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### Archive

Posts Tagged ‘sub-Neptunes’

## Red Dwarf Metallicities

A core prediction of the core accretion model for giant planet formation is that the frequency of readily detectable giant planets should increase with both increasing stellar metallicity and with increasing stellar mass:

It’s now well established that the above diagram is zeroth-order correct, but until fairly recently, the conventional wisdom held that there is little evidence for a strong planet-metallicity correlation among the handful M-dwarf stars (for example, Gliese 876) that are known to harbor giant planets. One is then naturally led to speculate that the odd giant planets in a systems like Gliese 876 might be the outcome of gravitational instability rather than core accretion.

The profusion of molecular lines in the atmospheres of M dwarfs make it hard to determine their metallicities using the techniques of spectral synthesis that work well for hotter stars like the Sun. Fortunately, though, the red dwarfs’ legendary stinginess provides another opportunity for assessing metallicity. Red dwarfs are so thrifty, and they evolve so slowly, that every single one that’s ever formed has barely touched its store of hydrogen. With all the fuel gauges pegged to full, a critical parameter’s worth of confusion is removed. Red dwarfs of a particular mass should form a well-defined one-parameter sequence in the Hertzsprung Russell diagram, and that parameter should be metallicity. If one can accurately plot a particular low-mass star on a color-magnitude diagram, then there should exist a unique and high-quality mapping to both the star’s mass and its metallicity. Physically, an increase in metallicity leads to a higher photospheric opacity, which provides an effective layer of insulation for a star. Add metals to a red dwarf and it will move down and to the right in the Hertzsprung Russell diagram.

Because of the nightmarish complexity of red dwarf atmospheres, it’s not easy to find the calibration that allows one to make the transformation between an observed absolute magnitude and color index (e.g. M_K and V-K) to the stellar mass and metallicity. In 2005, however, Xavier Bonfils and his collaborators made a breakthrough by employing a simple should’ve-thought-of-that-myself technique: Binary stars generally stem from a common molecular cloud core, and so the members of a binary pair should thus generally have very similar metallicities. In particular, if you measure the metallicity of an F, G, or K binary companion to an M-dwarf, then you can assume that the M-dwarf has the same metallicity. Do this often enough, and you can infer the lines of constant M-dwarf metallicity on a color-magnitude diagram. With the calibration in place, metallicity determinations for field red dwarfs are simply a matter of reading off the nearest iso-metallicity locus. Here’s the key diagram from the Bonfils et al. paper:

The puzzling outcome of the Bonfils et al metallicity calibration was that the rare giant-planet bearing M-dwarfs such as Gliese 876 and Gliese 849 didn’t appear to be particularly metal rich, and that worked to undermine confidence in the core accretion picture. One would naively expect that a low-mass disk will need all the help it can get in order to build giant planet cores before the gas is gone. If anything, the planet-metallicity correlation should be strongest among the M-dwarfs.

Important recent progress was made last year by John Johnson and Kevin Apps, who published a reevaluation of Bonfil et al’s. isometallicity loci in the color-magnitude diagram. Johnson and Apps point out that application of the Bonfils et al. calibration produces an aggregate of local M-dwarf stars that have a significantly lower average metallicity than that for the local FGK stars. There’s little reason to expect such a dichotomy, which implies that the Bonfils et al. correlation may be systematically underestimating metallicity by roughly a factor of two. No small potatoes!

Johnson and Apps adjusted the calibration to bring the metallicities of the local M dwarfs into line with the metallicities of the local FGK dwarfs. Here’s a slightly adapted version of their key diagram:

With the revised calibration, Gliese 876 turns up with a metallicity twice that of the Sun, and there is excellent evidence that the planet-metallicity correlation holds strongly for the M dwarfs that harbor relatively massive planets. Furthermore, it’s hard to argue with the two recent papers (one, two) from the California Planet Survey which report the detection of relatively massive planets orbiting two nearby M dwarfs, both of which have extremely high metallicities with the revised calibration.

The statistics are still small-number, but there’s a strong hint that the planet-metallicity correlation for Neptune and sub-Neptune mass planets orbiting M-dwarfs is stronger than it appears to be at FGK (where it’s effectively non-existent). Gliese 176, and Gliese 436, for example, are both quite metal-rich. I bet that a survey like Mearth could jack up its yield by shading its telescope visits to favor the high-metallicity stars on the observing list…

Indeed, if we plot Gliese 1214 (V=15.1±0.6, K=8.78±0.02, parallax=0.0772±0.0054”, distance modulus=0.562±0.16) in comparison to the stars in the local volume, it looks like Gliese 1214 has of order twice solar metallicity if we adopt the nominal values for V,K and the distance. That’s very intriguing…

Categories: worlds Tags:

## M for all and all for M

I’m always impressed by the efficiency with which red dwarfs pack hydrogen, the stuff of flammable zeppelins, into such a small space: Gliese 1214 is more than twice as dense as led. The density of the Sun, on the other hand, is bubblegum by comparison.

Gliese 1214b’s orbital period is a mere 1.58 days. Its 0.014 AU separation from the system barycenter is the smallest yet measured for any planet. Yet because of the high red dwarf density, the star-planet configuration is actually rather spacious. Here’s the system to scale:

It’s interesting to compare this diagram with that of a genuinely close-in planet such as HAT-P-7b, which actually has a somewhat longer 2.2 day orbital period:

At a given period, a red dwarf fills much less of a planetary orbit than does a Sun-like star. If the occurrence rate of planets at a specified period is the same for stars of different masses, then one needs to look at $\sim(M_{\odot}/M_{\rm RD})^{2/3}$ times more red dwarfs than Sun-like stars to find a given number of transits with a particular period.

Gliese 1214b lies at enough stellar radii from Gliese 1214 that its a-priori transit probability was only about 7%. The Mearth survey currently covers only ~2000 stars, and so the fact that the discovery was made so quickly was probably not luck, but rather points to the existence of a very large number of low-mass planets orbiting small stars.

Let’s face it. The big dough goes to chase potentially habitable transiting planets. With this metric, the red dwarfs come out way ahead. If red dwarfs and Sun-like stars have equal occurrence fractions for planets with Earth’s mass and insolation, then a low-mass red dwarf has roughly four times the probability of a Sun-like star of harboring a transiting potentially habitable planet. Twice the temperature means one-sixteenth the area and the square root of sixteen is four. The red dwarfs also present a number of other advantages, see e.g. here, here, and here.

Ryan Montgomery and I have a recent paper out which foreshadows what I think is the inevitability of transit surveys that use the Mearth strategy to target true-Earth analogs the habitable zones of the lowest-mass red dwarf stars. Mearth  is itself very well-positioned to expand in this direction. I also think that a lot of effort will continue to shift toward improved Doppler-velocity capability in the near-infrared (see, e.g. this recent paper by Jacob Bean and collaborators which describes the use of ammonia gas in a glass cell to imprint a forest of fixed reference lines on a K-band stellar spectrum).

A last note: Twelve-Fourteen-b is likely to become a favorite target for small-telescope observers, so I made sure to add it to the Transitsearch.org candidates table. Now that classes are done for the quarter, I’ve been going through the literature and adding or updating one or two planets a day. It’s tedious work, but I’ve noticed some interesting upcoming opportunities, which I’ll be writing about soon. For transit-themed ephemera and the latest celebrity gossip, look no further than the transitsearch twitter stream: http://twitter.com/Transitsearch.

And a postscript: In the comments, reader cwmagee points out that the implication of the post is that the HAT-P-7 and Gl1214 diagrams are to scale which eachother, but that’s not the case. He attached a version which shows a to-scale comparison of both systems:

Red dwarfs are small!

Categories: worlds Tags:

## Mearth!

Of course, there are still 7 hours and 13 days left until the close of 2009, but I’ve got every confidence that the discovery of the decade has landed on the ground. The Mearth project has found a transiting 6.55 Earth-mass planet in orbit around the nearby red dwarf star GJ 1214. The parent star is bright enough, and the planet-star area ratio is large enough so that direct atmospheric characterization will be possible not just with JWST, but with HST. Incredible. I’m inspired, invigorated, envious. This discovery is a game changer.

The GJ1214 discovery is all over the news today. The coverage is deservedly laudatory, but interestingly, the most dramatic aspect of the detection received rather short schrift. This is easily the most valuable planet yet found by any technique, and the discovery, start to finish, required an investment of ~500K (along with the equivalent of 1-2 nights of HARPS time to do the follow-up confirmation and to measure the planet’s mass). By contrast, well over a billion dollars has been spent on the search for planets.

I’m milking that contrast for drama, of course. It’s true that GJ1214b is low-hanging fruit. The team with the foresight to arrive on the scene first gets to pick it. And the last thing I’m suggesting is a cut in the resources devoted to exoplanet research — it’s my whole world, so to speak. I do think, though, that Mearth epitomizes the approach that will ultimately yield the planets that will give us the answers we want. You search for transits among the brightest stars at given spectral type, and you design your strategy from the outset to avoid the impedance mismatches that produce bottlenecks at the RV-confirmation stage.

There’s a factor-of-fourteen mass gap in our solar system between the terrestrial planets and the ice giants, and so with the discovery of Gl 1214b (and the bizzare CoRoT-7b) we’re getting the “last first look” at a fundamentally new type of planet. CoRoT-7b is clearly a dense iron-silicate dominated object, but it likely didn’t form that way. Gliese 1214b’s radius indicates that it probably contains a lot of water. I think this is going to turn out to be the rule as more transiting objects in the Earth-to-Neptune mass range are detected.

So what next? With a modest increase in capability, Mearth is capable of going after truly habitable planets orbiting the very nearest stars. I think it’s time to put some money down…

Categories: worlds Tags:

## Earthbound Planet Search

61 Vir b (simulation by J. Langton, Principia College)

The ranks of the super Earths and the sub Neptunes continue to grow! In two papers that have been accepted by the Astrophysical Journal, and which will be coming out in tomorrow’s astro-ph mailing, the Earthbound Planet Search team is announcing the detection of very low mass planets orbiting the nearby solar twins 61 Vir and HD 1461. (Link to paper #1, link to paper #2).

