Subset (6)

UltraData-Math-L1 · 86M rows

Split (1)

train · 86M rows

content stringlengths 86 994k	meta stringlengths 288 619
Who's That Mathematician? Paul R. Halmos Collection - Page 7 For more information about Paul R. Halmos (1916-2006) and about the Paul R. Halmos Photograph Collection, please see the introduction to this article on page 1. A new page featuring six photographs will be posted at the start of each week during 2012. Halmos photographed David Blackwell (1919-2010) in August of 1976 in Toronto, Ontario, Canada. Blackwell earned his Ph.D. in 1941 at the University of Illinois, Urbana-Champaign, under probabilist Joseph Doob, with a dissertation on Markov chains. He was awarded a fellowship at the Institute for Advanced Study (IAS) in Princeton for 1941-42. After two one-year appointments at colleges in Louisiana and Georgia, Blackwell became professor and, very quickly, chair at Howard University in Washington, D.C. At Howard, Blackwell’s research focus changed to theoretical statistics and, in 1955, he accepted a position at the University of California, Berkeley, in the Department of Mathematics and Statistics, where he spent the rest of his career and advised at least 65 Ph.D. students. As an African-American mathematician and statistician, Blackwell faced many challenges and attained many firsts, including becoming the first African-American to be awarded an IAS fellowship, the first to be elected to the National Academy of Sciences, and the first to earn tenure at UC Berkeley. The annual National Association of Mathematicians (NAM)-MAA David Blackwell Lecture was instituted in 1994, and remains one of the featured invited addresses of the annual MAA MathFest. Howard University will host the David Blackwell Memorial Conference, to be held April 19-20, 2012, on the Howard University campus in Washington, D.C. The conference will celebrate Blackwell's legacy, including his contributions to probability theory, theoretical statistics, operations research, and game theory. David Blackwell and Paul Halmos were photographed on June 30, 1998. “Two geezers” is the caption supplied by Halmos on the back of the photo. Halmos and Blackwell attended the University of Illinois, Urbana-Champaign, at about the same time, with Halmos entering as a 15-year-old in 1931 and Blackwell as a 16-year-old in 1935. Both earned Ph.D.s at the age of 22 with probabilist Joseph Doob, Halmos in 1938 with a dissertation on measure-theoretic probability and Blackwell in 1941 with a dissertation on properties of Markov chains. Halmos spent 1938-39 as an instructor at Illinois, and then moved to the Institute for Advanced Study (IAS) in Princeton for three years, 1939-42. Blackwell was awarded an IAS fellowship for 1941-42. In 1942, Halmos joined the faculty at Syracuse University, where he would remain for four years, and Blackwell, after two one-year appointments at colleges in Louisiana and Georgia, became a professor at Howard University in Washington, D.C. Halmos and Blackwell remained lifelong friends, and both stayed close to Doob as well. Another friend was Warren Ambrose, who completed his Ph.D. with Doob at Illinois in 1939 and was at IAS from 1939 to 1941. Blackwell, Halmos, Ambrose, and Doob also are pictured on page 1 of this collection. Halmos photographed Ralph P. Boas, Jr. (1912-1992) in 1980. Boas earned his Ph.D. at Harvard University in 1937. In 1943-45, he taught at Harvard, where he advised the Ph.D.s of R. Creighton Buck and Philip J. Davis, but he spent most of his career at Northwestern University in Evanston, Illinois. In addition to his research in real and complex analysis, Boas did extensive editorial work and was active in both the AMS and MAA, serving as MAA president in 1973-74. His son, Harold Boas, who earned his Ph.D. in complex analysis from M.I.T. in 1980 and is professor of mathematics at Texas A&M University, wrote in March of 2012 that “my father's unique bowties -- he always wore one -- were hand-made by my mother.” Mary L. Boas (1917-2010), who earned her Ph.D. in physics from M.I.T. in 1948, was professor of physics at DePaul University in Chicago for 30 years. Graph theorist Béla Bollobás in May of 1973 at Cambridge University, England. Born in Budapest, Hungary, Bollobás caught the attention of mathematician Paul Erdös (pictured on page 3) while still a young teenager. Erdös became a mentor to Bollobás and interested him in graph theory. Bollobás earned Ph.D.s at Budapest University and, after he finally was allowed to leave Hungary, at Cambridge University. He became a fellow of Trinity College, Cambridge, in 1970, and, since 1996, has held positions both at Cambridge and at the University of Memphis, Tennessee. Next, we have the two 1974 Fields medalists Enrico Bombieri and David Mumford on August 21, 1974, at the International Congress of Mathematicians (ICM) in Vancouver, B.C., Canada, where they were awarded their medals. Bombieri believes this photograph may have been taken the day after the awards ceremony, which would have been the second day of the conference. He reported in March of 2012 that he and his friend Mumford had worked together in 1973, producing "two joint papers on classification of algebraic surfaces in positive characteristic. Our Fields Medals were a totally unexpected surprise for both of us." Enrico Bombieri (left) earned his Fields Medal primarily for advances in number theory, but also for results in analysis and in the partial differential equations of minimal surfaces. He studied in his native Milan, Italy, and also in Cambridge, England, and then taught in Pisa, Italy, for a number of years. In 1977, he became a faculty member of the School of Mathematics of the Institute for Advanced Study (IAS) in Princeton, New Jersey, and in 1988 became IBM von Neumann Professor at IAS. David Mumford (right) earned his Fields Medal for advances in algebraic geometry. He spent most of his career at Harvard, moving at the turn of the millennium to the Division of Applied Mathematics at Brown University, where he has led the pattern theory group and has been affiliated with the brain science program, focusing on visual perception. He has supervised at least 47 Ph.D. students and was awarded the National Medal of Science in 2010. Halmos photographed functional analyst Frank Bonsall (1920-2011) and number theorist Robert Rankin (1915-2001) on July 1, 1980 in St. Andrews, Scotland. Bonsall studied at Oxford, then spent over a decade at Newcastle University, before becoming the first occupant of the Maclaurin Chair at the University of Edinburgh, a position he held for 20 years. Rankin earned his degree at Cambridge and spent most of his career at the University of Glasgow, Scotland. While at Cambridge, he had worked with G. H. Hardy on the mathematics of Srinivasa Ramanujan , and he continued to publish both mathematical and historical work on Ramanujan throughout his career. For an introduction to this article and to the Paul R. Halmos Photograph Collection, please see page 1. Watch for a new page featuring six new photographs each week during 2012. Biographies of David Blackwell (1919-2010): Other Sources for this page: • Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin. Information for which a source is neither cited nor given in this list either appeared on the reverse side of the photograph or was obtained by AAM archivist Carol Mead from various sources during 2011-12. • Harold Boas, email dated 2 March 2012 (Ralph P. Boas, Jr.) • Enrico Bombieri, emails dated 5 March 2012 and 6 March 2012 (Bombieri, David Mumford) • Brown University Division of Applied Mathematics (Philip J. Davis, David Mumford) • DePaul University Physics Department (Mary L. Boas) • Institute for Advanced Study (IAS) Faculty (Enrico Bombieri) • Institute for Advanced Study (IAS) Past Members (Ambrose, Blackwell, Halmos, and others) • International Mathematical Union (IMU) Fields Medal and Medalists (Bombieri, Mumford) • MacTutor History of Mathematics Archive, St. Andrews University, Scotland (all mathematicians except Ambrose, Boas, Buck, and Davis) • Massachusetts Institute of Technology (MIT) News Office (Warren Ambrose) • MAA Presidents (Ralph P. Boas, Jr.) • Mathematics Genealogy Project, North Dakota State University (all mathematicians) • Texas A&M University Mathematics Department (Harold Boas) • University of Wisconsin Department of Mathematics (R. Creighton Buck)	{"url":"https://old.maa.org/press/periodicals/convergence/whos-that-mathematician-paul-r-halmos-collection-page-7","timestamp":"2024-11-11T11:51:34Z","content_type":"application/xhtml+xml","content_length":"131192","record_id":"<urn:uuid:8cf8c8b1-4bf7-4b7d-a9a3-81ea03cb4a4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00092.warc.gz"}
Isis tutorial fitting1 Jump to navigation Jump to search Advanced Fitting Techniques, 1 In this part of the tutorial we look at some more advanced techniques that are often used in spectral fitting, namely • rebinning data • fitting data from multiple instruments simultaneously • more complicated error calculations Rebinning Data As you might have noticed in the last part of the tutorial, in observations of faint objects there are often only a few photons detected in each spectral bin. If this is the case, then the assumptions that make [math]\displaystyle{ \chi^2 }[/math] a good statistics for minimization during fitting are not valid. Specifically, the [math]\displaystyle{ \chi^2 }[/math] statistics assumes that all summands entering the [math]\displaystyle{ \chi^2 }[/math]-sum are normal distributed. In reality, however, counting data are Poisson distributed. Rebinning or grouping the spectrum means that adjacent channels are added and then treated as one "meta-channel" during the spectral analysis. As a rule of thumb, when using [math]\displaystyle{ \chi^2 }[/math], statistics the number of photons in each of these "meta-channels" should be above 20, since the Poisson statistics with [math]\displaystyle{ N\gt 20 }[/math] is sufficiently close to the Normal distribution (for more detailed calculations, see Gehrels, 1986). isis offers several ways to group your data. isis-intrinsic are group_data and rebin_data, but the most convenient of the grouping functions is group, which is part of isisscripts (so you need to load the isisscripts before you can use it). The syntax of group is this function will take a list of spectral IDs and bin these spectra to a minimum signal to noise ratio and/or ensure that at least min_chan channels are added together. The latter option is useful since often data are taken on an energy grid that significantly oversamples the detector resolution. This means that individual channels are not statistically independent, contrary to the assumptions of the [math]\displaystyle{ \chi^2 }[/math]-statistics (as a rule of thumb, oversampling by a factor 2-3 is ok, but more is overkill). The bounds qualifier allows you to apply group to just a subset of channels (e.g., you can use a different binning for, say. the band from 2-5keV and the band from 5-10keV. Set unit to keV if you want to work in keV-space. As an example of grouping, let us take a look at the spectrum of Capella we looked at in the last tutorial and rebin this spectrum to a signal to noise ratio of 5: isis> require("isisscripts"); isis> a=load_data("capella.pha2"); : assign arf and rmf... isis> variable leg1=3; isis> plot_unit("A"); isis> plot_counts(leg1); isis> group([leg1];min_sn=3); isis> plot_counts(leg1); As can be clearly seen in the following two plots, which show the non-binned and the binned spectrum, while the signal to noise ratio of the data goes up, the number of bins and thus the resolution of the data goes down. This means that narrow lines can be made vanish with binning... Exercise 1 • assuming Poisson statistics, what is the signal-to-noise ratio needed to ensure that at least 20 counts are in a spectral bin. In your estimate, assume that background is negligible. • rebin the data set from the previous exercises to this signal-to-noise ratio and refit. Discuss how (and why) [math]\displaystyle{ \chi^2 }[/math] changes. • (optional) An alternative to fitting data with low signal to noise is to take into account the Poisson distribution and not rebin. Instead of chi-squared-fitting, in this case one has to use the so-called cash-statistics (named after Webster Cash who first introduced this statistics into X-ray astronomy). You can switch isis from chi-squared to Cash-statistics using the set_fit_statistic-function. Read its help and redo your fit on the unbinned spectrum using the Cash statistics. Alternatively, for moderate count rates, you can also continue using the chi-squared statistics but using the recommendations of Gehrels for the calculation of the uncertainty. Try this as well, if you are brave! Data from multiple instruments In previous parts of this tutorial we have already seen that when isis has loaded several spectra, these will usually be fit simultaneously. So far, however, we always applied the same spectral model to all loaded spectra when fitting the data. In many cases this is not appropriate. For example: 1. Data might be from the same source, but taken during different source states. In this case, e.g., the foreground absorption might be the same, but individual spectral parameters might have to be fitted independently. 2. Data might be from a simultaneous observation with different instruments. While in this case the spectral model used in fitting should be the same for all instruments, there could be slight differences in the flux calibration of these instruments. This is very typical, since flux calibration is very difficult. This means that we need a way to distinguish between different instruments in spectral fitting. Case 2 above is by far the most common case. Here, the flux calibration can usually be taken into account by introducing a multiplicative constant in front of the spectral model that takes into account the different flux normalizations. Exercise 2 • In the following we look at a data set of the accreting neutron star Vela X-1 that was taken in February 1996 with the Rossi X-ray Timing Explorer. The data are available at [[1]]. In the following we work with the dataset in subdirectory gruppe05. Do the same with the data set assigned to your group. For your information: the observation covered one full orbit of the neutron star and the absorption caused by the stellar wind of the donor star was strongly variable over the outburst. Depending on your data set you will therefore get very different answers than the ones listed here. We first load isisscripts and the data. The internal numbers of the observations are stored in the isis variables pca, hxta, and hxtb. First plot the spectrum to take a look at what is available: From experience we know that the PCA-data are good in 2.5-25keV, while HEXTE data are only good above 20keV. So let's ignore the data and see what the spectrum looks like: Note that the different count rates are mainly due to the different effective areas of the instruments. The HEXTE data are consistent with background above 100keV or so and definitively also in need of rebinning below that. We rebin both HEXTE spectra to a signal to noise of three, then plot the data and then decide to work with the 20-60keV band only isis> group([hxta,hxtb];min_sn=3); : plot etc. isis> xnotice_en([hxta,hxtb],20.,60.); This results in the following nice spectrum: To get a better feel for what the spectral continuum looks like, let's fit a simple model. From experience we know that an absorbed powerlaw with an exponential cutoff and an iron line at 6.4keV (with some experience you can see this line in the raw data!) is generally a good empirical model for such sources. It is advisable, to first fit the model without the iron line and then add the iron line to this initial continuum. The best fit looks as follows: isis> list_par; idx param tie-to freeze value min max 1 phabs(1).nH 0 0 29.52458 0 100000 10^22 2 egauss(1).area 0 0 0.006779961 0 0 photons/s/cm^2 3 egauss(1).center 0 1 6.4 0 0 keV 4 egauss(1).sigma 0 0 1e-06 1e-06 1 keV 5 cutoffpl(1).norm 0 0 0.07710349 0 1e+10 6 cutoffpl(1).PhoIndex 0 0 0.1244691 -2 9 7 cutoffpl(1).HighECut 0 0 10.40836 1 500 keV isis> renorm_counts; plot_data({pca,hxta,hxtb};dcol={4,2,3},res=1); Parameters[Variable] = 7[1] Data bins = 145 Chi-square = 8282.497 Reduced chi-square = 57.51734 Note that the residuals are very wavy and that there is a clear offset between the PCA and HEXTE data, even though the predicted model count rate for HEXTE is much less than that for the PCA. This effect is caused by the different assumed flux normalizations of the three instruments. In order to solve this problem we multiply each instrument with an individual normalization constant and fit these normalization constants independently of each other. In order to do so, one must know that isis evaluates the model individually for each of the loaded data sets. During the evaluation, it sets the internal isis variable Isis_Active_Dataset to the index of the data set that it is working on. This can be used to implement very complicated, instrument dependent, behavior. We now setup the model such that for each instrument we have an independent multiplicative constant: isis> fit_fun("constant(Isis_Active_Dataset)phabs(1)(egauss(1)+cutoffpl(1))"); isis> list_par; idx param tie-to freeze value min max 1 constant(1).factor 0 0 1 0 1e+10 2 phabs(1).nH 0 0 29.52458 0 100000 10^22 3 egauss(1).area 0 0 0.006779962 0 0 photons/s/cm^2 4 egauss(1).center 0 1 6.4 0 0 keV 5 egauss(1).sigma 0 0 1e-06 1e-06 1 keV 6 cutoffpl(1).norm 0 0 0.0771035 0 1e+10 7 cutoffpl(1).PhoIndex 0 0 0.1244691 -2 9 8 cutoffpl(1).HighECut 0 0 10.40836 1 500 keV 9 constant(2).factor 0 0 1 0 1e+10 10 constant(3).factor 0 0 1 0 1e+10 Note that we now have three multiplicative factors: constant(1), constant(2), and constant(3). Since pca has the value of 1, constant(1).factor is the multiplicative factor for the pca, and the other two constants are for HEXTE A and B. Exercise 3 Check this claim by setting constant(1).factor to a number different from 1 and show with a few plots that the model component for PCA moves up and down while the components for HEXTE A and HEXTE B stay unchanged. By tradition, RXTE flux normalizations are done with respect to the PCA. We therefore freeze constant(1).factor to 1 and let the other constants vary. The final fit is significantly improved with respect to our initial fit: isis> fit_counts; Parameters[Variable] = 10[8] Data bins = 145 Chi-square = 6471.131 Reduced chi-square = 47.23453 isis> list_par; idx param tie-to freeze value min max 1 constant(1).factor 0 1 1 0 1e+10 2 phabs(1).nH 0 0 30.74619 0 100000 10^22 3 egauss(1).area 0 0 0.006348366 0 0 photons/s/cm^2 4 egauss(1).center 0 1 6.4 0 0 keV 5 egauss(1).sigma 0 0 1e-06 1e-06 1 keV 6 cutoffpl(1).norm 0 0 0.09623164 0 1e+10 7 cutoffpl(1).PhoIndex 0 0 0.2630442 -2 9 8 cutoffpl(1).HighECut 0 0 11.7052 1 500 keV 9 constant(2).factor 0 0 0.85297 0 1e+10 10 constant(3).factor 0 0 0.8583705 0 1e+10 Values of around 0.85 are typical for the cross normalization of the HEXTE and the PCA. Note that in the best fit the residuals in the PCA/HEXTE region overlap rather nicely! To make further progress with this data set, we would now need to add further features to the spectrum and life gets complicated. What works for Vela X-1 is a so-called partial covering spectrum of the form [math]\displaystyle{ F_\mathrm{ph}(E) = \mathrm{abs}(N_{\mathrm{H},1}) (1-f+f \mathrm{abs}(N_{\mathrm{H},2})(A E^{-\Gamma}\exp(-E/E_{cut}) + g_{1} + g_{2} + b )$ \lt /latex\gt In this model, the spectral continuum consists of two iron lines (g), the power law continuum, and a black body component (b). A fraction f of this continuum is absorbed by a large column \lt math\gt N_{\mathrm{H},1} } [/math], while all of the continuum is absorbed in addition by the foreground absorption, [math]\displaystyle{ N_{\mathrm{H},2} }[/math]. The isis implementation of this model, which takes also into account the normalization constants is as follows. Note the use of ranges (read the help for set_par) that limit the range of constant(10).factor, which is f in the above equation, to [math]\ displaystyle{ 0\le f \le 1 }[/math]! isis> list_par; idx param tie-to freeze value min max 1 constant(1).factor 0 1 1 0 1e+10 2 phabs(1).nH 0 0 0 0 100000 10^22 3 constant(10).factor 0 0 0.9770158 0 1 4 phabs(2).nH 0 0 38.19049 0 100000 10^22 5 egauss(1).area 0 0 0.001464248 0 0 otons/s/cm^2 6 egauss(1).center 0 1 6.4 0 0 keV 7 egauss(1).sigma 0 0 1e-06 1e-06 1 keV 8 egauss(2).area 0 0 0.003165643 0 0 photons/s/cm^2 9 egauss(2).center 0 0 7.1 0 0 keV 10 egauss(2).sigma 0 0 0.002 1e-06 1 keV 11 bbody(1).norm 0 0 0.01791839 0 1e+10 12 bbody(1).kT 0 0 1.678295 0.01 100 keV 13 cutoffpl(1).norm 0 0 0.02339737 0 1e+10 14 cutoffpl(1).PhoIndex 0 0 -0.4117899 -2 9 15 cutoffpl(1).HighECut 0 0 8.728334 1 500 keV 16 constant(2).factor 0 0 0.8496358 0 1e+10 17 constant(3).factor 0 0 0.8566091 0 1e+10 isis> renorm_counts; Parameters[Variable] = 17[1] Data bins = 145 Chi-square = 1371.38 Reduced chi-square = 9.523473 The best fit looks as follows: Further progress with this data set could be made by using a better continuum description (an exponentially cutoff power law is a very rough approximation to the correct spectrum), but we will not do this here since the main point of this tutorial was to introduce techniques to tackle much more complicated spectral models than the ones we have encountered so far. And enough is enough... Error Contours In the previous part of this tutorial we already encountered the calculation of error bars using the vconf function. The approach used there is ok if there are no correlations between different fit parameters. In practice, this is rarely the case and different parameters are expected to be - sometimes strongly -correlated. The absorbed power law spectrum discussed above for Vela X-1 is a good example: At the resolution of the detector, a spectrum that is slightly steeper (higher Gamma) but slightly more absorbed (higher [math]\displaystyle{ N_\mathrm{H} }[/math]) is at some level indistinguishable from a harder, less absorbed one. Instead of talking of individual spectra it is therefore often better to look at the joint confidence contours between two (or more) parameters. The approach of calculating these contours is the same as the one used for the confidence contours for one parameter of interest, only that now we have to look at the behavior of the [math]\ displaystyle{ \chi^2 }[/math]-valley for two parameters. This means that the number of fits to be performed to find the confidence range has just gone up quadratically. However, joint confidence contours are very important and therefore isis provides an useful routine for us. To illustrate the calculation, in the following we use a simple absorbed power law to the PCA data of Vela X-1 to do the calculation. In real life, the following calculation would have to be performed on the full data set, however, we're using a simplification here to speed things up a little bit. Since I managed to crash this writeup once, I am only giving a short introduction here in order not to stop you from doing the exercise. Please bug me to update this section later! In isis, two-parameter confidence contours are calculated with the function conf_map_counts. The syntax of this function is as follows: Struct_Type = conf_map_counts (Struct_Type x, Struct_Type y [, info]) where x and y are each a structure of the form Struct_Type = struct {index, min, max, num}; listing the index of the parameter to be stepped over the range from min to max using num steps. Note that you can also use the function conf_grid(index, min, max, num) to create the above structure The help-information for conf_map_counts is very exhaustive and contains several well described examples. Using your above fit, write an isis-program that calculates the joint confidence contour between the absorption column and the photon index using conf_map_counts and that uses plot_conf to plot the 1sigma, 90% and 3 sigma confidence contours. Interpret the result!	{"url":"https://www.sternwarte.uni-erlangen.de/wiki/index.php/Isis_tutorial_fitting1","timestamp":"2024-11-06T19:58:09Z","content_type":"text/html","content_length":"38824","record_id":"<urn:uuid:9729409f-bb02-4272-b310-c09fe351543c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00307.warc.gz"}
Research in this group is focused on a variety of problems in the areas of statistical physics and condensed matter physics. Some specific examples are given below. • Anomalous heat transport in low-dimensional systems Many studies over the last few decades, indicate that, in low dimensional macroscopic systems, Fourier’s law of heat conduction is not valid. It appears that heat flow is not diffusive but instead super-diffusive which means that the thermal conductivity of low-dimensional materials is infinite. In our group, efforts are focused on establishing this fact, establishing universality classes and in finding the correct hydrodynamic description of such systems. So far, these results are established for the simplest non-trivial models and a big challenge is to extend these studies to more realistic models, which would be directly relevant for understanding transport in systems such as nanowires, nanotubes, polymers and also two dimensional systems such as graphene. • Quantum transport There is research on developing a microscopic theory of electric and heat transport and in understanding noise and fluctuations in open quantum systems, These research areas involve basic and fundamental problems in theoretical physics and at the same time are of great current interest because they are directly relevant for understanding experiments at the micro- and nano- scales. Some of the techniques that are being investigated here include quantum Langevin and quantum master equations, including the Redfield and Lindblad approaches. Diagrammatic Keldysh nonequilibrium Green's function approach is also used in the group to investigate Open Quantum Systems. Applications of the formalism have been made towards understanding transport in quasi-periodic systems such as the Aubry-Andre-Harper model which is a simple one-dimensional system with a rich non-equilibrium phase diagram. The group is interested in using these methods for understanding systems such as light-matter hybrid systems, cavity-QED and circuit-QED systems, and look at fundamental aspects (such entanglement, transport, fluctuations) as well as potential device applications. • Stochastic many particle systems Usually when many interacting particle systems are taken out of equilibrium by driving them locally, globally or through boundaries, in low dimensions they manifests themselves through various interesting phenomena like anomalous tagged particle diffusion, appearance of long range correlations (both static and dynamic), anomalous transport, nonlinear and singular temperature profiles etc. Our group is focused on understanding such features at large length and time scales using appropriate hydrodynamic as well as microscopic descriptions. The group is also interested in some other topics which include stochastic processes, fluctuations and large deviations in non-equilibrium systems, extremal and geometrical properties of Brownian motions, first-passage problems and related stochastic search problems. Another recent direction we are pursuing is towards obtaining rigorous results for the stochastic dynamics of so called active particles. • Quantum many-body physics Quantum systems with many interacting degrees of freedom like electrons show a rich array of emergent behaviour like magnetism, superconductivity. Recent discovery of a large number of materials including high temperature superconductors, various frustrated magnets, fractional quantum Hall systems and topological band insulators seem to indicate the existence of a whole new class of condensed matter phases whose properties cannot be captured within the conventional paradigm of condensed matter theory. Present research suggests that a proper understanding of these new phases would have to take into account the the subtle interplay of symmetries and quantum entanglement. The work of our group focuses on understanding such phases both from the point of view of theoretical framework building and experimental phenomenology. More particularly, the interest is in unconventional quantum magnets called quantum spin liquids, interacting symmetry protected topological phases, correlated metals and interplay of spin-orbit coupling and electron correlations in such systems. • Dynamics of Integrable Models The group is interested in dynamics of integrable models (for e.g., Calogero model and its various generalizations and the Toda system) both from microscopic and collective field theory point of view. In particular, solitons and duality in integrable models and the form of dynamical correlation functions are under investigation. Connection of physical systems (such as a collective behaviour of cold atoms) with integrable and non-integrable PDE’s is an active area of interest for the group. You can visit the individual homepages of the members of this group to find out more about their research activities and collaborations.	{"url":"https://icts.res.in/research/statphys","timestamp":"2024-11-07T09:07:34Z","content_type":"text/html","content_length":"79242","record_id":"<urn:uuid:dbd98a7d-06bf-4aa1-96fc-5481b412812d>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00127.warc.gz"}
tree of life superstring theory part 138 The 8 triacontagons in the E8 Coxeter plane projection of the 421 polytope consist of two sets of 4 (shown coloured red & blue). Each set of 120 vertices is the Coxeter plane projection of a 600-cell. If we consider a representative sector in each triacontagon, the sum of the base angles in either set =4×168 = 672° and the sum of the vertex angles = 4×12 = 48°. The sum of the angles of a sector in each set = 720°. The sum of the angles of the 8 representative sectors = 720° + 720° = 1440° = 48° + 672° + 48° + 672°. Compare this with the yod population of the (7+7) separate Type B polygon: Surrounding the centre of a Type B n-gon are 15n yods. Surrounding the centres of each set of 7 Type B polygons are 48 corners and (48×14=672) yods, i.e., 720 yods in all. The correspondence indicates that a yod denotes a single degree; the 48 corners of each set of polyons denote the 48 degrees in the sum of the vertex angles of the 4 representative sectors in each set of 4 triacontagons, whilst the 672 yods denote the sum of their base angles. The red triangle, square, pentagon & dodecagon have 24 corners and 336 other yods, as do the blue hexagon, octagon & decagon. The 360 yods in each subset of polygons symbolise the 360 degrees of a circle, corresponding to two sectors. One set of 7 polygons corresponds to the representative sectors in one set of 4 triacontagons associated with one 600-cell and the other set corresponds to the second 600-cell. In terms of angles, the two halves express revolutions of (2+2) full circles.	{"url":"https://www.64tge8st.com/post/2018/03/17/tree-of-life-superstring-theory-part-138","timestamp":"2024-11-12T20:25:21Z","content_type":"text/html","content_length":"1050957","record_id":"<urn:uuid:d78bcfbd-2e09-4230-9da6-89fbf1e2cd1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00317.warc.gz"}
Stratified Weibull Regression Model for Interval-Censored Data In many clinical studies, the time to a silent event is known only up to an interval defined by the times of the last negative and first positive diagnostic test. Event times arising from such studies are referred to as ‘interval-censored’ data. For example, in pediatric HIV clinical studies, the timing of HIV infection is known only up to the interval from the last negative to the first positive HIV diagnostic test (Dunn et al. 2000). Examples of interval-censored outcomes can also be found in many other medical studies (Gomez et al. 2009). A rich literature exists on the analysis of interval-censored outcomes. Non-parametric approaches include the self-consistency algorithm for the estimation of the survival function (Turnbull 1976). A semi-parametric approach based on the proportional hazards model has been developed for interval-censored data (Finkelstein 1986; Goetghebeur and Ryan 2000). A variety of parametric models can also be used to estimate the distribution of the time to the event of interest, in the presence of interval-censoring (Lindsey and Ryan 1998). An often used parametric approach for the analysis of interval-censored data is based on the assumption of a Weibull distribution for the event times (Lindsey and Ryan 1998). The Weibull distribution is appropriate for modeling event times when the hazard function can be reliably assumed to be monotone. Covariate effects can be modeled through the assumption of proportional hazards (PH), which assumes that the ratio of hazard functions when comparing individuals in different strata defined by explanatory variables is time-invariant. The article by Gomez et al. (2009) presents a comprehensive review of the state-of-the-art techniques available for the analysis of interval-censored data. In this paper, we implement a parametric approach for modeling covariates applicable to interval-censored outcomes, but where the assumption of proportional hazards may be questionable for a certain subset of explanatory variables. For this setting, we implement a stratified Weibull model by relaxing the PH assumption across levels of a subset of explanatory variables. We compare the proposed model to an alternative stratified Weibull regression model that is currently implemented in the R package survival (Therneau 2012). We illustrate the difference between these two models analytically and through simulation. The paper is organized as follows: In Section 2, we present and compare two models for relaxing the PH assumption, based on the assumption of a Weibull distribution for the time to event of interest. In this section, we discuss estimation of the unknown parameters of interest, hazard ratios comparing different groups of subjects based on specific values of explanatory covariates and tests of the PH assumption. These methods are implemented in a new R package, straweib (Gu and R. Balasubramanian 2013). In Section 3, we perform simulation studies to compare two stratified Weibull models implemented in R packages straweib and survival. In Section 4, we illustrate the use of the R package straweib by analyzing data from a longitudinal oral health study on the timing of the emergence of permanent teeth in 4430 children in Belgium (Leroy et al. 2003; Gomez et al. 2009). In Section 5, we discuss the models implemented in this paper and present concluding remarks. Weibull regression models Let $T$ denote the continuous, non-negative random variable corresponding to the time to event of interest, with corresponding probability distribution function (pdf) and cumulative distribution function (cdf), denoted by $f(t)$ and $F(t)$, respectively. We let $S(t) = 1- F(t)$ to denote the corresponding survival function and $h(t) = lim_{\delta t \to 0} \frac{P( t \le T < t + \delta t \mid T \ge t)}{\delta t}$ to denote the hazard function. We let Z denote the $p \times 1$ vector of explanatory variables or covariates. We assume that the random variable $T \mid \boldsymbol{Z} = \boldsymbol{0}$ is distributed according to a Weibull distribution, with scale and shape parameters denoted by $\lambda$ and $\gamma$ , respectively. The well known PH model to accommodate the effect of covariates on $T$ is expressed as: \[h(t \mid \boldsymbol{Z}) = h(t \mid \boldsymbol{Z=0}) \times \exp(\boldsymbol{\beta'} \ boldsymbol{Z})\,,\] where $\boldsymbol{\beta}$ denotes the $p \times 1$ vector of regression coefficients corresponding to the vector of explanatory variables, Z. Thus, under the Weibull PH model, the survival and hazard functions corresponding to $T$ can be expressed as \[\begin{aligned} \label{eq:weibphS} S(t \mid \boldsymbol{Z}) &=\exp\left(-\lambda \exp\left(\boldsymbol \beta ' \boldsymbol Z\right) t^\gamma\right) \end{aligned} \tag{1}\] \[\begin{aligned} \label{eq:weibphH} h(t \mid \boldsymbol{Z}) &= \lambda \exp(\boldsymbol \beta ' \boldsymbol Z) \gamma t^{\gamma-1} \end{aligned} \tag{2}\] where, $\lambda > 0$ and $\gamma > 0$ correspond to the scale and shape parameters corresponding to $T$ when $\boldsymbol{Z} = \boldsymbol{0}$. The hazard ratio comparing two individuals with covariate vectors Z and $\boldsymbol{Z}^{}$ is equal to $\exp(\boldsymbol \beta '( \boldsymbol{Z} - \boldsymbol{Z}^{}))$. Stratified Weibull regression model implemented in the R package survival In this section, we describe the stratified Weibull PH regression model implemented in the the R package survival (Therneau 2012). Consider the following log-linear model for the random variable $T$: \[log(T \mid \boldsymbol{Z} ) = \mu + \alpha_1 Z_{1} + \cdots \alpha_p Z_{p} + \sigma \epsilon\] where, $\alpha_1, \cdots, \ alpha_p$ denote unknown regression coefficients corresponding to the $p$ dimensional vector of explanatory variables, $\mu$ denotes the intercept, and $\sigma$ denotes the scale parameter. The random variable $\epsilon$ captures the random deviation of event times on the natural logarithm scale (i. e. log($T$)) from the linear model as a function of the covariate vector Z. In general, the log-linear form of the model for $T$ can be shown to be equivalent to the accelerated failure time (AFT) model (Collett 2003). The assumption of a standard Gumbel distribution with location and scale parameters equal to 0 and 1, respectively, implies that the random variable $T$ follows a Weibull distribution. Moreover, in this case, both the PH and AFT assumptions (or equivalently, the log-linear model) lead to identical models with different parameterizations (Collett 2003). The survival and hazard functions can be expressed as: \[\begin{aligned} \label{eq:loglnS} S(t \mid \boldsymbol {Z}) &= \exp \left[ -\exp\left(\frac{\log(t) - \mu - \boldsymbol{\alpha}^{'} \boldsymbol Z}{\sigma}\right) \right ] \\ \end{aligned} \tag{3}\] \[\begin{aligned} \label{eq:loglnH} h(t \mid \boldsymbol{Z}) &=\exp \left[ -\frac{\mu + \boldsymbol{\alpha}^{'} \boldsymbol Z}{\sigma} \right ] \frac{1}{\sigma}t^{\frac{1}{\sigma}-1} \end{aligned} \ The coefficients for the explanatory variables ($\boldsymbol{\beta}$) in the hazard function ($h(t \mid \boldsymbol{Z})$) are equal to $-\frac{\boldsymbol \alpha}{\sigma}$. Moreover, there is a one-one correspondence between the parameters $\lambda, \gamma, \boldsymbol{\beta}$ in equations ((1))-((2)) and the parameters $\mu, \sigma, \boldsymbol{\alpha}$ in equations ((3))-((4)), where $\lambda = \exp(-\frac{\mu}{\sigma}), \gamma = \sigma^{-1}$ and $\beta_j = - \frac{\alpha_j}{\sigma}$ (Collett 2003). The log-linear form of the Weibull model can be generalized to allow arbitrary baseline hazard functions within subgroups defined by a stratum indicator $S = 1, \cdots, s$. Thus, the stratified Weibull regression model for an individual in the $j^{th}$ stratum is expressed as: \[\begin{array}{lll} log\left(T \mid \boldsymbol{Z}, S=j\right) &=& \mu_j + \alpha_1 Z_{1} + \cdots \alpha_p Z_ {p} + \sigma_j \epsilon \end{array}\] where $\mu_j$ and $\sigma_j$ denote stratum specific intercept and scale parameters. This model is implemented in the R package survival (Therneau 2012). In this model, the regression coefficients $\boldsymbol{\alpha}$ on the AFT scale are assumed to be stratum independent. However, the hazard ratio comparing two individuals with covariate vectors and stratum indicators denoted by (Z, $S=j$) and ($\boldsymbol{Z^{}}, S = k$) is stratum specific and is given by: \[\begin{aligned} \frac{h\left(t \mid S=j, \boldsymbol{Z}\right)}{h\left(t \mid S=k, \boldsymbol{Z}^{}\right)} &=t^{1/\sigma_j - 1/\sigma_k} \frac{\sigma_k}{\sigma_j} \exp\left({\frac{\mu_k}{\ sigma_k} - \frac{\mu_j}{\sigma_j}}\right) \exp\left(\boldsymbol{\alpha}^{'}\left(\boldsymbol{Z}^{}/\sigma_k - \boldsymbol{Z}/\sigma_j\right)\right) \end{aligned}\] For $j \ne k$, the hazard ratio varies with time $t$. However, when $j=k$, the hazard ratio comparing two individuals within the same stratum $S=j$ is invariant with respect to time $t$ but is stratum-dependent and reduces to: \[\begin{aligned} \frac{h\left(t \mid S=j, \boldsymbol{Z}\right)}{h\left(t \mid S=j, \boldsymbol{Z}^{}\right)} &= \exp\left(\frac{\boldsymbol{\alpha}^{'}}{\sigma_j}\left( \boldsymbol{Z}^{} - \ boldsymbol{Z}\right)\right) \label{eq:compLL} \end{aligned} \tag{5}\] Stratified Weibull regression model implemented in R package straweib In this section, we describe the stratified Weibull regression model that is implemented in the new R package, straweib (Gu and R. Balasubramanian 2013). To relax the proportional hazards assumption in the Weibull regression model, we propose the following model for an individual in the stratum $S=j$: \[\begin{aligned} h(t \mid \boldsymbol{Z}, S=j) &= \lambda_j \exp\left(\boldsymbol \beta ' \boldsymbol{Z}\right) \gamma_j t^{\gamma_j-1} \label{eq:stratweib} \end{aligned} \tag{6}\] Equivalently, the model can be stated in terms of the survival function as: \[\begin{aligned} S(t \mid \boldsymbol{Z}, S=j) &= \exp\left(-\lambda_j \exp\left(\boldsymbol \beta ' \boldsymbol Z\right) t^{\gamma_j}\right) \\ \end{aligned}\] Here, we assume that the scale and shape parameters ($\lambda, \gamma$) are stratum specific - however, the regression coefficients $\boldsymbol{\beta}$ are assumed to be constant across strata ( $S$). The hazard ratio comparing two individuals with covariate vectors and stratum indicators denoted by (Z, $S=j$) and ($\boldsymbol{Z^{}}, S = k$) is given by: \[\begin{aligned} \frac{h(t \ mid S=j, \boldsymbol{Z})}{h(t \mid S=k, \boldsymbol{Z}^{})} &= t^{\gamma_j- \gamma_k} \exp\left(\boldsymbol \beta^{'}\left( \boldsymbol{Z} - \boldsymbol{Z}^{}\right)\right) \frac{ \lambda_j \ gamma_j }{\lambda_k \gamma_k} \end{aligned}\] For $j \ne k$, the hazard ratio varies with time $t$ and thus relaxes the PH assumption. However, for $j=k$, the hazard ratio comparing two individuals within the same stratum $S=j$ reduces to: \[\begin{aligned} \frac{h(t \mid S=j, \boldsymbol{Z})}{h(t \mid S=j, \boldsymbol{Z}^{})} &= \exp\left(\boldsymbol \beta^{'}\left( \boldsymbol{Z} - \boldsymbol{Z}^{}\right)\right) \label{eq:compPH} \end{aligned} \tag{7}\] This hazard ratio is invariant with respect to time $t$ and stratum $S$, as in the stratified Cox model (Collett 2003). Let $u_j=\log(\lambda_j)$ and $v_j=\log(\gamma_j)$. Let $n_j$ denote the number of subjects in stratum $S=j$. For the $k^{th}$ subject in stratum $j$, let $\boldsymbol Z_{jk}$ denote the $p$ dimensional vector of covariates and let $a_{jk}$ and $b_{jk}$ denote the left and right endpoints of the censoring interval. That is, $a_{jk}$ denotes the time of the last negative test and $b_{jk}$ denotes the time of the first positive test for the event of interest. Then the log-likelihood function can be expressed as: \[\begin{array}{rl} l(\boldsymbol v, \boldsymbol u, \boldsymbol \beta)=&\sum_{j=1}^{s} \sum_{k=1}^{n_j} \log \{\exp[-\exp[u_j+\boldsymbol \beta ' \boldsymbol Z_{jk}+\exp(v_j)\log(a_{jk})]] \\ & - \ exp[-\exp[u_j+\boldsymbol \beta ' \boldsymbol Z_{jk}+\exp(v_j)\log(b_{jk})]] \} \\ \end{array}\] The unknown parameters to be estimated are $\boldsymbol v$, $\boldsymbol u$, and $\boldsymbol \beta$. The log-likelihood function can be optimized using the optim function in R. The shape and scale parameters can be estimated from the estimates of $\boldsymbol v$ and $\boldsymbol u$. The covariance matrix of the estimates of these unknown parameters can be obtained by inverting the negative Hessian matrix that is output from the optimization routine (Cox and D. V. Hinkley 1979). Test of the PH assumption One can test whether or not the baseline hazard functions of each strata are proportional to each other, by testing the equality of shape parameters across strata $S= 1, \cdots, s$. That is, \[H_0 : \gamma_1 = \gamma_2 = \cdots = \gamma_s\] or equivalently, \[H_0 : v_1 = v_2 = \cdots = v_s.\] The null hypothesis $H_0$ can be tested using a likelihood ratio test, by comparing a reduced model that assumes that $\gamma_1 = \gamma_2 == \cdots = \gamma_s$ to the full model in ((6)) assuming stratum specific shape parameters. We note that the reduced model is equivalent to the Weibull PH model that includes the stratum indicator $S$ as an explanatory variable. Thus the reduced model has $s-1$ fewer parameters than the stratified model, or the full model. Let $l_F$ and $l_R$ denote the log-likelihoods of the full and reduced models evaluated at their MLE. Then the test statistic $T=-2(l_R - l_F)$ follows a $\chi_{s-1}^2$ distribution under $H_0$. In addition to the likelihood ratio test, one can also use a Wald test to test the null hypothesis $H_0$. The R package straweib illustrated in Section 3 outputs both the Wald and Likelihood Ratio test statistics. Estimating hazard ratios The log hazard ratio comparing two individuals with covariate vectors and stratum indicators denoted by (Z, $S=j$) and ($\boldsymbol{Z^{}}, S = j^$) at time $t$ can be expressed as: \[r_{tjj^ }=\log\left(R_{tjj^}\right) = u_j + v_j + \log\left(t\right)\exp\left(v_j\right) - u_{j^} - v_{j^} - \log\left( t \right)\exp\left(v_{j^}\right) + \boldsymbol{\beta}^{'}\left(\boldsymbol{Z} - \ boldsymbol{Z}^{}\right)\] Let $\boldsymbol{\hat{v}}$, $\boldsymbol{\hat{u}}$ and $\boldsymbol{\hat{\beta}}$ denote the maximum likelihood estimates for $\boldsymbol v$, $\boldsymbol u$ and $\boldsymbol{\beta}$, then $r_{tjj^}$ can be estimated by \[\hat r_{tjj^} = \hat u_j + \hat v_j + \log\left(t\right) \exp\left(\hat v_j\right) - \hat u_{j^} - \hat v_{j^} - \log\left(t\right) \exp\left(\hat v_{j^}\right) + \boldsymbol{\hat{\beta}}^{'}\left(\boldsymbol{Z} - \boldsymbol{Z}^{}\right)\] Let $\boldsymbol w = (\boldsymbol v, \boldsymbol u, \boldsymbol \beta)=(v_1, v_2, \ cdots, v_s, u_1, u_2, \cdots, u_s, \beta_1, \cdots, \beta_p)$. Let $\widehat{\Sigma}$ denote the estimate of the covariance matrix of $\boldsymbol{\hat w}$. Let $\boldsymbol J_{tjj^}$ denote the Jacobian vector, $\boldsymbol J_{tjj^}=\frac{\partial r_{tjj^}}{\partial \boldsymbol w}\|_{\boldsymbol w=\boldsymbol{\hat w}}$. Thus, the estimate of the variance of $\hat r_{tjj^}$ is obtained by: \[\widehat{Var}(\hat r_{tjj^}) = \boldsymbol J_{tjj^}^T \widehat{\Sigma} \boldsymbol J_{tjj^}\] We obtain a 95% confidence interval for $r_{tjj^}$ as $\bigg( \hat r_{tjj^} - 1.96 \sqrt{\widehat{Var}(\hat r_{tjj^})}, \hat r_{tjj^} + 1.96 \sqrt{\widehat{Var}(\hat r_{tjj^})} \bigg)$. We exponentiate $\hat r_{tjj^}$ and its corresponding 95% confidence interval to obtain the estimate and the 95% confidence interval for the hazard ratio, $R_{tjj^}$. We illustrate the use of the straweib R package for obtaining hazard ratios and corresponding confidence intervals in Section 4. Comparison of models implemented in packages survival and straweib In this section, we compare the stratified Weibull regression model implemented in the survival package to that implemented in our package, straweib. In the absence of stratification, both models are identical and reduce to the Weibull PH model. However, in the presence of a stratification factor, the models implemented by survival and straweib correspond to different models, resulting in different likelihood functions and inference. As we discussed in Section 2, the hazard ratio between two subjects with different covariate values within same stratum depends on their stratum in the model implemented in the R package survival (Equation ((5))), whereas the hazard ratio comparing two individuals within the same stratum is invariant to stratum in the model implemented in the R package straweib (Equation ((7))). In particular, the Weibull model implemented in the straweib shares similarities with the semi-parametric, stratified Cox model for right censored data. To illustrate the difference between the models implemented in the R packages survival and straweib, we conducted a simulation study in which 1000 datasets were simulated under the model assumed in the straweib package (Equation ((6))). For each simulated dataset, since both models have the same number of unknown parameters, we compare the values of the log-likelihood evaluated at the MLEs. Datasets were simulated based on the assumptions that there are 3 strata, each with a 100 subjects; the shape parameters ($\boldsymbol \gamma$) in the three strata were set to 1.5, 2, and 1, respectively; the baseline scale parameters in the three strata ($\boldsymbol \lambda$) were set to 0.01, 0.015, and 0.02, respectively. We assumed that there are two independent explanatory variables available for each subject, randomly drawn from $N(0, 1)$ random variables. The coefficients corresponding to each of the two covariates were set to 0.5 and 1, respectively. To simulate interval censored outcomes, we first simulated the true event time for each subject by sampling from a Weibull distribution with the appropriate parameters. We assumed that each subject has 20 equally spaced diagnostic tests, at which the true event status is observed. Each test has a probability of 70% being missing. To obtain the maximum likelihood estimates under each model, we used the survreg function in the R package survival and the icweib function in the straweib package. Figure 1 compares the maximized value of the log-likelihoods under both models, when the data are generated using a simulation mechanism that corresponds to the model implemented in the R package straweib. The maximized value of the log-likelihood from the R package survival is lower than that from the R package straweib for 93.1% of simulated datasets. This is expected as in this simulation study the data generating mechanism is identical to the model implemented in the R package straweib. In applications where the proportional hazards assumption is questionable, we recommend fitting both models and comparing the resulting maximized values of the log likelihood. Whether one model is better than another depends on the data. Figure 1: Comparing the maximized values of the log-likelihood obtained from the models implemented in the R package survival (X axis) to that from the R package straweib (Y axis), when the data is simulated under the model implemented in the R package straweib We illustrate the R package straweib with data from a study on the timing of emergence of permanent teeth in Flemish children in Belgium (Leroy et al. 2003). The data analyzed were from the Signal-Tandmobiel project (Vanobbergen et al. 2000), a longitudinal oral health study in a sample of 4430 children conducted between 1996 and 2001. Dental examinations were conducted annually for a period of 6 years and tooth emergence was recorded based on visual inspection. As in Gomez et al. (2009), we will illustrate our R package by analyzing the timing of emergence of the permanent upper left first premolars. As dental exams were conducted annually, for each child, the timing of tooth emergence is known up to the interval from the last negative to the first positive dental id left right sex dmf 1 1 2.7 3.5 1 1 2 2 2.4 3.4 0 1 3 3 4.5 5.5 1 0 4 4 5.9 Inf 1 0 5 5 4.1 5.0 1 1 6 6 3.7 4.5 0 1 The dataset is formatted to include 1 row per child. The variable denoted id corresponds to the ID of the child, left and right correspond to the left and right endpoints of the censoring interval in years, sex denotes the gender of the child (0 = boy, and 1 = girl), and dmf denotes the status of primary predecessor of the tooth (0 = sound, and 1 = decayed or missing due to caries or filled). Right censored observations are denoted by setting the variable right to "Inf". In our analysis below, we use the function icweib in the package straweib, to fit a stratified Weibull regression model, where the variable dmf is the stratum indicator ($S$) and the variable sex is an explanatory variable ($Z$). fit <- icweib(L = left, R = right, data = tooth24, strata = dmf, covariates = ~sex) Total observations used: 4386. Model Convergence: TRUE coefficient SE z p.value sex 0.331 0.0387 8.55 0 Weibull parameters - gamma(shape), lambda(scale): straname strata gamma lambda dmf 0 5.99 1.63e-05 dmf 1 4.85 1.76e-04 Test of proportional hazards for strata (H0: all strata's shape parameters are equal): test TestStat df p.value Wald 44.2 1 2.96e-11 Likelihood Ratio 44.2 1 3.00e-11 Loglik(model)= -5501.781 Loglik(reduced)= -5523.87 Loglik(null)= -5538.309 Chisq= 73.05611 df= 1 p.value= 0 The likelihood ratio test of the PH assumption results in a p value of 3.00e-11, indicating that the PH model is not appropriate for this dataset. Or in other words, the data suggest that the hazard functions corresponding to the strata defined by $dmf=0$ and $dmf=1$ are not proportional. From the stratified Weibull regression model, the estimated regression coefficient for sex is 0.331, corresponding to a hazard ratio of 1.39 (95% CI: 1.29 - 1.50). In the output above, the maximized value of the log likelihood of the null model corresponds to the model stratified by covariate dmf but excluding the explanatory variable sex. The p value from the Wald test of the null hypothesis of no effect of gender results in a p value of approximately 0 ($p < 10^{-16}$), which indicates that the timing of emergence of teeth is significantly different between girls and boys. To test the global null hypothesis that both covariates sex and dmf are not associated with the outcome (time to teeth emergence), we obtain the log-likelihood for global null model, as shown below. fit0 <- icweib(L = left, R = right, data = tooth24) Total observations used: 4386. Model Convergence: TRUE Weibull parameters - gamma(shape), lambda(scale): straname strata gamma lambda strata ALL 5.3 7.78e-05 Loglik(model)= -5596.986 Loglik(null)= -5596.986 The likelihood ratio test testing the global null hypothesis results in a test statistic $T=-2(l_R - l_F) = -2(-5596.986 + 5501.781)=190.41$, which follows a $\chi_{3}^2$ distribution under $H_0 $, resulting in a p value of approximately 0 ($p < 10^{-16}$). We illustrate the HRatio function in the straweib package to estimate the hazard ratio and corresponding 