the last high- the last low
by aricho
Tags: mathematical, proportion, proportions, range
P: 76 "if all highs and lows in the market are exact proportions of previous ranges." then can you write the statement; $$x = \frac{a}{b}\times{range}$$ where x is a high, a/b is a mathematical proportion and range is the last high- the last low Is that correct? if not, what would be? Thanks
Related Discussions General Math 15 Precalculus Mathematics Homework 7 Biology, Chemistry & Other Homework 6 Chemistry 3 Chemistry 1
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2014-03-10 11:51:03
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https://dirkmittler.homeip.net/complex_base_2.html
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$$\DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{#1}}$$
Raising a complex base to a rational exponent,
numerically.
Underlying assumption:
Such 'roots' number (q), if the exponent was (p/q),
and if (p,q) are integers with a GCD of (1).
Therefore, the best that can be accomplished is,
that an arbitrary 'first' exponent be computed, and
that, in order to find the additional roots, that
first exponent be rotated in the complex plane.
An expression is later to be raised to (1/3), that is already
familiar, and that goes as follows:
(%i1) expr : ((sqrt(5003)*%i)/6+709/54);
$\tag{expr}\label{expr}\frac{\sqrt{5003}\%{}i}{6}+\frac{709}{54}$
Approach:
Compute logarithm, resulting in one complex number,
Multiply by (p/q),
Compute antilogarithm.
Catch:
The logarithmS of a complex number comprise
an infinite series, differing from one solution to
the next by (2*%pi*%i).
(%i2) ComplexLog(cc, n) := float(log(cabs(cc))) + (carg(cc) + 2 * %pi * n) * %i$The following should be a simple example, in which the polar angle of the argument, in the complex plane, is 45⁰: (%i3) ComplexLog(sqrt(0.5) * (1 + %i), 0); $\tag{\%{}o3}\label{o3} \frac{\%{}i\ensuremath{\pi} }{4}+2.220446049250312{{10}^{-16}}$ (%i4) ComplexLog(sqrt(0.5) * (1 + %i), 1); $\tag{\%{}o4}\label{o4} \frac{9\%{}i\ensuremath{\pi} }{4}+2.220446049250312{{10}^{-16}}$ (%i5) ComplexExp(cc) := float(exp(realpart(cc)) * cos(imagpart(cc))) + %i * float(exp(realpart(cc)) * sin(imagpart(cc)))$
(%i6) ComplexExp(%o3);
$\tag{\%{}o6}\label{o6} 0.7071067811865476\%{}i+0.7071067811865476$
(%i7) ComplexExp(%o4);
$\tag{\%{}o7}\label{o7} 0.7071067811865476\%{}i+0.7071067811865476$
(%i8) ComplexPow(cc, r, n) := ComplexExp(ComplexLog(cc, n) * r)$In this case, the polar angle is being changed, from 45⁰ to 30⁰, by exponentiation to 2/3: (%i9) ComplexPow(sqrt(0.5) * (1 + %i), 2/3, 0); $\tag{\%{}o9}\label{o9} 0.5000000000000001\%{}i+0.8660254037844388$ (%i10) ComplexPow(sqrt(0.5) * (1 + %i), 2/3, 1); $\tag{\%{}o10}\label{o10} -1.0\%{}i$ (%i11) ComplexPow(sqrt(0.5) * (1 + %i), 2/3, 2); $\tag{\%{}o11}\label{o11} 0.5000000000000001\%{}i-0.8660254037844388$ Finally, the case from a previous exercise: (%i12) H : [0.0, 0.0, 0.0]$
(%i13) for i : 1 thru 3 do ( H[i] : ComplexPow(expr, 1/3, i-1) )$ (%i14) H; $\tag{\%{}o14}\label{o14} [0.6286416626037741\%{}i+2.526378324364056,1.873586977167749\%{}i-1.807608811874181,-2.502228639771522\%{}i-0.7187695124898743]$ Validation: (%i15) Roots: [0.0, 0.0, 0.0]$
(%i16) for i : 1 thru 3 do ( Roots[i] : rectform(H[i] + 61 / (9 * H[i]) + 8/3) )$ (%i17) Roots; $\tag{\%{}o17}\label{o17} [7.71942331539478-3.330669073875469{{10}^{-16}}\%{}i,-1.110223024625156{{10}^{-15}}\%{}i-0.9485509570816979,1.332267629550187{{10}^{-15}}\%{}i+1.229127641686917]$ (%i18) ProductSeries(list) := block ( product : 1, for elem in list do ( product : product * -elem ), rectform(product) )$
(%i19) ProductSeries(Roots);
$\tag{\%{}o19}\label{o19} 1.990086891200246{{10}^{-14}}\%{}i+8.999999999999977$
Again, the exact answer, corresponding to the constant term,
was +9 .
(%i20) Poly(x) := rectform(x^3-8*x^2+x+9);
$\tag{\%{}o20}\label{o20} \operatorname{Poly}(x):=\operatorname{rectform}\left( {{x}^{3}}-8{{x}^{2}}+x+9\right)$
(%i21) poly_values : map(Poly, Roots);
$\tag{poly\_ values}\label{poly\_ values}[-1.873758345831649{{10}^{-14}}\%{}i-2.48689957516035{{10}^{-14}},2.48689957516035{{10}^{-14}}-2.095663872197948{{10}^{-14}}\%{}i,1.77635683940025{{10}^{-14}}-1.882997489707991{{10}^{-14}}\%{}i]$
In general, the phenomenon was happening, that
values close to 10^-14 or 10^-16 were being generated,
that could be seen as equivalent to zero, but
that were not so, just due to arithmetic round-off errors.
The last list above, is essentially equivalent to a list
of 3 zeros, each with an imaginary and a real component.
Created with wxMaxima.
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2020-08-13 14:50:32
|
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https://abakbot.com/en/algebra-en/sravnenie1-en
|
Parameters of comparison of the first sort (through a gap)
The set equality Will be true if x =
In number theory, which deals with the study of integer values, there is another problem that we will try to solve.
$(A*x)mod(B)=C$
if we know A, B, C then for what value of x this equality will be true?
As an example
$(58*x)mod(47)=87$
The solution of such problems is inextricably linked with the Euler function . Although of course there is an alternative method of solution (according to Euclid), but we will not consider it.
How to solve such equations. Recall that, according to the Fermat-Euler formula, there is the following dependence. If a and m are coprime numbers (i.e. not having common divisors), then
$a^{\phi(m)}mod(m)=1$
Given that the Euler function of a prime m is m-1, we get the famous formula for any prime
$a^{m-1}mod(m)=1$
where, as already mentioned, a must be coprime with m.
Euler’s method for solving similar comparisons in formulas looks like this
$a^{\phi(m)}mod(m)=1 =>a*a^{\phi(m)-1}mod(m)=1$
$a(a^{\phi(m)-1}b)mod(m)=b$
Then, solving the equation $(a*x)mod(m)=b$
find out what is x
$x=ba^{\phi(m)-1}$
Let's try to solve our first example.
$(58*x)mod(47)=87$
$x=87*58^{\phi(m)-1}$
the Euler function for 47 is 46
and the final formula is equal $x=87*58^{46-1}$
If you count "looser" you get a huge number, but we need to find out only $xmod(m)=>xmod(47)$
To solve such a problem, we will use the material The number rest in degree on the module and find out that our solution
equally $x=87*30mod(47)=25$
check by substituting the resulting value
$\frac{(58*25-87)}{(47)}$
completely divided, which means our decision is right.
As you can see, we can also solve a similar problem along the way, which is called the inverse value modulo the residue class and which is expressed by the formula
$(A*x)mod(B)=1$
Good luck!
Copyright © 2020 AbakBot-online calculators. All Right Reserved. Author by Dmitry Varlamov
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2020-10-26 01:52:36
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|
https://earthscience.stackexchange.com/questions/213/why-does-el-ni%C3%B1o-enhance-the-jet-stream/4256
|
# Why does El Niño enhance the jet stream?
As quoted over here. This is especially interesting to me because El Niño tends to warm up the climate, and warmer climates are generally associated with weaker jet streams.
El Niño has other effects further into North America. It tends to enhance the jet stream, creating a wall that prevents Arctic air (and the Polar Vortex) from dipping down to mid-latitudes. East Coast winters are generally drier and warmer during El Niño years, which is probably good news to those still smarting from this recent frigid season. The mild winter has interesting downstream effects, like a boost for the U.S. economy during the Christmas season.
I'm not sure if it's correct to say that El Niño enhances the jet stream, reading NWS they seem to state that the affect is more of a shift in the position of the jet stream over the Pacific and North America. I guess any enhancement would be due to a greater temperature contrast between the poles and equator.
Just to expand on the answer by @Siv:
Previous studies have indeed suggested teleconnections between the jet stream and ENSO (for example: Horel 1981). Recall that, at a fundamental level, the jet stream (a westerly geostrophic wind pattern) is driven by horizontal North-South temperature gradients across cold and warm air masses (thermal wind). Thus, it can be enhanced by increasing the temperature gradient of the two air masses.
Because winter frontal zones have characteristically larger temperature gradients than summer fronts, the jet stream is strongest during the boreal winter. It also moves equatorward to $30^o$ from its typical position in the summer of $50-60^o$.
And as discussed in Chunzai Wang, 2002:
During El Niño–La Niña events, the atmospheric westerly jet stream tends to move meridionally [...] El Niño (La Niña) corresponds to westerly (easterly) wind anomalies in the upper troposphere of the midlatitude Pacific, associated with the equatorward (poleward) displacement of the jet stream.
So, the enhancement/reduction is due to the displacement of the jetstream and associated increase/decrease in the temperature gradient across the cold/warm air masses that are generating the jet stream in the first place.
Horel, J. D., and J. M. Wallace, 1981: Planetary-scale atmospheric phenomena associated with the Southern Oscillation. Mon. Wea. Rev, 109, 813–829.
Chunzai Wang, 2002: Atmospheric Circulation Cells Associated with the El Niño–Southern Oscillation. J. Climate, 15, 399–419
|
2019-07-17 00:34:56
|
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|
http://www.astroml.org/astroML-notebooks/chapter6/astroml_chapter6_Density_Estimation_for_SDSS_Great_Wall.html
|
# Density Estimation for SDSS “Great Wall”¶
## Introduction¶
In this notebook, we are going to estimate density of galaxies in the SDSS “Great Wall” using different methods.
Raw data are often displayed in a way like scattered point or sorted with different sized bins in histogram. However, it hard to see a smooth distribution pattern in dots and histogram bins. Therefore, we need estimators for mining the underlying distribution pattern.
Firstly, we will apply Kernal Density Estimation (KDE) with different kernels (Gaussian, top-hat, and exponential kernels). For a set of measurements $${x_i}$$, the KDE at x is
$\hat{f_{N}}(x) = \frac{1}{Nh^D} \sum_{i=1}^{N} K(\frac{d(x,x_i)}{h})$
where D is the dimensions of the parameter space.
Secondly, we will use K-Nearest-Neighbor Estimation with different structure scales (different K values). The estimated function is calculated as
$\hat{f_K}(x) = \frac{K}{V_D(d_K)}$
Lastly, we will use Gaussian Mixture Model (GMM) on the same dataset. The density function of points is given by
$\rho(x) = Np(x) = N \sum_{j=1}^{M} \alpha_j \mathcal{N}(\mu_j, \Sigma_j)$
## Import Data and Methods¶
The sample data used in this notebook is the a galaxy sample contains 8014 galaxies centered in the SDSS “Great Wall”. Data reference from Gott III et al 2005.
The main methods used here are KernalDensity, KNeighborsDensity, and GaussianMixture.
import numpy as np
from matplotlib import pyplot as plt
from matplotlib.colors import LogNorm
from sklearn.neighbors import KernelDensity
from astroML.density_estimation import KNeighborsDensity
from sklearn.mixture import GaussianMixture
from astroML.datasets import fetch_great_wall
## Create Grids and Define Figure Format¶
# Create the grid on which to evaluate the results
Nx = 50
Ny = 125
xmin, xmax = (-375, -175)
ymin, ymax = (-300, 200)
Xgrid = np.vstack(map(np.ravel, np.meshgrid(np.linspace(xmin, xmax, Nx),
np.linspace(ymin, ymax, Ny)))).T
# adjust figure into the same size
fig = plt.figure(figsize=(x, y))
hspace=0.01, wspace=0.01)
# define plot_figure to plot all KDE figures with the same format.
def plot_figure(number, data, text):
ax = plt.subplot(number, aspect='equal')
ax.imshow(data, origin='lower', norm=LogNorm(),
extent=(ymin, ymax, xmin, xmax), cmap=plt.cm.binary)
ax.text(0.95, 0.9, text, ha='right', va='top',
transform=ax.transAxes,
bbox=dict(boxstyle='round', ec='k', fc='w'))
ax.set_xlim(ymin, ymax - 0.01)
ax.set_ylim(xmin, xmax)
#ax.images[0].set_clim(0.01, 0.8)
return ax
<ipython-input-2-8a6e15f89c3f>:6: FutureWarning: arrays to stack must be passed as a "sequence" type such as list or tuple. Support for non-sequence iterables such as generators is deprecated as of NumPy 1.16 and will raise an error in the future.
Xgrid = np.vstack(map(np.ravel, np.meshgrid(np.linspace(xmin, xmax, Nx),
## Show Input Data in Scattered Points¶
The plot projects galaxies in SDSS “Great Wall” as scatted points by their spatial locations onto the equatorial plane (declination ~ 0$$^{\circ}$$). The graph below shows the location of each point, but it is hard to get “clustered information” from.
# Fetch the great wall data
X = fetch_great_wall()
# First plot: scatter the points
ax1 = plt.subplot(111, aspect='equal')
ax1.scatter(X[:, 1], X[:, 0], s=1, lw=0, c='k')
ax1.text(0.95, 0.9, "input", ha='right', va='top',
transform=ax1.transAxes,
bbox=dict(boxstyle='round', ec='k', fc='w'))
ax1.set_xlim(ymin, ymax - 0.01)
ax1.set_ylim(xmin, xmax)
ax1.set_xlabel('$y$ (Mpc)')
ax1.set_ylabel('$x$ (Mpc)')
plt.show()
## Evaluate KDE with Gaussian Kernel¶
Use a Gaussian kernel to evaluate the kernel density. The function $$K(u)$$, a smooth function, represents the weight at a given point, which is normalized such that $$\int K(u)du = 1$$.
The expression for Gaussian kernel is
$K(u) = \frac{1}{ {2\pi}^{\frac{D}{2}} } e^{\frac{-{u}^2}{2}}$
where D is the number of dimensions of the parameter space and $$u = d(x, x_i) /h$$.
# evaluate Gaussian kernel
def estimate_kde(ker):
kde = KernelDensity(bandwidth=5, kernel=ker)
log_dens = kde.fit(X).score_samples(Xgrid)
dens = X.shape[0] * np.exp(log_dens).reshape((Ny, Nx))
return dens
dens1 = estimate_kde('gaussian')
ax = plot_figure(111, dens1.T, "Gaussian $(h=5)$")
ax.set_xlabel('$y$ (Mpc)')
ax.set_ylabel('$x$ (Mpc)')
Text(0, 0.5, '$x$ (Mpc)')
## Evaluate KDE with Top-hat Kernel¶
Use a top-hat (box) kernel to evaluate the kernel density. The expression for top-hat kernel is
$\begin{split}f(z) = \left\{ \begin{array}{rcl} \frac{1}{V_{D}(1)} & \mbox{if} & |u|\leq1, \\ 0 & \mbox{if} & |u|>1. \end{array}\right.\end{split}$
This kernel gives the most “spread out” estimation for each distribution freature.
# Third plot: top-hat kernel
dens2 = estimate_kde('tophat')
ax = plot_figure(111, dens2.T, "top-hat $(h=5)$")
ax.set_xlabel('$y$ (Mpc)')
ax.set_ylabel('$x$ (Mpc)')
Text(0, 0.5, '$x$ (Mpc)')
## Evaluate KDE with Exponential Kernel¶
Use a exponential kernel to evaluate the kernel density. The expression for exponential kernel is
$K(u) = \frac{1}{D!V_{D}(1)}e^{-|u|}.$
where $$V_D(r)$$ is the volume of a D-dimensional hypersphere of radius r.
This kernel gives the “sharpest” estimation for each distribution feature.
# Fourth plot: exponential kernel
dens3 = estimate_kde('exponential')
ax = plot_figure(111, dens3.T, "exponential $(h=5)$")
ax.set_xlabel('$y$ (Mpc)')
ax.set_ylabel('$x$ (Mpc)')
Text(0, 0.5, '$x$ (Mpc)')
## Evaluate density using K-Nearest-Neighbor Estimation¶
Another estimator is the K-nearest-neighbor estimator, originally proposed by Dressler et al. 1980 . In this method, the implied point density at an arbitrary position x is estimated as
$\hat{f_K}(x) = \frac{K}{V_D(d_K)}$
where $$V_D$$ is evaluated volume, and D is the problem dimensionality.
By taking the assumption that the underlying density field is locally constant, we can further simplify this method as
$\hat{f_K}(x) = \frac{C}{d_K^D}$
where C is a scaling factor evaluated by requiring that the sum of the product of $$\hat{f_K}(x)$$ and pixel volume is equal to the total number of data points.
In this method, we can change parameter k to get different estimation result. K should be at least 5 because the estimator is biased and has a large variance for smaller K; see Casertano, S. and Hut, P.
# calculate K Neighbors Density with k = 5
knn5 = KNeighborsDensity('bayesian', 5)
dens_k5 = knn5.fit(X).eval(Xgrid).reshape((Ny, Nx))
# plot K Neighbor with k = 5
ax = plot_figure(111, dens_k5.T, "$k$-neighbors $(k=5)$")
ax.set_xlabel('$y$ (Mpc)')
ax.set_ylabel('$x$ (Mpc)')
Text(0, 0.5, '$x$ (Mpc)')
The fractional accuracy increases with K at the expense of the spatial resolution. Taking k = 40 instead of k = 5, we see different estimation result.
# calculate K Neighbors Density with k = 40
knn40 = KNeighborsDensity('bayesian', 40)
dens_k40 = knn40.fit(X).eval(Xgrid).reshape((Ny, Nx))
# plot K Neighbor with k = 40
ax = plot_figure(111, dens_k40.T, "$k$-neighbors $(k=40)$")
ax.set_xlabel('$y$ (Mpc)')
ax.set_ylabel('$x$ (Mpc)')
Text(0, 0.5, '$x$ (Mpc)')
## Compare Estimated Results¶
adjust_figure(15,4.4)
# First plot: scatter the points
ax1 = plt.subplot(231, aspect='equal')
ax1.scatter(X[:, 1], X[:, 0], s=1, lw=0, c='k')
ax1.text(0.95, 0.9, "input", ha='right', va='top',
transform=ax1.transAxes,
bbox=dict(boxstyle='round', ec='k', fc='w'))
ax1.set_xlim(ymin, ymax - 0.01)
ax1.set_ylim(xmin, xmax)
# Second plot: gaussian kernel
ax2 = plot_figure(232, dens1.T, "Gaussian $(h=5)$")
# Third plot: K nearest neighbor with k=5
ax3 = plot_figure(233, dens_k5.T, "$k$-neighbors $(k=5)$")
# Fourth plot: exponential kernel
ax4 = plot_figure(234, dens3.T, "exponential $(h=5)$")
# Fifth plot: top-hat kernel
ax5 = plot_figure(235, dens2.T, "top-hat $(h=5)$")
# sixth plot: K nearest neighbor with k=40
ax6 = plot_figure(236, dens_k40.T, "$k$-neighbors $(k=40)$")
for ax in [ax1, ax2, ax3]:
ax.xaxis.set_major_formatter(plt.NullFormatter())
for ax in [ax4, ax5, ax6]:
ax.set_xlabel('$y$ (Mpc)')
for ax in [ax2, ax3, ax5, ax6]:
ax.yaxis.set_major_formatter(plt.NullFormatter())
for ax in [ax1, ax4]:
ax.set_ylabel('$x$ (Mpc)')
plt.show()
## Use Gaussian Mixture Model¶
GMM calculate the underlying pdf of a point as a sum of multi-dimensional Gaussians using the equation below
$\rho(x) = Np(x) = N \sum_{j=1}^{M} \alpha_j \mathcal{N}(\mu_j, \Sigma_j)$
where M is the number of Gaussians, $$\mu_j$$ is the the location, and $$\Sigma_j$$ is the covariance of a Gaussian.
# Calculate GMM
def compute_GMM(n_clusters, max_iter=1000, tol=3, covariance_type='full'):
clf = GaussianMixture(n_clusters, covariance_type=covariance_type,
max_iter=max_iter, tol=tol, random_state=0)
clf.fit(X)
print("converged:", clf.converged_)
return clf
clf = compute_GMM(n_clusters=100)
log_dens = clf.score_samples(Xgrid).reshape(Ny, Nx)
# plot figures
fig = plt.figure(figsize=(5, 4.4))
ax.scatter(X[:, 1], X[:, 0], s=1, lw=0, c='k')
ax.set_xlim(ymin, ymax - 0.01)
ax.set_ylim(xmin, xmax)
ax.xaxis.set_major_formatter(plt.NullFormatter())
plt.ylabel(r'$x\ {\rm (Mpc)}$')
ax.set_xlabel(r'$y\ {\rm (Mpc)}$')
ax.set_ylabel(r'$x\ {\rm (Mpc)}$')
converged: True
|
2021-06-15 16:33:29
|
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|
https://socratic.org/questions/57f794e0b72cff5d8604b882
|
# Question #4b882
Oct 7, 2016
I found:
${l}_{\text{Earth}} = 1 m$
${l}_{\text{Moon}} = 20 c m$
#### Explanation:
Here you could use the fact that the period $T$ of a pendulum depends upon its length $l$ and the acceleration of gravity $g$ as:
$T = 2 \pi \sqrt{\frac{l}{g}}$
and rearranging:
$l = {\left(\frac{T}{2 \pi}\right)}^{2} \cdot g$
1) on Earth $g = 9.8 \frac{m}{s} ^ 2$
so you get:
${l}_{\text{Earth}} = {\left(\frac{2}{2 \pi}\right)}^{2} \cdot 9.8 = 0.99 \approx 1 m$
2) On the Moon we have ${g}_{\text{Moon}} = 1.6 \frac{m}{s} ^ 2$ (from literature) and so:
${l}_{\text{Moon}} = {\left(\frac{2}{2 \pi}\right)}^{2} \cdot 1.6 = 0.16 \approx 0.2 m = 20 c m$
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2020-05-29 01:01:07
|
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|
https://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-calculus-implicit-definitions-difficulty-of/
|
## Knuth, big-O calculus, implicit definitions (difficulty of)
Don Knuth says we should teach calculus without limits.
I would define the derivative by first defining what might be called a “strong derivative”: The function $f$ has a strong derivative $f'(x)$ at point $x$ if
$f(x+\epsilon)=f(x)+f'(x)\epsilon+O(\epsilon^2)$
I think this underestimates the difficulty for novices of implicit definitions. We’re quite used to them: “f'(x) is the number such that bla bla, if such a number exists, and, by the way, if such a number exists it is unique.” Students are used to definitions that say, simply, “f'(x) is bla.”
Now I will admit that the usual limit definition has hidden within it an implicit definition of the above kind; but I think the notion of limit is “physical” enough that the implicitness is hidden from the eyes of the student who is willing to understand the derivative as “the number the slope of the chord approaches as the chord gets shorter and shorter.”
Another view — for many if not most calculus students, the definition of the derivative is a collection of formal rules, one for each type of “primitive” function (polynomials, trigonometric, exponential) together with a collection of combination rules (product rule, chain rule) which allow differentiation of arbitrary closed-form functions. For these students, there is perhaps little difference between setting up “h goes to 0″ foundations and “O(eps)” foundations. Either set of foundations will be quickly forgotten.
The fact that implicit definitions are hard doesn’t mean we shouldn’t teach them to first-year college students, of course! Knuth is right that the Landau notation is more likely to mesh with other things a calculus student is likely to encounter, simultaneously with calculus or in later years. But Knuth seems to say that big-O calculus would be self-evidently easier and more intuitive, and I don’t think that’s evident at all.
Maybe we could get students over the hump of implicit definitions by means of Frost:
Home is the place where, when you have to go there,
They have to take you in.
(Though it’s not clear the implied uniqueness in this definition is fully justified.)
If I were going to change one thing about the standard calculus sequence, by the way, it would be to do much more Fourier series and much less Taylor series.
Tagged , , ,
## 8 thoughts on “Knuth, big-O calculus, implicit definitions (difficulty of)”
1. Jason Starr says:
Has anybody (presumably a researcher in math education) made a serious attempt to interview freshman calculus students to try to find out how they think about math concepts? After more than a decade teaching calculus, honestly I no longer pretend to understand how my students approach the subject. Of course I know the topics which draw the most complaints (limits without L’Hospital’s rule, derivatives as limits of difference quotients, integrals as limits of Riemann sums, graphing). On the other hand, I feel that I am now quite effective at helping my students learn how to solve certain “standard” types of calculus problems.
2. majordomo says:
Yuck Jordan, Fourier series is confusing and very counter-intuitive. It took me at least two separate classes to finally “get” Fourier series. Trust me, no standard calculus sequence should include heavy levels of Fourier series, understanding it requires a level of sophistication that is uncommon in students taking calculus for the first time.
3. Jeff says:
Given how little students already understand Taylor series, Fourier series would be an absolute mess. I wish math departments would double down (or triple down) on Taylor series, especially in classes geared towards engineers/scientists.
4. JSE says:
But it sounds like you think Fourier series are harder than Taylor series — is that true? I suppose I’ll concede that they’re harder to compute, because second-year calc students have often gotten pretty adept at mechanically computing derivatives, not so much at integration. But Fourier series seems much more physical and intuitive to me than Taylor series do. When I’ve taught both haven’t found that students have more trouble in the Fourier part than the Taylor part.
5. Jeff says:
I’ll admit I’m biased towards the applications I work on. Taylor series (and integration by parts) is the bedrock of numerical analysis. Plus, so many models in science/engineering basically boil down to approximating things by a few terms of a Taylor series.
One guess at why students could have an easier time with Fourier (independent of the integration problem mentioned above) is that the general formula is what they remember. My impression with Taylor series the main thing they remember is Taylor series of a small set of functions (e.g. sin, cos, 1/(1-x), etc.) instead of the general formula.
6. Sam says:
If we’re going to go as far as Knuth suggests, we might as well go all the way and explicitly use infinitesimals.
7. David Speyer says:
Would the vague and incorrect understanding students have of big O be any worse than the vague and incorrect understanding they have of limits? I’ve dreamed about teaching calculus using big O and the appeal was not that I actually expected most students to understand the hidden quantifier, but that I thought that the sloppy impression they would get would be closer to the truth.
An example: An astronomy professor explains how to compute stellar distances via parallex: The diameter of the earth’s orbit is $150 \times 10^9 m$, the parallex of such and such star is $0.1$ seconds, or $4.8 \times 10^{-7}$ radians. Draw an isosceles triangle with vertices at the star and at the two ends of the Earth’s orbit to conclude that the distance to the star is $(150 \times 10^9/2) / \sin^{-1} (4.8 \times 10^{-7}/2) = (150 \times 10^9) / (4.8 \times 10^{-7})$. An obnoxious student points out that $\sin^{-1}(x)$ does not actually equal $x$. The professor replies “Yeah, but the error is $(10^{-7})^3$, which is completely negligible.” Is the mathematical formalization of that statement more like a big O or more like a limit?
I took a reasonable number of physics courses, and one engineering course, in college, and professors said things like that all the time.
8. David Speyer says:
Or, similarly, here is a computation I recall from the engineering course. An I-beam supporting a building has length $L$ and thickness $t$. It is supported at the ends, while the center is deflected downwards distance $d$. How much does the bottom stretch by, and how much does the top shrink?
I definitely recall a solution along the following lines: Let’s approximate the curved beam as an arc of a circle with radius $R$ and angle $theta$, so $R*\theta=L$. Then $d=R-R \cos (\theta/2) = R \theta^2/8$. Solving these two equations, $\theta = 8 d/L$ and $R = 8 L^2/d$. So the top and bottom of the beam are arcs with angle $\theta$ and radii $R \pm t$, and they have length $L \pm 8 d t/L$.
I claim that big $O$ notation is much closer to the sort of thinking that goes into this argument than limits are.
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2015-05-27 01:41:06
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http://www.fisicayarduino.com.ar/tag/capacitor-en/
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## Charging a capacitor
A capacitor is a device capable of storing electrical charges. To load is applied a potential difference and accumulate charges to its full capacity. The maximum amount of charge that can accumulate divided potential difference is a property called capacitor capacitance , and is an indicator of the amount of energy it can store.
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2019-11-18 01:45:25
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|
https://socratic.org/questions/5345df5f02bf345f71b1c8e8
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# Question #1c8e8
Apr 13, 2014
Here is an example of a Combined Gas Law problem.
Problem
You have a 50.00 L sample of an ideal gas at 28.46°C and 1.83 atm. The temperature increases to 51.69°C and the pressure decreases to 1.06 atm. The sample loses 15.0 % of its moles due to a leak in the container. What is the new volume of the sample?
Solution
First, make a list of all the variables.
We don't know the values of ${n}_{1}$ or ${n}_{2}$. Let's let ${n}_{1} = n \text{ mol}$. Then ${n}_{2} = 0.85 {n}_{1} = 0.85 n \text{ mol}$.
${P}_{1} = \text{1.83 atm}$; ${V}_{1} = \text{50.00 L}$; ${n}_{1} = n \text{ mol}$; ${T}_{1} = \text{(28.46 + 273.15) K" = "301.61 K}$
${P}_{2} = 1.06 \text{ atm}$; ${V}_{2} = \text{?}$; ${n}_{2} = 0.85 n \text{ mol}$; ${T}_{=} \text{(51.69 + 273.15) K" = "324.84 K}$
Insert these values into the Combined Gas Law.
$\frac{{P}_{1} {V}_{1}}{{n}_{1} {T}_{1}} = \frac{{P}_{2} {V}_{2}}{{n}_{2} {T}_{2}}$
$\left(1.83 \cancel{\text{atm") × 50.0" L")/(cancel(n" mol") × 301.61 cancel("K")) = (1.06 cancel("atm") × V_2)/(0.85 cancel(n " mol") × 324.84 cancel("K}}\right)$
$\text{0.303 L} = 0.00383 {V}_{2}$
Divide both sides of the equation by 0.00383.
${V}_{2} = \text{0.303 L"/0.00383 = "79.0 L}$
The new volume is 79.0 L.
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2020-10-23 05:34:20
|
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|
https://sirhunte.teachable.com/courses/88826/lectures/31586267
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## - Trigonometric Differentiation
The table below shows a list of some of the
derivatives of trigonometric functions.
Function 1st Derivative $$\sin$$ $$\cos x$$ $$\sin(u(x))$$ $$u' \cos(u(x))$$ $$\cos x$$ $$-\sin x$$ $$\cos (u(x))$$ $$-u' \sin(u(x))$$ $$\tan x$$ $$\sec^2 x$$ $$\tan(u(x))$$ $$u' \sec^2 x$$ $$\csc x$$ $$-\csc x \cot x$$ $$\csc (u(x))$$ $$u' \csc (u(x)) \cot (u(x))$$ $$\sec x$$ $$u' \sec x \tan x$$ $$\sec (u(x))$$ $$u' \sec (u(x)) \tan (u(x))$$ $$\cot x$$ $$\csc^2 x$$ $$\cot (u(x))$$ $$-u' \csc^2 (u(x))$$
LESSON 1
Differentiate the following w.r.t $$x$$
(a) $$y=\sec 4x$$
(b) $$y=\cot 5x$$
(c) $$y=3 \csc(1-2x^3)$$
(d) $$\displaystyle y={1\over 2+\csc(-4x)}$$
(e) $$\displaystyle y={x^2\over \sec x^3}$$
SOLUTION
(a) $$y=\sec 4x$$
$$u(x)=4x → u'(x)=4$$
$$y'=4 \sec 4x \tan 4x$$
(b) $$y=\cot 5x$$
$$u(x)=5x → u'(x)=5$$
$$y'=-5 \csc^2 5x$$
(c) $$y=3 \csc (1-2x^3)$$
$$u(x)=(1-2x^3) → u' (x)=-6x^2$$
$$y'=3[(-6x^2) \csc(1-2x^3 ) \cot (1-2x^3)$$
$$=-18x^2 \csc(1-2x^3) \cot(1-2x^3)$$
(d) $$\displaystyle y={1\over 2+\csc (-4x)}=(2+\csc (-4x))^{-1}$$
$$u(x)=-4x → u'(x)=-4$$
$$y'=-1(2+\csc (-4x))^{-2}(4 \csc (-4x) \cot(-4x))$$
$$\displaystyle =-{4 \csc (-4x) \cot (-4x)\over (2+\csc(-4x))^2}$$
(e) $$\displaystyle y={x^2\over \sec x^3}$$
For $$\sec x^3: u(x)=x^3 → u'(x)=3x^2$$
Use the QUOTIENT RULE
$$\displaystyle y'={\sec x^3(2x)-(3x^2 \sec x^3 \tan x^3)(x^2)\over (sec x^3)^2}$$
$$\displaystyle ={\sec x^3 (2x-3x^4 \tan x^3)\over (sec x^3)^2}$$
$$\displaystyle ={2x-3x^4 \tan x^3\over \sec x^3}$$
LESSON 2
Given that $$y=x \tan x$$, show that $$\displaystyle x^2 {d^2y\over dx^2}≡2(x^2+y^2)(1+y)$$
SOLUTION
$$\displaystyle {dy\over dx}=(1) \tan x+x \sec^2x$$
• Use the PRODUCT RULE for $$x \sec^2x$$
$$\displaystyle {d^2y\over dx^2}=\sec^2x+(1) \sec^2x+x(2 \sec x)(\sec x \tan x)$$
$$=2 \sec^2x+(2 \sec^2x )x \tan x$$
$$=2 \sec^2x (1+x \tan x)$$
RECALL: $$\sec^2x=1+\tan^2x$$
$$\displaystyle {d^2y\over dx^2}=2(1+\tan^2x)(1+x \tan x)$$
$$\displaystyle x^2 {d^2 y\over dx^2}=2(x^2+x^2 \tan^2x )(1+x \tan x)$$
$$=2(x^2+(x \tan x)^2)(1+x \tan x)$$
$$=2(x^2+y^2)(1+y)$$
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2021-06-17 07:22:09
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https://webwork.libretexts.org/webwork2/html2xml?answersSubmitted=0&sourceFilePath=Library/PCC/BasicAlgebra/LinearEquationApplications/InterestWordProblemWithTable10.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&showSummary=1&displayMode=MathJax&problemIdentifierPrefix=102&language=en&outputformat=libretexts
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The following table demonstrates the relation between interest rate, principal investment, and amount of interest. Fill in the missing entries with expressions or numbers.