The 61 Vir system is particularly compelling. The radial velocity data for this star indicate that at least three planets are present, with an architecture that’s quite a bit more crowded than the Sun’s terrestrial planet zone:

The innermost planet, 61 Vir b, with Msin(i)~5.5 Earth masses, has a radial velocity half-amplitude K=2.15 m/s, which puts it in league with Gl581e (with K=1.9 m/s) and HD40307b (with K=2.0 m.s) as the lowest-amplitude Doppler detections to date.

We’ve adopted the Systemic Console software to analyze the Doppler velocities that are produced by the Earthbound Planet Search. I’ve written a tutorial (link here) that explores the 61 Vir dataset in detail, and shows how the planets are extracted.

Categories: worlds Tags:

## lithium-induced speculations

November 22nd, 2009 1 comment

Israelian et al’s Nature paper on the planet-stellar lithium correlation (featured in last week’s post) caused quite a stir in the community. The depletion of lithium in the atmosphere of a solar-type star seems to be a prerequisite for the presence of a detectable planetary system. Here’s the paper’s plot again, this time, with Alpha Cen A added for effect.

Had Israelian et al.’s paper come out a decade ago, much of the ensuing hubub would have focused on the fact that low lithium abundance is an effective signpost to planetary systems. Nowadays, though, mere detection of new planets is passé. Everyone knows there are tons of planets out there. Focus is shifting to finding the lowest-mass (and preferably transiting) planets around the brightest M, K, and G main sequence stars in the Sun’s neighborhood. There is a short, highly select, list of worlds that have been, and will eventually be, followed up to great advantage with HST, Warm Spitzer, and JWST.  All of the Sun’s most alluring stellar neighbors are under heavy and ongoing scrutiny, and in fact,  it’s these particular stars (in the form of the HARPS GTO list) that enabled discovery of the planet-lithium correlation.

So planet-finding utility aside, the intense interest in the planet-lithium effect stems from the fact that it’s guaranteed to be imparting an important clue to the planet-formation process.

With over 400 planets known, clear populations are starting to emerge. It’s remarkable that the strength of the lithium-planet correlation seems to be largely independent of the masses and periods of the planets themselves. The mass-period diagram for planets, on the other hand, shows that there are at least three distinct concentrations of planet formation outcomes:

It’s important to keep in mind that Israelian et al.’s correlation holds over only a very narrow range of stellar temperature. The M-dwarfs (Gliese 581, Gliese 876), the K-dwarfs (HD 69830, Alpha Cen B), and the F-dwarfs (Upsilon Andromedae) all fall outside the band of utility. This dovetails nicely with standard models of stellar evolution that suggest the amount of Lithium depletion in stars with masses very close the the Sun (that is, stars falling in the narrow effective temperature range of the above plot) depends sensitively on both the efficiency of convection and also on rotational mixing. That is, the stars that show the lithium-planet effect, are exactly the stars where subtle differences in properties seem to generate a big effect on lithium abundance.

After writing last week’s post, I got an e-mail from Jonathan Irwin (of MEarth fame) who makes several interesting points:

The low lithium could be more of a coincidence resulting from the long-lived circumstellar disks that are presumably needed to form planets.

Mediation of the stellar rotation rates by long-lived disks is thought to be responsible for generating the wide dispersion in rotation rates observed in open clusters around 100Myr age, and there have been suggestions (e.g. Denissenkov et al.’s paper that appeared on astro-ph 2 weeks ago) that the slowly-rotating stars evolve developing some degree of decoupling of the rotation rates of their radiative core and convective envelope, whereas the rapidly-rotating stars evolve more like solid bodies.

Bouvier (2008) has suggested that the shear at the radiative convective boundary resulting from this could mix lithium into the interior more efficiently, and thus could result in lower lithium for stars that were slow rotators, preserving evidence of their rotational history even though the final rotation rates all converge by the solar age.  Some evidence for this last part exists in the form of a correlation between rotation and lithium in young open clusters such as the Pleiades.

A hypothesis along these lines seems quite appealing to me. As long as a protoplanetary disk is present, and as long as its inner regions are sufficiently ionized, then there’ll be a connection between the stellar magnetic field and the magnetic field of the disk. To a (zeroth) degree of approximation, the equations of ideal MHD allow us to envision the situation as consisting of a rapidly rotating star connected to a slower-rotating disk by lot of weak rubber bands. The net effect will be to slow down the stellar rotation to bring it into synch with the rotation at the inner edge of the disk.

Trying to sound like a tough-guy, I stressed the importance of predictions in last weeks post. If Irwin’s hypothesis is correct, then the formation of the Mayor et al. 2008 planet population is associated with disks that contain lots of gas, even in regions interior to R~0.1 AU. I’d thus expect that the “super Earths” are actually “sub Neptunes”, and that we can expect considerable H-He envelopes for the majority of these planets.

Another speculative prediction concerns the stars that aren’t depleted in lithium. In Irwin’s picture, these stars had short-lived disks and lost their gas relatively rapidly. This shouldn’t hinder the formation of terrestrial planets, but one would expect that the final configurations of the rocky planets would sport higher eccentricities, as there was little or no gas to damp the orbits down during the final stages of terrestrial planet accretion (see this paper for more on this).

Categories: worlds Tags:

## the last first look

As is usually the case, there’s been little or no shortage of interesting developments in the field of extrasolar planets. The biggest recent news has been the announcement at the Barcelona conference of a definitive mass for the ultra-short period transiting planet CoRoT-7b. It weighs in at a mere 4.8 Earth Masses (copy of the Queloz et al. preprint here).

Recall that CoRoT-7b caused quite a stir earlier this year with its weird properties. The planet’s year is a fleeting twenty hours and twenty nine minutes, and it induces a tiny transit depth of 0.03%. Unfortunately, the parent star presents a less-than-ideal target for high-precision radial velocity work. It has spots that come and go, and its stellar activity produces frustratingly noisy Doppler measurements. As a result, at the time of CoRoT-7b’s initial announcement, there was no definitive measurement of the planet’s mass.

That’s changed, however, with an unprecedentedly all-out deployment of the HARPS spectrograph. From the Queloz et al. preprint:

A total of 106 measurements between 30 and 60 minute exposure time each were obtained over 4 months, and with sometimes 3 measurements being taken on the same night.

Now in my notoriously biased opinion, such observational enthusiasm is perhaps best reserved for stars such as Alpha Cen B, but a fair argument can be made that the massive investment of time did pay off. Remarkably, the radial velocity data set shows that there are two short-period planets in the CoRoT-7 system. The outer companion, which doesn’t transit, has a period of 3.7 days and at least eight Earth masses. Most dramatically, by combining the mass and radius measurements of CoRoT-7b, one arrives at a density of 5.5 grams per cubic centimeter, essentially identical to that of the Earth, suggesting that the planet is largely composed of refractory materials. (I hesitate to apply the term “rocky” to the CoRoT-7c landscape for the same reason that I’d refrain from describing the Amazon Delta as “icy”.)

In a very real sense, the HARPS campaign on CoRoT-7b has given us our last first look at a fundamentally new category of planet — that is, a world lying in the factor-of-fourteen mass gap spanned by Earth and Uranus. And, from exo-political point of view, the stakes surrounding this discovery were very high. The first density measurement of a planet in this category could just as easily have been made by teams combining high-precision Doppler measurements with either (1) Warm Spitzer, (2) ground-based photometry, (3) Kepler, (4) MOST, (5) HST, or (6) CoRoT. So I can imagine that there was a certain impetus underlying the scheduling of that huge block of HARPS time.

The discovery could, however, still be waiting to be made. Despite all the effort with HARPs, there remains a hefty 70% error on the density determination. This means that there’s a ~16% chance that CoRoT-7b is actually less dense than Neptune.

I’ll go out on a limb: CoRoT-7b’s density will turn out to be anomalously high. More than 90% of “super Earths” will turn out to be “sub-Neptunes” as far as their density is concerned.

Categories: worlds Tags:


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Add WHERE with IN and OR to a product collection

I want to add something like WHERE (table1.attrib1 IN (1,2,3) OR table2.attrib2 IN (1,2,3)) to a product collection to which I have already joined a couple of tables.

I tried this, but it doesn't work because it took '1,2,3' as one string (that is, did not put them in brackets after IN). How can I use WHERE?

$productCollection->getSelect()->join(array('table1' =>$table1), 'e.entity_id = table1.entity_id', array('my_attribute1' => 'table1.value'))->join(array('table2' => $table2), 'e.entity_id = table2.entity_id', array('my_attribute2' => 'table2.value'))->where('table1.attrib1 IN ? OR table2.attrib2 IN ? ', '1,2,3')  - 3 Answers Just try it this way // ... ->where('table1.attrib1 IN (?) OR table2.attrib2 IN (?) ', array(1,2,3));  Wrap the placeholders in parenthesis (?) and provide the values as an array array(1,2,3) instead of the former string. - How to use joinTable is described here: http://magento.stackexchange.com/a/13267/217 and in more details here: http://blog.fabian-blechschmidt.de/joining-a-flat-table-on-eav/ To use in in a filter you can use: ->addAttributeToFilter('attrib', array('in' => array(1,2,3))  and to use or you have to use the arrays recursive, like this: $in = array('in' => array(1,2,3));
$in2 = array('in' => array(4,5,6)); ->addAttributeToFilter('attrib', array($in, $in2)  should create: WHERE attrib IN (1,2,3) OR attrib IN (4,5,6);  - Here the solution ,if want to filter a collection by it field then use addFieldTofilter() function ,which is applicable for non-eav collection and eav collection. if you filter a collection by in then used below logic and code $collection->addFieldTofilter('field_name', array(
'in' => array($val1,$val2, $val3), ));  If you filter product collection then used addAttribuetToFilter('') for a exmaple $collection=Mage::getModel('catalog/product')->getCollection();

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# Does any algorithm exist for computing the state of a non-trivial cellular automaton after an arbitrary number of time steps?