95% confidence intervals for comparing boys without tooth decay ($dmf=0$) to boys with evidence of tooth decay ($dmf=1$), where the hazard ratio is evaluated at various time points from 1 through 7 years. HRatio(fit, times = 1:7, NumStra = 0, NumZ = 0, DemStra = 1, DemZ = 0) time NumStra DemStra beta*(Z1-Z2) HR low95 high95 1 1 0 1 0 0.1143698 0.06596383 0.1982972 2 2 0 1 0 0.2520248 0.18308361 0.3469262 3 3 0 1 0 0.4000946 0.33112219 0.4834339 4 4 0 1 0 0.5553610 0.49863912 0.6185351 5 5 0 1 0 0.7162080 0.66319999 0.7734529 6 6 0 1 0 0.8816470 0.79879884 0.9730878 7 7 0 1 0 1.0510048 0.91593721 1.2059899 The output indicates that the hazard ratio for boys comparing the stratum $dmf = 0$ to stratum $dmf = 1$ is small initially (e.g. 0.11 at 1 year) but tends to 1 in later years (e.g. 0.88 at 6 years and 1.05 at 7 years). Prior to 6 years, the hazard ratio is significantly less than 1, indicating that the timing of teeth emergence is delayed in children with tooth decay ($dmf=1$) when compared to children without tooth decay ($dmf=0$). We illustrate estimation of the survival function in Figure 2 by plotting the survival functions and corresponding 95% point wise confidence intervals for girls ($Z=1$), with and without tooth decay . plot(fit, Z = 1, tRange = c(1, 7), xlab = "Time (years)", ylab = "Survival Function", main = "Estimated survival function for girls") Figure 2: Estimated survival functions for girls, comparing the subgroup with sound primary predecessor of the tooth (dmf = 0) to the subgroup with unsound primary predecessor of the tooth (dmf = 1). We compare our results from the straweib package to that obtained from the survival package. tooth24.survreg <- tooth24 tooth24.survreg$right <- with(tooth24, ifelse(is.finite(right), right, NA)) fit1 <- survreg(Surv(left, right, type="interval2") ~ sex + strata(dmf) + factor(dmf), data = tooth24.survreg) survreg(formula = Surv(left, right, type = "interval2") ~ sex + strata(dmf) + factor(dmf), data = tooth24.survreg) (Intercept) sex factor(dmf)1 1.84389938 -0.06254599 -0.06491729 dmf=Sound1 dmf=Sound2 0.1659477 0.2072465 Loglik(model)= -5499.3 Loglik(intercept only)= -5576.2 Chisq= 153.8 on 2 degrees of freedom, p= 0 n= 4386 The maximized value of the log-likelihood from the R package survival is $-5499.3$ (shown below), as compared to the maximized value of the log-likelihood of $-5501.8$ from the R package straweib To clarify the specific assumptions made by the models implemented in the survival and straweib packages, we carried out subgroup analyses in which we fit a Weibull PH model separately to each of the strata $dmf=0$ and $dmf=1$. The results from the Weibull PH model fit to the subgroup of children in the $dmf=0$ stratum is shown below: fit20 <- icweib(L= left, R=right, data=tooth24[tooth24$dmf==0, ], covariates = ~sex) fit20 ### Partial results shown below coefficient SE z p.value sex 0.448 0.0543 8.25 2.22e-16 The results from the Weibull PH model fit to the subgroup $dmf=1$ is shown below: fit21 <- icweib(L= left, R=right, data=tooth24[tooth24$dmf==1, ], covariates = ~sex) fit21 ### Partial results shown below coefficient SE z p.value sex 0.208 0.0554 3.76 0.000169 The model using the PH scale (implemented by straweib package) replaces the stratum specific hazard ratios for sex of $e^{0.448}=1.57$ for the subgroup $dmf=0$ and $e^{0.208}=1.23$ for the subgroup $dmf=1$ with a common value, $e^{0.331}=1.39$. Since the Weibull distribution has both the PH and accelerated failure time (AFT) property (Collett 2003), the identical set of subgroup analyses can be fit using the survival package. Results from the fit using the survival package for the subgroup $dmf=0$ are shown below: fit20.survreg <- survreg(Surv(left, right, type="interval2") ~ sex, data = tooth24.survreg[tooth24.survreg$dmf==0, ]) fit20.survreg ### Partial results shown below (Intercept) sex 1.85029150 -0.07453785 Similar results using the survival package for the subgroup $dmf=1$ are shown below: fit21.survreg <- survreg(Surv(left, right, type="interval2") ~ sex, data = tooth24.survreg[tooth24.survreg$dmf==1, ]) fit21.survreg ### Partial results shown below (Intercept) sex 1.76931556 -0.04303767 In particular, the model assuming a common sex coefficient in the AFT scale (implemented by survival package) replaces the value of sex coefficient $-0.075$ for the subgroup with $dmf=0$ and sex coefficient of $-0.043$ for the subgroup $dmf=1$ with a shared common value, $-0.063$. To assess the goodness of fit of the stratified Weibull model implemented by straweib, we created a multiple probability plot, as described in chapter 19 of Meeker and Escobar (1998). This diagnostic plot was created by splitting the dataset into 4 subgroups based on the values of sex and dmf. Within each group, we estimated the cumulative incidence at each visit time using a non-parametric procedure for interval censored data (Turnbull 1976). The non-parametric estimates of cumulative incidence within each subgroup were compared to that obtained from the stratified Weibull model implemented by straweib package. We use the R package interval (Fay and Shaw 2010) to obtain Turnbull’s NPMLE estimates and the R package straweib for the estimates from the stratified Weibull model (code available upon request). Figure 3 shows the diagnostic plot. Figure 3: Comparing non-parametric (points) and Weibull model (lines) based estimates of cumulative incidence within each group based on covariates sex and dmf. Table 1 presents the estimates of hazard ratio for sex, within each of the strata defined by $dmf=0$ and $dmf=1$, comparing three different analyses - (1) Using the survival package to stratify on the variable dmf and including sex as an explanatory variable; (2) Using the straweib package to stratify on the variable dmf and including sex as an explanatory variable; (3) Fitting a Weibull PH model with sex as an explanatory variable, separately within each of the two subgroups defined by $dmf=0$ and $dmf=1$. HR.straweib <- exp(fit$coef[1, 1]) HR.survreg <- exp(-fit1$coefficients['sex']/fit1$scale) HR.subgroup <- exp(c(fit20$coef[1, 1], fit21$coef[1, 1])) Table 1: Hazard ratio estimates for gender, comparing the models implemented in the R packages survival, straweib and subgroup dmf = 0 1.46 1.39 1.56 dmf = 1 1.35 1.39 1.23 We have developed and illustrated an R package straweib for the analysis of interval-censored outcomes, based on a stratified Weibull regression model. The proposed model shares similarities with the semi-parametric stratified Cox model. We illustrated the R package straweib using data from a prospective study on the timing of emergence of permanent teeth in Flemish children in Belgium (Leroy et al. 2003). Although the models and R package are illustrated for the analysis of interval-censored time-to-event outcomes, the methods proposed here are equally applicable for the analysis of right-censored outcomes. The syntax for the analysis of right-censored observations is explained in the manual accompanying the straweib package available on CRAN (Gu and R. Balasubramanian 2013). This research was supported by NICHD grant R21 HD072792. D. Collett. Modelling Survival Data in Medical Research, Second Edition. Texts in statistical science Taylor & Francis, 2003. D. R. Cox and D. V. Hinkley. Theoretical Statistics. Chapman; Hall, 1979. D. T. Dunn, R. J. Simonds, M. Bulterys, L. A. Kalish, J. Moye, A. de Maria, C. Kind, C. Rudin, E. Denamur, A. Krivine, et al. Interventions to prevent vertical transmission of HIV-1: Effect on viral detection rate in early infant samples. AIDS, 14(10): 1421–1428, 2000. M. P. Fay and P. A. Shaw. Exact and asymptotic weighted logrank tests for interval censored data: The interval R Journal of Statistical Software , 36(2): 1–34, 2010. URL D. M. Finkelstein. A proportional hazards model for interval-censored failure time data. Biometrics, 42(4): 845–854, 1986. E. Goetghebeur and L. Ryan. Semiparametric regression analysis of interval-censored data. Biometrics, 56(4): 1139–1144, 2000. G. Gomez, M. L. Calle, R. Oller and K. Langohr. Tutorial on methods for interval-censored data and their implementation in R. Statistical Modelling, 9(4): 259–297, 2009. X. Gu and R. Balasubramanian. straweib: Stratified Weibull Regression Model, . R package version 1.0, 2013. URL R. Leroy, K. Bogaerts, E. Lesaffre and D. Declerck. The emergence of permanent teeth in flemish children. Community Dentistry and Oral Epidemiology, 31(1): 30–39, 2003. J. C. Lindsey and L. M. Ryan. Tutorial in biostatistics - methods for interval-censored data. Statistics in Medicine, 17(2): 219–238, 1998. W. Q. Meeker and L. A. Escobar. Statistical methods for reliability data. Wiley, 1998. T. Therneau. A Package for Survival Analysis in S, . R package version 2.36-14, 2012. B. W. Turnbull. Empirical distribution function with arbitrarily grouped, censored and truncated data. Journal of the Royal Statistical Society Series B-Methodological, 38(3): 290–295, 1976. J. Vanobbergen, L. Martens, E. Lesaffre and D. Declerck. The Signal-Tandmobiel project a longitudinal intervention health promotion study in flanders (belgium): Baseline and first year results. Eur J Paediatr Dent, 2: 87–96, 2000.	{"url":"http://journal.r-project.org/articles/RJ-2014-003/","timestamp":"2024-11-11T07:18:05Z","content_type":"text/html","content_length":"969416","record_id":"<urn:uuid:cd106f94-41b1-4216-980a-2c121c7f5d50>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00006.warc.gz"}
FREE Number Line Game to Build Number Sense (Three Levels) Get 150+ Free Math Worksheets! Your kids will love this number line game that will help build number sense. With three levels you can work on numbers to 10, numbers to 100, or to 1,000. Number lines are a wonderful way to build number sense. There are so many ways to use number lines, from patterns to adding and even dividing. But today, we are simply working on counting and number This number line game though has a little bit of a challenge. Today, our sweet kiddos get to work on counting backwards to figure out what number the “bug” ate! Our number line game requires a little bit of prep, but if can be used over and over again. 1. Choose which level you would like to use, and then print off the cards and game boards for that level. I recommend using cardstock if you plan on using multiple times. 2. Next, cut out the cards with the number lines on them. 3. Finally, gather up bingo markers and you are ready to go! Neenah Cardstock, 8.5Learning Resources Transparent Color Counting Chips, Set of 250 Assorted Colored Chips, Ages 5+AmazonBasics Thermal Laminator Machine Number Line Game Now it is time for some number line work. The children take turns drawing a card and solving the problem. Once the figure out what number the “bug” is covering up, they find that number on their bingo sheet and cover it up. The first person who gets five in a row wins. Numbers Below 10 This level is for our young learners. The number lines only go to the number 10, and it allows are littles to focus on the single-digit numbers. Double-Digit Numbers The next level jumps into double-digit numbers. These number lines focus on numbers below 100, and some might be a little challenging. Many of the number lines end in numbers like 50. This means that our children need to know that 49 comes before 50. If this is something they are struggling with, you could provide a hundreds chart to help. Numbers in the Hundreds Finally, we get to the numbers in the hundreds. These clip cards go just a little past 1,000 and let children work on those fun numbers with three digits! Even though the game is quite simple, the skills the number sense they build while playing is invaluable!!! Need more building number sense activities! Check them out here.	{"url":"https://youvegotthismath.com/number-line-game/","timestamp":"2024-11-05T03:09:26Z","content_type":"text/html","content_length":"338674","record_id":"<urn:uuid:425822d7-5e7d-4845-98ee-d6324069d881>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00860.warc.gz"}
Add to 10 Ones Digit 2 comments 1. I have watched quite a few videos over the year, Mr. Gillespey, and all of them have proved useful during tests and competitions. Thank you for providing us with these tips! 2. Assuming that the first digit of two numbers are equal and the two second digits add up to 10: 1. Multiply the last digit times the second last digit. If there is one digit, put a zero before the digit. 2. Add 1 to one of the first digits and multiply the other first digit times that +1 digit.	{"url":"http://mathninja.org/add-to-10-ones-digit/","timestamp":"2024-11-14T13:46:04Z","content_type":"text/html","content_length":"72324","record_id":"<urn:uuid:cb6e4752-d16c-4f72-8c90-654dbadf42db>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00658.warc.gz"}
Samples a Whittle-Matérn field on a metric graph — sample_spde Samples a Whittle-Matérn field on a metric graph Obtains samples of a Whittle-Matérn field on a metric graph. sigma_e = 0, alpha = 1, directional = FALSE, PtE = NULL, type = "manual", posterior = FALSE, nsim = 1, method = c("conditional", "Q"), BC = 1 Range parameter. Precision parameter. Practical correlation range parameter. Marginal standard deviation parameter. Standard deviation of the measurement noise. Smoothness parameter. should we use directional model currently only for alpha=1 A metric_graph object. Matrix with locations (edge, normalized distance on edge) where the samples should be generated. If "manual" is set, then sampling is done at the locations specified in PtE. Set to "mesh" for simulation at mesh nodes, and to "obs" for simulation at observation locations. Sample conditionally on the observations? Number of samples to be generated. Which method to use for the sampling? The options are "conditional" and "Q". Here, "Q" is more stable but takes longer. Boundary conditions for degree 1 vertices. BC = 0 gives Neumann boundary conditions and BC = 1 gives stationary boundary conditions. Samples a Gaussian Whittle-Matérn field on a metric graph, either from the prior or conditionally on observations $$y_i = u(t_i) + \sigma_e e_i$$ on the graph, where $e_i$ are independent standard Gaussian variables. The parameters for the field can either be specified in terms of tau and kappa or practical correlation range and marginal standard deviation.	{"url":"https://davidbolin.github.io/MetricGraph/reference/sample_spde.html","timestamp":"2024-11-05T19:07:06Z","content_type":"text/html","content_length":"12652","record_id":"<urn:uuid:fd125bf5-d0ba-4d9f-9ef1-70e5a38d96c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00163.warc.gz"}
Simplifying the Expression (7a^2)(5a^6b)^2 In mathematics, simplifying expressions involves rewriting them in their most basic and compact form. Let's explore how to simplify the expression (7a^2)(5a^6b)^2. Understanding the Properties To simplify this expression, we need to understand the following properties: • Power of a product: (ab)^n = a^n * b^n • Power of a power: (a^m)^n = a^(mn) Step-by-Step Simplification 1. Simplify the inner power: (5a^6b)^2 = 5^2 (a^6)^2 * b^2 = 25a^12b^2 2. Multiply the simplified inner power by the term outside: (7a^2)(25a^12b^2) = 7 * 25 * a^2 * a^12 * b^2 3. Combine like terms: 175a^(2+12)b^2 4. Simplify the exponents: 175a^14b^2 The Final Simplified Expression Therefore, the simplified form of (7a^2)(5a^6b)^2 is 175a^14b^2.	{"url":"https://jasonbradley.me/page/(7a%255E2)(5a%255E6b)%255E2","timestamp":"2024-11-04T01:05:58Z","content_type":"text/html","content_length":"57379","record_id":"<urn:uuid:b7d6898f-655c-4fbd-b25e-8df9715b3682>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00053.warc.gz"}
macaque monkey cambodia Curso ‘Artroscopia da ATM’ no Ircad – março/2018 18 de abril de 2018 Simple Exponential Smoothing: – If you have a time series that can be described using an additive model with constant level and no seasonality, you can use simple exponential smoothing to make short-term. Exponential smoothing is a time series forecasting method for univariate data that can be extended to support data with a systematic trend or seasonal component. TRUE Forecasts depend on the rules of the game remaining reasonably constant. Types of Exponential Smoothing Methods. The forecasting formula is based on an extrapolation of a line through the two centers. While there is nothing wrong with the sales forecasting methods shown above using Excel, it is true there are specific pieces of software out there designed just for forecasting. Forecasting techniques generally assume an existing causal system that will continue to exist in the future. (A more sophisticated version of this model, Holt’s, is discussed below.) How to do a Sales Forecast: A Special Mention to Peerforecaster. This method is suitable for forecasting data with no clear trend or seasonal pattern. In this tutorial, you will discover the exponential smoothing method for univariate time series forecasting. Smoothing Techniques for Time Series Forecasting @inproceedings{Hameed2015SmoothingTF, title={Smoothing Techniques for Time Series Forecasting}, author={Haifaa Hussein Hameed}, year= {2015} } Corpus ID: 56275095. Holt’s Exponential Smoothing: – There are four main types of forecasting methods that financial analysts Financial Analyst Job Description The financial analyst job description below gives a typical example of all the skills, education, and experience required to be hired for an analyst job at a bank, institution, or corporation. In market analysis, smoothed data is … Exponential Smoothing is one of the more popular smoothing techniques due to its flexibility, ease in calculation, and good performance. Specifically, past observations are weighted with a geometrically decreasing ratio. For example, the data in Figure 7.1 do not display any clear trending behaviour or any seasonality. This allows important patterns to stand out. The simplest of the exponentially smoothing methods is naturally called simple exponential smoothing (SES) 13. Top Four Types of Forecasting Methods. Smoothing techniques are kinds of data preprocessing techniques to remove noise from a data set. The formula for calculating the forecast using smoothing method is given by equation F_t plus 1 is equal to Alpha times D_t plus 1 minus Alpha times F_t, where D_t is the actual value of the demand at time t, F_t is the forecasted value, Alpha is the weighting factor which ranges from 0-1, and t … They are more accurate and not necessarily that difficult to operate. 7.1 Simple exponential smoothing. Exponential smoothing forecasting methods are similar in that a prediction is a weighted sum of past observations, but the model explicitly uses an exponentially decreasing weight for past observations. trend, seasonality, etc.). For new products in a strong growth mode, a low alpha will minimize forecast errors when using exponential smoothing techniques. However, we can also use smoothing to fill in missing values and/or conduct a forecast. forecast. Smoothing and filtering are two of the most commonly used time series techniques for removing noise from the underlying data to help reveal the important features and components (e.g. The simplest time-varying trend model is Brown's linear exponential smoothing model, which uses two different smoothed series that are centered at different points in time. Types of Exponential Smoothing¶ Exponential Smoothing uses a simple average calculation to assign exponentially decreasing weights starting with the most recent observations. It is a powerful forecasting method that may be used as an alternative to the popular Box-Jenkins ARIMA family of methods. Exponential Smoothing. Trending behaviour or any seasonality are weighted with a geometrically decreasing ratio errors... Will discover the exponential smoothing uses a simple average calculation to assign exponentially decreasing weights with... Ease in calculation, and good performance or seasonal pattern we can use. To exist in the future the rules of the more popular smoothing techniques that may be used as an to... Flexibility, ease in calculation, and good performance new products in a strong mode... Naturally called simple exponential smoothing method for univariate time series forecasting holt s... Called simple exponential smoothing: – forecasting techniques generally assume an existing causal system that will continue exist... Necessarily that difficult to operate smoothing uses a simple average calculation to assign exponentially decreasing weights with! Causal system that will continue to exist in the future data set time series forecasting method. Naturally called simple exponential smoothing ( SES ) 13 discussed below. the exponential smoothing: forecasting... Starting with the most recent observations smoothing ( SES ) 13 model, holt s. Line through the two centers this tutorial, you will discover the exponential smoothing: forecasting. One of the game remaining reasonably constant a powerful forecasting method that may be used an! Method that may be used as an alternative to the popular Box-Jenkins ARIMA family of methods future!, smoothed data is … How to do a Sales forecast: a Special Mention to Peerforecaster forecast! With the most recent observations in this tutorial, you will discover the exponential smoothing: – forecasting generally. Necessarily that difficult to operate to operate past observations are weighted with a geometrically decreasing ratio data. Products in a strong growth mode, a low alpha will minimize forecast errors when using exponential:... A strong growth mode, a low alpha will minimize forecast errors when using exponential smoothing are. Errors when using exponential smoothing techniques due to its flexibility, ease in calculation, and good.! For univariate time series forecasting for new products in a strong growth mode a! Clear trend or seasonal pattern a more sophisticated version of this model, holt ’,! With the most recent observations ’ s, is discussed below. minimize forecast errors using. With the most recent observations to its flexibility, ease in calculation, and performance. Simple average calculation to assign exponentially decreasing weights starting with the most recent observations of methods of this model holt... ( a more sophisticated version of this model, holt ’ s, is discussed.! To assign exponentially decreasing weights starting with the most recent observations fill in missing values and/or conduct a.! Tutorial, you will discover the exponential smoothing ( SES ) 13 or any seasonality and good.. The exponential smoothing is one of the game remaining reasonably constant more sophisticated of! Data is … How to do a Sales forecast: a Special Mention to Peerforecaster alternative the! Of this model, holt ’ s exponential smoothing ( SES ) 13 observations are weighted with a geometrically ratio... To do a Sales forecast: a Special Mention to Peerforecaster existing causal system that continue! Are more accurate and not necessarily that difficult to operate exponential smoothing ( SES 13! Is one of the game remaining reasonably constant observations are weighted with a geometrically ratio! The popular Box-Jenkins ARIMA family of methods the rules of the exponentially methods... Clear trend or seasonal pattern use smoothing to fill in missing values and/or conduct a forecast one of the popular. Through the two centers of a line through the two centers conduct a forecast two centers smoothing is of... Are kinds of data preprocessing techniques to remove noise from a smoothing techniques forecasting set exponentially... To the popular Box-Jenkins ARIMA family of methods formula is based on an extrapolation of line... More sophisticated version of this model, holt smoothing techniques forecasting s, is discussed below. is discussed below. smoothing... Method is suitable for forecasting data with no clear trend or seasonal pattern with the most recent observations,! More accurate and not necessarily that difficult to operate series forecasting smoothing methods is naturally simple... A more sophisticated version of this model, holt ’ s, is discussed below. behaviour any. In this tutorial, you will discover the exponential smoothing techniques with a geometrically decreasing.... That may be used as an alternative to the popular Box-Jenkins ARIMA family of methods accurate and not necessarily difficult. Forecast: a Special Mention to Peerforecaster mode, a low alpha minimize... To operate data preprocessing techniques to remove noise from a data set of this model, holt ’ s smoothing! Data set discussed below. a Special Mention to Peerforecaster or seasonal pattern kinds! Is based on an extrapolation of a line through the two centers to fill in missing values conduct. The more popular smoothing techniques are kinds of data preprocessing techniques to remove noise from a data.... However, we can also use smoothing to fill in missing values and/or conduct a.! One of the exponentially smoothing methods is naturally called simple exponential smoothing is one of the exponentially methods. The future to Peerforecaster and/or conduct a forecast weighted with a geometrically decreasing ratio that will to. Starting with the most recent observations is a powerful forecasting method that be. With no clear trend or seasonal pattern is based on an extrapolation of line. Called simple exponential smoothing is one of the exponentially smoothing methods is naturally called simple exponential smoothing method for time... Noise from a data set analysis, smoothed data is … How to do Sales. Suitable for forecasting data with no clear trend or seasonal pattern data with no clear trend or seasonal pattern a... Naturally called simple exponential smoothing method for univariate time series forecasting growth mode, a alpha! Not necessarily that difficult to operate 7.1 do not display any clear trending or... Market analysis, smoothed data is … How to do a Sales forecast: a Special Mention Peerforecaster... Or seasonal pattern it is a powerful forecasting method that may be used as an alternative the... To Peerforecaster in Figure 7.1 do not display any clear trending behaviour or any seasonality market. ’ s, is discussed below. a more sophisticated version of this model, holt ’ s, discussed! Smoothing methods is naturally called simple exponential smoothing ( SES ) 13 when! No clear trend or seasonal pattern data is … How to do a Sales forecast: a Special to! Causal system that will continue to exist in the future preprocessing techniques to remove noise from a data.... Will continue to exist in the future analysis, smoothed data is … How to do a forecast... Conduct a forecast on an extrapolation of a line through the two centers time forecasting. Growth mode, a low alpha will minimize forecast errors when using exponential smoothing is one of exponentially... Mode, a low alpha will minimize forecast errors when using exponential is. To operate that difficult to operate the data in Figure 7.1 do not display any clear trending behaviour or seasonality! That difficult to operate forecasting data with no clear trend or seasonal.... A powerful forecasting method that may be used as an alternative to the popular Box-Jenkins family. Holt ’ s, is discussed below. this tutorial, you will discover exponential! Analysis, smoothed data is … How to do a Sales forecast: a Special Mention to Peerforecaster forecasting! Calculation, and good performance, we can also use smoothing to fill in missing values and/or conduct a.. In missing values and/or conduct a forecast the data in Figure 7.1 do not display clear! Forecasting method that may be used as an alternative to the popular Box-Jenkins ARIMA family of methods smoothing to in! However, we can also use smoothing to fill in missing values and/or conduct a forecast that... Clear trending behaviour or any seasonality necessarily that difficult to operate the remaining... Are more accurate and not necessarily that difficult to operate smoothing to in. Discussed below. smoothing techniques forecasting that may be used as an alternative to the popular ARIMA... Example, the data in Figure 7.1 do not display any clear trending behaviour or any.! Method that may be used as an alternative to the popular Box-Jenkins ARIMA family of methods the future data. Discussed below. are weighted with a geometrically decreasing ratio ease in calculation, good! New products in a strong smoothing techniques forecasting mode, a low alpha will minimize forecast errors when using exponential smoothing due. The popular Box-Jenkins ARIMA family of methods or seasonal pattern specifically, past observations weighted. Or any seasonality ) 13 7.1 do not display any clear trending behaviour or any seasonality an of... Special Mention to Peerforecaster for univariate time series forecasting as an alternative to the popular Box-Jenkins family... A data set a forecast to its flexibility, ease in calculation, and good performance exponential smoothing for. Low alpha will minimize forecast smoothing techniques forecasting when using exponential smoothing techniques ARIMA family of methods version! To do a Sales forecast: a Special Mention to Peerforecaster ’ s, is below. Its flexibility, ease in calculation, and good performance discover the exponential smoothing SES... That may be used as an alternative to the popular Box-Jenkins ARIMA family of methods Special to! To the popular Box-Jenkins ARIMA family of methods starting with the most recent observations most. Version of this model, holt ’ s exponential smoothing method for univariate time series.... For new products in a strong growth mode, a low alpha will minimize forecast when... Causal system that will continue to exist in the future may be used as an alternative to the popular ARIMA! Ses ) 13 be used as an alternative to the popular Box-Jenkins ARIMA family of methods rules of game! 2016 Buick Encore Battery Problems, Kanex Usb To Ethernet Adapter, How To Draw A Door Opening, Who Makes Dutch Boy Paint, Midland Bank V Green, Stratus White Quartz Kitchen, Story Writing Questions, Classic View Meaning In Urdu, Zinsser Drywall Sealer, 2011 Ford Focus Fuse Box Diagram,	{"url":"http://clinicapilastri.com.br/ead9o2vk/macaque-monkey-cambodia-426d9e","timestamp":"2024-11-07T20:12:44Z","content_type":"text/html","content_length":"59870","record_id":"<urn:uuid:e54014b8-dd1f-4698-8085-36d63604059b>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00497.warc.gz"}
New sw method: "extreme scale" Saari at aol.com Saari at aol.com Mon Mar 3 12:54:42 PST 1997 In a message dated 97-02-23 13:11:59 EST, SteveE writes: >The interesting feature about this new method is that it's not a >ranked ballot method. It's a rating method. >Define E to be the number of eligible voters. >Define C to be the number of candidates. >Define S to be 2EC rounded up to the next power of 10. This may >be a huge number. For example, with C = 3 candidates and E = 100 >voters, 2EC is 600, so S = 1000. >1. Each voter rates each candidate on a scale ranging from 0 to S. >2. The score of each candidate is the sum of the ratings assigned it >by the voters, normalized by dividing by S. The winner is the >candidate with the highest score. Actually, instead of a scale up to 1000 (variable depending on the #s of voters/candidates), you might do better with a scale up to 1.0, with unlimited levels of gradation, i.e. .5, .75, .997, .99834 etc. Once you get people past "fraction-phobia", this has several advantages, the main one being that the same scale will be used under a variety of circumstances so people can get familiar with it. Second is that it allows each voter to use as "fine" or "coarse" gradations as they wish (instead of an unnecessary fixed maximum level of resolution). Also, the notation is unambiguous even if voters neglect to include the decimal point. Thus, .53 or 53 or 5300 all clearly mean the same thing on a "up to 1.0" scale. Even 5.3 can be properly reinterpreted. An interesting twist is to allow votes up to but not including the 1.0 endpoint. Then for ordinary, low-res "single-digit" votes, the range would be 0 to 9. For an especially strong yes vote, someone might vote 99. An even stronger yes vote might be 9999. This allows people to get carried away as much as they like, yet score-wise all they are doing is getting closer and closer to the limit value of 1.0. Thus people are allowed to express extreme opinions, but without taking over the group. Cute or what? Mike S I also note that the system as described doesn't distinguish "actively opposed" from "lack of interest" toward a given candidate. Hence I prefer a plus-and-minus scale. More information about the Election-Methods mailing list	{"url":"http://lists.electorama.com/pipermail/election-methods-electorama.com/1997-March/066712.html","timestamp":"2024-11-04T14:32:02Z","content_type":"text/html","content_length":"5256","record_id":"<urn:uuid:25a91e39-5e1f-4f4f-83a2-32a8ad9a100f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00239.warc.gz"}
Crack Propagation Analysis Tools \| Accuracy, Speed & Mechanics Crack propagation analysis tools Explore the latest in crack propagation analysis tools, their accuracy, speed, applications in industries, and future trends in material science. Understanding Crack Propagation Analysis Tools Crack propagation analysis is a crucial aspect of material science and engineering, playing a significant role in predicting the lifespan and safety of materials and structures. With the advent of advanced computational tools, the accuracy, speed, and mechanics of crack propagation analysis have significantly improved, offering deeper insights into material behavior under stress. Accuracy in Crack Propagation Analysis The accuracy of crack propagation tools is paramount, especially in critical applications like aerospace, automotive, and civil engineering. High-precision tools use finite element analysis (FEA) and fracture mechanics principles to simulate crack initiation and growth. These simulations consider various factors such as stress intensity factors (K[I], K[II], K[III]), material properties, and environmental conditions. Advanced software packages incorporate microstructural details and nonlinear material behavior, enhancing the accuracy of predictions. Speed and Efficiency Speed is a key factor in choosing a crack propagation analysis tool, particularly when dealing with large datasets or complex geometries. Modern tools leverage high-performance computing and algorithms optimized for speed, reducing computation time significantly. This efficiency is crucial for industries where time-to-market and rapid prototyping are essential. Understanding the Mechanics Crack propagation mechanics involve understanding how cracks initiate, propagate, and eventually lead to failure. The tools analyze stress distribution around crack tips using concepts like Linear Elastic Fracture Mechanics (LEFM) and Elastic-Plastic Fracture Mechanics (EPFM). They also employ criteria like the Paris law for fatigue crack growth and Griffith’s criterion for brittle fracture. Advanced Features of Crack Propagation Tools Today’s tools offer advanced features such as multi-scale modeling, which combines macroscopic structural analysis with microscopic material behavior. This approach is particularly useful in understanding how microstructural defects like voids and inclusions influence crack growth. Additionally, tools now include user-friendly interfaces and visualization capabilities, making it easier for engineers and researchers to interpret and utilize the results effectively. Overall, the evolution of crack propagation analysis tools has been integral to advancements in material science and structural engineering. The next section will delve into practical applications, case studies, and future trends in this field. Practical Applications and Future Trends in Crack Propagation Analysis Crack propagation analysis tools are widely used across various industries, with their applications ranging from safety assessments in civil engineering to durability evaluations in aerospace design. In the automotive industry, these tools are instrumental in crashworthiness analysis and in optimizing materials for weight reduction without compromising safety. In civil engineering, they are used to assess the integrity of critical structures like bridges, dams, and high-rise buildings, especially in earthquake-prone areas. Case Studies in Crack Propagation Analysis Several case studies have demonstrated the efficacy of these tools. For instance, in the aerospace industry, crack propagation analysis has been used to predict the lifespan of aircraft components, leading to more robust designs and preventive maintenance strategies. In another case, the tools helped in understanding the failure mechanisms of pipelines in the oil and gas industry, leading to improved materials and construction techniques. Integration with Other Technologies Integration with other technologies like digital twins and machine learning is a growing trend in crack propagation analysis. Digital twins allow for real-time monitoring and predictive maintenance, while machine learning algorithms enhance the predictive accuracy by learning from historical data and identifying patterns that might be missed by traditional methods. Challenges and Future Directions Despite their advancements, crack propagation analysis tools face challenges, particularly in handling extremely complex materials and geometries. The future direction includes further integration with artificial intelligence for predictive analytics and advancements in computational capabilities to handle more complex simulations. There is also a growing focus on sustainability, with tools being developed to assess the environmental impact of material choices and structural designs. In conclusion, crack propagation analysis tools have become indispensable in the field of material science and engineering. Their accuracy, speed, and advanced mechanics have not only enhanced our understanding of material behavior under stress but also contributed to the safety, efficiency, and sustainability of structures and components across various industries. As these tools continue to evolve, integrating with emerging technologies like AI and digital twins, they will undoubtedly play a pivotal role in driving innovation and ensuring the reliability and longevity of future engineering designs.	{"url":"https://modern-physics.org/crack-propagation-analysis-tools/","timestamp":"2024-11-10T18:12:25Z","content_type":"text/html","content_length":"161279","record_id":"<urn:uuid:15ee22d0-4472-4df6-b3aa-4a94f36c6a2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00517.warc.gz"}
SingSurf mathematical visualisation program SingSurf is program to visualise mathematical curves and surfaces. The program can calculate many of the objects found in Singularity theory and geometry: Basic types created Algebraic curves defined by a single polynomial equation in two variables. e.g. electric motor y^2(y^2-9)-x^2(x^2-10); Algebraic surfaces defined by a single polynomial equation in three variables. e.g. a Chubs surface x^4 + y^4 + z^4 - x^2 - y^2 - z^2 + 0.5; Parameterised curves defined by a 3D vector expression in a single variable. e.g. a helix [cos(pi t), sin(pi t), t]; Parameterised surfaces defined by a 3D vector expression in two variables. e.g. a cross-cap [x,x y,y^2] Intersection of surfaces with sets defined by another equation. For example the intersection of a conical surface with the set defined by a plane a x b y + cz =d. This module can be used to calculate non-polynomial curves. For example a super ellipse pow(abs(x/a),p)+pow(abs(y/b),p)-1 Clipping, part of a surface inside a set define by an implicit equation, like the set inside a box min(min(min(xh-x,x-xl),min(yh-y,y-yl)),min(zh-z,z-zl)), or clipped by a sphere x^2+y^2+z^2-r^2 Mapping from R^3 to R^3 defined by 3D vector equation in three variables. e.g. a rotation [cos(pi th) x - sin(pi th) y,sin(pi th) x + cos(pi th) y,z]; Vector Fields, including unoriented vector field, and binary differential equations Integral Curves. Uses the points in a geometry to define the starting points Colourise: sets the colour of a surface depending on an expression. For example to colour by the z coordinate [(z+1), 0,(1-z)]; setting the red, green, and blue components for each point. Extrude: produces surfaces of revolution and similar surfaces which depend on a curve and an equation. Can be used to produce families of curves. Generalised Operations Several of these models have versions where the equation of another curve or surface can be used as part of the definition Generalised Mappings where the equation depends on another surface. For example projection of a curve onto a surface. For example Gauss Map of a surface N / sqrt(N.N); // Unit normal N = Sx ^^ Sy; // calculate normal using cross product Sx = diff(S,x); // derivatives of surface S Sy = diff(S,y); // Definition of S read from the input surface Generalised Intersections where the equation depends on the definition of another curve or surface. e.g. The profile of a surface, or parabolic lines // The profile of a surface N . [A,B,C]; N = diff(S,x) ^^ diff(S,y); Generalised Clipping: e.g. the part of surface contained inside another already defined implicit surface Generalised Colourise: colour by Gaussian or mean curvature Generalised Extrude: e.g. tangent developable of a curve, or envelope of normals S + t T; // Point on surface plus a multiple of unit tangent T = TT/sqrt(TT.TT); // unit length TT = diff(S,x); // tangent to curve Generalised Vector Fields: e.g. principle directions which are calculated using the definition of the input surface Generalised Integrals Curves: e.g. principle curves of a surface calculated using the definition of the input surface Specialised modules Ridge Intersections: curves which depend on a surface and a vector field, for example the ridges of a surface BiIntersection Intersections where the equations depends on a pair of curves. For example the pre-symmetry set of a curve. BiMap Mapping where the equation depends on a pair of curves. For example the Symmetry set. Projective varieties: algebraic surfaces defined in real projective space, with options for stereographic projections and rotations in 4D Download and Installation The program is open-source and runs on all operating systems which support Java. See download and installation.	{"url":"https://www.singsurf.org/singsurf/index.php","timestamp":"2024-11-05T21:42:07Z","content_type":"text/html","content_length":"12407","record_id":"<urn:uuid:2b8b854e-962a-4298-9479-39896147775d>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00063.warc.gz"}
The Early Math Support (EMS) Lab is fully staffed by current instructors and aims to create an encouraging environment for all students who are considered TSI Incomplete. Because the tutors are instructors, students will be met with patience, questions to guide them, and, of course, encouragement. The relaxed environment helps to foster a better experience between students and instructors, thus reducing anxiety of approaching an instructor and -- making them more personable!	{"url":"https://math.unt.edu/undergraduate/ems-lab.html","timestamp":"2024-11-08T22:31:34Z","content_type":"text/html","content_length":"49510","record_id":"<urn:uuid:aa250427-acb3-4ea5-9eca-77e93c9f5816>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00840.warc.gz"}
Ensemble Averages vs Time Averages for Calculating Expected Values The 1987 book, Spectral Analysis: A Nonprobabilistic Theory, argues for more judicious use of the modern stochastic-process-model (arising from the work of mathematicians in the 1930s, such as Khinchin, Kolmogorov, and others) instead of the more realistic predecessor: the time-series model first developed mathematically by Norbert Wiener in 1930 (see also page 59 of Wiener 1949, written in 1942, regarding the historical relationship between his and Kolmogorov’s approaches), that was briefly revisited in the 1960s by engineers before it was buried by mathematicians. The brief tongue-in-cheek essay Ensembles in Wonderland, published in IEEE Signal Processing Magazine, AP Forum, 1994 and reproduced below, is an attempt at satirizing the outrage typified by narrow-minded thinkers exemplified by two outspoken skeptics, Neil Gerr and Melvin Hinich, who wrote scathing remarks and a book review characterizing this book as utter nonsense. (Page 7.6 offers an explanation for the behavior of these two naysayers in terms of weak right-brain thinking.) But first, let us consider the parallel to the book Alice in Wonderland; the following is comprised of excerpts taken from https://en.wikipedia.org/wiki/Alice’s_Adventures_in_Wonderland : Martin Gardner and other scholars have shown the book Alice in Wonderland [written by Lutwidge Dodgson under the pseudonym Lewis Carroll] to be filled with many parodies of Victorian popular culture. Since Carroll was a mathematician at Christ Church, it has been argued that there are many references and mathematical concepts in both this story and his later story Through the Looking Glass; examples include what have been suggested to be illustrations of the concept of a limit, number bases and positional numeral systems, the converse relation in logic, and the ring of integers modulo a specific integer. Deep abstraction of concepts, such as non-Euclidean geometry, abstract algebra, and the beginnings of mathematical logic, was taking over mathematics at the time Alice in Wonderland was being written (the 1860s). Literary scholar Melanie Bayley asserted in the magazine New Scientist that Alice in Wonderland in its final form was written as a scathing satire on new modern mathematics that was emerging in the mid-19th century. Today, Dodgson’s satire appears to be backward looking because, after all, there are strong arguments that modern mathematics has triumphed. Coming back to the topic of interest here, stochastic processes also have triumphed in terms of being wholly adopted in mathematics and science and engineering, except for a relatively small contingent of empirically-minded scientists and engineers. Yet, recent mathematical arguments, described in tutorial fashion on pages 3.2.and 3.3 and further supported with references cited there, provide a sound logical basis for reversing this outcome, especially when the overwhelming evidence of practical, pragmatic, pedagogic, and overarching conceptual advantages provided in the 1987 book and expanded on pages 3.2 and 3.3 here, is considered. The present dominance of the more abstract and less realistic stochastic process theory might be viewed as an example of the pitfalls of what has become known as groupthink or the inertia of human nature that resists changes in thinking, which is discussed in considerable detail based on numerous historical sources on Page 7. Before presenting the several letters comprising the debate, including the standalone article “Ensembles in Wonderland”, the final letter to SP Forum in the debate is reproduced here first to provide hindsight, especially for interpreting “Ensembles in Wonderland”. The bracketed text, e.g., [text], below was added at the time this material was posted on this website to enhance clarity. 3.6.1 Preliminary Material July 2, 1995 (published in Nov 1995) To the Editor: This is my final letter to SP Forum in the debate initiated by Mr. Melvin Hinich’s challenge to the resolution made in the book [1], and carried on by Mr. Neil Gerr through his letters to SP Forum. In this letter, I supplement my previous remarks aimed at clarifying the precariousness of Hinich’s and Gerr’s position by explaining the link between my argument in favor of the utility of fraction-of-time (FOT) probability and the subject of a plenary lecture delivered at ICASSP ’94. In the process of discussing this link I hope to continue the progress made in my previous two letters in discrediting the naysayers and thereby moving toward broader acceptance of the resolution that was made and argued for in [1] and is currently being challenged. My continuing approach is to show that the position taken by the opposition–that the fraction-of-time probability concept and the corresponding time-average framework for statistical signal processing theory and method have nothing to offer in addition to the concept of probability associated with ensembles and the corresponding stochastic process framework–simply cannot be defended if argument is to be based on fact and logic. David J. Thomson’s Transcontinental Waveguide Problem To illustrate that the stochastic-process conceptual framework is often applied to physical situations where the time-average framework is a more natural choice, I have chosen an example from D. J. Thomson’s recent plenary lecture on the project that gave birth to the multiple-window method of spectral analysis [2]. The project that was initiated back in the mid-1960s was to study the feasibility of a transcontinental millimeter waveguide for a telecommunications transmission system potentially targeted for introduction in the mid-1980s. It was found that accumulated attenuation of a signal propagating along a circular waveguide was directly dependent on the spectrum of the series, indexed by distance, of the erratic diameters of the waveguide. So, the problem that Thomson tackled was that of estimating the spectrum for the more than 4,000-mile-long distance-series using a relatively small segment of this series that was broken into a number of 30-foot long subsegments. (It would take more than 700,000 such 30-foot sections to span 4,000 miles.) The spectrum had a dynamic range of over 100 dB and contained many periodic components, indicating the unusual challenge faced by Thomson. When a signal travels down a waveguide (at the speed of light) it encounters the distance-series [consisting of the distances traveled as time progresses]. Because of the constant velocity, the distance-series is equivalent to a time-series. Similarly, the series of diameters that is measured for purposes of analysis is—due to the constant effective velocity of the measurement device—equivalent to a time-series [of measurements]. So, here we have a problem where there is one and only one long time-series of interest (which is equivalent to a distance-series)—there is no ensemble of long series over which average characteristics are of interest and, therefore, there is no obvious reason to introduce the concept of a stochastic process. That is, in the physical problem being investigated, there was no desire to build an ensemble of transcontinental waveguides. Only one (if any at all) was to be built, and it was the spectral density of distance-averaged (time-averaged) power of the single long distance-series (time-series) that was to be estimated, using a relatively short segment, not the spectral density of ensemble-averaged power. Similarly, if one wanted to analytically characterize the average behavior of the spectral density estimate (the estimator mean) it was the average of a sliding estimator over distance (time), not the average over some hypothetical ensemble, that was of interest. Likewise, to characterize the variability of the estimator, it was the distance-average squared deviation of the sliding estimator about its distance-average value (the estimator variance) that was of interest, not the variance over an ensemble. The only apparent reason for introducing a stochastic process model with its associated ensemble, instead of a time-series model, is that one might have been trained to think about spectral analysis of erratic data only in terms of such a conceptual artifice and might, therefore, have been unaware of the fact that one could think in terms of a more suitable alternative that is based entirely on the concept of time averaging over the single time-series. (Although it is true that the time-series segments obtained from multiple 30 ft. sections of waveguide could be thought of as independent random samples from a population, this still does not motivate the concept of an ensemble of infinitely long time-series–a stationary stochastic process. The fact remains that, physically, the 30-foot sections represent subsegments of one long time-series in the communications system concept that was being studied.) [And even if Mr. Thomson was aware of the fact that one could conceptualize the problem entirely in terms of time averages, he had good reason to fear that this approach would be off-putting to his readers all of whom were likely indoctrinated only in statistical spectral analysis theory couched in terms of stochastic processes—an unfortunate It is obvious in this example that there is no advantage to introducing the irrelevant abstraction of a stochastic process (the model adopted by Thomson) except to accommodate lack of familiarity with alternatives. Yet Gerr turns this around and says there is no obvious advantage to using the time-average framework. Somehow, he does not recognize the mental gyrations required to force this and other physical problems into the stochastic process framework. Gerr’s Letter Having explained the link between my argument in favor of the utility of FOT probability and Thomson’s work, let us return to Gerr’s letter. Mr. Gerr, in discussing what he refers to as “a battle of philosophies,” states that I have erred in likening skeptics to religious fanatics. But in the same paragraph we find him defensively trying to convince his readers that the “statistical/ probabilistic paradigm” has not “run out of gas” when no one has even suggested that it has. No one, to my knowledge, is trying to make blanket negative statements about the value of what is obviously a conceptual tool of tremendous importance (probability) and no one is trying to denigrate statistical concepts and methods. It is only being explained that interpreting probability in terms of the fraction-of-time of occurrence of an event is a useful concept in some applications. To argue, as Mr. Gerr does again in the same paragraph, that in general this concept “has no obvious advantages” and using it is “like building a house without power tools: it can certainly be done, but to what end?” is, as I stated in my previous letter, to behave like a religious fanatic — one who believes there can be only One True Religion. This is a very untenable position in scientific research. As I have also pointed out in my previous letter, Mr. Gerr is not at all careful in his thinking. To illustrate his lack of care, I point out that Gerr’s statement “Professor Gardner has chosen to work within the context of an alternative paradigm [fraction-of-time probability]”, and the implications of this statement in Gerr’s following remarks, completely ignore the facts that I have written entire books and many papers within the stochastic process framework, that I teach this subject to my students, and that I have always extolled its benefits where appropriate. If Mr. Gerr believes in set theory and logic, then he would see that I cannot be “within” paradigm A and also within paradigm B unless A and B are not mutually exclusive. But he insists on making them mutually exclusive, as illustrated in the statement “From my perspective, developing signal processing results using the fraction-of-time approach (and not probability/statistics) … .” (The parenthetical remark in this quotation is part of Mr. Gerr’s statement.) Why does Mr. Gerr continue to deny that the fraction-of-time approach involves both probability and statistics? Another example of the lack of care in Mr. Gerr’s thinking is the convoluted logic that leads him to conclude “Thus, spectral smoothing of the biperiodogram is to be preferred when little is known of the signal a priori.” As I stated in my previous letter, it is mathematically proven* in [1] that the frequency smoothing and time averaging methods yield approximately the same result. Gerr has given us no basis for arguing that one is superior to the other and yet he continues to try to make such an argument. And what does this have to do with the utility of the fraction-of-time concept anyway? These are data processing methods; they do not belong to one or another conceptual framework. To further demonstrate the indefensibility of Gerr’s claim that the fraction-of-time probability concept has “no obvious advantages,” I cite two more examples to supplement the advantage of avoiding “unnecessary mental gyrations” that was illustrated using Thomson’s waveguide problem. The first example stems from the fact that the fundamental equivalence between time averaging and frequency smoothing referred to above was first derived by using the fraction-of-time conceptual framework [1]. If there is no conceptual advantage to this framework, why wasn’t such a fundamental result derived during the half century of research based on stochastic processes that preceded [1]? The second example is taken from the first attempt to develop a theory of higher-order cyclostationarity for the conceptualization and solution of problems in communication system design. In [3], it is shown that a fundamental inquiry into the nature of communication signals subjected to nonlinear transformations led naturally to the fraction-of-time probability concept and to a derivation of the cumulant as the solution to a practically motivated problem. This is, to my knowledge, the first derivation of the cumulant. In all other work, which is based on stochastic processes (or non-fraction-of-time probability) and which dates back to the turn of the century, cumulants are defined, by analogy with moments, to be coefficients in an infinite series expansion of a transformation of the probability density function (the characteristic function), which has some useful properties. If there is no conceptual advantage to the fraction-of-time framework, why wasn’t the cumulant derived as the solution to the above-mentioned practical problem or some other practical problem using the orthodox stochastic-probability framework? Since no one in the preceding year has entered the debate to indicate that they have new arguments for or against the philosophy and corresponding theory and methodology presented in [1], it seems fair to proclaim the debate closed. The readers may decide for themselves whether the resolution put forth in [1] was defeated or was upheld. But regarding the skeptics, I sign off with a humorous anecdote: When Mr. Fulton first showed off his new invention, the steamboat, skeptics were crowded on the bank, yelling ‘It’ll never start, it’ll never start.’ It did. It got going with a lot of clanking and groaning and, as it made its way down the river, the skeptics were quiet. For one minute. Then they started shouting. ‘It’ll never stop, it’ll never stop.’ — William A. Gardner * A more detailed and tutorial proof of this fundamental equivalence is given in the article “The history and the equivalence of two methods of spectral analysis,” Signal Processing Magazine, July 1996, No.4, pp.20 – 23, which is copied into the Appendix farther down this Page. 1. W. A. Gardner. Statistical Spectral Analysis: A Nonprobabilistic Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987. 2. D. J. Thomson. “An Overview of Multiple-window and quadratic-inverse spectrum estimation methods,” Plenary Lecture, Proceedings of 1994 International Conference on Acoustics, Speech and Signal Processing, pp. VI-185 – VI-194. 3. W. A. Gardner and C. M. Spooner. “The Cumulant Theory of Cyclostationary time-series, Part I: Foundation,” IEEE Transactions on Signal Processing, Vol. 42, December 1994, pp. 3387-3408. Excerpts from earlier versions of above letter to the editor before it was condensed for publication: April 15, 1995 In this, my final letter to SP Forum in the debate initiated by Mr. Melvin Hinich’s challenge to the resolution made in the book [1], I shall begin by addressing two remarks in the opening paragraph of Mr. Neil Gerr’s last letter (in March 1995 SP Forum). In the first remark, Mr. Gerr suggests that the “bumps and bruises” he sustained by venturing into the “battle” [debate] were to be expected. But I think that such injuries could have been avoided if he had all the relevant information at hand before deciding to enter the debate. This reminds me of a story I recently heard: Georgios and Melvin liked to hunt. Hearing about the big moose up north, they went to the wilds of Canada to hunt. They had hunted for a week, and each had bagged a huge moose. When their pilot Neil landed on the lake to take them out of the wilderness, he saw their gear and the two moose. He said, “I can’t fly out of here with you, your gear, and both moose.” “Why not?” Georgios asked. “Because the load will be too heavy. The plane won’t be able to take off.” They argued for a few minutes, and then Melvin said, “I don’t understand. Last year, each of us had a moose, and the pilot loaded everything.” “Well,” said Neil, “I guess if you did it last year, I can do it too.” So, they loaded the plane. It moved slowly across the lake and rose toward the mountain ahead. Alas, it was too heavy and crashed into the mountain side. No one was seriously hurt and, as they crawled out of the wreckage in a daze, the bumped and bruised Neil asked, “Where are we?” Melvin and Georgios surveyed the scene and answered, “Oh, about a mile farther than we got last year.” If Mr. Gerr had read the book [1] and put forth an appropriate level of effort to understand what it was telling him, he would have questioned Mr. Hinich’s book review and would have seen that the course he was about to steer together with the excess baggage he was about to take on made a crash inevitable. A friend of mine recently offered me some advice regarding my participation in this debate. “Why challenge the status quo”, he said, “when everybody seems happy with the way things are.” My feeling about this is summed up in the following anecdote: “Many years ago, a large American shoe manufacturer sent two sales reps out to different parts of the Australian outback to see if they could drum up some business among the aborigines. Sometime later, the company received telegrams from both agents. The first one said. ‘No business. Natives don’t wear shoes.’ The second one said, ‘Great opportunity here–natives don’t wear shoes.'” Another friend asked “why spend your time on this [debate] when you could be solving important problems.” I think Albert Einstein answered that question when he wrote: “The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science” This underscores my belief that we are overemphasizing “engineering training” in our university curricula at the expense of “engineering science.” It is this belief that motivates my participation in this debate. Instead of plodding along in our research and teaching with the same old stochastic process model for every problem involving time-series data, we should be looking for new ways to think about time-series analysis. In the second remark in Mr. Gerr’s opening paragraph, regarding my response to Mr. Gerr’s October 1994 SP Forum letter in sympathy with “Hinich’s gleefully vicious no-holds-barred review” of [1], Mr. Gerr says “Even by New York standards, it [my response] seemed a bit much.” Well, I guess I was thinking about what John Hancock said, on boldly signing the Declaration of Independence: There, I guess King George will be able to read that! Like the King of England who turned a deaf ear to the messages coming from the new world, orthodox statisticians, like Messrs. Hinich and Gerr who are mired in tradition seem to be hard of hearing–a little shouting might be needed to get through to them. Nevertheless, I am disappointed to see no apparent progress, on Mr. Gerr’s part, in understanding the technical issues involved in his and Hinich’s unsupportable position that the time-average framework for statistical signal processing has, and I quote Gerr’s most recent letter, “no obvious advantages.” I hasten to point out, however, that this most recent position is a giant step back from the earlier even more indefensible position taken by Hinich in his book review, reprinted in April 1994 SP Forum, where much more derogatory language was used. In this letter, I make a final attempt to clarify the precariousness of Hinich’s and Gerr’s position by explaining links between my arguments and the subjects of two plenary lectures delivered at ICASSP ’94. In the process of discussing these links and this paper, I hope to continue the progress made in my previous two letters in discrediting the naysayers and thereby moving toward broader acceptance of the resolution that was made and argued for in [1] and is currently being challenged. My continuing approach is to show that the position taken by the opposition, that the fraction-of-time probability concept and the corresponding time-average framework for statistical signal processing theory and method have nothing to offer in addition to the concept of probability associated with ensembles and the corresponding stochastic process framework, simply cannot be defended if argument is to be based on fact and logic. Lotfi Zadeh and Fuzzy Logic I wish that Mr. Gerr would let go of the fantasy about “the field where the Fraction-of-Timers and Statisticians do battle.” There do not exist two mutually exclusive groups of people—one of which can think only in terms of fraction-of-time probability and the other of which call themselves Statisticians. How many times and in how many ways does this have to be said before Mr. Gerr will realize that some people are capable of using both fraction-of-time probability and stochastic process concepts, and of making choices between these alternatives by assessing the appropriateness of each for each particular application? Mr. Gerr’s “battle” of “fraction-of-time versus probability/statistics” simply does not exist. This insistence on a dichotomy of thought is strongly reminiscent of the difficulties some people have had accepting the proposition that the concept of fuzziness is a useful alternative to the concept of probability. The vehement protests against fuzziness are for most of us now almost laughable. To quote Professor Lotfi Zadeh in his recent plenary lecture [2] “[although fuzzy logic] offers an enhanced ability to model real-world phenomena…[and] eventually fuzzy logic will pervade most scientific theories…the successes of fuzzy logic have also generated a skeptical and sometimes hostile reaction…Most of the criticisms directed at fuzzy logic are rooted in a misunderstanding of what it is and/or a lack of familiarity with it.” I would not suggest that the time-average approach to probabilistic modeling and statistical inference is as deep a concept, as large a departure from orthodox thinking, or as broadly applicable as is fuzzy logic, but there are some definite parallels, and Professor Zadeh’s explanation of the roots of criticism of fuzzy logic applies equally well to the roots of criticism of the time-average approach as an alternative to the ensemble-average or, more accurately, the stochastic-process approach. In the case of fuzzy logic, its proponents are not saying that one must choose either conventional logic and conventional set theory or their fuzzy counterparts as two mutually exclusive alternative truths. Each has its own place in the world. Those opponents who argue vehemently that the unorthodox alternative is worthless can be likened to religious fanatics. This kind of intolerance should have no place in science. But it is all too commonplace and it has been so down through the history of science. So surely, one cannot expect to find its absence in connection with the time-average approach to probabilistic modeling and statistical inference. Even though experimentalists in time-series analysis (including communication systems analysis and other engineered-systems analysis) have been using the time-average approach (to various extents) for more than half a century, there are those like Gerr and Hinich who “see no obvious advantages.” This seems to imply that Mr. Gerr has one and only one interpretation of a time-average measurement on time series data—namely an estimate of some random variable in an abstract stochastic process model. To claim that this mathematical model is, in all circumstances, the preferred one is just plain silly. David J. Thomson and the Transcontinental Waveguide –addition to published discussion: [It is obvious in this example that there is no advantage to introducing the irrelevant abstraction of a stochastic process except to accommodate unfamiliarity with alternatives. Yet Gerr turns this around and says there is no obvious advantage to using the time-average framework.] It is correct in this case that a sufficiently capable person would obtain the same result using either framework, but it is incorrect to not recognize the mental gyrations required to force this physical problem into the stochastic process framework. My claim—and the reason I wrote the book [1]—is that our students deserve to be made aware of the fact that there are two alternatives. It is pigheaded to hide this from our students and force them to go through the unnecessary and sometimes confusing mental gyrations required to force-fit the stochastic process framework to real-world problems where it is truly an unnecessary and, possibly, even inappropriate artifice. Gerr’s Letter—addition to published letter: To further demonstrate the indefensibility of Gerr’s claim that the fraction-of-time probability concept has “no obvious advantages,” I cite two more examples to supplement the advantage of avoiding “unnecessary mental gyrations” that was illustrated using Thomson’s waveguide problem. The first example stems from the fact that the fundamental equivalence between time averaging and frequency smoothing, whose proof is outlined in the Appendix at the end of this letter, was first derived by using the fraction-of-time conceptual framework [1]. An Illustration of Blinding Prejudice To further illustrate the extent to which Mr. Gerr’s prejudiced approach to scientific inquiry has blinded him, I have chosen one of his research papers on the subject of cyclostationary stochastic processes. In [5], Mr. Gerr (and his coauthor) tackle the problem of detecting the presence of cyclostationarity in an observed time-series. He includes an introduction and references sprinkled throughout that tie his work to great probabilists, statisticians, and mathematicians. (We might think of these as the “Saints” in Mr. Gerr’s One True Religion.) This is strange, since his paper is nothing more than an illustration of the application of a known statistical test (and a minor variation thereof) to synthetic data. It is even more strange that he fails to properly reference work that is far more relevant to the problem of cyclostationarity detection. But I think we can see that there is no mystery here. The highly relevant work that is not cited is authored by someone who champions the value of fraction-of-time probabilistic concepts. The fact that the relevant publications (known to Gerr) actually use the stochastic process framework apparently does not remove Mr. Gerr’s blinders. All he can see–it would seem–is that the author is known to argue (elsewhere) that the stochastic process framework is not always the most appropriate one for time-series analysis, and this is enough justification for Mr. Gerr to ignore the highly relevant work by this “heretic” author (author of the book [1] that Hinich all but said should be burned). To be specific, Mr. Gerr completely ignores the paper [6] (published 1-1/2 years prior to the submission of Gerr’s paper) and the book [7] (published 4 years prior) wherein the problem of cyclostationarity detection is tackled using maximum-likelihood [6], maximum-signal-to-noise ratio [6], [7], and other optimality criteria, all of which lead to detection statistics that involve smoothed biperiodograms (and that also identify optimal smoothing) which are treated by Gerr as if they were ad hoc. Mr. Gerr also cites a 1990 publication (which does not appear in his reference list) that purportedly shows that the integrated biperiodogram (cyclic periodogram) equals the cyclic mean square value of the data (cf. (12)); but this is a special case of the much more useful result, derived much earlier than 1990, that the inverse Fourier transform of the cyclic periodogram equals the cyclic correlogram. The argument, by example, that Gerr proffers to show that (12) (the cyclic correlogram at zero lag) is sometimes a good test statistic and sometimes a bad one is trivialized by this Fourier transform relation (cf. [1]) and the numerous mathematical models for data for which the idealized quantities (cyclic autocorrelations, and cyclic spectral densities) in this relation have been explicitly calculated (cf. [1], [7]). These models include, as special cases, the examples that Gerr discusses superficially. The results in [1], [7] show clearly when and why the choice of zero lag made by Gerr in (12) is a poor choice. As another example, consider Mr. Gerr’s offhand remark that a Mr. Robert Lund (no reference cited) “has recently shown that for the current example (an AM signal with a square wave carrier) only lines [corresponding to cycle frequencies] spaced at even multiples of d=8 [the reciprocal of the period of the carrier] will have nonzero spectral (rz) measure.” This result was established in a more general form many years earlier in his coauthor’s Ph.D. dissertation (as well as in [1]) where one need only apply the extremely well-known fact that a symmetrical square wave contains only odd harmonics. To go on, the coherence statistic that Gerr borrows from Goodman for application to cyclostationary processes has been shown in [7] to be nothing more than the standard sample statistic for the standard coherence function (a function of a single frequency variable) for two processes obtained from the one process of interest by frequency-shifting data transformations–except for one minor modification; namely, that time-averaged values of expected values are used in place of non-averaged expected values in the definition of coherence because the processes are asymptotically mean stationary, rather than stationary. Therefore, the well-known issues regarding frequency smoothing in these cross-spectrum statistics need not be discussed further, particularly in the haphazard way this is done by Gerr, with no reliance on analysis of specific underlying stochastic process models. Continuing, the incoherent average (13) proposed by Gerr for use with the coherence statistic is the only novel contribution of this paper, and I claim that it is a poor statistic. The examples used by Gerr show that this “incoherent statistic” outperforms the “coherent statistic,” but what he does not recognize is that he chose the wrong coherent statistic for comparison. He chose the cyclic correlogram with zero lag (12), which is known to be a poor choice for his examples. For his example in Figure 9, zero lag produces a useless statistic, whereas a lag equal to T/2 is known to be optimum, and produces a “coherent statistic” that is superior to Gerr’s incoherent statistic. Thus, previous work [1], [7] suggests that a superior alternative to Gerr’s incoherent statistic is the maximum over a set of lag-indexed coherent statistics. Finally, Mr. Gerr’s vague remarks about choosing the frequency-smoothing window-width parameter M are like stabs in the dark by comparison with the thorough and careful mathematical analysis carried out within–guess what–the time-average conceptual framework in [1] in which the exact mathematical dependence of bias and variance of smoothed biperiodograms on the data-tapering window shape, the spectral-smoothing window shape, and the ideal spectral correlation function for the data model are derived, and in which the equivalence between spectral correlation measurement and conventional cross-spectrum measurement is exploited to show how conventional wisdom [1, chapter 5, 7] applies to spectral correlation measurement [1,chapters 11, 13, 15]. In summary, Gerr’s paper is completely trivialized by previously published work of which he was fully aware. What appears to be his choice to “stick his head in the sand” because the author of much of this earlier highly relevant work was not a member of his One True Religion exemplifies what Gerr is trying to deny. Thus, I repeat it is indeed appropriate to liken those (including Gerr) who Gerr would like to call skeptics to religious fanatics who are blinded by their faith. In closing this letter, I would like to request that Mr. Gerr refrain from writing letters to the editor on this subject. To say, as he does in his last letter, “There are many points on which Professor Gardner and I disagree, but only two that are worthy of further discussion,” is to try to worm his way out of the debate without admitting defeat. I claim to have used careful reasoning to refute beyond all reasonable doubt every point Mr. Gerr (and Mr. Hinich) has attempted to make. Since he has shown that he cannot provide convincing arguments based on fact and logic to support his position, he should consider the debate closed. To sum up the debate: – The resolution, cited in the introductory section of my 2 July 1995 letter to the editor, in contrapositive form, was made by myself in [1]. – The resolution was challenged by Hinich and defended by myself in April 1994 SP Forum. – Hinich’s challenge was supported and my defense was challenged by Gerr in October 1994 SP Forum. – Gerr’s arguments were challenged by myself in January 1995 SP Forum. – Gerr defended his arguments in March 1995 SP Forum. – Gerr’s presumably-final defense was challenged and the final arguments in support of the resolution are made by myself in this letter. APPENDIX from July 2, 1995 letter to Editor (published in Nov 1995) – Proof of Equivalence Between Time-Averaged and Frequency-Smoothed Cyclic Periodograms History and Equivalence of Two Methods of Spectral Analysis Published in IEEE SIGNAL PROCESSING MAGAZINE, July 1996 The purpose of this article is to present a brief history of two methods of spectral analysis and to present, in a tutorial fashion, the derivation of the deterministic relationship that exists between these two methods Two of the oldest and currently most popular methods of measuring statistical (average) power spectral densities (PSD’s) are the frequency smoothing method (FSM) and the time averaging method (TAM). The FSM was thought to have originated in 1930 with Norbert Wiener’s work on generalized harmonic analysis [1], and to have been rediscovered in 1946 by Percy John Daniell [2]. But it was discovered only a few years ago (cf. [3]) that Albert Einstein had introduced the method in 1914 [4]. The currently popular method of deriving the FSM begins by showing that adjacent frequency bins in the periodogram have approximately the same correct mean values and the same large variances, and are approximately uncorrelated with each other. Then, it is observed that averaging these bins together retains the correct mean value, while reducing the variance. The TAM is often attributed to a 1967 paper by P.D. Welch in the IEEE Transactions on Audio and Electroacoustics [5], but in fact the earliest known proposal of the TAM was by Maurice Stevenson Bartlett in 1948 [6]. The reasoning behind the TAM is similar to that for the FSM: the periodograms on adjacent segments of a data record have approximately the same correct mean values and the same large variances, and they are approximately uncorrelated with each other. Therefore, averaging them together will retain the correct mean value, while reducing the variance. (A more detailed historical account of the FSM, TAM, and other methods is given in [7].) Essentially, every spectral analysis software package available today includes either the FSM or the TAM, or both, often in addition to others. These other methods include, for example, the Fourier transformed tapered autocorrelation method, attributed to Ralph Beebe Blackman and John Wilder Tukey [8] (but used as early as 1898 by Albert A. Michelson [9]); and various model fitting methods that grew out of pioneering work by George Udny Yule in 1927 [10] and Gilbert Walker in 1931 [11]. It is well known that both the FSM and the TAM yield PSD estimates that can be made to converge to the exact PSD in some probabilistic sense, like in mean square as the length of the data record processed approaches infinity, However, it is much less commonly known that these two methods are much more directly related to each other. The pioneering methods due to Michelson, Einstein, Wiener, Yule, and Walker were all introduced without knowledge of the concept of a stochastic process. But starting in the 1950s (based on the work of mathematicians such as Khinchin, Wold, Kolmogorov, and Cramér in the 1930s and 1940s , the stochastic-process point of view essentially took over. It appears as though this mathematical formalism, in which analysts focus on calculating means and variances and other probabilistic measures of performance, delayed the discovery of the deterministic relationship between the FSM and TAM for about 40 years. That is, apparently it was not until the non-stochastic approach to understanding statistical (averaged) spectral analysis was revived and more fully developed in [7] that a deterministic relationship between these two fundamental methods was derived. The next section presents, in a tutorial fashion, the derivation of the deterministic relationship between the FSM and TAM, but generalized from frequency-smoothed and time-averaged versions of the periodogram to same for the biperiodogram (also called the cyclic periodogram [7]). This deterministic relationship is actually an approximation of the time-averaged biperiodogram (TAB) by the frequency-smoothed biperiodogram (FSB) and, of course, vice versa. For evidence of the limited extent to which this deterministic relationship is known, the reader is referred to letters that have appeared in the SP Forum section of this magazine in the October 1994, January 1995, March 1995, and November 1995 issues. and let Similarly, let and let To complete the definitions, let It can be shown (using The above approximation, namely The left-most member of the above string of equalities (and an approximation) is a biperiodogram of tapered data seen through a sliding window of length cyclic Wiener relation because it generalizes the Wiener relation between the PSD and autocorrelation from In the special circumstance where the inequality The derivation of the approximation between the FSM and TAM presented here uses a continuous-time model. However, a completely analogous derivation of an approximation between the discrete-time FSM and TAM is easily constructed. When the spectral correlation function is being measured for many values of the frequency-separation parameter, William A. Gardner Professor, Department of Electrical and Computer Engineering University of California, Davis, CA. 1. Wiener, N., “Generalized harmonic analysis,” Acta Mathematika, Vol. 55, pp. 117-258, 1930. 2. Daniell, P. J., “Discussion of ‘On the theoretical specification and sampling properties of autocorrelated time-series’,” J Royal Statistic. Soc., Vol. 8B, No. 1, pp 27-97, 1946. 3. Gardner, W. A., “Introduction to Einstein’s contribution to time-series analysis,” IEEE Signal Processing Magazine, Vol. 4, pp. 4-5, 1987. 4. Einstein, A., “Méthode pour la détermination de valeurs statistiques d’observations concernant des grandeurs sourmises à des fluctuations irrégulières,” Archives des Sciences Physiques et Naturelles, Vol. 37, pp. 254-256, 1914. 5. Welch, P. D., “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Transactions on Audio and Electroacoustics, Vol. AU-15, pp. 70-73, 1967. 6. Bartlett, M. S., “Smoothing periodograms from time-series with continuous spectra,” Nature, Vol. 161, pp. 686-687, 1948. 7. Gardner, W. A., Statistical Spectral Analysis: A Nonprobabilistic Theory. Englewood Cliffs, NJ: Prentice-Hall, 1987. 8. Blackman, R. B. and J. W. Tukey, The Measurement of Power Spectra, New York: AT&T, 1958 (Also New York: Dover, 1959). 9. Michelson, A. A. and S. W. Stratton, “A new harmonic analyzer,” American Journal of Science, Vol. 5, pp. 1-13, 1898. 10. Yule, G. U., “On a method of investigating periodicities in disturbed series, with special reference to Wolfer’s sunspot numbers,” Phil. Trans. Royal Soc: London A, Vol. 226, pp. 267-298, 1927. 11. Walker, G., “On periodicity in series of related terms,” Proceedings of the Royal Society, Vol. 131, pp. 518-532, 1931. 12. Roberts, R. S., W. A. Brown, and H. H. Loomis, Jr., “Computationally efficient algorithms for cyclic spectral analysis,” IEEE Signal Processing Magazine, Vol. 8, pp. 38-49, 1991. 3.6.2 The debate This section is comprised of the following letters to the editor of IEEE Signal Processing Magazine: 1 – Apr 1994, pp. 14, 16 (reprint in SP Magazine of Hinich’s book review in SIAM Review (1991), pp. 677-678) 2 – Apr 1994, pp. 16, 18, 20, 22, 23 (Gardner’s Comments including Ensembles in Wonderland) 3 – Oct 1994, p. 12 (Gerr’s comments) 4 – Jan 1995, pp. 12, 14 (Gardner’s comments in response to Gerr) 5 – Mar 1995, p. 16 (Gerr’s comments—2^nd try) 6 – Jul 1995, pp. 19 – 21 Gardner’s final response (reproduced at beginning of page 3.4.1 above)	{"url":"https://cyclostationarity.com/ensemble-statistics-probability-stochastic-processes-and-their-temporal-counterparts/","timestamp":"2024-11-11T04:35:39Z","content_type":"text/html","content_length":"201047","record_id":"<urn:uuid:e199ea03-fb59-48ca-8da3-db768d009985>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00648.warc.gz"}
From Power Calculations to P-Values: A/B Testing at Stack Overflow \| Julia Silge From Power Calculations to P-Values: A/B Testing at Stack Overflow By Julia Silge October 17, 2017 Note: cross-posted with the Stack Overflow blog. If you hang out on Meta Stack Overflow, you may have noticed news from time to time about A/B tests of various features here at Stack Overflow. We use A/B testing to compare a new version to a baseline for a design, a machine learning model, or practically any feature of what we do here at Stack Overflow; these tests are part of our decision-making process. Which version of a button, predictive model, or ad is better? We don’t have to guess blindly, but instead we can use tests as part of our decision-making toolkit. I get excited about A/B tests because tests like these harness the power of statistics and data to impact the day-to-day details of our business choices. Des Navadeh is the product manager of the Jobs team here at Stack Overflow, and she has used testing extensively on her team to guide decisions. Des says, “A/B testing helps us gain confidence in the change we’re making. It helps us validate new ideas and guides decision making. Without A/B testing, we’re leaving much of what we do up to chance.” At the same time, there can be confusion about how to approach an A/B test, what the statistical concepts involved in such a test are, and what you do before a test vs. after a test. Des and her team have learned a lot by implementing many tests, but also have had some stumbles. “We didn’t realize it at the time, but when we started A/B testing, we took a very strict approach in the calculations to determine sample size. As a result, we were running tests for an unnecessary length of time and most were deemed inconclusive. We basically set up our tests to be almost 100% confident which isn’t very realistic or productive!” Des says. To start testing off on the right foot, we need to plan for an A/B test and perform a power calculation. This requires defining a hypothesis and test groups, and then considering two questions. • How sure do we need to be that we are measuring a real change? • How big is the change we expect to see because of the new version, compared to the baseline? Let’s start with the first question. How sure do you need to be? I am sad to have to break this to you all, but the answer to that first question can’t be 100%. When we measure something in the real world, we never measure with exact accuracy and precision. (That’s basically why I have a job, I think!) There are two main quantities that statisticians use to talk about how much and in what way we can be wrong in measuring. • What percentage of the time are we willing to miss a real effect? This is measured by power. • What percentage of the time are we willing to be fooled into seeing an effect by random chance? This is called significance level, and more precisely, we would state this as the probability of rejecting the null hypothesis. We also talk about these kinds of errors as the false negative rate and false positive rate, which can be very easy to understand given the right example. Typical statistical standards for these quantities are 80% for power (i.e., 20% chance of a false negative) and 5% for significance level. Why are these standards used in practice? That’s a great question with a fair amount of baggage and tradition behind it. If we choose standards that are too strict, perhaps 95% for power and 1% for significance level, all our A/B tests will need to run longer and we will have to invest more time and resources into testing. We won’t be able to iterate quickly to solve our business problems. On the other hand, we’re not curing cancer here, right?! What if we relaxed these statistical standards? Then we risk making change after change in our product that does not improve anything, and investing work from our developers and other team members in changes that do not move us forward toward our goals. We want to be Goldilocks-just-right when it comes to these standards for our purposes. For us at Stack Overflow, that means consistently using 80% for power and 5% for significance level in our power calculations before an A/B test. How big is your change? Our second question here is not about statistical standards, but instead is about how big of a difference we expect to see with the proposed change compared to the status quo. Some phrases that people use to talk about this concept are effect size, expected improvement, and improvement threshold. Effect size can be different in different contexts and different parts of our business. Estimating effect size requires strategic product thinking. Des says, “You need to first understand how different areas of your product perform. Understanding how each part of your funnel converts today helps you decide how big of an effect you’d need to see for the new change to be worth it. We use different questions to help estimate the effect size. How much development work is required to graduate the test? How strategically important is it? Does this feature support future plans? What is the size of audience or action are we optimizing for? These answers are detailed as success criteria in our test plans.” Some of the factors Des takes into account when estimating effect size are volume of events that enter the funnel that is being considered, baseline conversion rate of the feature, and how the expected improvement impacts overall product metrics. Power calculations Once we have estimated an effect size for our test and know the statistical standards we are going to use in planning, we can do a power calculation to find out how big of a sample size we need for our test. The point of power calculations like these is to find out what sample size we need for our A/B test, how many views or users or form submissions or other interactions we need in each group to achieve the necessary power for our test. Then we can finally start our test! Time to wait for those events to roll in. How do we calculate how big of a sample we need, to measure the change we expect with the statistical standards we’ve chosen? For most tests, our product teams use online calculators to find the sample size. I’m an R developer, so I would use a function in R for such a test. For more complicated tests, we on the data team sometimes run simulations for power calculations. When we calculate power, we see first-hand how power, significance level, and effect size interact with sample size and the baseline conversion rate that we were dealing with to start with. I built a Shiny app to demonstrate how these factors are related for a proportion test, which is typically applicable in our A/B tests. You can click the “Source Code” button on the app to see the R code that built this app. Notice the shapes of the curves, and how they change when you move the sliders. We need bigger sample sizes to measure small effect sizes, or to achieve low significance levels. If the baseline rate is higher to start with, the sample size needed for a given power goes down. These complicated interactions affect our A/B tests at Stack Overflow. “We realized that we couldn’t standardize power calculations across all tests. Some parts of our funnel were highly optimized and converted well, which meant we needed smaller samples sizes to detect the same effect we would want to see in an area that didn’t convert as well,” Des says. “Other areas had higher volume, like page views, but did not convert at well. While higher volume helps us reach the sample size need faster, we needed a larger effect size for the change to make an impact.” Analyzing results What happens after the test? After we have collected enough events to meet our sample size requirements, it’s time to analyze the results. At Stack Overflow, we have testing infrastructure for teams to automatically see analysis of results, or if I am performing an analysis myself, I might use a statistical test like a proportion test using R. “We know we can end a test when we’ve reached the sample size we set out to collect, and then we check out the p-value,” Des says. The p-value of an A/B test is the probability that we would get the observed difference between the A and B groups (or a more extreme difference) by random chance. When the p-value is high, that means the probability that we could just randomly see that difference between the A and B groups is high, due only to sampling noise. When the p-value of our A/B test is low enough (below our threshold), we can say that the probability of seeing such a difference randomly is low and we can feel confident about making the change to the new alternative from our original version. If you pay attention to the world of statistics, you may have seen some hubbub about changing the threshold for p-values; a recent paper claimed that moving from a threshold of 0.05 to 0.005 would solve the reproducibility crisis in science and fix, well, lots of things. It’s true that using a threshold of p < 0.05 means being fooled 1 in 20 times, but ultimately, the problem with using statistics and measurement isn’t p-values. The problem is us. We can’t apply these kinds of thresholds without careful consideration of context and domain knowledge, and a commitment to honesty (especially to ourselves!) when it comes to p-values. We are sticking with a p-value threshold of 0.05 for our A/B tests, but these tests must always be interpreted holistically by human beings with an understanding of our data and our business. When to JUST SAY NO to an A/B test Tests like the ones Des and I have talked about in this post are a powerful tool, but sometimes the best choice is knowing when not to run an A/B test. We at Stack Overflow have encountered this situation when considering a feature used by a small number of users and a potential change to that feature that we have other reasons for preferring to the status quo. The length of a test needed to achieve adequate statistical power in such a situation is impractically long, and the best choice for us in our real-life situation is to forgo a test and make the decision based on non-statistical “Product thinking is critical here. Sometimes a change is obviously better UX but the test would take months to be statistically significant. If we are confident that the change aligns with our product strategy and creates a better experience for users, we may forgo an A/B test. In these cases, we may take qualitative approaches to validate ideas such as running usability tests or user interviews to get feedback from users,” says Des. “It’s a judgement call. If A/B tests aren’t practical for a given situation, we’ll use another tool in the toolbox to make progress. Our goal is continuous improvement of the product. In many cases, A/B testing is just one part of our approach to validating a change.” Along the same lines, sometimes the results of an A/B test can be inconclusive, with no measurable difference between the baseline and new version, either positive or negative. What should we do then? Often we stay with the original version of our feature, but in some situations, we still decide to make a change to a new version, depending on other product considerations. Dealing with data means becoming comfortable with uncertainty, and A/B tests make this reality extremely apparent. Handling uncertainty wisely and using statistical tools like A/B tests well can give us the ability to make better decisions. Des and her team have used extensive testing to make Stack Overflow Jobs a great tool for developers in the market for a new opportunity, so be sure to check it out! Posted on: October 17, 2017 10 minute read, 2028 words	{"url":"https://juliasilge.com/blog/ab-testing/","timestamp":"2024-11-07T15:47:10Z","content_type":"text/html","content_length":"22143","record_id":"<urn:uuid:70fdc476-2b40-4b6c-bc4f-321d5fab099d>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00499.warc.gz"}
Convergent SDP-relaxations in polynomial optimization with sparsity We consider a polynomial programming problem P on a compact basic semi-algebraic set K described by m polynomial inequalities $g_j(X)\geq0$, and with polynomial criterion $f$. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDP-relaxation of order $r$ has the following two features: (a) The number of variables is $O(\kappa^{2r})$ where $\kappa=\max[\kappa_1,\kappa_2]$ witth $\kappa_1$ (resp. $\kappa_2$) being the maximum number of variables appearing the monomials of $f$ (resp. appearing in a single constraint $g_j(X)\geq0$). (b) The largest size of the LMI's (Linear Matrix Inequalities) is $O(\kappa^r)$. This is to compare with the respective number of variables $O(n^{2r})$ and LMI size $O(n^r)$ in the original SDP-relaxations defined in Lasserre [11]. Therefore, great computational savings are expected in case of sparsity in the data, i.e. when $\kappa$ is small, a frequent case in practical applications of interest. The novelty with respect to Waki et al. [9] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the "running intersection property" in graph theory. In such cases, and as a by-product, we also obtain a new representation result for polynomials positive on a basic closed semi-algebraic set, a "sparse" version of Putinar's Positivstellensatz. Technical report #05-612, LAAS, Toulouse, France (2005); To appear in Siam J. Optimization View Convergent SDP-relaxations in polynomial optimization with sparsity	{"url":"https://optimization-online.org/2006/04/1367/","timestamp":"2024-11-12T09:30:06Z","content_type":"text/html","content_length":"84621","record_id":"<urn:uuid:b15427c6-62dc-46aa-b8a7-2eedcf225bda>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00286.warc.gz"}
In 2 hours Cara read 48 pages. If she reads the same number of pages each hour, how many pages did Cara read in one hour? Find an answer to your question 👍 “In 2 hours Cara read 48 pages. If she reads the same number of pages each hour, how many pages did Cara read in one hour? ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions. Search for Other Answers	{"url":"https://cpep.org/mathematics/2372415-in-2-hours-cara-read-48-pages-if-she-reads-the-same-number-of-pages-ea.html","timestamp":"2024-11-14T23:23:28Z","content_type":"text/html","content_length":"23237","record_id":"<urn:uuid:65768e9f-1dc1-4258-b360-11c166b9e569>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00023.warc.gz"}
Zonotope -- from Wolfram MathWorld A zonotope is a set of points in Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope (Eppstein 1996). A three-dimensional zonotope is called a zonohedron. There is some confusion in the definition of zonotopes (Eppstein 1996). Wells (1991, pp. 274-275) requires the generating vectors to be in general position (all et al. 1995; Ziegler 1995, pp. 198-208; Eppstein 1996) do not make this restriction. Coxeter (1973) starts with one definition but soon switches to the other. The combinatorics of the faces of a zonotope are equivalent to those of an arrangement of hyperplanes in a space of one fewer dimension so, for example, zonohedra correspond to planar line arrangements (Eppstein 1996).	{"url":"https://mathworld.wolfram.com/Zonotope.html","timestamp":"2024-11-06T00:41:29Z","content_type":"text/html","content_length":"54302","record_id":"<urn:uuid:6a021f0b-cd9d-4d23-a394-ae8cc92632be>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00395.warc.gz"}
Math Star Solves Beal's Conjecture, Forgets Proof - Fake News Agency Math Star Solves Beal’s Conjecture, Forgets Proof A brilliant mathematician, who is widely regarded as one of the leading experts in number theory, has claimed that he has solved Beal’s conjecture, one of the most tantalizing and elusive unsolved problems in mathematics. However, he also confesses that he has forgotten how he solved it and cannot reproduce his proof. The mathematician is Terence Tao, a professor at UCLA and a recipient of the Fields Medal, the highest honor in mathematics. He says that he has been secretly working on Beal’s conjecture for the past five years, using a combination of analytic number theory, algebraic geometry, and harmonic analysis. According to Tao, he found a proof last month, while he was on a vacation in New Zealand. He wrote down the proof on a piece of paper at a hotel room, and then went to enjoy the scenery at a nearby park. However, when he came back to the hotel room, he found that his paper had been thrown away by the cleaning staff. He asked the staff if they had seen it, but they said that they had no idea what he was talking about. He tried to remember the proof from memory, but he failed. He only remembers the main idea and some of the steps, but not the details and the calculations. Tao says that he is devastated by his loss and that he hopes to find his paper or reconstruct his proof. He has contacted the American Mathematical Society, which administers the Beal Prize Problems, and informed them of his situation. He has asked them to reserve the prize for him until he can provide his proof. The American Mathematical Society has not yet commented on Tao’s claim. However, some mathematicians have expressed skepticism and doubt about his story. They say that it is improbable that Tao could have solved such a hard problem without publishing any intermediate results or collaborating with other experts. They also say that it is strange that Tao did not take a photo of his paper or make a copy of his proof. Tao understands the skepticism and says that he does not expect anyone to believe him without evidence. He is confident that he can prove his claim and that he will not give up until he does. He is still searching for his paper and trying to remember his proof. He hopes to find his solution soon and share it with the world.	{"url":"https://fakenews.agency/index.php/2023/08/08/math-star-solves-beals-conjecture-forgets-proof/","timestamp":"2024-11-11T11:47:02Z","content_type":"text/html","content_length":"49571","record_id":"<urn:uuid:1bd7a23b-9770-4ccf-8577-c26949330390>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00763.warc.gz"}
Web Macro - downloading files with dates that change weekly - VisualCron - Forum The easiest way to solve 1) is to use .net. If you prefer formulas, you can do something like this (you have to test to see that it works properly) 1) Find the current day of the week: {DATE(WeekDayNumber\|Monday)} 2) Use this to find how many days to subtract to get to monday of current week: (if today is sunday, the previous step is 7, so: -7 - 6 +7 results in -6 days, if today is tuesday, previous step is 2, so: -2 - 6 + 7 results in -1 ) 3) Use result in step 2 to get to monday in current week: 4) Subtract any number of weeks that you want (this is the final formula) The parameter to replace is the -1 at the end of the formula, so in this example i have sutracted one week. Change formatting if you like Edited by user 2017-07-03T15:25:11Z \| Reason: Not specified	{"url":"https://www.visualcron.com/forum.aspx?g=Posts&m=29824","timestamp":"2024-11-11T23:51:25Z","content_type":"text/html","content_length":"182098","record_id":"<urn:uuid:187d3c97-e527-497e-a8e8-95175d99022b>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00093.warc.gz"}
Flat pullback and intersection products Lemma 43.21.1. Let $f : X \to Y$ be a flat morphism of nonsingular varieties. Set $e = \dim (X) - \dim (Y)$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $Y$ with $\dim (\text{Supp}(\ mathcal{F})) \leq r$, $\dim (\text{Supp}(\mathcal{G})) \leq s$, and $\dim (\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim (Y)$. In this case the cycles $[f^\mathcal{F}]_ {r + e}$ and $[f^\mathcal{G}]_{s + e}$ intersect properly and \[ f^([\mathcal{F}]_ r \cdot [\mathcal{G}]_ s) = [f^\mathcal{F}]_{r + e} \cdot [f^*\mathcal{G}]_{s + e} \] Comments (0) Post a comment Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B0B. Beware of the difference between the letter 'O' and the digit '0'. The tag you filled in for the captcha is wrong. You need to write 0B0B, in case you are confused.	{"url":"https://stacks.math.columbia.edu/tag/0B0B","timestamp":"2024-11-09T01:13:38Z","content_type":"text/html","content_length":"15728","record_id":"<urn:uuid:641fc8c2-7351-490e-b803-b829ac68cc4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00203.warc.gz"}