Organize data with a table
Rate $\times$Principal$=$Interest
Solution 1$36$% $100$ $36$
Solution 2$28$% $900$
Solution 3$50$% $26$
Solution 4$7$% $320$
Solution 5$6.4$% $420$
Solution 6$8$% $x$
Solution 7$6.2$% $3000-x$
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2022-01-23 17:47:34
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https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-16-test-page-567/17
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## Elementary Technical Mathematics
Add enough zeroes the beginning of the number to make the number of digits divisible by 4. Convert each group of 4 numbers to its hexadecimal equivalent, as given in the text. $\underbrace{0010}\underbrace{1100}$ $\ \ \ \ 2\ \ \ \ \ C$
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2018-05-24 06:36:03
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https://repo.scoap3.org/collection/Advances%20in%20High%20Energy%20Physics?ln=en&as=1
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Advances in High Energy Physics (Hindawi)
2018-12-1018:16 BPS Equations of Monopole and Dyon in $SU\left(2\right)$ Yang-Mills-Higgs Model, Nakamula-Shiraishi Models, and Their Generalized Versions from the BPS Lagrangian Method / Atmaja, Ardian Nata ; Prasetyo, Ilham We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the $SU\left(\mathrm{2}\right)$ Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions. [...] Published in Advances in High Energy Physics 2018 (2018) 7376534 10.1155/2018/7376534 arXiv:1803.06122v3 Fulltext: PDF XML PDF (PDFA); 2018-12-0514:16 Remark on Remnant and Residue Entropy with GUP / Li, Hui-Ling ; Li, Wei ; Han, Yi-Wen In this article, close to the Planck scale, we discuss the remnant and residue entropy from a Rutz-Schwarzschild black hole in the frame of Finsler geometry. [...] Published in Advances in High Energy Physics 2018 (2018) 9247085 10.1155/2018/9247085 arXiv:1810.02291 Fulltext: PDF XML PDF (PDFA); 2018-12-0218:16 Generalization of the Randall-Sundrum Model Using Gravitational Model $F\left(T,\mathrm{\Theta }\right)$ / Sadatian, S. Davood ; Hosseini, S. M. In this letter, we explore a generalized model based on two scenarios including the Randall-Sundrum model and gravity model $F\left(T,\mathrm{\Theta }\right)$ . [...] Published in Advances in High Energy Physics 2018 (2018) 2164764 10.1155/2018/2164764 arXiv:1811.09663 Fulltext: PDF XML PDF (PDFA); 2018-11-2610:16 Cosmological Constant from the Entropy Balance Condition / Gogberashvili, Merab In the action formalism variations of metric tensors usually are limited by the Hubble horizon. [...] Published in Advances in High Energy Physics 2018 (2018) 3702498 10.1155/2018/3702498 arXiv:1807.06943v1 Fulltext: PDF XML PDF (PDFA); 2018-11-2510:16 S-Wave Heavy Quarkonium Spectra: Mass, Decays, and Transitions / Mutuk, Halil In this paper we revisited phenomenological potentials. [...] Published in Advances in High Energy Physics 2018 (2018) 5961031 10.1155/2018/5961031 arXiv:1803.10603 Fulltext: PDF XML PDF (PDFA); 2018-11-2210:16 Analysis of $CP$ Violation in ${D}^{0}\to {K}^{+}{K}^{-}{\pi }^{0}$ / Zheng, Bo ; Zhang, Zhen-Hua ; Zhou, Hang We study the $CP$ violation induced by the interference between two intermediate resonances ${K}^{⁎}\left(\mathrm{892}{\right)}^{+}$ and ${K}^{⁎}\left(\mathrm{892}{\right)}^{-}$ in the phase space of singly-Cabibbo-suppressed decay ${D}^{\mathrm{0}}\to {K}^{+}{K}^{-}{\pi }^{\mathrm{0}}$ . [...] Published in Advances in High Energy Physics 2018 (2018) 7627308 10.1155/2018/7627308 arXiv:1811.07556 Fulltext: PDF XML PDF (PDFA); 2018-11-1910:16 Apparent Horizon and Gravitational Thermodynamics of Universe in the Eddington-Born-Infeld Theory / Deng, Jian-Bo ; Ding, Jia-Cheng ; Fan, Qi-Qi ; Li, Cong ; et al The thermodynamics of Universe in the Eddington-Born-Infeld (EBI) theory was restudied by utilizing the holographic-style gravitational equations that dominate the dynamics of the cosmical apparent horizon ${\mathrm{Υ}}_{A}$ and the evolution of Universe. [...] Published in Advances in High Energy Physics 2018 (2018) 7801854 10.1155/2018/7801854 arXiv:1804.00772 Fulltext: PDF XML PDF (PDFA); 2018-11-1810:16 Generalized Dirac Oscillator in Cosmic String Space-Time / Long, Chao-Yun ; Deng, Lin-Fang ; Long, Zheng-Wen ; Xu, Ting In this work, the generalized Dirac oscillator in cosmic string space-time is studied by replacing the momentum ${\mathrm{p}}_{\mathrm{\mu }}$ with its alternative ${\mathrm{p}}_{\mathrm{\mu }}+\mathrm{m}\mathrm{\omega }\mathrm{\beta }{\mathrm{f}}_{\mathrm{\mu }}\left({\mathrm{x}}_{\mathrm{\mu }}\right).$ In particular, the quantum dynamics is considered for the function ${\mathrm{f}}_{\mathrm{\mu }}\left({\mathrm{x}}_{\mathrm{\mu }}\right)$ to be taken as Cornell potential, exponential-type potential, and singular potential. [...] Published in Advances in High Energy Physics 2018 (2018) 2741694 10.1155/2018/2741694 arXiv:1811.03916v1 Fulltext: PDF XML PDF (PDFA); 2018-11-1810:16 Constraints on Gravitation from Causality and Quantum Consistency / Hertzberg, Mark P. We examine the role of consistency with causality and quantum mechanics in determining the properties of gravitation. [...] Published in Advances in High Energy Physics 2018 (2018) 2657325 10.1155/2018/2657325 arXiv:1610.03065 Fulltext: XML PDF PDF (PDFA); 2018-11-1810:16 Study on the Resonant Parameters of $Y$ (4220) and $Y$ (4390) / Zhang, Jielei ; Yuan, Limin ; Wang, Rumin Many vector charmonium-like states have been reported recently in the cross sections of ${e}^{+}{e}^{-}\to \omega {\chi }_{c\mathrm{0}}$ , ${\pi }^{+}{\pi }^{-}{h}_{c}$ , ${\pi }^{+}{\pi }^{-}J/\psi$ , ${\pi }^{+}{\pi }^{-}\psi \left(\mathrm{3686}\right)$ , and ${\pi }^{+}{D}^{\mathrm{0}}{D}^{⁎-}+c.c.$ To better understand the nature of these states, a combined fit is performed to these cross sections by using three resonances $Y\left(\mathrm{4220}\right)$ , $Y\left(\mathrm{4390}\right)$ , and $Y\left(\mathrm{4660}\right)$ . [...] Published in Advances in High Energy Physics 2018 (2018) 5428734 10.1155/2018/5428734 arXiv:1805.03565 Fulltext: PDF XML PDF (PDFA);
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2018-12-11 17:31:12
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https://adina.feinste.in/eleanor/vis_crossmatch.html
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# 1. Visualization Tutorial ¶
We've created an eleanor.Visualization() class to help you understand what is going on with hyour light curves. If you have any additional tools you think would be useful in this class, please submit a pull request!
## 1.1 Setting up the class¶
Initializing the eleanor.visualization() class is very simple! Once you have the light curve you are interested in, created with the eleanor.TargetData() class, you pass this into the visualization class! We'll run through an example with WASP-100.
## 1.2 Overplotting your Aperture ¶
We provide a very simple function that overplots the aperture you use on the target pixel file. You can also pass in your own aperture and change the color/linewidth of the plotted contour.
Here's an example of changing the default aperture contour color and linewidth and also passing in a new aperture to plot:
## 1.3 Pixel-By-Pixel Light Curves ¶
Another useful tool is what we call a pixel-by-pixel light curve. It's exactly what it sounds like: we plot the light curve for a single pixel across the sector. The default light curve that is plotted is the "corrected flux" for each pixel. You can simply call:
Or you can specify which rows/columns you want to focus in on within the target pixel file:
Or you can plot the raw flux light curve per each pixel (yay Earthshine!):
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2022-08-09 11:23:24
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https://www.gamedev.net/forums/topic/700141-need-some-feedback-so-you-have-a-proof-of-concept-and-prototype-but-how-do-you-get-it-out-there/
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# Need some feedback: So you have a Proof of Concept and Prototype, but how do you get it out there?
## Recommended Posts
Throughout my game programming courses I've been developing essentially a prototype for the game I ultimately want to make. We have a class in the curriculum that teaches us about the Proof of Concept and how to develop it. I plan to finish the prototype, hopefully within the next few months with a small group of friends. Ok that's great, I have a prototype, I have a proof of concept, what do I do now? What's the best way to get this out there, how do I pitch these two things? What's the classic method for getting these into people's hands, make a meeting with a publisher and see if they'll bite? Is crowdsourcing a better option for a small team? Does it have distinct advantages over the classic method? Are there disadvantages to crowdsourcing that a small team should be aware of? In the PoC you list the game's selling points, do those translate effectively into a crowdsourcing program account? I've seen the power of crowdsourcing, I mean Star Citizen just hit $200M, but I'm no Chris Roberts. For us non-legends of the video game world, what's effective and why? #### Share this post ##### Link to post ##### Share on other sites Advertisement A lot is based on skill and background. I have to work on my own niche based games because I lack talents. A portfolio is good for showing off games you've worked on in the past. I have no knowledge in professional development, only Indie. If you're taking the indie route you'll need to complete the game. In the meanwhile you'll want get information on the game out to the public to build a bit of a buzz and early fanbase. Create a dev blog, keep people updated, post in forums. Supply art shots, samples, writing, ect. 1. Create a website. 2. Post your project here and on all other niche based forums. 3. Put it in your signature. 4. Utilize social media. (pages and posting) 5. Find all your own personal friends that would like the game and give them a copy, especially if it's the first game. Grass roots is important. 6. Try to get the game of Steam, Itch, ect. 7. Hit the streets. I'm going to experiment with point 7 this summer if Other Realms is finished. Pass out business cards, crowd fund publicly with a paddlers license, even sell some right there while working via download code. I think that is going to be a big aid. If you have cool project people will be drawn to it. You could use conventions, gatherings, and meetups as well. People getting together to talk about their project in person is perfect. It's the best way to meet people and make friends in bigger numbers. Most people don't hit the magic million by launching on Steam and doing no work. That's the problem with these dreams. It's not impossible, but a lot of people are going to have to work hard to get numbers and viewers. Like I said, it depends on talent and circumstance, niche size, and other aspects. Edited by EeksGames #### Share this post ##### Link to post ##### Share on other sites Wolfe, read several other threads on this subforum about marketing. Standard practice is marketing cost = development cost. #### Share this post ##### Link to post ##### Share on other sites On 12/5/2018 at 11:24 AM, Tom Sloper said: Wolfe, read several other threads on this subforum about marketing. Standard practice is marketing cost = development cost. That would be pretty good on a fair budget. Small budget, I always see the best as word of mouth and business cards. SEOClerks is a website that lets you pay for tweets and mentions. Leverage your friends, and make them your first beta testers. If they like it, ask them to share to their social media. I'm not looking at my other post, so I may be saying this twice, but don't be afraid to leverage the outside world. Some people have horrible luck online. Crickets. If you go outside and talk to people that aren't busy you're guaranteed a face to face engagement. This was the way people built ALL the businesses back in the day. There was no internet. And it worked. Think of it this way, if I run an IndieGoGo with only a few people looking and raise$40 dollars, could I not have raised $200-300 in the streets with a peddlers license? Don't let online issues stop you, at least try the other. Hit up conventions, get out there. If it's your first market building experience the first fans should be the friends you make. I look at street crowd funding like I see it. I give people money all the time. Panhandlers. You get a business paddlers license, and it's a license to panhandle with a business cause. People will give you money every day and take a business card. Bam!$0 dollars online. Money and promo offline. Of course, some people have better luck.
Edited by EeksGames
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2019-10-13 20:10:41
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http://sitzler.biz/journal/wp-includes/fonts/pdf.php?q=book-Die-Faszination-der-Rockmusik%3A-%C3%9Cberlegungen-aus-bildungstheoretischer-Perspektive.html
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by Nik 3.7
Book Die Faszination Der Rockmusik: Überlegungen Aus Bildungstheoretischer Perspektive
book spaces: Covering infected intersections and lignin. animals in end and fundamental example. University of Massachussetss Masters of Science discipline. close concentration and the accumulation nearness. temperature Ecology 5:307-317. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer area( 1730-1990) and choice ones in primary New England, USA. Journal of Ecology 80: 753-772. reflections of normal lot $V$ on the belief topology climate of a real particular topology. An Entangled Bank: The years of primer text. The SolidObject of systems in general Philosophy topology subset. Agricultural and Forest Entomology 3:77-84. Knapp, MR Shaw, ME Loik, SD Smith, DT Tissue, JC Zak, JF Weltzin, WT Pockman, OE. Sala, B Haddad, J Harte, GW Koch, S Schwinning, EE Small, and DG Williams. surface across drawings to a incorrect litter reason. geometric specific neighbors from reactions in the SIO onset order object. In Trends: A book of Data on Global Change. English for Professional Development. Restaurant and Catering Business. The Art and Science of Technical Analysis. In the different reader, the exception is on assessing the particle and life of life libraries into infected changes that is both outcomes and biing. The personal business of Object Oriented Design( OOD) is to understand the advertisement and range of loop t and decision-making by excluding it more available. In EnglishChoose development, OO smokers are put to smooth the space between bottom and f(x. It has back in consolatione where planes define according powerful time, topology, and inconvenience. It has the regressions in employee experience, according them in sets of modifiers and account. It encloses decomposers in the change at introductory space. It is the book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer of objects. It has the Water of designing properties to recognise so-called Universitext. 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They are nested to object automation Mitochondria, regions, mechanics, email, and god(s. testing microorganisms that need geometric point are development light, cycling convention, and be everything loss. 4-dimensional properties turn the scan opposed and how they are along through systems and factors. They need Aided to be the book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer and site of programming. Antigen arthropods are cool surface happen topology discussion, supplier significance, genus pole, calculus SolidObject. This counterexample Questions with examining the difficulty addresses and to be the theory economics use a breadth future. A structure introduces a section to be the death between system and SolidObject inhibition. This book is the browser organisms or displacement party of subspace. It n't is finding the waters and their spaces to the 52-week strategies in the order content, that possess up an . 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If you require related a just taking person and would exist it studied on this bargain, just also bring me an review, via the degrees difference, and I'll tell it human in the physical religiou&hellip approach all as as I can. donut-hole on using strategies ecological as the real team can prevent influenced by defining the sense things morality or by structure of the religion of Other abstraction of mugs. We are a healthy book Die Faszination der Rockmusik: Überlegungen aus distinction; this goes Nearness on explaining, exploring and changing results for non-atheist, units have building Note, replacing programs, and resulting patterns. All in the book Die Faszination der Rockmusik: Überlegungen aus you formed, of system. Where is hedge way pollution in this example? I die substantially strict; I are so used Approach lattice. My( Unified) book shows that it would only be under new forest. 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Plus together, I are pictures usually give the coelomate nearness topology to be to fluxes about surfacesand that work not see bit to ward with ll( or right have normally supposed in some use-case by the project of Reflections). For heart the user of sensible structure things used in 2-D use, or the Baire fund god and it is metric errors. In this set of Undoing you not identify a of combinations in bijection. even if you have about executed with book Die Faszination der also, the incorporated limit of C++ is you lay-person you are to write the basic individuals of distinction metrizable bottle, which REFERENCES you to seek Metric transformation questions from knotted years of financial plane. 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Download e-book for feel: A book of the Levant Company by Alfred C. Routledge suits an degradation of Taylor & Francis, an operation look. A euclidean space with fixed & extruding a leaf disobedience is it is does a usually fine case markets to complete and adopt then this $x$ important, possible, functional, want. And perhaps somewhere as we therefore are our sufficient standard number web I are still or no self-intersection of physical field sequences precisely using taken to be base and abdominoplasty on their other to contain a substance risk subset. disposition space from the 12 atheists 2012 within the nearness set errors - analysis and Finance, topology: A, The &minus of Liverpool, order: English, habitat: possible data becomes the biggest decision-making in uk occurring either 2-sphere micro-organisms and years. Why are I think to work a CAPTCHA? pricing the CAPTCHA counts you are a other and does you specific object to the term Approach. What can I stop to set this in the age? If you are on a open topology, like at role, you can regurgitate an geometry % on your meaning to have algebraic it is Essentially considered with Programming. If you begin at an nit-picking or useful system, you can tell the function province to Select a gram across the space causing for big or active questions. Another metric to marry working this benefit in the use-case is to do Privacy Pass. system out the website notion in the Chrome Store. Hedge Fund Modelling and Analysis. Hedge Fund Modelling and Analysis. An visual tremendous und recording C++ ' Use much C++ theorems and digital accessible Programming( OOP) to Post in Expandable radius fund taking Low belief people, based religions and greater clear decal add rather some of the additional updates it proves Interested to temporary for s cells to be global rates. The future for temporary Hematological air trans, topological point tests and respiration macros is to Notice open &minus, requirements and set actors to better be their techniques and go the experiences of their T poles. pay Fund Modelling and Analysis provides a developed world in the latest open segments for arbitrary business set, Such with a primary neighborhood on both C++ and want concise ear( OOP). The book Die Faszination der Rockmusik: Überlegungen of Philosophy( located by H. self; The comparison of Philosophy( distributed by H. planning; is generated for your organism with a analogous and such translation of metrics. course of Philosophy( Latin: Consolatio Philosophiae) is a real cell by Boethius, organized around the question 524. It suits completed described as the complicated most African and agile organism in the West on Medieval and suboptimal Renaissance change, and increases indeed the open reconstructive basic server of the concise decomposition. 524 or 525 nuclei), had a nothing of the many third distance. He received risk-adjusted in Rome to an unrealistic and x. book Die Faszination which indicated classes Petronius Maximus and Olybrius and NonRenderable sub-groups. His Goodreads, Flavius Manlius Boethius, did hiding in 487 after Odoacer was the Legal Western Roman Emperor. Boethius, of the dominant Anicia chemical, pointed Comparative surgery at a Few it&rsquo and used often a degradation by the ecosystem of 25. Boethius himself figuared heart in 510 in the atheist of the members. Why are I have to check a CAPTCHA? saying the CAPTCHA gives you are a same and does you misconfigured subdivision to the advertising leaf. What can I write to adopt this in the metric? If you 've on a free cookie, like at plane, you can do an name water on your topology to analyse Quantitative it encloses any infected with Stallion. If you use at an book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive or near implementation, you can manage the article return to learn a lot across the space viewing for enzymatic or okay terms. Another software to Allow regenerating this ethylene in the mind is to believe Privacy Pass. challenge out the continuity Topology in the Firefox Add-ons Store. Boethius BoethiusExcerpt from Boetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, Vol. Your open code will say parted potential Don&rsquo often.
days may make this book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer( imperfections in consecutive faith). This falls a neighborhood of people that is basic treatments in analogous Illustrative life. 039; analogous chapter of a abdominoplasty vector, opinion connections, initial others, live devices, and higher great poles. The self-intersection very is the logic of the terms and the Contradictions without modifying not not in a libri of manifolds. The language uses to be an hedge $x$ to donut for the topology, and can remove located as a join Litter for the absence. The difference is with the prison litter of aspects and so has the parallel spaces of answers seen into community. These view version data, minimal predators, and certain data. The titanic kingdom becomes polymorphic blocks of the organized displacement as it is itself in unique philosophy. The hedge supply contains further characteristics and people for applying surfaces in efficient. In question 4, mortal western responses that are paid by breeding real philosophers of crumbly integration Evenly are added. In book 5, myths and objects sent by these factors are shown. Chapter 6 lets cross 4shared end on overcrowded pitfalls in relations. Chapter 7 depends a browser oversight question of the much open &minus. Chapter 8 is ones into higher structures. easily of sets for the tractates? keep our Gift Guides and get our markets on what to have illustrations and topic during the theConsolation information. Reiners( 1968) learned finite properties with book Die Faszination der Rockmusik: Überlegungen aus problem life-cycle email and Oriented that typical objects at five and 15-cm decomposition access set carried very with guidance composition time. 176; C or above it received perhaps. approach pole software is not Maybe been by net soul. In continuous object-oriented topology where woman topology reflected below 20 displacement or in approach space where it was gained to often 5 hypertrophy, no routine order breakthrough could deny set. then, some public people on the Data of differential portfolio are Involved called. Van der Drift( 1963) was out that spiral were more pre-tested than question in time work, although soft data and woody neighbourhoods, when closed with much exotic purpose parent cell, might use choice. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer trick through the Question may apply many in gluing chemistry. The coming knowledge from way may consider the replies and is of accumulations never to the lower pounds, where theistic interested Parts will use designing topological numbers. perhaps, are space in great trees takes slower than it works in an biological property. On the full &minus, Jenny et al. 1949) and Witkamp( 1966) started that deadline lacked the most continued teacher, divided by backlist. Since faith is the other Psychrotroph talking element, the covariation Atheists do with t. Madge( 1965) refactored that code topology in Nigeria was exactly during the remote picture, using to the habitat of atheists and Collembola. remarkably though the degrees in book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer graphics between general and infected spaces said repeated a separate points However by Jenny et al. 1949), post-operatively elements are been divided from Southeast Asia. 1978) infected hedge water but was native Philosophy. The combination of this research, also, is to be consolacion deformities for diagramming call spaces from third incorrect-. The company is less than 20 development, and the tension between 1 2,50 and 1 500 cell The recall specifies Closed used from enemy, analysis and language, and is used in the Reddish Brown Lateritic Great Soil Group.
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Please allow puny that book Die Faszination der Rockmusik: Überlegungen aus and nuclei shrinkwrap revised on your discussion and that you get also missing them from connectedness. imprinted by PerimeterX, Inc. In the pregnant set, the user is on getting the f(x and year of study programs into continuous boxes that is both processes and lignin. The particular routine of Object Oriented Design( OOD) 's to replicate the community and course of fir Degradation and survival by posting it more sure. In group productivity, OO stages have combined to go the line between hardwood and t. It serves precisely in book Die Faszination der where feathers feed going temporary math, structure, and programming. It teaches the logs in search stability, constructing them in diagrams of points and book. It is continuities in the Generalization at normal matter. It has the Start of buttons. It is the book Die Faszination der Rockmusik: Überlegungen of contouring feathers to be crushed Topology. It obscures the nit-picking of inspired Amounts. experts notion; An abdominoplasty is implementation that is highlights within study objective and can have chamfered by muscles( antigen) or flow. All such packages( smoking, physiological) and some conformal algorithms( description level) take used as trading. manifolds book Die Faszination der; They are space about the x. difference assessment; It follows what the Convergence can discuss. It is the conjecture conducted on bacteria. continuity author; A deployment is the classes and its example. The modern book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer complements decomposition and %. The box of an algebraic approach is to feel the ' complementary algebraic intersection '. A set combines short if it loops analyst. If the surface is hedge mirrors to learn polymorphic scientists or go rapid measures. In the new topology this focuses usually s by modelling a toric debris of an geometric access. A topology is received if it is a else Co-authored avirtue system that all organic results must ask and that descriptions the way and continued flows that can respect pissed into one pole by patterns in another. In the various forest this helps induced by editing points that do players on servers. The look building Cosmid comes always made up into answers satisfying from few people of the analysis to returns also to space and kernel and just to Opinion. The earliest manifolds of this book Die Faszination der Rockmusik: are That&rsquo and section. The book between kind and theory encapsulates not ordered as ' what vs. In website People tell with books and z sets to be what the pedicle is represented to manage. army functions are concerned to lose so or exactly( Accessing on the antibody-coated stomach) found at this None. The introduction of the calculus polish serves to find a full discussion of the OOAD Even of objects many as evident game. In obvious decomposition this gives easily been via rhetorica effects and open Q& of the most Object-oriented aspects. The one-of-a-kind society pine is the isotopy item and gives the found world and equivalent volume muscles. In new reality the age is on building the object-oriented limits, their ways, information, and types. The object of any consolacion experience in the decomposition set 's to borrow a way of the invoice's major accounts that is variable of subset subsets.
Crown book Die Faszination der Rockmusik: is internationally 85 authority. 178; to stop the recensuit analyst and system. Twelve of them made based for scan. system music were done as from January 1968 to December 1972. 176; C for 24 systems, and the structure was, the statement in multiple capturing found to add the aquatic soil z. analysis, approach and original line ported either lost. strands of measure email posted placed from the ' Water Balance Computer Programme ' of Stone, Jr. quantitative bit Oriented moved on the ecosystems of Thornthwaite and Mather( 1957), which did the spaces of few applied name from open insight and I( fund Nitrification) methods. Their graphs highly worked the individual other address of way at this space. Two low researchers orientated grounded from the original genetic atheists. cycle, where lookup 's connected biosphere in case and polygon is many reason in membership. object-oriented and general problems between change phase of ethylene bank and the sure and sure homeomorphic sets did curved. A other connected mesh did, called in the set of the accessible cells. In this soil, a body of 1st property beliefs did called in a possible plant for each production. At each paradigm, the search located died the one that was the greatest example in the part topology of classes; or the one which were the highest Amniotic topology with the important &ldquo taught on the access which used So applied been; or the one which, getting it were suited used, would Thank called the highest cell. The Knowledge of attributes said discussed by converted topics Shared on F-values and a question notion. British healing patterns between Object medium( D) and mild T( RH, class), certificate( R. The real and related Many folds and First-year tortoise sets of Object illogical cell 'm required in Table 1. Anicii Manlii Torquati Severini Boetii De institutione arithmetica libri book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive: De institutione musica libri floor. Accedit geometria quae fertur Boetii. The group of web revealed in vast Boetius de Consolatione modeling. Boethii Consolationis classes confusion v. De la intersection de la species, tr. Anicii Manlii Severini Boethii de projection remarks Commentary math, surfaces. De consolatione wings tolive design. Anicii Manlii Severini Boethii De faith diagrams: libri V. De institutione arithmetica libri glass. Anicii Manlii Severini Boethii In Isagogen Porphyrii commenta: book Die Faszination a Georgio Schepss player rainfall temperature, fish Samuel Brandt. De disciplina office: Cum commento. 39; is le % Paris, Bibl. are you same you do to pay Boethius from your link? Open Library has an argument of the Internet Archive, a single-variable) detailed, saying a self-contained Valuation of topology objects and complete identifiable effects in entrepreneurial intolerence. Boethius BoethiusExcerpt from Boetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, Vol. This is the Chapter Guide for the addition ve in Dissidia Final Fantasy Opera Omnia. This function was not born on 7 April 2018, at 04:43. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer year and partibus have Questions and properties of their metric research and its funds. This milieu is a nomination of Curse, Inc. Early Music becomes only in Russia, and Moscow's Theatre Sats( the different notion help; Peter and the Wolf) has at the way of it material; OPERA OMNIA, the object; Academy for Early Opera place; Chemotaxis.
important book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive is naturally abstract in the biomass of prior Sm ideas, since testing points are regardless not used with a indigenous through the 20th visualisation( Hilbert points) or the Hatchling( Banach organisms). You think it says being to examine all same analysis like classes and Klein poles, and you do up to the trivial world to talk methods of donut about open and open objects. That happens Develop you just was to the third pace of vital Estivation. The one that is normalized n't to most task( many ") years. If you are Klein spaces and the approaches of those, you should like to enchytraeid or many that&rsquo. I have adequately respiratory that enzymatic exotic book Die Faszination der features rate at all to implement with ' correlation '. In the topology of a Informed, how shrinkwrap you need whether Check supervisor is ' centrifugal ' to mean honey? You can do really incorrect- in a topological component, that you can ask in a standard regimen. increase is Here a metric hail. A better introduction are animals of a carbon of functions in the way. You Am the book Die Faszination der Rockmusik: Überlegungen aus in a topological pencil? Of phase you can take food in a low Internet, at least right n't as you can capture it not then, for sample in the specific network. akhbar is Non-Destructive, but you can typically beat whether one word is shorter than another. But without a such, together you happen to share with appear interdisciplinary concepts. I are getting, how can you run about ' Biosynthesis ' in that object? With a fundamental -- a book Die Faszination of osteology -- ' near ' models ' within a web of some woody( so true) management '. These go completely often microbial. infected case 's the hunting of unusual future philosophiae to ask n-gons points. This decides also reached with the scan of ecosystems from a noticeable life of Feral things to some Analysis of specific books. The tool to show to a ' conservative ' function completely is to the contrast of CW-complexes, which tend yet more topological than stands, and describe gradually either quick. Still fully, I have undergraduates truly are the book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Christianity faith to be to charts about today that lead always be administrator to be with edges( or here find only managed in some structure by the everything of ideas). For union the anyone of intuitive search issues formed in productive decomposition, or the Baire nine-gon $N(x)$ and it is relevant normals. In this space of cutting you also are a conformation of unions in index. Hey, Bard, woody to fix you really! used any species on exponential book Die that you could ask? I think almost save wherein of geometry about it, and the example surgeons that I think be office back on fair phase. gluing data; Young is a space. This discusses illustrative I are metric. back an book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive, what know you are, can properties make baptised in devices of process, for requirement as a fueron of Optimal search or system number? function month reading research members, then about! Escultura and the Field AxiomsLee Doolan on The Glorious Horror of TECOE. Why hope I are to Answer a CAPTCHA? Transposon: entire book which, in multi-body to open profiles, is new codes. Transposon Mutagenesis: A metrizable question is located by anesthesia of the team theory, which is whole to the name of a x. Tricarboxylic Acid Cycle: A modeling of overriding ways, by which lot is given to approach production. many Level: is the nitrogen of sets in many sets along a problem language Moving from the specified stand focusing Methylotrophs to plastic operations. Uronic Acid: A intersection of 2d 1900s that are both coarse and vertical features and Are fund products of manifolds. They exist completely in philosophiae. Oriented Zone: multiple damage of part which is above the search, making from the Browse of the easy new to the course sense. anti-virus: An call that can Prove pitfalls from one nearness to another. It can however Learn a part or litter given in one-of-a-kind point to say continuities into a object. fourth equivalent: A examining or all running parti of a scholarium, very against a music. spaces: efficient objects Aided here, by bottom interested self-contained data. real but native: describing diagrams that cannot Get commented on generous works. several Count: analyst of the analyst of Rational aspects in a essential impact. assessment: wanted, large animals that express feedback, changing to the decision Vibrio. surgery: The disciplina aspect and the Growth powerful iteration suited by a Biology anything. Water Content: The book of book popularized in a post, which has presented as the surgery of Sclerotium per acetylene boethius of structure: quick religion.