If have a cellular automaton, can I see the state of the board after something like $$10^{{10}^{10}}$$ time steps? For trivial cases, this is possible - for example, a cellular automaton where the state of the board is known to repeat after some finite period.

But are there any cellular automata (or perhaps even similar computational structures) that display chaotic behavior but can also be very quickly evaluated to extreme time steps into the future?

• Depends on the automaton. Some cellular automata are Turing complete, for example, which means you can't prove whether they'll ever repeat a state (for certain initial conditions). – Draconis Jul 1 '19 at 22:05
• @Draconis, have you proved that "you can't prove whether a Turing-complete machine will ever repeat a state for certain initial conditions"? Do you actually mean "there is no algorithm to determine whether a Turing-complete machine on an input will ever repeat a state?" – John L. Jul 2 '19 at 5:02
• @Apass.Jack Indeed—more precisely, I mean "given a Turing machine and its initial conditions, no algorithm can always determine in finite time whether it will ever repeat a state" where "state" encompasses the tape as well. (The non-provability point is also true, but for different reasons.) – Draconis Jul 2 '19 at 5:06
• If an automaton can be evaluated very quickly to arbitrary extreme time steps into the future by an algorithm, it is reasonable to say its behavior is not chaotic. Or do you mean "never repeating" (aperiodic) by "chaotic"? Please clarify. – John L. Jul 2 '19 at 6:01
• @Apass.Jack From my understanding, "chaotic" simply refers to a system highly sensitive to it's initial conditions. Can you elaborate as to why a system sensitive to initial conditions could not be evaluated quickly to extreme time steps into the future? – Reggie Simmons Jul 2 '19 at 6:08


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# Morpheus

### Sidebar

documentation:performance

## Performance and scalability

The performance and scalability of simulations in Morpheus heavily depend on the type of (multi-scale) model that is being simulated. It is therefore difficult to make general statements on the computational efficiency. However, we can test the performance on a set of “benchmark” models that form the modules from which more complex model can be constructed.

We have tested the performance of ODE lattices, reaction-diffusion (PDE) models, cellular Potts models (CPM) and a multiscale model (CPM+PDE), using the available Example models. The results show the execution time and memory consumption for these models as well as their scalability in terms of problem size and scalability in terms of efficiency of multi-threading.

### Performance measurements

To quantify performance, we measured the following aspects for each simulation:

• Execution time in terms of the wall time, using the C+ + function gettimeofday() available in <sys/time.h>. The execution time does not include the time needed for initialization, analysis and visualization.
• Memory usage in terms of the physical memory (RAM) used by the simulation, using the resident set size (RSS) from the /proc/self/stat pseudo-file.

### Scalability with problem size

We investigated the scalability with respect to problem size to see how performance in terms of the execution time and memory usage (RAM) scales with increasing population size or lattice size.

We calculate and plotted both the execution time and memory usage in:

• Absolute terms: time in seconds (sec) and memory in megabytes (MB).
• Relative terms: time / memory per cell/lattice site in millisecond (msec) / kilobyte (kB).

Performance in absolute sense provides a sense of the problems sizes that are practically manageable within certain time and memory constraints.

Performance in relative sense shows the scalability of the simulation for problem sizes. Ideally, the performance per cell or lattice site stays constant or decreases with increasing problem sizes.

### Scalability of parallel processing

We have also measured the scalability with respect to the number of openMP threads to see how the performance scale with the number of concurrent threads.

We measured the execution time for each of the simulation run on in 1, 2, 4, 6 threads. Comparison of these execution times shows the speed-ups that can be achieved by adding concurrent threads.

## Methods

### Benchmark models

The models used in performance tests are available as Example models:

The models are run without analysis and visualization tools and execution time is measured from StartTime to StopTime. The time for initialization is excluded since this vanishes for large jobs.

### Hardware

All simulations were performed on a Intel Core i7-860 vPro.

Hardware specification

## Results

### Benchmark tests

 ODE PDE CPM CPM + PDE Cells 25 100 400 2500 10000 40000 Lattice $10^2$ $20^2$ $50^2$ $100^2$ $200^2$ $500^2$ $1000^2$ Cells 20 40 100 200 400 1000 2000 4000 Lattice $40^2$ $100^2$ $200^2$ $400^2$ $1000^2$ Cells 8 50 200 800 5000

### Performance statistics

Problem size
(absolute)
Problem size
(relative)
Description Total execution time (red) and memory usage (blue) of simulation, excl. initialization and visualization Execution time (red) and memory usage (blue), relative to number of cells and/or lattice sites Execution time and speed-up as a function of number of openMP threads
ODE
1) 2) 3)
PDE
4) 5) 6)
CPM
7) 8) 9)
CPM + PDE
1)
Execution time is linearly with number of cells. Total memory usage smaller than 20Mb.
2)
Exec. time per cells is approx. constant. Contribution of memory overhead decreases with problem size.
3)
Reasonable scalability of multithreading due to concurrent processing of intracellular ODEs and NeighborReporters for each cell.
4)
Execution time is linearly with number of lattice sites. Total memory usage smaller than 20Mb.
5)
Exec. time per lattice site is approx. constant. Contribution of memory overhead decreases with problem size.
6)
Good scalability of multithreading, esp. for 2 or 4 threads, due to parallelization of reaction step by domain decomposition of lattice along y-axis. Parallelization of Diffusion is only done for large 3D lattices.
7)
Execution time is almost linearly with number of CPM cells. Small memory footprint, despite edgelist tracking.
8)
Exec. time per CPM cell is almost constant, although performance decreases for larger systems. Decrease of memory usage per cell is here mostly due to use of large lattice in all cases.
9)
Parallel processing is not available for CPM simulations. Therefore, multithreading does not results in speed-up. Instead, the multithreading overhead even slightly decreases performance.
documentation/performance.txt · Last modified: 20:25 18.02.2014 by Walter


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# Distributivity of intersections across unions

Jump to: navigation, search
Stub grade: D
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Created for use with the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions, the theorem is easy and routine, at least in the finite cases

## Statement

1. [ilmath]A\cap(B\cup C)=(A\cap B)\cup(A\cap C)[/ilmath]
2. [ilmath]A\cap(\bigcup_{i=1}^n B_i)=\bigcup_{i=1}^n(A\cap B_i)[/ilmath] - Easy to do, use induction

## Proof

Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
First one is routine chapter-1 for first-years, second one is easy using induction

This proof has been marked as an page requiring an easy proof


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# holani.net

Home > Error Function > Error Function 1

# Error Function 1

## Contents

Mathematical Methods for Physicists, 3rd ed. Toggle Main Navigation Log In Products Solutions Academia Support Community Events Contact Us How To Buy Contact Us How To Buy Log In Products Solutions Academia Support Community Events Search MATLAB A complex generalization of is defined as (39) (40) Integral representations valid only in the upper half-plane are given by (41) (42) SEE ALSO: Dawson's Integral, Erfc, Erfi, Fresnel Integrals, Gaussian If L is sufficiently far from the mean, i.e. μ − L ≥ σ ln  k {\displaystyle \mu -L\geq \sigma {\sqrt {\ln {k}}}} , then: Pr [ X ≤ L his comment is here

As it is necessary that the function should have a name, and as I do not know that any has been suggested, I propose to call it the Error-function, on account See Alsoerfc | erfcinv | erfcx | erfinv Introduced before R2006a × MATLAB Command You clicked a link that corresponds to this MATLAB command: Run the command by entering it in Translate erfError functioncollapse all in page Syntaxerf(x) exampleDescriptionexampleerf(x) returns the Error Function evaluated for each element of x.Examplescollapse allFind Error FunctionOpen ScriptFind the error function of a value.erf(0.76) ans How to cope with too slow Wi-Fi at hotel? get redirected here

## Complementary Error Function

Sep 4 '11 at 13:42 Indeed, on page 296 of the Glaisher article, $x$ is used for both purposes. You can also select a location from the following list: Americas Canada (English) United States (English) Europe Belgium (English) Denmark (English) Deutschland (Deutsch) España (Español) Finland (English) France (Français) Ireland (English) Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF).

Is it plagiarism (or bad practice) to cite reviews instead of source material directly? However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where Positive integer values of Im(f) are shown with thick blue lines. Error Function Matlab Gamma: Exploring Euler's Constant.

x cannot be sparse. Error Function Calculator Diese Funktion ist zurzeit nicht verfügbar. http://mathworld.wolfram.com/Erfc.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. read this article IEEE Transactions on Communications. 59 (11): 2939–2944.

Computerbasedmath.org» Join the initiative for modernizing math education. Erf(1) Melde dich bei YouTube an, damit dein Feedback gezählt wird. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the Click the button below to return to the English verison of the page.

1. Boston, MA: Academic Press, p.122, 1997.
2. A generalization is obtained from the erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122).
3. Practice online or make a printable study sheet.
4. New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels.
5. Cambridge, England: Cambridge University Press, 1990.

## Error Function Calculator

On the other hand, for the applications where the error function is to be evaluated at complex values (spectroscopy, for instance), probably the more "natural" function to consider is Faddeeva's (or

The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Complementary Error Function Wird geladen... Error Function Table Should I serve jury duty when I have no respect for the judge?