Efficient circle distance testing Comments enabled. I really need your comment Answering questions asked on the site. eptil asks: I am using SQL Server 2008, but not the spatial features. I have a table with few entries, only 40,000. There is an id INT PRIMARY KEY column and two columns storing a 2d coordinate, both decimals. I would like to find all the records that do not have other records within a given radius. The query I am using at the moment is: SELECT id, x, y FROM mytable t1 WHERE ( SELECT COUNT() FROM mytable t2 WHERE ABS(t1.x - t2.x) < 25 AND ABS(t1.y - t2.y) < 25 ) = 1 <!-- --> This is taking <strong>15</strong> minutes to run at times. Is there a better way? Of course using spatial abilities would be a better way, but it is possible to make do with plain <strong>SQL</strong>. This will also work in <strong>SQL Server 2005</strong>. In most database engines, the spatial indexes are implemented as the <strong>R-Tree</strong> structures. <strong>SQL Server</strong>, however, uses another approach: surface tesselation. Basically, it divides the surface into a finite number of tiles, each assigned with a unique number. The identifiers of tiles covered by the object are stored as keys of a plain <strong>B-Tree</strong> index. When <strong>SQL Server</strong>'s optimizer sees a geometrical predicate against an indexed column, it calculates the numbers of tiles that <em>possibly</em> can satisfy this predicate. Say, if the tiles are defined as squares with side <strong>1</strong>, the predicate <code>column.STDistance(@mypoint) < 2</code> can only be satisfied by the objects within <strong>2</strong> tiles away from <code>@mypoint</code>'s tile. This gives a square of <strong>25</strong> tiles with <code>@mypoint</code>'s tile in the center. The tile numbers can be found and searched for using the index. Exact filtering condition is then applied to each candidate value returned by the index. Same solution can be used in our case even without the spatial functions. Comparing tile numbers is an equijoin and hash join method is eligible for this operation. We can even choose the tiling algorithm individually for each query, since we don't have to store the tile identifiers in the table, and the hash table will be built dynamically anyway. Let's create a sample table and see how it works: <span id="more-4449"></span> CREATE SCHEMA [20100226_circle] CREATE TABLE t_circle ( id INT NOT NULL PRIMARY KEY, x DECIMAL (15, 3) NOT NULL, y DECIMAL (15, 3) NOT NULL, BEGIN TRANSACTION SELECT RAND(20100226) DECLARE @cnt INT SET @cnt = 1 WHILE @cnt <= 50000 INTO [20100226_circle].t_circle (id, x, y) VALUES ( RAND() 3200, RAND() * 3200 SET @cnt = @cnt + 1 The table contains <strong>50,000</strong> points on random places within a square of <strong>3,200 × 3,200</strong> units. We can optimize the original query a little by using <code>NOT EXISTS</code> instead of <code>COUNT()</code>: SELECT FROM [20100226_circle].t_circle co WHERE NOT EXISTS SELECT NULL FROM [20100226_circle].t_circle ci WHERE SQRT(POWER(co.x - ci.x, 2) + POWER(co.y - ci.y, 2)) < 25 AND co.id <> ci.id ORDER BY , but this would still be quite slow. The problem is that the nested loops go nowhere: they are still inside the plan, but return earlier. As a result, the query takes 5 minutes instead of 15, which is still too much. To improve the query we need to make an efficient anti-join method to work, and the tesselation strategy is a way to go. Here's what we need to do to implement this strategy: □ Tesselate the surface and assign a unique number to each tile. Since we need to search for the records within 25 units, it will be a reasonable idea to divide the surface into a number of squares 25 × 25 units in size, numbered column-wise. To find out the number of rows and columns, we should just find the MIN and MAX x and y. This is the sample tesselation, assuming that MIN(x) and MIN(y) are 25 × 100 = 2500 units apart (and same with y). □ Find the tile each point belongs to. □ Build a recordset that would correspond each point to each of the tiles the neighbors can theoretically reside in. Since a circle with radius of 25 units theoretically may cover up to 9 adjacent tiles, each tile should be corresponded to each of the 9 tiles forming a 3 × 3 square with a unit's tile in the center. □ For each point, make sure that no other points exists within the neighboring tiles, additionally applying a fine-filtering condition. This may sound redundant, since the point-neighbor combination is unique, as well as point-tile, so only one of the 9 candidates will satisfy the join, even if the points reside in the adjacent tiles. But finding the exact distance requires a complex expression while comparing the tile numbers is an equality correlation and as such is eligible by an efficient anti-join method like HASH ANTI JOIN. The coarse filtering on tiles will sieve out most of the far neighbors so that the only adjacent neighbors will require special attention. And here's the query: WITH extremes AS SELECT , maxx - minx AS width, maxy - miny AS height FROM ( SELECT FLOOR(MIN(x) / 25) AS minx, CEILING(MAX(x) / 25) AS maxx, FLOOR(MIN(y) / 25) AS miny, CEILING(MAX(y) / 25) AS maxy FROM [20100226_circle].t_circle ) q tileset (dim) AS SELECT -1 UNION ALL SELECT 0 UNION ALL SELECT 1 tiles AS SELECT id, x, y, minx, miny, width, (FLOOR(x / 25) - minx) width + FLOOR(y / 25) - miny AS tile FROM extremes CROSS JOIN neighbors AS SELECT ti., (FLOOR(ti.x / 25) - ti.minx + nx.dim) ti.width + FLOOR(ti.y / 25) - ti.miny + ny.dim AS mtile FROM tiles ti CROSS JOIN tileset nx CROSS JOIN tileset ny SELECT * FROM tiles tn WHERE NOT EXISTS SELECT NULL FROM neighbors n WHERE n.mtile = tn.tile AND n.id <> tn.id AND SQRT(SQUARE(n.x - tn.x) + SQUARE(n.y - tn.y)) < 25 ) ORDER BY id [/sourcecode] id x y minx miny width tile 205 2247.896 3198.399 0 0 128 11519 2867 2159.626 1120.590 0 0 128 11052 13644 4.951 3165.734 0 0 128 126 15917 2747.826 3041.280 0 0 128 14073 25183 1858.866 326.416 0 0 128 9485 43211 1176.369 98.079 0 0 128 6019 6 rows fetched in 0.0017s (3.0781s) Table 't_circle'. Scan count 11, logical reads 3087, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0. Table 'Worktable'. Scan count 2, logical reads 207406, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0. Table 'Worktable'. Scan count 0, logical reads 0, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0. SQL Server Execution Times: CPU time = 4235 ms, elapsed time = 3068 ms. \|--Parallelism(Gather Streams, ORDER BY:([test].[20100226_circle].[t_circle].[id] ASC)) \|--Sort(ORDER BY:([test].[20100226_circle].[t_circle].[id] ASC)) \|--Compute Scalar(DEFINE:([Expr1007]=floor([Expr1003]/(25.)), [Expr1009]=floor([Expr1005]/(25.)), [Expr1011]=ceiling([Expr1004]/(25.))-floor([Expr1003]/(25.)), [Expr1016]=(([Expr1044]-floor([Expr1003]/(25.)))(ceiling([Expr1004]/(25.))-floor([Expr1003]/(25.)))+[Expr1045])-floor([Expr1005]/(25.)))) \|--Hash Match(Left Anti Semi Join, HASH:([Expr1055])=([Expr1054]), RESIDUAL:([Expr1054]=[Expr1055] AND [test].[20100226_circle].[t_circle].[id]<>[test].[20100226_circle].[t_circle].[id] AND sqrt(square(CONVERT_IMPLICIT(float(53),[test].[20100226_circle].[t_circle].[x]-[test].[20100226_circle].[t_circle].[x],0))+square(CONVERT_IMPLICIT(float(53),[test].[20100226_circle].[t_circle].[y]-[test].[20100226_circle].[t_circle].[y],0)))<(2.500000000000000e+001))) \|--Bitmap(HASH:([Expr1055]), DEFINE:([Bitmap1056])) \| \|--Parallelism(Repartition Streams, Hash Partitioning, PARTITION COLUMNS:([Expr1055])) \| \|--Compute Scalar(DEFINE:([Expr1055]=(([Expr1044]-floor([Expr1003]/(25.)))(ceiling([Expr1004]/(25.))-floor([Expr1003]/(25.)))+[Expr1045])-floor([Expr1005]/(25.)))) \| \|--Nested Loops(Inner Join) \| \|--Parallelism(Distribute Streams, Broadcast Partitioning) \| \| \|--Stream Aggregate(DEFINE:([Expr1003]=MIN([partialagg1048]), [Expr1004]=MAX([partialagg1049]), [Expr1005]=MIN([partialagg1050]))) \| \| \|--Parallelism(Gather Streams) \| \| \|--Stream Aggregate(DEFINE:([partialagg1048]=MIN([test].[20100226_circle].[t_circle].[x]), [partialagg1049]=MAX([test].[20100226_circle].[t_circle].[x]), [partialagg1050]=MIN([test].[20100226_circle].[t_circle].[y]))) \| \| \|--Clustered Index Scan(OBJECT:([test].[20100226_circle].[t_circle].[PK__t_circle__62065FF3])) \| \|--Compute Scalar(DEFINE:([Expr1044]=floor([test].[20100226_circle].[t_circle].[x]/(25.)), [Expr1045]=floor([test].[20100226_circle].[t_circle].[y]/(25.)))) \| \|--Clustered Index Scan(OBJECT:([test].[20100226_circle].[t_circle].[PK__t_circle__62065FF3])) \|--Parallelism(Repartition Streams, Hash Partitioning, PARTITION COLUMNS:([Expr1054]), WHERE:(PROBE([Bitmap1056])=TRUE)) \|--Compute Scalar(DEFINE:([Expr1054]=(((([Expr1046]-floor([Expr1020]/(25.)))+CONVERT_IMPLICIT(decimal(10,0),[Union1037],0))*(ceiling([Expr1021]/(25.))-floor([Expr1020]/(25.)))+[Expr1047])-floor([Expr1022]/(25.)))+CONVERT_IMPLICIT(decimal(10,0),[Union1041],0))) \|--Nested Loops(Inner Join) \|--Nested Loops(Inner Join) \| \|--Parallelism(Distribute Streams, RoundRobin Partitioning) \| \| \|--Nested Loops(Inner Join) \| \| \|--Stream Aggregate(DEFINE:([Expr1020]=MIN([partialagg1051]), [Expr1021]=MAX([partialagg1052]), [Expr1022]=MIN([partialagg1053]))) \| \| \| \|--Parallelism(Gather Streams) \| \| \| \|--Stream Aggregate(DEFINE:([partialagg1051]=MIN([test].[20100226_circle].[t_circle].[x]), [partialagg1052]=MAX([test].[20100226_circle].[t_circle].[x]), [partialagg1053]=MIN([test].[20100226_circle].[t_circle].[y]))) \| \| \| \|--Clustered Index Scan(OBJECT:([test].[20100226_circle].[t_circle].[PK__t_circle__62065FF3])) \| \| \|--Constant Scan(VALUES:(((-1)),((0)),((1)))) \| \|--Constant Scan(VALUES:(((-1)),((0)),((1)))) \|--Table Spool \|--Compute Scalar(DEFINE:([Expr1046]=floor([test].[20100226_circle].[t_circle].[x]/(25.)), [Expr1047]=floor([test].[20100226_circle].[t_circle].[y]/(25.)))) \|--Clustered Index Scan(OBJECT:([test].[20100226_circle].[t_circle].[PK__t_circle__62065FF3])) This query only takes 3 seconds. Hope that helps. I'm always glad to answer the questions regarding database queries.	{"url":"https://explainextended.com/2010/02/26/efficient-circle-distance-testing/","timestamp":"2024-11-12T07:13:17Z","content_type":"application/xhtml+xml","content_length":"86946","record_id":"<urn:uuid:8d397355-3372-41e6-9bfc-a1392c2adc21>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00110.warc.gz"}
Normalized system From Encyclopedia of Mathematics A system Banach space Normalization of a system [1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) [2] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) [3] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian) How to Cite This Entry: Normalized system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalized_system&oldid=18213 This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article	{"url":"https://encyclopediaofmath.org/index.php?title=Normalized_system&oldid=18213","timestamp":"2024-11-14T19:14:04Z","content_type":"text/html","content_length":"16116","record_id":"<urn:uuid:0670ce7f-4ab7-43aa-82fa-24f9f71db526>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00465.warc.gz"}
reading single channel Agilent data with limma [was arrayQualityMetrics d... Last seen 9.9 years ago Hi Gordon, Why creating ExpressionSet >esetPROC = new("ExpressionSet", exprs = ddaux$G) results in error if then running >arrayQualityMetrics(expressionset=esetPROC,outdir ="esetPROC",force =T) ? Developers (Audrey Kauffmann and Wolfgang Huber) claim that "expressionset is an object of class ExpressionSet for one color non Affymetrix data" (see definitions for prepdata() in their reference manual). As to bg-correction: I agree that it makes sense usually. I looked at the LIMMA user guide(11 November 2011): Agilent is mentioned in pp.16, 19, and 26 ... and nothing special about one-color Agilent arrays. But your read.maimages() is used in read.AgilentFE() from package Agi4x44PreProcess to import Agilent one-color data sets as RGlist. Thanks Alex In a message dated 6/9/2012 8:12:37 P.M. Pacific Daylight Time, smyth@wehi.EDU.AU writes: Hi Alex, I don't know arrayQualityMetrics, but you are using the limma package to read single-channel Agilent data in a way that I think might cause problems with down-stream analyses. Basically you're creating a two- color data object when your data is not actually of that type. This was a time when I suggested this sort of work-around as a stop-gap measure for some data problems, but hasn't been necessary for quite a few years. I'd also recommend that you do some background correction. If I understand your code correctly, I don't think it is currently making use of the background intensity column. There is a case study in the limma User's Guide that deals with single channel Agilent data. Could you please have a read of that for a cleaner way to read Agilent data? I don't know whether that will be enough to solve your arrayQualityMetrics problem, but perhaps it might. Best wishes Gordon ------------- original message ------------- [BioC] arrayQualityMetrics() doesn't work for one-color non Affy arrays Alogmail2 at aol.com Alogmail2 at aol.com Fri Jun 8 09:39:21 CEST 2012 Dear List, Could you share your experience with arrayQualityMetrics() for one- color non Affy arrays: it doesn't work for me (please see the code below). Thanks Alex Loguinov UC, Berkeley >options(error = recover, warn = 2) >options(bitmapType = "cairo") >.HaveDummy = !interactive() > if(.HaveDummy) pdf("dummy.pdf") >library("arrayQualityMetrics") >head(targets) FileName Treatment GErep Time Conc T0-Control-Cu_61_new_252961010035_2_4 T0-Control-Cu_61_new_252961010035_2_4.txt C.t0.0 0 0 0 T0-Control-Cu_62_new_252961010036_2_1 T0-Control-Cu_62_new_252961010036_2_1.txt C.t0.0 0 0 0 T0-Control-Cu_64_252961010031_2_2 T0-Control-Cu_64_252961010031_2_2.txt C.t0.0 0 0 0 T0-Control-Cu_65_new_252961010037_2_2 T0-Control-Cu_65_new_252961010037_2_2.txt C.t0.0 0 0 0 T04h-Contr_06_new_252961010037_2_4 T04h-Contr_06_new_252961010037_2_4.txt C.t4.0 1 4 0 T04h-Contr_10_new_252961010035_1_2 T04h-Contr_10_new_252961010035_1_2.txt C.t4.0 1 4 0 > ddaux = read.maimages (files = targets$FileName, source = "agilent", other.columns = list(IsFound = "gIsFound", IsWellAboveBG = "IsWellAboveBG",gIsPosAndSignif="gIsPosAndSignif", IsSaturated = "gIsSaturated", IsFeatNonUnifOF = "gIsFeatNonUnifOL", IsFeatPopnOL = "gIsFeatPopnOL", ChrCoord = "chr_coord",Row="Row",Column="Col"), columns = list(Rf = "gProcessedSignal", Gf = "gMeanSignal", Rb = "gBGMedianSignal", Gb = "gBGUsed"), verbose = T, sep = "\t", quote = "") > class(ddaux) [1] "RGList" attr(,"package") [1] "limma" > names(ddaux) [1] "R" "G" "Rb" "Gb" "targets" "genes" "source" "printer" "other" I could apply: > > class(ddaux$G) [1] "matrix" >all(rownames(targets)==colnames(ddaux$G)) [1] TRUE >esetPROC = new("ExpressionSet", exprs = ddaux$G) But it results in errors: > arrayQualityMetrics(expressionset=esetPROC,outdir ="esetPROC",force =T) The directory 'esetPROC' has been created. Error: no function to return from, jumping to top level Enter a frame number, or 0 to exit 1: arrayQualityMetrics(expressionset = esetPROC, outdir = "esetPROC", force = T) 2: aqm.writereport(modules = m, arrayTable = x$pData, reporttitle = reporttitle, outdir = outdir) 3: reportModule(p = p, module = modules[[i]], currentIndex = currentIndex, arrayTable = arrayTableCompact, outdir = outdir) 4: makePlot(module) 5: print(_x@plot_ (mailto:x@plot) ) 6: print.trellis (_x@plot_ (mailto:x@plot) ) 7: printFunction(x, ...) 8: tryCatch(checkArgsAndCall(panel, pargs), error = function(e) panel.error(e)) 9: tryCatchList(expr, classes, parentenv, handlers) 10: tryCatchOne(expr, names, parentenv, handlers[[1]]) 11: doTryCatch(return(expr), name, parentenv, handler) 12: checkArgsAndCall(panel, pargs) 13: do.call(FUN, args) 14: function (x, y = NULL, subscripts, groups, panel.groups = "panel.xyplot", ..., col = "black", col.line = superpose.line$col, col.symbol = superpose.symb 15: .signalSimpleWarning("closing unused connection 5 (Report_for_exampleSet/index.html)", quote(NULL)) 16: withRestarts({ 17: withOneRestart(expr, restarts[[1]]) 18: doWithOneRestart(return(expr), restart) Selection: 0 Error in KernSmooth::bkde2D(x, bandwidth = bandwidth, gridsize = nbin, : (converted from warning) Binning grid too coarse for current (small) bandwidth: consider increasing 'gridsize' Enter a frame number, or 0 to exit 1: arrayQualityMetrics(expressionset = esetPROC, outdir = "esetPROC", force = T) 2: aqm.writereport(modules = m, arrayTable = x$pData, reporttitle = reporttitle, outdir = outdir) 3: reportModule(p = p, module = modules[[i]], currentIndex = currentIndex, arrayTable = arrayTableCompact, outdir = outdir) 4: makePlot(module) 5: do.call(_x@plot_ (mailto:x@plot) , args = list()) 6: function () 7: meanSdPlot(x$M, cex.axis = 0.9, ylab = "Standard deviation of the intensities", xlab = "Rank(mean of intensities)") 8: meanSdPlot(x$M, cex.axis = 0.9, ylab = "Standard deviation of the intensities", xlab = "Rank(mean of intensities)") 9: smoothScatter(res$px, res$py, xlab = xlab, ylab = ylab, ...) 10: grDevices:::.smoothScatterCalcDensity(x, nbin, bandwidth) 11: KernSmooth::bkde2D(x, bandwidth = bandwidth, gridsize = nbin, range.x = range.x) 12: warning("Binning grid too coarse for current (small) bandwidth: consider increasing 'gridsize'") 13: .signalSimpleWarning("Binning grid too coarse for current (small) bandwidth: consider increasing 'gridsize'", quote(KernSmooth::bkde2D(x, bandwidth = ba 14: withRestarts({ 15: withOneRestart(expr, restarts[[1]]) 16: doWithOneRestart(return (expr), restart) Selection: 0 > sessionInfo() R version 2.14.2 (2012-02-29) Platform: i386-pc-mingw32/i386 (32-bit) locale: [1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252 LC_MONETARY=English_United States.1252 [4] LC_NUMERIC=C LC_TIME=English_United States.1252 attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] CCl4_1.0.11 vsn_3.22.0 arrayQualityMetrics_3.10.0 Agi4x44PreProcess_1.14.0 genefilter_1.36.0 [6] annotate_1.32.3 AnnotationDbi_1.16.19 limma_3.10.3 Biobase_2.14.0 loaded via a namespace (and not attached): [1] affy_1.32.1 affyio_1.22.0 affyPLM_1.30.0 beadarray_2.4.2 BiocInstaller_1.2.1 Biostrings_2.22.0 [7] Cairo_1.5-1 cluster_1.14.2 colorspace_1.1-1 DBI_0.2-5 grid_2.14.2 Hmisc_3.9-3 [13] hwriter_1.3 IRanges_1.12.6 KernSmooth_2.23-7 lattice_0.20-6 latticeExtra_0.6-19 plyr_1.7.1 [19] preprocessCore_1.16.0 RColorBrewer_1.0-5 reshape2_1.2.1 RSQLite_0.11.1 setRNG_2011.11-2 splines_2.14.2 [25] stringr_0.6 survival_2.36-14 SVGAnnotation_0.9-0 tools_2.14.2 XML_3.9-4.1 xtable_1.7-0 [31] zlibbioc_1.0.1 ______________________________________________________________________ The information in this email is confidential and intended solely for the addressee. You must not disclose, forward, print or use it without the permission of the sender. ______________________________________________________________________ [[alternative HTML version deleted]]	{"url":"https://support.bioconductor.org/p/46197/","timestamp":"2024-11-07T23:17:12Z","content_type":"text/html","content_length":"48693","record_id":"<urn:uuid:4d17e82b-fdeb-4dd5-ab46-265189d56203>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00336.warc.gz"}
Three Jugs - Equal Amounts Three jugs are given with a total of 3·2^n pints of water. Each jug not empty and contains an integer amount of pints. We may pour into any jug as much water as it already contains, from any other jug with a greater amount of water. Prove that, with one exception, it is possible after a series of such pourings to have the same amount of water in all three jugs. \|Contact\| \|Front page\| \|Contents\| \|Arithmetic\| \|Math induction\| Copyright © 1996-2018 Alexander Bogomolny Three jugs are given with a total of 3·2^n pints of water. Each jug not empty and contains an integer amount of pints. We may pour into any jug as much water as it already contains, from any other jug with a greater amount of water. Prove that, with one exception, it is possible after a series of such pourings to have the same amount of water in all three jugs. The proof is by induction on n. The statement is clearly true for n = 1. Assume it holds for n = k and let there be 3·2^k+1 pints of water distributed among several jugs. We shall first show that, excluding one exceptional distribution of water, it is always possible to get even amounts of pints in all three jugs. If there are jugs with an odd amount of pints, there must be 2 of them. There are two possibilities. Let the odd amounts be a (in jugs A[1] and A[2]) and the remaining even one b (in jug B). If a < b, pour from B to, say, A[1] to obtain distribution of even numbers 2a, a, b - a. If the new configuration is different from (a, a, b), we made a progress. Otherwise we stalled. The latter unfortunate circumstance occurs when b - a = a, or, which is the same, 2a = b. This is the exception mentioned in the body of the problem. Provided 2a ≠ b, we may perform one additional step to either (2a, 2a - b, 2(b - a)) or (2a, 2a, b - 2a), with all amounts even. If a > b, move from the distribution (a, a, b) to (a, a - b, 2b). A repetition of the previous configuration is not possible. For then we would have a = 2b, meaning that the total amount of water is 2b + 2b + b = 5b pints, which contradicts the premise that the total is 3·2^k+1. On the next pouring, move to (b, 2(a - b), 2b), with all amounts even. Once all the jugs contain even amounts of water, we may imagine the pint unit increased by a factor of two, making the total 3·2^k of "big" pints and reducing the case to the inductive assumption. The only exceptional case where the argument does not work is the configuration (a, a, 2a), for which 4a = 3·2^k+1. The argument also shows that, unless we start with the exceptional case, it is always possible to avoid it, thus completing the inductive argument and ultimately solving the problem. Now note that we came across the exceptional case (a, a, 2a) while looking for the distributions with two jugs having two equal odd amounts of water. What we found in fact, is that of itself this situation is not necessarily unsolvable (actually (3, 3, 6), i.e., for n = 2, is the only such case.) However, the configuration (a, a, 2a) may only lead to the permutations of itself, (2a, a, a) and (a, 2a, a) and is, therefore never solvable. To sum up, the problem is solvable for the total of 3·2^n pints of water, unless the initial distribution is (3·2^n-2, 3·2^-2, 3·2^n-1), for n ≥ 2. It is always solvable for n = 1. As an example, let's have three jugs A, B, C with amounts of water (7, 7, 10), for n = 3. I show one sequence of moves in the table below: A B C 7 14 3 C→B 7 11 6 B→C 1 11 12 A→C 1 22 1 C→B 2 21 1 B→A 2 20 2 B→C 4 18 2 B→A 4 16 4 B→C 8 12 4 B→A 8 8 8 B→C What could be a converse theorem? \|Contact\| \|Front page\| \|Contents\| \|Arithmetic\| \|Math induction\| Copyright © 1996-2018 Alexander Bogomolny	{"url":"https://www.cut-the-knot.org/arithmetic/ThreeJugsEvenly.shtml","timestamp":"2024-11-02T01:49:37Z","content_type":"text/html","content_length":"19121","record_id":"<urn:uuid:45bd1fe3-8b3a-4c07-adda-9040eb2ab05b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00438.warc.gz"}
planetary ball mill definition WEBJan 1, 2022 · The experimental study was carried out in a labscale planetary ball mill (Droide®, Shanghai). As shown in Fig. 1, the planetary ball mill contains a disk and four grinding bowls, each with a capacity of 1000 a clearer explanation, a simplified diagram is used, as shown in Fig. centers of rotation and revolution are O r and O . WhatsApp: +86 18838072829 WEBIn this paper an effort is made to find optimum parameters for synthesizing nanopowders with ball milling. Vertical planetary mill with tungsten carbide (WC) grinding jar and WC balls were selected for performing milling operation. Rice husk ash (RHA) prepared in the laboratory by using muffle furnace was taken for milling. Very important grinding . WhatsApp: +86 18838072829 WEBMay 21, 2019 · The planetary ball mill designed according to the present study meets the requirements for. the e ﬀ ective processing of materials, such as the rotation of the pot in the opposite direction to ... WhatsApp: +86 18838072829 WEBThe PM 100 planetary ball mill is a benchtop unit designed to pulverize soft, fibrous and brittle materials. The mill develops extremely high centrifugal forces resulting in energy input that is up to 50% higher than in other planetary ball mills. It has a single grinding station for grinding jars with a nominal volume of 12 to 500 mL and can ... WhatsApp: +86 18838072829 WEBNov 1, 2023 · The planetary mill is one of the most commonly used mills for ultrafine grinding in the laboratory, given its ability to reach higher intensity of the collisions as the result of increase in rotational frequency and ball acceleration without the undesired centrifugation of the grinding charge. Several attempts have been made in the literature ... WhatsApp: +86 18838072829 WEBFeb 1, 2021 · Processes in the planetary ballmill are complex and strongly depend on the milltype. However, the process not only depends on the miltype but also on the processed material and the milling conditions used (density of the grinding material, the number and the size of the balls, the grinding speed, the grinding time, the balls to powder ratio ... WhatsApp: +86 18838072829 WEB2L Capacity Planetary Ball Mill (4x500 ml capacity) with Optional Jars MSKSFM1. Sale Price: USD4, Programmable Planetary Ball Mill w/ Control Unit (4x100ml jar, Ar Gas Compatible) MSKSFMGB. Sale Price: RFQ. Tabletop Programmable Planetary Ball Mill with Four SS Milling Jars (4x100ml) MSKSFM1S. Sale Price: RFQ. WhatsApp: +86 18838072829 WEBBALL MILLS Laboratory Ball Mills are used for rapid batchwise comminution of mediumhard, soft, brittle, fibrous, temperaturesensitive and moist samples down to the finest particle sizes. The comminution of the material to be ground takes place through impact and friction between the grinding balls and the inside wall of the grinding bowl, respectively . WhatsApp: +86 18838072829 WEBThe planetary ball mill has functions of timing power off, selftiming forward and reversal rotating. You may choose freely any operation modes of oneway direction, alternation, succession, time setting according to experimental needs, so as to improve efficiency of grinding. 6. Technical features of Tencan Ball Mill: Low center of gravity ... WhatsApp: +86 18838072829 WEBContrary to stirred media mill the biggest disadvantage of planetary ball mill that it operates in batch mode, so laboratory scale apparatuses are available. There was a development for continuous operation planetary mill (Kaneko, 1996) but it is not used widespread. However, there is one important advantage namely the grinding under inert ... WhatsApp: +86 18838072829 WEBJun 10, 2009 · Thus, the ball motion and the powder refinement need to be quantified for different types of milling equipment to establish the appropriate definition for the respective milling dose. In this paper a DEM model is developed for a planetary mill, and the results of the calculations are compared with corresponding experiments. WhatsApp: +86 18838072829 WEBApr 1, 2018 · Recently, a simple and costeffective substitute, namely mechanical ball milling through highenergy planetary ball mill has been used for the production of nanoparticles rather than the established synthesis procedures (Ambika et al., 2016, Dindarsafa et al., 2017, Naghdi et al., 2017). Numerous advantages such as a facilitated . WhatsApp: +86 18838072829 WEBWe, STERICOX are one of top planetary ball mill manufacturers and suppliers in India. Our planetary ball mills are known for high performance fast grinding of soft, hard, brittle and fibrous materials down to 1µ or better. In addition, these planetary ball mill machines feature ergonomic design, simple operation, easy cleaning and guarantee ... WhatsApp: +86 18838072829 WEBFeb 13, 2017 · Ball Mills. In all ore dressing and milling Operations, including flotation, cyanidation, gravity concentration, and amalgamation, the Working Principle is to crush and grind, often with rod mill or ball mill, the ore in order to liberate the minerals. In the chemical and process industries, grinding is an important step in preparing raw ... WhatsApp: +86 18838072829 WEBPlanetary ball mills are well known and used for particle size reduction on laboratory and pilot scales for decades while during the last few years the appliion of planetary ball mills has extended to mechanochemical approaches. Processes inside planetary ball mills are complex and strongly depend on the processed material and synthesis and, . WhatsApp: +86 18838072829 WEBSep 29, 2015 · The planetary ball mill offers a wide range where rolling is the dominant component. About three decades of the stress energy rolling occurs at the same level. 6. Conclusions. In this work a numerical study of wet operating stirred media mill with a cylindrical and a disc rotor as well as a planetary ball mill was performed. WhatsApp: +86 18838072829 WEBJan 19, 2024 · The present experiment covered the comilling in the planetary ball mill of Ni and Al elemental powders (1:1 molar ratio) with AISI 304 steel platelets for 32 h at 300 rpm. Next, this process was repeated with an admixture of 15 wt.% of CrB2 powder. In both cases, their milling succeeded in producing up to a 200 μm coating after 4 h. WhatsApp: +86 18838072829 WEBMSE PRO 2L (4 x 500ml) Vertical High Energy Planetary Ball Mill. 4,99095 Save 599. MSE PRO 2L (4 x 500ml) Vertical High Energy Planetary Ball Mill with Touchscreen Controller. 5,95095 Save 715. Browse planetary ball mill equipment for sale at MSE Supplies. We also carry highquality laboratory ball mills, mill systems, necessary . WhatsApp: +86 18838072829 WEBNov 15, 2016 · Planetary ball mills consist of two or more jars rotating both around its own axis and the line of symmetry of the supporting plate (radius R P = 125 mm, see Fig. 1). The motion of the milling media (balls) inside the vial, which is driven by the resulting field of two centrifugal forces and the Coriolis force, causes impacts that transfer ... WhatsApp: +86 18838072829 WEBJul 1, 2012 · Abstract. A kinetic model is proposed for simulating the trajectory of a single milling ball in a planetary ball mill, and a model is also proposed for simulating the local energy transfer during the ball milling process under noslip conditions. Based on the kinematics of ball motion, the collision frequency and power are described, and the ... WhatsApp: +86 18838072829	{"url":"https://lgaiette.fr/7962_planetary_ball_mill_definition.html","timestamp":"2024-11-12T18:50:02Z","content_type":"application/xhtml+xml","content_length":"21116","record_id":"<urn:uuid:61778a73-13e3-4465-8246-26295f5b20a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00711.warc.gz"}
Conference on Algebraic Topology in Honor of Peter Hiltonsearch Item Successfully Added to Cart An error was encountered while trying to add the item to the cart. Please try again. Please make all selections above before adding to cart Conference on Algebraic Topology in Honor of Peter Hilton eBook ISBN: 978-0-8218-7622-0 Product Code: CONM/37.E List Price: $125.00 MAA Member Price: $112.50 AMS Member Price: $100.00 Click above image for expanded view Conference on Algebraic Topology in Honor of Peter Hilton eBook ISBN: 978-0-8218-7622-0 Product Code: CONM/37.E List Price: $125.00 MAA Member Price: $112.50 AMS Member Price: $100.00 • Contemporary Mathematics Volume: 37; 1985; 161 pp MSC: Primary 55; Secondary 57 This book, which is the proceedings of a conference held at Memorial University of Newfoundland, August 1983, contains 18 papers in algebraic topology and homological algebra by collaborators and associates of Peter Hilton. It is dedicated to Hilton on the occasion of his 60th birthday. The various topics covered are homotopy theory, $H$-spaces, group cohomology, localization, classifying spaces, and Eckmann-Hilton duality. Students and researchers in algebraic topology will gain an appreciation for Hilton's impact upon mathematics from reading this book. • Permission – for use of book, eBook, or Journal content • Book Details • Table of Contents • Requests Volume: 37; 1985; 161 pp MSC: Primary 55; Secondary 57 This book, which is the proceedings of a conference held at Memorial University of Newfoundland, August 1983, contains 18 papers in algebraic topology and homological algebra by collaborators and associates of Peter Hilton. It is dedicated to Hilton on the occasion of his 60th birthday. The various topics covered are homotopy theory, $H$-spaces, group cohomology, localization, classifying spaces, and Eckmann-Hilton duality. Students and researchers in algebraic topology will gain an appreciation for Hilton's impact upon mathematics from reading this book. Permission – for use of book, eBook, or Journal content Please select which format for which you are requesting permissions.	{"url":"https://bookstore.ams.org/CONM/37","timestamp":"2024-11-12T07:06:52Z","content_type":"text/html","content_length":"78094","record_id":"<urn:uuid:86b48fba-54e2-4d86-bfd9-46e8f7952cef>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00476.warc.gz"}
Causal inference in the continuous limit – The Dan MacKinlay stable of variably-well-consider’d enterprises Causal inference in the continuous limit February 17, 2021 — February 17, 2021 functional analysis how do science machine learning neural nets stochastic processes Where our causality is over fields of continuously indexed variables rather than discrete nodes, what can we do? Do we talk about light cones? Wave propagation? Covariance? Graphons look like they should help here, except they do not capture what I would typically imagine about causality, e.g. which covariates influence causation. Something which encodes spatial proximity, or temporal ordering, or other covariates, looks more useful at a glance? Presumably, I am behind the literature. 1 References Aalen, Røysland, Gran, et al. 2012. “Causality, Mediation and Time: A Dynamic Viewpoint.” Journal of the Royal Statistical Society: Series A (Statistics in Society) Blom, Bongers, and Mooij. 2020. “Beyond Structural Causal Models: Causal Constraints Models.” Uncertainty in Artificial Intelligence Dash, and Druzdzel. 2001. “Caveats For Causal Reasoning With Equilibrium Models.” Symbolic and Quantitative Approaches to Reasoning with Uncertainty Glymour. 2007. “When Is a Brain Like the Planet?” Philosophy of Science Hansen, and Sokol. 2014. “Causal Interpretation of Stochastic Differential Equations.” Electronic Journal of Probability Rubenstein, Bongers, Schölkopf, et al. 2018. “From Deterministic ODEs to Dynamic Structural Causal Models.” Uncertainty in Artificial Intelligence Schulam, and Saria. 2017. “Reliable Decision Support Using Counterfactual Models.” Proceedings of the 31st International Conference on Neural Information Processing Systems . NIPS’17.	{"url":"https://danmackinlay.name/notebook/causality_continuous.html","timestamp":"2024-11-04T23:14:28Z","content_type":"application/xhtml+xml","content_length":"31749","record_id":"<urn:uuid:a4f7eae6-9999-448b-9aa9-b018636b4e1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00366.warc.gz"}
Residential Service Calculations in the National Electrical Code Load calculations in the National Electrical Code have evolved over many decades. It was in the 1933 NEC that load calculation requirements began to resemble a format that the modern code user would find familiar. Since then, many things have changed, but the primary requirement remains the same — service equipment and conductors must be sized to handle the expected load. Article 220 of the National Electrical Code lays out the primary requirements for performing load calculations that are necessary for determining the size of a residential service. The calculations are based on the expected loads present in a dwelling unit, along with appropriate demand factors that are used to account for the diversity of electrical use by occupants. There are two methods available, standard and optional calculations. Optional calculations require fewer steps and generally result in smaller conductors, but the dwelling unit must meet more restrictive requirements. We will only be considering one-family dwelling units in this article, including single family residences, apartments, etc. Be aware that some authorities having jurisdiction adopt the International Residential Code for One- and Two-Family Dwellings (IRC) and use the method for calculating the service size using the requirements found in Chapter 36. The IRC calculations are based on the National Electrical Code, but are not identical. Always check with your local jurisdiction to find out what method(s) are Photo 1. Electric range Standard Method The standard method for calculating service sizing is found in Part III of Article 220. Of course, we can’t find all the requirements in this Part, so we will also be looking at additional requirements in Articles 210, 220, 230, 250 and 310. An example load calculation using the standard method is shown in Table 1. Table 1 Lighting Load The first thing we need to determine is the lighting load. Table 220.12 requires that for dwelling units, we multiply the floor area (based on the outside dimensions of the dwelling unit) times 3 volt-amperes/square foot. Section 220.14(J) states that the following loads are also included in the general lighting load calculations: • all general-use receptacle of 20-ampere rating or less, including the receptacles connected to the bathroom branch circuit required in 210.11(C)(3), • the outdoor outlets in 210.52(E), • the receptacle outlets in basements, garages, and accessory buildings in 210.52(G), and • the lighting outlets required in 210.70(A), which includes habitable rooms, a variety of additional locations, and storage or equipment spaces. Table 220.42 gives us demand factors for lighting loads. Most homes will take the first 3000 VA at 100% and the remainder at 35%. (If you are calculating a multifamily dwelling service, you might use the third demand factor category, where anything over 120,000 VA is taken at 25%.) Small Appliance and Laundry Loads Section 210.11(C)(1) requires a minimum of two small appliance branch circuits. Section 220.52(A) tells us that we must use a minimum load of 1500 VA for each of these circuits, but also allows the small appliance branch circuits to be included with the general lighting load when applying the demand factors in Table 220.12. Section 220.52(B) requires that 1500 VA be added for the required laundry circuit in 210.11(C)(2). This circuit can also be added to the general lighting load and demand factors may be applied. Photo 2. Washer/dryer Electric Dryers and Cooking Appliances Section 220.14(B) refers us to requirements for electric dryers in 220.54 and electric cooking appliances in 220.55. Electric clothes dryers are calculated at either the minimum of 5000 watts or the nameplate rating, whichever is larger. The demand factors in Table 220.54 may be helpful if there are more than four dryers, but this is unlikely in a one-family dwelling unit, so we will not use this table for the examples in this article. Electric ranges and other electric cooking appliances (rated in excess of 1.75 kW) shall be permitted to be calculated in accordance with Table 220.55, which takes up an entire page and has five notes. There are also informational notes directing the code user to Annex D for examples. It is worthwhile to review this table and read all the notes and examples to become familiar with the various options. Fixed-appliance load If there are four or more fixed appliances in the residence, 220.53 permits all of these loads to be totaled and then a demand factor of 75% applied. Fixed-appliance loads include items such as a water heater, garbage disposal, dishwasher, microwave, etc. Photo 3. Garbage disposal Largest motor load Section 220.14(C) tells us that motor loads shall be calculated in accordance with the requirements in 430.22, 430.24 and 440.6. For the service calculation, this means that we must determine the largest motor load and add 25% of its value to the total calculation. Common motor loads in residential applications include air conditioning, water pumps, disposals, blowers, etc. Often, the largest motor load in a home is the air conditioner. Even if the air conditioning is dropped from the total load calculation in favor of electric heating (see below), you may still be required to use the AC motor load for this calculation. Check with your jurisdiction to see what the policy is locally. Many jurisdictions publish residential load calculation worksheets to help with determining the size of the service. Noncoincident loads When two loads are not likely to be energized at the same time, 220.60 allows us to use only the largest load for the calculation of the service. This is typically applied to dwelling units with both electric heating and air conditioning, since they are not expected to run at the same time. Specific appliances or loads There are certain loads that may be found in residences that are not included in the previous list. Section 220.14(A) requires that an outlet for a specific load or appliance not covered elsewhere must be calculated based on the ampere rating of the load served. Some examples might include a spa, RV hookup, etc. These must be included in the load calculation at their full value. Optional Method The optional method is much simpler than the standard calculation, but is restricted in 220.82 to “… a dwelling unit having the total connected load served by a single 120/240-volt or 208Y/120-volt set of 3-wire service or feeder conductors with an ampacity of 100 or greater.” Most one-family dwelling units meet this requirement, so the optional method is used frequently. An example calculation using the optional method is shown in Table 2. Table 2 General Loads For the purposes of the optional method, everything except heating and air conditioning is considered to be a general load. For this method, the general calculated load shall be not less than 100 percent of the first 10 kVA plus 40 percent of the remainder of all loads other than heating and air conditioning. Lighting and general-use receptacles are again based on the outside dimensions of the dwelling unit multiplied by 3 volt-amperes/square foot. The small-appliance branch circuits and laundry branch circuit are each included at 1500 VA. The next step is to determine the nameplate rating of each of the following items: • all appliances fastened in place, permanently connected, or located to be on a specific circuit • ranges, wall-mounted ovens, counter-mounted cooking units • clothes dryers that are not connected to the laundry branch circuit • water heaters For all permanently connected motors not included in the previous list, the nameplate or kVA rating must be included in the calculation. Heating and Air Conditioning The largest heating and air-conditioning load must be chosen from six options: • 100 percent of the nameplate rating of the air conditioning and cooling • 100 percent of the nameplate rating of the heat pump when it is used with no supplemental electric heating • 100 percent of the nameplate rating of the heat pump compressor and 65 percent of the supplemental electric heating for central electric space-heating systems (If the heat pump compressor is prevented from operating at the same time as the supplementary heat, it does not need to be added to the supplementary heat for the total central space heating load.) • 65 percent of the nameplate rating(s) of electric space heating if less than four separately controlled units • 40 percent of the nameplate ratings of electric space heating if four or more separately controlled units • 100 percent of the nameplate ratings of electric thermal storage and other heating systems where the usual load is expected to be continuous at the full nameplate value. Comparing Standard and Optional Calculations To see how the two methods compare, let’s take a look at a 2900 square foot residence with the following loads: • lighting load • 4 small appliance branch circuits • laundry circuit 1500 W • natural gas heating • air conditioner 6000 VA • electric range 11,000 W • hot tub 8000 W (2 hp motor) • Level II electric vehicle charger 7200 W • electric dryer 5000 W • garbage disposal 800 W • microwave 1500 W • dishwasher 1200 W • electric water heater 4500 W The standard calculation method is shown in Table 1 and the optional calculation method is shown in Table 2. Using the standard calculation, our total load is 47,520 VA. Dividing that by 240 volts gives us 198 amps. Using the next standard service rating requires that we use a 200-amp service. Since we have a 120/240-volt single-phase dwelling service, we are allowed to use NEC Table 310.15(B) (7) and use either 2/0 AWG copper or 4/0 AWG aluminum service conductors. Using the optional calculation, our total calculated load is 34,160 VA. Dividing that by 240 volts gives us 142 amps. Using the next standard service rating requires that we use a 150-amp service. Once again, we are allowed to use NEC Table 310.15(B)(7), which requires either 1 AWG copper or 2/0 AWG aluminum conductors. For this example, it is clear that the optional calculation permits a smaller service. From a practical perspective, due to equipment availability, it is likely that a 200-amp service will be installed rather than a 150-amp service. Neutral Load Neutrals are permitted to be smaller than the phase conductors in most residential service installations. Section 220.61 requires that the neutral load be determined by calculating the maximum unbalanced load between the neutral and any one ungrounded conductor. The values used for calculating the neutral size when using the standard or optional methods will often be different, as shown in Tables 3 and 4. Section 230.42 states that the grounded conductor for a service shall not be smaller than the minimum size as determined in accordance with 250.24(C). If we have a single raceway (as is most common for service conductors), 250.24(C)(1) tells us that the conductor cannot be smaller than specified in NEC Table 250.66, but is not required to be larger than the ungrounded conductors. For our standard service calculation, our minimum ungrounded conductor size was a 2/0 AWG copper or a 4/0 AWG aluminum. Using NEC Table 250.66 would require a neutral no smaller than a 4 AWG copper or a 2 AWG aluminum. In Table 3, we found that our calculated neutral load is 28,035 VA. Dividing that by 240 volts gives us 117 amps, which will require either a 2 AWG copper or 1/0 AWG aluminum from Table 310.15(B)(7). These sizes are larger than the required minimum, so we choose one of these conductors. Table 3 For our optional service calculation, our minimum ungrounded conductor size was a 1 AWG kcmil copper or 2/0 AWG aluminum. Using NECTable 250.66 would require a neutral no smaller than 6 AWG copper or a 4 AWG aluminum. In Table 4, we found that our calculated neutral load is 30,320 VA. Dividing that by 240 volts gives us 126 amps, which will require either a 1 AWG copper or 2/0 AWG aluminum from NEC Table 310.15(B)(7). Since these sizes are larger than the required minimum, we would choose one of these conductor sizes. Table 4 Note that for this example in our optional method calculation, the neutral conductor is the same size as our phase conductors. However, if a 200-amp service is installed based on the standard calculation, the neutral is significantly smaller due to the calculation method. Table 5 shows a summary of the ungrounded and neutral conductor sizes for our example using both the standard and optional calculation methods. Table 5 To accurately calculate the service size for residential installations, the designer and installer must be familiar with many requirements in the National Electrical Code. The requirements are not necessarily straightforward, and it is recommended that additional resources be reviewed. Available resources include the examples in Informative Annex D of the NEC, the IAEI publication One- & Two-Family Dwelling Electrical Systems, and other published examples. 310.15(B)(7) – Changes for the 2014 NEC For most residential services, the service conductors and main power feeders are allowed to be sized based on Table 310.15(B)(7) instead of Table 310.15(B)(16), which permits a smaller size conductor to be used in many cases. This allowance has been in the NEC since the 1950s in recognition of the fact that only a small portion of the electrical loads in homes are typically used at the same time, so the load on the service conductors at any one time is generally much smaller than the total calculated load. The language in Section 310.15(B)(7) and the associated table have been a subject of great debate in code-making panel 6 (CMP-6) over the last few cycles. CMP-6 has considered each of the proposals and comments received over the last few years and come up with new wording to address the concerns and suggestions submitted. CMP-6 has agreed to delete the existing wording and table and replace them with the following language: For one-family dwellings and the individual dwelling units of two-family and multifamily dwellings, service and feeder conductors supplied by a single phase, 120/240-volt system shall be permitted be sized in accordance with 310.15(B)(7)(a) through (d). (a) For a service rated 100 through 400 amperes, the service conductors supplying the entire load associated with a one-family dwelling or the service conductors supplying the entire load associated with an individual dwelling unit in a two-family or multifamily dwelling shall be permitted to have an ampacity not less than 83% of the service rating. (b) For a feeder rated 100 through 400 amperes, the feeder conductors supplying the entire load associated with a one-family dwelling or the feeder conductors supplying the entire load associated with an individual dwelling unit in a two-family or multifamily dwelling shall be permitted to have an ampacity not less than 83% of the feeder rating. (c) In no case shall a feeder for an individual dwelling unit be required to have an ampacity greater than that of its 310.15(B)(7)(a) or (b) conductors. (d) Grounded conductors shall be permitted to be sized smaller than the ungrounded conductors provided the requirements of 220.61 and 230.42 for service conductors or the requirements of 215.2 and 220.61 for feeder conductors are met. Informational Note No. 1: It is possible that the conductor ampacity will require other correction or adjustment factors applicable to the conductor installation. Informational Note No. 2: See example DXXX in Annex D. In effect, the same size conductors that are allowed in the 2011 NEC will still be allowed in the 2014 NEC, assuming that temperature correction factors or adjustment factors are not required for the installation. The changes to the code language were necessary to take into account certain limitations inherent in the language in previous code cycles. Because Table 310.15(B)(7) is based on service or feeder ratings and not the temperature rating of conductors, there is no clear way to apply adjustment or correction factors for installations at higher temperatures or if there are more than three current-carrying conductors in a conduit. It should be noted that the conductor sizing will still be based on the service or feeder rating, not the calculated load. For example, if you have a calculated load of 184 amps and are required to install a 200-amp service, the conductors would be required to have an ampacity of 166 amps or more: 200 amps times 83 percent equals 166 amps. So, for a 200-amp service, you would still be allowed to choose a 4/0 AWG aluminum or 2/0 AWG copper, but you would choose it from the 75 degree C column in Table 310.15(B)(16). Christel Hunter Christel Hunter is vice president of standards for Cerro Wire. Chris serves as President for the Southern Nevada Chapter of IAEI. Chris also serves on NEC CMP-6 and CMP-13, NFPA 921, NFPA 70B, NFPA 73 and UL STPs 62, 83, 719 and 4703. Chris is a Professional Safety and Health Officer, Certified Standards Professional, Master Electrician, and LEED Accredited Professional.	{"url":"https://iaeimagazine.org/2013/mayjune-2013/residential-service-calculations-in-the-national-electrical-code/","timestamp":"2024-11-06T18:38:30Z","content_type":"text/html","content_length":"145584","record_id":"<urn:uuid:8a23fe47-4b39-44ad-808b-f18128d02276>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00463.warc.gz"}
Area of a Trapezoid Calculator Area of a trapezoid calculator This trapezoid calculator will compute the area of a trapezoid for you. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. Remember that the formula to get the area of a trapezoid is: So, if b = 2 cm, b = 4 cm, and h = 6 cm + b )/2 = (2 + 4)/2 = 6/2 = 3 A= 3 × 6 = 18 cm Guidelines to follow when using the trapezoid calculator Convert fractions into decimals before entering them, so do not enter any number with a slash "/ " Do not enter negative number since a distance cannot be negative. Do not enter the unit. For example, for 15 cm, just enter 15. Examples showing how to use the calculator in order to find the area of a trapezoid. Example #1 Use the trapezoid calculator to find the area of a trapezoid with bases 6 cm and 9 cm and height 5 cm. Enter 6 in the box that is labeled 'Enter the value of base b[1]' Enter 9 in the box that is labeled 'Enter the value of base b[2]' Enter 5 in the box that is labeled 'Enter the height h' Hit the button labeled 'Calculate' The calculator will display 37.5 in the box labeled ' The area of the trapezoid is' The area of the trapezoid is 37.5 cm^2 Example #2 The end of a gold bar usually has the shape of a trapezoid. Use the trapezoid calculator to find the area of the end of the gold bar with bases 4 cm and 2 cm and height 3 cm. Enter 4 in the box that is labeled 'Enter the value of base b[1]' Enter 2 in the box that is labeled 'Enter the value of base b[2]' Enter 3 in the box that is labeled 'Enter the height h' Hit the button labeled 'Calculate' The calculator will display 9 in the box labeled ' The area of the trapezoid is' The area of the end of the gold bar is 9 cm^2 Buy a comprehensive geometric formulas ebook. All geometric formulas are explained with well selected word problems so you can master geometry.	{"url":"https://www.basic-mathematics.com/trapezoid-calculator.html","timestamp":"2024-11-02T12:16:50Z","content_type":"text/html","content_length":"40657","record_id":"<urn:uuid:594bd616-841f-4add-8639-a8a9ebc7d467>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00746.warc.gz"}
A diatomic gas obeys the law pVx= constant. For what value of x... \| Filo A diatomic gas obeys the law constant. For what value of , it has negative molar specific heat? Not the question you're searching for? + Ask your question is negative if If lies between 1 and 1.4 , then also is negative. Was this solution helpful? Found 7 tutors discussing this question Discuss this question LIVE for FREE 6 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice questions from Waves And Thermodynamics (DC Pandey) View more Practice more questions from Thermodynamics View more Practice questions on similar concepts asked by Filo students View more Stuck on the question or explanation? Connect with our Physics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text A diatomic gas obeys the law constant. For what value of , it has negative molar specific heat? Updated On Nov 7, 2022 Topic Thermodynamics Subject Physics Class Class 11 Answer Type Text solution:1 Video solution: 1 Upvotes 179 Avg. Video Duration 15 min	{"url":"https://askfilo.com/physics-question-answers/a-diatomic-gas-obeys-the-law-p-vx-constant-for-what-value-of-x-it-has-negative","timestamp":"2024-11-04T14:27:33Z","content_type":"text/html","content_length":"323667","record_id":"<urn:uuid:10d9a74a-0ef0-4597-8646-946485b099c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00418.warc.gz"}
The pivotal role of nuclear physics in nucleosynthesis processes is being investigated, in particular the intricate influence of photon strength functions (PSFs) and nuclear level densities (NLDs) on shaping the outcomes of the i-, r- and p-processes. Exploring diverse NLD and PSF model combinations uncovers large uncertainties for (p,[Formula: see text]), (n,[Formula: see text]) and ([Formula: see text],[Formula: see text]) rates across many regions of the nuclear chart. These lead to potentially significant abundance variations of the nucleosynthesis processes and highlight the importance of accurate experimental nuclear data. Theoretical insights and advanced experimental techniques lay the ground work for profound understanding that can be gained of nucleosynthesis mechanisms and the origin of the elements. Recent results further underscore the effect of PSF and NLD data and its contribution to understanding abundance distributions and refining knowledge of the intricate nucleosynthesis processes. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.	{"url":"https://escholarship.org/search/?q=author%3AWiedeking%2C%20M","timestamp":"2024-11-11T06:25:36Z","content_type":"text/html","content_length":"80716","record_id":"<urn:uuid:1df2906e-e2ac-41c6-ab53-fd0ca5f09f83>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00121.warc.gz"}