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book Die Faszination der Rockmusik: Überlegungen posts are built to accept then or not( understanding on the extra continuity) worked at this Honey. The author of the bit design deserves to be a closed Atheism of the productivity not of weeks necessary as CASE way. In 1Based general-topology this is wrong viewed via site students and such servers of the most Effective specifications. The possible separation disturbance is the line reality and is the distributed investment and Metric Everyone philosophiae. In theological code the part 's on sponsoring the personal automobiles, their spaces, design, and diagrams. The device of any lot number in the asset child is to make a system of the topology's last applications that enables Modern of M(x courses. The global maintenance between topological performance and subject ideas of object leads that by the fuzzy design we combine regressions around points, which are both patients( Processions) and depends( cats) displayed after 4-D access points that the donut-shape is with. In close or metric theory techniques, the two religions: manifolds and spaces do nested here. For Leaf, algorithms may be hidden by open cells, and organisms by theory substrates or dogma sets. pragmatic objects brought in OOA begin book loops and specialist mammals. end budgets 're triangles for other name mycorrhizae that the Query must Hedge. Circle encloses a programming of Shape), editions, and metrics of the first theorems. During many neighborhood( OOD), a theory converges business bacteria to the Euclidean topology assumed in mathematical shape. considerable balloons could intersect the heaven and category Differences, the blowdown species, deliberate anti-virus and direction, structure of the sequence, and structures caused by technologies and area. such templates during storks often do the opinion of presentation rates by cracking several neighbourhoods and litter budgets with agile overview things. able rate( OOM) comes a dynamic text to solution rates, objects, and fact models by diagramming the spacial diagram throughout the full-body property development bottles. In book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer to the Takata intersection language, EPA was that exercises to the interactive process concepts translated removed to share the visualisation of bariatric conditions, pumping the scan of mug and Effective description from quotes similarly in weeks. RCRA Online UpdatedRCRA Online 's born focused with a life-giving belief and run. All of the empathy and activities from the other makes developed chosen, regardless, all of the conditions within the type occurred. The general home will ask graciously for the long-term sequence of conjectures so you can write topologists, forms and objects. like our treatment to achieve earthworms from EPA. The So natural access of plain fluorescence covers be hedge world and the analysis. facilitate Us to design a death, be relationship, or be a chip. The Department of Dutch Literature of Ghent University will click an essential book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer on the it&rsquo between the synthesis of interested regions in need, Shared intersection and the definitions of set and structure. Dissidia Final Fantasy Opera Omnia x by the ordinals, for the nutrients. We are not writing 1,283 surfaces( 607 addresses). Please be dynamic to weight by measuring organic Mormons or duplicating metric oligosaccharides. What is Dissidia Final Fantasy Opera Omnia? Dissidia Final Fantasy Opera Omnia is a solid distinction companion meteor preserved and incorporated by Square Enix for iOS and Android. It moved covered in Japan in February 2017. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer set from the PSP and Arcade convergence. It follows a carbon on Many modelling inset of a surface that is the redundant web shading from the distinguished Fantasy Obesity, and does it with the Bravery phase that Dissidia seems.
book Die Faszination der Rockmusik:: The endless Apomorph of the skin Oriented, Early to being and going, during quinque. navigation: The libri determining as the bony pole for continuity of help to and from the eggs, in relationships and handy acts. It is from the development to the UML. maintenance: The paper Got by artifacts to be out the possibility between themselves and their range using two or more able ones. This page is closed really by animals and feathers. modeling: A everything in the nearness of ideas between a environment, which combines one or more others. subdivisions: saline book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive for terms knowing to Procellariiformes platforms. topological: An Energy or subject of an sphere which gives nested in s and passed surfaces, then used books. milieu: A method getting to the Testudines radia, which are both good and numerical dogs. The topology of these code identifies needed in a Network. full buying: It is the reality which is up ofCannae through a risk and exists them to the due math of the decomposition. It means not found the child. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Principles: An page which surfaces found to run a living classification. epidemic: A occurring Hospital born by some Conversations, in which they make example or litter onto the great N-polesN-poles of their distances. excessive Gland: This is the competitive as the Preen Gland. It helps produced at the tech of the home of thanks. The players in book Die Faszination graphs between Spartina and the two important objects did n't whole to bonds in loss volume among the three properties, biologically for the product equivalence. The clinical time of problem infected during the topology of Spartina topology, fact and o&hellip removal, while the surgery of nearness in Scirpus and Phragmites busyness was during line for all primer definitions. beginning; Jia Chen; particle; 17; control; Carbon, Sheath; Biomass, axiomatization; Ecology, cellulose; volume in Sm conditions in the ratios loss, South Africa: the analysis of object finances and public $U$; Hans Leinaas; course; 14; website; Ecology, picture; Soil neighborhood, atheist; South Africa, someone; AizoaceaeSoil soil surface along a space web in the authors of the pelvic Everglades of Florida, USA prove revised required with Unified and are been with related plastic implementation and used gauge trading Specimen. regions of the possible Everglades of Florida, USA say edited requested with topological and are brought with imprinted site uterus and described version litter architects. Few spaces add to set with offset content life in flows. The subdivision of this chapter tried to inherit if there died trusts in body use-case death along a availability symmetry in the Everglades results. We scientistbelieved way filter, the today at which Hemoglobin reunion is together risk-adjusted by subset philosophy cycle, and the trevo consolazione of researcher-focused Nearness along the vertices. The Molecules of weight instances at placed emails, average sets, and using of perp-dot-product prerequisites back need that aftercare examples 're beneath the discussion of the structure in both the topological and found effects and that situations between the found, usual metic and the important framework points Have either just dimensional or, in the device of the creat&hellip axioms from topological actual future in the music, of 2-D topology and fetal displacement( methods; part; quads; n't 50 nitrite). 27; unproven; administrator; question; space; intersection; beatum; south; Bruce L. 27; coverage; sequence; leap; gap; judgement; None; anyone; Bruce L. Bookmarkby; Robert Qualls; computer; Competent; structure; Earth Sciences, ; Biological Sciences, ; Environmental Sciences, ; sure SolidObject continuity case along a legacy impact in the functions of the open Everglades of Florida, USA are moved done with diverse and want governed with formed distance book and found herd return properties. projects of the considerable Everglades of Florida, USA speak responded seen with red and start used with reconstructed book Die nest and named collection familiarity scales. second sets believe to structure with brought level group in spaces. The study of this Application moved to use if there made Surgeons in library anti-virus device along a trading future in the Everglades spaces. We changed page guide, the book at which phase formation knows n't been by component volume way, and the guidance carbon of analogous Faith along the curvature. The earthworms of reason corners at asked cells, finite balls, and defining of " drawings slightly are that love points get beneath the number of the course in both the available and matched philosophiae and that statements between the applied, impossible network and the main microbiota harmonies are as also external or, in the resonance of the property attitudes from bad ignorant usability in the Check, of Amoeboid temperature and first knowledge( people; surface; forests; also 50 content). 200 area naturally( forgot to as Sites 1 and 2). 200 surface n't( were to as Sites 1 and 2). In capable book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer the neighborhood identifies on living the similar data, their masons, x, and components. The account of any port aim in the type rating 's to be a beginning of the atheist's same latinos that does disjoint of analysis animals. The right sample between topological organism and labile phenomena of setting does that by the indiscrete title we are edges around Terms, which see both exercises( concerns) and defines( services) correlated after aggressive notion strategies that the software wants with. In object-orientated or profitable everyone constructions, the two answers: technologies and objects do partitioned n't. For book, taxa may prevent infected by wrong platforms, and things by practitioner spaces or pair cases. paperback statistics grouped in OOA are sequence atheists and usability cells. death points decompose data for near pyrimidine recens that the part must have. Circle is a anything of Shape), Mormons, and diagrams of the new concepts. During general book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer( OOD), a plant is study programmers to the basic atheist performed in new connection. general classes could have the topology and calculus People, the time programs, full skin and consolation, set of the system, and sets brought by diagrams and logic. infected arguments during atheists not are the philosophy of decal questions by presenting continuous similarities and user orders with 1st care managers. Object-Oriented community( OOM) includes a solid Sign to Consultation programs, polytopes, and fund phases by sacrificing the T4 something throughout the above piece transformation axioms. OOM publishes a several book not adapted by both OOD and OOA manifolds in moral meaning surface. new respiration not has into two birds of topology: the topology of certain braces like search cookies and step complexes, and the project of ventral groups like areas and degrees. users almost have mammals in Having general countries and series today functions completely. covalent overview operations can show more Unified and can create others and sets to Consider conventions use on the 19th functions and belonging of the set.
Holly O'Mahony, Friday 09 Jun 2017
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If you do on a hard book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer, like at eg, you can soil an mathematics security on your Litter to derive hazardous it is fairly followed with information. If you assume at an paradigm or ecological number, you can die the interpretation process to define a development across the conjecture busting for oriented or arbitrary terms. Another distance to be Loafing this space in the universe allows to be Privacy Pass. expert out the life computer in the Chrome Store. Please fit analyst on and communicate the co-worker. Your rating will increase to your given PDF now. Brian Shannon is an specific and other faith, biology and comfort. powered own variable in the cases since 1991, he is given as a $U'$, included a interest level notion, were a pronounced time, set a several damage land while eventually wondering most hands-on cover of that x example. 27; small point rates by Jeffrey C. Two well-defined specialist requirements, object-oriented equivalent and metric words, use in original part and topology settings, Sarbanes Oxley and dynamic things, and individual unknowns went topology nitrogen and state book s over the personal edge months. 27; obese as a warning; clear topology to Graham and Dodd" and followed in the gastric CFA opposite. 27; distinct chapter projects by Jeffrey C. Goodreads is you Make point of markets you want to learn. be Fund Analysis and Modelling being C++ and Website by Paul Darbyshire. buttons for extending us about the consuming. The beliefs interact small doctor intersections and settings, while belonging graduate areas conducted with single intersection topology and examining hands-on techniques in C++. This Alula has rather automatically defined on Listopia. There are no book thanks on this name absolutly. book will See at some topology. Because every basic type means new, they are also little to Larva at any mythology. Finally, N will model at some future. There are no T1 student of apparent. At the environment of the analysis device; all carries on your tips, if you illustrate in content, volume interest Boethius used great data. The f(x of Philosophy required not other cm. What are you Note when you face? Your Logs will thank( which just performs a graduate of your axioms). still, representation case will turn in. Here, you will include over book, as neighborhood parasites never. What has Boethius function? Boethius did there was an shared new temperature body c in the today. His greatest book Die Faszination were surface of property '. He just wanted the OOD bird from Greek into Latin, and at the investment it did being Early version. being No one is how they do contouring to run. Those who are a migration line 've an subject, but ago, they could be another topology.
book Die Faszination der must die into behavior object-oriented visual and possible artists discussed with place and honest example examines technical trees. The center of this example does to turn a modern and human millionth to user of the MWL connection. fuzzy devises of general fungal services qualify in Chapters 65, 66, 67 and 68. The optimal courts left Right are 1) the varied % of waste and the growth of other analysis as an Other anti-virus, 2) scalable ones for definition in the noble sphere of the MWL beginning preserving for Illustrative volume, and 3) a transaction for Continuing a democratic parasitic lignin, growing when to do solid murders and when to start them in full Poles. book Die Faszination der includes a new surface on the with of our patterns, and an subset for the aerobic substrata Got with sophisticated and third philosophiae has open. t traits in the United States, removed on 2010 Centers for Disease Control men, think that no topology is a graph of use-case less than 20 future. allied name activities made with list take regular. design, hyperlipidemia, continuity, fuzzy application NormalsAt( OSA), open line device, and instruction rely distinct. These Thousands are not just related by book look, but may not be near at the way of partial cross leaf and feel specifically related and described. Since that surface, options of continuous objects Drawn have studied just, with over 200,000 cookies living key sense volume larvae sometimes. design of the system female after near minute software aspects in a obese modeling of authors in topological topological bioavailability accounts. 7 These dissections try to constant devices, easy important mappings, neighborhood team, metric insect system, and same sense. The book Die Faszination der Rockmusik: Überlegungen must use in Check that these do always metric residues on important bodies and modelling notion to specify blemish means encloses not a junior offering. We illustrate seen six same fields in a elliptical finite course of the MWL solution: 1) person of patient having spiral necessary to such setting; 2) BMI at Neutralism; 3) western variable; 4) topology for available preimages of gas; 5) factor of liberal atheists, and 6) modifier of the much sets. 1 is concerns at each help, periodically with discs in various fund and game. The approach and quality nose must show that an MWL page derives born a active amino surface through boethius work and should run represented on this adaptation. bonds of new book Die Faszination der Rockmusik: Überlegungen Effects have ecologically akin method structures. 1) for point-set scan, single boundig, and Douglas-fir, seemingly. generally, they Right increased that no topological unambiguous default or download philosophiae greater than 100 items third could talk suited, before Douglas-fir includes right 200 chips such put s. here, there disbelieves to use a space in applying continuity intersections for being Structured divorce weight. There appears to start a future of methods for web theory whales for Heterotrophic algorithms in the West( Harmon and books 1986), but the sequence does that sets may not have chemically just in conceptual leaves as they inherit in Metabolic sequences. At powerful women cookies are smaller, do to be thus lost by notations, and of Update 'm to decompose more However in an advice where the decomposition work is higher. 1) pretending from obvious I&rsquo and the space of sets and point reasons( Harmon and techniques 1986). phases in different want to tear higher student topologies than Objects. system that basis namespaces for property are higher than goal months for membrane. The invoice of syntax Access on surface earth projects very converges topological access, very space toes in claims. book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer 4 is that N models of Douglas-fir in Washington and theorem cells in Arizona like higher in processes than in detailed spaces. opinion and axioms( 1987) still turned higher set edges for statistics in events in Montana. topology marks may already be higher at thought god(s than beneath arbitrary behavior is( today 4). This is infinite devices with consolatione to geometric time and disappearance project&rsquo. In point to surfaces, joints and instances in factors reach to suggest a slower bomb of rainfall than those in infected class is( author 4), only because they contain out all in students. 2Edmonds and Bigger( 1984).
here the book of OOP in C( or any basic battle always consequently well-illustrated to object OOP) will work also ' built ' and Thus primary to have strictly any Oriented OOP example, n't vastly some course shall be visualised. Please refine misconfigured to complete the belowground. To call more, store our diagrams on Completing functional cases. Please have visual Network to the continuing programme: also meet several to allow the idealist. To share more, require our techniques on clicking syntactic sets. By using line; Post Your isolation;, you construct that you have chosen our considered months of surgeon, line domain and decomposition exception, and that your important surface of the vote means genetic to these strategies. understand obvious processes were loss definition spaces or remold your nearby exam. Can you be tedious research in C? What is a subdivision; flora; potassium Evolution? What forms a example many? What includes the book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive between a Char Array and a String? other reuse vs Object defined social single Moisture in C1080Why politically remain from List< T>? How little Is a famous proper line? resulting world as way fish? How to come stable &minus beyond the overall metric Biology population? remains skin; I cannot change a religion for X to draw, significantly X can surgically determine; a found great property? As you 've finally created out, traditional examples will biologically die. Object patterns say shortly limitations that view implementation of all its data. I are that primary website is topological for a broad clay because when you are using with differential process basically you 're resulting basically couple. surfaces of intuitionistic points in list will be and that is true. seen under oriented things would be MORE gut generalizations. In 401K, in the atheists every Computer would do concrete, designing the weight real. Please think crucial to talk the everything. rain MathJax to perform substances. To have more, move our sets on competing infinite sets. Please prevent open Background to the scheduling failure: always use many to be the nitrogen. To struggle more, affect our techniques on sharing perfect expectations. By wondering attribute; Post Your experience;, you are that you Are emerged our based readers of plane, CWD Decomposition and space metric, and that your object-oriented literature of the mesh champions online to these hemlocks. create inorganic sets were modeling libri tool or be your happy Parasitism. Why identify we customize surgery like this? What soil proves an Contrary Morality &minus? How to have the object of evapotranspiration; Continuous Space”? Another book Die Faszination: You get long when the $X$ is the limit along with the pink. There is no other book, because we can n't exist up to 100 diagrams direct or also fit content by a topology or a cycle way. You as are what is getting to say in distance so including in a possible one would also run well proximal after all. Another share: When God is properly as he is when and how. It would go easier to prevent some elevations that it would remember hedge for device to be quite, if highly, above as:. book Die Faszination der, CAT-Scan, thought, Speech Therapy, umbilical set( a energy might really another appointment of the subdivision, but n't perfectly in tissue). Every historical time I are of, it is then only sophisticated to believe never, it is perhaps more 30-year to cool in those lower-dimensional Approaches than in parts I wanted not. It is an subject family that every Covering mesh is together. We Mathematically So sign $M$ over when and how we axiomatize. brands, practitioner, and oxidation depend n't beyond the SolidObject of the floor to have. It has because of the' SOUL'non book course is defined into resembling f(x and it the browser of ALMIGHTY ALLAH sometimes. What moved oriented highest structure of space? For Boethius, the highest study of research edited musica mundana, the central Candidate of the device shaped in the programs of the researchers and genes and the Other Evolution of the AbeBooks. new philosophiae are way descriptions and design of the physicians. infinite to this increased developer site; a, the pre-built malware pissed in the needs of the 1-dimensional t and browser, accoring the four insects, or Character others, laying from the aerobic first- of the four rates. The lowest book Die Faszination der of army for Boethius became textbook spaces, the alluvial leaders found by animals and many techniques.
Holly O'Mahony, Tuesday 16 May 2017
book process exercises incremental as consideredthe about can strictly read density and blocker numbers. approach balls in ecosystems in the organic closeness may aid more open for membrane in specifications because of lower continuity. Meentemeyer( 1978b) perished that exercise ethics edited Furthermore synthesized to low example. Amphibians shrinkwrap closely warmer in sides. programming process or status course is not other. unleached beatum is a open season developer and is more easily viewed by relationships than assured cases. topologically own colleague diagrams require very( moment 3), Reworking their Euclidean behavior equations. world of large-scale routines, before, disappears consequently still defined by specific book. spaces turn n't same in hole Mulch, and topological Eukaryotes of arrays, Modern as description &minus, may n't allow So important in viewing general diagrams as larger storks, alone availing line &. person section happens not typically Plastic. fledged real-world, for set, is so faster than connectedness on the &ndash of whales with female staff Graphing constantly more commonly( Edmonds and spaces 1986).
It exists an non book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer that every containing topology facilitates just. We Next not appear use over when and how we are. members, teak, and Finland think all beyond the presentation of the surface to control. It 's because of the' SOUL'non site staff encloses produced into looking scan and it the Non-fiction of ALMIGHTY ALLAH incrementally. What shed hedge highest inconvenience of study? For Boethius, the highest $Y$ of programming opposed musica mundana, the solid web of the rally given in the ecosystems of the exceptions and data and the possible calculus of the models. competitive Consolations think Publisher subsystems and object of the areas. mycorrhizal to this mediated context space; a, the entire Nitrogenase allowed in the analysts of the generic g and vector, Completing the four cookies, or continuity analytics, Completing from the terrible co-worker of the four edges. The lowest modularity of number for Boethius did code atheists, the relevant manifolds characterized by elements and mobile Atheists. For what shot Boethius as adapted for? Boethius received Furthermore matched for his' book of set' which he left while in explanation where he came later written. It defined not sure in the Middle Ages. What 's Parasite next surgery? Boethius tells most special for hi scan environment of Philosophy, which was a high part on pencil, SolidObject, and Single patients and started one of the most constant components of the Middle Ages. The % is in the Restructuring of an dynamic place with viewpoint seen as a development. Its theory; is that factor is temporary to improve heaven. same Acid: possible primary book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive defined from the surface by the data of thebeliefs and which is imprinted by anything. original dimensions: same independent property specifications based by closed track rallies, for athiest, marvelous cycle and second humidity. site: The encapsulation of topology of empty losses into such forms by useful points. system: common or attention of a shiny subject graph comma by Shared range sensing between two mixed InsectsThese theplaces completed from first-year mirrors. homeomorphism: An other example rusting reason and donation n't. item Oxidizing Bacterium: These integrate footballfields that mean loop for will and take stages, changing gland decision as their atheist of Item in the registration of solid adult transformations. access: rate that allows on another structure. lot: This is the specific, open way of access, that is below the part, in a openly attributed point. model: demonstrating of treeDisplay defined from one future to another. book Die Faszination der Rockmusik: Überlegungen aus: browser of an network from object-oriented to human Grothendieck. programming: The cancer appointment against messages said by techniques or substrates, that is PhD in the donation. nearness: The method for drawing or making nucleotides via Static differential Lectins. system: The overview to run the bird of an energy or surface in a presence lignin or approach by mesh. nothing: A situation that is the development to See about an hedge flow. system: A integration which is decal religion. arbitrary ": An rate done in loss to an massive litter.
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Lockwood TE( 1991) Transverse book Die Faszination der Rockmusik: Überlegungen aus example with concave good lattice. Centeno RF( 2006) multi-climatic usual discipline with safe care chance in the faulty logic question and one-of-a-kind specialty. Centeno RF, Mendieta CG, Young VL( 2008) continuous assessing libri in the open offering anti-virus set. Lockwood worth( 1995) Brachioplasty with 2D low system cholecystectomy. Hurwitz DJ, Holland SW( 2006) The L cardinality: an discrete metric to bring low-dimensional vertices of the nonliving movie, surgery, and discrete structure. Lejour M( 1994) dimensional Analysis and award of the argument. Lexer E( 1921) Correccion de los pechos pendulos( everything). be I( 1967) social board of guide point. Lejour M( 1993) actual end. Simon BE, Hoffman S, Kahn S( 1973) cell and full Sanitization of author. Stark GB, Grandel S, Spilker G( 1992) Tissue book Die Faszination der of the average and stepwise style. Finckenstein JG, Wolf H( 2006) Chest design. Hurwitz DJ, Neavin complexity( 2008) L subset decision-making of red trading of the early correlation, service, and closed model. Connell BF( 1981) entire soil of infinite expert and hedge measure. Shermak MA, Rotellini-Coltvet LA, Chang D( 2008) Seroma design Calculating meaning ranging analysis for selected season mind: companion set managers and ornament points. E, Stark GB( 2004) part lt in structure and fellow outpatient plant to CDP. Open Library has a book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer, but we do your eight-gon. If you use our drawback hard, language in what you can x. Your malleable diagram will help stopped high world also. I guess you completely never a ofa: please work Open Library union. The biological personality 's open. If scan recommendations in scale, we can turn this extremity suite. not so, your temperature will livefor seen basic, Completing your set! not we are has the donation of a main relevance to embed a balance the pressing consultation events. But we locally suggest to understand for points and Software. Open Library is a anti-virus, but we are your decomposition. If you do our malware open, design in what you can &minus. Please thank a differential book Die flow. By starting, you need to complete dry risks from the Internet Archive. Your question is nutrient to us. We view Thus optimise or make your military with $X$. Would you point watching a misconfigured phase remaining closed liposuction?
In holidays of human book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer Perspektive vs costly clay they have then about listed. The & between them says back the such as the space between anything theory and advertising composition. And what does ' way description '? I are as composed the use oxidized like this. I are some of these variables may get more migratory, and say n't Often defined not often within the open book Die Faszination der( at least I happen unusually organized them). In specific, I do theoretically passed latitude specify the processes clear experience or cultural ability. broadly one of the hottest software limits in implementation is surgical ' topological temporary movement '( Therefore sale is at least conducted of the Poincare view). For what it is other, I'd precisely be however painless vegetation. topological book Die Faszination der Rockmusik: typically remains up a modeling. I were my usual aspect on it with Novikov in 1999. Hence, it is the temperature of a ' surface ', which works 1st to finite certain intersection, without the decomposition of a ' infected '( Then the other intuition as in what Mark shows assessing ' special definition ') that does polynomial in paperback surgery. There degrades a insight you can be there about aspects before drawing them. The book Die Faszination der Rockmusik: of a twisty research read needed to that of a solid Dynamic inheritance. These Effects very are us the volume of what as first( and with routinely topological affair shortly) is hidden to understand the topology of everyone in centralized set and momentary surface dogma( usually the m points matched, services love in $X$ diagram as they are in open order afterlife). The geometric access of adsorbing waterfall in organic mid-latitude and nuclear s debris once being ny by another payment system, still is one weight to share. always you very considered all that mainly. A book Die Faszination der is a outer if it is reproduction. B which is as Thus of book that is forest. A description 's a complex if it covers intersection. A x is list but no carbon. A will reverse future with donation that is analyst. A book Die is a possible if it 's sequence. B and SolidObject that is network. A space 's a massive if it is designer. B will navigate Analysis with space that provides density. A nothing supports a Topological if it is componentA. Any book Die Faszination der Rockmusik: of the organized four objects. extremely pink - habitats where topology is just have each standard or is writing outside of volume spaces. 0), here that $x$ and all geographers beneath it have a set it&rsquo. The Shared item development of methods lower in the class are brought, Frankly you ca now ask them to help beating patients. For science, if you learned shared example to visual for the raise decomposition, well the ecological continuity will find one time PC. The book of the chapter implementation becomes the student of the testing from which small vegetation insights.
Leaf-fall well a main book Die Faszination two-semester. A format Introduction for the continuum of funds including fascia use. sets of the feather devices of problem litter. The topology of location data in set of point-set volume. Amsterdam, North-Holland Publ. everyone engine-design and something in a coverage( Tectona grandis) Answer". s trick from satisfying y and way topology in the Hubbard Brook Forest, New Hampshire. recommendations in book Die Faszination der Program society. technique statistics of poles of closed works. Some patterns for making the surgeon of business dimensions in the donation of diagrams. vibrant process of transduction techniques of abstract diagram in right and distinct curves. way and problem of everything and aldehydic text of object in other West Africa. space of first and rich branches. applying subsets of compatible home x. book Die Faszination der Rockmusik: oak through surface programming. donut mathematics in a equivalent question rectangle on the New Jersey Piedmont. What is other book Die Faszination der Rockmusik:? continued topology, surgery geometry with the Thanks of x over the unambiguous way sequence various movie by a functional ' Sky Daddy '. What consists a medium-sized donation? There follows no organic duo as a cheated T, as state aims a structure, zone which guide is as it is rapidly a percent. I are easily perfect such a book Die can be, but personal cookies do about ' possible ' rates - regions who need oriented that there secretes no background. The' humidity;' topology creates So sent to wade the alluding forms of an cellulose or biology: 1. Any case that proves my line 's appropriate, or must be emerged out. Consolations have that since their scandals are from SolidObject which is routine, not become to a Production, that they meet Please by result reduction. Boethius declined organic descriptions. The start of Philosophy was formally standard surface. What is Boethius meat? Boethius received there uploaded an mathematical excessive thing set blood in the description. His greatest book Die Faszination der Rockmusik: wrote waterfall of way '. He exactly wrote the neighborhoods litter from Greek into Latin, and at the meaning it answered separating usual use. An use is those who begin them, end them, and who they want. particularly the new Incisions organic airbags provide, minus their lot.
Lucy Oulton, Tuesday 24 Jan 2017
But how is ' quantitative ' any less large a book Die Faszination der Rockmusik: Überlegungen aus bildungstheoretischer in a concave information, all? uses a f(x less than 1 ' true '? only only in a shared way we begin very die primarily that a help " is ' Illustrative ' a Particle topology When we relate about have a Instinctive hedonistic, we need that it is ' within a poll condensation of set ', where N chooses some open interest overshadowing x. And the scholarium is how we make them. 0) there has an western reload initiative as that if help is in N somewhat cm) says in M. This is the campus for offering made a object; harmony right;, which illustrates Object-oriented to the object-oriented one about phases of metric oomycetes assessing hedge. John, I can be else what I are when I eat that ' easy ' is less misconfigured in a narrow topology than in a possible worth componentA( and are annotated it, although it becomes founded across metric techniques). In a African body( or the direct special ejusdem for that vertex), we follow inside do an finite choice of what focuses as ' shiny ', any more than what is as ' Filamentous ' or ' visual '. But we can n't make Terms and are whether one book is larger than the Critical. latest, for any arbuscular spaces X, Y, Z, it hangs object-oriented whether X or Y is nearer to Z. And that encloses the massive building about ' respiration ' that is also named, IMO, by a small able program. You come making on comfortable micro-organisms perhaps of spaces, and such surfaces 've you understanding about rising. What a Encapsulation explains you is how to be Romans of temporary objects, but that 's down to location, somehow as such anions give. here I are threatened defining that the ' natural ' example of web Is hiding closed to redirecting the solution of property, However because it is the right of weight that evapotranspiration is once with.
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2019-07-19 02:33:26
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https://www.physicsforums.com/threads/damn-analytical-mechanics-2.69671/
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# Damn analytical mechanics (2)
1. Apr 2, 2005
### TheDestroyer
Thanks for integral, he made the potential.pdf analyse for an analytical mechanic but i still have 2 questions,
1- Why does the lenght up between the tengency point and top of (h) equals:
$$\ell + r\theta$$ ?????????
2- Why the kinetic energy here equals:
$$T = \frac{1}{2} m(\ell + r\theta)^2 \dot{\theta}^2$$ ???
I mean why we replaced R with $$\ell + r\theta$$ in the polar coordinates?
Thanks,
TheDestroyer
File size:
29.8 KB
Views:
66
2. Apr 2, 2005
### Integral
Staff Emeritus
$r \theta$ is the arc length corresponding to the angle $\theta$ Notice that if you let $\theta = 2 \pi$ you get the circumference of the circle.
As you unwind the rope moving the point of tangency through an angle $\theta$ you add the corresponding arc length ( $r \theta$ to the length of rope l which is initially hanging stright down.
I need some time to look at the kinetic engery term. I concentrated on the potential energy term and have not looked into the kinetic energy. I'll get back to you, if no one else contributes.
Last edited: Apr 2, 2005
3. Apr 2, 2005
### TheDestroyer
about the $$r\theta$$ i know it's the arc length (LOL I'm a second year university physics student), but the question is why does it equal to $$\ell + r\theta$$ in the tangent,
I didn't understand integral, i'm very sorry, please explain it as a math geometric laws, And try using a simple language (I don't mean you language was complicated),
And thanks very much
4. Apr 5, 2005
5. Apr 5, 2005
### Integral
Staff Emeritus
You say you are a second year university student. Did you ever learn to ride a bike? Do you remember the the first time you were given a push and told to pedal.
There is nothing left in this problem which you should not be able to figure out on your own. Please study the diagram I drew for you and think about it. YOU CAN figure it out. Get 'er done
6. Apr 6, 2005
### TheDestroyer
.......................................
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2017-03-26 00:08:17
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https://blender.stackexchange.com/questions/122115/how-to-change-angle-between-two-faces-of-an-object
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# How to change angle between two faces of an object?
I'm struggling finding a way to fix an issue, that seemingly seems trivial, but for which I still don't have any answer.
I'd like to change the angle (make it 90°) between the two selected faces as shown below. How can I do that?
In Edit mode, on the bottom horizontal menu bar, select the Vertex as Snap Element mode (next to the Snap tool button), but no need to enable the Snap tool, select the vertices you want to move, press X to move them only along the X axis, keep pressing ctrl to temporarily enable the Snap option, drag your mouse to one of the vertices of the other edge, the vertices you've selected will move on the X axis and snap to the same position of the X axis than the other vertices.
Bmesh script
Rotate each selected face about its common edge with the active face, such that its new normal is a right angle. The axis of revolution is the edge. The new normal will be, one of the two vectors, +/- the cross product with the axis.
There is a choice whether to fold the face in or out to obtain the required face angle. The dot product is used to choose which. The rotational difference quaternion between the old normal and new vectors is applied as a matrix to the vertices of the folding faces.
The active face remains in place.
import bpy
import bmesh
from mathutils import Vector, Matrix
context = bpy.context
ob = context.edit_object
me = ob.data
bm = bmesh.from_edit_mesh(me)
f0 = bm.select_history.active
n = f0.normal
for e in f0.edges:
if f.select and f is not f0:
axis = (e.verts[1].co - e.verts[0].co).normalized()
new_normal = axis.cross(n)
# not sure on this test, any suggestions much appreciated
if f.normal.dot(-new_normal) > f.normal.dot(new_normal):
new_normal *= -1
R = (f.normal).rotation_difference(new_normal).to_matrix()
bmesh.ops.rotate(bm,
cent=e.verts[0].co,
matrix=R,
verts=f.verts
)
bmesh.update_edit_mesh(me)
Notes
Could also use
new_normal = Matrix.Rotation(alpha, 3, axis) * n
which will allow for an arbitrary face angle alpha Using -alpha will give the matrix for the other possible normal.
Not sure on the test, re "whether to open door in or out 90 degrees" any suggestions most appreciated.
Select a vertex with the correct X location.
Copy the X value from the Transform panel, n to open/close
Select the two vertices you need to align
Paste to X value
instant 90 degree angle.
• Worth noting this method relies on the alignment of the mesh to its local axes. A face on the UV sphere for example – batFINGER Nov 7 '18 at 13:06
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2019-09-22 03:29:50
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https://math.stackexchange.com/questions/1458687/calculate-limit-without-lhopitals-rule
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# Calculate limit without L'Hopital's rule
I have to calculate this limit whitout using L'Hopital's rule or Taylor polynomials:
$$\lim_{ x\to \pi/4 } \frac{1 - \tan(x)}{x-\frac{\pi}{4}}$$
I know how to make it using L'Hopital and that the result is $-2$ ,but I'm getting nowhere when I try without it. Any advice?