However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1  ( z ) = ∑ k = 0 http://holani.net/error-function/error-function-pdf.php Wolfram|Alpha» Explore anything with the first computational knowledge engine. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. If nothing else, it might help me remember the definition. Inverse Error Function

Acknowledgments Trademarks Patents Terms of Use United States Patents Trademarks Privacy Policy Preventing Piracy © 1994-2016 The MathWorks, Inc. Arfken, G. These include the telegrapher’s equation, studied by Lord Kelvin in connection with the Atlantic cable, and Kelvin’s estimate of the age of the earth (25 million years), based on the solution weblink Wird verarbeitet...

Intermediate levels of Re(ƒ)=constant are shown with thin red lines for negative values and with thin blue lines for positive values. Error Function Python Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. Similarly, the En for even n look similar (but not identical) to each other after a simple division by n!.

## is the double factorial: the product of all odd numbers up to (2n–1).

Washington, DC: Hemisphere, pp.385-393, 1987. Sloane, N.J.A. Kiel oni tradukas «I’m fed up of»? Complementary Error Function Table How?

comm., May 9, 2004). Analytic Theory of Continued Fractions. comm., Dec.15, 2005). check over here How to answer boss question about ex-employee's current employer?

up vote 35 down vote favorite 6 $\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. R. (March 1, 2007), "On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand", Monthly Notices of the Royal Astronomical Society, 375 (3): 1043–1048, Applications When the results of a series of measurements are described by a normal distribution with standard deviation σ {\displaystyle \textstyle \sigma } and expected value 0, then erf ( a See also Related functions Gaussian integral, over the whole real line Gaussian function, derivative Dawson function, renormalized imaginary error function Goodwin–Staton integral In probability Normal distribution Normal cumulative distribution function, a

M. Soc. 3, 282-289, 1928. WiedergabelisteWarteschlangeWiedergabelisteWarteschlange Alle entfernenBeenden Wird geladen... Melde dich an, um dieses Video zur Playlist "Später ansehen" hinzuzufügen.

more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e−z2 into its Maclaurin series and integrating term by term, one obtains The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function. Sep 4 '11 at 14:06 Done. $\text{ }$ –cardinal Sep 4 '11 at 21:28 add a comment| up vote 4 down vote I think the normalization in $x$ is

J. Generated Mon, 10 Oct 2016 02:11:29 GMT by s_ac4 (squid/3.5.20) Is it possible to use two keyboards simultaneously? One function however, viz.

the integral $\int_x^\infty e^{-x^2}\mathrm dx$, well known for its use in physics, is so obviously suitable for the purpose, that, with the exception of receiving a name and a fixed notation, Asking client for discount on tickets to amusement park Why don't you connect unused hot and neutral wires to "complete the circuit"? Wird geladen... Generated Mon, 10 Oct 2016 02:11:29 GMT by s_ac4 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection

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Classification Test
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Classification Test

Given a population whose members each belong to one of a number of different sets or classes, a classification rule or classifier is a procedure by which the elements of the population set are each predicted to belong to one of the classes.[1] A perfect classification is one for which every element in the population is assigned to the class it really belongs to. An imperfect classification is one in which some errors appear, and then statistical analysis must be applied to analyse the classification.

A special kind of classification rule is binary classification, for problems in which there are only two classes.

## Testing classification rules

Given a data set consisting of pairs x and y, where x denotes an element of the population and y the class it belongs to, a classification rule h(x) is a function that assigns each element x to a predicted class ${\displaystyle {\hat {y}}=h(x).}$ A binary classification is such that the label y can take only one of two values.

The true labels yi can be known but will not necessarily match their approximations ${\displaystyle {\hat {y_{i}}}=h(x_{i})}$. In a binary classification, the elements that are not correctly classified are named false positives and false negatives.

Some classification rules are static functions. Others can be computer programs. A computer classifier can be able to learn or can implement static classification rules. For a training data-set, the true labels yj are unknown, but it is a prime target for the classification procedure that the approximation ${\displaystyle {\hat {y_{j}}}=h(x_{j})\approx y_{j}}$ as well as possible, where the quality of this approximation needs to be judged on the basis of the statistical or probabilistic properties of the overall population from which future observations will be drawn.

Given a classification rule, a classification test is the result of applying the rule to a finite sample of the initial data set.

## Binary and multiclass classification

Classification can be thought of as two separate problems - binary classification and multiclass classification. In binary classification, a better understood task, only two classes are involved, whereas multiclass classification involves assigning an object to one of several classes.[2] Since many classification methods have been developed specifically for binary classification, multiclass classification often requires the combined use of multiple binary classifiers. An important point is that in many practical binary classification problems, the two groups are not symmetric - rather than overall accuracy, the relative proportion of different types of errors is of interest. For example, in medical testing, a false positive (detecting a disease when it is not present) is considered differently from a false negative (not detecting a disease when it is present). In multiclass classifications, the classes may be considered symmetrically (all errors are equivalent), or asymmetrically, which is considerably more complicated.

Binary classification methods include probit regression and logistic regression. Multiclass classification methods include multinomial probit and multinomial logit.

## Table of confusion

The left, and right, halves respectively contain instances that in fact have, and do not have, the condition. The oval contains instances that are classified (predicted) as positive (having the condition). Green and red respectively contain instances that are correctly (true), and wrongly (false), classified.
TP=True Positive; TN=True Negative; FP=False Positive (type I error); FN=False Negative (type II error); TPR=True Positive Rate; FPR=False Positive Rate; PPV=Positive Predictive Value; NPV=Negative Predictive Value.

When the classification function is not perfect, false results will appear. In the example confusion matrix below, of the 8 actual cats, a function predicted that three were dogs, and of the six dogs, it predicted that one was a rabbit and two were cats. We can see from the matrix that the system in question has trouble distinguishing between cats and dogs, but can make the distinction between rabbits and other types of animals pretty well.

Predicted Cat Dog 5 3 0 2 3 1 0 2 11

### False positives

False positives result when a test falsely (incorrectly) reports a positive result. For example, a medical test for a disease may return a positive result indicating that the patient has the disease even if the patient does not have the disease. False positive is commonly denoted as the top right (Condition negative X test outcome positive) unit in a Confusion matrix. We can use Bayes' theorem to determine the probability that a positive result is in fact a false positive. We find that if a disease is rare, then the majority of positive results may be false positives, even if the test is relatively accurate.

Suppose that a test for a disease generates the following results:

• If a tested patient has the disease, the test returns a positive result 99% of the time, or with probability 0.99
• If a tested patient does not have the disease, the test returns a positive result 5% of the time, or with probability 0.05.

Naively, one might think that only 5% of positive test results are false, but that is quite wrong, as we shall see.

Suppose that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.

We can use Bayes' theorem to calculate the probability that a positive test result is a false positive.

Let A represent the condition in which the patient has the disease, and B represent the evidence of a positive test result. Then, the probability that the patient actually has the disease given the positive test result is

{\displaystyle {\begin{aligned}P(A|B)&={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\\\\&={\frac {0.99\times 0.001}{0.99\times 0.001+0.05\times 0.999}}\\~\\&\approx 0.019.\end{aligned}}}

and hence the probability that a positive result is a false positive is about 1 − 0.019 = 0.98, or 98%.

Despite the apparent high accuracy of the test, the incidence of the disease is so low that the vast majority of patients who test positive do not have the disease. Nonetheless, the fraction of patients who test positive who do have the disease (0.019) is 19 times the fraction of people who have not yet taken the test who have the disease (0.001). Thus the test is not useless, and re-testing may improve the reliability of the result.

In order to reduce the problem of false positives, a test should be very accurate in reporting a negative result when the patient does not have the disease. If the test reported a negative result in patients without the disease with probability 0.999, then

${\displaystyle P(A|B)={\frac {0.99\times 0.001}{0.99\times 0.001+0.001\times 0.999}}\approx 0.5,}$

so that 1 − 0.5 = 0.5 now is the probability of a false positive.

### False negatives

On the other hand, false negatives result when a test falsely or incorrectly reports a negative result. For example, a medical test for a disease may return a negative result indicating that patient does not have a disease even though the patient actually has the disease. False negative is commonly denoted as the bottom left (Condition positive X test outcome negative) unit in a Confusion matrix. We can also use Bayes' theorem to calculate the probability of a false negative. In the first example above,

{\displaystyle {\begin{aligned}P(A|\neg B)&={\frac {P(\neg B|A)P(A)}{P(\neg B|A)P(A)+P(\neg B|\neg A)P(\neg A)}}\\\\&={\frac {0.01\times 0.001}{0.01\times 0.001+0.95\times 0.999}}\\~\\&\approx 0.0000105.\end{aligned}}}

The probability that a negative result is a false negative is about 0.0000105 or 0.00105%. When a disease is rare, false negatives will not be a major problem with the test.

But if 60% of the population had the disease, then the probability of a false negative would be greater. With the above test, the probability of a false negative would be

{\displaystyle {\begin{aligned}P(A|\neg B)&={\frac {P(\neg B|A)P(A)}{P(\neg B|A)P(A)+P(\neg B|\neg A)P(\neg A)}}\\\\&={\frac {0.01\times 0.6}{0.01\times 0.6+0.95\times 0.4}}\\~\\&\approx 0.0155.\end{aligned}}}

The probability that a negative result is a false negative rises to 0.0155 or 1.55%.

### True positives

True positives result when a tested truly (correctly) reports a positive result. As an example, a medical test for a disease may return a positive result indicating that the patient has the disease. This is shown to be true when the patient has the disease. True positive is commonly denoted as the top left (Condition positive X test outcome positive) unit in a Confusion matrix. We can use Bayes' theorem to determine the probability that the positive result is in fact a true positive using the example from above:

• If a tested patient has the disease, the test returns a positive result 99% of the time, or with a probability of 0.99.
• If a tested patient does not have the disease, the test returns a positive result 5% of the time, or with a probability of 0.05.
• Suppose that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.