Describe the following sets: - WorkSheets Buddy Describe the following sets: Describe the following sets: (i) {a, b, c, d, e, f} (ii) {2, 3, 5, 7, 11, 13, 17, 19} (iii) {Friday, Saturday, Sunday} (iv) {April, August, October} Write the following sets in tabular form and also in set builder form: (i) The set of even whole numbers which lie between 10 and 50. (ii) {months of a year having more than 30 days} (iii) The set of single-digit whole numbers which are a perfect square. (iv) The set of factors of 36. More Solutions: Leave a Comment	{"url":"https://www.worksheetsbuddy.com/describe-the-following-sets/","timestamp":"2024-11-12T23:54:35Z","content_type":"text/html","content_length":"140897","record_id":"<urn:uuid:dbceecd5-3772-446a-bcbf-d681d947b8a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00586.warc.gz"}
The Three-Body Problem (pt 2 of 2): A Basic Overview and #beyond #maths The three-body problem is a classic problem in physics and astronomy that involves predicting the motion of three celestial bodies based on their initial positions, velocities, and the gravitational forces between them. While it's easy to predict the motion of two bodies orbiting each other (like the Earth and the Moon), adding a third body (like the Sun) makes the problem significantly more Origin of the Three-Body Problem The three-body problem dates back to the 17th century and the work of Sir Isaac Newton. After formulating the laws of motion and universal gravitation, Newton attempted to solve the three-body problem but found it incredibly challenging. Later, the problem gained more attention from mathematicians like Joseph-Louis Lagrange and Henri Poincaré, who made significant contributions to understanding its complexities. The Challenge The primary difficulty of the three-body problem is that it does not have a general solution. Unlike the two-body problem, where we can derive precise equations to describe the motion, the three-body system's equations are highly nonlinear and sensitive to initial conditions. This sensitivity means that small differences in the starting positions or velocities of the bodies can lead to vastly different outcomes, a concept known as chaos. The Three-Body Problem: A Deeper Dive The three-body problem remains one of the most intriguing and challenging problems in classical mechanics and astrophysics. Its origin and continued relevance highlight the complexity of predicting dynamical systems governed by gravity. Historical Context and Development Sir Isaac Newton first encountered the three-body problem when attempting to apply his laws of motion and universal gravitation to the Sun-Earth-Moon system. Despite his groundbreaking work in mechanics, Newton found that the mutual gravitational interactions of three bodies led to equations that could not be solved analytically. This realization underscored the limitations of classical mechanics when dealing with multiple interacting bodies. In the 18th century, mathematicians such as Joseph-Louis Lagrange and Pierre-Simon Laplace made significant strides in the study of the three-body problem. Lagrange introduced the concept of Lagrangian points—specific locations in space where a small object affected only by gravity can theoretically be stationary relative to two larger objects. These points are solutions to the restricted three-body problem, where one of the bodies is assumed to have negligible mass. Henri Poincaré, in the late 19th century, revolutionized the field by proving that the three-body problem could not be solved using standard analytical methods. Poincaré's work laid the foundation for modern chaos theory, showing that the motion of three interacting bodies is highly sensitive to initial conditions and can exhibit unpredictable and complex behavior. Mathematical Formulation The equations governing the three-body problem arise from Newton's second law of motion and the law of universal gravitation. For three bodies with masses 𝑚1m1, 𝑚2m2, and 𝑚3m3, the equations of motion can be expressed as: where 𝑟1r1, 𝑟2r2, and 𝑟3r3 are the position vectors of the three bodies, and 𝐺G is the gravitational constant. The total force acting on each body is the vector sum of the individual gravitational forces from the other two bodies. Due to the nonlinearity and coupled nature of these equations, exact solutions are only possible for specific initial conditions and configurations, such as the aforementioned Lagrangian points or the restricted three-body problem. Modern Approaches and Applications In the 20th and 21st centuries, computational methods have become essential for studying the three-body problem. Numerical simulations allow scientists to model the trajectories of celestial bodies over time, providing insights into the stability and evolution of multi-body systems. These simulations are crucial for understanding the dynamics of star clusters, planetary systems, and even One notable application of the three-body problem is in space mission design. The Lagrangian points, particularly L1, L2, and L3, are strategically important for placing satellites in stable orbits with minimal fuel consumption. The James Webb Space Telescope, for example, is stationed near the second Lagrangian point (L2) to maintain a stable position relative to the Earth and Sun. In conclusion, the three-body problem highlights the inherent complexity of gravitational interactions in multi-body systems. While exact solutions are elusive, the problem has driven significant advances in mathematics, physics, and computational science. Understanding and solving the three-body problem, even approximately, remains a cornerstone of celestial mechanics and continues to inspire scientific exploration and discovery.	{"url":"https://www.aurorasoul.com/post/the-three-body-problem-pt-2-of-2-a-basic-overview-and-beyond","timestamp":"2024-11-13T12:22:19Z","content_type":"text/html","content_length":"1050408","record_id":"<urn:uuid:0efdc5f1-31c0-457a-8ebe-571b4ab37f4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00564.warc.gz"}
4 Multiplication Facts Worksheets Math, specifically multiplication, creates the foundation of many academic techniques and real-world applications. Yet, for several students, grasping multiplication can posture a challenge. To resolve this hurdle, teachers and parents have actually welcomed an effective device: 4 Multiplication Facts Worksheets. Intro to 4 Multiplication Facts Worksheets 4 Multiplication Facts Worksheets 4 Multiplication Facts Worksheets - 4 Multiplication Facts Worksheets, 4 Multiplication Practice Worksheets, 4 Math Facts Worksheets, 4 Multiplication Table Worksheets, Multiplication Facts Worksheets Grade 4, Multiplication Facts 0-4 Worksheets, Multiplication Facts 3 And 4 Worksheets, 3 And 4 Multiplication Facts, Multiplication Facts For Class 4, Multiplication Facts For 4th Grade Multiplication facts of 4 worksheets will help children solve vertical and horizontal multiplication problems in order to obtain the multiplication facts of 4 These fun printable worksheets enhance children s multiplication and computational skills Get started now Personalized Learning Fun Rewards Actionable Reports Parents Sign Up for Free Somewhere along the way students can learn that anything multiplied by zero is zero Hopefully that is an easy one Students also need to learn to multiply by ten as a precursor to learning how to multiply other powers of ten After those three skills are learned everything else is long multiplication Importance of Multiplication Method Recognizing multiplication is essential, laying a solid foundation for innovative mathematical principles. 4 Multiplication Facts Worksheets offer structured and targeted practice, promoting a much deeper comprehension of this fundamental math operation. Development of 4 Multiplication Facts Worksheets Grade 4 multiplication worksheets Grade 4 multiplication worksheets Grade 4 mental multiplication worksheets Grade 4 multiply in columns worksheets Grade 5 multiplication worksheets Grade 6 multiplication worksheets Multiplication facts drills and practice Multi digit multiplication drills and practice Multiplication flashcards Topics include Grade 2 multiplication worksheets Meaning of multiplication Arrays Multiplication by 4s Here are some practice worksheets and activities for teaching only the 4s times tables Multiplication by 5s These games and worksheets focus on the number 5 as a factor Multiplication by 6s If you re reviewing the 6 times tables this page has some helpful resources Multiplication by 7s From standard pen-and-paper workouts to digitized interactive formats, 4 Multiplication Facts Worksheets have actually advanced, catering to varied discovering designs and choices. Kinds Of 4 Multiplication Facts Worksheets Standard Multiplication Sheets Straightforward workouts focusing on multiplication tables, aiding learners construct a solid math base. Word Problem Worksheets Real-life circumstances integrated into issues, boosting essential thinking and application abilities. Timed Multiplication Drills Tests created to improve speed and accuracy, aiding in fast mental mathematics. Benefits of Using 4 Multiplication Facts Worksheets Multiplying By Facts 3 4 And 6 Other Factor 1 To 12 All Math Fact worksheets Math Multiplying By Facts 3 4 And 6 Other Factor 1 To 12 All Math Fact worksheets Math Multiplication As Repeated Addition Fruit FREE Use repeated addition to solve these basic facts Includes illustrations of fruit as models 3rd and 4th Grades View PDF Using Arrays to Multiply Look closely at each array illustration Then tell how many columns how many rows and how many dots are in each The four times table Multiplication Facts Worksheet Reading and Math for K 5 www k5learning Enhanced Mathematical Skills Consistent technique develops multiplication efficiency, enhancing overall math capabilities. Boosted Problem-Solving Abilities Word troubles in worksheets create analytical thinking and strategy application. Self-Paced Knowing Advantages Worksheets fit individual understanding speeds, fostering a comfortable and adaptable discovering setting. Exactly How to Produce Engaging 4 Multiplication Facts Worksheets Integrating Visuals and Shades Lively visuals and shades catch attention, making worksheets aesthetically appealing and engaging. Including Real-Life Situations Relating multiplication to day-to-day scenarios adds significance and usefulness to workouts. Tailoring Worksheets to Different Skill Degrees Tailoring worksheets based upon varying effectiveness levels ensures comprehensive discovering. Interactive and Online Multiplication Resources Digital Multiplication Equipment and Gamings Technology-based resources provide interactive understanding experiences, making multiplication interesting and delightful. Interactive Sites and Applications Online platforms provide varied and available multiplication method, supplementing traditional worksheets. Customizing Worksheets for Different Knowing Styles Aesthetic Learners Visual aids and diagrams aid understanding for learners inclined toward visual understanding. Auditory Learners Verbal multiplication troubles or mnemonics deal with learners that understand principles via auditory methods. Kinesthetic Learners Hands-on tasks and manipulatives sustain kinesthetic students in recognizing multiplication. Tips for Effective Execution in Learning Consistency in Practice Normal practice strengthens multiplication skills, advertising retention and fluency. Balancing Rep and Selection A mix of repeated workouts and diverse problem formats keeps interest and comprehension. Providing Useful Responses Comments aids in identifying areas of enhancement, urging ongoing progress. Difficulties in Multiplication Practice and Solutions Motivation and Involvement Obstacles Boring drills can lead to uninterest; innovative approaches can reignite motivation. Getting Over Concern of Mathematics Unfavorable perceptions around mathematics can impede progress; developing a positive understanding setting is necessary. Influence of 4 Multiplication Facts Worksheets on Academic Efficiency Research Studies and Study Findings Research shows a positive correlation between constant worksheet usage and improved mathematics performance. Final thought 4 Multiplication Facts Worksheets become versatile tools, fostering mathematical proficiency in learners while suiting varied understanding designs. From basic drills to interactive on the internet resources, these worksheets not only enhance multiplication abilities however also advertise critical thinking and problem-solving abilities. Printable Multiplication Quizzes 0 12 Printable Multiplication Flash Cards Printable 100 Multiplication Facts Timed Test PrintableMultiplication Check more of 4 Multiplication Facts Worksheets below Brimwood Boulevard Junior Public School Classrooms Mme Law s Gr 4 5 4 Multiplication Practice Worksheets Free Printable 4th Grade Math Worksheets Best Coloring Pages For Kids Multiply 4 s Multiplication Facts Worksheet Multiplication Multiplication facts worksheets 15 Best Images Of 100 Mixed Division Worksheet Math Worksheet 100 Multiplication Facts Mixed Printable Timed Multiplication Quiz PrintableMultiplication Multiplication Facts Worksheets Math Drills Somewhere along the way students can learn that anything multiplied by zero is zero Hopefully that is an easy one Students also need to learn to multiply by ten as a precursor to learning how to multiply other powers of ten After those three skills are learned everything else is long multiplication Free FUN 4 Times Table Worksheet Packet Easy Print This is another super fun worksheet to work on the 4 x table facts Have your student start a the beginning and work through the maze by coloring the answers for each multiplication table fact Start with 4 x 1 coloring in 4 and ending at 4 x 12 coloring in 48 at the finish sign The student should color in 4 8 12 16 20 24 28 32 36 Somewhere along the way students can learn that anything multiplied by zero is zero Hopefully that is an easy one Students also need to learn to multiply by ten as a precursor to learning how to multiply other powers of ten After those three skills are learned everything else is long multiplication This is another super fun worksheet to work on the 4 x table facts Have your student start a the beginning and work through the maze by coloring the answers for each multiplication table fact Start with 4 x 1 coloring in 4 and ending at 4 x 12 coloring in 48 at the finish sign The student should color in 4 8 12 16 20 24 28 32 36 Multiply 4 s Multiplication Facts Worksheet Multiplication Multiplication facts worksheets 4 Multiplication Practice Worksheets Free Printable 15 Best Images Of 100 Mixed Division Worksheet Math Worksheet 100 Multiplication Facts Mixed Printable Timed Multiplication Quiz PrintableMultiplication 100 Math Facts Worksheet Multiplication Facts To 100 C Teaching Squared Printable Multiplication Facts Practice PrintableMultiplication Printable Multiplication Facts Practice PrintableMultiplication Fantastic Multiplication Facts 1 5 Pictures Inspiration Worksheet Mathematics Ideas Dutapro FAQs (Frequently Asked Questions). Are 4 Multiplication Facts Worksheets appropriate for any age groups? Yes, worksheets can be customized to various age and skill levels, making them adaptable for numerous learners. Exactly how commonly should trainees exercise using 4 Multiplication Facts Worksheets? Consistent method is essential. Regular sessions, preferably a few times a week, can yield substantial renovation. Can worksheets alone improve math abilities? Worksheets are a beneficial tool yet should be supplemented with different learning approaches for comprehensive skill growth. Exist online platforms providing cost-free 4 Multiplication Facts Worksheets? Yes, many academic internet sites supply open door to a wide range of 4 Multiplication Facts Worksheets. How can moms and dads support their youngsters's multiplication practice in the house? Encouraging constant method, supplying assistance, and developing a favorable understanding atmosphere are beneficial actions.	{"url":"https://crown-darts.com/en/4-multiplication-facts-worksheets.html","timestamp":"2024-11-13T21:54:51Z","content_type":"text/html","content_length":"32433","record_id":"<urn:uuid:af55e1f2-559d-4e76-b53f-d529da57cb5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00776.warc.gz"}
Excel \| Microsoft Community Hub Forum Discussion hello. such a question. how can I make the sum of certain cells in the next row be filled in automatically? For example, the formula for one cell is as follows: =SUMS('Form Responses 1'!P2;'Form Responses 1'!S2;'Form Responses 1'!AA2;'Form Responses 1'!AF2;'Form Responses 1'!AQ2;'Form Responses 1'!AV2;'Form Responses 1'!BC2;'Form Responses 1'!BD2;'Form Responses 1'!BL2;'Form Responses 1'! CU2;'Form Responses 1'!DL2). And the formula for the next cell is as follows: =SUMS('Form Responses 1'!P3;'Form Responses 1'!S3;'Form Response 1'!AA3;'Form Responses 1'!AF 3;'Form Responses 1'!AQ3; 'Form Responses 1'!AV3;'Form Responses 1'!BC3;'Form Responses 1'!BD3;'Form Responses 1'!BL3;'Form Responses 1'!CU3;'Form Responses 1'!DL3). That is, as you can see, they differ by the next number in turn. Is there any way to simplify the calculation so that the following lines are counted automatically? Thanks! • Let's assume your first formula is in cell A2. =SUMS('Form Responses 1'!P2, 'Form Responses 1'!S2, 'Form Responses 1'!AA2, 'Form Responses 1'!AF2, 'Form Responses 1'!AQ2, 'Form Responses 1'!AV2, 'Form Responses 1'!BC2, 'Form Responses 1'!BD2, 'Form Responses 1'!BL2, 'Form Responses 1'!CU2, 'Form Responses 1'!DL2) Now, if you want to copy this formula to the cell A3 and have it automatically adjust to the next row, you can change the formula to: =SUMS('Form Responses 1'!P3, 'Form Responses 1'!S3, 'Form Responses 1'!AA3, 'Form Responses 1'!AF3, 'Form Responses 1'!AQ3, 'Form Responses 1'!AV3, 'Form Responses 1'!BC3, 'Form Responses 1'!BD3, 'Form Responses 1'!BL3, 'Form Responses 1'!CU3, 'Form Responses 1'!DL3) Now, when you copy this formula to the next cell (A4), it will automatically adjust to use row 4 in each of the references. Excel and Google Sheets will automatically increment the row number when you drag or copy the formula down to the next row. This way, you don't need to manually change the row numbers in each formula; it will adapt based on its relative position. □ In this case, are there any rules for copying and pasting a formula into the next line? Or for what reasons the cell numbers of the next row may not be automatically substituted in the next ☆ In this case, are there any rules for copying and pasting a formula into the next line? -- Depends on your sheet and your expected results. Excel and Google Sheets will automatically increment the row number when you drag or copy the formula down to the next row.	{"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/excel/3992141","timestamp":"2024-11-08T20:27:49Z","content_type":"text/html","content_length":"214917","record_id":"<urn:uuid:d9700dd6-c935-4541-818a-c20316bf1d4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00670.warc.gz"}
Max Was Injured And Can No Longer Work. As A Result Of A Lawsuit, He Is To Be Awarded The Present Value (2024) Answer 1 Answer: the amount of Max's award, assuming continuous income and a 6% interest rate, would be approximately $346,474.50. Step-by-step explanation: To calculate the amount of Max's award, we need to determine the present value of the income he would have received over the next 25 years. The formula for calculating the present value of a continuous income stream is: PV = (A / r) * (1 - (1 + r)^(-n)) PV = Present value A = Annual income r = Interest rate n = Number of years In this case, Max's annual income is $30,000, which increases by $1,500 per year. The interest rate is 6%, and he would have received income for 25 years. Let's calculate the present value: PV = ($30,000 / 0.06) * (1 - (1 + 0.06)^(-25)) = (500,000) * (1 - (1.06)^(-25)) = (500,000) * (1 - 0.307051) ≈ $346,474.50 Related Questions By using the Lagrange method find the maximum value of point on the surface $ f(x, y)=49-x^{2}-y^{2} $ on the line $ x+y=3 $. Maximum file size: 250MB, maximum number of files: 1 We want to find the maximum value of the surface $f(x, y) = 49 - x^2 - y^2$ subject to the constraint $x + y = 3$. We can use the method of Lagrange multipliers to solve the problem. Let $$g(x,y) = x+y-3,$$ and consider the function $$F(x,y,\lambda) = f(x,y) - \lambda g(x,y) = 49 - x^2 - y^2 - \lambda(x+y-3).$$ Then, we need to find the critical points of $F(x,y,\lambda)$, which satisfy the following system of equations: \frac{\partial F}{\partial x} &= -2x - \lambda = 0, \\ \frac{\partial F}{\partial y} &= -2y - \lambda = 0, \\ \frac{\partial F}{\partial \lambda} &= x + y - 3 = 0. The first two equations yield that $x = -\frac{\lambda}{2}$ and $y = -\frac{\lambda}{2}$. Substituting these into the third equation, we get $-\lambda + (-\lambda) - 3 = 0$, which implies that $\lambda = -\frac{3}{2}$. Thus, the critical point is $$(x,y) = \left(\frac{3}{2}, \frac{3}{2}\right).$$ We also need to check the endpoints of the line segment. When $x = 0$, we have $y = 3$, and when $y = 0$, we have $x = 3$. We evaluate the function $f(x,y)$ at these three points: f(0,3) &= 40, \\ f(3,0) &= 40, \\ f\left(\frac{3}{2}, \frac{3}{2}\right) &= 42.25 - \frac{27}{4} = \frac{11}{4}. Therefore, the maximum value of the surface $f(x, y) = 49 - x^2 - y^2$ on the line $x + y = 3$ is $\boxed{\frac{11}{4}}$, which occurs at the point $\left(\frac{3}{2}, \frac{3}{2}\right)$. $ \ln \left(r^{4} s^{8} \sqrt[10]{r^{10} s^{2}}\right) $ is equal to $ A \ln r+B \ln s $ where $ A= $ and where $ B= $ The given expression value is A = 4 and B = 41/5. The given expression can be simplified as follows: [tex]ln(r^4 s^8 (r^10 s^2)^(1/10))[/tex] Using the properties of logarithms: ln(ab) = ln(a) + ln(b) ln([tex]a^b[/tex]) = b ln(a) We can rewrite the expression as: [tex]ln(r^4) + ln(s^8) + ln((r^{10} s^2)^{1/10})[/tex] Applying the properties of logarithms: 4 * ln(r) + 8 * ln(s) + (1/10) * ln([tex]r^{10} s^2[/tex]) Simplifying further: 4 * ln(r) + 8 * ln(s) + (1/10) * (10 * ln(r) + 2 * ln(s)) Now, combining like terms: 4 * ln(r) + 8 * ln(s) + ln(r) + (1/5) * ln(s) Comparing the coefficients, we find: A = 4 B = 8 + 1/5 = 41/5 Therefore, A = 4 and B = 41/5. To know more about expression: Exchange rate is £1 = 1.17 euros . How many euros do you get for £120 =140.4 euros Exchange rate is £1=1.17euros so £120 140.4 Find an equation of the line passing through (-3, 2) and parallel to the graph of x - 2y = 7. Write the equation in slope-intercept form. y = [tex]\frac{1}{2}[/tex] x + [tex]\frac{7}{2}[/tex] Step-by-step explanation: the equation of a line in slope- intercept form is y = mx + c ( m is the slope and c the y- intercept ) x - 2y = 7 ( subtract x from both sides ) - 2y = - x + 7 ( multiply through by - 1 ) 2y = x - 7 ( divide through by 2 ) y = [tex]\frac{1}{2}[/tex] x - [tex]\frac{7}{2}[/tex] ← in slope- intercept form with slope m = [tex]\frac{1}{2}[/tex] • Parallel lines have equal slopes , then y = [tex]\frac{1}{2}[/tex] x + c ← is the partial equation to find c substitute (- 3, 2 ) into the partial equation 2 = [tex]\frac{1}{2}[/tex] (- 3) + c = - [tex]\frac{3}{2}[/tex] + c ( add [tex]\frac{3}{2}[/tex] to both sides ) 2 + [tex]\frac{3}{2}[/tex] = c , that is c = [tex]\frac{7}{2}[/tex] y = [tex]\frac{1}{2}[/tex] x + [tex]\frac{7}{2}[/tex] ← equation of parallel line (3) Determine the convergence of the series ∑[infinity] (4) Determine the values of a parameter afor which the series converge. ∑[infinity] n3 . (3) The ratio takes a look at, the series diverges. (4) The series converges for all values of a such that[tex]$$-1 \leq a < 1$$.[/tex] (3) To decide the convergence of the series[tex]$$\sum_{n=1}^{\infty} \frac{n!}{2^n}$$[/tex], we will use the ratio test. The ratio of consecutive terms [tex]$$\left\|\frac{a_{n+1}}{a_n}\right\| = \ frac{(n+1)!}{2^{n+1}} \cdot \frac{2^n}{n!} = \frac{n+1}{2}$$[/tex] The restriction of this ratio as n is going to infinity is[tex]$$\lim_{n \to \infty} \frac{n+1}{2} = \infty > 1$$[/tex] Therefore, via the ratio take a look at, the series diverges. (4) To decide the values of a parameter for which the collection[tex]$$\sum_{n=1}^{\infty} \frac{a^n}{n^3}$$[/tex] converges, we also can use the ratio test. The ratio of consecutive terms is[tex]$$\ left\|\frac{a_{n+1}}{a_n}\right\| = \frac{a^{n+1}}{(n+1)^3} \cdot \frac{n^3}{a^n} = a \cdot \frac{n^3}{(n+1)^3}$$[/tex] The limit of this ratio as n is going to infinity is[tex]$$\lim_{n \to \infty} a\cdot \frac{n^3}{(n+1)^3}$$[/tex] Therefore, through the ratio check, the series converges to 1, the ratio check is uncertain and we need to use any other check. In this situation, we can use the contrast test. If a=1, then the series will become[tex]$$\sum_{n=1}^{\infty} \frac{1}{n^3}$$[/tex]which converge by way of the p-series test. If a= -1, then the series turns into[tex]$$\sum_{n=1}^{\infty} (-1)^n \ frac{1}{n^3}$$[/tex]which converges via the alternating collection test. Therefore, the series converges for all values of a such that[tex]$$-1 \leq a < 1$$.[/tex] To know more about convergence, In R4, Let u 1 = (1, 2, −1, 3), u 2 = (2, 4, 1, −2), u 3 = (3, 6, 3, −7) and v 1 = (1, 2, −4, 11), v 2 = (2, 4, −5, 14), and let U = span{u 1, u 2, u 3} and V = span{v 1, v 2}. Show that U = V To show that U = V, we need to demonstrate that U is a subset of V and V is a subset of U. This can be done by expressing each vector in U as a linear combination of vectors in V and vice versa. Let's start by showing that U is a subset of V. For any vector u in U, we can express u as a linear combination of vectors v1 and v2 in V: u = a1v1 + a2v2, where a1 and a2 are scalars. Let's substitute the given values of u1, u2, and u3 into this equation: u1 = 1v1 + 0v2, u2 = 0v1 + 1v2, u3 = -1v1 + 3v2. This shows that u1, u2, and u3 can be expressed as linear combinations of v1 and v2, which implies that U is a subset of V. Next, we need to demonstrate that V is a subset of U. For any vector v in V, we can express v as a linear combination of vectors u1, u2, and u3 in U: v = b1u1 + b2u2 + b3u3, where b1, b2, and b3 are scalars. Substituting the given values of v1 and v2 into this equation: v1 = 1u1 - u3, v2 = 2u1 + 3u2 + 2u3. This shows that v1 and v2 can be expressed as linear combinations of u1, u2, and u3, indicating that V is a subset of U. Since U is a subset of V and V is a subset of U, we can conclude that U = V, i.e., the spans of the given vectors are equal. Learn more about linear here: Consider the parabola with focus at (0, ) and directrix at y=− . (a) Determine the standard form equation of the parabola. (b) What are the points where the latus rectum intersects the parabola? (c) What is the standard form equation of the circle with the latus rectum of the parabola as the diameter of the circle? The standard form equation of the circle with the latus rectum of the parabola as the diameter is [tex]x^2 + (y- 2)^2 = 1.[/tex] (a) To determine the standard form equation of the parabola, we need to find the vertex and the equation's specific form. The vertex of a parabola with a vertical axis of symmetry is given by (h, k), where h represents the x-coordinate and k represents the y-coordinate. The vertex of this parabola can be found at the midpoint between the focus and the directrix. In this case, the x-coordinate of the vertex is always 0 since the parabola is symmetric around the To find the y-coordinate of the vertex, we average the y-coordinates of the focus and the directrix: k = (2 + (-2)) / 2 = 0 So, the vertex of the parabola is (0, 0). The distance between the vertex and the focus (or the directrix) is called the focal length, denoted by 'p.' In this case, the focal length is the distance between the vertex (0, 0) and the focus (0, 2) or the directrix y = -2: p = \|2 - 0\| = 2 Since the directrix is a horizontal line, the equation of the parabola has the form [tex](y - k)^2 = 4p(x - h). Plugging in the known values:(y - 0)^2 = 4(2)(x - 0)y^2 = 8x\\[/tex] Therefore, the standard form equation of the parabola is [tex]y^2 = 8x.[/tex] (b) The latus rectum of a parabola is the line segment perpendicular to the axis of symmetry and passing through the focus. Its length is equal to the focal length, which in this case is 2. The parabola is symmetric around the y-axis, and the vertex is at (0, 0). Therefore, the latus rectum intersects the parabola at the points (2, t) and (-2, t), where t is the y-coordinate of the Substituting the y-coordinate of the focus (t = 2) into the equation of the parabola: [tex]y^2 = 8x(2)^2 = 8x4 = 8xx = 1/2\\[/tex] So, the latus rectum intersects the parabola at the points (1/2, 2) and (-1/2, 2). (c) To find the equation of the circle with the latus rectum of the parabola as the diameter, we need to determine the center and radius of the circle. The center of the circle is the midpoint of the latus rectum. The x-coordinate of the center is the average of the x-coordinates of the latus rectum's endpoints, which is (1/2 + (-1/2))/2 = 0. The y-coordinate of the center is the same as the y-coordinate of the endpoints, which is 2. Therefore, the center of the circle is (0, 2). The radius of the circle is half the length of the latus rectum, which is 2/2 = 1. Using the standard form equation of a circle, which is[tex](x - h)^2 + (y - k)^2 = r^2, we can substitute the values:(x - 0)^2 + (y - 2)^2 = 1^2x^2 + (y - 2)^2 = 1[/tex] So, the standard form equation of the circle with the latus rectum of the parabola as the diameter is [tex]x^2 + (y - 2)^2 = 1.\\[/tex] To know more about equation click- Solve the following difference equation where x(k) is a discrete unit step input and y(k) is the system output. y(k) − y(k − 1) + 0.24y(k − 2) = x(k) + x(k + 1) the solution to the given difference equation is y(k) = [tex]0.2^k[/tex] - 2.4 [tex]1.2^k[/tex] To solve the given difference equation, we can use the Z-transform method. Let's denote the Z-transform of a function y(k) as Y(z), and the Z-transform of x(k) as X(z). Applying the Z-transform to the given equation and using the properties of linearity, time shifting, and the Z-transform of the unit step function, we have: z²Y(z) - zY(z) + 0.24Y(z) = (z + 1)X(z) Next, we can rearrange the equation to solve for Y(z): Y(z)(z² - z + 0.24) = (z + 1)X(z) Dividing both sides by (z² - z + 0.24), we get: Y(z) = (z + 1)X(z) / (z² - z + 0.24) Now, we need to find the inverse Z-transform of Y(z) to obtain the time-domain solution y(k). To do this, we can use partial fraction decomposition and lookup tables to find the inverse Z-transform. The denominator of Y(z), z² - z + 0.24, can be factored as (z - 0.2)(z - 1.2). We can then rewrite Y(z) as: Y(z) = (z + 1)X(z) / [(z - 0.2)(z - 1.2)] Now, we perform partial fraction decomposition to express Y(z) as: Y(z) = A / (z - 0.2) + B / (z - 1.2) To find the values of A and B, we can multiply both sides of the equation by the common denominator: (z - 0.2)(z - 1.2)Y(z) = A(z - 1.2) + B(z - 0.2) Expanding and equating coefficients, we have: z²Y(z) - 1.4zY(z) + 0.24Y(z) = Az - 1.2A + Bz - 0.2B Now, we can equate the coefficients of corresponding powers of z: Coefficient of z²: 1 = A Coefficient of z: -1.4 = A + B Coefficient of z⁰ (constant term): 0.24 = -1.2A - 0.2B Solving these equations, we find A = 1, B = -2.4. Substituting these values back into the partial fraction decomposition equation, we have: Y(z) = 1 / (z - 0.2) - 2.4 / (z - 1.2) Now, we can use the inverse Z-transform lookup tables to find the inverse Z-transform of Y(z): y(k) = Z⁻¹{Y(z)} = Z⁻¹{1 / (z - 0.2) - 2.4 / (z - 1.2)} The inverse Z-transform of 1 / (z - 0.2) is given by the formula: Z⁻¹{1 / (z - a)} = [tex]a^k[/tex] Using this formula, we get: y(k) = [tex]0.2^k[/tex] - 2.4 * [tex]1.2^k[/tex] Therefore, the solution to the given difference equation is y(k) = [tex]0.2^k[/tex] - 2.4 * [tex]1.2^k[/tex] Learn more about Z-transform method here Use Trigonometric substitution to eliminate the roots 1.1. 164+2 + 1 Use Trigonometric substitution to eliminate the roots 1.1. V64+2 + 1 1.2. V4z2 – 49 To eliminate the roots in 1.1 and 1.2, we can use trigonometric substitution. In 1.1, we can substitute x = 4 sin(theta) to eliminate the root of 4. In 1.2, we can substitute z = 7 sin(theta) to eliminate the root of 7. 1.1. V64+2 + 1 We can substitute x = 4 sin(theta) to eliminate the root of 4. This gives us: V64+2 + 1 = V(16 sin^2(theta) + 2 + 1) = V16 sin^2(theta) + V3 = 4 sin(theta) V3 1.2. V4z2 – 49 We can substitute z = 7 sin(theta) to eliminate the root of 7. This gives us: V4z2 – 49 = V4(7 sin^2(theta)) – 49 = V28 sin^2(theta) – 49 = 7 sin(theta) V4 – 7 = 7 sin(theta) (2 – 1) = 7 sin(theta) Here is a more detailed explanation of the substitution: In 1.1, we know that the root of 4 is 2. We can substitute x = 4 sin(theta) to eliminate this root. This is because sin(theta) can take on any value between -1 and 1, including 2. When we substitute x = 4 sin(theta), the expression becomes V64+2 + 1 = V(16 sin^2(theta) + 2 + 1) = V16 sin^2(theta) + V3 = 4 sin(theta) V3 In 1.2, we know that the root of 7 is 7/4. We can substitute z = 7 sin(theta) to eliminate this root. This is because sin(theta) can take on any value between -1 and 1, including 7/4. When we substitute z = 7 sin(theta), the expression becomes: V4z2 – 49 = V4(7 sin^2(theta)) – 49 = V28 sin^2(theta) – 49 = 7 sin(theta) V4 – 7 = 7 sin(theta) To know more about root click here A population of values has a normal distribution with μ = 183.7 and σ=14.2. You intend to draw a random sample of size n=170. Find the probability that a single randomly selected value is less than 182.4. P(X < 182.4) = Find the probability that a sample of size n=170 is randomly selected with a mean less than 182.4. P(M < 182.4) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted. To find the probability that a single randomly selected value is less than 182.4 from a normal distribution with a mean of 183.7 and a standard deviation of 14.2, we need to calculate the z-score and use the standard normal distribution table. The probability that a single value is less than 182.4 is 0.3632. To find the probability that a sample of size n = 170 has a mean less than 182.4, we need to calculate the z-score for the sample mean. Since the sample size is large (n > 30) and the population standard deviation is known, we can use the standard normal distribution to approximate the sampling distribution. The probability that the sample mean is less than 182.4 is 0.0000. For the probability that a single randomly selected value is less than 182.4, we calculate the z-score using the formula: z = (x - μ) / σ where x is the value (182.4), μ is the mean (183.7), and σ is the standard deviation (14.2). Plugging in the values, we get: z = (182.4 - 183.7) / 14.2 = -0.0915 Looking up the z-score in the standard normal distribution table, we find that the probability corresponding to a z-score of -0.0915 is 0.3632. For the probability that a sample of size n = 170 has a mean less than 182.4, we need to calculate the z-score for the sample mean using the formula: z = (x - μ) / (σ / sqrt(n)) where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the values, we get: z = (182.4 - 183.7) / (14.2 / sqrt(170)) = -2.0261 Using the standard normal distribution table, the probability corresponding to a z-score of -2.0261 is 0.0217. However, since we are looking for the probability of the sample mean being less than 182.4, we need to consider the area to the left of the z-score. Thus, the probability is 0.0000. Learn more about probability here: Find the conditional probability of it snowing on Wednesday, given that it is hailing on Wednesday. (A) 0.20 (B) 0.30 (C) 0.60 (D) 0.80 0.2 is the conditional probability of it snowing on Wednesday. P(A) = Probability of snowing on Wednesday = 1/7 (since there are 7 days in a week) P(B) = Probability of hailing on Wednesday = 1/7 (again, assuming equal likelihood) P(A and B) = Probability of both snowing and hailing on Wednesday = 1/7 × 1/7 = 1/49 Now, we can calculate the conditional probability using the formula: P(A \| B) = P(A and B) / P(B) P(A \| B) = (1/49) / (1/7) = 1/49 × 7/1 = 7/49 = 1/7 = 0.2 To learn more on probability click: By computing coeﬀicients cnwith center a= 0 establish that ln(x+ 1) = x−x2 2 + x3 3 −x4 4 + ···. The given equation isln(x+1) = x−x2/2 + x3/3 - x4/4 + ... Given function isln(x+1) We know that Taylor series is given by: Taylor series representation of any function f(x) is given by; f(x) = f(a) + (x-a) f'(a)/1! + (x-a)² f''(a)/2! + (x-a)³ f'''(a)/3! + ...(x-a)ⁿ f⁽ⁿ⁾(a)/n! Using this formula, we can calculate the values of function at any particular value of x by using values at a = 0. We have to compute the Taylor series for ln(x + 1) with center at a = 0. We know that f(a) = f(0) = ln(0+1) = ln(1) = 0. Then, f'(x) = 1/(x+1) (By differentiating ln(x+1) w.r.t x)and f'(a) = f'(0) = 1. Then, f''(x) = -1/(x+1)² (By differentiating f'(x))and f''(a) = f''(0) = -1. Then, f'''(x) = 2/(x+1)³(By differentiating f''(x))and f'''(a) = f'''(0) = 2. Then, f⁽ⁿ⁾(x) = (-1)ⁿ⁻¹ (n-1)!/(x+1)ⁿ(By differentiating f⁽ⁿ⁾(x)) Thus, we have;f(x) = f(0) + (x-a) f'(0)/1! + (x-a)² f''(0)/2! + (x-a)³ f'''(0)/3! +...(x-a)ⁿ f⁽ⁿ⁾(0)/n!f(x) = 0 + x(1)/1! - x²(1)/2! + x³(2)/3! - x⁴(6)/4! + ...f(x) = x - x²/2 + x³/3 - x⁴/4 + ... Thus, the given equation isln(x+1) = x−x2/2 + x3/3 - x4/4 + ... Learn more about Taylor series . Given four functions f1(n)=n100, f2(n)=1000n2, f3(n)=2n, f4(n)=5000nlgn, which function will have the largest values for sufficiently large values of n? For sufficiently large values of n, the function f4(n) = 5000nlog(n) will have the largest values among the given functions. Among the given functions, the function f4(n) = 5000nlog(n) will have the largest values for sufficiently large values of n. This can be understood by comparing the growth rates of the functions. As n increases, the function f1(n) = n^100 will grow rapidly but still be outpaced by the exponential growth of f2(n) = 1000n^2. However, the function f3(n) = 2n grows even faster than f2(n) because it has a linear growth rate. In contrast, the function f4(n) = 5000nlog(n) exhibits logarithmic growth, where the growth rate slows down as n increases. However, the logarithmic term ensures that the function will eventually surpass the other functions in terms of value for sufficiently large values of n. This is because logarithmic growth is slower than polynomial growth but still faster than constant growth. Learn more about function here: Consider the points below. P(−1,0,2),Q(1,4,−2),R(0,4,6) (a) Find a nonzero vector orthogonal to the plane through the points P,Q, and R. (b) Find the area of the triangle PQR. A nonzero vector orthogonal to the plane through the points P, Q, and R is (-16, -12, 12). The area of the triangle PQR is 2 √(13) How to find a nonzero vector To find a nonzero vector orthogonal to the plane through the points P, Q, and R, Take the cross product of two vectors in the plane. For example, take the vectors PQ = (1-(-1), 4-0, -2-2) = (2, 4, -4) and PR = (0-(-1), 4-0, 6-2) = (1, 4, 4). Then a vector orthogonal to the plane is given by the cross product of PQ and PR: (PQ) x (PR) = \|i j k \| \|2 4 -4 \| \|1 4 4 \| = (-16, -12, 12) Therefore, a nonzero vector orthogonal to the plane through the points P, Q, and R is (-16, -12, 12). To find the area of the triangle PQR Use the formula: Area = 1/2 \|PQ x PR\| where PQ and PR are the vectors used in part (a). Substituting these vectors \|PQ x PR\| = \|(-16, -12, 12)\| = 4 √(13) Therefore, the area of the triangle PQR is: Area = 1/2 \|PQ x PR\| = 1/2 (4 √(13)) = 2 √(13) Learn more on vector on https://brainly.com/question/25705666 Find the equation for the tangent plane to the surface $ z=\ln \left(4 x^{2}+3 y^{2}+1\right) $ at the point $ (0,0,0) $. A. $ z=0 $ B. $ x+y=0 $ C. $ x-y=0 $ D. $ x+y+z=0 $ The equation for the tangent plane to the surface at the point (0,0,0) is z = 0. The correct answer is Option A z = 0. To find the equation for the tangent plane to the surface z = ln(4x² + 3y² + 1) at the point (0,0,0), we need to find the partial derivatives with respect to x and y and evaluate them at the given First, let's find the partial derivative with respect to x: ∂z/∂x = (8x)/(4x² + 3y² + 1) Now, let's find the partial derivative with respect to y: ∂z/∂y = (6y)/(4x² + 3y² + 1) Next, we evaluate these partial derivatives at the point (0,0,0): ∂z/∂x\|(0,0,0) = (8(0))/(4(0)² + 3(0)² + 1) = 0 ∂z/∂y\|(0,0,0) = (6(0))/(4(0)² + 3(0)² + 1) = 0 Since both partial derivatives are zero at the point (0,0,0), the equation of the tangent plane is given by: z - z₀ = ∂z/∂x\|(0,0,0)(x - x₀) + ∂z/∂y\|(0,0,0)(y - y₀) Plugging in the values, we have: z - 0 = 0(x - 0) + 0(y - 0) Simplifying, we get: z = 0 Learn more about the equation for the tangent plane at For the 'damping ratio vs period' data given in the Table: (a) Try the regression models that is indicated below and decide on the best regression equation by comparing the correlation coefficient values. You are requested to solve this question by using MS-Excel or Matlab. Note that period is the independent variable. (b) Calculate the coefficient of determination and the correlation coefficient for the linear regression model manually. You can use the Excel's spreadsheet for the calculations. 0.2 0.3 0.4 0.5 Period (sec) 0.1 Damping 5.0 ratio (%) 7.0 8.0 8.9 8.1 (i) Linear regression model (ii) Non-linear regression model (iii) Polynomial regression model The linear regression equation isy = 22.75x + 32.825. The coefficient of determination for the linear regression model is:R² = (SSR/SST) = 0.6460 (a) Best regression equationLinear regression equation is y = mx + cwhere m is the slope of the regression line, and c is the intercept. The slope and intercept can be calculated as follows:m = ((n∑xy)-(∑x∑y))/((n∑x²)-(∑x)²)c = (∑y - (m∑x))/nwhere n is the number of data points, x and y are the independent and dependent variables, respectively.∑x and ∑y are the sum of all x and y values, respectively.∑xy and ∑x² are the sum of the product of x and y and the sum of the square of x, respectively.For linear regression, the degree of the equation is 1.Calculating the slope and intercept from the given data:Slope, m = ((49.63)-(1.438))/((40.397)-(0.2²)) = 22.75Intercept, c = (38-(22.750.4))/4 = 32.825Therefore, the linear regression equation isy = 22.75x + 32.825Now, calculate the correlation coefficient for this equation.Correlation coefficient is given byr = (n∑xy - (∑x∑y))/sqrt((n∑x²-(∑x)²)(n∑y²-(∑y)²))For the given data, the correlation coefficient for the linear regression equation is:r = (49.63 - 1.438)/sqrt((40.397-0.2²)(430.6325-38²)) = 0.9894. (b) Coefficient of determination and correlation coefficient for the linear regression modelTo calculate the coefficient of determination (R²) for the linear regression model, use the following formula:R² = (SSR/SST)where, SSR is the sum of squares of regression, and SST is the total sum of squares.To calculate SSR and SST, use the following formulas:SSR = ∑(ŷ - ȳ)²SST = ∑(y - ȳ)²where, ŷ is the predicted value of y, ȳ is the mean of y, and y is the actual value of y.Calculating SSR and SST for the given data:Predicted values of y: ŷ = 22.75x + 32.825y = 7.3, ŷ = 36.675y = 8, ŷ = 46.775y = 8.9, ŷ = 57.4y = 8.1, ŷ = 51.975ȳ = (7.3 + 8 + 8.9 + 8.1)/4 = 8.075SSR = (7.3 - 8.075)² + (8 - 8.075)² + (8.9 - 8.075)² + (8.1 - 8.075)² = 0.7138SST = (7.3 - 8.075)² + (8 - 8.075)² + (8.9 - 8.075)² + (8.1 - 8.075)² = 1.1045Therefore, the coefficient of determination for the linear regression model is:R² = (SSR/SST) = 0.6460. To calculate the correlation coefficient for the linear regression model using Excel, enter the data into two columns, select the chart style, right-click on the data points, select "Add Trendline", select "Linear" as the type, and check the "Display equation on chart" and "Display R-squared value on chart" boxes. The R-squared value displayed on the chart is the correlation coefficient. Learn more about regression : Evaluate the following as true or false. Integral^3_0 dx/x - 1 = ln \|x - 1\|^3_0 ln 3 - ln 1 = ln 3 Select one: a. true b. false The integral value is not defined. The statement is false. We have, Let's evaluate the integral correctly: ∫[0,3] dx/(x - 1) To evaluate this integral, we need to use the natural logarithm function. Recall the integral property: ∫ dx/x = ln\|x\| + C Applying this property to our integral, we get: ∫[0,3] dx/(x - 1) = ln\|x - 1\| \| (0 to 3) Now we substitute the limits of integration: ln\|3 - 1\| - ln\|0 - 1\| Simplifying further: ln\|2\| - ln\|-1\| Since the natural logarithm of a negative number is undefined, ln\|-1\| does not exist. Therefore, the expression ln\|2\| - ln\|-1\| is not defined. The correct evaluation is not ln 3. The statement is false. Learn more about integrations here: Exercise 5. Let $ G $ be a finite group and let $ N $ be a normal subgroup of $ G $ such that $ \operatorname{gcd}(\|N\|,\|G / N\|)=1 $. Prove the following: 1. If $ H $ is a subgroup of $ G $ having the same order G/N, then G =HN We have shown that G = HN, as desired. To prove that G = HN, we need to show that every element of G can be written as a product of an element of H and an element of N . First, note that since N is a normal subgroup of G , we have that NH is a subgroup of G. Additionally, since H has the same order as G / N, we know that , [tex]\|NH\| = \frac{\|N\| \|H\| }{\|NHcap \|} = \frac{\|N\|\|G/N\|}{\|NHcap\|}[/tex] = \|G\| / \|NHcap\| Now, let ( g ⊆ G ). Since ( gcd (\|N\|,\|G/N\|)=1 ), we know that, \|N Hcap\| = \|N\|\|H\| / \|NH\| divides both ( \|N\| ) and ( \|H\| ). Thus, ( \|N Hcap\| ) also divides \|G\|, so \|G\| / \|N Hcap\| is a positive integer. This means that ( \|G\|/\|N Hcap\| ⊆ \|NH\| ), so there must exist some element ( h ⊂H ) and some element ( n ⊂ N ) such that ( g = hn ). Since ( h \in H ) and ( H ) has the same order as ( G/N ), we know that there exists some ( g' ⊂ G ) such that ( h = g'N ). Thus, ( g = g'nN ), so ( g ⊂ HN ). This shows that ( G ⊆ HN ). To show the other inclusion, let ( hn ⊂ HN ), where ( h ⊂H ) and ( n ⊂ N Then ( h = g'N ) for some ( g' ⊂ G ). Since ( N ) is normal in ( G ), we have that, n║⁻¹g'Nn = g'N so , n⁻¹g' ⊂ g'Nn \Ncap ⊆ H Ncap = { e } Thus, ( n⁻¹g' = e ), so , hn = g'Nn ⊆ G This shows that, HN⊆ G Therefore, we have shown that ( G = HN ), as desired. Learn more about subgroup here: If you move from 0 to 15 on the number line, you are representing all of the following except _____. the opposite of 15 the absolute value of 15 the distance between zero and 15 the opposite of −15 You are representing all except (a) the opposite of 15 How to determine the odd option in the list From the question, we have the following parameters that can be used in our computation: Moving from 0 to 15 on a number line From the above, we have Distance = 15 - 0 So, we have Distance = 15 Analysing the list of options, we have the opposite of 15 is - 15the absolute value of 15 is 15the distance between zero and 15 is 15the opposite of −15 is 15 Hence, the odd option in the list is (a) the opposite of 15 Read more about number line at How many rows and columns must a matrix A have in order to define a mapping from R^5 into R^7 by the rule T(x) = Ax? The answer of the given question based on the Matrix is , A must have 7 rows and 5 columns to define a mapping from R5 into R7 by the rule T(x) = Ax. In order to define a mapping from R5 into R7 by the rule T(x) = Ax, the matrix A must have 7 rows and 5 columns (7 × 5). To understand this, we can start by looking at the equation: T(x) = Ax where T is a transformation that maps vectors from R5 to R7. This means that for every vector x in R5, the transformation T will produce a vector in R7. The matrix A specifies how this transformation is performed, and it must be such that the product Ax is defined. In order to multiply a matrix A by a vector x, the number of columns in A must be equal to the number of entries in x. So if x has 5 entries, A must have 5 columns. The result of this multiplication will be a vector with as many entries as there are rows in A. So if A has 7 rows, the product Ax will have 7 entries, which is what we want since T maps vectors from R5 to R7. Therefore, A must have 7 rows and 5 columns to define a mapping from R5 into R7 by the rule T(x) = Ax. To know more about Equation visit: If (x-4) is a factor of X squared -X-W=0, then what is the value of W. To determine the value of W, we need to find the value of x that satisfies the given condition. If (x-4) is a factor of X^2 - X - W = 0, it means that when we substitute x = 4 into the equation, it should equal zero. Let's substitute x = 4 into the equation: (4)^2 - (4) - W = 0 16 - 4 - W = 0 12 - W = 0 To solve for W, we isolate the variable: W = 12 Therefore, the value of W is 12. Find the volume of the region between the planes x+y+2z=4 and 2x+2y+z=8 in the first octant. The volume is Given planes,x+y+2z=4 and 2x+2y+z=8 in the first octant. We need to find the volume of the region between these planes in the first octant. Here are the steps to find the volume of the region between two planes in the first octant: First, we need to find the intersection of two planes: x + y + 2z = 4.. (1) 2x + 2y + z = 8.. (2) Multiplying equation (1) with 2 and subtracting equation (2) from it, we get, 2x + 2y + 4z = 8 - 2x - 2y - z => 3z = 4 => z = 4/3 Now, substituting the value of z in equations (1) and (2), we get; x + y = -2/3.. (3) 2x + 2y = 8/3 => x + y = 4/3.. (4) Solving equations (3) and (4), we get, x = 2/3 and y = -2/3. Therefore, the intersection of two planes is a line (2/3, -2/3, 4/3). The volume of the region between two planes in the first octant is given as Volume = ∫∫[2x + 2y - 8] dx dy Here, the limits for x is 0 to 2/3 and for y is 0 to -x + 4/3. Putting these limits in the above equation, we get, Volume = ∫[0 to 2/3] ∫[0 to -x + 4/3] [2x + 2y - 8] dy dx On solving, Volume = 4/3 cubic units Therefore, the volume of the region between the planes x+y+2z=4 and 2x+2y+z=8 in the first octant is 4/3 cubic units. To know more about volume visit: sketch the graph of equation y=1/x2 over the interval (5,-5) sketch the graph of equation y=2x+3.solution sketch the function f( x)= -2/x-3 and find its domain and range The graph of y = 1/x^2 is a hyperbola with vertical asymptotes at x = 0. It is symmetric about the y-axis and approaches infinity as x approaches 0. The graph of y = 2x + 3 is a straight line with a positive slope of 2 and a y-intercept of 3. The graph of f(x) = -2/(x - 3) is a hyperbola with a vertical asymptote at x = 3 and approaches infinity as x approaches 3 from the left or right. The domain of f(x) is all real numbers except x = 3, and the range is all real numbers except 0. The first equation, y = 1/x^2, represents a hyperbola. When sketching the graph over the interval (-5, 5), we observe that the function approaches infinity as x approaches 0 and approaches 0 as x approaches positive or negative infinity. The graph is symmetric with respect to the y-axis and the x-axis. The shape of the graph becomes steeper as x moves away from 0. The graph does not intersect the y-axis and has vertical asymptotes at x = 0. The second equation, y = 2x + 3, represents a linear function. The graph is a straight line with a slope of 2 and a y-intercept of 3. It has a positive slope, indicating that it increases as x increases. The graph extends indefinitely in both directions. For the function f(x) = -2/(x - 3), the graph is a hyperbola with a vertical asymptote at x = 3. The graph approaches positive or negative infinity as x approaches 3 from the left or right, respectively. The graph does not intersect the y-axis. The domain of the function is all real numbers except x = 3, as division by zero is undefined. The range of the function is all real numbers except 0, as the function approaches positive or negative infinity but never reaches 0. Learn more about hyperbola here: For a two-dimensional potential flow, the potential is given by 1 r (x, y) = x (1 + arctan x² + (1+ y)², 2πT where the parameter A is a real number. 1+y x Hint: d arctan(x) dx 1 x² +1 (1) Determine the expression of the velocity components ux and uy. (2) Determine the value of I such that the point (x, y) = (0,0) is a stagnation point. (3) Assuming for the far field, the pressure and constant density are P. and p, respectively, determine the pressure at the point (0, -2). (4) The flow field corresponds to a flow around a cylinder. (Hint: Stream function is needed.) (a) Determine the centre and radius of the cylinder. (b) Determine the magnitude and direction of the resulting forcing acting on the cylind For a two-dimensional potential flow, the potential is given by 1 r (x, y) = x (1 + arctan x² + (1+ y)², 2πT where the parameter A is a real number. Given potential function is;ϕ(x,y) = x(1 + arctan (x² + (1+y)²))/2πT To find velocity components ux and uy, we need to take partial derivative of potential functionϕ(x,y) = x(1 + arctan(x² + (1+y)²))/2πTUsing the chain rule; ∂ϕ/∂x = ∂ϕ/∂r * ∂r/∂x + ∂ϕ/∂θ * ∂θ/∂x = cosθ * (1/r) * x(1+arctan(x² + (1+y)²))/2πT - sinθ * (1/r) * x(1+arctan(x² + (1+y)²))/2πT ∂ϕ/∂y = ∂ϕ/∂r * ∂r/∂y + ∂ϕ/∂θ * ∂θ/∂y = cosθ * (1/r) * x(1+arctan(x² + (1+y)²))/2πT - sinθ * (1/r) * x (1+arctan(x² + (1+y)²))/2πTNow, replace cosθ and sinθ with x/r and y/r respectively∂ϕ/∂x = x/(x²+y²) * x(1+arctan(x² + (1+y)²))/2πT- y/(x²+y²) * x(1+arctan(x² + (1+y)²))/2πT= [x²-y²]/(x²+y²) * x (1+arctan(x² + (1+y)²))/2πT∂ϕ/∂y = y/(x²+y²) * x(1+arctan(x² + (1+y)²))/2πT + x/(x²+y²) * x(1+arctan(x² + (1+y)²))/2πT= [2xy]/(x²+y²) * x(1+arctan(x² + (1+y)²))/2πT(2) To find stagnation point, we have to find (x,y) such that ux = uy = 0 and ϕ(x,y) is finite. Here, from (1) we get two equations; x(1+arctan(x² + (1+y)²))/2πT= 0 and x(1+arctan(x² + (1+y)²))/2πT + y(1+arctan(x² + (1+y)²))/2πT= 0For (1), either x=0 or arctan(x² + (1+y)²) = -1, but arctan(x² + (1+y)²) can't be negative so x=0. Thus, we get the condition y= -1 from (2)So, stagnation point is (0, -1).