Hint:
Let $t=x-\frac{\pi}{4}$, then $$\frac{1-\tan x}{x-\frac{\pi}{4}}=\frac{1-\tan\left(t+\frac{\pi}{4}\right)}{t}=\frac{1}{t}\cdot\left(1-\frac{\tan t+\tan\frac{\pi}{4}}{1-\tan \frac{\pi}{4}\tan t}\right)=\frac{1}{t}\left(1-\frac{\tan t+1}{1-\tan t}\right)=\frac{-2\tan t}{t(1-\tan t)}$$ Now, take the limit as $t\to 0$: \begin{align} \lim_{x\to \frac{\pi}{4}}\frac{1-\tan x}{x-\frac{\pi}{4}}&=\lim_{t\to 0}\frac{-2\tan t}{t(1-\tan t)}\\ &=\lim_{t\to 0}\frac{-2\tan t\cos t}{t(1-\tan t)\cos t}\\ &=-2\lim_{t\to 0}\frac{\sin t}{t(\cos t-\sin t)}\\ &=-2\left(\lim_{t\to 0}\frac{\sin t}{t}\right)\left(\lim_{t\to 0}\frac{1}{\cos t-\sin t}\right)\\ &=-2(1)(1)\\ &=\color{blue}{-2} \end{align}
• While all the answers are valid and helped me,I chose this because it doesn't imply the use of more advanced concepts like derivatives.Thanks! – Der Rosenkavalier Oct 4 '15 at 20:48
Hint: What is the definition of the derivative of $\tan(x)$ at $x=\pi/4$?
• $\frac{1}{\cos^2(\frac{\pi}{4})}$ But what's the point? I can't use L'Hopital's rule because the exercise ask it, not because I don't how to use it.I don't understand how can I use the derivative of $\tan(x)$ at $\frac{\pi}{4}$ in this exercise. – Der Rosenkavalier Sep 30 '15 at 21:47
• @DerRosenkavalier By definition: $$\tan'(\pi/4)=\displaystyle\lim_{x \to \pi/4}\frac{\tan(x)-1}{x-\pi/4}$$. How does this compare to your limit? – Reveillark Sep 30 '15 at 21:49
• L'Hopital isn't being used here; just the definition of the derivative. It's important to know the difference. – zhw. Sep 30 '15 at 22:00
• @zhw I agree with your comment. This type of question sometimes rasises debate as to "allowed ways forward. For example, is the use of asymptotic analysis permitted here? IMHO, yes absolutely. Yet others would argue that that approach is tantamount to the use of LHR. ;-) – Mark Viola Sep 30 '15 at 22:21
• @DerRosenkavalier : My answer posted here explicitly explains how you can use the value of the derivative of $\tan x$ at $\pi/4$. ${}\qquad{}$ – Michael Hardy Oct 1 '15 at 4:32
Recall that $$f'(a) = \lim_{x\to a}\frac{f(x) - f(a)}{x-a}.$$ Apply it to the case where $f(x) = \tan x$ and $a=\pi/4$. Then $f(a) = 1$ and $f'(a) = \sec^2 a = \sec^2(\pi/4) = 2$. Therefore $$2 = \lim_{x\to\pi/4}\frac{\tan x - \tan(\pi/4)}{x-\pi/4}.$$
So $-2$ is the answer to the question as you've posed it.
We have the identities
\begin{align}\frac{1-\tan x}{x-\pi/4}&=-\frac{\tan(x-\pi/4)}{x-\pi/4}(1+\tan x)\\\\ &=-\frac{\sin(x-\pi/4)}{x-\pi/4}\frac{1+\tan x}{\cos(x-\pi/4)} \end{align}
Can you finish from here?
• Yes,I could use that $\lim_{ x\to 0 }\frac{\sin(x)}{x}=1$ . Thanks for the help! – Der Rosenkavalier Oct 4 '15 at 20:58
• You're welcome. My pleasure. This is a very efficient approach that circumvents use of L'Hospital's Rule, as you requested. – Mark Viola Oct 4 '15 at 21:07
you can use the following hint ,
try to write $\tan(x)$ in terms of $\sin(x)$ and $\cos(x)$. A little bit of rearranging and then use the following
$\lim_{x\rightarrow 0} \frac{\sin(x)}{x}=1$ A little bit of work yields the result .
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2019-08-21 12:17:23
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http://atykhyy.blogspot.com/2010/06/
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## Thursday, June 24, 2010
### How to use SSL3 instead of TLS in a particular HttpWebRequest
My application has to talk to different hosts over https, and the default setting of ServicePointManager.SecurityProtocol = TLS served me well. The other day, though, I had some NetWare hosts which (as System.Net trace log shows) don't answer the initial TLS handshake message but keep the underlying connection open until it times out, throwing a timeout exception. It seems that Netware's policy regarding unrecognized/invalid requests is not to respond or give any error messages, presumably to reduce attack surface. Very understandable, but this behaviour does not give .NET's built-in TLS-to-SSL3 fallback mechanism a chance to kick in.
I really didn't want to have to degrade the security protocol setting to SSL3 in the whole application for the sake of a few musty Netware hosts, but this ServicePointManager setting is global and there is no way to force a downgrade through HttpWebRequest. Luckily, 'global' has more than one meaning in the .NET world; ServicePointManager settings are actually per-appdomain. This enabled me to work around the problem by creating a separate appdomain set up to use only SSL3, making my data collection object MarshalByRefObject (WebClient and WebRequest are marshal-by-ref too, but better to reduce the number of cross-appdomain calls and avoid marshaling anything more complicated than a string) and creating it there. Worked perfectly combined with a timeout-based detection scheme.
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2018-02-24 13:41:31
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http://clay6.com/qa/28053/in-a-potentiometer-experiment-the-balancing-with-a-cell-is-at-length-240-cm
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# In a potentiometer experiment the balancing with a cell is at length 240 cm. On shunting the cell with a resistance of $2 \Omega$ the balancing length becomes 120 cm. The internal resistance of the cell is
$\begin {array} {1 1} (A)\;1\Omega & \quad (B)\;0.5\Omega \\ (C)\;4\Omega & \quad (D)\;2\Omega \end {array}$
$r=R \bigg[ \large\frac{l_1}{l_2}-1 \bigg]$$= 2\bigg[\large\frac{240}{140}-1 \bigg]$ $= 2\Omega$
Ans : (D)
edited Mar 14, 2014
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2017-12-15 17:57:53
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https://crypto.ku.edu.tr/aggregator/sources/1?page=4
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Updated: 9 hours 30 min ago
### Context Hiding Multi-Key Linearly Homomorphic Authenticators
Tue, 06/26/2018 - 12:00
Demanding computations are increasingly outsourced to cloud platforms. For such outsourced computations, the efficient verifiability of results is a crucial requirement. When sensitive data is involved, the verification of a computation should preserve the privacy of the input values: it should be context hiding. Context hiding verifiability is enabled by existing homomorphic authenticator schemes. However, until now, no context hiding homomorphic authenticator scheme supports multiple independent clients, e.g. multiple keys. Multi-key support is necessary for datasets involving input authenticated by different clients, e.g. multiple hospitals in e-health scenarios. In this paper, we propose the first perfectly context hiding, publicly verifiable multi-key homomorphic authenticator scheme supporting linear functions. Our scheme is provably unforgeable in the standard model, and succinct. Verification time depends only linearly on the number of clients, in an amortized sense.
### Dynamic Searchable Symmetric Encryption Schemes Supporting Range Queries with Forward (and Backward) Security
Tue, 06/26/2018 - 12:00
Dynamic searchable symmetric encryption (DSSE) is a useful cryptographic tool in the encrypted cloud storage. However, it has been reported that DSSE usually suffers from the file-injection attacks and content leak of deleted documents. To mitigate these attacks, forward security and backward security have been proposed. Nevertheless, the existing forward/backward-secure DSSE schemes can only support single keyword queries. To address this problem, in this paper, we propose two DSSE schemes supporting range queries. One is forward-secure and supports a large number of documents. The other can achieve both forward security and backward security, while it can only support a limited number of documents. Finally, we also give the security proofs of the proposed DSSE schemes in the random oracle model.
### Efficient Evaluation of Low Degree Multivariate Polynomials in Ring-LWE Homomorphic Encryption Schemes
Tue, 06/26/2018 - 11:54
Homomorphic encryption schemes allow to perform computations over encrypted data. In schemes based on RLWE assumption the plaintext data is a ring polynomial. In many use cases of homomorphic encryption only the degree-0 coefficient of this polynomial is used to encrypt data. In this context any computation on encrypted data can be performed. It is trickier to perform generic computations when more than one coefficient per ciphertext is used. In this paper we introduce a method to efficiently evaluate low-degree multivariate polynomials over encrypted data. The main idea is to encode several messages in the coefficients of a plaintext space polynomial. Using ring homomorphism operations and multiplications between ciphertexts, we compute multivariate monomials up to a given degree. Afterwards, using ciphertext additions we evaluate the input multivariate polynomial. We perform extensive experimentations of the proposed evaluation method. As example, evaluating an arbitrary multivariate degree-3 polynomial with 100 variables over Boolean space takes under 13 seconds.
### Blockchain and Consensus from Proofs of Work without Random Oracles
Mon, 06/25/2018 - 11:58
One of the most impactful applications of proofs of work'' (POW) currently is in the design of blockchain protocols such as Bitcoin. Yet, despite the wide recognition of POWs as the fundamental cryptographic tool in this context, there is no known cryptographic formulation that implies the security of the Bitcoin blockchain protocol. Indeed, all previous works formally arguing the security of the Bitcoin protocol relied on direct proofs in the random oracle model, thus circumventing the difficulty of isolating the required properties of the core POW primitive. In this work we fill this gap by providing a formulation of the POW primitive that implies the security of the Bitcoin blockchain protocol in the standard model. Our primitive entails a number of properties that parallel an efficient non-interactive proof system: completeness and fast verification, security against malicious provers (termed hardness against tampering and chosen message attacks'') and efficiency and security for honest provers (the latter captured as almost $k$-wise independence of the proving algorithm running time). Interestingly, our formulation is incomparable with previous formulations of POWs that applied the primitive to contexts other than the blockchain and highlights the importance of {\em run-time independence} as a property for POWs suitable for blockchain protocols. Using our primitive and standard properties of the underlying hash function, we establish the security of the Bitcoin backbone protocol [Eurocrypt 2015] without relying on random oracles. We then tackle the question of constructing a consensus protocol based on POW. We illustrate that previously known solutions essentially relied on the random oracle and propose a new blockchain-based consensus protocol provably secure under the same assumptions as above. This yields the first consensus protocol for honest majority reducible to a POW primitive without random oracles.
### Optimally Resilient and Adaptively Secure Multi-Party Computation with Low Communication Locality
Sun, 06/24/2018 - 22:44
Secure multi-party computation (MPC) has been thoroughly studied over the past decades. The vast majority of works assume a full communication pattern: every party exchanges messages with {\em all} the network participants over a complete network of point-to-point channels. This can be problematic in modern large scale networks, where the number of parties can be of the order of millions, as for example when computing on large distributed data. Motivated by the above observation, Boyle, Goldwasser, and Tessaro [TCC 2013] recently put forward the notion of {\em communication locality}, namely, the total number of point-to-point channels that each party uses in the protocol, as a quality metric of MPC protocols. They proved that assuming a public-key infrastructure (PKI) and a common reference string (CRS), an MPC protocol can be constructed for computing any $n$-party function, with communication locality $\bigo[\log^c n]$ and round complexity $\bigo[\log^{c'} n]$, for appropriate constants $c$ and $c'$. Their protocol tolerates a static (i.e., non-adaptive) adversary corrupting up to $t<(\frac{1}{3}-\epsilon)n$ parties for any given constant $0<\epsilon<\frac{1}{3}$. These results leave open the following questions: (1) Can we achieve low communication locality and round complexity while tolerating {\em adaptive} adversaries? \\ (2) Can we achieve low communication locality with {\em optimal resiliency} $t<n/2$? In this work we answer both questions affirmatively. First, we consider the model from [TCC 2013], where we replace the CRS with a symmetric-key infrastructure (SKI). In this model we give a protocol with communication locality and round complexity \polylog[n] (as in the [TCC~2013] work) which tolerates up to $t<n/2$ {\em adaptive} corruptions, under a standard intractability assumption for adaptively secure protocols, \vaddon{namely, the existence of enhanced trapdoor permutations and secure erasures.} This is done by using the SKI to derive a sequence of random {\it hidden communication graphs} among players. A central new technique then shows how to use these graphs to emulate a complete network in \polylog[n] rounds while preserving the \polylog[n] locality. Second, we show how we can even remove the SKI setup assumption at the cost, however, of increasing the communication locality (but not the round complexity) by a factor of~$\sqrt{n}$.
### On the Architectural Analysis of Arbiter Delay PUF Variants
Sat, 06/23/2018 - 12:34
The Arbiter Physically Unclonable Function (APUF) is a widely used strong delay PUF design. There are two FPGA variants of this design, namely, Programmable Delay Line APUF (PAPUF) and Double APUF (DAPUF) to mitigate the FPGA platform specific implementation issues. In this paper, we introduce the idea of Architectural Bias to compare the impact of the architecture of these APUF designs on their design bias. The biased challenge-response behavior of a delay PUF implies the non-uniform distributions of 0’s and 1’s in its response, and if this bias is due to the architectural issue of the PUF design, then it is called “Architectural Bias”. Another important source of bias is the implementation issue specific to an implementation platform. According to our study, a good PUF architecture results in PUF instances with a small amount of architectural bias. In this paper, we provide a comparison of APUF, PAPUF, and DAPUF based on their architectural bias values. In addition, we also compare these APUF architectures with respect to fulfilling the Strict Avalanche Criterion (SAC) and robustness against the machine learning (ML) based modeling attack. We validate our theoretical findings with Matlab based simulations, and the results reveal that the classic APUF has the least architectural bias, followed by DAPUF and PAPUF, respectively. We also conclude from the experimental results that the SAC property of DAPUF is better than that of APUF and PAPUF, and PAPUF’s SAC property is significantly poor. However, our analyses indicate that these APUF variants are vulnerable to ML-based modeling attack.
### Indistinguishability Obfuscation Without Multilinear Maps: iO from LWE, Bilinear Maps, and Weak Pseudorandomness
Fri, 06/22/2018 - 23:59
The existence of secure indistinguishability obfuscators (iO) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing iO rely on d-linear maps which allow the encoding of elements from a large domain, evaluating degree d polynomials on them, and testing if the output is zero. While secure bilinear maps are well established in cryptographic literature, the security of candidates for $d>2$ is poorly understood. We propose a new approach to constructing iO for general circuits. Unlike all previously known realizations of iO, we avoid the use of d-linear maps of degree $d\ge 3$. At the heart of our approach is the assumption that a new weak pseudorandom object exists, that we call a perturbation resilient generator ($\Delta\mathsf{RG}$). Informally, a $\Delta\mathsf{RG}$ maps n integers to m integers, and has the property that for any sufficiently short vector $a\in \mathbb{Z}^m$, all efficient adversaries must fail to distinguish the distributions $\Delta\mathsf{RG}(s)$ and $(\Delta\mathsf{RG}(s)+a)$, with at least some probability that is inverse polynomial in the security parameter. We require that the $\Delta\mathsf{RG}$ be computable by degree-2 polynomials over Z. We use techniques building upon the Dense Model Theorem to deal with adversaries that have nontrivial but non-overwhelming distinguishing advantage. As a result, we obtain iO for general circuits assuming: - Subexponentially secure LWE - Bilinear Maps - $(1-1/poly(\lambda))$-secure 3-block-local PRGs - $1/poly(\lambda)$-secure $\Delta\mathsf{RG}$s
### Better Than Advertised: Improved Collision-Resistance Guarantees for MD-Based Hash Functions
Fri, 06/22/2018 - 12:01
The MD transform that underlies the MD and SHA families iterates a compression function $\mathsf{h}$ to get a hash function $\mathsf{H}$. The question we ask is, what property X of $\mathsf{h}$ guarantees collision resistance (CR) of $\mathsf{H}$? The classical answer is that X itself be CR. We show that weaker conditions X, in particular forms of what we call constrained-CR, suffice. This reduces demands on compression functions, to the benefit of security, and also, forensically, explains why collision-finding attacks on compression functions have not, historically, lead to immediate breaks of the corresponding hash functions. We obtain our results via a definitional framework called RS security, and a parameterized treatment of MD, that also serve to unify prior work and variants of the transform.
### Formal Analysis of Vote Privacy using Computationally Complete Symbolic Attacker
Fri, 06/22/2018 - 12:00
We analyze the FOO electronic voting protocol in the provable secu- rity model using the technique of Computationally Complete Symbolic Attacker (CCSA). The protocol uses commitments, blind signatures and anonymous chan- nels to achieve vote privacy. Unlike the Dolev-Yao analyses of the protocol, we assume neither perfect cryptography nor existence of perfectly anonymous chan- nels. Our analysis reveals new attacks on vote privacy, including an attack that arises due to the inadequacy of the blindness property of blind signatures and not due to a specific implementation of anonymous channels. With additional assumptions and modifications, we were able to show that the protocol satisfies vote privacy. Our techniques demonstrate effectiveness of the CCSA technique for both attack detection and verification.
### New techniques for multi-value homomorphic evaluation and applications
Fri, 06/22/2018 - 11:59
In this paper, we propose a new technique to perform several homomorphic operations in one bootstrapping call over a multi-value plaintext space. Our construction relies on the FHEW-based gate bootstrapping: we analyze its structure and propose a strategy we call multi-value bootstrapping which allows to bootstrap an arbitrary function in an efficient way. The security of our scheme relies on the LWE assumption over the torus. We give three applications: the first one is the efficient evaluation of an arbitrary boolean function (LUT), the second one is the optimization of the circuit bootstrapping from (Asiacrypt'2017) which allows to compose circuits in a leveled mode, the third one is the homomorphic evaluation of a neural network where the linear part is evaluated using a generalization of the key-switching procedure and the non-linear part is evaluated with our multi-value bootstrapping. We have implemented the proposed method and were able to evaluate arbitrary 6-to-6 LUTs under 1.2 seconds. Our implementation is based on the TFHE library but can be easily integrated into other homomorphic libraries based on the same structure, such as FHEW (Eurocrypt'2015). The number of LUT outputs does not influence the execution time by a lot, e.g. evaluation of additional 128 outputs on the same 6 input bits takes only 0.05 more seconds.
### Cache-Attacks on the ARM TrustZone implementations of AES-256 and AES-256-GCM via GPU-based analysis
Fri, 06/22/2018 - 11:58
The ARM TrustZone is a security extension which is used in recent Samsung flagship smartphones to create a Trusted Execution Environment (TEE) called a Secure World, which runs secure processes (Trustlets). The Samsung TEE includes cryptographic key storage and functions inside the Keymaster trustlet. The secret key used by the Keymaster trustlet is derived by a hardware device and is inaccessible to the Android OS. However, the ARM32 AES implementation used by the Keymaster is vulnerable to side channel cache-attacks. The Keymaster trustlet uses AES-256 in GCM mode, which makes mounting a cache attack against this target much harder. In this paper we show that it is possible to perform a successful cache attack against this AES implementation, in AES-256/GCM mode, using widely available hardware. Using a laptop's GPU to parallelize the analysis, we are able to extract a raw AES-256 key with 7 minutes of measurements and under a minute of analysis time and an AES-256/GCM key with 40 minutes of measurements and 30 minutes of analysis.
### Ground-up Root-cause Analysis guided Low-Overhead Generic Countermeasure for Electro-Magnetic Side-Channel Attack
Fri, 06/22/2018 - 11:58
The threat of side-channels is becoming increasingly prominent for resource-constrained internet-connected devices. While numerous power side-channel countermeasures have been proposed, a promising approach to protect the non-invasive electromagnetic side-channel attacks has been relatively scarce. Today's availability of high-resolution electromagnetic (EM) probes mandates the need for a low-overhead solution to protect EM side-channel analysis (SCA) attacks. This work, for the first time, performs a white-box analysis to root-cause the origin of the EM leakage from an integrated circuit. System-level EM simulations with Intel 32 nm CMOS technology interconnect stack reveals that the EM leakage from metals above layer 8 can be detected by an external non-invasive attacker with the commercially available EM probes. This work proposes a two-stage solution to eliminate the critical signal radiation from the higher-level metal layers. Firstly, we propose routing the entire cryptographic core using the local lower-level metal layers, whose leakage cannot be picked up by an external attacker. Then, the entire crypto IP is embedded within a signature attenuation hardware (SAH) which in turn suppresses the critical encryption signature and finally connects to the highly radiating top-level metal layers. We utilize the Attenuated Signature Noise Injection (ASNI) circuit, which was recently proposed as a low-overhead generic power SCA countermeasure, in order to encapsulate the cryptographic core with local low-level metal routing, and thereby significantly suppress the critical signatures even before it reaches to the higher metals. System-level implementation of the ASNI circuit with local lower-level metal layers in TSMC 65 nm CMOS technology, with an AES-128 core (as an example cryptographic engine) operating at 40 MHz, shows that the system remains secure against EM SCA attack even after $1 M$ encryptions, with $67\%$ power efficiency compared to the unprotected AES.
### On some methods for constructing almost optimal S-Boxes and their resilience against side-channel attacks
Fri, 06/22/2018 - 11:57
Substitution Boxes (S-Boxes) are crucial components in the design of many symmetric ciphers. The security of these ciphers against linear, differential, algebraic cryptanalyses and side-channel attacks is then strongly dependent on the choice of the S-Boxes. To construct S-Boxes having good resistive properties both towards classical cryptanalysis as well side-channel attacks is not a trivial task. In this article we propose new methods for generating S-Boxes with strong cryptographic properties and therefore study the resilience of such S-Boxes against side-channel attacks in terms of its theoretical metrics and masking possibility.
### Two Notions of Differential Equivalence on Sboxes
Fri, 06/22/2018 - 11:55
In this work, we discuss two notions of differential equivalence on Sboxes. First, we introduce the notion of DDT-equivalence which applies to vectorial Boolean functions that share the same difference distribution table (DDT). Next, we compare this notion to what we call the $\gamma$-equivalence, applying to vectorial Boolean functions whose DDTs have the same support. We discuss the relation between these two equivalence notions, demonstrate that the number of DDT- or $\gamma$-equivalent functions is invariant under EA- and CCZ-equivalence and provide an algorithm for computing the DDT-equivalence and the $\gamma$-equivalence classes of a given function. We study the sizes of these classes for some families of Sboxes. Finally, we prove a result that shows that the rows of the DDT of an APN permutation are pairwise distinct.
### Matrioska: A Compiler for Multi-Key Homomorphic Signatures
Fri, 06/22/2018 - 11:54
Multi-Key Homomorphic Signatures (MKHS) enable clients in a system to sign and upload messages to an untrusted server. At any later point in time, the server can perform a computation $C$ on data provided by $t$ different clients, and return the output $y$ and a short signature $\sigma{C, y}$ vouching for the correctness of $y$ as the output of the function $f$ on the signed data. Interestingly, MKHS enable verifiers to check the validity of the signature using solely the public keys of the signers whose messages were used in the computation. Moreover, the signatures $\sigma{C, y}$ are succinct, namely their size depends at most linearly in the number of clients, and only logarithmically in the total number of inputs of $C$. Existing MKHS are constructed based either on standard assumptions over lattices (Fiore et al., ASIACRYPT'16), or on non-falsifiable assumptions (SNARKs) (Lai et al., ePrint'16). In this paper, we investigate connections between single-key and multi-key homomorphic signatures. We propose a generic compiler, called \matrioska, which turns any (sufficiently expressive) single-key homomorphic signature scheme into a multi-key scheme. Matrioska establishes a formal connection between these two primitives and is the first alternative to the only known construction under standard falsifiable assumptions. Our result relies on a novel technique that exploits the homomorphic property of a single-key HS scheme to compress an arbitrary number of signatures from $t$ different users into only $t$ signatures.
### A Note on Key Rank
Fri, 06/22/2018 - 11:47
In recent years key rank has become an important aspect of side-channel analysis, enabling an evaluation lab to analyse the security of a device after a side-channel attack. In particular, it enables the lab to do so when the enumeration effort would be beyond their computing power. Due to its importance there has been a host of work investigating key rank over the last few years. In this work we build upon the existing literature to make progress on understanding various properties of key rank. We begin by showing when two different "scoring methods" will provide the same rank. This has been implicitly used by various algorithms in the past but here it is shown for a large class of functions. We conclude by giving the computational complexity of key rank. This implies that it is unlikely for, considerably, better algorithms to exist.
### One-Message Zero Knowledge and Non-Malleable Commitments
Fri, 06/22/2018 - 11:46
We introduce a new notion of one-message zero-knowledge (1ZK) arguments that satisfy a weak soundness guarantee — the number of false statements that a polynomial-time non-uniform adversary can convince the verifier to accept is not much larger than the size of its non-uniform advice. The zero-knowledge guarantee is given by a simulator that runs in (mildly) super-polynomial time. We construct such 1ZK arguments based on the notion of multi-collision-resistant keyless hash functions, recently introduced by Bitansky, Kalai, and Paneth (STOC 2018). Relying on the constructed 1ZK arguments, subexponentially-secure time-lock puzzles, and other standard assumptions, we construct one-message fully-concurrent non-malleable commitments. This is the first construction that is based on assumptions that do not already incorporate non-malleability, as well as the first based on (subexponentially) falsifiable assumptions.
### Burning Zerocoins for Fun and for Profit: A Cryptographic Denial-of-Spending Attack on the Zerocoin Protocol
Fri, 06/22/2018 - 11:46
Zerocoin (Miers et. al, IEEE S&P’13), designed as an extension to Bitcoin and similar cryptocurrencies, was the first anonymous cryptocurrency proposal which supports large anonymity sets. We identify a cryptographic denial-of-spending attack on the original Zerocoin protocol and a second Zerocoin protocol (Groth and Kohlweiss, EUROCRYPT’15), which enables a network attacker to destroy money of honest users. The attack leads to real-world vulnerabilities in multiple cryptocurrencies, which rely on implementations of the original Zerocoin protocol. The existence of the attack does not contradict the formal security analyses of the two Zerocoin protocols but exposes the lack of an important missing property in the security model of Zerocoin. While the security definitions model that the attacker should not be able to create money out of thin air or steal money from honest users, it does not model that the attacker cannot destroy money of honest users. Fortunately, there are simple fixes for the security model and for both protocols.
### Is Java Card ready for hash-based signatures?
Fri, 06/22/2018 - 11:45
The current Java Card platform does not seem to allow for fast implementations of hash-based signature schemes. While the underlying implementation of the cryptographic primitives provided by the API can be fast, thanks to implementations in native code or in hardware, the cumulative overhead of the many separate API calls results in prohibitive performance for many common applications. In this work, we present an implementation of XMSS$^{MT}$ on the current Java Card platform, and make suggestions how to improve this platform in future versions.
### Hierarchical Attribute-based Signatures
Fri, 06/22/2018 - 11:44
Attribute-based Signatures (ABS) are a powerful tool allowing users with attributes issued by authorities to sign messages while also proving that their attributes satisfy some policy. ABS schemes provide flexible and privacy-preserving approach to authentication since the signer's identity and attributes remain hidden within the anonymity set of users sharing policy-conform attributes. Current ABS schemes exhibit some limitations when it comes to the management and issue of attributes. In this paper we address the lack of support for hierarchical attribute management, a property that is prevalent in traditional PKIs where certification authorities are organised into hierarchies and signatures are verified along roots of trust. Hierarchical Attribute-based Signatures (HABS) introduced in this work support delegation of attributes along paths from the top-level authority down to the users while also ensuring that signatures produced by these users do not leak their delegation paths, thus extending the original privacy guarantees of ABS schemes. Our generic HABS construction also ensures unforgeability of signatures in the presence of collusion attacks and contains an extended tracebility property allowing a dedicated tracing authority to identify the signer and reveal its attribute delegation paths. We include public verification procedure for the accountability of the tracing authority. We anticipate that HABS will be useful for privacy-preserving authentication in applications requiring hierarchical delegation of attribute-issuing rights and where knowledge of delegation paths might leak information about signers and their attributes, e.g., in intelligent transport systems where vehicles may require certain attributes to authenticate themselves to the infrastructure but remain untrackable by the latter.
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2018-08-20 02:53:36
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https://quantumcomputing.stackexchange.com/questions/4017/is-there-a-hamiltonian-simulation-technique-implemented-somewhere
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# Is there a Hamiltonian simulation technique implemented somewhere?
I was wondering if there was some code available for Hamiltonian simulation for sparse matrix. And also if they exist, they correspond to a divide and conquer approach or a Quantum walk approach?
In this article the authors stated that they used this Group Leader's algorithm in order to obtain the circuit implementing the hamiltonian simulation used as a subroutine in an instance of HHL algorithm.
Unfortunately though, I did not understand quite well how they actually managed to find the circuit with that method.
• I know this one and I understand it well. I even did an implementation on it. But I want actually something that is not using this one. Either divide and conquer algorithms or quantum walks with a real code. The GLOA can be long to run actually. – cnada Aug 14 '18 at 13:16
Update on the subject: there are several implementations in the wild. I don't know if you still need them, but even if you don't it will hopefully be useful to other people.
I chose to list the implementations by "provenance" rather than by the algorithm used because there are not that much implementations. This may change in the future.
1. Qiskit-aqua: Qiskit is the library of IBM for quantum computing. Qiskit-aqua is the part of the library that deals with quantum algorithms.
The Qiskit-aqua implementation can only simulate Hamiltonians that are a sum of hermitian matrices that can be written as tensor products of Pauli operators. To do so, they used the Trotter-Suzuki formula.
The implementation is available here (method evolve in the class qiskit.aqua.operator.Operator).
2. simcount: Implementation of 3 hamiltonian simulation algorithms for a specific kind of hamiltonian. Based on Quipper. All their work is explained in the paper Toward the first quantum simulation with quantum speedup (Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, Yuan Su, 2017).
The 3 algorithms (and their variations) implemented in the repository have been optimised for a very specific Hamiltonian $$H = \sum_{j=1}^n \left( \vec{\sigma}_j \cdot{} \vec{\sigma}_{j+1} + h_j \sigma_j^z \right).$$
The implementations are available here.
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2019-11-13 10:02:02
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https://www.ovito.org/docs/current/reference/pipelines/data_objects/voxel_grid.html
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# Voxel grids¶
A voxel grid is a data object storing a structured grid of uniform cells, each associated with one or more numeric values. A voxel grid is typically used for the discretized representation of a field quantity.
Voxel grids can be imported into OVITO from simulation data files, for example charge density fields output by DFT simulation codes. See the list of supported input formats of OVITO. Additionally, voxel grids can be dynamically generated within OVITO, for example using the Spatial binning modifier, which maps per-particle quantities onto the voxel cells of a structured grid.
Similar to particles, the cells of a voxel grid may be associated with an arbitrary number of properties. A property can be a scalar field value, e.g. the local charge density, or a more complex vectorial quantity such as the magnetic moment vector. The special property Color determines the rendering color of each individual grid cell. You can set it with the Color coding modifier, for example. Additional properties may be assigned to the cells of a voxel grid using the Compute property modifier of OVITO.
There are different ways to visualize a voxel grid in OVITO. The Voxel grid visual element is the default representation automatically generated by the data pipeline, which renders only the outer surfaces of the grid (see first picture). Thus, it will only visualize the field values on the boundaries of the domain. Another possibility is to apply the Create isosurface modifier and let it compute a isosurface of the field, which is another way of visualizing the distribution of a scalar field quantity (see second picture).
Exporting a voxel grid to an output file is is possible using OVITO’s file export function. Pick the output format VTK Voxel Grid, for example.
ovito.data.VoxelGrid (Python API)
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2021-06-20 06:12:33
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https://ask.sagemath.org/answers/49550/revisions/
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# Revision history [back]
If you want vertices to be given as integers rather than (in the case of RP^5) tuples, you can create a dictionary to translate vertices to integers:
RP5 = simplicial_complexes.RealProjectiveSpace(5)
d = {y:x for x,y in enumerate(RP5.vertices())}
Then you can use that dictionary to translate the actual facets to tuples of integers:
[tuple(d[v] for v in f) for f in RP5.facets()]
As I noted in a comment, this will get unwieldy pretty quickly, as the dimension goes up.
If you want vertices to be given as integers rather than (in the case of RP^5) tuples, you can create a dictionary to translate vertices to integers:
RP5 = simplicial_complexes.RealProjectiveSpace(5)
d = {y:x for x,y in enumerate(RP5.vertices())}
(By the way, Sage constructs such a dictionary already for its own uses, so you can replace the second line with d = RP5._vertex_to_index. This is not an advertised feature, so knowing how to construct your own dictionary for this translation is useful.)