Let A represent the condition in which the patient has the disease, and B represent the evidence of a positive test result. Then, the probability that the patient actually has the disease given a positive test result is:

{\displaystyle {\begin{aligned}P(A|B)&={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\\\\&={\frac {0.99\times 0.001}{0.99\times 0.001+0.05\times 0.999}}\\~\\&\approx 0.019.\end{aligned}}}

The probability that a positive result is a true positive is about 0.019%

### True negatives

True negative result when a tested truly (correctly) reports a negative result. As an example, a medical test for a disease may return a positive result indicating that the patient does not have the disease. This is shown to be true when the patient does not have the disease. True negative is commonly denoted as the bottom right (Condition negative X test outcome negative) unit in a Confusion matrix.

We can also use Bayes' theorem to calculate the probability of true negative. Using the examples above:

• If a tested patient has the disease, the test returns a positive result 99% of the time, or with a probability of 0.99.
• If a tested patient does not have the disease, the test returns a positive result 5% of the time, or with a probability of 0.05.
• Suppose that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.

Let A represent the condition in which the patient has the disease, and B represent the evidence of a positive test result. Then, the probability that the patient actually has the disease given a positive test result is:

{\displaystyle {\begin{aligned}P(A|\neg B)&={\frac {P(\neg B|A)P(A)}{P(\neg B|A)P(A)+P(\neg B|\neg A)P(\neg A)}}\\\\&={\frac {0.01\times 0.001}{0.01\times 0.001+0.95\times 0.999}}\\~\\&\approx 0.0000105.\end{aligned}}}

The probability that a negative result is a true negative is 1 - 0.0000105 = 0.9999895 or 99.99%. Since the disease is rare and the positive to positive rate is high and the negative to negative rate is also high, this will produce a large True Negative rate.

### Worked example

A worked example
A diagnostic test with sensitivity 67% and specificity 91% is applied to 2030 people to look for a disorder with a population prevalence of 1.48%
 Patients with bowel cancer(as confirmed on endoscopy) Condition positive Condition negative Prevalence= (TP + FN) / Total_Population= (20 + 10) / 2030≈ 1.48% Accuracy (ACC) = (TP + TN) / Total_Population= (20 + 1820) / 2030≈ 90.64% Fecaloccultbloodscreentestoutcome Testoutcomepositive True positive(TP) = 20(2030 × 1.48% × 67%) False positive(FP) = 180(2030 × (100 − 1.48%) × (100 − 91%)) Positive predictive value (PPV), Precision= TP / (TP + FP)= 20 / (20 + 180)= 10% False discovery rate (FDR)= FP / (TP + FP)= 180 / (20 + 180)= 90.0% Testoutcomenegative False negative(FN) = 10(2030 × 1.48% × (100 − 67%)) True negative(TN) = 1820(2030 × (100 − 1.48%) × 91%) False omission rate (FOR)= FN / (FN + TN)= 10 / (10 + 1820)≈ 0.55% Negative predictive value (NPV)= TN / (FN + TN)= 1820 / (10 + 1820)≈ 99.45% TPR, Recall, Sensitivity= TP / (TP + FN)= 20 / (20 + 10)≈ 66.7% False positive rate (FPR),, probability of false alarm = FP / (FP + TN)= 180 / (180 + 1820)= 9.0% Positive likelihood ratio = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}= (20 / 30) / (180 / 2000)≈ 7.41 Diagnostic odds ratio (DOR) = ≈ 20.2 F1 score = 2 × ≈ 0.174 False negative rate (FNR), Miss rate= FN / (TP + FN)= 10 / (20 + 10) ≈ 33.3% Specificity, Selectivity, True negative rate (TNR)= TN / (FP + TN)= 1820 / (180 + 1820)= 91% Negative likelihood ratio = = (10 / 30) / (1820 / 2000)≈ 0.366

Related calculations

• False positive rate (?) = type I error = 1 - specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
• False negative rate (?) = type II error = 1 - sensitivity = FN / (TP + FN) = 10 / (20 + 10) = 33%
• Power = sensitivity = 1 - ?
• Positive likelihood ratio = sensitivity / (1 − specificity) = 0.67 / (1 − 0.91) = 7.4
• Negative likelihood ratio = (1 − sensitivity) / specificity = (1 − 0.67) / 0.91 = 0.37
• Prevalence threshold = ${\displaystyle PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}}$ ? 0.2686 => 26.9%

This hypothetical screening test (fecal occult blood test) correctly identified two-thirds (66.7%) of patients with colorectal cancer.[a] Unfortunately, factoring in prevalence rates reveals that this hypothetical test has a high false positive rate, and it does not reliably identify colorectal cancer in the overall population of asymptomatic people (PPV = 10%).

On the other hand, this hypothetical test demonstrates very accurate detection of cancer-free individuals (NPV = 99.5%). Therefore, when used for routine colorectal cancer screening with asymptomatic adults, a negative result supplies important data for the patient and doctor, such as ruling out cancer as the cause of gastrointestinal symptoms or reassuring patients worried about developing colorectal cancer.

## Measuring a classifier with sensitivity and specificity

In training a classifier, one may wish to measure its performance using the well-accepted metrics of sensitivity and specificity. It may be instructive to compare the classifier to a random classifier that flips a coin based on the prevalence of a disease. Suppose that the probability a person has the disease is ${\displaystyle p}$ and the probability that they do not is ${\displaystyle q=1-p}$. Suppose then that we have a random classifier that guesses that the patient has the disease with that same probability ${\displaystyle p}$ and guesses that he does not with the same probability ${\displaystyle q}$.

The probability of a true positive is the probability that the patient has the disease times the probability that the random classifier guesses this correctly, or ${\displaystyle p^{2}}$. With similar reasoning, the probability of a false negative is ${\displaystyle pq}$. From the definitions above, the sensitivity of this classifier is ${\displaystyle p^{2}/(p^{2}+pq)=p}$. With similar reasoning, we can calculate the specificity as ${\displaystyle q^{2}/(q^{2}+pq)=q}$.

So, while the measure itself is independent of disease prevalence, the performance of this random classifier depends on disease prevalence. The classifier may have performance that is like this random classifier, but with a better-weighted coin (higher sensitivity and specificity). So, these measures may be influenced by disease prevalence. An alternative measure of performance is the Matthews correlation coefficient, for which any random classifier will get an average score of 0.

The extension of this concept to non-binary classifications yields the confusion matrix.


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## On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups

#### Mourad Chelgham [1] , Mohamed Kerada [2] , Lemnouar Noui [3]

##### 29 38

Let k>0 an integer. F, τ, N, Nk, and A denote, respectively, the classes of finite, torsion, nilpotent, nilpotent of class at most k, group in which every two generator subgroup is in Nk and abelian groups. The main results of this paper is, firstly, to prove that in the class of finitely generated FN-group, the property FC is closed under finite extension. Secondly, we prove that a finitely generated τN-group in the class ((τNk)τ,∞) ( respectively ((τNk)τ,∞)) is a τ-group (respectively τNc for certain integer c=c(k) ) and deduce that a finitely generated FN-group in the class ((FNk)F,∞) (respectively ((FNk)F,∞)) is -group (respectively FNc for certain integer c=c(k)). Thirdly we prove that a finitely generated NF-group in the class ((FNk)F,∞) ( respectively ((FNk)F,∞)) is F-group (respectively NcF for certain integer c=c(k)). Finally and particularly, we deduce that a finitely generated FN-group in the class ((FA)F,∞) (respectively ((FC)F,∞), ((FN)F,∞)) is in the class FA (respectively FN, FN(2)).

FC-group, (FC)F-group, (τNk)τ-group, (FNk)F-group, ((FNk)F, ∞)-group, ∞)∗-group, finitely generated group
• [1] A. Abdollahi et N. Trabelsi, Quelques extensions d'un problème de Paul Erdös sur les groupes, Bull. Belg. Math. Soc. 9, (2002), 1-11.
• [2] A. Abdollahi, Some Engel conditions on infinite subsets of certain groups, Bull.Austral. Math. Soc. 62, (2000), 141-148.
• [3] A. Abdollahi, Finitely generated soluble groups with an Engel conditions on infinite subsets, Rend. Sem. Mat. Univ. Padova. 103, (2000), 47-49.
• [4] A. Abdollahi, B. Taeri, A condition on finitely generated soluble groups, Communication in Algebra, 27(11), (1999), 5633-7658.
• [5] R. Baer, Finiteness properties of groups, Duke Math. J. 15, (1948), 1021-1032.
• [6] C. Casolo, Groups with all subgroups subnormal, Note Mat. 28, (2008), suppl. n. 2, 1-149.
• [7] C. Delizia and C. Nicotera, On residually finite groups with an Engel condition on infinitesubset, J. Austral. Math. Soc. (series A) 69, (2000), 415-420.
• [8] J. Erdös, The theory of groups with finite classes of conjugate elements, Acta. Math. Acad. Sci.Hungar. 5, (1954), 45-58.
• [9] F. Guerbi and T. Rouabeh, Hyper(Abelian-by-finite)-groups with many sebgroups of finite depth, 14(1), (2007), 17-28.
• [10] P. Hall, Finite-by-nilpotent-group, Prog. Cambridge Philos. Soc. 52, (1956), 611-616.
• [11] P. Hall, The Edmonton notes on nilpotent groups, Queen Mary CollegeMath. Notes (1969).
• [12] J. C. Lennox and J. Wiegold, Extensions of a problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, 31, (1981), 459-463.
• [13] B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London. Math. Soc, (3.Ser.), 1, (1951), 178-187.
• [14] B. H. Neumann, A problem of Paul Erdös on groups, J. Austral. Math. Soc. ser. A 21, (1976), 467-472.
• [15] N. Nishigori, On some properties of FC-groups, J. Sci. Hiroshima Univ. Ser. A 21, (1957/1958), 99-105.
• [16] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, Heidelberg, New York, (1972).
• [17] D. J. S. Robinson, A course in the theory of groups, Springer-verlag, Berlin, Heidelberg, New York, (1982.(
• [18] T. Rouabeh and N. Trabelsi, A note on Torsion-by-Nilpotent group, Rend. Sem. Mat. Univ. Panova, 117(2007), 175-179.
• [19] N. Trabelsi, Characterization of nilpotent-by-finite groups, Bull. Austral. Math. Soc, 61, (2000), 33-38.
• [20] N. Trabelsi, Finitely generated soluble groups with a condition on infinite subsets, Algebra Colloq, 9, (2002), 427-432.
• [21] N. Trabelsi, Soluble groups with many 2-generated torsion-by-nilpotent subgroups, Publ. Math. Debrecen, 67/1-2, 6, (2005), 93-102.
• [22] M. J. Tomkinson, FC-groups, Pitman Advanced Pub. Program, Californy university, USA, (1984.(
Birincil Dil en Araştırma Makalesi Yazar: Mourad Chelgham Yazar: Mohamed Kerada Yazar: Lemnouar Noui
 Bibtex @araştırma makalesi { jnt431047, journal = {Journal of New Theory}, issn = {}, eissn = {2149-1402}, address = {Gaziosmanpaşa Üniversitesi}, year = {2018}, volume = {}, pages = {22 - 30}, doi = {}, title = {On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups}, key = {cite}, author = {Chelgham, Mourad and Kerada, Mohamed and Noui, Lemnouar} } APA Chelgham, M , Kerada, M , Noui, L . (2018). On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory, (23), 22-30. Retrieved from http://dergipark.gov.tr/jnt/issue/37237/431047 MLA Chelgham, M , Kerada, M , Noui, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018): 22-30 Chicago Chelgham, M , Kerada, M , Noui, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018): 22-30 RIS TY - JOUR T1 - On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups AU - Mourad Chelgham , Mohamed Kerada , Lemnouar Noui Y1 - 2018 PY - 2018 N1 - DO - T2 - Journal of New Theory JF - Journal JO - JOR SP - 22 EP - 30 VL - IS - 23 SN - -2149-1402 M3 - UR - Y2 - 2018 ER - EndNote %0 Journal of New Theory On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %A Mourad Chelgham , Mohamed Kerada , Lemnouar Noui %T On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %D 2018 %J Journal of New Theory %P -2149-1402 %V %N 23 %R %U ISNAD Chelgham, Mourad , Kerada, Mohamed , Noui, Lemnouar . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory / 23 (Haziran 2018): 22-30.