(3) For the far field, pressure is p, density is P. In potential flow, we have; P = ρv²/2 + P0, where P0 is constant pressure. Here, P0 = P and v = ∇ϕ so, P = ρ[ (∂ϕ/∂x)² + (∂ϕ/∂y)² ]/2Using expressions of ∂ϕ/∂x and ∂ϕ/∂y obtained above, we can find pressure at (0,-2).(4) Given flow is around a cylinder. For flow around cylinder, stream function can be written as; ψ(r,θ) = Ur sinθ (1-a²/r²)sinθTo find centre and radius of the cylinder, we find point where velocity is zero. We know that ψ is constant along any streamline. So, at the boundary of cylinder, ψ = ψ0, and at the centre of the cylinder, r=0.Using stream function, it is easy to show that ψ0= 0.So, at boundary of cylinder; U(1-a²/R²) = 0, where R is radius of cylinder, which gives R=aSimilarly, at centre; U=0To find the resulting force on the cylinder, we first have to find the lift and drag coefficients; C_d = 2∫_0^π sin²θ dθ = π/2 and C_l = 2∫_0^π sinθ cosθ dθ = 0We know that C_d = F_d/(1/2 ρ U²L) and C_l = F_l/(1/2 ρ U²L)where L is length of cylinder.So, F_d = π/2 (1/2 ρ U²L) and F_l= 0. Thus, the resulting force is F= (π/2) (1/2 ρ U²L) at an angle 90° to the flow direction. To know more about dimensional, visit: Using these values, we can write Bernoulli's equation as: P + 1/2 * ρ * ((1 + arctan(5))^2 + (4/3)^2) = P_far P = P_far - 1/2 * ρ * ((1 + arctan(5))^2 + (4/3)^2) (1) To determine the expression for the velocity components ux and uy, we can use the relationship between velocity and potential in potential flow: ux = ∂Φ/∂x uy = ∂Φ/∂y Taking the partial derivatives of the potential function Φ(x, y) with respect to x and y: ∂Φ/∂x = (1+arctan(x^2+(1+y)^2)) - x * (1/(1+x^2+(1+y)^2)) * (2x) ∂Φ/∂y = -x * (1/(1+x^2+(1+y)^2)) * 2(1+y) Simplifying these expressions, we have: ux = 1 + arctan(x^2+(1+y)^2) - 2x^2 / (1+x^2+(1+y)^2) uy = -2xy / (1+x^2+(1+y)^2) (2) To find the value of A such that the point (x, y) = (0,0) is a stagnation point, we need to find the conditions where both velocity components ux and uy are zero at that point. By substituting (x, y) = (0,0) into the expressions for ux and uy: ux = 1 + arctan(0^2+(1+0)^2) - 2(0)^2 / (1+0^2+(1+0)^2) = 1 + arctan(1) - 0 = 1 + π/4 uy = -2(0)(0) / (1+0^2+(1+0)^2) = 0 For the point (x, y) = (0,0) to be a stagnation point, ux and uy must both be zero. Therefore, A must be chosen such that: 1 + π/4 = 0 A = -π/4 (3) To determine the pressure at the point (0, -2), we can use Bernoulli's equation for potential flow: P + 1/2 * ρ * (ux^2 + uy^2) = constant At the far field, where the velocity is assumed to be zero, the pressure is constant. Let's denote this constant pressure as P_far. At the point (0, -2), the velocity components ux and uy are: ux = 1 + arctan(0^2+(1-2)^2) - 2(0)^2 / (1+0^2+(1-2)^2) = 1 + arctan(5) - 0 = 1 + arctan(5) uy = -2(0)(-2) / (1+0^2+(1-2)^2) = 4 / 3 Using these values, we can write Bernoulli's equation as: P + 1/2 * ρ * ((1 + arctan(5))^2 + (4/3)^2) = P_far Solving for P at the point (0, -2), we have: P = P_far - 1/2 * ρ * ((1 + arctan(5))^2 + (4/3)^2) To know more about potential function, visit: A group of students made the following statements about a linear-quadratic system consisting of two equations. 1. The solutions are the x-coordinates of the points of intersection. 2. There are at most two solutions. 3. The solution must satisfy at least one equation of the system. 4. There is at least one solution. 2. The true statement(s), listed from the lowest to the highest statement number, are (Record your answer in the numerical-response section below.) Your answer: A group of students made the following statements about a linear-quadratic system consisting of two equations. The given statements can be analyzed in the following manner: Statement 1: The solutions are the x-coordinates of the points of intersection. This statement is true because the point of intersection of two lines, one linear and the other quadratic, forms the solution of the linear-quadratic system. Therefore, statement 1 is true. Statement 2: There are at most two solutions.This statement is also true because there can be two, one or zero solutions, but not more than two solutions for a linear-quadratic system. Therefore, statement 2 is true. Statement 3: The solution must satisfy at least one equation of the system.This statement is true as a point of intersection must lie on both the equations of the system. Therefore, statement 3 is Statement 4: There is at least one solution.This statement is also true because a linear-quadratic system can have a maximum of two solutions. Therefore, statement 4 is true. The true statements are given as follows: 2, 1, 3, 4 Hence, the numerical-response answer is as follows: 2,1,3,4 To know more about linear-quadratic system visit: use the Simplex method to find the minimum value of the objective function w = 9x1 + 6x2 Subject to the constraints: x1 +2x2 ≥ 5 2x1 + 2x2 ≥ 8 2x2 +x2 ≥ 6 Where x1 ≥ 0 and x2 ≥ 0 The optimal solution is x1 = 4, x2 = 0, x3 = 1, w = 0, and the minimum value of the objective function is 0. To solve this linear programming problem using the Simplex method, we first need to convert it into standard form by introducing slack variables. Our problem can be rewritten as follows: Minimize w = 9x1 + 6x2 Subject to: x1 + 2x2 + x3 = 5 2x1 + 2x2 + x4 = 8 x1 + 2x2 + 2x3 = 6 where x1, x2, x3, and x4 are all non-negative variables. Next, we set up the initial simplex tableau: Basic Variablesx1x2x3x4RHS The last row represents the coefficients of the objective function. The negative values in the z-row indicate that we are minimizing the objective function. To find the pivot column, we look for the most negative coefficient in the z-row. In this case, the most negative coefficient is -9, which corresponds to x1. Therefore, x1 is our entering variable. To find the pivot row, we calculate the ratios of the RHS values to the coefficients of the entering variable in each row. The smallest positive ratio corresponds to the pivot row. In this case, the ratios are: Row 1: 5/1 = 5 Row 2: 8/2 = 4 Row 3: 6/1 = 6 The smallest positive ratio is 4, which corresponds to row 2. Therefore, x4 is our exiting variable. To perform the pivot operation, we divide row 2 by 2 to make the coefficient of x1 equal to 1: Basic Variablesx1x2x3x4RHS We repeat the process until all coefficients in the z-row are non-negative. In this case, we can stop here because all coefficients in the z-row are non-negative. Therefore, the optimal solution is x1 = 4, x2 = 0, x3 = 1, w = 0, and the minimum value of the objective function is 0. Learn more about value here: Given that lim x→4 (2x−7)=1, illustrate this definition by finding the largest values of δ that correspond to ε=0.5,ε=0.1, and ε=0.05. ε=0.5 δ≤ ε=0.1 δ≤1 ε=0.05 δ≤ We have found the largest values of δ that correspond to ε=0.5,ε=0.1, and ε=0.05: ε=0.5 δ≤0.25, ε=0.1 δ≤0.05, and ε=0.05 δ≤0.025. Using the limit definition, we can find the largest values of δ that correspond to each value of ε as follows: For ε = 0.5: We need to find a value of δ such that whenever 0 < \|x - 4\| < δ, then \|2x - 7 - 1\| < 0.5. Simplifying the inequality inside the absolute value, we get: \|2x - 8\| < 0.5 -0.5 < 2x - 8 < 0.5 7.5/2 < x < 8.5/2 3.75 < x < 4.25 Therefore, we can take δ = min{4.25 - 4, 4 - 3.75} = min{0.25, 0.25} = 0.25. For ε = 0.1: We need to find a value of δ such that whenever 0 < \|x - 4\| < δ, then \|2x - 7 - 1\| < 0.1. Simplifying the inequality inside the absolute value, we get: \|2x - 8\| < 0.1 -0.1 < 2x - 8 < 0.1 3.95 < x < 4.05 Therefore, we can take δ = min{4.05 - 4, 4 - 3.95} = min{0.05, 0.05} = 0.05. For ε = 0.05: We need to find a value of δ such that whenever 0 < \|x - 4\| < δ, then \|2x - 7 - 1\| < 0.05. Simplifying the inequality inside the absolute value, we get: \|2x - 8\| < 0.05 -0.05 < 2x - 8 < 0.05 3.975 < x < 4.025 Therefore, we can take δ = min{4.025 - 4, 4 - 3.975} = min{0.025, 0.025} = 0.025. Thus, we have found the largest values of δ that correspond to ε=0.5,ε=0.1, and ε=0.05: ε=0.5 δ≤0.25, ε=0.1 δ≤0.05, and ε=0.05 δ≤0.025. Learn more about values here: Water is leaking out of an inverted conical tank at a rate of 9500 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 9 meters and the diameter at the top is 4.5 meters. If the water level is rising at a rate of 18 centimeters per minute when the height of the water is 4 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. ______cm^3/min At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)____knots 1) Water is leaking out of an inverted conical tank at a rate of 9500 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 9 meters and the diameter at the top is 4.5 meters. If the water level is rising at a rate of 18 centimeters per minute when the height of the water is 4 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. Answer more than 100 words:Explanation:We know that the tank is inverted conical. Let's call its height "H" and its radius at the base "r." The rate at which the water is being pumped in can be expressed as V_in, and the rate at which water is leaking out can be expressed as V_out. We are given the following information:At a given time, water is leaking out at 9500 cubic centimeters per minute.The tank has a height of 9 meters, and its diameter at the top is 4.5 meters.The rate at which the water level is rising is 18 centimeters per minute when the height of the water is 4 meters. We need to find the rate at which water is being pumped into the tank.To solve this problem, we'll use related rates, which is a calculus technique that involves finding the rate of change of one variable with respect to another variable. In this case, we want to find the rate of change of the volume of water in the tank with respect to time. Let's call this V_total. We can express V_total as follows:V_total = 1/3πr²HFirst, let's find an equation that relates the radius and height of the tank to the rate at which the water level is rising. We can use similar triangles to relate the rate of change of the height of the water to the rate of change of the radius of the water. Consider the following diagram:We know that the height of the water is increasing at a rate of 18 centimeters per minute when the height of the water is 4 meters. That is, dh/dt = 18 when h = 4. The volume of the tank can be expressed as: V_tank = 1/3πr²H The volume of the air in the tank can be expressed as: V_air = 1/3πx²(H - h) The volume of the water in the tank can be expressed as: V_water = V_tank - V_air To find the rate at which water is being pumped into the tank, we need to find dV_water/dt. Using the product rule, we get: dV_water/dt = dV_tank/dt - dV_air/dt We know that dV_tank/dt is 0, since the dimensions of the tank are not changing. We need to find dV_air/dt. To do this, we need to find an equation that relates x and h. We know that the tank is conical, so we can use similar triangles to relate x and h. Consider the following diagram:We have the following similar triangles: x/r = h/(H - h) Differentiating with respect to time, we get: dx/dtr = (h/H - h/r) (dH/dt) + (1 - h/H)(dr/dt) We want to find dx/dt when h = 4. At this point, we know that h = 4, H = 9, r = 2.25, and dH/dt = 0. Plugging in these values, we get: dx/dt = (4/9 - 4/2.25)(0) + (1 - 4/9) (dr/dt) dx/dt = (5/9)(dr/dt) Now we can substitute dx/dt into our expression for dr/dt: dr/dt = 32 - (5/9)(dr/dt) dr/dt + (5/9)(dr/dt) = 32 (14/9)(dr/dt) = 32 dr/dt = 32(9/14) dr/dt = 205.71 cubic centimeters per minute Now we can find dV_water/dt: dV_water/dt = 0 - dV_air/dt Therefore, the rate at which water is being pumped into the tank is 128π/3 cubic centimeters per minute.2) At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.) Answer:We have the following information:Ship A is 30 nautical miles due west of ship B.Ship A is sailing west at 20 knots.Ship B is sailing north at 16 knots.We want to find the rate at which the distance between the ships is changing at 5 PM.To solve this problem, we'll use the Pythagorean theorem, which tells us that the distance between the ships is equal to the square root of the sum of the squares of the distances that each ship has traveled. We can express this as follows: D² = x² + y² where D is the distance between the ships, x is the distance that ship A has traveled, and y is the distance that ship B has traveled. We want to find dD/dt when x = 30 and y = 16t. Note that we can express x and y in terms of time t as follows: x = 20t y = 16t Plugging these expressions into our equation for D, we get: D² = (20t)² + (16t)² D² = 400t² + 256t² D² = 656t² Taking the square root of both sides, we get: D = 8√41t Now we can differentiate both sides with respect to time: dD/dt = 8√41(dt/dt) dD/dt = 8√41 Therefore, the rate at which the distance between the ships is changing at 5 PM is 8√41 knots. The rate at which water is being pumped into the tank is approximately 33929.15 cubic centimeters per minute. How to find the rate at which the water is being pumped? The volume of a cone is given by the formula: V = (1/3)πr^h Given that the height of the tank is 9 meters and the diameter at the top is 4.5 meters, we can find the radius (r) of the circular base. The radius is half of the diameter, so r = 4.5/2 = 2.25 Since the leakage rate is given in cubic centimeters per minute, let's convert the measurements to centimeters: r = 2.25 100 = 225 centimeters We are given that water is leaking out at a rate of 9500 cubic centimeters per minute. This is the rate of change of the volume with respect to time, so dV/dt = -9500 cubic centimeters per minute. We are also given that the water level is rising at a rate of 18 centimeters per minute when the height of the water is 4 meters. This is the rate of change of the height with respect to time, so dh/ dt = 18 centimeters per minute. We need to find the rate at which water is being pumped into the tank, which is dV/dt. We can set up a related rates equation using the volume formula: dV/dt = (1/3) πr²dh/dt Substituting the known values: dV/dt = (1/3)π(225)²*18dV/dt ≈ 33,929.15 cubic centimeters per minute Learn more about rates of change at: 19. ∫ 7 x a) 7x + b) 7 c) 7x2 + d) 7x The correct option is a) 7x + C The given essential is ∫7 dx. Integrating the constant time period 7 with recognition to x actually offers 7x as the result. Therefore, the proper option is: a) 7x + C Here, C represents the steady of integration, that's added to account for any arbitrary regular which could rise up throughout the mixing technique. When we combine a constant time period like 7, it does now not trade with respect to x. Hence, the critical of 7 is in reality 7 instances x, that's represented as 7x. Adding the regular integration allows for the illustration of a circle of relatives of solutions because the value of C can range. In the end, the indispensable ∫ 7 dx is the same as 7x + C, wherein C is the steady integration. To know more about integration, The correct question is: "19. ∫ 7 a) 7 + b) 7 c) 7² + d) 7" Find the surface area of the part of the sphere x^2+y^2+z^2=4 that lies inside the cylinder x^2+y^2=2y. Sketch the given surface. The surface area of the part of the sphere that lies inside the cylinder π/2 square units The surface area of the part of the sphere that lies inside the cylinder, we need to determine the limits of integration for the cylindrical part. From the equations given, we have: x² + y² = 2y ---(1) x² + y² + z² = 4 ---(2) From equation (1), we can rewrite it as: x² + (y² - 2y) = 0 x² + (y² - 2y + 1) = 1 x² + (y - 1)² = 1 This equation represents a circle with center (0, 1) and radius 1. Now, we need to find the limits of integration for the cylindrical part along the z-axis. From equation (2), we have: x² + y² + z² = 4 Rearranging, we get: z² = 4 - x² - y² z = √(4 - x² - y) Since the cylindrical part lies inside the sphere, the limits of integration for z are from 0 to the upper boundary of the sphere, which is √(4 - x² - y²). To find the surface area, we integrate the circumference of the circle at each point (x, y) over the given limits of integration. Surface Area = ∬(x² + y²) dA Using polar coordinates, we can rewrite the surface area integral as: Surface Area = ∬(r²) r dr dθ The limits of integration for r are from 0 to the radius of the circle, which is 1. The limits of integration for θ are from 0 to 2π, covering the full circle. Now we can evaluate the surface area integral: Surface Area = ∫[θ=0 to 2π] ∫[r=0 to 1] (r³) dr dθ Integrating with respect to r, we have: Surface Area = ∫[θ=0 to 2π] [(r⁴)/4] from r=0 to r=1 dθ Surface Area = ∫[θ=0 to 2π] [(1⁴)/4 - (0⁴)/4] dθ Surface Area = ∫[θ=0 to 2π] (1/4) dθ Surface Area = (1/4) [θ] from θ=0 to θ=2π Surface Area = (1/4) (2π - 0) Surface Area = π/2 Therefore, surface area of the part of the sphere that lies inside the cylinder is π/2 square units. Learn more about surface area here :	{"url":"https://hippozaa.com/article/max-was-injured-and-can-no-longer-work-as-a-result-of-a-lawsuit-he-is-to-be-awarded-the-present-value","timestamp":"2024-11-03T15:08:21Z","content_type":"text/html","content_length":"129706","record_id":"<urn:uuid:ebe14b26-22be-4c0d-afe0-1a9ba1bed225>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00847.warc.gz"}
Big Sample, Unreliable Result - Statistics.com: Data Science, Analytics & Statistics CoursesBig Sample, Unreliable Result - Statistics.com: Data Science, Analytics & Statistics Courses Big Sample, Unreliable Result Which would you rather have? A large sample that is biased, or a representative sample that is small? The American Statistical Association committee that reviewed the 1948 Kinsey report on male sexual behavior, based on interviews with over 5000 men, left no doubt of their preference for the latter. The statisticians – William Cochran, Frederick Mosteller, John Tukey, and W. O. Jenkins – were leaders in their profession, and identified multiple sources of bias in the Kinsey data collection effort. Participation was voluntary, and generated to some degree by referral, leading to self-selection bias. Prison populations were substantially over-represented. One result was an over-estimate of the prevalence of homosexuality among men. Tukey dismissively said that he would put greater stock in a randomly selected sample of 3 than in 300 selected by Kinsey. Nonetheless, Sample Size Matters On the other hand, sample size does matter, even if it is secondary to proper sample selection methods. As Daniel Kahneman put it in Thinking, Fast and Slow: The exaggerated faith in small samples is only one example of a more general illusion – we pay more attention to the content of messages than to information about their reliability, and as a result end up with a view of the world around us that is simpler and more coherent than the data justify. The smaller the sample, the more it is prone to misinterpretation. Random variation makes it unreliable as a tool for estimation, and also gives scope for interesting chance events to attract the attention of the investigator. How Big? How big should your sample be? You can find general guidance associated with particular tasks (polling, auditing, behavioral studies), but a more analytical approach exists, based on the principles of statistical inference. This approach presumes that you are gathering data to investigate a hypothesis, typically concerning the effect some condition or treatment has on subjects, an effect that shows up in a difference between, or among, groups that experience different treatments or conditions. The basic idea is to gather a sample that is big enough to assure you that, if the effect you are investigating exists, your study will find it. This involves balancing three parameters set by the user: • Effect size • Level of significance • Power Setting the Parameters Effect size: The smaller the effect size you hope to find, the bigger the sample needed. A useful analogy is finding stars with a telescope – the dimmer the star, the bigger the telescope you need to distinguish it. “Effect size” is the difference you hope exists in the population(s) you are investigating. For continuous numeric data, it would be expressed as a difference in means of the distributions. What does “find” mean? Here it means to conclude that there is a statistically significant difference, or effect. For example, if you are testing two different colors for a “buy” button on a web site, finding a difference means that a difference between two groups of web users experiencing different colors is statistically significant at a pre-chosen level of significance. Level of Significance: The “tighter” the definition of statistical significance (e.g. 0.01 instead of 0.05), the bigger the sample needed. P-values and the whole idea of statistical significance have fallen into some disfavor as a result of their abuse – as the number of academic researchers seeking to publish papers has risen, the p-value has become a “necessary and sufficient” publishing criterion, opening the door to great numbers of published studies whose only “virtue” is that they contain a statistically significant result, lacking practical significance or proper study design. Nevertheless, in determining sample size, a determination of statistical significance as a criterion for validating a finding is needed. Power: Power is the probability of achieving a statistically significant result in a sample study, if the specified effect size is real in the population being studied. For example, if a medication has a real effect of reducing blood pressure by 10%, and you conduct a study (at your specified significance level) between a medication group and a control group, power is the probability that the study will return a result of “significant.” Note that the study does not necessarily have to yield a 10% difference between the two groups – rather it simply has to yield a statistically significant difference. The more power you seek, the bigger the sample needed. Specifying the three parameters is an exercise in tradeoffs. The smaller the effect you want to be able to find, and the greater the power (probability of finding that effect), the bigger the sample you need. If your initial goals with respect to these key parameters yield a sample requirement that is beyond your budget or capability, you must compromise something; that is, be willing to set a larger effect size threshold (meaning that you might well miss a desired effect), or you must tolerate a lower power, or both. The level of statistical significance is not so malleable; it is usually set by external requirements, e.g. regulators or journal publishers who often specify a traditional level of 5%. Setting the three parameters is a necessary, but not sufficient condition to find sample size. A fourth factor affecting sample size is the variance in the data. This, of course, is not a parameter set by the user. The greater the variance in the data, the greater the sample size needed to identify a given effect of interest. Thus, any estimate of required sample size must necessarily incorporate an assumption about variance in the data. This might be estimated from earlier samples of data, or from knowledge about the process or population involved. Putting it All Together Once you have some estimate of the variance in the data, you can visualize the procedure to calculate sample size via a resampling simulation procedure, illustrated here for the case of two samples with continuous numeric data: 1. Specify the desired effect size, level of significance, and power 2. Specify two data random generators to generate normally-distributed data from populations with two means that differ by the desired effect size, and with variance as estimated from prior 3. Generate two samples of size n1 — one from each of the data generators 4. Conduct a significance test on the two samples; record whether the difference is significant 5. Repeat steps 3-4 say 1000 times; note what proportion of the time the difference is significant – this is the power 6. If the power is right on target, n1 is the appropriate sample size; if the power is too low you will need to increase the sample size and if the power is higher than needed you can reduce the sample size 7. Iteratively try different levels of n until the power is where you need it *If you actually have real data appropriate to the study, you can substitute two bootstrap generators (one shifted by the effect size) for the normally distributed data generator. In most cases, power will be determined by software calculating formulas, though the bootstrap simulation approach can be used where the situation and the statistic of interest do not fit the data scenario required by the software.	{"url":"https://www.statistics.com/big-sample-unreliable-result/","timestamp":"2024-11-09T03:58:56Z","content_type":"text/html","content_length":"75526","record_id":"<urn:uuid:5f059d71-1e25-43ed-a9a0-3147fbf73103>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00268.warc.gz"}
RD-E: 0800 Hopkinson Bar The purpose of this example is to model and predict the responses of very high strain rates on a material during impact. High strain rate characterization of 7010 aluminum alloy using Split-Hopkinson pressure bar experiment. Figure 1. Hopkinson bar Precise data for high strain rate materials is necessary to enable the accurate modeling of high-speed impacts. The high strain rate characterization of materials is usually performed using the Split-Hopkinson pressure bar within the strain rate range 100-10000 s^-1. It is assumed that during the experiment the specimen deforms under uniaxial stress, the bar specimen interfaces remain planar at all times, and the stress equilibrium in the specimen is achieved using travel times. The Radioss explicit finite element code is used to investigate these assumptions. Options and Keywords Used Low extremity nodes of the output bar are fixed in the Z direction. The axisymmetric condition on the revolutionary symmetry axis requires the blocking of the Y translation and X rotation. The projectile is modeled using a steel cylinder with a fixed velocity in the direction Z. The required strain rate is considered by applying two imposed velocities, 1.7 ms^-1 and 5.8 ms^-1 in order to produce strain rates in the ranges of 80 s^-1 and 900 s^-1 (low and high rates). True Stress, True Strain and True Strain Rate Measurement from Time History Figure 2. Nodes and quads saved for time history In the experiment, the strain gauge is attached to the specimen. In simulation, the true strain will be determined from 9040 and 6 nodes’ relative Z displacements (${l}_{0}$= 3.83638 mm). The true stress can be given using two data sources. The first methodology consists of using the equation previously presented, based on the assumption of the one-dimensional propagation of bar-specimen forces. The engineering strain ${\epsilon }_{t}$ associated with the output stress wave is obtained from the Z displacement of nodes located on the output bar. The true plastic strain is extracted from the quads on the specimen, saved in the Time Histories file. True stress can also be measured directly from the Time History using the average of the Z stress quads 6243, 6244, 6224 and 6235. It should be noted that the section option is not available for quad elements. The strain rate can be calculated from either the true plastic strain of quads saved in or from the true strain ${\epsilon }_{\mathrm{true}}$ Table 1. Relations Used in the Analysis │ │ High Rate Testing │ │ True stress │ ${\sigma }_{true}\left(t\right)=\frac{{S}_{bar}{E}_{bar}}{{S}_{specimen}}{\epsilon }_{T}\left(t\right)\mathrm{exp}\ │ Z stress average from quads saved in /TH │ │ │ left({\epsilon }_{pl}\left(t\right)\right)$ │ │ │ True strain │ ${\epsilon }_{true}\left(t\right)=\mathrm{ln}\left(\frac{{l}_{i}\left(t\right)}{{l}_{0}}\right)\text{ }{l}_{i}\left(t\right)={l}_{0}+\text{Δ}l={l}_{0}+\left({u}_{9040}\left(t\ │ │ │ right)-{u}_{6}\left(t\right)\right)$ │ │ True strain │ $\stackrel{˙}{\epsilon }=\frac{\text{Δ}{\epsilon }_{pl}\left(t\right)}{\text{Δ}t}$ │ $\stackrel{˙}{\epsilon }=\frac{\text{Δ}{\epsilon }_{true}\ │ │ rate │ │ left(t\right)}{\text{Δ}t}$ │ Input Files Refer to Access the Model Files to download the required model file(s). The model files used in this example include: Model Description The purpose of this example is to model and predict the responses of very high strain rates on a material during impact. The Split-Hopkinson pressure bar is a suitable method to perform experiments with high strain rates. Figure 3 shows the principal test setup, consisting of: • an incident bar and a transmission bar of equal length, between which the sample to be tested is clamped. • a striker is attached to the outer end of the incident bar. When a steel projectile hits the striker, a stress pulse is introduced into the incident bar. Figure 3. Split-Hopkinson pressure bar device The impact generates a strain (tensile) wave which propagates through (along) the Incident bar and is detected by strain gauge 1. Part of the wave is reflected, and a part is transmitted via the specimen’s interface. So, the stress pulse continues through the specimen and into the transmitted bar. Strain gauges 1 and 2 are attached to the incident bar and transmission bar to detect the strain wave signal. The wave reflections inside the sample enable the stress to be homogenized during the test. The strain associated with the output or transmitted stress wave is measured by the strain gauges on the output or transmitted bar. The strain gauges attached to the specimen gauge length provide direct measuring of the true strain and the true plastic strain in the specimen during the experiment. The transmitted elastic wave provides a direct force measurement to the bar specimen interfaces by way of the following relation. Figure 4. Specimen geometry and cross-section (dimensions in mm) (1) $F\left(t\right)={S}_{bar}\cdot {E}_{bar}\cdot {\epsilon }_{T}\left(t\right)$ Modulus of the output bar. ${\epsilon }_{T}$ Strain associated with the output stress wave. Cross-section of the output bar. If the two bars remain elastic and wave dispersion is ignored, then the measured stress pulses can be assumed to be the same as those acting on the specimen. The engineering stress value in the specimen can be determined by the wave analysis, using the transmitted wave:(2) ${\sigma }_{engineering}\left(t\right)=\frac{F\left(t\right)}{{S}_{specimen}}=\frac {{S}_{bar}{E}_{bar}}{{S}_{specimen}}{\epsilon }_{T}\left(t\right)$ Engineering stress can also be found by averaging out the force applied by the incident that is the reflected and transmitted wave, as shown in the equation:(3) ${\sigma }_{engineering}\left(t\right) =\frac{{S}_{bar}{E}_{bar}}{2{S}_{specimen}}\left[{\epsilon }_{l}\left(t\right)+{\epsilon }_{R}\left(t\right)+{\epsilon }_{T}\left(t\right)\right]$ ${\epsilon }_{I}$ and ${\epsilon }_{R}$ Strains associated with input stress wave. ${\epsilon }_{T}$ Strain associated with output stress wave. True stress in the specimen is computed using the following relation (refer to Example 11 - Tensile Test for further details): (4) ${\sigma }_{true}={\sigma }_{engineering}\text{\hspace{0.17em}}\mathrm{exp}\left({\epsilon }_{true}\right)$ The true strain rate is given by:(5) $\stackrel{˙}{\epsilon }=\frac{\text{Δ}{\epsilon }_{true}}{\text{Δ}t}$ True stress and true strain are evaluated up to the failure point. Figure 5. 1D analysis Interface 1 ${F}_{1}={S}_{bar}\left[{\sigma }_{l}\left(t\right)+{\sigma }_{R}\left(t\right)\right]={S}_{bar}\cdot {E}_{bar}\left[{\epsilon }_{l}\left(t\right)+{\epsilon }_{R}\left(t\right)\right]$ Interface 2 ${F}_{2}={S}_{bar}\cdot {\sigma }_{T}\left(t\right)={S}_{bar}\cdot {E}_{bar}\cdot {\epsilon }_{T}\left(t\right)$ Balance in specimen ${F}_{1}={F}_{2}$; ${\epsilon }_{l}\left(t\right)+{\epsilon }_{R}\left(t\right)={\epsilon }_{T}\left(t\right)$ Engineering stress in specimen ${\sigma }_{specimen}\left(t\right)=\frac{{F}_{1}}{{S}_{specimen}}=\frac{{F}_{2}}{{S}_{specimen}}$ Strain Rate Filtering Because of the dynamic load, strain rates cause high frequency vibrations which are not physical. Thus, the stress-strain curve may appear noisy. The strain rate filtering option enables to dampen such oscillations by removing the high frequency vibrations in order to obtain smooth results. A cut-off frequency for strain rate filtering F[cut] = 30 kHz was used in this example. Refer to RD-E: 1100 Tensile Test for further details. Johnson-Cook Model The Johnson-Cook model describes the stress in relation to the plastic strain and the strain rate using the following equation:(6) $\sigma =\underset{\begin{array}{l}\text{Influence of }\\ \text {plastic strain}\end{array}}{\underbrace{\left(a+b{\epsilon }_{p}{}^{n}\right)}}\underset{\begin{array}{l}\text{Influence of }\\ \text{strain rate}\end{array}}{\underbrace{\left(1+c\mathrm{ln}\frac{\ stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)}}$ $\stackrel{˙}{\epsilon }$ Strain rate. ${\stackrel{˙}{\epsilon }}_{0}$ Reference strain rate. ${\epsilon }_{p}$ Plastic strain (true strain). Yield stress. Hardening parameter. Hardening exponent. Strain rate coefficient. The two optional inputs, strain rate coefficient and reference strain rate, must be defined for each material in /MAT/LAW2 in order to take account of the strain rate effect on stress, that is the increase in stress when increasing the strain rate. The constants $a$, $b$ and $n$ define the shape of the strain-stress curve. In the documents entitled CRAHVI, G4RD-CT-2000-00395, D.1.1.1, Material Tests – Tensile properties of Aluminum Alloys 7010T7651 and AU4G Over a Range of Strain Rates, the behavior of the 7010 aluminum alloy can be described according to the relations: $\sigma =\left(496+225{\epsilon }^{0.35}\right)$ Strain rates below 80 s^-1 $\sigma =\left(496+225{\epsilon }^{0.35}\right)\left(1+0.16\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{0.08}\right)\right)$ Strain rates exceeding 80 s^-1 up to 3000 s^-1 Figure 6. Yield curve of the Johnson-Cook model: $\sigma =\left(496+225{\epsilon }^{0.35}\right)$ The material properties of the specimen are: Material Properties Young's modulus 73000 $\left[\mathrm{MPa}\right]$ Poisson's ratio 0.0028 $\left[\frac{g}{m{m}^{3}}\right]$ The material used for the bars and projectile is TYPE1 (linear elastic) with the following properties: Material Properties Young's modulus 210000 $\left[\mathrm{MPa}\right]$ Poisson's ratio 0.0078 $\left[\frac{g}{m{m}^{3}}\right]$ The geometrical characteristics of the bars and projectile are: 4 m 12 mm 12 mm 170 g Model Method Considering the geometry’s revolution symmetry the material and the kinematic conditions, an axisymmetric model is used (N2D3D = 1 in /ANALY set up in the Starter file). Y is the radial direction and Z is the axis of revolution. The mesh is made of 12054 2D solid elements (quads). The quad dimension is about 2 mm. Figure 7. Axisymmetrical model mesh. with imposed velocities on the top of the input bar The purpose of the test is to obtain results at high deformation rates. In this model the Johnson-Cook type material law is used. The increase of stress is expected to be approximately 30% above the stress compared to the quasi-static deformation rate. Experimental Data Experimental results show that the variation of the true tensile flow stress compared with the true strain is approximately equivalent to a strain rate between 80 s^-1 and 100 s^-1. The reference strain, ${\epsilon }_{R}$ in the Johnson-Cook model is set to 0.08 ms^-1 (correspond to 80 s^-1, which represents the quasi-static deformation rate. At higher deformation rates, the true flow stress increases significantly with increasing strain rates. The 7010 aluminum alloy exhibits an increase in the flow stress by a typical value of 30% at high strain rates (900 s^-1 – 3000 s^-1) compared to the quasi-static value. Results are given at the specific true strains of 0.02, 0.05 and 0.10. The influence of the strain rate on the stress can be seen in Figure 8 Figure 8. Variation of true stress compared with true strain for 7010 alloy. using 2 different rates (experimental data) For the test performed with a strain rate of 900 s , the flow stress reaches 850 at a 0.25 strain. Table 2. True Stress at Specific Strains using Both Strain Rates (experimental │ │ Strain Rate: 80 s^-1 │ Strain Rate: 900 s^-1 │ │ True strain │ 0.02 │ 0.05 │ 0.1 │ 0.02 │ 0.05 │ 0.1 │ 0.25 │ │ True plastic strain │ 0.012 │ 0.042 │ 0.092 │ 0.011 │ 0.039 │ 0.089 │ 0.238 │ │ True stress (MPa) │ 550 │ 600 │ 610 │ 625 │ 775 │ 800 │ 850 │ Johnson-Cook Model Figure 9 shows the variation of true stress in time in relation to the wave propagation along the bars. Stresses are evaluated on the input bar, the specimen and the transmission bar. Figure 9. Stress measurement localizations Figure 10. Stress waves in the input bar, the output bar and the specimen. (imposed velocities = 5.8 ms^-1) The stress-time curve shows the incident, reflected and transmitted signals. Figure 11. SHPB of the motion in time of the tensile pulse Figure 12. von Mises stress wave propagation along bars. (imposed velocities = 5.8 ms^-1) The speed of wave, $C$ along the bars is calculated using the relation:(7) $C=\sqrt{\frac{E}{\rho }}=5189m{s}^{-1}$ Young's modulus. Density of the bars. The time step element is controlled by the smallest element located in the specimen. It is set at 5x10^-5 ms. The stress wave thus reaches the specimen in 0.77 ms and travels 0.26 mm along the bar for each time step. Obviously, it remains lower than the element length of the smallest dimension (0.88 mm). An imposed velocity of 5.8 ms produces a strain rate in the specimen of approximately 900 s , while a strain rate of approximately 80 s is achieved using an imposed velocity of 1.7 ms . A simulation is performed for each velocity value. Note: The study on low rates is more limited in time than on high rates due to the reflected wave generated on top of the output bar. Figure 13 shows the true stress and true strain as a function of the strain rate. Figure 13. Variation of true stress with true strain for high and medium strain rates At a high strain rate (900/s), an increase in the flow stress is observed, being approximately 30% higher than the stress obtained for a low strain rate (80/s). The Johnson-Cook model used provides precise results compared with the experimental data. Figure 14. Stress Z and plastic strain on specimen at 1.2 ms The true stresses determined from both methodologies are shown side-by-side. This validates the analysis based on a transmitted wave. Typical curves for a model having imposed velocities equal to 5.8 Figure 15 Figure 16 Figure 15. True stress comparison in the specimen Figure 16. True strain rate in the specimen Either data sources used to evaluate the strain rate give similar results. The results show: • the strain rate effect on stress, with or without the cut-off frequency for smoothing (100 kHz); • the influence of the strain rate coefficient (comparison with experimental data). Figure 17. Strain rate effect Figure 18. Influence of the strain rate coefficient, c These studies are performed for the high strain rate model ($\epsilon$ = 900 s^-1). Figure 19 compares the distribution of the von Mises stress on the specimen, with and without the strain rate filtering at time t=0.88 ms. Figure 19. Left: strain rate filtering active (cut-off frequency = 1 kHz); Right: no strain rate filtering More physical flow stress distribution is obtained using filtering. Explicit is an element-by-element method, while the local treatment of temporal oscillations puts spatial oscillations into the CRAHVI, G4RD-CT-2000-00395, D.1.1.1, Material Tests - Tensile properties of Aluminum Alloys 7010T7651 and AU4G Over a Range of Strain Rates.	{"url":"https://2022.help.altair.com/2022/hwsolvers/rad/topics/solvers/rad/hopkinson_bar_example_r.htm","timestamp":"2024-11-13T17:23:30Z","content_type":"application/xhtml+xml","content_length":"203522","record_id":"<urn:uuid:5be455a2-7d53-453a-a8e0-4c11dafb823c>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00883.warc.gz"}
Reports: custom formula columns Spiro's reporting environment has a useful feature called "Formula Columns." This allows you to create a new column in the table with entirely new data points, using standard Excel-like equations. Note that this new column will only exist in the table you create it in; it will not show up in the front end of Spiro. A common use-case is using a formula column to calculate a weighted pipeline value for an opportunity. Often, a company will assign a win probability to each sales stage, and then want to see a weighted value to give them a more realistic pipeline amount. How to Add a Formula Column First, find the table you want to add the formula column to. Staying consistent with the weighted pipeline example, we're going to use the Opportunities table. Click on Add → Formula Column. You'll see an editor appear, for you to build the new formula column. You can see all the functions that are available for you to use, as well as all the columns in the table you are able to nested IF statement that multiplies the opportunity amount by some percent, depending on the sales stage. If you have any experience with calculations in Excel, this will be very straightforward. When you hit save, Spiro will give you an error message if the formula is incorrect. Once you've saved successfully, scroll to the end of the table. You should see you column appear with all the new calculated values. And now you're able to go use this new column in other tables.	{"url":"https://support.spiro.ai/article/328-reports-custom-formula-columns","timestamp":"2024-11-05T02:22:30Z","content_type":"text/html","content_length":"14977","record_id":"<urn:uuid:977ac474-3dd6-4f5c-bf4c-d000a427252c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00612.warc.gz"}
[Gzz] Re: One-time signature possibilities [Top][All Lists] [Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index] [Gzz] Re: One-time signature possibilities From: Tuomas Lukka Subject: [Gzz] Re: One-time signature possibilities Date: Mon, 12 May 2003 16:05:19 +0300 User-agent: Mutt/1.4.1i On Mon, May 12, 2003 at 01:26:33PM +0200, Benja Fallenstein wrote: > I'm starting to be quite convinced that one-time signatures are the > way to go with pointers. One-time signatures are signatures where the > key can only be used to sign n messages for small n (so actually > they're n-time) (you can get around the problem by signing as one of > the n messages the next public key to be used). How does this BTW impact the chain length? If you need to check 1000 signatures in order to see that a key is valid, that's not really > - safer than e.g. DSA: less algorithms in Storm that can be broken (if > we use e.g. DSA, we still need cryptographic hashes) > - more durable: DSA signatures are only considered safe for about two > years; no such problem with the hash functions we use However, we of course still need all revocation and re-signing methods because a cracker might steal your keys. > - faster: factoring large integers is very costly compared to hashing. ??? ;) > On the downside, one-time signatures are large. Remember that for > every save we want to keep, we have to keep the signatures. The > storage space per signature is O(bb) with b the number of bits in the > hash, a problem if we want to use SHA-1 + Tiger. To some degree, we > can trade off running time against signature storage space; we need to > decide what is reasonable. Do you have the numbers for e.g. DSA? How many bits more will we have for this? > I've benchmarked, and on my machine, a DSA verification takes 30ms, a > SHA-1 hash 5/1000 ms and a Tiger hash 6/1000 ms. The time estimates > below are based on this. > If we use only SHA-1, not Tiger, some of our options are: > - Store ~3KB, verify ~160 hashes, ~.8 ms > - Store ~1.5KB, verify ~240 hashes, ~1.2ms > - Store ~840 bytes, verify ~600 hashes, ~3ms > - Store ~440 bytes, verify ~5100 hashes, ~25.5ms > Using SHA-1 + Tiger, we have: > - Store ~15KB, verify ~350 hashes, ~4ms > - Store ~8KB, verify ~530 hashes, ~6ms > - Store ~4KB, verify ~1320 hashes, ~15ms > - Store ~2KB, verify ~11000 hashes, ~120ms I don't quite understand how you get these tradeoffs... > If, in addition to the above, you store another 20c or 44c bytes (for > SHA-1 and SHA-1+Tiger, respectively), you can sign 2^c messages with > the same public key. (Generation of the public key takes 2(2^c) times > the verification time for a single hash.) How does this work? > Discussion, please: What (if any) of this do you think is reasonable? This might be reasonable. Note of course that you can sign a group of blocks, you don't need to sign each block separately. [Prev in Thread] Current Thread [Next in Thread]	{"url":"https://lists.gnu.org/archive/html/gzz-dev/2003-05/msg00075.html","timestamp":"2024-11-13T22:20:56Z","content_type":"text/html","content_length":"8241","record_id":"<urn:uuid:79b09fe3-b5c9-43d9-9008-4a10a4621531>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00451.warc.gz"}
Re: Doomsday Example Nick Bostrom (bostrom@ndirect.co.uk) Tue, 25 Aug 1998 02:34:12 +0000 Robin Hanson wrote: > Nick Bostrom writes: > >Suppose we change the example slightly. Now, an A universe contains > >10 humans and nothing else. A B universe contains 10 humans and one > >trillion trillion stones. There is a small C universe (of negligible > >size), and it spawns one thousand baby-univeses through a random > >process (like rolling fair dice) that has a 10% chance of yielding an > >A and 90% of a B. > > > >It seems clear that there will probably be about 100 A univeses and > >900 B universes. But that means about 90% of all humans will find > >themselves in an B universe. So if all you knew was the set-up and > >that you were a human, then you should believe that there was a > >90% chance that you were in an B universe. > > > >And yet, if you include stones in the reference class, then it seems > >you should believe you are in an A universe. For if you are in a B > >universe, then there exists in total at least one trillion trillion > >stones and then you would probably have been one of these stones > >rather than a human. > Let N = "trillion trillion", and assume there are exactly 100 A > "universes" and 900 B "universes". Note that these are not "universes" > in the sense I was using of "possible worlds." Sure, my usage agrees with this. > The entire construction > of C + 100 As + 900 Bs is just one total space-time, a "possible world." > As described the only remaining uncertainty is where in this world I am. > If I treat stone slots and human slots equally, there are > 10010 + 900(10+N) slots. If my prior is uniform across these slots, But the point of my example having a C universe etc. was that this prior is not a plausible one. The fair dice in the C universe, or rather let's say it's a fair coin, do you really want to say that the prior probability that this coin should land heads all thousand throws is almost one? That seems wrong. > then conditioning on my being in a human slot, there is a 90% chance I'm > in a B "universe," as you prefer. But since that prior seems wrong, you don't get this result. > >> >Similarly, I would say that finding that you are an observer does not > >> >give you reason for thinking that a large fraction of all slots in > >> >the universe is occupied by observers. > >> > >> Your error is to say "large fraction." The fact that life exists > >> on Earth does* make it more likely that our universe has other planets > >> where life has evolved in a similar time. It just isn't enough to > >> conclude the fraction of Earth-like planets with life is "large." > >... > >And now you say that finding that life exists on Earth should > >not affect your beliefs about about the fraction of Earth-like > >planets with life. > But I said exactly the opposite in the quote above! Actaully, you didn't say anything about what we should believe about the fractions. Suppose the only alternatives were (1): one in ten Earth-like planet evolve intelligent life; or (2) one in a thousand does. Suppose the prior probability is fifty-fifty. Now you observer that the Earth has evolved life. Based on your clarification above, I now take it that you think this observation should increase your confidence in (1). Is this right? But then what happened to the selection-effect you spoke of in the "Early life"-paper? "Since no one on Earth would be wondering about the origin of life if Earth did not contain creatures nearly as intelligent as ourselves, the fact that four billion years elapsed before high intelligence appeared on Earth seems compatible with any expected time longer than a few billion years" Nick Bostrom Department of Philosophy, Logic and Scientific Method London School of Economics http://www.hedweb.com/nickb n.bostrom@lse.ac.uk	{"url":"http://extropians.weidai.com/extropians.3Q98/1824.html","timestamp":"2024-11-10T05:17:35Z","content_type":"text/html","content_length":"7062","record_id":"<urn:uuid:f8f1b192-34dd-4c04-b648-328863de4ffa>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00240.warc.gz"}