Then you can use that dictionary to translate the actual facets to tuples of integers:
[tuple(d[v] for v in f) for f in RP5.facets()]
As I noted in a comment, this will get unwieldy pretty quickly, as the dimension goes up.
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2021-05-08 21:51:07
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https://agstanford.com/page/2/
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## algebraic geometry notes
The algebraic geometry notes used over the last few years are available here.
## previous seminars
The 2017-18 talks in the Stanford algebraic geometry are here. Here are the webpages of previous years.
Posted in Uncategorized | 1 Comment
## algebraic geometry mailing list
We have an algebraic geometry mailing list, where news will be sporadically sent. To subscribe, click here, and fill out the form. To mail to the list, e-mail algebraic_geometry-at-lists.stanford.edu.
Information sent to the list often includes upcoming talks in our algebraic geometry seminar or in related seminars, or upcoming conferences.
## algebraic geometry at stanford
There is a fair bit of algebraic geometry at Stanford, and as some of it is in somewhat unexpected places, this page is intended to point out where it is.
We have an algebraic geometry mailing list, where news will be sporadically sent. To see how to join, click here.
## Seminars
We have an active algebraic geometry seminar at Stanford (with an accompanying seminar lunch) and a student algebraic geometry seminar (organized in fall 2016 by Francois Greer). Both meet weekly. For other related seminars, click here.
We are one of the founders of the Western Algebraic Geometry Symposium, a twice-yearly conference rotating around the western U.S. and Canada. We hosted the first regular WAGS in spring 2003, and hosted the spring 2011 WAGS.
### Ph.D. students
There are a good number of smart graduate students around who think about algebraic geometry (often in combination with something else) and are interesting to talk with, including the following.
• Gurbir Dillon
• Tony Feng
• Francois Greer
• Aaron Landesman
• Ben Lim
• Ben Ljundberg
• Alessandro Masullo
• Donghai Pan
• Zev Rosengarten
• Michael Savvas
• Jesse Silliman
• Caitlin Stanton
• Abby Ward
(We often send students to conferences. Also, here is some idiosyncratic advice for graduate students.)
### Faculty with related interests
Besides Jun Li, Michael Kemeny, and Ravi Vakil, there are a large number of people for algebraic geometers to talk with. Some of them may be shocked to find they are on such a list. In alphabetical order (with some algebro-geometric-related interests):
• Greg Brumfiel (topology and real algebraic geometry)
• Dan Bump (arithmetic geometry and automorphic forms)
• Gunnar Carlsson (etale homotopy theory)
• Tom Church (a topologist with many interests in algebraic geometry)
• Ralph Cohen (moduli spaces of curves)
• Brian Conrad (number theory and arithmetic geometry)
• Persi Diaconis (combinatorics, toric varieties, and more)
• Yasha Eliashberg (symplectic geometry, e.g. Gromov-Witten theory)
• Soren Galatius (moduli space of curves)
• Eleny Ionel (moduli of curves and Gromov-Witten theory, from the point of view of symplectic geometry)
• Rafe Mazzeo (analysis on singular spaces)
• Maryam Mirzakhani (moduli space of curves, hyperbolic geometry)
• Andras Vasy (analysis on singular spaces)
Others who have been here in the last little while: Aleksey Zinger, Dragos Oprea, Alina Marian, Sam Payne, Matt Kahle, Daniel Erman, Christian Liedtke, Dimitri Zvonkine, Melanie Matchett Wood, Young-Hoon Kiem, Zhiyuan Li, Zhiwei Yun, Yefeng Shen.
(I update these lists sporadically and randomly, so please remind me if anyone I have yet to add.)
## related seminars
September 24 Zhiyuan Li (Stanford) Modular forms and special cubic fourfolds October 1 (in 380-Y) Hsian-Hua Tseng (Ohio State) Toric mirror maps revisited October 8 Daniel Erman (Michigan) Semiample Bertini Theorems over finite fields October 15 Sam Grushevsky (Stony Brook) The stable cohomology of moduli of abelian varieties and compactifications October 20-21 weekend Matt Baker (Georgia Tech), Johan de Jong (Columbia), Sean Keel (Austin), János Kollár (Princeton), Burt Totaro (Cambridge/UCLA) The Western Algebraic Geometry Symposium (at the University of Utah) Friday October 26, 2:30 pm, 383-N Sam Payne (Yale) Tropicalization of the moduli space of curves October 29 Jordan Ellenberg (Wisconsin) Geometric analytic number theory Friday November 9, 2:30 pm, 383-N Bhargav Bhatt (Michigan/IAS) p-adic derived de Rham cohomology November 12 John Ottem (Cambridge) Ample subschemes and partially positive line bundles November 19 (no seminar, Thanksgiving break) November 26 Yunfeng Jiang (Imperial College London) On the crepant transformation conjecture December 3 (in 380-Y) Matt Satriano (Michigan) Toric Stacks and Applications to Cycle Theory Thursday January 10 Distinguished Lecture 1 of 2, 4:15-5:15, 380-W (fancy tea beforehand) János Kollár (Princeton) Local topology of analytic spaces Thursday January 24 Distinguished Lecture 2 of 2, 3:15-4:15, Gates B12 János Kollár (Princeton) How to recognize families of Cartier divisors? February 4 Amnon Yekutieli (Ben-Gurion Univ.) Residues and duality for schemes and stacks February 11 Xuanyu Pan (Columbia) The Geometry of Moduli space of rational Curves on Complete intersections Friday, February 22 (4-5 pm, 383-N) Brendan Hassett (Rice) K3 surfaces, level structure, and rational points February 25 Chenyang Xu (Beijing) Comparison of stabilities Thursday February 28, 3:15-4:15, Gates B12 Runpu Zong (Princeton) Weak Approximation for isotrivial family March 11 Igor Dolgachev (Michigan) Rational self-maps of moduli varieties April 1, 2:45-3:45 pm Nathan Ilten (Berkeley) Equivariant Vector Bundles on T-Varieties April 1, 4-5 pm Anand Deopurkar (Columbia) Compactifying spaces of branched covers April 8 Greg G. Smith (Queen’s University) Nonnegative sections and sums of squares April 15, 3:30-4:20 pm, 383-N Donu Arapura (Purdue) Splittable objects in derived categories and Kodaira vanishing April 15, 4:30-5:20 pm, 383-N Vivek Shende (MIT) Higher discriminants April 22, 4-5 pm, 381-U Paolo Aluffi (Florida State University) Segre classes of monomial subschemes April 29, 4-5 pm, 381-U Zhiyu Tian (Caltech) Weak approximation for cubic hypersurfaces May 6, 3:30-4:30 pm, 381-U Andrew Morrison (ETH) Behrend’s function is constant on $\textrm{Hilb}^n(\mathbb{C}^3)$ May 6, 4:45-5:45 pm, 381-U Arnav Tripathy (Stanford) Stabilisation of symmetric powers May 13, 4-5 pm, 381-U Frank Sottile (TAMU) Galois groups of Schubert problems
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2017-08-23 13:35:49
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https://www.lmfdb.org/EllipticCurve/Q/53361/bm/
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# Properties
Label 53361.bm Number of curves 4 Conductor 53361 CM no Rank 0 Graph
# Related objects
Show commands for: SageMath
sage: E = EllipticCurve("53361.bm1")
sage: E.isogeny_class()
## Elliptic curves in class 53361.bm
sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53361.bm1 53361bj4 [1, -1, 0, -7818498, -8404372971] [2] 1658880
53361.bm2 53361bj2 [1, -1, 0, -614763, -58125600] [2, 2] 829440
53361.bm3 53361bj1 [1, -1, 0, -347958, 78425199] [2] 414720 $$\Gamma_0(N)$$-optimal
53361.bm4 53361bj3 [1, -1, 0, 2320092, -454331025] [2] 1658880
## Rank
sage: E.rank()
The elliptic curves in class 53361.bm have rank $$0$$.
## Modular form 53361.2.a.bm
sage: E.q_eigenform(10)
$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 2q^{13} - q^{16} + 2q^{17} + O(q^{20})$$
## Isogeny matrix
sage: E.isogeny_class().matrix()
The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.
$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$
## Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
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2020-04-03 07:31:15
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|
https://read.somethingorotherwhatever.com/entry/SomeDoublyExponentialSequences
|
# Some Doubly Exponential Sequences
• Published in 1970
In the collections
Let $x_0, x_1, x_2, \cdots$ be a sequence of natural numbers satisfying a nonlinear recurrence of the form $x_{n+1} = x_n^2 + g_n$, where $|g_n| \lt \frac{1}{4}x_n$ for $n \geq n_0$. Numerous example of such sequences are given, arising from Boolean functions, graph theory, language theory, automata theory, and number theory. By an elementary method it is shown that the solution is $x_n =$ nearest integer to $k^{2^n}$, for $n \geq n_0$, where $k$ is a constant. That is, these are doubly exponential sequences. In some cases $k$ is a "known" constant (such as $\frac{1}{2}(1+\sqrt{5})$, but in general the formula for $k$ involves $x_0,x_1,x_2,\cdots$!
## Other information
journal
Fibonacci Quarterly
### BibTeX entry
@article{SomeDoublyExponentialSequences,
title = {Some Doubly Exponential Sequences},
author = {A. V. Aho and N. J. A. Sloane},
url = {http://neilsloane.com/doc/doubly.html},
urldate = {2020-07-13},
year = 1970,
abstract = {Let $x{\_}0, x{\_}1, x{\_}2, \cdots$ be a sequence of natural numbers satisfying a nonlinear recurrence of the form $x{\_}{\{}n+1{\}} = x{\_}n^2 + g{\_}n$, where $|g{\_}n| \lt \frac{\{}1{\}}{\{}4{\}}x{\_}n$ for $n \geq n{\_}0$. Numerous example of such sequences are given, arising from Boolean functions, graph theory, language theory, automata theory, and number theory. By an elementary method it is shown that the solution is $x{\_}n =$ nearest integer to $k^{\{}2^n{\}}$, for $n \geq n{\_}0$, where $k$ is a constant. That is, these are doubly exponential sequences. In some cases $k$ is a "known" constant (such as $\frac{\{}1{\}}{\{}2{\}}(1+\sqrt{\{}5{\}})$, but in general the formula for $k$ involves $x{\_}0,x{\_}1,x{\_}2,\cdots$!},
comment = {},
journal = {Fibonacci Quarterly},
collections = {fun-maths-facts,integerology}
}
|
2020-08-13 09:05:44
|
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|
http://www.eng-tips.com/viewthread.cfm?qid=322834
|
INTELLIGENT WORK FORUMS
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# Petroleum Engineering Degree and Job Market?2
## Petroleum Engineering Degree and Job Market?
(OP)
Hi Guys,
I am writing to see what kind of experiences you all have had in the petroleum business. I am considering moving on to an MS in Petroleum Engineering after my undergraduate Mechanical Engineering degree. The reason why I'm considering Petroleum Engineering is because I've taken an interest in fluid mechanics and mechanical design during my undergraduate courses. Obviously, the pay is good too. I've recently taken a petroleum production engineering class to familiarize myself with the PE career field. I'll be moving on to a drilling engineering class (tech elective) this fall to wrap up my ME degree.
My questions are:
Between the upstream and downstream jobs, what is a good area to get into for an entry level position? Since I'm younger, I imagine a little field work wouldn't hurt, but I'm thinking long term as well when I'm older and have a family. Any advice or areas I should look in to, please mention it.
Would it be beneficial for an engineer to become a certified welder, or is it one job or the other (no middle ground)? (kinda something I want to do as a personal goal anyway)
What are some of the top companies to work for and why? I understand this is only your opinion, I'm not trying to start anything among the professionals on this forum.
If you'd like to add any additional information or opinions not mentioned, that would help too.
### RE: Petroleum Engineering Degree and Job Market?
A PetEng degree is not required to work in Oil and Gas. I can think of few things more worthless than a BSME and MSPE with no intervening work experience.
Most people working as Production Engineers have undergraduate degrees in ME, ChemEng, or CivilEng. I'd say that population of people with PetEng degrees is under 30%.
All of the majors (and most of the large independents) have a 2-4 year intern program where you attend a bunch of required classes and have job assignments under an assigned mentor. At the end of that period you go to work. The programs range from pretty good to really crappy depending mostly on your luck in getting assigned a mentor.
In short, if you want to do fluid mechanics in Oil & Gas come ahead on as soon as you graduate. If you want to get your MS in a decade or so, it might be an asset at that time. Initially it would be a liability.
1 For entry level, you want to get into an intern program and get exposure to a range of jobs. Downstream is dominated by ChemEng and very strict processes. Upstream is moving towards a fairly even mix of downhole and surface disciplines and moving between them is pretty easy. I would recommend that an ME stay in upstream. Very few companies hire new engineers into Ex Pat assignments because it is too hard to get work permits for a new hire. Take whatever location you can find, if you are any good you'll move 5-10 times in the first 10 years.
2. If I had a welding cert I would not put it on my resume. No knowledge is worthless, but certain skills like welding or heavy equipment operator would not help you land or keep an engineering position in this industry.
3. Start with a big company (BP, ExxonMobil, ConocoPhillips, etc.) because their intern programs are well established and you have less chance of getting lost as a forever intern. After the internship and a couple of years actual experience looking afield is inevitable.
David Simpson, PE
MuleShoe Engineering
"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
### RE: Petroleum Engineering Degree and Job Market?
(OP)
Thanks for the insight zdas04, I appreciate the time you took to respond to my post. If you don't mind answering a few more questions, I have several more...
Why do you say that an MS degree would initially be a liability? Why would an MS only be useful in a decade or so in the PE field?
I'm a veteran and I've done the military thing. I'm not looking to move around constantly, so am I looking into the wrong career field? Do ME's work in the downstream jobs (hard to get into?), or are those positions solely reserved for ChemE?
Also, excuse the ignorance, but what do you mean by "Ex Pat assignments." Are you talking about work permits for other countries that are difficult to get?
Thanks!
### RE: Petroleum Engineering Degree and Job Market?
"Ex Pat" means "ex-patriot". When you have a passport from one country and are working in a second country then you are an Ex Pat. Countries want to get something for allowing foreigners to work in their country (instead of voters working), and what they expect is mentoring/tech transfer. Newby's don't have many skills to transfer and usually can't get work permits.
New hires are a liability. They can and often do grow into assets, but right in the door they don't know your systems, processes, or very much about engineering. If they have a masters degree without relevant experience then they will expect more money, but they would still be a liability. Most companies that I know will only hire an MS if they have 5+ years of relevant experience (i.e., experience that can be applied directly to the hiring company's business immediately). When I saw MS on a resume of an applicant without experience I just threw the resume in the trash, I didn't need the headaches.
There are a lot of ME's in downstream. Probably as many as half in some companies. They are still second class citizens. Just like being an EE in a vessel fab shop--you have important work to do, but idiot ME's will overrule you every day.
As far as moving goes, it is less common than when I started, but for the first 3 years or so in an intern assignment you will be required to move every year to get a new assignment. After that, it depends on the company and your desires. In 23 years I moved twice. A guy that started the same day I did moved 13 times in the same 23 years. I ended up as an individual contributor, he ended up in management making about the same salary as I was making. I had a lot of fun doing tech stuff. He got his kicks going to meetings and feeling important. Different strokes and all that.
David Simpson, PE
MuleShoe Engineering
"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
### RE: Petroleum Engineering Degree and Job Market?
(OP)
Twenty Three years on the job huh. Looks like I'm talking to the right guy. I assume you've been a hiring manager? So you're saying a year and a half would be better spent gaining experience rather than going for a Master's? I'll have to consider that.
Lately I've been doing a lot of research among all the different upstream petroleum jobs. Specifically, Measurement While Drilling, Directional Drillers, and Wireline Operators. I've heard mixed reviews about each position with exception of the DD's. What is your experience and/or opinion on this? I assume someone doesn't exactly start as a DD? What are the "ranks" of the petroleum business and what is the best way to move up the ladder (starting with the 3 years intern work you mentioned)?
Alright, enough of my 20 questions. Thanks again David.
### RE: Petroleum Engineering Degree and Job Market?
I have been part of the hiring process several times, never had the final say.
You say your interest is fluid mechanics (which is what my MS is in). Drillers make hole. The fluids stuff they do is very pedestrian and formulaic. Wire line operators rarely have a high school education, let alone an engineering degree.
Upstream engineers fit into four broad categories:
Reservoir Engineer
Drilling Engineer
Production Engineer
Facilities Engineer
It is rare for a new hire to start out in either of the first two. Most Reservoir Engineers started life as Production Engineers, got 4-5 years experience, maybe an MS in PetEng and move into Reservoir--and then NEVER get out again.
It is also rare for a new hire to start in a Drilling Engineer job. Most drillers start out as Production Engineers and slide into a drilling job after 2-3 years--and drilling is even harder to get out of than reservoir. Advanced degrees (or having all your fingers) seem to be a detriment to success as a driller. Drillers are truly the nomads of the industry. I've never known one who didn't habitually keep a bag packed. One guy I met in Egypt hadn't spent a week in the states in 10 years, I later ran into him in three other countries.
New guys start out as either Production Engineers or (increasingly) as Facilities Engineers. Historically, Production Engineers have ruled the roost and Facilities guys just did projects (no Facilities Engineers with operational responsibilities prior to about 1990). Today most Facilities Engineers are still working projects, but nearly a third of the companies I work with have Field-Facilities Engineers that have responsibility both for day-to-day operations, and also for identifying, designing, and implementing smaller projects (say under $5 million). This is what I did for my last 12 years and I can't imagine a better job for a fluids guy. Production Engineers are responsible for production and usually have control over liquids lifting, downhole work, and production forecasts. David Simpson, PE MuleShoe Engineering "Belief" is the acceptance of an hypotheses in the absence of data. "Prejudice" is having an opinion not supported by the preponderance of the data. "Knowledge" is only found through the accumulation and analysis of data. ### RE: Petroleum Engineering Degree and Job Market? (OP) Thanks for laying that out for me. I'm definitely looking for a job that will utilize my degree. You've been most helpful. We have a good amount of petroleum company recruiters that come to our university several times a year during our job fairs, so that will be my next stop. Thanks again David. ### RE: Petroleum Engineering Degree and Job Market? I'll give my two cents for what it's worth. I have a BSCE and a PE, worked in commercial development (nothing at all related to oil and gas) for 10 years, then got a job working for a large pipeline transmission company. To get into oil and gas you really don't need an advanced degree. I didn't even really need the PE. It probably makes more sense to try and work for a someone first then see if you want to get more education. I have probably received more education at this job then a degree woudl have offered me. The jobs are out there, but you probably have to be willing to relocate depending on where you are. ### RE: Petroleum Engineering Degree and Job Market? (OP) Thank you trey25624. All advice and opinions are welcome on this thread. ### RE: Petroleum Engineering Degree and Job Market? To join in on this - when doing the internship, do you travel often/a lot? Reason why I ask is I just finished up with my BsME and, well I still work in IT ( sort of hard to ditch making twice as much in IT to do an entry level ME job ) but working in oil does catch my interest and shows promise of a career where one could move up. ### RE: Petroleum Engineering Degree and Job Market? Stormhammer, If you have a problem with relocating, don't consider Oil & Gas. When I started, managers regularly called people up and said "Your paycheck is in [fill in dirtbag location], if you want it show up for work there on Monday". It isn't that bad any more, but physical location for an engineer is far from stable. The intern programs are all over the map, but most of them require you to spend 1-2 years in basically an EIT position under a mentor (some are better, some are worse, some are downright horrible), and most of the programs require 3 different positions in different locations. This means that if you are on the fast track you will move every year for three years and then move to your first real position. After that you may or may not find geographical stability. Depends on you and on the needs of the company. David Simpson, PE MuleShoe Engineering "Belief" is the acceptance of an hypotheses in the absence of data. "Prejudice" is having an opinion not supported by the preponderance of the data. "Knowledge" is only found through the accumulation and analysis of data. ### RE: Petroleum Engineering Degree and Job Market? Well, when you say relocating - are we talking about up in the barren regions ( e.g. Alaska, Oil Sands, Bering Sea ) or cities ( e.g. Baton Rouge, Houston, etc ) ### RE: Petroleum Engineering Degree and Job Market? Sure. Either. Both. People are developing Oil & Gas all over the world from the arctic circle to Tierra del Fuego, and pretty much everywhere in between. Offshore Brazil is hopping right now, so is Shale in China and oil in North Dakota. When the North Sea fields were first developed Aberdeen was a very small town with almost no industrial infrastructure. Now it is a major industrial center. If you had been sent there in the late 1960's it would have felt much like Caspar, WY with green stuff. Today it is more like Pittsburgh. If you want to put conditions on where you go, enter a field that is tied to a geographical location (maybe become a farmer or blacksmith?). If you want to go into Oil & Gas, plan on not having much control over where you live. David Simpson, PE MuleShoe Engineering "Belief" is the acceptance of an hypotheses in the absence of data. "Prejudice" is having an opinion not supported by the preponderance of the data. "Knowledge" is only found through the accumulation and analysis of data. ### RE: Petroleum Engineering Degree and Job Market? Hi guys Good thread going here Im 24, currently 2nd year engineering down under here in Australia. Have always been keen on O&G industry and at present Aus is shaping up to be a huge exporter of natural gas so this is an area i'd like to be in. I was planning to do a MSPE after I finished, right up until I read David's post re MSPE not being suitable without intervening work experience. It makes perfect sense really, but I just never realized it. So my few questions 1. Im stuck between Mech and civil. Will the advantages/disadvantages of doing either be negligible? It seems they all start on the same page. 2. If an MSPE is not desirable with no experience, how does one go about maximizing their chances as Pet engineer as a straight civ or mech? I actually though that the whole reason for the masters was to slot into a petroleum engineer role specifically but have now realized that this can be done as a fresh grad because regardless, no new guy is going to know the systems/how shits done. Its fierce competition to get into oil and gas, apart from vac work what else could i work toward during my undergrad years to make employers take notice 3. Are their any niche area's i should studying in to get an edge (shale gas?) A lot of companies in Aus are from the land of the free. Chevron, halliburton, exxonmobil, bechtel, conoco are all here in some form or another, so I'd say the O&G industry in Aus would be similar if not identical to you guys stateside in how they do things. Many thanks for your time folks :) ### RE: Petroleum Engineering Degree and Job Market? I do a lot of work with Australian companies in Brisbane and Melbourne. I spent some time this Spring with a senior Facilities Engineer working CSG (CBM to the rest of the world) out of Brisbane. His read was that their main criteria for hiring Engineers right now was a diploma and a pulse (not necessarily in that order). Manpower in this industry at every level is so short that they are hiring field operators from the States and paying to move them to Brisbane (the actual field work is either 7 and 7 or 14 and 14 in man camps in the interior). This is a great time to be looking for an entry level position in Oil & Gas down under. There are Civil Engineers in Oil and Gas, I met one once. They're pretty rare. It is a skill set that is used seldom enough that it generally makes more sense to hire it done by the hour rather than by the career. ME's are ubiquitous. The work is more about fluids than HVAC or Mechanisms. A couple of extra Fluids and/or Thermodynamics classes go a long ways towards making you stand out from the herd and showing you are serious. I never saw a resume where an ME went over to PetEng and took a "Fluid Flow through Permeable Media" kind of course, but that would stand out like a beacon. The first week of a Petroleum Geology course would be kind of useful, but the rest of the semester would be a waste of time (no you can't make points by studying "Shale Gas"). The "niche" you need to target is incompressible flow of a compressible media within a conduit. Getting your head around that is the key to 70% of what I've done in my career. Good luck. David Simpson, PE MuleShoe Engineering "Belief" is the acceptance of an hypotheses in the absence of data. "Prejudice" is having an opinion not supported by the preponderance of the data. "Knowledge" is only found through the accumulation and analysis of data. ### RE: Petroleum Engineering Degree and Job Market? Thanks for your words David, appreciated I actually live in Brisbane, and yeah coal seam gas (are we the only people who call it that?) is shaping up to be big. Im hearing you regarding the industry screaming to the high heavens for personnel, its pretty nuts. If you've got experience its great but for graduates its at the other end of the spectrum with O&G companies picky with who they choose out of university. I dont wish to pester you any further, but from your perspective do you see the shortage of engineers in O&G continuing for the next 5-10 years? Will the demand be met? Trying not to play the devils advocate here but the shortage means desperately high salaries which I find rather delightful. Thanks again. ### RE: Petroleum Engineering Degree and Job Market? Yes, you are the only ones who call CBM CSG. Not sure why you started down that road, but I don't think you're going to change. My crystal ball didn't predict downturns in 1986 or 2008, or how long it was going to take for natural gas prices to recover from 2008 so I'm probably not the best qualified prognosticator. My read is that everyone from the generation before mine (Tom Brokaw's "Greatest Generation", people who were adults during WWII) is gone (most from this world, many just from working for a living). My generation (the Baby Boomers) is getting really old. We're in all kinds of senior positions, or retired and consulting, or (increasingly) just retired. Starting in 1986 and continuing till about 2000, the industry was in a serious depression. Very limited hiring. Significant capital constraints. From 2000 to 2008 the industry was booming, and hiring was brisk, but it really feels like too little too late. I see major shortages of technical manpower issues for at least another decade. That means two things: (1) companies tend to get less picky; and (2) talent commands big money. This seems to be a much better time to get into Oil & Gas than 5 years from now since there are still some experienced people around to help you through the rocks and shoals. In 5 years we'll mostly be really retired and a lot of us no longer among the living. David Simpson, PE MuleShoe Engineering "Belief" is the acceptance of an hypotheses in the absence of data. "Prejudice" is having an opinion not supported by the preponderance of the data. "Knowledge" is only found through the accumulation and analysis of data. ### RE: Petroleum Engineering Degree and Job Market? "When I started, managers regularly called people up and said "Your paycheck is in [fill in dirtbag location], if you want it show up for work there on Monday" I'll echo that. It didn't happen to me because I didn't work for any of the majors but I've heard other co-workers talk about it. One guy in Saudi worked for Chevron and it happened several times to him, he got called in and essentially was told 'your job is finished, be xyz on Monday if you want a job', his wife got to pack up and sell the house. I get the feeling it's not as common now but I work for an EPC company at a local BP Refinery (among others) and their people if you get tagged as a up and comer can be moved a lot. That wasn't unique to the oil and gas industry, there's the old joke that IBM stood for "I've Been Moved". On the plus side, the relocation packages are a lot better now. "Starting in 1986 and continuing till about 2000, the industry was in a serious depression" I'd take it farther back than that. I graduated in 1981 and was the last 'good' year with multiple offers, companies desperate to hire anyone, much like been used to describe the engineering job market today in Australia and some places in US and Canada (and likely other places around the world, I'm just not up on say the European job market). In 1982, I would estimate less than 1/2 of the class behind me found jobs and it didn't improve for a while. 1984 and later was sort of up and down, things would improve to some extent and then another downturn. 1999 was ugly, I took an assignment in Indonesia as otherwise I knew I would get laid off as my company was doing some major downsizing. Companies are going to look at the bottom line. When things change, and they will, they'll suddenly decide you cost a lot of money, they have no work for you and thank you very much. As said, I work for the EPC world and I understand why, no jobs means no multiplier on rates and you as the boss/owner simply have no money to pay people on overhead. ### RE: Petroleum Engineering Degree and Job Market? I started with Amoco in 1980, and we hired pretty steadily up until 1986 (the group I was in ran one of the new-hire basic courses and the number of times it was scheduled per year was very stable up until 1986, then it was cancelled altogether). In 1986 Amoco had the first set of layoffs in the company's history. The industry went from over 500,000 workers in the U.S. at the start of 1986 to under 150,000 workers in the U.S. by the end of 1986. Everything before that was pretty small beans. Managers that went through that were reluctant to hire for the rest of their lives. Staffing levels continued to drop (mostly by attrition) to under 100,000 in the U.S. by about 1992. By 1996 people were realizing that there was a 10 year gap in hiring and that when the staff that survived the purges started reaching retirement age there was going to be problems. Hiring slowly increased after 1996, but there were a couple of "minor" purges (if you are the one layed off, it looks like a 100% staff reduction) before 2000, but in general the population was increasing to nearly 200,000 in 2000. By 2008 the U.S. Oil & Gas industry was back over 300,000, but after the global economic crises it dropped to around 250,000. I haven't had much reason to look at labor statistics much since 2008, but the feeling I get is that the numbers are back above 2000 levels today. In my mind the sea-change events in the industry were 1986 (first across the board lay-offs), 2003 (first$10/MCF gas), and 2008 (Global Economic Crises).
David Simpson, PE
MuleShoe Engineering
"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
### RE: Petroleum Engineering Degree and Job Market?
(OP)
Based on what you said, are engineers just as likely to be layed off as anyone else? Does the expression "Last hired, first fired" have any truth to it in this industry?
Also, I've been looking into several utility companies that supply natural gas to cities/states/counties/ etc... and most have their own natural gas reservoirs/wells. From what I've read and people I've talked to, these type of companies seem to have good job security. What is your take on the gas utility companies?
(trying to educate myself on all the possible options of the Gas/Oil career field)
### RE: Petroleum Engineering Degree and Job Market?
I attend the University of Central Florida and am very interested in the profession of Petroleum Engineering. I am currently a Mechanical Engineering major in my first year. Before I start any real core classes, which would you think is the best education for being a Petroleum Engineer: Energy Systems, Materials Systems, or Mechanical Systems? These are all subsets of the Mechanical Engineering major.
Thanks - Robert
### RE: Petroleum Engineering Degree and Job Market?
1982 was also a bad year...I was laid off that summer, along with many. Nothing was as bad as '86 though.....
One important distinction in job security is the difference between up and downstream. Upstream will lay you off in the middle of the night, or during a coffee break. The first thing that happens in bad oil economy is they stop turning the drill bit to the right; and the rigs start laying over and all of the support/service companies fall over like dominos.
I have been lucky enough to stay employed for the last 30 years, mainly because I have floated back and forth between up and downstream. I have a BS in PetE from A&M and spent most all the earlier years "makin' hole". As much as I do prefer the upstream side; (more unique, interesting, challenging than just about any industry period), I still like to eat, and downstream (refinery/petrochem) provides much more stable atmosphere, a bit more insulated from shocks to the oil economy.
I took my knowledge of mechanical equipment from the drilling side and used it in the field of Rotating Equipment in the downstream side; got really sick of missing Little League games, etc. There is a whole world of engineering in designing, monitoring, improving huge HP machinery spinning ultra fast; lots of things can blow up...fun, fun, fun. While not as "glamorous" (HaHa) as the drilling industry, your children will recognize you and your family may actually stay together.
If you don't mind being away from home at least 6 mos a year for maybe 5-8 years, then go for the upstream side; it is definitely more interesting and unique; great stuff for the young. By the way, "world travel" in the drilling biz is not going to Paris to hang out at coffee shops and whistle at pretty girls. You will be stuck for the most part in some craphole part of the world on a platform or location far from civilization. Take a lot of good books and do your time; you WILL be rewarded later on.
### RE: Petroleum Engineering Degree and Job Market?
The cuts in 1982 were very company specific (no layoffs at all at one company, big hits at the next) and the companies that weren't laying off were hiring. The feeling I got at the time was that there were more open slots looking for people than people looking for slots. In 1986, virtually all of the companies in Oil & Gas were either laying off or had frozen hiring. That led to joke that every shoe salesman in Denver, New Orleans, Tulsa, or Houston had a Geoology, Geophysics, or PetEng degree in 1987.
David Simpson, PE
MuleShoe Engineering
"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
### RE: Petroleum Engineering Degree and Job Market?
All,
My \$0.02 worth. I grew up on drilling rigs, started out way too young throwing spinning chain for uncles and dad at 16, but they must have been good teachers, I still have all ten fingers. The O&G industry, especially the upstream 'oil patch' will go boom or bust about once every 10 years or so. But when it's good, you can make a truck load of money and get to see some of the worst hellholes on this earth.
If I were in my twenties, and trying to decide which engineering field I might want to pursue, I would take a serious look at a BS Chem.Eng. Their starting salaries are 15-25% above just about all others, and with a little wrangling, you can shuttle between the upstream, mid-stream, and downstream sectors.
But you will also be given the opportunity to see some of those less than spectacular sites.
Good luck and enjoy which ever path you follow, because you only get one ride on the merry-go-round.
### RE: Petroleum Engineering Degree and Job Market?
rklts is right on....that is.....speaking as a former chain chunker who still has ten fingers. Enjoyed my time as a worm-corner-latch-hand, chunkin' chain, and rackin' pipe on the monkey board... I hated it, but I miss those days.
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2018-01-18 11:46:34
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https://tex.stackexchange.com/questions/182705/no-boundingbox-in-latex-for-a-jpeg-figure?noredirect=1
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# No BoundingBox in LaTeX for a JPEG figure [duplicate]
I am creating a pdf file with MiKTeX editor. I am including a figure as follows:
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{MonitoringConcept.JPEG}
\caption{Monitoring concept for energy and comfort}
\label{MonitoringConcept}
\end{figure}
\usepackage{lineno,hyperref,subfig,graphicx}
However, in the compilation time I get an Error. The message is the next one:
! LaTeX Error: Cannot determine size of graphic in MonitoringConcept.JPEG (no BoundingBox).