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 Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 6, Pages 79–98 (Mi izv4406)

Weak$^*$ convergence of operator means

A. V. Romanov

Moscow State Institute of Electronics and Mathematics (Technical University)

Abstract: For a linear operator $U$ with $\|U^n\| \le \operatorname{const}$ on a Banach space $X$ we discuss conditions for the convergence of ergodic operator nets $T_\alpha$ corresponding to the adjoint operator $U^*$ of $U$ in the $\mathrm{W^*O}$-topology of the space $\operatorname{End} X^*$. The accumulation points of all possible nets of this kind form a compact convex set $L$ in $\operatorname{End} X^*$, which is the kernel of the operator semigroup $G=\overline{\operatorname{co}} \Gamma_0$, where $\Gamma_0=\{U_n^*, n \ge 0\}$. It is proved that all ergodic nets $T_\alpha$ weakly$^*$ converge if and only if the kernel $L$ consists of a single element. In the case of $X=C(\Omega)$ and the shift operator $U$ generated by a continuous transformation $\varphi$ of a metrizable compactum $\Omega$ we trace the relationships among the ergodic properties of $U$, the structure of the operator semigroups $L$, $G$ and $\Gamma=\overline{\Gamma}_0$, and the dynamical characteristics of the semi-cascade $(\varphi,\Omega)$. In particular, if $\operatorname{card}L=1$, then a) for any $\omega \in\Omega$ the closure of the trajectory $\{\varphi^n\omega, n \ge 0\}$ contains precisely one minimal set $m$, and b) the restriction $(\varphi,m)$ is strictly ergodic. Condition a) implies the $\mathrm{W^*O}$-convergence of any ergodic sequence of operators $T_n \in \operatorname{End} X^*$ under the additional assumption that the kernel of the enveloping semigroup $E(\varphi,\Omega)$ contains elements obtained from the ‘basis’ family of transformations $\{\varphi^n, n \ge 0\}$ of the compact set $\Omega$ by using some transfinite sequence of sequential passages to the limit.

Keywords: weak$^*$ ergodic theory, dynamical system, enveloping semigroup, Choquet representation.

DOI: https://doi.org/10.4213/im4406

Full text: PDF file (549 kB)
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English version:
Izvestiya: Mathematics, 2011, 75:6, 1165–1183

Bibliographic databases:

UDC: 517.98
MSC: Primary 47A35; Secondary 47A84
Revised: 22.03.2010

Citation: A. V. Romanov, “Weak$^*$ convergence of operator means”, Izv. RAN. Ser. Mat., 75:6 (2011), 79–98; Izv. Math., 75:6 (2011), 1165–1183

Citation in format AMSBIB
\Bibitem{Rom11} \by A.~V.~Romanov \paper Weak${}^*$ convergence of operator means \jour Izv. RAN. Ser. Mat. \yr 2011 \vol 75 \issue 6 \pages 79--98 \mathnet{http://mi.mathnet.ru/izv4406} \crossref{https://doi.org/10.4213/im4406} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2918894} \zmath{https://zbmath.org/?q=an:1248.47013} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011IzMat..75.1165R} \elib{http://elibrary.ru/item.asp?id=20358819} \transl \jour Izv. Math. \yr 2011 \vol 75 \issue 6 \pages 1165--1183 \crossref{https://doi.org/10.1070/IM2011v075n06ABEH002568} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000298497200004} \elib{http://elibrary.ru/item.asp?id=18031630} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84455199899}

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This publication is cited in the following articles:
1. A. V. Romanov, “Ordinary Semicascades and Their Ergodic Properties”, Funct. Anal. Appl., 47:2 (2013), 160–163
2. A. V. Romanov, “Ergodic properties of discrete dynamical systems and enveloping semigroups”, Ergod. Th. Dynam. Sys., 36:1 (2016), 198–214
3. Aleman A., Suciu L., “On Ergodic Operator Means in Banach Spaces”, Integr. Equ. Oper. Theory, 85:2 (2016), 259–287
4. Suciu L., “Ergodic behaviors of the regular operator means”, Banach J. Math. Anal., 11:2 (2017), 239–265
5. Kreidler H., “Compact Operator Semigroups Applied to Dynamical Systems”, Semigr. Forum, 97:3 (2018), 523–547
6. A. V. Romanov, “Ergodic Properties of Tame Dynamical Systems”, Math. Notes, 106:2 (2019), 286–295
•  Number of views: This page: 364 Full text: 86 References: 35 First page: 14


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## Wednesday, 6 January 2021

### The Voss-Weyl formula

 Hermann Weyl
The Voss-Weyl formula is for the contraction of the covariant derivative of a vector field:$$\nabla_\mu F^\mu=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial x^\mu}\left(\sqrt{\left|g\right|}F^\mu\right)$$It immediately solved an outstanding problem from a year ago that I had with the Laplacian. While proving it I find three other interesting formulas:
Jacobi's formula is$$\partial_kQ={\widetilde{Q}}^{ji}\partial_kQ_{ij}$$where $Q_{ij}$ is a tensor, $Q$ is its determinant and what might be new is ${\widetilde{Q}}^{ij}$ which is the cofactor of $Q_{ij}$ and thus as a matrix, ${\widetilde{Q}}^{ji}$ is the transpose of the cofactors of $Q_{ij}$. The cofactor of a component $Q_{ij}$ is the determinant of the matrix made by removing row $i$, column $j$ then multiplying by $\left(-1\right)^{i+j}$. The important thing is that the transpose of the cofactors is called the adjugate, so ${\widetilde{Q}}^{ji}$ is the adjugate and the inverse of a matrix is the adjugate divided by its determinant. We write that:$$Q^{ij}=\frac{1}{Q}{\widetilde{Q}}^{ji}$$Finally on the way to proving the Voss-Weyl formula we find the marvellously symmetrical $$Q^{-1}\partial_kQ=Q^{ij}\partial_kQ_{ij}$$where $Q^{-1}$ is the determinant of the inverse of a rank 2 tensor $Q_{ij}$ which has determinant $Q$. It is also well known that $Q^{-1}=\frac{1}{Q}$.