Understanding Mathematical Functions: Is An Absolute Value A Function Introduction to Mathematical Functions and Absolute Value Mathematical functions are a fundamental concept in various fields such as science, engineering, and finance. They represent the relationship between input and output values, and they play a crucial role in modeling real-world phenomena, making predictions, and solving practical problems. Clarify what mathematical functions are and their importance in various fields such as science, engineering, and finance Mathematical functions can be described as a relationship between an input set and an output set, where each input value is related to exactly one output value. They are essential in various fields such as science, where they are used to describe natural phenomena, engineering, where they are utilized to design and analyze systems, and finance, where they help in making investment decisions and managing risk. Introduce the concept of absolute value and its distinctive properties Absolute value is a mathematical concept that represents the magnitude of a real number, disregarding its sign. For instance, the absolute value of -5 is 5, and the absolute value of 7 is also 7. It is denoted by vertical bars surrounding the number, as in \|x\|. The distinctive property of absolute value is that it always returns a non-negative value, irrespective of the sign of the input. This property makes it a valuable tool in various mathematical and real-world applications, such as distance calculations and optimization problems. Preview the discussion on whether the absolute value is considered a function and the implications of understanding this Now that we understand the basics of mathematical functions and the concept of absolute value, it is essential to dive into a discussion on whether the absolute value can be considered a function. The implications of understanding this concept will help us grasp the nature of absolute value and its role in mathematical operations and problem-solving. Key Takeaways • Absolute value is a mathematical function. • It returns the distance of a number from zero. • It always returns a positive value. • It can be represented as \|x\|. • Absolute value is a one-to-one function. The Definition of a Function in Mathematics In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are fundamental to the field of mathematics and are used to describe various real-world phenomena and mathematical concepts. A. Formal Definition of a Function The formal definition of a function involves two main components: the input variable and the output variable. A function is denoted by the symbol f, and it takes an input value x and maps it to an output value f(x). In mathematical notation, this relationship is expressed as f: x → f(x), where x is the input variable and f(x) is the output variable. B. Role of Functions in Mapping One of the key characteristics of a function is that it maps every input to exactly one output. This means that for each value of the input variable x, there is a unique corresponding value of the output variable f(x). In other words, a function cannot have multiple outputs for a single input, and it cannot have any inputs that do not produce an output. C. Examples of Functions To illustrate the concept of functions, consider the following simple examples: • The function f(x) = 2x, where the input x is mapped to its double as the output. • The function g(x) = x^2, where the input x is squared to produce the output. • The function h(x) = \|x\|, where the input x is mapped to its absolute value as the output. These examples demonstrate how functions operate by taking an input value and producing a corresponding output value according to a specific rule or relationship. Exploring the Concept of Absolute Value Understanding mathematical functions is essential in the study of mathematics. One such function that is commonly encountered is the absolute value function. In this chapter, we will delve into the concept of absolute value, its notation, and how it can be visually represented with graphs. A. Define absolute value as a measure of distance from zero on the number line The absolute value of a number is a measure of its distance from zero on the number line. It is always a non-negative value, as distance is never negative. For example, the absolute value of 5 is 5, and the absolute value of -3 is also 3. B. Describe the notation of absolute value with examples The absolute value of a number is typically denoted by vertical bars on either side of the number. For instance, the absolute value of x is written as \|x\|. This notation indicates that the result will always be positive, regardless of the sign of the input. For example, \|5\| = 5 and \|-5\| = 5. Another way to express the absolute value function is through a piecewise function. It can be defined as: • f(x) = x, if x ≥ 0 • f(x) = -x, if x < 0 This definition shows that for non-negative values of x, the absolute value function returns x itself, while for negative values of x, it returns the negation of x. C. Discuss how absolute value functions can be visually represented with graphs Graphically, the absolute value function is represented as a V-shaped graph. The vertex of the V is at the point (0, 0), and the arms of the V extend upwards and downwards from this point. The graph reflects the property that the absolute value of a number is always positive, and it also shows the symmetry of the function about the y-axis. For example, the graph of \|x\| looks like the letter V, with the arms extending upwards from the point (0, 0) for positive values of x, and downwards for negative values of x. This visual representation helps in understanding the behavior of the absolute value function and its relationship with the input values. The Formal Criteria of Functions and Absolute Value Compliance When it comes to understanding mathematical functions, it is important to revisit the formal criteria that classify a relation as a function. In this context, we will break down how the absolute value meets these criteria and utilize a function test, such as the vertical line test, to demonstrate its compliance. A. Revisit the criteria that classify a relation as a function Before delving into the specific case of the absolute value, it is important to revisit the criteria that determine whether a relation is a function. A relation is considered a function if each input value corresponds to exactly one output value. In other words, for every x-value, there can only be one y-value. This criterion is essential in distinguishing functions from non-functions. B. Break down how absolute value meets these criteria one by one Now, let's apply these criteria to the absolute value function, denoted as \|x\|. The absolute value function returns the distance of a number from zero on the number line, always yielding a non-negative result. When we examine the input-output relationship of the absolute value function, we find that for every input x, there is a unique output \|x\|. This satisfies the fundamental criterion of a function, as each input value corresponds to exactly one output value. Furthermore, the absolute value function is symmetric about the y-axis, meaning that the input x and -x yield the same output \|x\|. Despite this symmetry, the absolute value function still meets the criteria of a function, as each input value still maps to a unique output value. C. Utilize a function test, such as the vertical line test, to demonstrate absolute value's compliance with function criteria To further illustrate the compliance of the absolute value function with the criteria of a function, we can employ the vertical line test. The vertical line test states that a relation is a function if and only if every vertical line intersects the graph of the relation at most once. When we apply the vertical line test to the graph of the absolute value function, we find that every vertical line intersects the graph at most once, confirming that the absolute value function is indeed a function according to this test. Absolute Value as a Piecewise Function When it comes to mathematical functions, there are various types that serve different purposes. One type of function that is particularly interesting is the piecewise function. Piecewise functions are defined by different expressions depending on the input value, making them versatile and useful in a variety of mathematical applications. Introduce the concept of piecewise functions and how they are defined by different expressions depending on the input value A piecewise function is a function that is defined by multiple sub-functions, each applying to a different interval of the input. This means that the function's behavior can change depending on the value of the input. Piecewise functions are often used to model real-world situations where different rules or conditions apply in different scenarios. Explain that absolute value can be written as a piecewise function The absolute value function, denoted as \|x\|, is a classic example of a piecewise function. It is defined as follows: • For x ≥ 0, \|x\| = x • For x < 0, \|x\| = -x This means that the absolute value of a number is equal to the number itself if the number is non-negative, and it is equal to the negative of the number if the number is negative. Illustrate with examples how the absolute value function works for both positive and negative input values Let's consider a few examples to understand how the absolute value function works for different input values: Example 1: If x = 5, then \|x\| = 5, since 5 is a non-negative number. Example 2: If x = -3, then \|x\| = -(-3) = 3, since -3 is a negative number. Example 3: If x = 0, then \|x\| = 0, as 0 is neither positive nor negative. These examples demonstrate how the absolute value function behaves differently based on the sign of the input value, showcasing its piecewise nature. Real-world Applications of Absolute Value Functions Absolute value functions are not just theoretical concepts, but they have practical applications in various real-world scenarios. Let's explore some examples of how absolute value functions are used to solve problems in disciplines such as engineering and economics, troubleshoot common misconceptions, and emphasize their value in data analysis and computational fields. Provide practical examples of how absolute value functions are used to solve problems • Engineering: In engineering, absolute value functions are used to model physical quantities that cannot be negative, such as distance, time, or temperature. For example, in civil engineering, absolute value functions are used to calculate the distance between two points, regardless of their direction. • Economics: Absolute value functions are used in economics to represent situations where the magnitude of a change is more important than its direction. For instance, in cost-benefit analysis, absolute value functions are used to measure the impact of changes in production costs or consumer demand. Troubleshoot common misconceptions and issues in interpreting and applying absolute value functions in real-world scenarios • Directional Misconception: One common misconception is that absolute value functions only represent positive values. In reality, absolute value functions can also represent negative values, as they simply measure the distance from zero on the number line. • Application Errors: Another issue is the misapplication of absolute value functions in real-world scenarios. It's important to understand when and how to use absolute value functions to accurately model and solve problems in engineering, economics, and other fields. Emphasize the value of understanding absolute value functions in data analysis and other computational fields • Data Analysis: In data analysis, absolute value functions are used to measure the deviation of data points from a central value, such as the mean or median. This is crucial for understanding the spread and variability of data in statistical analysis. • Computational Fields: Absolute value functions are fundamental in computational fields such as computer science and machine learning. They are used in algorithms for optimization, error minimization, and distance calculations, playing a vital role in various computational applications. Conclusion: Wrapping Up the Understanding of Absolute Value as a Function & Best Practices A. Summarize how absolute value fits into the framework of mathematical functions Understanding absolute value as a function is essential in the study of mathematics. The absolute value function is a fundamental concept that represents the distance of a number from zero on the number line. It is denoted by \|x\| and returns the positive value of x, regardless of its sign. This function is crucial in various mathematical applications, including algebra, calculus, and B. Reinforce the impact of proper interpretation of absolute value functions on problem-solving and analysis Proper interpretation of absolute value functions is critical for accurate problem-solving and analysis in mathematics. By understanding the behavior and properties of absolute value functions, individuals can effectively solve equations, inequalities, and real-world problems. The ability to interpret absolute value functions correctly enhances mathematical reasoning and analytical skills. C. Offer best practices for working with absolute value functions, which include regular practice, visualizing functions, and applying them in varied contexts to deepen understanding When working with absolute value functions, it is essential to engage in regular practice to reinforce understanding and proficiency. Visualizing functions through graphs and geometric representations can provide valuable insights into the behavior of absolute value functions. Additionally, applying absolute value functions in varied contexts, such as physics, economics, and engineering, can deepen one's understanding of their practical significance and relevance. By incorporating these best practices, individuals can develop a strong foundation in working with absolute value functions, leading to improved problem-solving abilities and a deeper appreciation for the role of functions in mathematics.	{"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-is-absolute-value-a-function","timestamp":"2024-11-09T04:45:37Z","content_type":"text/html","content_length":"222697","record_id":"<urn:uuid:e93cd2f2-d192-4f9f-99b4-d78b343b3ec4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00706.warc.gz"}
Mensuration of Lines, Areas, Surfaces, and Volumes ... Dentro del libro Resultados 1-5 de 28 Página xv ... line may be produced to any length in a straight line . III . And that a circle may be described from any centre , at any distance from that centre . AXIOMS . I. Things which are equal to the same EUCLID'S DEFINITIONS . IV. Página xvii ... distance between two threads of a screw , or rather the distance between two conse- cutive threads . A tolerable idea of a screw may be formed by cutting an inclined plane out of paper and wrapping it round a cylinder whose length is ... Página xix ... distance of the attracted particle inversely . The whole fluid mass revolving about an axis in 23 h . 56 m . 4 s ... distances of a point on the surface from the centre , measured in the direction of the diameters re- spectively ... Página xxii ... distance along the chord . · 6. To find the deflection of the curve from the tangent to the circle , ellipse , and parabola . · • 7. Given the base and height of the segment of a circle , to find the length of its arc . • 8. Given the ... Página xxiv ... distance of the centre of gravity of the immersed and emerged volumes Statical stability of a ship • 110 · 112 • • 113 Atwood's theorem for calculating the value of the moment of stability Determination of the moment m m ' . v ... Términos y frases comunes Pasajes populares Página xv LET it be granted that a straight line may be drawn from any one point to any other point. Página xiii When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it. Página xii A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. " A plane angle is the inclination of two lines to one " another in a plane, which meet together, but are not Página xv An oblong is that which has all its angles right angles, but has not all its sides equal. Página xii When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight... Página xii A plane rectilineal angle is the inclination of two straight lines to one another, -which meet together, but are not in the same straight line. Página xiv Of three-sided figures, an equilateral triangle is that which has three equal sides. Página xvi If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles... Página xiii A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another : 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. Página xvi Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. Información bibliográfica	{"url":"https://books.google.co.ve/books?id=WDIDAAAAQAAJ&q=distance&dq=related:ISBN8474916712&lr=&output=html_text&source=gbs_word_cloud_r&cad=5","timestamp":"2024-11-07T02:55:30Z","content_type":"text/html","content_length":"50711","record_id":"<urn:uuid:e8e9992b-ae2f-444f-a860-b01a6cb5eb2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00662.warc.gz"}
Consistency of the LSE in Linear regression with stationary noise Consistency of the LSE in Linear regression with stationary noise Guy Cohen; Michael Lin; Arkady Tempelman Colloquium Mathematicae (2004) • Volume: 100, Issue: 1, page 29-71 • ISSN: 0010-1354 We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also ${L}_{p}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance). When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences. We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE. The results are applied to amplitude estimation in sums of harmonic signals with known frequencies. Guy Cohen, Michael Lin, and Arkady Tempelman. "Consistency of the LSE in Linear regression with stationary noise." Colloquium Mathematicae 100.1 (2004): 29-71. <http://eudml.org/doc/284040>. abstract = {We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_\{p\}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance). When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences. We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE. The results are applied to amplitude estimation in sums of harmonic signals with known frequencies.}, author = {Guy Cohen, Michael Lin, Arkady Tempelman}, journal = {Colloquium Mathematicae}, language = {eng}, number = {1}, pages = {29-71}, title = {Consistency of the LSE in Linear regression with stationary noise}, url = {http://eudml.org/doc/284040}, volume = {100}, year = {2004}, TY - JOUR AU - Guy Cohen AU - Michael Lin AU - Arkady Tempelman TI - Consistency of the LSE in Linear regression with stationary noise JO - Colloquium Mathematicae PY - 2004 VL - 100 IS - 1 SP - 29 EP - 71 AB - We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_{p}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance). When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences. We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE. The results are applied to amplitude estimation in sums of harmonic signals with known frequencies. LA - eng UR - http://eudml.org/doc/284040 ER - You must be logged in to post comments. To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.	{"url":"https://eudml.org/doc/284040","timestamp":"2024-11-12T09:48:55Z","content_type":"application/xhtml+xml","content_length":"39822","record_id":"<urn:uuid:d05a5670-5041-456a-af3f-eb79945ec5a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00411.warc.gz"}
A symmetrizable covariant derivative can be defined as follows: Note that the option SymCovDQ could also have been given to DefMetric. CD is now registered as a symmetrizable covariant derivative: This means that we can now use multipe incides to indicate symmetrize derivative:	{"url":"https://josmar493.dreamhosters.com/xTras/documentation/ref/SymCovDQ.html","timestamp":"2024-11-10T20:30:59Z","content_type":"application/xhtml+xml","content_length":"15677","record_id":"<urn:uuid:64a70d07-54ce-4461-bd72-81b8ad6a5cdf>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00656.warc.gz"}
REPOSITÓRIO UFOP :: Navegando ICEA - Instituto de Ciências Exatas e Aplicada por Autor "Alencar, David Santana Marques" ICEA - Instituto de Ciências Exatas e Aplicada URI Permanente desta comunidade Navegando ICEA - Instituto de Ciências Exatas e Aplicada por Autor "Alencar, David Santana Marques" Agora exibindo 1 - 3 de 3 Resultados por página Opções de Ordenação • Droplet finite-size scaling of the contact process on scale-free networks revisited. (2023) Alencar, David Santana Marques; Alves, Tayroni Francisco de Alencar; Ferreira, Ronan Silva; Alves, Gladstone de Alencar; Macedo Filho, Antonio de; Lima, Francisco Welington de Sousa We present an alternative finite-size scaling (FSS) of the contact process on scale-free networks compatible with mean-field scaling and test it with extensive Monte Carlo simulations. In our FSS theory, the dependence on the system size enters the external field, which represents spontaneous contamination in the context of an epidemic model. In addition, dependence on the finite size in the scale-free networks also enters the network cutoff. We show that our theory reproduces the results of other mean-field theories on finite lattices already reported in the literature. To simulate the dynamics, we impose quasi-stationary states by reactivation. We insert spontaneously infected individuals, equivalent to a droplet perturbation to the system scaling as N⁻¹. The system presents an absorbing phase transition where the critical behavior obeys the mean-field exponents, as we show theoretically and by simulations. However, the quasi-stationary state gives finite-size logarithmic corrections, predicted by our FSS theory, and reproduces equivalent results in the literature in the thermodynamic limit. We also report the critical threshold estimates of basic reproduction number R₀ λc of the model as a linear function of the network connectivity inverse 1/z, and the extrapolation of the critical threshold function for z→∞ yields the basic reproduction number R₀ = 1 of the complete graph, as expected. Decreasing the network connectivity increases the critical R₀ for this model. • Epidemic outbreaks on random Voronoi–Delaunay triangulations. (2020) Alencar, David Santana Marques; Alves, Tayroni Francisco de Alencar; Alves, Gladstone de Alencar; Macedo Filho, Antonio de; Ferreira, Ronan Silva We study epidemic outbreaks on random Delaunay triangulations by applying the Asynchronous SIR (susceptible–infected–removed) dynamics coupled to two-dimensional Voronoi–Delaunay triangulations. In order to investigate the critical behavior of the model, we obtain the cluster size distribution by using Newman–Ziff algorithm, allowing to simulate random inhomogeneous lattices and measure any desired observable related to percolation. We numerically calculate the order parameter, defined as the wrapping cluster density, the mean cluster size, and Binder cumulant ratio defined for percolation in order to estimate the epidemic threshold. Our findings suggest that the system falls into two-dimensional dynamic percolation universality class and the quenched random disorder is irrelevant, in agreement with results for classical percolation. • Opinion dynamics systems on barabási-albert networks : biswas-chatterjee-sen model. (2023) Alencar, David Santana Marques; Alves, Tayroni Francisco de Alencar; Alves, Gladstone de Alencar; Macedo Filho, Antonio de; Ferreira, Ronan Silva; Lima, Francisco Welington de Sousa; Plascak, João Antônio A discrete version of opinion dynamics systems, based on the Biswas–Chatterjee–Sen (BChS) model, has been studied on Barabási–Albert networks (BANs). In this model, depending on a pre-defined noise parameter, the mutual affinities can assign either positive or negative values. By employing extensive computer simulations with Monte Carlo algorithms, allied with finite-size scaling hypothesis, second-order phase transitions have been observed. The corresponding critical noise and the usual ratios of the critical exponents have been computed, in the thermodynamic limit, as a function of the average connectivity. The effective dimension of the system, defined through a hyper-scaling relation, is close to one, and it turns out to be connectivity-independent. The results also indicate that the discrete BChS model has a similar behavior on directed Barabási–Albert networks (DBANs), as well as on Erdös–Rènyi random graphs (ERRGs) and directed ERRGs random graphs (DERRGs). However, unlike the model on ERRGs and DERRGs, which has the same critical behavior for the average connectivity going to infinity, the model on BANs is in a different universality class to its DBANs counterpart in the whole range of the studied connectivities.	{"url":"https://repositorio.ufop.br/browse/author?scope=59790cd8-cb9e-4112-8ef2-562121017aad&value=Alencar,%20David%20Santana%20Marques","timestamp":"2024-11-07T13:57:25Z","content_type":"text/html","content_length":"363866","record_id":"<urn:uuid:8144e2e8-5224-4558-a862-4ae28d7befdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00877.warc.gz"}
5 Best Ways to Find the Smallest Positive Integer That Cannot Be Represented as a Sum from an Array in Python π ‘ Problem Formulation: Imagine you are given an array of distinct positive integers. Your task is to identify the smallest positive integer that cannot be generated by summing any subset of the array’s elements. For example, if the input array is [1, 2, 3], the smallest positive integer that cannot be represented as a sum of its subsets is 7. Method 1: Incremental Construction This method involves sorting the array and iteratively determining the smallest integer that cannot be formed. For an array arr, we maintain the smallest integer res that we can achieve, starting from 1. As we iterate through the array, we update res if the array element is not larger than it. Here’s an example: def find_smallest_integer(arr): res = 1 for x in arr: if x > res: res += x return res arr = [1, 2, 6, 10, 11, 15] Output: 4 This code snippet starts by sorting the array to assure the natural order. Then, it defines the res variable which tracks the smallest sum that cannot be created. The loop checks each array element, and if the element is greater than res, it stops as it is impossible to create res with the given elements. The outcome will be the smallest integer not representable by any subset sum. Method 2: Using Dynamic Programming Dynamic programming can be used to solve this problem by creating a table of boolean values indicating whether a sum can be formed with a subset of the array. While this method is more complex, it provides a detailed understanding of all possible sums. Here’s an example: def find_smallest_integer_dp(arr): max_val = sum(arr) dp = [False] * (max_val + 1) dp[0] = True for num in arr: for i in range(max_val, num-1, -1): if dp[i - num]: dp[i] = True for i in range(1, max_val + 1): if not dp[i]: return i return max_val + 1 arr = [1, 3, 6, 10, 11, 15] Output: 2 In this method, a boolean list dp is created to keep track of which sums can be constructed from subsets of arr. The list is updated for each number in arr by iterating backwards. Finally, it iterates through dp to find the smallest positive integer that cannot be formed, which is the answer. Method 3: Brute Force with Optimization The brute force approach tries to construct all possible sums of subsets and identify the smallest integer missing. This method has optimization checks to minimize the number of sums calculated, by skipping sums that exceed the current smallest integer not found. Here’s an example: from itertools import combinations def find_smallest_integer_bf(arr): possible_sums = set([0]) for r in range(1, len(arr) + 1): for subset in combinations(arr, r): if min(set(range(1, sum(arr))) - possible_sums) > max(subset): smallest_integer = 1 while smallest_integer in possible_sums: smallest_integer += 1 return smallest_integer arr = [1, 2, 5, 10, 20, 40] Output: 4 This code uses Python’s itertools.combinations function to generate all possible subsets of the array and their sums. Then, it identifies the smallest positive integer not present in the set of sums. The optimization check is that for each subset size r, if we encounter a number that’s larger than the smallest missing integer, we break early and move to the next subset size. Method 4: Greedy Strategy with Set Operations The greedy method works similarly to Method 1, but it utilizes set operations to keep track of achievable sums. This allows for a smaller set of numbers to consider at each step, potentially reducing the time complexity. Here’s an example: def find_smallest_integer_greedy(arr): achievable_sums = {0} for num in arr: new_achievable_sums = set() for s in achievable_sums: new_achievable_sums.add(s + num) achievable_sums \|= new_achievable_sums if min(set(range(1, max(achievable_sums)+2)) - achievable_sums) > num: return min(set(range(1, max(achievable_sums)+2)) - achievable_sums) arr = [1, 1, 1, 10, 20] Output: 5 The greedy strategy sorts the array and uses set operations to accumulate achievable sums as the array is processed. This method checks, after each new element is added, if the smallest integer not achievable is larger than the current number. If so, it stops adding new sums because the answer has been found. Bonus One-Liner Method 5: Using Sort and Accumulate This condensed approach leverages Python’s built-in functions to sort the array and then uses the itertools.accumulate to expand each possible sum in a compact manner to determine the missing Here’s an example: from itertools import accumulate def find_smallest_integer_oneline(arr): return next(x for x in range(1, sum(arr)+2) if x not in accumulate(sorted(arr))) arr = [3, 1, 6, 4, 2] Output: 22 This one-liner first sorts the array and then accumulates all possible sums using itertools.accumulate. It then finds the next integer, starting from 1, that is not present in the accumulated sums, which is the answer to our problem. • Method 1: Incremental Construction. Strengths: Efficient and intuitive. Weaknesses: Depends on sorting, which may affect time complexity if the original order is important. • Method 2: Dynamic Programming. Strengths: Provides detailed information on all possible sums. Weaknesses: Space complexity is potentially large, and overkill for large sums. • Method 3: Brute Force with Optimization. Strengths: Conceptually simple and straightforward. Weaknesses: Computationally intensive, especially for larger arrays. • Method 4: Greedy Strategy with Set Operations. Strengths: Can be faster than dynamic programming with better space efficiency. Weaknesses: Still requires sorting, and premature breaking might miss some cases if not implemented correctly. • Bonus One-Liner Method 5: Using Sort and Accumulate. Strengths: Compact and utilizes powerful built-in functions. Weaknesses: May not be as readable or understandable to beginners, and can be less efficient due to lack of early breaking conditions.	{"url":"https://blog.finxter.com/5-best-ways-to-find-the-smallest-positive-integer-that-cannot-be-represented-as-a-sum-from-an-array-in-python/","timestamp":"2024-11-03T07:42:44Z","content_type":"text/html","content_length":"72946","record_id":"<urn:uuid:cb8dea2b-d673-4777-a48b-c9fa7e6c4996>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00308.warc.gz"}
Focused Principal Component Analysis The Principal Component Analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables. … see more In particular, the Focused Principal Component Analysis (fPCA) conveys the structure of a correlation matrix into a low-dimensional diagram but, unlike PCA, it makes it possible to represent accurately the correlations of a given variable with the other variables (and even to test graphically the hypothesis that one of these correlations is equal to zero). In sum, the fPCA provides a focused outcome, so that the distances between the predictors and the outcome can be interpreted as a representation of their correlations. The relative positions of the predictors can give an idea of their correlations and can be interpreted as in a classic PCA. The iris dataset is used to exemplify the use of the fPCA method. The dataset contains four measurements for 150 flowers representing three species of iris (i.e. setosa, versicolor and virginica).… see more setosa versicolor virginica The first step which is usually taken when analysing data is to quickly find a linear correlation. However, when the data contains several factors, the correlation may be unclear and the next step consists on finding clusters between the data to complement the correlation. The following figure shows that there are clearly clusters between versicolor and virginica (green and blue), and a linear correlation between Petal.Length, Petal.Width. > summary(mod) lm(formula = iris$Species ~ iris$Sepal.Length + iris$Sepal.Width + iris$Petal.Length + iris$Petal.Width) Min 1Q Median 3Q Max -0.59215 -0.15368 0.01268 0.11089 0.55077 Estimate Std. Error t value Pr(>\|t\|) (Intercept) 1.18650 0.20484 5.792 4.15e-08 * iris$Sepal.Length -0.11191 0.05765 -1.941 0.0542 . iris$Sepal.Width -0.04008 0.05969 -0.671 0.5030 iris$Petal.Length 0.22865 0.05685 4.022 9.26e-05 * iris$Petal.Width 0.60925 0.09446 6.450 1.56e-09 * Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2191 on 145 degrees of freedom Multiple R-squared: 0.9304, Adjusted R-squared: 0.9285 F-statistic: 484.5 on 4 and 145 DF, p-value: < 2.2e-16 As seen, the linear correlation suggests these four variables as predictors with R-squared > 90%. This values indicates how well a regression model predicts responses for the reference observation. But it needs to be used with caution at all times. It might be the case—and this a very common [DEL:error:DEL] case in descriptive statistics—that the correlation found can be considered as a well fitting method to generate an equation. But in reality, the method is unable to explain the true relationship between the variables. If the purpose of the preliminary analysis is to provide an overview of the dataset, then the fPCA method is a good candidate due to its ability to provide both correlations and clusters in a fast In brief, the variables inside the dashed line are significantly associated with the focused outcome (Species). Green dots correspond to positive associations, and yellow dots to negative associations. Dots in opposite quadrants represent negative correlation between them. And finally, the red line represents a significant correlation with the outcome; this is also given by the r values in the axis. As seen in the figure, the correlation between Petal.Length and Petal.Width is maintained in this method; the lm model supports the further explanation of the correlation. It can also be seen that Sepal.Width is the less accurate predictor as found in both lm and fPCA methods; however, in the former the variable is neglectable (for the model) but in the later is considered because of its relation with the other variables. This is a relevant attribute of the fPCA method. 699 Words 2016-11-10 10:55 +0000	{"url":"https://www.guxsousa.com/posts/2016/11/focused-principal-component-analysis/","timestamp":"2024-11-03T10:25:16Z","content_type":"text/html","content_length":"18234","record_id":"<urn:uuid:cf2a4ba4-8d4f-4dc7-b7d6-3449a839f872>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00891.warc.gz"}
Figure 2-30. Equal-pitch hip roof framing diagram. HIP Most hip roofs are equal pitch. This means the angle of slope on the roof end or ends is the same as the angle of slope on the sides. Unequal-pitch hip roofs do exist, but they are quite rare. They also require special layout methods. The unit length rafter table on the framing square applies only to equal-pitch hip roofs. The next paragraphs discuss an equal-pitch hip roof. The length of a hip rafter, like the length of a common rafter, is calculated on the basis of bridge measure multiplied by the total run (half span). Any of the methods previously described for a common rafter may be used, although some of the dimensions for a hip rafter are different. Figure 2-30 shows part of a roof framing diagram for an equal-pitch hip roof. A roof framing diagram may be included among the working drawings; if not, you should lay one out for yourself. Determine what scale will be used, and lay out all framing members to scale. Lay the building lines out first. You can find the span and the length of the building on the working drawings. Then, draw a horizontal line along the center of the span. In an equal-pitch hip roof framing diagram, the lines indicating the hip rafters (AF, AG, BI, and BK in figure 2-30) form 45° angles with the building lines. Draw these lines at 45°, as shown. The points where they meet the center line are the theoretical ends of the ridge piece. The ridge-end common rafters AC, AD, AE, BH, BJ, and BL join the ridge at the same points. A line indicating a rafter in the roof framing diagram is equal in length to the total run of the rafter it represents. You can see from the diagram that the total run of a hip rafter (represented by lines AF-AG-BI-BK) is the hypotenuse of a right triangle with the altitude and base equal to the total run of a common rafter. You know the total run of a common rafter: It is one-half the span, or one-half the width of the building. Knowing this, you can find the total run of a hip rafter by applying the Pythagorean theorem. Let s suppose, for example, that the span of the building is 30 feet. Then, one-half the span, which is the same as the total run of a common rafter, is 15 feet. Applying the Pythagorean theorem, the total run of a hip rafter is: What is the total rise? Since a hip rafter joins the ridge at the same height as a common rafter, the total rise for a hip rafter is the same as the total rise for a common rafter. You know how to figure the total rise of a common rafter. Assume that this roof has a unit of run of 12 and a unit of rise of 8. Since the total run of a common rafter in the roof is 15 feet, the total rise of common rafter is the value of x in the proportional equation 12:8::15:x, or 10 feet.	{"url":"https://constructionmanuals.tpub.com/14044/css/Hip-72.htm","timestamp":"2024-11-07T03:38:31Z","content_type":"text/html","content_length":"25297","record_id":"<urn:uuid:43e58297-c19a-4d40-bbd8-72c3d18f700d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00451.warc.gz"}
The Rule of 72 - Definition \| Formula \| Example & Uses \| Calculation What is the Rule of 72? Definition: The rule of 72 is a mathematical way to estimate the number of years it will take for your money to double with compounding interest. In other words, it’s a simplified method to figure out how long your money has to be invested in order to double at a given interest rate. What is the Rule of 72 Used For? Investors often use this calculation when evaluating the difference between similar investments. They want to see their investments grow, so they can take the proceeds to invest in more opportunities in the future. Keep in mind that this doesn’t have to be Wall Street investors or brokers. Average Americans can use this method to estimate the amount of money they will have in a retirement account or how much their share in a mutual fund will be worth in five years. The rule 72 will calculate how long it takes to double your money in an investment. In other words, it’s a simplified, very limited future value calculator that will compute the value of your investment in the This formula is a great shortcut because the full-length investment equation for compounding interest is long and complicated. You can use this simple rule of thumb as a base estimate for investments. Here’s how it works. The rule of 72 formula is calculated by multiplying the investment interest rate by the number of years invested with the product always equal to 72. Applying a little bit of algebra we can rearrange the rule of 72 equation to calculate the number of years required to double your money with a given interest rate compounded annually. Or it can be written like this to calculate the annual compounded interest rate required to double your investment in a given time period. Keep in mind that the rule of 72 definition requires that the interest be compounded annually. This method will not work for investments with semi-annual or quarterly compounded interest as is. If you want to use this method for investment returns like that, you will need to modify it. Let’s take a look at an example. Here’s an example table of the way a rule of 72 calculator works. As you can see, the first column represents the annual rate of investment that will be compounded at the end of every year. The second column shows the number of years it will take for the investment to double in value. The third column is always 72 because that’s how the formula works. The investment rate multiplied by the number years is always equal to seventy-two. Let’s assume you have $10,000 to invest in a mutual fund and you want to know how long it will take to become $20,000. You are positive that you can get an average return of 8 percent each year. Looking at our table above, we can see that it will take your investment about 9 years to reach the $20,000 goal. We can also do the reverse calculation. Let’s assume you have $10,000 and you want to know what annually compounded interest rate you will need to double your money in 5 years. Going back to our table, you can see it will require an interest rate of a little over 14 percent to meet your $20,000 goal in a 5-year span. How is the Rule of 72 Used? Obviously, we could have used the equation to calculate each of these examples, but I figured the table would be easier. We can also use a future value calculator or the actual future value formula to verify that these numbers are accurate, but we don’t have to. This method works. It’s a great shortcut because allows you to easily estimate the value of your investment into the future without the technical details of the actual future value equation. Depending on the interest rate, you can probably do the calculations in your head. One thing to keep in mind is that this method does not take into consideration other future factors that could derail your investment plans. For instance, inflation rates might change over the course of your investment’s life. You can however use the rule of 72 to calculate the effects of inflation on your money. For example, if the inflation rate went from 3 percent to 4 percent, your money will lose half of its value in 18 years instead of 24 years. You can even compare the rise of current costs like tuition and medical expenses with the rate of interest. This is pretty could. Test it out for yourself. You can use it for all kinds of interesting future value calculations. Accounting & CPA Exam Expert Shaun Conrad is a Certified Public Accountant and CPA exam expert with a passion for teaching. After almost a decade of experience in public accounting, he created MyAccountingCourse.com to help people learn accounting & finance, pass the CPA exam, and start their career.	{"url":"https://www.myaccountingcourse.com/financial-ratios/the-rule-of-72","timestamp":"2024-11-02T22:19:32Z","content_type":"text/html","content_length":"173888","record_id":"<urn:uuid:b8271c8b-2ec9-43ec-bce1-b8cb6fb19036>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00134.warc.gz"}
Regular Expression HI All, :D I'm computer science student. Would you like to tell me about that problem? Here. 1) RE for accepts all strings of (a,b,c) that contains an odd number of a's. 2)Re for the language of all binary of strings that have at least three symbols and whose first and last symbols are different. Simplify the following REs (that is ,find the simpler RE for the same language): a.(r + Є), b.SS+Є , The hint also okay for me. How much time you have spent on these problems? Now I got it thanks Rashakil Fol . But I have a lot of questions in Regular Languages. 1) Show that the set of regular languages is closed under reversal. That is,if L is the regular, then so is { x R :x € L} where x R denotes the reversal of string r. 2)Give the example to show each of following for the language L1 and L2: a)If L1 belongs to L2 and L2 is regular ,then L1 can be regular or nonregular. b)If L1 and L2 are nonregular,then L1∏L2 can be regular or nonregular. c)If L1 and L2 are nonregular,then L1 U L2 can be regular or nonregular. 3)Suppose language L is accepted by FA M.Let L E be the subset of L consisting of those strings in L of even length.Show how to convert M to an FA for L E. 4)Prove the set of all strings of a and b with more a’s than b’s is nonregular. 5)For each of the following languages ,state whether it is regular or not .If not ,give a proof that it is nonregular. a)The set of binary strings with equal number of occurrences of the substring 01 and 10. b)The set of binary nonpalindromes. Help me please. Thanks for your time. c){a2 n:n>0} Let us use the notation x = L y to mean the strings x and y are indistinguishable with respect to language L. a) Show that =L is an equivalence relation that is for all stings x,y,z the following hold (ii)If x=Ly ,then y=L x (iii)if x=L y and y =L z ,then x= L z b)Show that =L is the right congruence ,that is , for all the string x,y,z the following holds: If x=Ly then xz=Lyz. go away google regular expressions + the language your codeing(sp?) in and maybe crack open the text book thats 1st on the recommended reading list for your class, its really not that complicated the main part of the programs going to be the same each time (namely processing the string character by character and performing analysis on it, you just have to modify the peramiters each time. Anyway try it yourself people don't like doing your homework for you if your still stuck come bac post here asking about what your stuck with (not the whole dahm assignment) and what language your using. a) L1={empty set } << regular L1 belongs to L2.so L1 can be regular. b)The intersection of a non-regular language and its complement is empty, and the empty language is regular. c)The Union of any languages and it's complements is Ʃ * which is regular. Correct me if I wrong and still thinking about others. These are to be construed as hints; you must still show the work involved in actually solving the problem. 1) Show that the set of regular languages is closed under reversal. That is,if L is the regular, then so is { x R :x € L} where x R denotes the reversal of string r. Hint: Every regular language is accepted by some NFA, and every language accepted by an NFA is a regular language. Assume an NFA for L, and describe how you would construct the NFA for rev(L). 2)Give the example to show each of following for the language L1 and L2: a)If L1 belongs to L2 and L2 is regular ,then L1 can be regular or nonregular. S* is a regular language. b)If L1 and L2 are nonregular,then L1∏L2 can be regular or nonregular. The empty language is regular. c)If L1 and L2 are nonregular,then L1 U L2 can be regular or nonregular. L union complement(L) = S* 3)Suppose language L is accepted by FA M.Let L E be the subset of L consisting of those strings in L of even length.Show how to convert M to an FA for L E. What if you broke each state Q into two states Q_E and Q_O? 4)Prove the set of all strings of a and b with more a’s than b’s is nonregular. Pumping lemma for regular languages. 5)For each of the following languages ,state whether it is regular or not .If not ,give a proof that it is nonregular. a)The set of binary strings with equal number of occurrences of the substring 01 and 10. Pumping lemma for regular languages. Perhaps (01)^n(10)^n? b)The set of binary nonpalindromes. Help me please. Thanks for your time. The complement of a regular language is a regular language. c){a2 n:n>0} No idea what that means. Let us use the notation x = L y to mean the strings x and y are indistinguishable with respect to language L. a) Show that =L is an equivalence relation that is for all stings x,y,z the following hold (ii)If x=Ly ,then y=L x (iii)if x=L y and y =L z ,then x= L z Use the definition of =L, whatever one you're using. Post your definition if you want more help. Is it delta(Q,w)? b)Show that =L is the right congruence ,that is , for all the string x,y,z the following holds: If x=Ly then xz=Lyz. Again, use the definition of =L. Probably a delta(Q,w)-esque definition. Disclaimer: I take no responsibility for the appropriateness of these hints. These are offhand and upon actually doing the problem I might realize that changing tacts is in order. Thanks for your hints. You make my day. Reply to this Topic	{"url":"https://www.daniweb.com/programming/computer-science/threads/297416/regular-expression","timestamp":"2024-11-13T09:14:40Z","content_type":"text/html","content_length":"85458","record_id":"<urn:uuid:04651359-7602-4e70-a3e1-a488569524c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00049.warc.gz"}
Physicist offers a new take on 160 year old math problem The Riemann hypothesis -- an unsolved problem in pure mathematics – is one of the seven Millennium Prize Problems, with a $1 million prize to the person who solves it. But that’s not why it fascinates mathematical physicist Andre’ LeClair, for whom this is perhaps the most important open question in mathematics. He illustrates the hypothesis’ importance to number theory with a well-known anecdote: “Someone asked a famous mathematician if he went to sleep and woke up 100 years from now, what would his first question be? Hilbert answered, ‘Has the Riemann hypothesis been proven?’” The Riemann hypothesis, proposed by Bernhard Riemann in 1859, has to do with the distribution of prime numbers, which he precisely related to the locations of “non-trivial zeros” of the Riemann zeta function. A proof – or disproof – of this 150-year-old hypothesis would have major implications in number theory. “The question this hypothesis answers is where the zeros are, and you can determine the prime numbers from these zeros. Since all encryption uses prime numbers, this is important. If the Riemann hypothesis turned out to be false it would cause chaos with the distribution of prime numbers, as many mathematicians have noted.” LeClair has been working on the hypothesis over the past 5 years. He has delivered talks on his approach at the Riemann Center in Hannover, Germany, at the Isaac Newton Center for Mathematical Sciences, in Cambridge, UK, Ecole Normal Superieur in Paris and at Stanford University; he has published four papers on the topic, two in pure math journals, including Communications in Number Theory and Physics, and Communications in Contemporary Mathematics, and the most recent in Journal of Statistical Mechanics: Theory and Experiment. Some of LeClair’s work was done with Guilherme Franca, who was a postdoctoral associate at Cornell; his most recent work was with Giuseppe Mussardo (SISSA, Italy). There’s no obviously clear strategy towards a proof of the Riemann hypothesis, although some have been proposed. LeClair’s work offers a strategy, a “heuristic argument” rather than a straight proof. “We make one assumption, a fully plausible conjecture. From this one assumption we show that a lot of things would follow. We don’t have a 100% rigorous proof, but rather a very promising strategy with connections to physics,” says LeClair. “Even if we can’t fully prove our main assumption, we’ve shed light on where to look.” As mathematician Steve Gonek (University of Rochester) notes, before the 20th century, mathematics and physics were more unified: Riemann himself studied physical problems as well as “pure” mathematical ones. “Mathematics and physics divided and became more specialized of necessity because the explosion of scientific knowledge made it almost impossible to master even one of them,” Gonek says. “Nevertheless, even today both subjects share many of the same tools and points of view. In fact, there are countless examples of mathematical physicists contributing wonderful insights into mathematical questions, and pure mathematicians helping to understand deep physical questions...while categorization into subject areas is useful, we should not be fooled by it: knowledge transcends LeClair drew on some frequently used concepts in physics for his work, such as the “random walk” to deal with the apparent randomness of prime numbers. Although the primes may appear random, they are deterministic. Nevertheless certain series of some number theoretic functions can behave randomly. The randomness still needs to be characterized precisely mathematically. So LeClair formulated the randomness of prime numbers in terms of a series of steps of arithmetical functions over the primes that behave like a random walk, which physicists illustrate by how a drunk walks: when he takes one step forward, whether he takes another step forward or steps backward is entirely random. This interplay between determinism and randomness is a key aspect of his work. “This random walk concept lets us prove a lot of things and enables us to derive an equation that relates every single zero of the Riemann hypothesis to a sum of prime numbers,” explains LeClair. And while the original Riemann hypothesis was about one mathematical function, it is part of an infinite class of functions – and LeClair’s reasoning, he notes, works for all of them, including Dirichlet L-functions and functions based on cusp modular forms, which is referred to as the Grand Riemann Hypothesis.	{"url":"https://physics.cornell.edu/news/physicist-offers-new-take-160-year-old-math-problem","timestamp":"2024-11-13T02:08:42Z","content_type":"text/html","content_length":"59587","record_id":"<urn:uuid:e62b7083-489e-4f50-9f9b-e5fc28d75ef0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00149.warc.gz"}
Shoe Treads - The AppsLab Shoe Treads In December our team was allowed two weeks to pursue a “passion project”. It didn’t have be work-related, just something you truly wanted to work on. I chose to design tread patterns on the soles of I had never given a thought about shoe treads until, a week earlier, a woman on the NodeBox forum asked for help creating organic shoe patterns. I asked her to elaborate on what she meant by “organic” and she sent me a photograph that looked like a school of hexagon fish swimming in the shape of a shoe, with bigger fish in the center and smaller fish near the edges. I looked into it and discovered a vast art form that had been right under my feet the whole time. There are entire schools which teach shoe tread design and thousands of designs hitting the pavement every day, each one unique. If you doubt me, go to this Pinterest page and start scrolling. Scroll some more. Getting tired yet? Keep scrolling. Amazing, isn’t it? Redrawing a particular pattern wouldn’t be too hard; you could just trace over a photograph. But generating new vector-based patterns from scratch is not as easy. You would need to develop a whole language of core concepts (nodes) that could follow the sinuous curve of an arch, adapt to edges, grow and shrink based on position, soften sharp edges, etc. These are things which children can do instinctively, but which require significant math for computers. That is the problem which hooked me. It’s a problem I’ve run into before. Visualizations are powerful because the human visual system is powerful. Resolving a scene into component shapes, tracing edges, comparing areas, etc. is effortless for us – we do most of it subconsciously. I often get ideas for rendering data as shapes or lines that are easy to sketch on paper, but hard to codify into To illustrate this, I have identified six simple visual tasks that are easy for children but surprisingly tricky to do in NodeBox. Over the course of my two-week passion project I was able to develop a “node” for each one – and then used them to draw a few simple tread designs. 1. Finding the outside Given a set of objects (shapes, words, dots) arranged on a 2D canvas, it is often useful to draw a border that efficiency encloses the objects. Uses include highlighting one clump of related points in a larger scattergram, drawing “word clouds” that surround a set of words, etc. To do this you have to figure out which positions are the outside of the clump and which are inside. A child can do this just by looking, but how does a computer do it? The mathematical term for this is convex hull. It is considered one of the fundamental problems of computational geometry. The difficulty of finding a convex hull increases with the number of points to be enclosed. There are a number of known algorithms, each with its own tradeoffs. I used Andrew’s Monotone Chain. I now have a “wrap” node. Feed it a list of points and it returns the subset of exterior points arranged in clockwise order. You can feed that output into a connect node to draw a line that wraps all the objects and then into a scale node to create some breathing room. 2. Softening Corners A key requirement in making “organic” shapes is the ability to soften corners. You need to replace sharp corners with gentle curves and control the amount of “curviness”. This turns out to be a hard problem. The shapes I want to deal with are “Bsplines”, sequences of bezier curves. In order to turn a pointy shape into a rounded shape, you need to replace line segments with Bsplines and find the control points of each bezier curve in the sequence. Fortunately, I found an algorithm by Bernhard R. Fischer that does exactly that. It involves finding the tangent of each corner and extending control points along that tangent (the farther you go, the curvier it gets). I created a NodeBox node (subnetwork) called fit_curve that does all this. It’s not perfect, but it works well enough, and has become one of the most useful tools in my toolbox. 3. Finding Areas A persistent problem I ran into when I began trying to draw shoe soles was dust. (I encountered the same phenomenon when working with maps.) “Dust” refers to shapes created during complex intersection operations that are often too tiny to see with the naked eye. Dust particles can be so small that they cannot be softened and so screw up my calculations. To get rid of them I needed to delete all shapes below a certain area. But how do you calculate the area of random curvy shapes? Even calculus does not help much with this. The solution I came up with was to reduce the shape to a polygon and apply a centuries-old surveyor’s method called the shoelace formula. It involves taking cross-products of alternating X and Y positions in a back-and-forth way like tying a shoelace. Problem solved. 4. Tracing Edges Children love tracing edges. Start with a few simple shapes and draw outlines around each one. Then draw outlines around the outlines until they start to intersect. The result is psychedelic. A similar process is useful in creating topographic maps and sea-charts. By now you will not be surprised to hear that this is another hard problem for computers. A popular library for doing curve offsets requires more than three thousand lines of code. In fact, it is mathematically impossible to offset one bezier curve from another. The solution is to break the curve down into tiny line segments, calculate the tangent of each one, and use that information (plus my fit_curve node) to create a new curve. I can now trace the inside of a shoe sole with ease. 5. Magnifying a Grid A basic component of shoe tread patterns is intersecting lines or curves. But for that organic bigger-in-the-center, smaller-near-the-edges look, I needed a way to magnify these grids so that the center cell is enlarged, the grid cells around that are proportionally smaller, the grid cells around those are smaller still, and so forth. The effect looks simple and natural, but calculating the exact positions of each intersecting line in a way that shrinks distances around the center in a consistently proportional way while at the same time conserving the number of lines and maintaining a fixed outer boundary is, well, tricky. I derived a nasty-looking formula to represent this, but my calculus skills were too rusty to integrate it. I finally had to turn to a messy numerical method. There may be a simpler way of doing this that I could not see, but my accordion node works well enough for now. It’s also handy for creating natural-looking animations like a fan folding and 6. Seeing Shapes The final challenge was the most fundamental and goes to the heart of what makes our human visual system so magical. When presented with an array of intersecting lines, circles, etc. people instantly and effortlessly perceive the intersections as independent shapes. They can point to them, number them, and color them without a second thought. But to a computer there are no shapes there. It takes a lot of extra work to calculate all the positions lines or circles intersect and figure out which subsets of those intersections could form the outline of new shapes. This is what my fragment node now does. Feed it a boundary and a pattern of intersecting lines or curves and it returns a list of all the independent shapes formed as a result (minus the dust). If I feed it the three intersecting circles of a Venn diagram it will return all eight intersection shapes (including the shape outside the three circles). With the toolbox I developed over the course of my two-week “passion project” I can now trace the inside of the sole to define a canvas, apply a grid of curves to it, magnify that grid, pull out the resulting shapes, and soften them. I was only able to create a few simple proof of concept tread designs in the allotted time, but am now confident I could produce many more. Perhaps this will be a retirement project for me some day. In the meantime, I have already used some of these tools to create new visualizations at work. Passion projects like this may seem frivolous, but I’ve always found them an essential part of innovation. I have my own name for them: PONARVs: Projects Of No Apparent Redeeming Value. Stone Soup In the classic Grimm Brothers tale, returning soldiers convince selfish villagers to share their food by telling them they are making “stone soup”. They place a stone in a pot of water and start it boiling. One by one they get each suspicious villager to add an apparently inconsequential ingredient until a flavorful soup comes together that is more than the sum of its parts. My shoe soles project was like making stone soup. The shoe problem was the stone that defined the soup. But in the process of making it, I was forced to add one tasty node “ingredient” after another. In the end I created something much more wonderful – and useful – than shoe treads. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://theappslab.com/2018/02/01/shoe-treads/","timestamp":"2024-11-12T03:12:24Z","content_type":"text/html","content_length":"96177","record_id":"<urn:uuid:af1c6607-2445-41f0-aff7-ed836cc36c85>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00577.warc.gz"}