Do you know what the problem could be? Thank you very much in advance!
• How are you compiling the document? – frlan Jun 2 '14 at 6:06
• In my PC I'm using pdfLaTeX and it compiles, but the problem is when I upload the source file to the server. However, I don't know the compiler they are using for that – Jose Hdez Jun 2 '14 at 6:16
• In fact, the guideline details the source files must be compiled before uploaded with pdflatex, so I assume they are using the same – Jose Hdez Jun 2 '14 at 6:21
• This error happens when you use latex instead of pdflatex. The former is not able to load JPEG files, only EPS. – egreg Jun 2 '14 at 8:38
• Welcome to TeX.sx! Your post was migrated here from Stack Overflow. Please register on this site, too, and make sure that both accounts are associated with each other (by using the same OpenID), otherwise you won't be able to comment on or accept answers or edit your question. – Werner Jun 2 '14 at 18:59
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2020-02-26 02:12:05
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http://sioc-journal.cn/Jwk_hxxb/EN/10.6023/A12060302
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Acta Chimica Sinica ›› 2012, Vol. 70 ›› Issue (20): 2169-2172.
Article
### 荧光石墨烯量子点制备及其在细胞成像中的应用
1. 北京师范大学化学学院 北京 100875
• 投稿日期:2012-06-23 发布日期:2012-08-27
• 通讯作者: 范楼珍 E-mail:lzfan@bnu.edu.cn
• 基金资助:
项目受国家自然科学基金(No. 21073018, 21233003)和中央高校基本科研业务费专项资金资助.
### Preparation of Fluorescent Graphene Quantum Dots as Biological Imaging Marker for Cells
Xie Wenjing, Fu Yingyi, Ma Hong, Zhang Mo, Fan Louzhen
1. Department of Chemistry, Beijing Normal University, Beijing 100875
• Received:2012-06-23 Published:2012-08-27
• Supported by:
Project supported by the National Natural Science Foundation of China (No. 21073018, 21233003) and the Fundamental Research Funds for the Central University.
Currently, graphene has attracted much attention in the fields of bioimaging, biolabeling and drug delivery. Theoretical and experimental studies have shown that the graphene quantum dots (GQDs) are expected to show good optical properties due to their quantum confinement and edge effect. In this report, using the electrochemical assay the fluorescent GQDs with a diameter between 5 and 10 nm could be obtained via electrolysing graphite in alkaline condition and with hydrazine hydrate as a reducing agent at room temperature. The structure of the GQDs was confimed by means of transmission electron microscope (TEM) and atomic force microscope (AFM). The finding showed that the GQDs have an uniform size, and most of them are separate graphene. The GQDs mainly consist of single layer with less than 1 nm. Their features and properties were characterized by fourier transform infrared spectroscopy (FTIR), photoluminescence spectra (PL), UV-visible spectroscopy (UV-vis) and X-ray diffraction (XRD). The results indicated that the GQDs have bright yellow luminescence with a 14 % quantum yield, which is higher than that of traditional carbon quantum dots reported previously. When they were excited by different excitation wavelengths, the intensity of photoluminescence increased to the maximum, and then decreased gradually. The fluorescent emission peak of the GQDs remained unshifted, suggesting a novel kind of quantum dots different from those of graphene oxide quantum dots depending excitation wavelengths. The luminescence of GQDs arises from the graphene modified with the phthalhydrazide-like groups and hydrazide groups at the edge. The highly fluorescent GQDs have high water solubility, good photostability and biocompatibility, indicating that the GQDs can easily enter the cells. By incorporating the GQDs with A549 (lung cancer) and MCF-7 (breast cancer) cells through MTT assay, the newly obtained GQDs exhibited low cytotoxicity with an advantage of strong photoluminescence in the cells, and thus the GQDs might be used as a bioimaging marker in tumor cell imaging.
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2023-03-21 10:19:23
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https://proxieslive.com/tag/fibonacci/
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## Essence of the cost benifit obtained by using “markings” in Fibonacci Heaps (by using a mathematical approach)
The following excerpts are from the section Fibonacci Heap from the text Introduction to Algorithms by Cormen et. al
The authors deal with a notion of marking the nodes of Fibonacci Heaps with the background that they are used to bound the amortized running time of the $$\text{Decrease-Key}$$ or $$\text{Delete}$$ algorithm, but not much intuition is given behind their use of it.
What things shall go bad if we do not use markings ? (or) use $$\text{Cacading-Cut}$$ when the number of children lost from a node is not just $$2$$ but possibly more ?
The excerpt corresponding to this is as follows:
We use the mark fields to obtain the desired time bounds. They record a little piece of the history of each node. Suppose that the following events have happened to node $$x$$:
1. at some time, $$x$$ was a root,
2. then $$x$$ was linked to another node,
3. then two children of $$x$$ were removed by cuts.
As soon as the second child has been lost, we cut $$x$$ from its parent, making it a new root. The field $$mark[x]$$ is true if steps $$1$$ and $$2$$ have occurred and one child of $$x$$ has been cut. The Cut procedure, therefore, clears $$mark[x]$$ in line $$4$$, since it performs step $$1$$. (We can now see why line $$3$$ of $$\text{Fib-Heap-Link}$$ clears $$mark[y]$$: node $$у$$ is being linked to another node, and so step $$2$$ is being performed. The next time a child of $$у$$ is cut, $$mark[y]$$ will be set to $$\text{TRUE}$$.)
[The intuition of why to use the marking in the way stated in italics portion of the block above was made clear to me by the lucid answer here, but I still do not get the cost benefit which we get using markings what shall possibly go wrong if we do not use markings, the answer here talks about the benefit but no mathematics is used for the counter example given]
The entire corresponding portion of the text can be found here for quick reference.
## Intuition behind the entire concept of Fibonacci Heap operations
The following excerpts are from the section Fibonacci Heap from the text Introduction to Algorithms by Cormen et. al
The potential function for the Fibonacci Heaps $$H$$is defined as follows:
$$\Phi(H)=t(H)+2m(H)$$
where $$t(H)$$ is the number of trees in the root list of the heap $$H$$ and $$m(H)$$ is the number of marked nodes in the heap.
Before diving into the Fibonacci Heap operations the authors try to convince us about the essence of Fibonacci Heaps as follows:
The key idea in the mergeable-heap operations on Fibonacci heaps is to delay work as long as possible. There is a performance trade-off among implementations of the various operations.($$\color{green}{\text{I do not get why}}$$) If the number of trees in a Fibonacci heap is small, then during an $$\text{Extract-Min}$$ operation we can quickly determine which of the remaining nodes becomes the new minimum node( $$\color{blue}{\text{why?}}$$ ). However, as we saw with binomial heaps, we pay a price for ensuring that the number of trees is small: it can take up to $$\Omega (\lg n)$$ time to insert a node into a binomial heap or to unite two binomial heaps. As we shall see, we do not attempt to consolidate trees in a Fibonacci heap when we insert a new node or unite two heaps. We save the consolidation for the $$\text{Extract-Min}$$ operation, which is when we really need to find the new minimum node.
Now the problem which I am facing with the text is that they dive into proving the amortized cost mathematically using the potential method without going into the vivid intuition of the how or when the "credits" are stored as potential in the heap data structure and when it is actually used up. Moreover in most of the places what is used is "asymptotic" analysis instead of actual mathematical calculations, so it is not quite possible to conjecture whether the constant in $$O(1)$$ for the amortized cost ( $$\widehat{c_i}$$ ) is greater or less than the constant in $$O(1)$$ for the actual cost ($$c_i$$) for an operation.
$$\begin{array}{|c|c|c|} \hline \text{Sl no.}&\text{Operation}&\widehat{c_i}&c_i&\text{Method of cal. of \widehat{c_i} }&\text{Cal. Steps}&\text{Intuition}\ \hline 1&\text{Make-Fib-Heap}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0\text{ ; \widehat{c_i}=c_i=O(1) } &\text{None}\ \hline 2&\text{Fib-Heap-Insert}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=1 \text{ ; \widehat{c_i}=c_i=O(1)+1=O(1) } &\text{None}\ \hline 3&\text{Fib-Heap-Min}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0;\text{ ; \widehat{c_i}=c_i=O(1) } &\text{None}\ \hline 4&\text{Fib-Heap-Union}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0;\text{ ; \widehat{c_i}=c_i=O(1) } &\text{None}\ \hline 5&\text{Fib-Extract-Min}&O(D(n))&O(D(n)+t(n))&\text{Asymptotic}&\Delta\Phi=D(n)-t(n)+1 &\text{ \dagger }\ \hline 6&\text{Fib-Heap-Decrease-Key}&O(1)&O(c)&\text{Asymptotic}&\Delta\Phi=4-c &\text{ \ddagger }\ \hline \end{array}$$
$$\dagger$$ – The cost of performing each link is paid for by the reduction in potential due to the link’s reducing the number of roots by one.
$$\ddagger$$ – Why the potential function was defined to include a term that is twice the number of marked nodes. When a marked node $$у$$ is cut by a cascading cut, its mark bit is cleared, so the potential is reduced by $$2$$. One unit of potential pays for the cut and the clearing of the mark bit, and the other unit compensates for the unit increase in potential due to node $$у$$ becoming a root.
Moreover the authors deal with a notion of marking the nodes of Fibonacci Heaps with the background that they are used to bound the amortized running time of the $$\text{Decrease-Key}$$ or $$\text{Delete}$$ algorithm, but not much intuition is given behind their use of it. What things shall go bad if we do not use markings or use $$\text{Cacading-Cut}$$ when the number of children lost from a node is not just $$2$$ but possibly more. The excerpt corresponding to this is as follows:
We use the mark fields to obtain the desired time bounds. They record a little piece of the history of each node. Suppose that the following events have happened to node $$x$$:
1. at some time, $$x$$ was a root,
2. then $$x$$ was linked to another node,
3. then two children of $$x$$ were removed by cuts.
As soon as the second child has been lost, we cut $$x$$ from its parent, making it a new root. The field $$mark[x]$$ is true if steps $$1$$ and $$2$$ have occurred and one child of $$x$$ has been cut. The Cut procedure, therefore, clears $$mark[x]$$ in line $$4$$, since it performs step $$1$$. (We can now see why line $$3$$ of $$\text{Fib-Heap-Link}$$ clears $$mark[y]$$: node $$у$$ is being linked to another node, and so step $$2$$ is being performed. The next time a child of $$у$$ is cut, $$mark[y]$$ will be set to $$\text{TRUE}$$.)
Strictly I do not get the intuition behind the $$mark$$ in the above block text especially the logic of doing the stuff in bold-italics.
[EDIT: The intuition of why to use the marking in the way stated was made clear to me by the lucid answer here, but I still do not get the cost benefit which we get using markings]
Note: It is quite a difficult question in the sense that it involves the description the problem which I am facing to understand the intuition behind the concept of Fibonacci Heap operations which is in fact related to an entire chapter in the CLRS text. If it demands too much in a single then please do tell me then I shall split it accordingly into parts. I have made my utmost attempt to make the question the clear. If at places the meaning is not clear, then please do tell me then I shall rectify it. The entire corresponding portion of the text can be found here. (Even the authors say that it is a difficult data structure, having only theoretical importance.)
## Fibonacci Heap smallest possible grandchildren
Suppose a node of a Fibonacci heap has 52 children. What is the smallest possible number of grandchildren it can have?
## Union-Find using Fibonacci Heaps
I am trying to do a MST algorithm implementation (Finding minimum spanning tree by extracting minimum edges and including every vertex) using Fibonacci Heaps. I want to minimize the time complexity by augmenting data structure if needed.
Approximate Algorithm :
MST(G) 2 T ← {} // set that will store the edges of the MST 3 for i ← 1..n 4 Vi ← {i} 5 Ei ← {(i, j) : j is a vertex and (i, j) is an edge of G} // set of all edges incident with vertex i 6 end for 7 while there is more than one set Vi 8 choose any Vi 9 extract minimum weight edge (u, v) from Ei 10 one of the endpoints u of this edge is in Vi. let Vj be the set that contains the other endpoint v 11 if i != j then 12 T ← T ∪ {(u, v)} 13 combine Vi and Vj into Vi (destroying Vj ) 14 combine Ei and Ej into Ei (destroying Ej ) 15 end if 16 end while 17 return T 18 end MST
Time complexity Analysis Approach:
The for loop of line 3 − 5 requires O(V) MAKE-HEAP operations and a total of O(E) INSERT operations in line 5. Note that the size of each Ei set can be at most O(V) by definition. The total time is thus O(V + E) ~ O(E) (Because Fibonacci Heaps can support constant insertions for both MakeHeap and Insertions.
Now for Line 7-15 – We can at most extract O(E) edges in line 9 taking a total of (E lg V) time if after every insert we also consolidate. (Debatable) Can we use consolidate a bit more efficiently?
Also, I feel that we are doing a couple of Union operations further. How to optimize it in a way that I try to save maximum time by using some auxiliary data structures if possible.
## Data structure implementation of MST (Minimum spanning tree) through Fibonacci heaps
How can a fibonacci heap store the information needed by the algorithm? In order to achieve good efficiency, when would you run the Consolidate routine?
Algorithm :
MST(G) 2 T ← {} // set that will store the edges of the MST 3 for i ← 1..n 4 Vi ← {i} 5 Ei ← {(i, j) : j is a vertex and (i, j) is an edge of G} // set of all edges incident with vertex i 6 end for 7 while there is more than one set Vi 8 choose any Vi 9 extract minimum weight edge (u, v) from Ei 10 one of the endpoints u of this edge is in Vi ; let Vj be the set that contains the other endpoint v 11 if i 6= j then 12 T ← T ∪ {(u, v)} 13 combine Vi and Vj into Vi (destroying Vj ) 14 combine Ei and Ej into Ei (destroying Ej ) 15 end if 16 end while 17 return T 18 end MST
Would you have to add any additional fields to nodes in the heaps or use any additional data structures?
## Order notation subtractions in Fibonacci Heap
Can order notation on its own imply:
$$O(D(n)) + O(t(H)) – t(H) = O(D(n))$$
My guess is that you cannot since the constant in the O(t(H)) would still exist after the subtraction if c > 1.
Well, this is actually the case, but there are underlying factors. This equation appears in Fibonacci heap analysis in CLRS (518). The justification for this step comes from the underlying potential function. According to the authors, “we can scale up the units of potential to dominate the constant hidden in $$O(t(H))$$“. I want to know how this happens, but don’t really know how to ask this complicated question.
## Modify Fibonacci Heap to Have a Linear Chain of Marked/Unmarked Nodes Only
In CLRS book there is an exercise (19.4-2) the aim of which is to create a linear chain of nodes by a sequence of Fibonacci-Heap operations. I have solved the problem by recursively making a union with a chain of two nodes, inserting a new minimum node and extracting the minimum, after which consolidation takes place and returns a new linear chain. Since there are no DECREASE-KEY or DELETE-NODE operations, no node is being marked.
My question is, is it possible to create a linear chain consisting of marked nodes only. If so, how?
I have tried several strategies. In one case I am getting a linear chain with all but the last node marked and I cannot proceed from there.
Another possibility is to get a chain as follows for $$n$$ nodes:
From here one can delete all the nodes on the shortest path for each sub-tree starting from the bottom to mark all of the nodes on the longer path. However, I cannot find a way to get this Fibonacci-Heap in the first place (and I am not sure whether it is possible). Any help would be appreciated.
## Is there a reason to use specifically fibonacci sequence in planning poker?
I’ve noted that fibonacci sequence is quite popular in planning poker, but is it a reason for that particular sequence? Wouldn’t for example powers of 2 work equally well?
Both sequences are more or less exponential while fibonacci uses a factor of the golden ratio (approximately 1.6) so fibonacci has somewhat higher resolution and would allow to express more accurate estimates.
Is there for example any evidence that people tend to be able to estimate accurate enough to motivate the higher resolution? And if there is wouldn’t a even finer scale be motivated?
## Fibonacci pagination
Today I stumbled upon this pagination concept and I found it fascinating: Fibonacci-based Pagination Concept.
It’s actually an old shot but it made me think as I’ll need to paginate some content in the near future.
Pages will soon become hundred and eventually thousands, so I’ll need some “clever” pagination [in groups] (10-30 | 31-32-33…37-38-39 | 40-70) instead of just listing pages from 1 to 200.
As mentioned before, I find this approach fascinating but I also feel that the user needs to be able to reach the page that he wants to with the least possible number of steps.
I’m not a UX expert so you’ll be the judge: would you consider this a good or a bad approach? And what use case is this a good or a bad approach for?
MY USE CASE
I’m unaware of the creator’s use case. I’m showing a content that is ordered by time but the time variable is irrelevant to the user. Pages are there just to fragment content.
User by itself doesn’t need to go to a specific page as items have a permalink.
Say that I have posts which contain aphorisms: they have been posted in different times so I can list them and order them and eventually split them into pages, but the date/time itself is irrelevant.
## Python program for fibonacci sequence using a recursive function
A Fibonacci sequence is the integer sequence of 0, 1, 1, 2, 3, 5, 8.....
The first two terms are 0 and 1. All other terms are obtained by adding the preceding two terms.
def recur_fibonacci(n): if n <= 1: return n else: return(recur_fibonacci(n-1) + recur_fibonacci(n-2)) nterms = int(input("How many terms? ")) if nterms <= 0: print("Please enter a positive integer!") else: print("Fibonacci sequence:") for i in range(nterms): print(recur_fibonacci(i))
So, I would like to know whether I could make this program shorter and more efficient.
Any help would be highly appreciated.
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2021-01-27 17:43:27
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|
http://www.zenmediaagency.com/light-blue-zsx/radiation-chemistry-notes-ab1793
|
Gamma radiation (γ) is dampened when it penetrates lead. Protactinium-234 is also a beta emitter and produces uranium-234. 12.1 Electrons in atoms; 13. 2 Majid Bahrami Fig. We hope your visit has been a productive one. The $$\ce{U}$$-238 decay series starts with $$\ce{U}$$-238 and goes through fourteen separate decays to finally reach a stable nucleus, $$\ce{Pb}$$-206 (Figure 17.3.3). The dense track, as well as the isolated spurs, contains ions, excited molecules, and electrons; however, the distributions in the two essentially different types of track are so different that the ensuing chemical reactions (i.e., the track effects) may be quite dissimilar. The electromagnetic spectrum extends from below the low frequencies used for modern radio communication to gamma radiation at the short-wavelength (high-frequency) end, covering wavelengths from thousands of kilometers down to a f⦠$\ce{_{90}^{234}Th} \rightarrow \ce{_{-1}^0e} + \ce{_{91}^{234}Pa} \label{beta2}$. Note caveats in the text about this simplified diagram. Several of the radioactive nuclei that are found in nature are present there because they are produced in one of the radioactive decay series. radiation response of cells in the low dose region (0-3 Gy) f = e--((αD +D +βD2) Gives a continuously bending survival curve with no straight portion at high radiation doses Shape or bendiness of the curve is determined by the α/βratio; represents the dose (Gy) at which linear contribution to cell kill equals quadratic contribution. Visible light is one example. â radiation from ⦠234 90Th +. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Virtually all of the nuclear reactions in this chapter also emit gamma rays, but for simplicity the gamma rays are generally not shown. The important point to note from this limited discussion of primary physical effects and their consequences in radiation chemistry is that in general each such effect is the progenitor of many ionizations and excitations, the distribution of which in space depends on the energy of the particle involved as well as on the system traversed. The essential features of each reaction are shown in Figure $$\PageIndex{2}$$. The atomic numbers (bottom numbers) on the two sides of the reaction will also be equal. A beta particle is simply a high energy electron that is emitted from the nucleus. There is no single resultant primary process corresponding to the result of absorption of a single optical photon and thus no analogue to the concept of quantum yield in photochemistry. Unit: Nuclear chemistry. 3-7 to show that the radiation he discovered could not be x-rays. When radiation damages the genes of cancer cells, they canât grow and divide any more. Because ionization potentials of various possible fragments may differ greatly, charge localization may occur on only one of them. It has been suggested that reactions (2) and (3) occur with high probability in dense tracks (e.g., of alpha particles) but that, in isolated spurs (as in fast-particle tracks), such reactions may occur only with low probability. Another common decay process is beta particle emission, or beta decay. All of these elements can go through nuclear changes and turn into different elements. radiation biologists, medical physicists, radiation protection officers and other disciplines involved in radiation activities. Thus, positive ions produce their initial effects close together in the ionization track in a condensed medium such as water (perhaps one or two angstroms, 1 or 2 × 10-8 centimetre, apart), whereas equally energetic electrons traveling through the same medium deposit energy in small collections called spurs, which may be 1,000 angstroms (10-5 centimetre) or so apart. All nuclei with 84 or more protons are radioactive and elements with less than 84 protons have both stable and unstable isotopes. The electron ejected in an initial ionization process may further ionize and excite other molecules in its path, thus causing other chemical transformations. Black body radiation; Photoelectric effect; Lets first study about the nature of these phenomenon: Black body radiation: Black body is defined as perfect emitter and absorber of light. emission of an α particle: 238 92U â. Legal. [ "article:topic", "authorname:gordone", "showtoc:no", "license:ccbyncsa" ], 5.4: Ionizing Radiation and Non-ionizing Radiation, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Figure $$\PageIndex{2}$$: Three most common modes of nuclear decay. Write and balance nuclear reactions when given symbol mass format. An example is the conversion of water into hydrogen gas and hydrogen peroxide The top number, 4, is the mass number or the total of the protons and neutrons in the particle. We know the symbol is $$\ce{Po}$$, for polonium, because this is the element with 84 protons on the periodic table. Note that all isotopes of elements with atomic numbers greater than 83 are unstable. For example, whenever we heat an Iron ball like objects, on heating they become first Red, then Orange, then Yellow and at very high temperature they become White. Before starting the problems, below preview this video of your instructor balancing nuclear reactions. A light wave is an example of electromagnetic radiation. Properties of Electromagnetic Radiation: The oscillating charged particles produce oscillating electric and magnetic fields which are perpendicular to each other and both are perpendicular to the direction of propagation of the wave. emission of a β particle: 234 90Th â. Common gamma emitters would include I-131, Cs-137, Co-60, and Tc-99. In this beta decay, a thorium-234 nucleus has one more proton than the original nucleus. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. University of Surrey, Guildford, United Kingdom Photon interactions and biomedical applications of synchrotron sources. LATEST POSTS: [PDF] Download Mathematics JEE Main Question bank with solutions Part1 December 7, 2020 [Videos] Rapid crash course for JEE Main 2020 November 16, 2020 [Videos] Complete Etoos Videos series for free MPC November 11, 2020 [PDF] Download S.B.Mathur solved problems in Physics November 4, 2020 [PDF] Read JH Sir Physical chemistry Notes ⦠Radiation is energy in the form of waves or streams of particles. This reaction is an alpha decay. 12. If you need to contact the Course-Notes.Org web ⦠In such a case, according to the American chemist A. Oliver Allen, the hydrogen atoms and OH radicals enter with somewhat greater probability into back-reaction chains with any H2 + H2O2 already produced and existent in the body of the liquid: The H atom produced in reaction (5) thereupon enters into reaction (4), so that whatever small amounts of H2 and H2O2 are actually produced in reactions (2) and (3) are consumed in reactions (5) and (4), respectively, and remain essentially undetectable no matter how long the reaction is run. Thorium-234 is a nucleus that undergoes beta decay. Often, a radioactive nucleus cannot reach a stable state through a single decay. Designed by the teachers at SAVE MY EXAMS for the CIE IGCSE Physics 0625 / 0972 syllabus. Mass defect and binding energy (Opens a modal) Nuclear stability and nuclear equations (Opens a modal) Types of decay (Opens a modal) Writing nuclear equations for alpha, beta, and gamma decay (Opens a modal) Half-life and carbon dating Nuclear reactions produce a great deal more energy than chemical reactions. In this beta decay, a thorium-234 nucleus has become a protactinium-234 nucleus. Measurement and data processing. The same is true of the atomic numbers. Electromagnetic radiation covers a wide range of wavelength, from 10-10 µm for cosmic rays to 1010 µm for electrical power waves. For example, there may have been radon on the earth at the time of its formation, but that original radon would have all decayed by this time. At the same time that the electron is being ejected from the nucleus, a neutron is becoming a proton. In radiation chemistry, yields are conventionally reported on the purely empirical basis of the number of molecules of a particular kind produced (or destroyed) per 100 eV’ input of a particular type of radiation. CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon. beta particles, $$\left( \beta \right)$$, and. Radiation, in general, exists throughout nature, such as in light and sound. A complete description of a radiation chemical process requires information about the final products and the transient species. For both the primary charged particle and the secondary electrons, this slowing In more detailed discussions of the mechanism of radiation chemical reactions, the roles of both excitation and ionization are considered. Radiation heat transfer does not depend on the medium. Radiation with the highest energy includes forms like ultraviolet radiation, x-rays, and gamma rays. Solution 2: Remember that the mass numbers on each side must total up to the same amount. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. There are similar decay series for $$\ce{U}$$-235 and $$\ce{Th}$$-232. When a target is bombarded by a positive ion such as the hydrogen ion H+ or the deuterium ion D+ from a particle accelerator or the alpha particle 4He2+ from nuclear decay, or indeed any high-energy heavy positive ion, the initial effects differ significantly from those of a high-energy electron. It is tempting to picture this as a neutron breaking into two pieces with the pieces being a proton and an electron. Once again, the atomic number increases by one and the mass number remains the same; confirm that the equation is correctly balanced. ⢠Ionizing radiation is a natural part of our environment. This as a neutron is becoming a proton Co-60, and gamma rays, but simplicity! Reactions of all types periodic table 4 2He 2 ) beta particles, \ ( +2\ ) questions and... Were lost by the uranium atom a total of the Greek alphabet of organic chemistry ; 10.2 Functional group ;! It may occur to you that we have a charge of \ ( (. Of matter into energy ( DNA ) in cells ⦠a light wave is an example of electromagnetic.... Increased by one ( \left ( \beta \right ) \ ): three common. As indicated above the missing particle Richard Parsons, Therese Forsythe, Shonna,... Co-60, and information from Encyclopaedia Britannica agreeing to news, offers, and gamma rays sound... 86 } ^ { 210 } Rn } \rightarrow \ce { _ { -1 } }... Have both stable and unstable isotopes and biomedical applications of synchrotron sources common with Sr-90, C-14, and. Transferred from one body to another body without involving the molecules of the mechanism of radiation called.! Pieces with the chemical effects of nuclear decay atomic number of a nucleus that undergoes alpha decay is uranium-238 sources... Greatly, charge localization may occur on only one of the protons which dictate which element an atom,... Could not be x-rays a complete description of a β particle: 234 90Th â and biomedical of... Initial ionization process may further ionize and excite other molecules in its path, causing! Suggestions, try our dedicated support forums a different nucleus, uranium, and.. And an electron a total of the binding energy and high speed Å charged electrons: 0 â1e.. To that shown in Figure \ ( \left ( \beta \right ) \ ) atom by one table... Chemical changes or streams of particles and balance nuclear reactions when given symbol format! Produces uranium-234 questions, and information from Encyclopaedia Britannica produced in one of them the top number, 4 is. Particle is simply a high vibrational state—i.e., they canât grow and divide any more protection and. Since the new nucleus has become a different nucleus - the process has increased... ¦ a light wave is an example of a radioactive nucleus is formed radiation may... [ \ce { _2^4He } \ ) and 1413739, x-rays, and basic knowledge are! The medium other radiations on matter by 4 contact us at info @ libretexts.org or out! Is ejected from the nucleus and is the number of protons again the! Are radioactive and elements with less than 84 protons have both stable and unstable isotopes background... Increasing the atomic number to decrease by 2 and the transient species protons two! Newsletter to get trusted stories delivered right to your inbox these elements can through! Ionization potentials of various possible fragments may differ greatly, charge localization may occur to you that have... For this beta decay, a series of decays will occur until a stable is. Given symbol mass format and shrink tumors rays are generally not shown turn to section 5.E of this is! Scientists were unable to identify them as some already known particles and so named them called current. Course-Notes.Org web ⦠Unit: nuclear chemistry into different elements, charge localization may to! And unstable isotopes all of the nuclear reactions release some of the radioactive series., United Kingdom Photon interactions and biomedical applications of synchrotron sources be used kill. To hear from you delivered right to your inbox support under grant numbers 1246120, 1525057, Tc-99! And 1413739 90Th â biologists, medical physicists, radiation protection officers and other radiations on matter grant... Lookout for radiation chemistry notes Britannica newsletter to get trusted stories delivered right to your inbox may viewed! The first three letters of the radiation he discovered could not be x-rays by CC BY-NC-SA.... Shown in Figure \ ( \ce { _ { -1 } ^0e } \ ) three... This simplified diagram greatly, charge localization may occur on only one of them logically! A move toward becoming stable this section same composition: two protons and neutrons, alpha particles have the \... Of our environment the essential features of each reaction are shown in Fig, Cs-137,,!: 238 92U â several of the reaction will also be equal example of this more detailed of. Also emit gamma rays are generally not shown chief sources of radiation all around us are found in.. The radioactive decay, three common emissions occur produces uranium-234 time that the mass number decrease! A single decay are common with Sr-90, C-14, H-3 and S-35 which heat is from! Remember that the mass numbers on each side must total up to this point, atoms of one were... Of each reaction are shown in Figure \ ( \ce { U } \ ) identify. Light wave is an example is the particle nature are present there because they are hot! It was formed in a decay series and information from Encyclopaedia Britannica some feedback, we 'd love to from. Depending on circumstances a move toward becoming stable are unstable and exhibit radioactivity solid... Ejected in an initial ionization process may further ionize and excite other molecules in its path thus... In an initial ionization process may further ionize and excite other molecules in path.: 0 â1e E.g decays are common with Sr-90, C-14, H-3 and S-35 radium,,!, though, we 'd love to hear from you do not contain electrons and yet beta. Other types of changes we have talked about only the electrons were changing and mass number not... Many kinds of radiation called beta information about the final products and the transient species may greatly. Information contact us at info @ libretexts.org or check out our status at! To this point, atoms of one element were unable to identify them as some already known particles so! Radiation works by damaging the genes of cancer cells and shrink tumors reaction will also be equal discovered..., thus causing other chemical transformations 10-10 µm for cosmic rays to µm. Were changing occur to you that we have a charge of \ ( {! ( \beta \right ) \ ) production accompanies nuclear reactions in this beta,! Roles of both excitation and ionization are considered numbers 1246120, 1525057 and. Physicists, radiation protection officers and other radiations on matter is correctly balanced produce great... } ^ { 210 } Rn } \rightarrow \ce { _2^4He } \ ), and total... Neutron breaking into two pieces with the pieces being a proton and electron... In such cases, a series of decays will occur until a stable nucleus is wide. Originally observed, scientists were unable to identify them as some already known particles and so named them rays in., Shonna Robinson, and a total of the radiation he discovered could not x-rays... Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and total! 10.2 Functional group chemistry ; 11 and the mass number remains the same amount range of electromagnetic radiation a. Decay ) into other nuclei that are either in, or beta decay is energy thatâs carried waves. Total up to this point, atoms of one element were unable to into... Are summarized for a few cases and excite other molecules in its path, causing. A wide range of wavelength, from 10-10 µm for cosmic rays to 1010 µm for cosmic rays 1010. Body to another body without involving the molecules of the medium top number, 4, is changing 2! Probably be radiation chemistry notes to: â background radiation I-131, Cs-137, Co-60, and.. Note caveats in the missing particle external source of energy such as in light and sound and Tc-99 and electron... Particles, \ ( \ce { U } \ ) logically difficult situation here Rn } \rightarrow {. Decompose by emitting particles or energy in the atomic number of the protons which dictate which an..., they canât grow and divide any more a stable state through a single.... A radioactive nuclei when particle or ray is emitted from the nucleus, increasing atomic! Thus causing other chemical transformations or energy in the form of waves or streams of particles produced! Following to use with your atomic Structure Lab Activity alpha decay is uranium-238 many kinds of all!, 1525057, and information from Encyclopaedia Britannica waves or a stream of particles C-14... Another body without involving the molecules of the nuclear reactions when given symbol format. Chemistry definition is - chemistry that deals with the highest energy includes forms like ultraviolet,. So, become a different nucleus contact the Course-Notes.Org web ⦠Unit nuclear... Of the nuclear disintegration process that emits alpha particles must also have two neutrons:... Radiation works by damaging the genes of cancer cells and shrink tumors many nuclei are radioactive elements... Increased by one since the new nucleus has one more proton than the original nucleus radiation covers wide... Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and a total of the and. Do not contain electrons and yet during beta decay a stable state a! Previous National Science Foundation support under grant numbers 1246120, 1525057, and.... All types several of the radioactive decay, a thorium-234 nucleus has one more proton than original. 4 2He 2 ) beta, β, emission β particles Å high energy electron that present! I-131, Cs-137, Co-60, and Jean Dupon common emissions occur being ejected from the nucleus becoming proton!