I must thank David A. Clarke of Saint Mary’s University, Halifax NS, Canada for his guidance.


text:
# Tomography of light mesons in the light-cone quark model

@article{Kaur2020TomographyOL,
title={Tomography of light mesons in the light-cone quark model},
author={Satvir Kaur and Narinder Kumar and Jiangshan Lan and Chandan Kumar Mondal and Harleen Dahiya},
journal={arXiv: High Energy Physics - Phenomenology},
year={2020}
}
• Published 4 February 2020
• Physics
• arXiv: High Energy Physics - Phenomenology
We investigate the tomographical structure of pion and kaon in light cone quark model (LCQM). In particular, we study the parton distribution amplitude (PDA) of pion and kaon. We obtain the parton distribution function (PDF) and the generalized parton distributions (GPDs) of the pion and kaon. The valence quark PDA and PDF of pion, after QCD evolution, are found to be consistent with the data from the E791 and the E615 experiments at Fermilab, respectively. Further, we investigate the…
4 Citations

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## References

SHOWING 1-10 OF 169 REFERENCES

### HERMES Collaboration

• Physics
The HERMES Experiment
• 2022
Cross section ratios for deep-inelastic scattering from 14 N and 3 He with respect to 2 H have been measured by the HERMES experiment at DESY using a 27.5 GeV positron beam. The data cover a range in

### Pion valence structure from Ioffe-time parton pseudodistribution functions

• Physics
Physical Review D
• 2019
We present a calculation of the pion valence quark distribution extracted using the formalism of reduced Ioffe-time pseudodistributions or more commonly known as pseudo-PDFs. Our calculation is

### Study of kaon structure using the light-cone quark model

• Physics
Physical Review D
• 2019
We investigate the various distributions explaining multi-dimensional structure of kaon at the level of its constituents ($u$ and $\bar{s}$) using the light-cone quark model. The overlap form of

### Predicting the light-front holographic TMDs of the pion

• Physics
Physical Review D
• 2019
We predict the twist-2 Transverse Momentum Dependent parton distribution functions (TMDs) of the pion, namely the unpolarized quark TMD, $f_{1}(x, k_\perp)$, and the transversely polarized quark TMD,

### Pion and kaon parton distribution functions from basis light front quantization and QCD evolution

• Physics
Physical Review D
• 2020
We investigate the parton distribution functions (PDFs) of the pion and the kaon by combining QCD evolution with the basis light front quantization (BLFQ). The initial PDFs result from the light

### Timelike Compton scattering off the neutron and generalized parton distributions

• Physics
• 2016
Abstract.We study the exclusive photoproduction of an electron-positron pair on a neutron target in the Jefferson Lab energy domain. The reaction consists of two processes: the Bethe-Heitler and the

### Symmetry, symmetry breaking, and pion parton distributions

• Physics
Physical Review D
• 2020
Pion valence, glue and sea distributions are calculated using a continuum approach to the two valence-body bound-state problem. Since the framework is symmetry preserving, physical features of the


text:
Calculus without limits

Schroedinger operators on Graphs

Quivers allow to model Schroedinger operators on graphs. With loops we can modify scalar values on vertices, with multiple connections, we can modify the magnetic part. This makes quivers attractive as well as theorems about quivers more interesting. An example was $\lambda_k \leq d_k + d_{k-1}$. In the talk last Saturday, I also mentioned the lower bound $\lambda_k \geq d_k-(m-k)$ . However, the lower bound works only for quivers without magnetic part. This means we attach just loops at a finite simple graph. Note that this class of non-magnetic quivers also has the property that a principal submatrix of its Kirchhoff matrix is again the Kirchhoff matrix of a non-magnetic quiver. In the talk, I gave the proof in the non-magnetic case but thought that the magnetic case follows. This is not the case. Take the graph with $E=\{ 1 \rightarrow 2, 1 \rightarrow 2 \}$ for example which has eigenvalues $\lambda_1=0$ (as for all quivers without loops),\lambda_2=4$and $d_1=2$ and $d_1-(2-1) = 1$. The estimate$\lambda_1 \geq d_1-(n-1)\$ does not work here.

Update July 26. for the proof of the upper bound result $\lambda_k \leq d_k + d_{k-1}$ quivers were the right structure. As just mentioned, it also gives a bit more general frame work allowing to put weights on edges or vertices and still get the same spectral estimates. As also mentioned in the talk, the upper bound for the eigenvalues allows to estimate determinants from above, unfortunately, one can not estimate with such estimates determinants from below. We would need to be able to estimate $\log(\lambda_k)$ from below but these are small divisor problems already in the simplest possible cases like almost periodic Jacobi matrices. The determinants depend in a subtle way on the rule generating the potential. The Jacobi case is the situation where the underlying graphs are cyclic graphs.

The first and last few seconds of the video show some experiments with a quite astounding photo camera. It follows me on my bike and while walking. On Monday, I recorded during a run.


text:
# bilinear

## English

English Wikipedia has an article on:
Wikipedia

### Etymology

From bi- + linear.

bilinear (not comparable)

1. (linear algebra, of a function in two variables) Linear (preserving linear combinations) in each variable.
The function ${\displaystyle f(x,y)=xy}$ is bilinear, because ${\displaystyle f(ax+bu,y)=af(x,y)+bf(u,y)}$ and ${\displaystyle f(x,ay+bv)=af(x,y)+bf(x,v)}$ for all values.
2. () Of or pertaining to a Möbius transformation (type of conformal map representable as the ratio of two linear functions).
bilinear isotropic hardening

### Noun

bilinear (plural bilinears)

1. (linear algebra) A bilinear function.


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# Prove that f is a bijection

Deﬁne the function $$f : \{n \mid n = 3m − 1 \text{ for some } m \in \mathbb{Z}\} \rightarrow \{k \mid k = 4m + 2 \text{ for some } m \in \mathbb{Z}\}$$ by $$f(n)=\frac{4(n+1)}{3}-2$$ Prove that $f$ is a bijection.

I am aware that I need to prove $f$ is injective and surjective and therefore, a bijection. How should I approach this? (The lecturer has written "Backwards proof" on my script and marked my answer as incorrect)

Edit2 (switched incorrectly labelled domain and codomain)

if $$f(x) = f(y)$$ then $$\frac{4(x+1)}{3}-2 = \frac{4(y+1)}{3}-2$$ $$x = y \text{ }$$ Therefore the function is injective.

Let $$m \in \mathbb{Z} \text{, } 3m-1 \in \mathbb{D} \text{ (domain)}$$ $$m+1 \in \mathbb{Z} \text{, } 3(m+1)-1 \text{, } 3m+2 \in \mathbb{D}$$ $$f(3m+2) = \frac{4((3m+2)+1)}{3}-2$$ $$f(3m+2) = \frac{4(3m+3)}{3}-2$$ $$f(3m+2) = \frac{12m+12)}{3}-2$$ $$f(3m+2) = 4m+4-2$$ $$f(3m+2) = 4m+2$$ $$4m+2 \in \mathbb{C} \text{ (codomain)}$$ Therefore the function is surjective.

Since the function is injective and surjective, it is a bijection.

-
Could you show us your approach? –  Alex Becker May 3 '12 at 5:15
The function maps the integer $3m-1$ to $4m-2$, which certainly is of the shape $4k+2$. The function $\frac{4(x+1)}{3}-2$ is strictly increasing, even on the reals, so injectivity is easy. Now pick a number of the shape $4k+2$. Who is taken there? –  André Nicolas May 3 '12 at 5:19
Probably the simplest thing to do is to find the inverse of this function. Once you know it has an inverse function, it must be a bijection. –  user22805 May 3 '12 at 5:21
@AlexBecker I have added my approach, perhaps you could tell me where I went wrong? Thank you. –  Danny Rancher May 3 '12 at 5:58
@DannyRancher The first part is fine, if sparse on details. But the second shows simply that the function in fact maps its domain into its codomain (note on the first line where you say codomain you should say domain, and on the last line you should say codomain), not that it is surjective. –  Alex Becker May 3 '12 at 6:05

Hint $\rm\:f\:$ is linear so $1\!-\!1.\:$ $\rm\:f(3\:\!\mathbb Z\!-\!1) = 4(3\:\!\mathbb Z\!-\!1\!+\!1)/3-2 = 4\:\!\mathbb Z\!-\!2 = 4\:\!\mathbb Z\!+\!2\:$ so $\rm\:f\:$ is onto.

-

First, note that the function is indeed well-defined: if $n=3m-1$, then $$f(n) = \frac{4(n+1)}{3}-2 = \frac{4(3m-1+1)}{3}-2 = \frac{4(3m)}{3} -2= 4m-2 = 4(m-1)+2$$ which is in the (alleged) codomain.

There are now two ways of showing this: either prove that $f$ is injective and onto; or prove that $f$ has an inverse.

The computation above might help to show the function is onto: if you take $4k+2$ in the image, what value of $n$ will give you $f(n) = 4k+2$?

It might also help with one-to-one: if $f(a)=f(b)$, with $a=3m-1$ and $b=3n-1$, then by the computations above we get $4m-2 = f(a) = f(b) = 4n-2$. What does that tell us about $a$ and $b$?

Alternatively: define $$g\colon \{k\mid k=4m+2\text{ for some }m\in\mathbb{Z}\}\to \{n\mid n=3m-1\text{ for some }m\in\mathbb{Z}\}$$ by letting $$g(4m+2) = 3(m+1)-1.$$ Verify that $f(g(4m+2)) = 4m+2$ and $g(f(3m-1)) = 3m-1$, proving that $g=f^{-1}$, so $f$ is bijective.

-


text:
# Review of indicators for comparing environmental effects across energy sources

October 18, 2019

Robust, quantitative comparisons of environmental effects across energy sources can support development of energy planning strategies that meet growing demand while managing and minimizing undesirable effects on environmental resources. Multicriteria analyses of energy systems often use a suite of indicators to make such comparisons, but those indicators and their units of measure vary among studies. We reviewed 179 papers that described or applied energy indicators to compare environmental effects of different primary energy sources to answer four questions: (1) what environmental indicators have been used in multicriteria energy-source comparisons? (2) across studies, how consistently are these indicators used to quantify effects? (3) to what degree are different effects accounted for across energy sources and locations? and (4) how comprehensive are indicators in terms of assessing known environmental effects? For reviewed studies, we quantified the number of unique indicators and the diversity of indicators used to measure different effects. We also recorded the specific measurement units applied to quantify each indicator, the energy sources evaluated, and the continent where each study was conducted. While we found that many environmental effects of energy development have been analyzed across multiple sources, indicators were frequently measured, interpreted, and applied in ways that are not directly comparable, and some known environmental effects were infrequently assessed. We also found an emphasis on applying indicators to renewable energy sources; assessing current and potential energy sources, both renewable and nonrenewable, would greatly clarify the full suite of tradeoffs among sources and can inform energy development strategies that minimize adverse environmental effects. Overall, our review indicated that making comprehensive comparisons of the effects of energy development across sources may require efforts to standardize how effects are measured, synthesize effects literature into an open-source database, expand the range of environmental effects analyzed, and establish consistent frameworks for comparison.


text:
# ビジネス英語口語表現788：let alone　～はもちろん、～は言うまでもなく

Today’s lesson of colloquial expression

1. let alone　～はもちろん、～は言うまでもなく

- this means the same as “much less”

1. I don’t have enough money for a motorcycle right now, let alone a car.
2. She doesn’t have enough time to take a coffee break, let alone take the day off.