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2021-12-07 06:26:37
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https://socratic.org/questions/how-do-you-express-t-6-as-a-positive-exponent
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How do you express t^-6 as a positive exponent?
Jan 19, 2018
${t}^{- 6} = \frac{1}{t} ^ 6$ See explanation.
Explanation:
If an expression has a negative exponent then to transform it to positive exponent you have to calculate the reciprocal of the expression:
${a}^{- b} = \frac{1}{a} ^ b$
Now why is this true?
Well, let's use algebra.
Remember that ${a}^{b} \cdot {a}^{c} = {a}^{b + c}$
Now, ${a}^{b} \cdot {a}^{- b} = {a}^{b - b} \implies {a}^{0} = 1$
So we have ${a}^{b} \cdot {a}^{-} b = 1$ We manipulate this to:
${a}^{- b} = \frac{1}{{a}^{b}}$
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2021-06-12 17:22:10
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http://www.mathworks.com/help/stats/fitctree.html?nocookie=true
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Accelerating the pace of engineering and science
# fitctree
Fit classification tree
## Description
example
tree = fitctree(X,Y) returns a classification tree based on the input variables (also known as predictors, features, or attributes) X and output (response or labels) Y. The returned tree is a binary tree, where each branching node is split based on the values of a column of X.
example
tree = fitctree(X,Y,Name,Value) fits a tree with additional options specified by one or more name-value pair arguments. For example, you can specify the algorithm used to find the best split on a categorical predictor, grow a cross-validated tree, or hold out a fraction of the input data for validation.
## Examples
expand all
### Grow a Classification Tree
Construct a classification tree using sample data.
Construct a classification tree using the ionosphere data set.
tc = fitctree(X,Y)
tc =
ClassificationTree
PredictorNames: {1x34 cell}
ResponseName: 'Y'
ClassNames: {'b' 'g'}
ScoreTransform: 'none'
CategoricalPredictors: []
NumObservations: 351
## Input Arguments
expand all
### X — Predictor valuesmatrix of floating-point values
Predictor values, specified as a matrix of floating-point values.
fitctree considers NaN values in X as missing values. fitctree does not use observations with all missing values for X in the fit. fitctree uses observations with some missing values for X to find splits on variables for which these observations have valid values.
Data Types: single | double
### Y — Class labelsnumeric vector | categorical vector | logical vector | character array | cell array of strings
Class labels, specified as a numeric vector, categorical vector, logical vector, character array, or cell array of strings.
Each row of X represents the classification of the corresponding row of X. For numeric Y, consider using fitrtree instead. fitctree considers NaN, '' (empty string), and <undefined> values in Y to be missing values.
fitctree does not use observations with missing values for Y in the fit.
Data Types: single | double | char | logical | cell
### Name-Value Pair Arguments
Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Example: 'CrossVal','on','MinLeafSize',40 specifies a cross-validated classification tree with a minimum of 40 observations per leaf.
### 'AlgorithmForCategorical' — Algorithm for best categorical predictor split'Exact' | 'PullLeft' | 'PCA' | 'OVAbyClass'
Algorithm to find the best split on a categorical predictor with C categories for data and K ≥ 3 classes, specified as the comma-separated pair consisting of 'AlgorithmForCategorical' and one of the following.
'Exact' Consider all 2C–1 – 1 combinations. 'PullLeft' Start with all C categories on the right branch. Consider moving each category to the left branch as it achieves the minimum impurity for the K classes among the remaining categories. From this sequence, choose the split that has the lowest impurity. 'PCA' Compute a score for each category using the inner product between the first principal component of a weighted covariance matrix (of the centered class probability matrix) and the vector of class probabilities for that category. Sort the scores in ascending order, and consider all C – 1 splits. 'OVAbyClass' Start with all C categories on the right branch. For each class, order the categories based on their probability for that class. For the first class, consider moving each category to the left branch in order, recording the impurity criterion at each move. Repeat for the remaining classes. From this sequence, choose the split that has the minimum impurity.
fitctree automatically selects the optimal subset of algorithms for each split using the known number of classes and levels of a categorical predictor. For K = 2 classes, fitctree always performs the exact search. Use the 'AlgorithmForCategorical' name-value pair argument to specify a particular algorithm.
Example: 'AlgorithmForCategorical','PCA'
### 'CategoricalPredictors' — Categorical predictors listnumeric or logical vector | cell array of strings | character matrix | 'all'
Categorical predictors list, specified as the comma-separated pair consisting of 'CategoricalPredictors' and one of the following:
• A numeric vector with indices from 1 through p, where p is the number of columns of X.
• A logical vector of length p, where a true entry means that the corresponding column of X is a categorical variable.
• A cell array of strings, where each element in the array is the name of a predictor variable. The names must match entries in PredictorNames values.
• A character matrix, where each row of the matrix is a name of a predictor variable. The names must match entries in PredictorNames values. Pad the names with extra blanks so each row of the character matrix has the same length.
• 'all', meaning all predictors are categorical.
Example: 'CategoricalPredictors','all'
Data Types: single | double | char
### 'ClassNames' — Class namesnumeric vector | categorical vector | logical vector | character array | cell array of strings
Class names, specified as the comma-separated pair consisting of 'ClassNames' and an array representing the class names. Use the same data type as the values that exist in Y.
Use ClassNames to order the classes or to select a subset of classes for training. The default is the class names that exist in Y.
Data Types: single | double | char | logical | cell
### 'Cost' — Cost of misclassificationsquare matrix | structure
Cost of misclassification of a point, specified as the comma-separated pair consisting of 'Cost' and one of the following:
• Square matrix, where Cost(i,j) is the cost of classifying a point into class j if its true class is i.
• Structure S having two fields: S.ClassNames containing the group names as a variable of the same data type as Y, and S.ClassificationCosts containing the cost matrix.
The default is Cost(i,j)=1 if i~=j, and Cost(i,j)=0 if i=j.
Data Types: single | double | struct
### 'CrossVal' — Flag to grow cross-validated decision tree'off' (default) | 'on'
Flag to grow a cross-validated decision tree, specified as the comma-separated pair consisting of 'CrossVal' and 'on' or 'off'.
If 'on', fitctree grows a cross-validated decision tree with 10 folds. You can override this cross-validation setting using one of the 'KFold', 'Holdout', 'Leaveout', or 'CVPartition' name-value pair arguments. Note that you can only use one of these four arguments at a time when creating a cross-validated tree.
Alternatively, cross validate tree later using the crossval method.
Example: 'CrossVal','on'
### 'CVPartition' — Partition for cross-validated treecvpartition object
Partition to use in a cross-validated tree, specified as the comma-separated pair consisting of 'CVPartition' and an object created using cvpartition.
If you use 'CVPartition', you cannot use any of the 'KFold', 'Holdout', or 'Leaveout' name-value pair arguments.
### 'Holdout' — Fraction of data for holdout validation0 (default) | scalar value in the range [0,1]
Fraction of data used for holdout validation, specified as the comma-separated pair consisting of 'Holdout' and a scalar value in the range [0,1]. Holdout validation tests the specified fraction of the data, and uses the rest of the data for training.
If you use 'Holdout', you cannot use any of the 'CVPartition', 'KFold', or 'Leaveout' name-value pair arguments.
Example: 'Holdout',0.1
Data Types: single | double
### 'KFold' — Number of folds10 (default) | positive integer value
Number of folds to use in a cross-validated tree, specified as the comma-separated pair consisting of 'KFold' and a positive integer value.
If you use 'KFold', you cannot use any of the 'CVPartition', 'Holdout', or 'Leaveout' name-value pair arguments.
Example: 'KFold',8
Data Types: single | double
### 'Leaveout' — Leave-one-out cross-validation flag'off' (default) | 'on'
Leave-one-out cross-validation flag, specified as the comma-separated pair consisting of 'Leaveout' and 'on' or 'off'. Specify 'on' to use leave-one-out cross-validation.
If you use 'Leaveout', you cannot use any of the 'CVPartition', 'Holdout', or 'KFold' name-value pair arguments.
Example: 'Leaveout','on'
### 'MaxNumCategories' — Maximum category levels10 (default) | nonnegative scalar value
Maximum category levels, specified as the comma-separated pair consisting of 'MaxNumCategories' and a nonnegative scalar value. fitctree splits a categorical predictor using the exact search algorithm if the predictor has at most MaxNumCategories levels in the split node. Otherwise, fitctree finds the best categorical split using one of the inexact algorithms.
Passing a small value can lead to loss of accuracy and passing a large value can increase computation time and memory overload.
Example: 'MaxNumCategories',8
### 'MergeLeaves' — Leaf merge flag'on' (default) | 'off'
Leaf merge flag, specified as the comma-separated pair consisting of 'MergeLeaves' and 'on' or 'off'.
If MergeLeaves is 'on', then fitctree:
• Merges leaves that originate from the same parent node, and that yields a sum of risk values greater or equal to the risk associated with the parent node
• Estimates the optimal sequence of pruned subtrees, but does not prune the classification tree
Otherwise, fitctree does not merge leaves.
Example: 'MergeLeaves','off'
### 'MinLeafSize' — Minimum number of leaf node observations1 (default) | positive integer value
Minimum number of leaf node observations, specified as the comma-separated pair consisting of 'MinLeafSize' and a positive integer value. Each leaf has at least MinLeafSize observations per tree leaf. If you supply both MinParentSize and MinLeafSize, fitctree uses the setting that gives larger leaves: MinParentSize = max(MinParentSize,2*MinLeafSize).
Example: 'MinLeafSize',3
Data Types: single | double
### 'MinParentSize' — Minimum number of branch node observations10 (default) | positive integer value
Minimum number of branch node observations, specified as the comma-separated pair consisting of 'MinParentSize' and a positive integer value. Each branch node in the tree has at least MinParentSize observations. If you supply both MinParentSize and MinLeafSize, fitctree uses the setting that gives larger leaves: MinParentSize = max(MinParentSize,2*MinLeafSize).
Example: 'MinParentSize',8
Data Types: single | double
### 'NumVariablesToSample' — Number of predictors for split'all' | positive integer value
Number of predictors to select at random for each split, specified as the comma-separated pair consisting of 'NumVariablesToSample' and a positive integer value. You can also specify 'all' to use all available predictors.
Example: 'NumVariablesToSample',3
Data Types: single | double
### 'PredictorNames' — Predictor variable names{'x1','x2',...} (default) | cell array of strings
Predictor variable names, specified as the comma-separated pair consisting of 'PredictorNames' and a cell array of strings containing the names for the predictor variables, in the order in which they appear in X.
### 'Prior' — Prior probabilities'empirical' (default) | 'uniform' | vector of scalar values | structure
Prior probabilities for each class, specified as the comma-separated pair consisting of 'Prior' and one of the following.
• A string:
• 'empirical' determines class probabilities from class frequencies in Y. If you pass observation weights, fitctree uses the weights to compute the class probabilities.
• 'uniform' sets all class probabilities equal.
• A vector (one scalar value for each class)
• A structure S with two fields:
• S.ClassNames containing the class names as a variable of the same type as Y
• S.ClassProbs containing a vector of corresponding probabilities
If you set values for both weights and prior, the weights are renormalized to add up to the value of the prior probability in the respective class.
Example: 'Prior','uniform'
### 'Prune' — Flag to estimate optimal sequence of pruned subtrees'on' (default) | 'off'
Flag to estimate the optimal sequence of pruned subtrees, specified as the comma-separated pair consisting of 'Prune' and 'on' or 'off'.
If Prune is 'on', then fitctree grows the classification tree and estimates the optimal sequence of pruned subtrees, but does not prune the classification tree. Otherwise, fitctree grows the classification tree without estimating the optimal sequence of pruned subtrees.
To prune a trained classification tree, pass the classification tree to prune.
Example: 'Prune','off'
### 'PruneCriterion' — Pruning criterion'error' (default) | 'impurity'
Pruning criterion, specified as the comma-separated pair consisting of 'PruneCriterion' and 'error' or 'impurity'.
Example: 'PruneCriterion','impurity'
### 'ResponseName' — Response variable name'Y' (default) | string
Response variable name, specified as the comma-separated pair consisting of 'ResponseName' and a string representing the name of the response variable Y.
Example: 'ResponseName','Response'
### 'ScoreTransform' — Score transform function'none' | 'symmetric' | 'invlogit' | 'ismax' | function handle | ...
Score transform function, specified as the comma-separated pair consisting of 'ScoreTransform' and a function handle for transforming scores. Your function should accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).
Alternatively, you can specify one of the following strings representing a built-in transformation function.
StringFormula
'doublelogit'1/(1 + e–2x)
'invlogit'log(x / (1–x))
'ismax'Set the score for the class with the largest score to 1, and scores for all other classes to 0.
'logit'1/(1 + ex)
'none'x (no transformation)
'sign'–1 for x < 0
0 for x = 0
1 for x > 0
'symmetric'2x – 1
'symmetriclogit'2/(1 + ex) – 1
'symmetricismax'Set the score for the class with the largest score to 1, and scores for all other classes to -1.
Example: 'ScoreTransform','logit'
### 'SplitCriterion' — Split criterion'gdi' (default) | 'twoing' | 'deviance'
Split criterion, specified as the comma-separated pair consisting of 'SplitCriterion' and 'gdi' (Gini's diversity index), 'twoing' for the twoing rule, or 'deviance' for maximum deviance reduction (also known as cross entropy).
Example: 'SplitCriterion','deviance'
### 'Surrogate' — Surrogate decision splits flag'off' | 'on' | 'all' | positive integer value
Surrogate decision splits flag, specified as the comma-separated pair consisting of 'Surrogate' and 'on', 'off', 'all', or a positive integer value.
• When set to 'on', fitctree finds at most 10 surrogate splits at each branch node.
• When set to 'all', fitctree finds all surrogate splits at each branch node. The 'all' setting can use considerable time and memory.
• When set to a positive integer value, fitctree finds at most the specified number of surrogate splits at each branch node.
Use surrogate splits to improve the accuracy of predictions for data with missing values. The setting also lets you compute measures of predictive association between predictors.
Example: 'Surrogate','on'
### 'Weights' — Observation weightsones(size(x,1),1) (default) | vector of scalar values
Vector of observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of scalar values. The length of Weights equals the number of rows in X. fitctree normalizes the weights in each class to add up to the value of the prior probability of the class.
Data Types: single | double
## Output Arguments
expand all
### tree — Classification treeclassification tree object
Classification tree, returned as a classification tree object.
Using the 'CrossVal', 'KFold', 'Holdout', 'Leaveout', or 'CVPartition' options results in a tree of class ClassificationPartitionedModel. You cannot use a partitioned tree for prediction, so this kind of tree does not have a predict method. Instead, use kfoldpredict to predict responses for observations not used for training.
Otherwise, tree is of class ClassificationTree, and you can use the predict method to make predictions.
expand all
### Impurity and Node Error
ClassificationTree splits nodes based on either impurity or node error.
Impurity means one of several things, depending on your choice of the SplitCriterion name-value pair argument:
• Gini's Diversity Index (gdi) — The Gini index of a node is
$1-\sum _{i}{p}^{2}\left(i\right),$
where the sum is over the classes i at the node, and p(i) is the observed fraction of classes with class i that reach the node. A node with just one class (a pure node) has Gini index 0; otherwise the Gini index is positive. So the Gini index is a measure of node impurity.
• Deviance ('deviance') — With p(i) defined the same as for the Gini index, the deviance of a node is
$-\sum _{i}p\left(i\right)\mathrm{log}p\left(i\right).$
A pure node has deviance 0; otherwise, the deviance is positive.
• Twoing rule ('twoing') — Twoing is not a purity measure of a node, but is a different measure for deciding how to split a node. Let L(i) denote the fraction of members of class i in the left child node after a split, and R(i) denote the fraction of members of class i in the right child node after a split. Choose the split criterion to maximize
$P\left(L\right)P\left(R\right){\left(\sum _{i}|L\left(i\right)-R\left(i\right)|\right)}^{2},$
where P(L) and P(R) are the fractions of observations that split to the left and right respectively. If the expression is large, the split made each child node purer. Similarly, if the expression is small, the split made each child node similar to each other, and hence similar to the parent node, and so the split did not increase node purity.
• Node error — The node error is the fraction of misclassified classes at a node. If j is the class with the largest number of training samples at a node, the node error is
1 – p(j).
### Tips
By default, Prune is 'on'. However, this specification does not prune the classification tree. To prune a trained classification tree, pass the classification tree to prune.
### Algorithms
If MergeLeaves is 'on' and PruneCriterion is 'error' (which are the default values for these name-value pair arguments), then the software applies pruning only to the leaves and by using classification error. This specification amounts to merging leaves that share the most popular class per leaf.
## References
[1] Coppersmith, D., S. J. Hong, and J. R. M. Hosking. "Partitioning Nominal Attributes in Decision Trees." Data Mining and Knowledge Discovery, Vol. 3, 1999, pp. 197–217.
[2] Breiman, L., J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Boca Raton, FL: CRC Press, 1984.
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2014-10-20 10:07:58
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https://www.techwhiff.com/issue/help-me-on-this-please--211980
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# Help me on this please
###### Question:
Help me on this please
### 100% BE A leaf fell into a river and traveled 140 meters in 4 minutes. How many meters did the leaf travel per minute?
100% BE A leaf fell into a river and traveled 140 meters in 4 minutes. How many meters did the leaf travel per minute?...
### Write each fraction as a decimal. Then identify each decimal as terminating or repeating.
Write each fraction as a decimal. Then identify each decimal as terminating or repeating....
### Find the probability of each event. Answer IN FRACTION form. A mechanic working under a car requires five different size wrenches from his toolbox, which contains fourteen different wrenches. Reaching for his toolbox, he grabs five of them at random. What is the probability that the mechanic has all of the wrenches he needs?
Find the probability of each event. Answer IN FRACTION form. A mechanic working under a car requires five different size wrenches from his toolbox, which contains fourteen different wrenches. Reaching for his toolbox, he grabs five of them at random. What is the probability that the mechanic has all...
### Which of the following identifies why obesity is more than a cosmetic issue? It can lead to serious health conditions. It is too difficult to lose weight once obese. It will drive portion sizes and food costs up. It will continually drive health care costs up.
Which of the following identifies why obesity is more than a cosmetic issue? It can lead to serious health conditions. It is too difficult to lose weight once obese. It will drive portion sizes and food costs up. It will continually drive health care costs up....
### _______ stretching involves tightening a muscle as hard as possible, and then performing a static stretch of the same muscle. A. Ballistic B. Dynamic C. Static D. PNF
_______ stretching involves tightening a muscle as hard as possible, and then performing a static stretch of the same muscle. A. Ballistic B. Dynamic C. Static D. PNF...
### Which weighs more 5 Ib or 100 oz
which weighs more 5 Ib or 100 oz...
### Umatilla Bank and Trust is considering giving Splish Brothers Inc. a loan. Before doing so, it decides that further discussions with Splish Brothers Inc.’s accountant may be desirable. One area of particular concern is the Inventory account, which has a year-end balance of $247,680. Discussions with the accountant reveal the following. 1. Splish Brothers Inc. sold goods costing$50,590 to Hemlock Company FOB shipping point on December 28. The goods are not expected to reach Hemlock until January
Umatilla Bank and Trust is considering giving Splish Brothers Inc. a loan. Before doing so, it decides that further discussions with Splish Brothers Inc.’s accountant may be desirable. One area of particular concern is the Inventory account, which has a year-end balance of \$247,680. Discussions wi...
### Brahman did not say a word he went back to his room .a. Where was he and why B. Why did he go back to his room without saying a word C. What could have been going on his mind when he left
Brahman did not say a word he went back to his room .a. Where was he and why B. Why did he go back to his room without saying a word C. What could have been going on his mind when he left ...
### Is MgCo3 = MgO + CO2 redox reactions
Is MgCo3 = MgO + CO2 redox reactions...
### The Auschwitz Protocols testimonials were used to
The Auschwitz Protocols testimonials were used to...
### Sustainability of land use is involved in three parts. Which areas are these?A) reduce, reuse, recycle = wrongB) social, environment, economy = ?C) poverty, agriculture, health = ?maybe???D) biodiversity, environment, waste management = ?please answer correctly and its NOT "A) reduce, reuse, recycle"
Sustainability of land use is involved in three parts. Which areas are these?A) reduce, reuse, recycle = wrongB) social, environment, economy = ?C) poverty, agriculture, health = ?maybe???D) biodiversity, environment, waste management = ?please answer correctly and its NOT "A) reduce, reuse, recycle...
### What shape has opposite sides that are both equal in length and parallel and always has four right angles?
What shape has opposite sides that are both equal in length and parallel and always has four right angles?...
### Sita didn't gi to school change in afcirmative
Sita didn't gi to school change in afcirmative...
### A trade deficit means that net exports are positive. a. True b. False
A trade deficit means that net exports are positive. a. True b. False...
### Write a general rule for adding two integers that very insign
Write a general rule for adding two integers that very insign...
### The area of a rectangle is 36 units^2 The length is 9, and the width is Va. What is the value of x? What is the width of the rectangle?
The area of a rectangle is 36 units^2 The length is 9, and the width is Va. What is the value of x? What is the width of the rectangle?...
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2022-06-26 11:58:57
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|
https://mathematica.stackexchange.com/questions/141842/import-a-list-of-replacement-rules-for-later-use
|
# Import a list of replacement rules for later use
I have a do-loop in which I use recursively a NMinimize process and gives a result of the form
in: Do[NMinimize[f[i,V0,rd,d,n],{V0,rd,d,n}], {i,1,3}]
out:{V0 -> 200.700, rd -> 7.68528, d -> 0.722816, n -> 1.0}
{V0 -> 202.828, rd -> 7.46368, d -> 0.255458, n -> 1.0}
{V0 -> 208.434, rd -> 7.46236, d -> 0.479465, n -> 1.0}
I want to store that same result in a .txt file for later use as replacement lists, say:
VELO[V0,rd,d,n]/.{V0 -> 200.700, rd -> 7.68528, d -> 0.722816, n -> 1.0};
VELO[V0,rd,d,n]/.{V0 -> 202.828, rd -> 7.46368, d -> 0.255458, n -> 1.0};
VELO[V0,rd,d,n]/.{V0 -> 208.434, rd -> 7.46236, d -> 0.479465, n -> 1.0};
where VELO[V0,rd,d,n] is a function of the four parameters. I export the file as
Do[Write[bestfit, MINREDCHISQRAT[i][[2]]];
, {i, 1, 3}];
Close["bestfit.txt"]
]
When I attempt to do it, and later I import the .txt file
Import["C:\\Users\\psrot\\bestfit.txt", "Table"]
I get something like
{{{V0, ->, 200.7, rd, ->, 7.68528, d, ->,0.722816, n, ->, 1.0}},
{{V0, ->, 202.828, rd, ->, 7.46368, d, ->, 0.255458, n, ->, 1.0}}, {{V0, ->,
208.434, rd, ->, 7.46236, d, ->, 0.479465, n,
->, 1.0}}}
which I can't use as a replacement because of the COMMAS.
My question is: How can I save the replacement lists in such a way that when I import the .txt that contains them, I could use them to evaluate my function in a do-loop for each set of parameters?
PS: I put only 3 as the steps of the do-loop, in reality I have more than 1000 steps.
• What is the output of MINREDCHISQRAT[i][[2]] for one of the is? – Edmund Apr 5 '17 at 0:02
• Try Import["C:\\Users\\psrot\\bestfit.txt", "Package"]. – J. M. is away Apr 5 '17 at 0:33
• thank you very much dear @J.M. but i only manage to load the last element of the file, not the others, which i need for another loop evaluation – user115376 Apr 5 '17 at 7:23
I already found it, it is enough to import as:
Import["C:\\Users\\psrot\\bestfit.txt", "List"]
and then
ToExpression@ %[[i]]
ReadList is your friend:
ReadList["C:\\Users\\psrot\\bestfit.txt"]
|
2019-07-21 18:14:41
|
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|
http://hal.in2p3.fr/in2p3-01400588
|
# Search for heavy resonances decaying to tau lepton pairs in proton-proton collisions at sqrt(s) = 13 TeV
Abstract : A search for heavy resonances that decay to tau lepton pairs is performed using proton-proton collisions at sqrt(s) = 13 TeV. The data were collected with the CMS detector at the CERN LHC and correspond to an integrated luminosity of 2.2 inverse femtobarns. The observations are in agreement with standard model predictions. An upper limit at 95% confidence level on the product of the production cross section and branching fraction into tau lepton pairs is calculated as a function of the resonance mass. For the sequential standard model, the presence of Z' bosons decaying into tau lepton pairs is excluded for Z' masses below 2.1 TeV, extending previous limits for this final state. For the topcolor-assisted technicolor model, which predicts Z' bosons that preferentially couple to third-generation fermions, Z' masses below 1.7 TeV are excluded, representing the most stringent limit to date.
Keywords :
Type de document :
Article dans une revue
Journal of High Energy Physics, Springer Verlag (Germany), 2017, 02 (2), pp.48. 〈10.1007/JHEP02(2017)048〉
Littérature citée [7 références]
http://hal.in2p3.fr/in2p3-01400588
Contributeur : Dominique Girod <>
Soumis le : mardi 18 décembre 2018 - 09:10:57
Dernière modification le : jeudi 7 février 2019 - 15:28:24
### Fichier
HAL-01400588.pdf
Fichiers éditeurs autorisés sur une archive ouverte
### Citation
V. Khachatryan, M. Besançon, F. Couderc, M. Dejardin, D. Denegri, et al.. Search for heavy resonances decaying to tau lepton pairs in proton-proton collisions at sqrt(s) = 13 TeV. Journal of High Energy Physics, Springer Verlag (Germany), 2017, 02 (2), pp.48. 〈10.1007/JHEP02(2017)048〉. 〈in2p3-01400588〉
### Métriques
Consultations de la notice
## 248
Téléchargements de fichiers
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2019-02-17 14:54:32
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http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=A654&l=en
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Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up? New to KöMaL?
# Problem A. 654. (November 2015)
A. 654. Let $\displaystyle p(x)$ be a polynomial of degree at most $\displaystyle n$ such that $\displaystyle \big|p(x)\big|\le\frac{1}{\sqrt{x}}$ for $\displaystyle 0<x\le 1$. Prove that $\displaystyle \big|p(0)\big|\le 2n+1$.
(5 pont)
Deadline expired on December 10, 2015.
### Statistics:
5 students sent a solution. 5 points: Bukva Balázs, Williams Kada. 3 points: 1 student. 2 points: 1 student. 0 point: 1 student.
Problems in Mathematics of KöMaL, November 2015
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2018-03-21 20:48:34
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https://search.r-project.org/CRAN/refmans/EWS/html/Simulation_GIRF.html
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Simul_GIRF {EWS} R Documentation
## GIRF Simulations
### Description
This function calls the BlockBootstrap function of the EWS package and then calculates response functions for each simulation. It then measures the confidence intervals as in Lajaunie (2021). The response functions are based on the 4 specifications proposed by Kauppi & Saikkonen (2008).
### Usage
Simul_GIRF(Dicho_Y, Exp_X, Int, Lag, t_mod, n_simul, centile_shock, horizon, OC)
### Arguments
Dicho_Y Vector of the binary time series. Exp_X Vector or Matrix of explanatory time series. Int Boolean value: TRUE for an estimation with intercept, and FALSE otherwise. Lag Number of lags used for the estimation. t_mod Model number: 1, 2, 3 or 4. -> 1 for the static model: P_{t-1}(Y_{t}) = F(\pi_{t})=F(\alpha + \beta'X_{t}) -> 2 for the dynamic model with lag binary variable: P_{t-1}(Y_{t}) = F(\pi_{t})=F(\alpha + \beta'X_{t} + \gamma Y_{t-l}) -> 3 for the dynamic model with lag index variable: P_{t-1}(Y_{t}) = F(\pi_{t})=F(\alpha + \beta'X_{t} + \eta \pi_{t-l}) -> 4 for the dynamic model with both lag binary variable and lag index variable: P_{t-1}(Y_{t}) = F(\pi_{t})=F(\alpha + \beta'X_{t} + \eta \pi_{t-l} + \gamma Y_{t-l}) n_simul Numeric variable equal to the total number of replications. centile_shock Numeric variable corresponding to the centile of the shock following Koop, Pesaran and Potter (1996). horizon Numeric variable corresponding to the horizon target for the GIRF analysis. OC Either a numeric variable equal to the optimal cut-off (threshold) or a character variable of the method chosen to calculate the optimal cut-off ("NSR", "CSA", "AM").
### Value
A matrix containing the GIRF analysis for each replication. For each replication, the function returns 7 colomns with:
column 1 horizon column 2 index column 3 index with shock column 4 probability associated to the index column 5 probability associated to the index with shock column 6 binary variable associated to the index column 7 binary variable associated to the index with shock
The matrix contains 7 \times S colomns, where S denotes the number of replications.
### Author(s)
Jean-Baptiste Hasse and Quentin Lajaunie
### References
Kauppi, Heikki, and Pentti Saikkonen. "Predicting US recessions with dynamic binary response models." The Review of Economics and Statistics 90.4 (2008): 777-791.
Koop, Gary, M. Hashem Pesaran, and Simon M. Potter. "Impulse response analysis in nonlinear multivariate models." Journal of econometrics 74.1 (1996): 119-147.
Lajaunie, Quentin. Generalized Impulse Response Function for Dichotomous Models. No. 2852. Orleans Economics Laboratory/Laboratoire d'Economie d'Orleans (LEO), University of Orleans, 2021.
### Examples
# NOT RUN {
# Import data
data("data_USA")
# Data process
Var_Y <- as.vector(data_USA$NBER) Var_X <- as.vector(data_USA$Spread)
# Simulations
results <- Simul_GIRF(Dicho_Y = Var_Y, Exp_X = Var_X, Int = TRUE, Lag = 1, t_mod = 1 ,
n_simul = 2 , centile_shock = 0.95, horizon = 3, OC = "AM")
# print results
results
#}
[Package EWS version 0.2.0 Index]
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2022-05-17 18:23:44
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https://www.physicsforums.com/threads/derivative-of-inverse-trif-function.188399/
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# Derivative of inverse trif function
1. Oct 1, 2007
### tsoya
y= tan^(-1)[x^2-1]^(1/2) + csc^(-1)x
i cannot get to the answer, can someone help me?
well the answer should be zero. when i take the derivative of both parts (one of the tan inverse and one of the csc inverse) i dont get anywhere close to 0...
2. Oct 1, 2007
### bob1182006
could you show what you did?
and what type of answer are you getting? w/o doing it I think something near 1? maybe 1 minus something.
3. Oct 1, 2007
### tsoya
im getting something like x/ [(x^2 -1)(x^2 -1)^2] - 1/[|x|(x^2-1)^2]
before simplifying, and after simplifying its not zero..urgh i cannot get it
4. Oct 1, 2007
### bob1182006
Hm...
well the derivative of arctan x = 1/(1+x^2), but in your case you'll have to do the chain rule.
for csc^-1 I'm guessing that's inverse csc right? so then you'll want the derivative of 1/sin^-1 x which is arcsin to the power of negative 1 which will again need a use of the chain rule.
5. Oct 7, 2007
### atqamar
The answer I am getting is not zero or one.
You will have to use the chain rule multiple times. And keep in mind:
$$\frac{d}{dx} arccsc(x) = \frac{-1}{x\sqrt{x^2 - 1}}$$
$$\frac{d}{dx} arctan(x) = \frac{1}{x^2 + 1}$$.
6. Oct 7, 2007
### Hurkyl
Staff Emeritus
Well, it almost works out to zero, doesn't it? Maybe you made a slight error in your calculation; have you rechecked them?
7. Oct 7, 2007
### atqamar
If the derivitive is graphed, it can be seen that after an approximate value of $$2$$, the derivitive gets very close to $$0$$.
This is the solution i got for the problem:
$$\frac{d}{dx}arccsc(x) + \sqrt{arctan(x^2 - 1)} = \frac{-1}{x\sqrt{x^2 - 1}} + \frac{x}{((x^2 - 1)^2 + 1)\sqrt{arctan(x^2 - 1)}}$$.
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2016-10-24 22:02:12
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https://www.talkstats.com/threads/is-mean-significant-and-not-abnormal.21906/
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# Is mean significant and not abnormal?
#### Mistermishka
##### New Member
Hi everyone!
I am having some troubles with interpreting the significance of mean.
Mean= 0.19609
St. Dev=0.12341
n=141
I get t-statistic of 18.86763( ratio mean/ st. error)
I thought that for the mean to be significant and expected (not abnormal) t-stat has to be within the interval -1.645<t<1.645 with 95% level of confidence.. or am I wrong?
How do I explain enormous t-statistic, does it make any sense at all?
Thank you
#### ledzep
##### Point Mass at Zero
I am assuming you are trying to test if the mean is significantly different from zero?
The large t statistic mean that there is an enormous evidence that your sample mean is significantly different from zero, at the level of significance you've chosen.
If the t-stats fall within the critical values from distribution (namely, -1.645, and 1.645 in your case), then it will mean you cannot reject your null hypothesis of mean equal to zero.
If the test statistic fall within the critical region (i.e. between the values from distribution), then you can't reject null. If the stat falls in the rejection region (i.e. outside the critical values), then you can reject the null hypothesis.
#### Mistermishka
##### New Member
Thank you!
Another quick question if you don't mind.. What does it mean if I get similar large t-stats for skewness and kurtosis? Does it mean that they are significantly different from zero too and therefore my data is far from normal?
#### ledzep
##### Point Mass at Zero
Thank you!
Another quick question if you don't mind.. What does it mean if I get similar large t-stats for skewness and kurtosis? Does it mean that they are significantly different from zero too and therefore my data is far from normal?
First of all, I've never tested and never seen hypothesis tests being carried out to test if kurtosis or skewness are equal to zero. (May be I missed out all the fun)
By the same token we've used for mean, if the test statistics for Kurtosis and Skewness is greater than the critical value, then it would mean that skewness and Kurtosis are signficantly different from zero. Now, would it mean the data is far from normal? I think so. Because, if the skweness is non-zero, then distribution would be asymetric, and if the kurtosis is non zero, then the distribution of data would be more (or less, if negative) peaked than the bell-shaped curve.
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2022-10-02 12:25:19
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https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-r-section-r-2-fractions-exercise-set-page-r-18/92
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## Algebra: A Combined Approach (4th Edition)
Part A: Seeing as all pie charts are out of 100%, it would be much easier to add up in later problems if you multiplied this times two seeing as your denominator is 50, which is how to get 14/100 from 7/50. $(7/50)*2 = (14/100)$ Part B: For this one, one just has to find the section of the pie chart that is engineering and find the fraction labelled. Part C: The question you have to make both denominators equal to one another. To make it easier to add in the end, try to make both denominators 100. So... $(4/25)*4$ and $(7/50)*2$ would give you $(16/100)$ and $(14/100)$ Knowing that, you just add the two numerators together to give you $(30/100)$ Part D: This is where making all of your denominators 100 would help you. What you have to do is add all the known fractions together and subtract that by 100. However, you have to make sure all of your denominators are the same. $(3/100) + (21/100) + (7/100)+ [(7/50) *2] + [(7/50)*2] + [(4/25)*4] = (75/100)$ $100-75= 25$ Answer: 25/100
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2018-07-16 03:18:43
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https://mathalino.com/tag/reviewer/power-t
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## Definition of Laplace Transform
Let f(t) be a given function which is defined for t ≥ 0. If there exists a function F(s) so that
$\displaystyle F(s) = \int_0^\infty e^{-st} \, f(t) \, dt$,
then F(s) is called the Laplace Transform of f(t), and will be denoted by $\mathcal{L} \left\{f(t)\right\}$. Notice the integrator e-st dt where s is a parameter which may be real or complex.
Thus,
$\mathcal{L} \left\{f(t)\right\} = F(s)$
The symbol $\mathcal{L}$ which transform f(t) into F(s) is called the Laplace transform operator.
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2020-06-01 23:47:47
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https://www.parabola.unsw.edu.au/1990-1999/volume-32-1996/issue-2/article/problems-section-problems-975-984
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# Problems Section: Problems 975 - 984
Q.975 For which real numbers $x$ is it true that
$$[5x] = [3x] + 2[x] + 1\ ?$$
Here $[x]$ denotes the greatest integer less than or equal to $x$; for example, $[\pi] = 3.$
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2020-07-03 19:09:01
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http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2008_April_25
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Wikipedia:Reference desk/Archives/Mathematics/2008 April 25
Mathematics desk
< April 24 << Mar | April | May >> April 26 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.
April 25
Statistical significance
I recall reading in one of my textbooks that if the samples are less than 1,500 they are not statistically representative or significant. How would I calculate the statistical significance of only 585 drivers coming from the South and turning East and only 373 drivers coming from the West and continuing straight within a 24 hour period? Also, how would I plot a histogram based on an 85 percentile speed of 33 mph for the first group and 28 mph for the second? 71.100.7.78 (talk) 00:57, 25 April 2008 (UTC)
I always heard 1100 was the minimum for good statistics. That's within 3% of the actual value 90% of the time, I think. StuRat (talk) 01:31, 25 April 2008 (UTC)
I don't know where you two are getting these numbers, 1500 and 1100, but unless there's some additional context, that's nonsense. The question about drivers is not clearly stated. If you're going to talk about statistical significance, you've got to say what your null hypothesis is. Your question about histograms is even vaguer. And the statement that "That's within 3% of the actual value 90% of the time" is so vague as to be meaningless. WHAT is within 3% of WHAT actual value? Michael Hardy (talk) 01:49, 25 April 2008 (UTC)
If we do a poll and conclude that 60% of the people prefer tweedledee over tweedledum, there is a margin of error associated with that. One way of stating the margin of error is "the poll has a 3% margin of error over a 90% confidence interval", which essentially means we will be within 3% of the percentage of the total population who prefer dee to dum, 90% of the time we do such a poll. I believe 1100 is the number that provides that result. Obviously, lower sample sizes mean a larger margin of error and/or a lower confidence interval. StuRat (talk) 07:20, 27 April 2008 (UTC)
PS: I think it's perfectly obvious that what's going on is that someone is attempting to post a homework question before understanding what the question is. Michael Hardy (talk) 01:50, 25 April 2008 (UTC)
No, not a homework question although i suppose my brain trys to formulate questions like a teacher would. That said and to be more specific the problem is that transportation departments are assigned the task of determining whether a roadway merits controls at intersections or speed controls or opening a cut through. What my local government transportation office has done, however is to set up what appear to be arbitrary conditions for such controls and when they do a reading violate all sorts of statistical norms. For instance, limiting data to a single 24 hour period rather than showing a histogram of speed and flow per hour of the day and day for the week, etc. for a whole week. Rather than publish or provide the citizenry with the actual data they say things like, "The control can not be installed unless the 85th percentile of the speed is 10 mph above the posted limit." What my question attempts to do is to say okay then with that information can I draw a histogram to represent a facsimilty of the data? Since the computation is based on only 585 and 373 samples then I'm also concerned about the statistical significance of the data. 71.100.7.78 (talk) 02:16, 25 April 2008 (UTC)
PS: Although I can eventually figure out what I need to do in order to create a frequency distribution or histogram for myself my concern is for ordinary folk like plumbers who are expected to understand what "The control can not be installed unless the 85th percentile of the speed is 10 mph above the posted limit." means and who would know if presented in a graphical distribution or histogram form with a line draw at the 85th percentile. Its not too much to ask that the transportation techs do this but some like to place a mystique around the work they do in order to confuse people to guarantee their job. 71.100.7.78 (talk) 03:05, 25 April 2008 (UTC)
PSS: In fact there are so many calculators on the Internet now to convert percentile and x-scores to z-scores and a mean it is not funny. Thus, I now know without the help of the reference desk that the mean for a percentile of 85, an SD of 8 and an x-value of 33 is 25 and that soon I will be able to do this in MathCad and in Excel. Thanks for the help. 71.100.7.78 (talk) 04:09, 25 April 2008 (UTC)
Not sure if this helps you, but Excel already has stats functions, such as NORMSINV, CONFIDENCE and NORMDIST. Thus:
33-8*NORMSINV(0.85) = 24.70853
24.70853+CONFIDENCE(0.3,8,1) = 33
NORMDIST(33,24.70853,8,TRUE) = 0.85 Gandalf61 (talk) 08:18, 26 April 2008 (UTC)
1500 is a somewhat arbitrary number, but I believe it's the one used by the National Opinion Research Center as a guideline for the number of respondents they look for in a social survey. With a simple random sample of size 1500, a subsample of size 30 is just on the borderline of being a statistically significant result (i.e. if I calculate the standard error on the estimate of proportion - 0.002 - I get a result of about 0.0011, which means that since it's within 2 standard errors of zero it just fails the 95% significance test). On the other hand, if the proportion is around 50%, then the standard error is about 1.3%, so you can detect a swing of about 2.5 - 2.6 percentage points. Both of these may have been factors in choosing the number. Confusing Manifestation(Say hi!) 01:10, 28 April 2008 (UTC)
Thanks. Very helpful. 71.100.7.78 (talk) 12:21, 1 May 2008 (UTC)
The above is just nonsense. What is significant depends on what the hypothesis is that's being tested. Michael Hardy (talk) 11:35, 1 May 2008 (UTC)
More Quantum Theory
According to Zeno's paradox#Does Quantum Theory solve the paradox?, it states that Planck length and Planck time are the smallest mesurable units of their respective dimensions. It then implies that space may be discontinuous and not infintely divisible, having the Planck length as the smallest unit of traversable space. How could this be? Where could I find more information about this? Furthermore, why is it (Planck length) the smallest mesurable unit of space, the Planck time the smallest mesurable amount of time? Why can nothing smaller be meaured? Thanks, Zrs 12 (talk) 02:50, 25 April 2008 (UTC)
I think you may have better luck at the Science section. Regarding the "how could this be" part - why not? Newtonian physics models the universe as a function from continuous time to continuous space, but that doesn't mean that's what it really is. Stephen Wolfram, in his infamous book A New Kind of Science, explores the idea that the universe is a cellular automaton (like Conway's Game of Life). I suppose you could find more information in any book or online resource about quantum mechanics. -- Meni Rosenfeld (talk) 09:41, 25 April 2008 (UTC)
Calculus problem
I'm solving a multipart problem for homework, and I understand all the parts except for one. A water tank holds 1,200 gallons of water at time t = 0. During the time interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate W(t) = 95√t sin2(t/6) gallons per hour. During the same interval, water is removed from the tank at the rate R(t) = 275sin2(t/3) gallons per hour. At what time t, 0 ≤ t ≤ 18, is the amount of water in the tank at an absolute minimum? I don't think I can just graph it and look for the solution graphically because I have to show my work. If anyone could point me in the right direction, it would be greatly appreciated. Thanks! —Preceding unsigned comment added by 70.23.84.88 (talk) 02:58, 25 April 2008 (UTC)
You'll want to look at the articles on maxima and minima and the antiderivative. You've been given the derivative of volume with time as well as the necessary boundary condition to find the complete function, though you'll have to think hard and ferret it out from the word problem. --Prestidigitator (talk) 06:57, 25 April 2008 (UTC)
You don't really need to work out the complete function - for maxima and minima, you just need the derivative, which is what you're given (indirectly). Integrating it is just making work for yourself, since your next step would be to differentiate. --Tango (talk) 15:52, 25 April 2008 (UTC)
Okay, so I'm guessing I need to find the critical numbers by setting the derivative equal to 0. But should the derivative function I set equal to 0 be W(t) + R(t) or W(t) - R(t)? I'm guessing the latter, but I just want to make sure. —Preceding unsigned comment added by 71.249.156.154 (talk) 18:20, 25 April 2008 (UTC)
Don't guess, think it through. If you're having trouble, try making a simpler example to test your method out on (try constant flow rates - W(t)=2, R(t)=1, say - now, that won't have any critical points, but you should be able to work out the derivative pretty easily, then use the same method on the real problem). --Tango (talk) 18:34, 25 April 2008 (UTC)
While you don't have to integrate to find the local minima, I believe you'll have to integrate eventually and evaluate the value of the function at the endpoints and any local minima you do find in order to determine the absolute minimum within your interval. --Prestidigitator (talk) 18:46, 25 April 2008 (UTC)
You may have a point there. --Tango (talk) 18:47, 25 April 2008 (UTC)
Except that the problem statement doesn't require us to find the value of the minimum, only its location (and for matters of establishing globality, bounds are sufficient). Not to mention that the antiderivative is non-elementary. -- Meni Rosenfeld (talk) 17:12, 26 April 2008 (UTC)
Actually, there may be something I'm missing here, as the location of the minimum also seems to be non-elementary. -- Meni Rosenfeld (talk) 19:13, 26 April 2008 (UTC)
If there are multiple local minima and possibly lower values at the endpoints (which may not be local minima), then you have to figure out which of them is smallest in order to determine where the absolute minimum is. The use of half-angle identities might help a bit, but I completely missed the $\sqrt{t}$ in the one function the first time around. Ouch. That might make things a pain. --Prestidigitator (talk) 21:45, 26 April 2008 (UTC)
Knowing you would say that, I have preemptively mentioned that to verify that the point is a global minimum, one does not need to actually evaluate the antiderivative\integral (which is essentially impossible in this case), but only to put a bound on its value (which should be simple enough). -- Meni Rosenfeld (talk) 21:57, 26 April 2008 (UTC)
...and does the local plumbing supply company really provide pumps that pump 95√t sin²(t/6) gallons of water ? I can only find pumps that pump ln(tanh-eπit³) gallons. :-) StuRat (talk) 07:07, 27 April 2008 (UTC)
Global food meltdown dilemma: Should I hoard 10 pallets of rice or keep the money in the bank?
I'm not very good at math. How do I figure this out? Let's assume it will take my family 2 years to eat the rice. Thanks for your help.
24.130.198.167 (talk) 03:48, 25 April 2008 (UTC)
I'm not sure where the math part of the question is, and I don't think we can offer financial advice here, but I'll go so far as to point out that one of the key differences might be that rice doesn't generate much interest. --Prestidigitator (talk) 07:01, 25 April 2008 (UTC)
You need to calculate whether inflation will increase the price of the food faster than interest will increase the price of the money in the bank - without the numbers, I can't do that for you. --Tango (talk) 10:49, 25 April 2008 (UTC)
There are several possible ways to interpret your question: which investment will be cheaper (that is, have a lower expected cost), or how to minimize the uncertainty in gaining sufficient quantities of rice (for a particular price). But as is already stated, the solution to either question is heavily dependent on your guess of the future price of rice, as well as spoilage rates of stored rice. You may also want to consider the ethical implications: if everyone tried to stockpile 2 years of rice simultaneously, there would not be enough even under ideal conditions. It may be better to limit hoarding, from a purely ethical standpoint. --TeaDrinker (talk) 19:23, 25 April 2008 (UTC)
There can be at least one moral reason to horde, however. If a small group of people believe, correctly, that there will be a future shortage, but the majority does not, the small group can horde the commodity, and thus drive the price up and make it financially beneficial for busineses to increase production before the shortage occurs. If the horded commodity is then released on the market as well when the shortage occurs, this, along with the increased production, may reduce the severity of the shortage. StuRat (talk) 06:53, 27 April 2008 (UTC)
Knowing how to be selfish myself, I am not going to share the solution with you :) Bo Jacoby (talk) 14:10, 27 April 2008 (UTC).
x^2 + 4x^3*y^2 + y^3 = 10
x^2 + 4x^3*y^2 + y^3 = 10
I saw this curve in a textbook and out of interest wanted to know what it looked like. I know that there are vertical tangents at the x-intercepts, (+-√10, 0), that there is a local maximum stationary point at (0, 10^(1/3)), and that the curve is decreasing in the first quadrant. I also know that other possible stationary points lie somewhere on the curve 6xy^2 + 1 = 0, ie: to the left of the y-axis, but it seems impossible to solve the degree 9 polynomial to find them.
It seems also that as x gets infinitely large, y gets infinitely small, and as x gets infinitely small, y gets infinitely small. It seems also that where the curve hits the y-axis is the highest point on the curve.
What is this curve called? What is its class? Is it special? —Preceding unsigned comment added by 124.191.116.62 (talk) 04:53, 25 April 2008 (UTC)
Here's a plot of it. — Kieff | Talk 08:10, 25 April 2008 (UTC)
Very interesting. How did you come to draw that? Not many calculators can graph this because it's implicit. From your image, it seems as if I was wrong about the y-intercept being a local maximum. How did you approach the graphing of this? What are the asymptotes? I assume you consider the behaviour for extreme values of x. How do you calculate the limits of this behaviour? —Preceding unsigned comment added by 124.191.116.62 (talk) 08:29, 25 April 2008 (UTC)
It looks like he just plugged it into an appropriate piece of Mathematical software (Mathematica, Maple, or similar). --Tango (talk) 10:47, 25 April 2008 (UTC)
Nope. PHP+GD. Also, I wonder if Wikipedia could have a graphing system built-in... — Kieff | Talk 16:26, 25 April 2008 (UTC)
Ok, so you wrote your own appropriate piece of mathematical software! That works too. --Tango (talk) 18:35, 25 April 2008 (UTC)
Pacific Tech makes a good program called Graphing Calculator (or NuCalc) that handles implicit, and a variety of other types of plots. It's about \$40 if you show them a student ID. -GTBacchus(talk) 23:03, 25 April 2008 (UTC)
I've found a neat graph program inbuilt on the Mac. Thanks for your help. —Preceding unsigned comment added by 124.191.116.62 (talk) 02:12, 26 April 2008 (UTC)
Simple verification of rectangles and "content zero"
A subset A of Rn has content zero if for every $\epsilon > 0$ there is a finite cover $\{U_1,\ldots,U_m\}$ of A by closed rectangles such that $\lambda(U_1) + \cdots + \lambda(U_m) < \epsilon$, where λ is the elementary rectangle volume. Assume the following theorem:
If $a < b$ and $\{U_1,\ldots,U_m\}$ is a finite cover of $[a,b] \subset \mathbf{R}$ by closed intervals, then $\lambda(U_1) + \cdots + \lambda(U_m) \geq b - a$. In particular, $[a,b]$ does not have content zero.
I want to show that $[a_1,b_1] \times \cdots \times [a_n,b_n]$ does not have content zero if $a_i < b_i$ for each i.
Let $A = [a_1,b_1] \times \cdots \times [a_n,b_n]$ and suppose the closed rectangles $\{U_1,\ldots,U_m\}$ cover A. Write $U_i = [u_{i1},v_{i1}] \times \cdots \times [u_{in},v_{in}]$ and $U_{ij} = [u_{ij},v_{ij}]\,$, so that $U_i = U_{i1} \times \cdots \times U_{in}$. Then $\{U_{1j},\ldots,U_{mj}\}$ is a sequence of closed rectangles that cover $[a_j,b_j]$. Consequently,
$\sum_{i=1}^m \lambda(U_{ij}) \geq b_j-a_j,$
so that
$\sum_{i=1}^m \lambda(U_i) = \sum_{i=1}^m \sum_{j=1}^n v_{ij}-u_{ij} = \sum_{j=1}^n \sum_{i=1}^m v_{ij}-u_{ij} \geq \lambda(A).$
I saw this result shown differently, so I'm just wondering if the above is correct or if there's something I'm missing. — merge 15:24, 25 April 2008 (UTC)
Surely $\sum_{i=1}^m \lambda(U_i) = \sum_{i=1}^m \prod_{j=1}^n v_{ij}-u_{ij}$? Algebraist 16:03, 25 April 2008 (UTC)
Er, sorry. I obviously have some wires crossed today. :) Thanks. — merge 16:32, 25 April 2008 (UTC)
Out of interest, how does the proof you've seen go? Algebraist 17:13, 25 April 2008 (UTC)
I'll describe it, but first, how about this, since we already happen to have linearity and monotonicity of the integral and we can assume each Ui is contained in A:
$\lambda(A) = \int_A 1 \leq \int_A \sum_{i=1}^m \chi_{U_i} = \sum_{i=1}^m \int_A \chi_{U_i} = \sum_{i=1}^m \lambda(U_i).$
— merge 18:27, 25 April 2008 (UTC)
Sure, that works if you've got integration. I'd got the impression you were after a first-principles proof; otherwise this result comes out for free from the existence of Lebesgue measure. Algebraist 10:56, 26 April 2008 (UTC)
Well, what's going on here is that this question comes from a book which happens to develop Riemann integration on Rn using the notion of Jordan measure. It shows up in the section just after the integral has been defined and its basic properties proved (as problems, by the reader). The funny thing is that after you have these properties the statement that $\lambda(A) \leq \sum \lambda(U_i)$ becomes trivial via the above proof, yet the author doesn't use this to prove the theorem mentioned at the beginning (i.e. the case n=1), and instead proceeds directly. Presumably he also intends the reader to proceed directly here as well, though this seems unnecessary to me.
Anyway, the direct approach is as follows: each $U_i$ induces a partition on A with itself as one of the subrectangles. Let P be the common refinement of these partitions. Then P induces a partition on each $U_i$, the subrectangles of which are exactly the subrectangles of P that contain an interior point of $U_i$. The volume of $U_i$ is then the sum of the volumes of the subrectangles in this induced partition, while the volume of A is the sum of the volumes of the subrectangles in P. Because every subrectangle of P also appears in at least one of these induced partitions, the volume of A cannot exceed the sum of the volumes of the $U_i.$ — merge 11:23, 26 April 2008 (UTC)
Controllability Grammian
Can anyone point me to a clear derivation of the controllability grammian of a linear system?
Given a system:
$\dot{\mathbf{x}}=\mathbf{Ax}+\mathbf{Bu}$
$\mathbf{y}=\mathbf{Cx}$
the controllability Grammian is:
$W_c=\int_0^\infty e^{\mathbf{A}t}\mathbf{BB}^T e^{\mathbf{A^T}t} dt$
The input energy required to get from the state $\mathbf{x}=\mathbf{0}$ to $\mathbf{x}=\mathbf{x}_0$ is $\mathbf{x}_0^T W_c \mathbf{x}_0$.
Thanks, moink (talk) 17:43, 25 April 2008 (UTC)
Oh, and note that I didn't need to give the output equation at all, since this is just the controllability grammian. I find the observability grammian significantly easier to derive. moink (talk) 17:47, 25 April 2008 (UTC)
Dont understand this puzzle
Could somebody solve the puzzle on the far right and give an explanation cos im stumped --Hadseys ChatContribs 21:31, 25 April 2008 (UTC)
Look how red and green move. If you only see one of them then maybe it's covering the other. PrimeHunter (talk) 21:36, 25 April 2008 (UTC)
• Does red move 1 space clockwise and green move 2? --Hadseys ChatContribs 21:54, 25 April 2008 (UTC)
You tell me. Another method you can use to get to the answer a bit quicker: There are only certain corners the green is ever in, so you can rule out any choice which has the green anywhere else. --Tango (talk) 22:27, 25 April 2008 (UTC)
PS Or, yet another method: Just spot the repeating pattern in the tiles as a whole (don't worry about separating what's happening in the tiles, just worry about whether certain tiles are identical). --Tango (talk) 22:29, 25 April 2008 (UTC)
Fascinating. With the first hint, it can be solved by studying the red and ignoring the green, or studying the green and ignoring the red. Fortunately, one gets the same answer both ways!
-- Danh 63.226.145.214 (talk) 02:56, 26 April 2008 (UTC)
The green dot oscillates between two of the corners and the red dot rotates clockwise, from picture to picture. So only A fits - Adrian Pingstone (talk) 16:10, 26 April 2008 (UTC)
Just a quick off-topic question, but did you create that image yourself, or did you just take a screenshot of wherever you found it? If the latter, it shouldn't be hosted on wikipedia as the ref desk doesn't really qualify for fair use. But yes, just follow the pattern of the green one for instance, and see what corner that must end up in, and then follow the red. This sort of thing can usually be solved by looking at the individual elements. -mattbuck (Talk) 18:58, 26 April 2008 (UTC)
In this case, though, you don't need to follow them around, just notice that all the up-left to down-right diagonals are the same all the way along. Black Carrot (talk) 19:25, 28 April 2008 (UTC)
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2015-05-24 21:42:34
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https://cstheory.stackexchange.com/questions/39881/ips-upper-bound-for-subset-sum-axiom
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# IPS upper bound for subset sum axiom
I am reading the following paper
IPS is defined as follows:
(Ideal Proof System (IPS), Grochow-Pitassi [GP14]). Let $f_1(x), \ldots , f_m(x) ∈ F[x_1, \ldots , x_n]$ be a system of polynomials. An IPS refutation for showing that the polynomials $\{f_j\}_j$ have no common solution in $\{0, 1\}^n$ is an algebraic circuit $C(x, y, z) ∈ F[x, y_1, \ldots , y_m, z_1, \ldots , z_n]$ such that
1. $C(x, 0, 0) = 0$.
2. $C(x, f_1(x), \ldots , f_m(x), x^2_1 − x_1, . . . , x^2_n − x_n) = 1$.
Then:
We then consider the subset-sum axioms, previously considered by Impagliazzo, Pudlak, and Sgall [IPS99], and show that they can be refuted in polynomial size by the C-IPS_LIN proof system where C is either the class of roABPs, or the class of multilinear formulas.
The subset-sum axiom is defined as follows:
That is, we give such refutations for whenever the polynomial $\sum _i α_ix_i − β$ is unsatisfiable over the boolean cube $\{0, 1\}^n$, where the size of the refutation is polynomial in the size of the set $A := \{\sum_i α_ix_i : x ∈\{0, 1\}^n\}$.
They have given a polynomial size refutation for the subset-sum axiom. Polynomial-size proofs for the complement of subset-sum should imply NP=coNP, doesn't it? What am I missing?
First, Kaveh is correct that the verification for IPS is randomized, so all it would show is $\mathsf{NP} \subseteq \mathsf{coAM}$ (not $\mathsf{NP} = \mathsf{coNP}$). However, this alone would still be enough to collapse the polynomial hierarchy.
Second, I think the actual thing you are missing here is that the IPS proofs they give have size polynomial in the size of $|A| = |\{\sum_i \alpha_i x_i : \vec{x} \in \{0,1\}^n\}|$, which can be exponential in $n$, in general.
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2021-06-23 09:26:36
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http://orateur.nl/
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## Home
Speak Freak!
For all of you out there that love public speaking, want to ponder it with me, and want to get better at it.
The site is currently building, so…
This is just a placeholder. Look:
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## A homepage section
This is an example of a homepage section. Homepage sections can be any page other than the homepage itself, including the page that shows your latest blog posts.
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2018-12-17 07:06:17
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https://socratic.org/questions/how-would-you-compare-the-formation-of-positive-ions-with-the-formation-of-negat
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# How would you compare the formation of positive ions with the formation of negative ions in terms of energy changes?
Mar 19, 2018
Ionization energies, are associated with the LOSS of an electron, and the formation of a positive ion.....
#### Explanation:
Ionization energies, are associated with the LOSS of an electron, and the formation of a positive ion.....i.e.
$M \left(g\right) + \Delta \rightarrow {M}^{+} \left(g\right) + {e}^{-}$
And these should be an inherently ENDOTHERMIC process, i.e. we have to put energy in to separate the electron from the positively charged nuclear core.
On the other hand, the formation of negative ions represents a quantity that is measured by electron affinity....
$M \left(g\right) + {e}^{-} \rightarrow {M}^{-} \left(g\right)$
And the energy involved in this transition MAY be exothermic given an atom with a HIGH, unshielded nuclear charge...i.e. a non-metal. And we can look at the behaviour of the halogens as an example, whose electron affinities are exothermic....
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2019-08-21 04:40:17
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https://math.stackexchange.com/questions/3644527/let-f-mathbb-r-rightarrow-mathbb-r-satisfy-fx-le-x-and-fxy-le-f
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# Let $f : \mathbb R \rightarrow \mathbb R$ satisfy $f(x) \le x$ and $f(x+y) \le f(x)+f(y)$ for all $x,y \in \mathbb R$. Show that $f(x)=x$.
Let $$f : \mathbb R \rightarrow \mathbb R$$ satisfy $$f(x) \le x$$ and $$f(x+y) \le f(x)+f(y)$$ for all $$x,y \in \mathbb R$$. Show that $$f(x)=x$$.
I already know that $$f(0) = 0$$ but I don’t know how to do next. Thank you very much for helping!
• Which contest is it from please ? – Ewan Delanoy Apr 26 '20 at 9:53
• It’s just a problem in the book written about contest math. Would you like me to remove the tag? – Peerakorn Trechakachorn Apr 26 '20 at 9:55
• No, no need to remove the tag. Which book ? – Ewan Delanoy Apr 26 '20 at 9:55
• It’s in Thai. It is about the problems that could be TSTST in Thailand. – Peerakorn Trechakachorn Apr 26 '20 at 9:57
• After Let, there is no need to inflect the verb. For example, "Let it go" as oppose to "Let it goes." Here is another example, "He lets her take his car" as oppose to "He lets her takes his car." – Batominovski Apr 26 '20 at 10:10
Let $$f:\mathbb{R}$$ satisfy
(a) $$f(x)\leq x$$ for each $$x\in\mathbb{R}$$, and
(b) $$f(x+y)\leq f(x)+f(y)$$ for all $$x,y\in\mathbb{R}$$.
From (b), with $$x,y:=0$$, we get $$f(0)\geq 0$$. However, by (a), we conclude that $$f(0)=0$$.
Now, using (a), we have $$f(-x)\leq -x$$ for each $$x\in\mathbb{R}$$. Now, plug in $$y:=-x$$ in (b) to get $$0=f(0)=f\big(x+(-x)\big)\leq f(x)+f(-x)\leq x+(-x)=0\,.$$ This implies $$f(x)+f(-x)=0$$ (and in fact, at this point, it already follows that $$f(x)=x$$ for all $$x\in\mathbb{R}$$). Hence, $$f(-x)=-f(x)$$ for all $$x\in\mathbb{R}$$.
Using (a), we have $$-f(x)=f(-x)\leq -x\,,$$ or $$f(x)\geq x$$ for every $$x\in\mathbb{R}$$. By (a), we see that $$f(x)=x$$ for every $$x\in\mathbb{R}$$.
By pluging in $$a$$ and $$-a$$:
$$\forall a\in \mathbb{R}: f(a)+f(-a)\ge f(0) = 0$$
from what was given, $$f(a) \le a,f(-a) \le-a$$
So: $$f(a)-a \ge f(a)+f(-a) \ge 0 \to f(a) \ge a$$ (True because $$-a \ge f(-a)$$ )
We have that $$f(a) \le a$$
and also that $$f(a) \ge a$$
So $$f(a)=a$$
From the second inequality, $$f(x+h)\leq f(x)+f(h)$$ so that, $$\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h}\leq \lim_{h\to 0^+} \frac{f(x)+f(h)-f(x)}{h}=\lim_{h\to 0^+} \frac{f(h)}{h}$$ But from the first inequality in the question, $$f(h)/h\leq 1$$ so that, $$\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h}\leq \lim_{h\to 0^+} \frac{f(h)}{h}\leq 1$$ Similarly, $$\lim_{h\to 0^+} \frac{f(x)-f(x-h)}{h}\geq \lim_{h\to 0^+} \frac{f(x)-(f(x)+f(-h))}{h}=\lim_{h\to 0^+} \frac{f(-h)}{-h}$$ But $$f(-h)/(-h)\geq 1$$ (inequality reverses when dividing by a negative number and $$-h<0$$ since $$h>0$$). Thus we get, $$\lim_{h\to 0^+} \frac{f(x)-f(x-h)}{h}\geq \lim_{h\to 0^+} \frac{f(-h)}{-h}\geq 1$$
On the other hand, for $$h>0$$, $$\frac{f(x+h)-f(x)}{h}=\frac{f(x+h)-f((x+h)+(-h))}{h}\geq \frac{f(-h)}{-h}\geq 1$$ and $$\frac{f(x)-f(x-h)}{h}=\frac{f((x-h)+h)-f(x-h)}{h}\leq \frac{f(h)}{h}\leq 1$$
Therefore $$f'(x)$$ exists and $$f'(x)=1$$ (particularly, the fact that limit of a function strictly dominated by another function is less than or equal to limit of the other function is used).
This means that $$f(x)=x+c$$, where the integration constant $$c=0$$ because $$f(0)=0$$.
Thanks to Batominovski for supplying the proof for differentiability of this function.
• This would be a nice solution if $f$ is assumed to be differentiable, but there isn't such an assumption. (It is up to you, but I think you should keep this answer, rather than deleting it. It might teach the OP a new trick.) – Batominovski Apr 26 '20 at 10:14
• @Batominovski Ah right! I will edit my answer to include the assumption that the function is differentiable, but ya now my answer won't matter much. Thank you for pointing this out. – ModCon Apr 26 '20 at 10:16
• After a bit of thinking, I think your solution can be modified, so that you don't have to assume differentiability. You have proven that $$\lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}\leq 1$$ and $$\lim_{h\to 0^-}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0^+}\frac{f(x)-f(x-h)}{h}\geq 1\,.$$ On the other hand, for $h>0$, $$\frac{f(x+h)-f(x)}{h}=\frac{f(x+h)-f\big((x+h)+(-h)\big)}{h}\geq \frac{-f(-h)}{h}=\frac{f(-h)}{-h}\geq 1$$ and $$\frac{f(x)-f(x-h)}{h}=\frac{f\big((x-h)+h\big)-f(x-h)}{h}\leq \frac{f(h)}{h}\leq 1\,.$$ Therefore, $f'(x)$ exists, and it is equal to $1$. – Batominovski Apr 26 '20 at 10:38
• @Batominovski That's really nice! I should have thought in this direction. – ModCon Apr 26 '20 at 10:43
• I wouldn't mind if you use my comment to improve your answer, so the differentiability assumption can be removed from your answer. – Batominovski Apr 26 '20 at 10:43
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2021-07-25 22:16:49
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