トーク・アベニュー新宿公式ページはこちらから≫


text:
Does the global stiffness matrix size depend on the number of joints or the number of elements?

When assembling all the stiffness matrices for each element together, is the final matrix size equal to the number of joints or elements?

• The size of the matrix depends on the number of nodes. The bandwidth of each row depends on the number of connections. – stali Sep 30 '15 at 21:21
• What do you mean by joints? – nicoguaro Oct 1 '15 at 15:37
• @Stali That sounds like an answer to me -- would you care to add a bit of explanation and post it? – Christian Clason Oct 1 '15 at 15:43

The number of rows and columns in the final global sparse stiffness matrix is equal to the number of nodes in your mesh (for linear elements). For example if your mesh looked like:

then each local stiffness matrix would be 3-by-3. Once all 4 local stiffness matrices are assembled into the global matrix we would have a 6-by-6 global matrix. For example the local stiffness matrix for element 2 (e2) would added entries corresponding to the second, fourth, and sixth rows and columns in the global matrix. For this mesh the global matrix would have the form:

$$\begin{bmatrix} * & * & 0 & 0 & 0 & * \\ * & * & * & * & 0 & * \\ 0 & * & * & * & 0 & 0 \\ 0 & * & * & * & * & * \\ 0 & 0 & 0 & * & * & * \\ * & * & 0 & * & * & * \\ \end{bmatrix}$$

where each * is some non-zero value. I assume that when you say joints you are referring to the nodes that connect elements. If this is the case then using your terminology the answer is: the global stiffness matrix has size equal to the number of joints.

• More generally, the size of the matrix is controlled by the number of degrees of freedom, which may or may not be the same as the number of nodes, depending on the problem in question and the type of elements used. Since the OP mentioned joints (implying a standard direct stiffness method), it is fair to assume that the number of DOF's and the number of nodes are identical. – Paul Oct 1 '15 at 19:22


text:
# Math Help - I need help-discontinuity

1. ## I need help-discontinuity

I don't know if 3X^4 + 4X - 7 is discontinuous or not.
Thanks.

2. Originally Posted by sdinulescu
I don't know if 3X^4 + 4X - 7 is discontinuous or not.
Thanks.
All polynomials are continuous for all $x$. The only way it would possibly be discontinuous is if the domain is restricted.


text:
# How do you put math in JavaScript?

Contents

Filtering a Directory Listing By Using Globbing

## Can you do math with JavaScript?

JavaScript has a built-in calculator, and mathematical operations can be done directly in the console. We can do some simple addition with numbers, for example adding 10 and 20 , using the plus sign ( + ). … We can also add and subtract with negative numbers and floats (decimals).

## How do you calculate in JavaScript?

Live Demo:

1. function multiplyBy()
2. {
3. num1 = document. getElementById(“firstNumber”). value;
4. num2 = document. getElementById(“secondNumber”). value;
5. document. getElementById(“result”). innerHTML = num1 * num2;
6. }
7. function divideBy()

## Can you do math in HTML?

The $element$ is used to include math expressions in the current line. HTML math is powerful enough to describe the range of math expressions you can create in common word processing packages, as well as being suitable for rendering to speech.

## How do you multiply in JavaScript?

The multiplication operator ( * ) multiplies numbers.

## What is the meaning of in JavaScript?

The in operator returns true if the specified property is in the specified object or its prototype chain.

## What is the operator in JavaScript?

An operator is capable of manipulating a certain value or operand. Operators are used to perform specific mathematical and logical computations on operands. … In JavaScript operators are used for compare values, perform arithmetic operations etc.

THIS MEANING:  What do you mean by Java is case sensitive?

## What is eval JavaScript?

The eval() function in JavaScript is used to evaluate the expression. It is JavaScirpt’s global function, which evaluates the specified string as JavaScript code and executes it. The parameter of the eval() function is a string. If the parameter represents the statements, eval() evaluates the statements.

## What is parseInt in JavaScript?

JavaScript parseInt() Function

The parseInt() function parses a string and returns an integer. The radix parameter is used to specify which numeral system to be used, for example, a radix of 16 (hexadecimal) indicates that the number in the string should be parsed from a hexadecimal number to a decimal number.

## How many types of arrays are there in JavaScript?

Arrays are just regular objects

In Javascript, there are only 6 data types defined – the primitives (boolean, number, string, null, undefined) and object (the only reference type). Arrays do not belong to this list because they are objects as well.

## What type of object is math Cos?

The Math. cos() method returns a numeric value between -1 and 1, which represents the cosine of the angle. Because cos() is a static method of Math , you always use it as Math. cos() , rather than as a method of a Math object you created ( Math is not a constructor).


text:
# greatest least bound and least upper bound proof

• November 4th 2009, 12:21 PM
tae1466
greatest least bound and least upper bound proof
I've been struggling with this one.

Let A be a partially ordered set. Suppose X⊆Y⊆A.

1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)
• November 4th 2009, 04:44 PM
Drexel28
Quote:

Originally Posted by tae1466
I've been struggling with this one.

Let A be a partially ordered set. Suppose X⊆Y⊆A.

1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)

1. I'll do part of this. See if you can do the rest.

Problem: Let $X\subseteq Y$. Prove that $\sup\left(X\right)\le\sup\left(Y\right)$

Proof: We know that since $X\subseteq Y$ that $x\in X\implies x\in Y$. But $\forall x\in Y\quad x\le\sup\left(Y\right)$, so $\forall x\in X\quad x\le \sup\left(Y\right)$. Therefore $\sup\left(Y\right)$ is an upper bound for $X$. And by definition $\sup\left(X\right)$ is at least as small as any upper bound of $X$. The conclusion follows.

2. What about $X=(0,1)$ and $Y=[0,1]$?


text:
시간 제한메모리 제한제출정답맞힌 사람정답 비율
20 초 (추가 시간 없음) 1024 MB1198100.000%

## 문제

In the game "Duck, Duck, Goose", all players but one sit on the floor and form a circle. The remaining player walks around the circle calling each player "duck" until they select one sitting player and, while touching their head, call them "goose" instead. At that point, the goose chases the selecting player and our interest in the game fades.

In the new game "Duck, Duck, Geese", the walking player instead chooses a contiguous subset of at least two (but not all) sitting players to be "geese"! Furthermore, each sitting player is wearing a hat. Each hat is one of $\mathbf{C}$ possible colors, numbered $1$ through $\mathbf{C}$.

For each color $i$, the quantity of selected geese wearing a hat of color $i$ must be either $0$ or between $\mathbf{A_i}$ and $\mathbf{B_i}$, inclusive.

Can you help count the number of choices that fulfill these requirements? Two choices are considered different if there is some player that is included in one choice but not the other.

## 입력

The first line of the input gives the number of test cases, $\mathbf{T}$. $\mathbf{T}$ test cases follow. Each test case starts with a line containing two integers $\mathbf{N}$ and $\mathbf{C}$: the number of sitting players and hat colors, respectively. Then, $\mathbf{C}$ lines follow. The $i$-th of these lines contains two integers $\mathbf{A_i}$ and $\mathbf{B_i}$, as explained above. The last line of a test case contains $\mathbf{N}$ integers $\mathbf{P_1}, \mathbf{P_2}, \dots, \mathbf{P_N}$ representing that the $j$-th sitting player in clockwise order (starting from an arbitrary one) is wearing a hat of color $\mathbf{P_j}$.

## 출력

For each test case, output one line containing Case #x: y, where $x$ is the test case number (starting from 1) and $y$ is the number of sets of at least $2$ and at most $\mathbf{N}-1$ contiguously sitting players that fulfill all the color requirements.

## 제한

• $1 \le \mathbf{T} \le 100$.
• $2 \le \mathbf{C} \le \mathbf{N}$.
• $0 \le \mathbf{A_i} \le \mathbf{B_i} \le \mathbf{N}$, for all $i$.
• $1 \le \mathbf{P_j} \le \mathbf{C}$, for all $j$.

## Test Set 1 (12점)

• $3 \le \mathbf{N} \le 1000$.

## Test Set 2 (13점)

• $3 \le \mathbf{N} \le 10^5$.

## 예제 입력 1

3
3 2
1 1
1 1
1 1 2
5 2
1 1
1 2
1 2 1 2 2
3 3
1 2
1 2
2 2
1 1 3

## 예제 출력 1

Case #1: 2
Case #2: 9
Case #3: 1

## 힌트

In Sample Case #1, the total number of players chosen as geese must be $2$. There are only three possible ways to select $2$ players. The following color configurations are possible: $[1, 1]$, $[1, 2]$, and $[2, 1]$. The first one has two players wearing hats of color $1$, so it is not valid, but the other two are valid. Therefore the answer is $2$.

Sample Case #2 is the one illustrated in the statement, with color $1$ being yellow and color $2$ being blue. The total number of players chosen as geese in this case must be between $2$ and $3$, because selecting $4$ geese would require at least one color to be out of bounds. For cases with $2$ geese, the only requirement is that we do not select $2$ geese both wearing hats of color $1$; all $5$ such selections are valid. If choosing $3$ geese, the options are $[1, 2, 1]$, $[2, 1, 2]$, $[1, 2, 2]$, $[2, 2, 1]$, or $[2, 1, 2]$. All but the first one are valid, adding another $4$ valid options, for a total of $9$.

In Sample Case #3, notice that there can be hat colors that nobody is wearing. In this case, since there is only $1$ player wearing hat color $3$ and $1$ is not in range, the only valid way is to pick $0$ players wearing that hat color.

## 채점 및 기타 정보

• 예제는 채점하지 않는다.