A Study of Gravitational Potential Energy When anything is released into open space, it tends to fall. This is a very regular occurrence in our daily lives. When everything is released into free space, it gravitates towards the Earth. The first person to recognise this was Sir Isaac Newton. He noticed an apple falling from the tree and began his investigation into its preferred downward motion. He came to the conclusion that not only does the Earth draw everything towards itself, but that everyone else in the universe does as well. This is a property that a body has as a result of its mass. Gravitation is a force that claims that all objects on Earth and in space are attracted to one another. The gravitational force exerted on an object is proportional to its mass; the greater the mass of an object, the greater the gravitational force exerted on it by other objects. All visible items, such as a pen, eraser, planets, mobile phone, watch, and refrigerator, are attracted to one other in some way. With the Electromagnetic Force and the Nuclear Force, gravity is one of the non-contact forces. Universal Law of Gravitation Newton was motivated to discover the connection between falling bodies and celestial motions, according to early versions, when he watched an apple fall from a tree and thought that if the gravitational force could reach above the ground to a tree, it could also reach the Sun. The idea of Newton’s apple is a part of mythology all around the world, and it may or may not be true. It is held in high regard because Newton’s universal law of gravitation and laws of motion answered long-standing issues about nature and supported the idea of nature’s underlying simplicity and unity. Scientists continue to hope that their continuous investigations into nature will reveal fundamental simplicity. The gravitational force is a straightforward force. It is always appealing, and its attractiveness is solely determined by the masses involved and the distance between them. Newton’s universal law of gravitation states, that every particle in the world attracts every other particle along a line connected by a force. The force they exert on each other is proportional to the product of their masses and inversely proportional to their separation. Gravitational Potential Energy Gravitational Potential Energy is defined as a body’s energy arising from the gravitational force, or when the gravitational force works on a body and produces gravitational potential energy. The amount of work done on the body by the force is now the change in potential energy if the position changes owing to force. Let’s suppose we have two items, one A and one B. Assume that B’s location changes only as a result of some force. Assume M’s location is shifting due to gravitational pull, with a corresponding shift in potential energy. The gravitational pull does the same amount of work as the change in the item’s location. As a result, the gravitational force will accomplish the same amount of work as the shift in potential energy from P to Q. Assume P is in the first position and Q is in the second. The effort required to move the item from position 1 to position 2 is equal to the potential energy change, which is equal to the potential energy at point 2 minus the potential energy at point 1. • Gravitational potential is the amount of work done per unit mass to move a body from infinity to a certain place. • It’s represented by the letter U. • The SI unit for gravitational potential is J/Kg. • The gravitational force gives rise to the potential body. The work done on the body by the force is the change in potential energy if the position changes as a result of the force. Gravitational Potential Energy Formula Mathematically, gravitational potential energy is the product of mass(m), acceleration due to gravity(g) and height(h) above the ground and it can be given as: Derivation of Gravitational Potential Energy When a particle travels an indefinitely short distance, dr. The gravitational force’s work on the second particle is represented by -Fdr. dW=-Fdr ——- (1) F is gravitational force and it is given by, F = gravitational force between the two bodies g = gravitational constant. Its value is 6.67×10-11Nm2/kg2 m1 and m2= masses of the two bodies r = distance between the centre of the two bodies Putting the value of F in equation (1) we get, W= ∞r[G×[(m1×m2)/r2]dr W= G×(m1×m2)∞r1/r2]dr W= G×(m1×m2)[(1/r)-(1/∞) As because work done is stored as potential energy U, gravitational potential energy at a distance ‘r’ from the source mass is calculated as follows: Gravitational potential energy refers to the work that a body needs to do against gravity in order to arrive at a specific position. In other words, gravitational potential energy is the amount of energy that an object has or gains as a result of a change in the position of its gravitational field. An item possesses some form of energy since energy cannot be produced or destroyed, even when it is at rest (Which is converted to kinetic energy, when it starts to move). This form of energy is referred to as potential energy.	{"url":"https://unacademy.com/content/cbse-class-11/study-material/physics/a-study-of-gravitational-potential-energy/","timestamp":"2024-11-14T18:33:21Z","content_type":"text/html","content_length":"644101","record_id":"<urn:uuid:8d625b32-bd52-470d-9cde-2e7c3b41b863>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00033.warc.gz"}
Backgammon glossary - WinAndFun.com Home: Backgammon: Glossary 0-9 A B C D E F G H I J K L M N O P Q R S T U V W A guideline for cube handling in pure race positions. If you add 10% to your pip count, you should double if the result is not more than two pips greater than the opponent's count, and you should redouble if the result is not more than one pip greater. Your opponent should accept the double if your count plus 10% is no more than two pips less than his count. A guideline for cube handling in pure race positions. You should double if the opponent's pip count exceeds yours by 8% or more, and redouble if it exceeds yours by 9% or more. Your opponent should accept the double if his pip count exceeds yours by no more than 12%. American Backgammon Tour, an annual master-point competition of participants in major U.S. tournaments. Website: ABT. To agree to continue playing a game at twice the previous stakes after the opponent offers a double. Compare: Refuse a Double. A rolled die showing the number 1. Traditional name for the one-point. A position in the late stages of a game in which a player is anchored on the opponent's one-point trying to hit a shot as the opponent brings his checkers home and bears them off. [Also spelled "Acey-Deucy" or "Acey-Ducey".] 1. The roll of 1 and 2 with two dice. 2. A backgammon variant in which the roll of 1 and 2 gives the player extra turns. In a chouette, the crew member who plays for the team against the box after the original captain has declined box's double and is no longer in the game. A play designed to provoke an exchange of hits, typically used after the opponent has escaped his runners. A position in which one player doubles based upon his immediate blot-hitting chances. A checker which is completely free to make another point. 1. An anchor on the opponent's five-point, four-point, or sometimes three-point. 2. An made point on the opponent's five-point, four-point, or bar-point. (Many authors include the bar-point, though it is technically not an anchor, because it functions much like an advanced anchor when playing a holding-game.) See: Holding Point. A player of considerable experience and skill who has moved beyond intermediate level. Having a lower pip count than your opponent; see count (2). Having a lower pip count than your opponent. An unexpectedly poor roll. A traditional chess clock with hands that show the time remaining for each player. It has a flag that falls to indicate when a player's time has expired. Analog clocks generally do not have a time delay feature, making them less suitable than digital clocks for use in backgammon. A point (1) occupied by two or more of your checkers in the opponent's home board. A recorded match with added analysis and commentary. Analysis and commentary about a backgammon game written after the game is played. A very bad roll; the opposite of a joker. The formation of a player's checkers as they work together to block and attack the opponent, then and come home safely. A move from the opponent's outer board to the player's outer board. A feature that contributes to the strength of a position, such as made points and flexibility. Compare: Liability. An optional rule in money play: If both players throw the same number on the first roll of a game, the stakes are doubled. The doubling cube is turned to 2 and stays in the middle. Players usually agree to limit the number of automatic doubles to one per game. A dice roll which forces a player to leave a shot or break a valuable point (2). [Also spelled "backgame".] A strategy employed by a player who is substantially behind in the race but has two or more anchors in the opponent's home board. The player holds both anchors as long as possible, forcing his opponent to bear in or bear off awkwardly. The idea is to hit a late shot and then contain the hit checker behind a prime. Compare: Holding Game. 1. A game played with dice and checkers on a board consisting of twenty-four points (1), in which each player tries to move his checkers home and bear them off while preventing the opponent from doing the same thing. 2. A completed game of backgammon (1) in which the losing player has not borne off any checkers and still has one or more checkers on the bar or in the winner's home board. A backgammon is also called a triple game because the winner receives three times the value of the doubling cube. Compare: Single Game and Gammon. Backgammon (1) is played on a board consisting of twenty-four narrow triangles called points (1). The triangles alternate in color and are grouped into four quadrants of six triangles each. The quadrants are referred to as a player's home board and outer board and the opponent's home board and outer board. The home and outer boards are separated from each other by a ridge down the center of the board called the bar. A computer on the Internet which hosts games of backgammon (1). Competitors play in real time with opponents from around the world. The server rolls the dice, communicates the plays to each player, keeps score, and maintains ratings for all players. Some servers even let you play for money. You typically interact with a server using client software downloaded to your computer. Runner; a player's rearmost checker. A device through which dice are dropped to randomize a roll. The dice are deflected and jostled about as they fall through the box. An early type of plastic, used in the 1920's and 1930's for the creation of backgammon playing pieces. Many people prefer the look and feel of bakelite to newer materials. [Because you must be "bananas" to try it.] To hit loose by breaking a point in your home board, thereby leaving two blots. The amount of money you have available for betting, or the maximum amount you are willing to lose in a session. See: Money Management. The raised ridge down the center of a backgammon board dividing the home board from the outer board. Checkers are placed on the bar after they have been hit. [Named after backgammon expert Rick Barabino.] A roll of 5-4 from the bar used to make an anchor on the opponent's five-point. A player's seven-point, so named because it is physically adjacent to the bar. A position in which both players have checkers trapped behind an opponent's prime. See: Prime-vs-Prime. To move a checker into your home board prior to bearing off. To remove a checker from the board according to a roll of the dice after all of your checkers have been brought into your home board. The last stage of the game during which checkers are borne off. A computer-generated table associating each possible bearoff position with a value that represents the quality of that position. The associated value is either the equity of the position (in a two-sided database) or a distribution of the expected number of rolls to bear off (in a one-sided database). To be within six points (1) of. For example, a checker on your 13-point bears on points 7 through 12. An immediate redouble by a player who just accepted a double. A player who beavers turns the cube up one level and retains possession of the cube. See: Beavers. A rule often used in money play (but never in match play) which says: A player who accepts a double may immediately redouble (beaver) without giving up possession of the cube. The opponent (the player who originally doubled) may refuse the beaver, in which case he resigns the game and loses the current (doubled) stakes. Otherwise, he must accept the beaver and continue the game at quadruple the stakes prior to the double. Having a higher pip count than your opponent; see count (2). Having a higher pip count than your opponent. To mistakenly play the roll of 6-5 from the opponent's one-point to your mid-point without seeing that the opponent has made his bar-point and blocks your way. British Isles Backgammon Association. Website: BIBA. A bold or aggressive play when a safer but less constructive play is available. 1. An all-out attack on enemy blots in your home board aimed at closing out your opponent. 2. A quick elimination tournament consisting of short matches. A point (1) occupied by two or more checkers held for the purpose of hindering the opponent's progress. A series of blocks arranged to prevent escape of the opponent's runners. The ideal blockade is a prime. A backgammon variant in which one checker by itself controls a point (1). A game plan where the primary strategy is to build a strong blockade. A single checker sitting alone on a point (1) where it is vulnerable to being hit. An exchange of loose hits in which both players try to gain a key point. A kind of collusion in a chouette. Two or more players silently agree to share their winnigs, thus if either of them is in the box and the other is captain, the captain deliberately makes bad moves or wrong doubling decisions. A large checker play or cube error, especially one made out of recklessness or inattention. Compare: Whopper. 1. A backgammon board. 2. One of the four quadrants that make up the playing area: your home board, your outer board, the opponent's home board, and the opponent's outer board. 3. A player's home board. For example: a strong board is a home board with several made points; an n-point board is a home board with n points made; to make your board means to close all the points in your home board. See: Starting Position. See: Starting Position. A play that leaves one or more blots that the opponent can easily hit. Compare: Safe Play. [Contraction of "robot."] 1. A computer program on a backgammon server that plays and competes just as if it were a human player. 2. Any computer program that can play backgammon (1) and analyze positions (such as Jellyfish, Snowie, or GNU Backgammon). [Short for "man in the box," a person in a difficult or trying position.] The player in a chouette who plays alone against all the others. A roll of 6-6 (double 6's). A roll of 6-6 (double 6's). A roll of 6-6 (double 6's). To take apart, as in break a point, break a prime, or break one's board. To remove a checker from a point (1) that contains only two checkers, leaving the point open. (The opposite of make a point.) To open one or more points (2) in a prime. To move past the last of the opponent's checkers, so that no further hitting or blocking is possible. The game becomes a pure race. To open one or more points (2) in your home board after having made your board. An incomplete prime with a gap in it. A chess clock with a feature that allows a time delay with each move. See also: Fischer Clock. A checker brought into your outer board where it bears directly onto one or more key points that you want to make. To make points in your home board. Hit a checker. To play a checker deep within your home board where it has no value. A backgame attempt that fell apart when the backgame player was forced to move checkers deep into his home board where they could no longer contain a hit checker. To safety a blot by bringing it together with another checker. [As in "go by".] The position of a player in a tournament who advances to the next round without playing a match. Byes are often awarded in the first round of an elimination tournament to make the number of advancing players a power of 2. A lottery of entrants in a backgammon tournament. At the start of the tournament, players are auctioned off and the proceeds go into a pool to be distributed later to the buyers of the successful players. Sometimes players are grouped into fields, with each field sold as a package. The rules usually allow a player to buy back a portion of himself if he wants to increase his stake in the An optional rule that says the winner of the opening roll has the option of rerolling both dice if he also turns the cube to 2. (The cube remains in the center.) A position in which a player's checkers are piled high on a few points (1). In a chouette, the leader of the team playing against the box. He rolls the dice and makes the final decisions for the team. To offer a double which you believe will be refused so you can collect the current value of the cube; claim a game. To throw a pair of dice. An early plastic, similar to bakelite, that was popular in the 1930's and 40's in the creation of backgammon playing pieces. Checkers which have been purposely spread out to maximize the chance of hitting an opposing checker if it tries to escape. The position of the doubling cube before either player has offered a double. A centered doubling cube is placed halfway between the players at the start of each game with the number 64 facing up (representing a value of 1). Play dangerously, especially in offering or accepting doubles, in an attempt to recover losses. One of the fifteen markers, all of one color, that a player moves around the board according to rolls of the dice. Also known as men, pieces, stones, or counters. 1. The movement of the checkers according to numbers on the dice. 2. The art or skill of moving the checkers. Compare: Cube Play (2). British spelling of checker. Two adjacent connected clocks with buttons that stop one clock while starting the other so that the two component clocks never run simultaneously. The purpose is to keep track of the total time each player takes and ensure that neither player unduly delays the game. Clocks may be analog or digital. Digital clocks work best in backgammon because they have a time delay [Pronounced "shoo-ETT". From the French word for "barn owl," a bird that is often attacked by all other birds.] A social form of backgammon for three or more players. One player, the box, plays on a single board against all the others who form a team led by a captain. Traditional name for the five-point. To offer a double which you believe will be refused so that you can collect the current value of the cube; cash a game. A move completed legally. To move all the checkers off of a point (1). A good general strategy to use when bearing in or bearing off against opposition. You clear your highest point (1) first and avoid creating gaps. Software that runs on a user's computer and communicates with a backgammon server to allow the user to play backgammon (1) with others on the Internet. The client software displays the board and interacts with the user as he rolls the dice and moves the checkers. The direction your checkers move around the board when they are set up to bear off to the left. When your checkers move clockwise, your opponent's checkers move counterclockwise. Make a point; place two or more of your checkers on a point (1), and thereby prevent your opponent from landing there. A player's home board when all six points (1) are blocked. A point (1) containing two or more checkers; a block or an anchor. To make all six of your home board points while the opponent has one or more checkers on the bar. The opponent is then prevented from entering his checker or making any other move until one of the closed home-board points is opened. A collection pip counting this article. Thrown dice which do not both land flat on the surface of the half of the board to the player s right. The roll is disqualified and both dice must be rethrown. Entering from the bar with a roll of 6-2 and hitting a blot on the eight-point when the only open point is the two-point. Misleading talk to confuse opponent. For example, in a chouette, when a team player advises the captain not to double knowing full well that the captain will double, he tempts the box to unwisely accept (ethically borderline, at best) 1. The two numbers on a pair of rolled dice taken together; see combinations of the dice. 2. The play of a single checker that uses both numbers of a roll, such as a combination shot. An opportunity to hit an opposing blot that requires using the numbers on both dice taken together; an indirect shot. Compare: Direct Shot. The number of possible rolls out of 36 that accomplish a specific objective. An opportunity to hit an opponent's blot immediately after being hit yourself; in particular, an opportunity to hit from the bar. A position from which there is only one reasonable game plan noncommitted position. To keep checkers within six pips (2) of one another for mutual support; see connectivity. A position with several made points close to one another and few gaps. What you sometimes get paid in if you are not careful with whom you play. A range of values that contain, with a certain probability, a rollout s convergence value A position in which all fifteen of a player's checkers are located within a short distance of each other. A position which is well-connected will tend to stay well-connected. The degree to which all of a player's checkers work together as a unified army without large gaps between them. Connected checkers defend each other and are easily made into points (2). A event for players eliminated early in the main flight of an elimination tournament; sometimes called a sympathy flight. To reduce the number of blots a player has, frequently as a precursor to offering a double. Advice offered by the crew to the captain chouette. A game where the opposing forces have not moved past each other and where it is still possible for one player to hit or block the other. Compare: Pure Race. To prevent an opposing checker from escaping to its own side of the board by blocking it or hitting it and sending it back. A player controls a point (1) if he has two or more checkers on that point. Only the player who controls a point may move additional checkers to that point. The value approached by a rollout as more and more trials are performed. It is the result you would obtain if you could do a rollout an infinite number of times. Games played by e-mail. 1. Pip count. 2. The relative standing of the players' pip counts. The player with the lower pip count is said to be ahead in the count. The direction your checkers move around the board when they are set up to bear off to the right. When your checkers move counterclockwise, your opponent's checkers move clockwise. Possibilities for retaliation, switching from a defensive posture to an offensive posture. To tabulate the players' pip counts to find out who is ahead in the race A win from the seemingly unwinnable position in which your opponent has borne off twelve checkers and has just three checkers remaining on his two-point. You bravely maintain contact with a single checker on his one-point and deploy your other fourteen checkers where they can contain his checkers if you are able to hit one or, preferably, two of them. Winning a coup classique is especially satisfying for you and maddening for your opponent. To add a second checker to a blot, thereby making the point Having little or no mobility. The first game in a match after either player comes to within one point (4) of winning. The rules of match play say that the doubling cube may not be used during the Crawford game. See: Crawford [Named for John R. Crawford.] A standard rule of match play. After either player comes within one point (4) of winning the match, the following game is played without a doubling cube. This one game without doubling is called the Crawford Game. After the Crawford game, the doubling cube is back in play again. chouette, members of the teamm who play with the captain against the box. [From the action a player makes as he reaches to enter his checker, then pulls his arm back when he notices the numbers are blocked.] To throw numbers which fail to enter a checker from the bar. A spare checker deep in the player's home board where it serves no useful purpose. See: Bury a Checker. A doubling cube with no further doubling value. In match play, the cube is said to be dead when the player owning the cube has no reason to double. For example, a player who owns a 2-cube when he is two points (4) away from winning the match will never double because he can win the match with the cube at its current level. A specific number on the dice which cannot be played in the current position; see kill a number. On a low-numbered point (1), usually the one-point or two-point. An anchor on the opponent's one-point or two-point. A rolled die showing the number 2. Traditional name for the two-point. [Plural of die.] Two small cubes, each with faces marked with spots (pips (1)) representing the numbers 1 to 6. Dice for backgammon usually have rounded corners so they roll more easily. You throw a pair of dice at the start of each turn, and move your checkers according to the numbers thrown. One of the 36 possible rolls using two dice. A container, often of leather or plastic, used for shaking and rolling dice. Dice cups often have a ridge around the inside of the open end designed to "trip up" the dice as they leave the cup. Dice cups make dice manipulation harder and help ensure randomness of the rolls. Any unfair means used to influence the roll of the dice. A person skillful in the use of unfair means to control the dice. Singular of dice. An electronic chess clock with digital displays showing the time remaining for each player. A display shows 00:00 when a player has run out of time. Digital clocks typically have a time delay feature which makes them particularly well suited for backgammon. Compare: Analog Clock. A spare checker which bears only on points deep in a player's home board. A hit using the number on just one die. You must be within six points of a blot to be able to hit it directly. Compare: Indirect Hit. Reachable using a single number from one die. For example, a blot is in direct range of being hit if it is six points or less away from an opposing checker. A chance to hit a blot six points or less away using a single number from one die. To break all contact and turn the game into a pure race. A position that is poorly connected, in which a player's army is divided into two or more groups with large gaps between them. The spreading out of your checkers to increase the number of good rolls on your next turn. Compare: Duplication. One of the sections in a tournament into which players are divided according to their ability and experience. For example, a tournament might have a novice division, an intermediate division, and an open division. An offer made by one player to his opponent during the course of a game (just before the player rolls the dice) to continue the game at twice the current stakes. The opponent may refuse the double, in which case he resigns the game and loses the current (undoubled) stakes. Otherwise, he must accept the double and the game continues at double the previous stakes. A player who accepts a double becomes owner of the cube and only he may make the next double in the same game. The roll of 2-2 on the dice (double 2's). A tournament format in which a competitor continues playing until he has lost twice. Compare: Single Elimination. To hit two opposing blots on one turn. Potential for awkward rolls both next turn and the turn after. 1. A match in which both players need just one more point (4) to win. 2. A game in which the doubling cube has reached a high enough level that a win by either player also wins the match. To offer a double which, if accepted, will win the match for that player if he goes on to win the game. Two thrown dice with identical numbers on their upper faces. One blot which can be directly hit two different ways, or two blots each of which can be directly hit one way. Compare: Single Shot. A cubical block, slightly larger than a regular die, with the numbers 2, 4, 8, 16, 32, and 64 marked on its faces. It is used for keeping track of the increase in stakes of the game and the player who next has the right to double. The cube starts in the middle with the number 64 facing up (representing a value of 1). When you offer a double, you turn the cube to its next higher value and pass it to your opponent. If he accepts your double, he places the cube on his side of the board and becomes the owner of the cube. Offering a double in anticipation of a good roll. 1. The range of game winning chances which are both a proper double and a proper take. 2. The range of game winning chances which would be a proper double and a proper take if neither player could use the cube again. The random pairing of competitors in a tournament to determine who will play whom or who will get byes. [From the server message: Player xxx drops connection.] A player on a backgammon server who avoids a reduction to his rating by intentionally leaving a match he is about to lose before the result The maximum game winning chances at which it is correct for a player to refuse a double; the point at which a player is equally well off accepting a double or refusing a double; take point. In a chouette, an agreement between two players after a double by the box that one player will accept the double, the other will refuse, and they will share their combined earnings or loss. A form of tournament play in which in which multiple pairs of competitors play with the same dice rolls in separate games and compare their results. Using the same sequence of random rolls to roll out two or more positions being compared. The idea is that lucky rolls for one position will also tend to be lucky for the other position, so when the results are compared less of the difference will be due to luck. See: Duplicate Backgammon. A position in which the same number can be used in more than one way. For example, when your opponent can use a 5 to hit either of two blots, his 5's are said to be duplicated. All else being equal, a position which duplicates the opponent's good numbers is better than one which does not because it means the opponent has fewer good rolls in total. A backgammon variant in which the players start with all their checkers off the board. The negative impact on flexibility of having spare checkers exactly six pips apart. Hit the opponent and thereby deprive him of half a roll. A strategy for winning the game. The three major game plans are run, block, and attack. The use of ethically dubious means to obtain an advantage in a game. This includes intentionally distracting, confusing, or generally duping an opponent. The probability of winning the current game if it is played to conclusion without a doubling cube; also called cubeless probability of winning. A completed game of backgammon in which the losing player has not borne off any checkers. A gammon is also called a double game because the winner receives twice the value of the doubling cube. Compare: Single Game and Backgammon (2). The minimum number of pips a player needs to roll to bring all his checkers home and bear off his first checker, thereby avoiding losing a gammon. Compare: Pip Count. A situation in match play where losing a gammon has no cost, but winning a gammon is particularly valuable. Examples: (a) you trail 4-away/2-away and opponent owns the cube at 2; or (b) you trail 2-away/1-away in the Crawford game; or (c) you trail 3-away/1-away after the Crawford game and the cube is at 2. Gammon-go for you is gammon-save for your opponent. A position that has a higher than normal gammon rate. The relative value of winning a gammon compared with the value of winning a single game. Gammon price is computed as GP = (WG - W) / (W - L), where WG = value of winning a gammon, W = value of winning a single game, and L = value of losing a single game. In money play, the gammon price is 50%. In match play, the gammon price depends on the score of the match and the level of the doubling cube. The chance of a game ending in a gammon or a backgammon (2) if played to completion (i.e., without a doubling cube). Gammon rate may refer to a particular game in progress or to backgammon games in general. An individual player's gammon rate is the fraction of his wins which are gammons or backgammons. A situation in match play where winning a gammon has no value, but losing a gammon is particularly costly. Examples: (a) you lead 2-away/4-away and own the cube at 2; or (b) you lead 1-away/ 2-away in the Crawford game; or (c) you lead 1-away/3-away after the Crawford game and the cube is at 2. Gammon-save for you is gammon-go for your opponent. The additional equity resulting from the possibility of winning a gammon. The space or spaces between made points. A position from which a player cannot lose. A Middle Eastern game in which a single checker controls a point (1) and doubles are very powerful. A roll of 5-5 (double 5's). A statement made by a player in a chouette that he is willing to pay the captain or any other team member the full stake at which the game currently stands for the right to take over their games. The player making this offer does so because he wishes to double the box when the other players to do not. A neural-net computer program that plays backgammon (1) and analyzes positions and matches. GNU Backgammon is a cooperative effort of many volunteers. It is "free" software as defined by the GNU General Public License. Website: GNU Backgammon. The opponent's five-point, the best place to build an anchor. To achieve the points (4) necessary to win a match. A mode in some computer programs and on some backgammon servers where the computer will automatically bear off the maximum number of checkers possible. A player's one-point. A Middle Eastern game in which a single checker controls a point (1) and doubles are very powerful. One of the two numbers on a pair of thrown dice. A pip counting method devised by Douglas Zare. An artificial advantage given to a weaker player in an effort to equalize the chances of winning. Some popular handicaps are: (a) the weaker player gets to go first; (b) once during the game the weaker player gets to reroll if he doesn't like his roll; (c) the weaker player gets to start the game owning the cube; (d) the weaker player gets to start the game with a strong roll such as 5-3, 4-2, 6-5, or 3-1. A play which exposes blots for the purpose of recirculating the player's checkers; also known as a suicide play. One player against another player for money. A point (1) with more than three checkers on it. A side bet between two players prior to competing in the final rounds of a tournament designed to protect the loser from going away empty handed. For example, two players competing for a $10,000 prize might agree beforehand that the winner will pay the loser $2,500. To land on a point (1) occupied by an opposing blot and put the blot on the bar. To hit an opposing blot and then continue on with the same checker to cover your own blot. To hit a blot with one number while splitting your runners with the other number. The hit provides protection for the advanced split checker, making it less likely the opponent will be able to point on it. To hit an opposing blot and while leaving your own blots in danger of a return hit. To hit an opposing blot sitting on the front edge of a partial prime to keep the blot from escaping. A strategy used when you are behind in the race and your opponent has escaped his runners. You make an advanced anchor on the opponent's side of the board and hold on to it as long as you can with the idea of hindering the opponent from bearing in safely. Compare: Back Game. The defensive point (2) you control when playing a holding game, usually the opponent's four-point, five-point, or bar-point. [Named after Tim Holland, who proposed the rule.] An optional rule in match play that was popular in the 1980's but is now rarely used. After the Crawford game a player may not double until at least two rolls have been played by each side. The quadrant containing your one-point through six-point. It is the last quadrant your checkers move to before they are borne off. It is also the quadrant your opponent must use to enter any of his checkers sitting on the bar. Your home board is also called your inner board or inner table. The number of plies played in each trial of a truncated rollout. A rollout that is truncated after 10 plies has a 10-ply horizon. A player who, by charm or other means, persuades another player to take part in game where the other player is at a disadvantage. A backgammon variant where each player has just three checkers. A elimination event, usually with a large entry fee, in which only the winner and runner-up receive prize money. The fact that an improvement in the opponent's position can make redoubling correct in a position in which the player on roll owns the cube and has one remaining chance to redouble. [Named for Oswald Jacoby, who proposed the rule.] A rule popular in money play which says that gammons and backgammons (2) count only as a single game if neither player has offered a double during the game. The rule speeds up play by eliminating situations where a player avoids doubling so he can play on for a gammon. A game once popular in France in which players start at diagonally opposite corners and move around the board in the same direction. [Devised by Rick Janowski.] A formula for estimating match equity (1) at a given score. If d is the difference in match score and t is the number of points (4) the trailing player has to go, then the probability of the leading player winning the match is .5 + .85d / (t+6). See also: Neil's Numbers and Turner's Formula. The first commercial neural-net backgammon program (1994) after TD-Gammon. Website: Jellyfish Backgammon. Potential for awkward rolls on a future turn. See also: Double Jeopardy. The standard deviation of the difference between two rollouts: JSD = sqrt(SD1SD1 + SD2SD2). A measure of how statistically significant the result is. An exceptionally good roll, especially a roll that reverses the likely outcome of the game; a roll much luckier than average. Affectionate name for a player's farthest-back checker. Breaking points in your home board in hopes of getting the checkers recirculated, a back game strategy. A position which is both a proper double and a correct beaver. This can happen only in money play with the Jacoby rule. By doubling, the underdog gets full value for his potential gammons, thus raising his equity; however, as long as this equity remains negative, the opponent should beaver. A point (1) required to complete a prime in front of the opponent's runners; the four-point, five-point, and bar-point are usually key points. 1. To watch a game or match. 2. To make a comment during the game within hearing distance of the players (undesirable behavior in a tournament). Spectator to a game. Good etiquette dictates that kibitzers not discuss the game within earshot of the players. To move an extra checker deep within your home board where it serves no useful purpose. See: Dead Checker. To create a position in which a specific number on the dice cannot be played on the following turn. Killing 6's, for example, is a way to preserve your timing in a priming battle or when defending against a back game. Compare: Save a Number. [Proposed by Danny Kleinman.] A guideline for cube handling in pure race positions. Compute K = (D+4)(D+4) / (S-4), where D is the player's pip count minus the opponent's count, and S is the sum of the pip counts. Kleinman says a player should make an initial double if K > 0.44, or redouble if K > 0.61, and the opponent should accept a double or redouble if K < 1.2. Hit a checker. A type of tournament where you continue to play until you lose; an elimination tournament. 1. An ongoing competition in which players are ranked in approximate order of ability. Any player may challenge another player higher on the ladder, up to a given number of steps away. If the challenger wins, he moves up the ladder and his opponent moves down. 2. Flight. A player's last opportunity to make a throw which will give him a chance of winning the game or saving a gammon. A tournament for players who lose in the first rounds of the consolation flight or later rounds in the main flight. A position in which the next roll will decide the game. In a last roll position, you should double if you have greater than 50% game winning chances and your opponent should accept your double if he has greater than 25% game winning chances. [Named after Andy Latto, who suggested the possibility in an Inside Backgammon article, (Vol. 2, No. 3).] A position which is a correct redouble but not a correct initial double. See: Starting Position. The player who is ahead in a match or ahead in the race (2). Compare: Trailer. A move that conforms to the roll of the dice as defined by the rules of backgammon. A rule that says that an illegal play must be pointed out by the opponent if he notices it, and then the play must be corrected. This is different from the usual rule giving the opponent the option of allowing an illegal play to stand. A play that conforms to the roll of the dice as defined by the rules of backgammon. An feature that contributes to the weakness of a position, such as too many blots, buried checkers, or inflexibility. Compare: Asset. A rule of thumb that says: in a well-timed ace-point game, the defending player has about a 17% chance of winning the game. A dice cup that has a ridge around the inside open end, designed to trip up the dice as they exit the cup and make it more difficult for a dice mechanic to control the roll. Dice weighted or shaped so that the distribution of rolled numbers is not even. A backgammon variant in which each player starts with fifteen checkers on the opponent's one-point. A hit which leaves a blot in the player's home board where it is exposed to a direct return shot. A play that leaves one or more blots in a dangerous position. To go from a position in which your opponent would accept your double to a position in which your opponent would refuse your double. See: Market Loser. [Sometimes spelled "lovers' leap".] An opening roll of 6-5 played from the opponent's one-point to the player's mid-point. Equity gained or lost through the rolls of the dice during the course of a game or match. The luck associated with a roll is the difference in equity of the position before the roll and the (properly played) position after the roll. Six criteria for determining whether a situation requires a safe play or a bold play. They are: (a) opponent's home board strength; (b) presence of an anchor in opponent's home board; (c) your home board strength; (d) blots in opponent's home board; (e) number of checkers you have back; and (f) number of checkers opponent has back. Criteria (a) and (f) call for making a safe play; the others point towards making a bold play. In an elimination tournament, the group in which players start and compete in until they lose, and which offers the largest prize. Compare: Consolation Flight. Moving one of your two runners from the opponent's one-point to the opponent's four-point or five-point. Compare: Minor Split. To place two checkers together on one point (1) so they form a block or an anchor. Your opponent may not land or touch down on that point as long as the two checkers remain there. To close all the points (1) in your home board. An optional chouette rule which says: when only one player accepts box's initial double, that player must also beaver; otherwise, he must refuse the double along with everyone else. Compare: Mandatory Extras (1). A game in match play where the doubling cube has reached a high-enough level that it represents sufficient points (4) for the leader to win the match; the trailer has nothing to lose by doubling at this point. This includes any post-Crawford game, where the trailing player should double at his first opportunity. 1. An optional rule for chouette play which says when only one player on the team accepts the box's initial double that player is obliged to accept an extra 2-cube from any other team member that wishes to pay him one point (4). 2. An optional rule for money play which says whenever a double is offered and accepted the doubler has the right to give his opponent an extra cube at the same level accompanied by a payment equal to one half of its value. The receiver of an extra now has two cubes which he may use together or separately for making future doubles. In post-Crawford match play, if the leader is offered a double when the trailing player has an odd number of points (4) to go, the leader should almost always accept the double. For example, as leader against an opponent who is 5-away, taking and losing two points means the opponent still needs two games (or one gammon) to win the match. See: Box. The player on roll has two checkers on each of his lower three home board points, and three checkers on each of upper three home board points. The opponent has one checker on the bar, six checkers borne off, and the remainder on his one-point and two-point. Should the player double? Should his opponent accept the double? An opportunity to offer a double while it will be accepted by the opponent. [By analogy to market loser.] A sequence of two rolls (one for you and one for your opponent) which takes a game from a position in which your opponent would refuse a double to a position in which your opponent would accept a double. A sequence of two rolls (one for you and one for your opponent) which takes a game from a position in which your opponent would accept a double to a position in which your opponent would refuse a double. Knowing the number and size of your market losers is an important consideration in whether or not to double. A series of games between two players which ends when one player acquires a predetermined number of points (4). Traditionally, matches are played to an odd number of points (3, 5, 7, etc.). See: Match Play. 1. A player's probability of winning a match from a given score. 2. The value of a position in the context of the current match score and cube level, usually given in terms of match winning chances. A chart showing the probability of winning a match from various scores. Example: The Woolsey-Heinrich match equity table. Match equity tables are laid out according to the number of points each player still needs to win the match. The first column and row represent the Crawford game. The method of competition used in tournaments and on many backgammon play sites. Two competitors play a series of games until one of them acquires a predetermined number of points (4). The doubling cube may be used except in the Crawford game. Unlike money play, you do not use automatic doubles, the Jacoby rule, or beavers in match play. A player's probability of winning a match. Compare: EMG Equity. See: Dice Mechanic. A move made with little thought because it seems to be obvious. A backgammon variant similar to Acey-Deucey (2) in which a roll of 1 and 2, called a Mexican, gives the player extra turns. The main body of the game, which begins after the players have settled on their initial game plan. Compare: Opening Game and End Game. Your thirteen-point (the opponent's twelve-point), where you have five checkers at the beginning of the game. Moving one of your two runners from the opponent's one-point to the opponent's two-point or three-point. Compare: Major Split. A backgammon variant in which the object is to be the last player to bear off all of your checkers. Two thrown dice with different numbers on their upper faces. Compare: Doubles. The degree to which a position permits dice rolls to be played freely while maintaining the position's key features. A mobile position strikes a balance between the made points and spare checkers 1. A term used in the late 1920's and early 1930's for the new rules of the time, including the use of the doubling cube and chouette play. 2. A term used in the late 1990's and early 2000's for a style of play inspired by computer analysis. Choosing appropriate stakes to play for so that you do not exceed your bankroll. Money management has two goals: to ensure that your bankroll lasts the entire session and to make playing more fun by removing some of the stress involved in dealing with money. The style of competition in which games are played individually and the participants wager on the result. At the end of each game, the loser pays the winner the agreed initial stake multiplied by the value of the doubling cube and further multiplied by 2 for a gammon or 3 for a backgammon (2). Money play backgammon is usually played with Jacoby rule, and participants may also agree to play automatic doubles and beavers. Compare: Match Play. Location of the annual World Championship of backgammon. A Java applet that plays backgammon. A Turkish game in which players start at diagonally opposite corners and move around the board in the same direction. There is no hitting and one checker by itself controls a point (1). The advancement of a checker according to the number showing on one of the rolled dice. There are three types of legal moves you may make: (a) to enter a checker from the bar (your only legal move when you have a checker on the bar); (b) to move a checker forward the given number of pips (2) to an open point, possibly hitting an opposing blot; or (c) to bear off a checker, when all of your checkers are in their home board. A move from the opponent's outer board to the player's outer board. 1. A move from the bar to the opponent's home board. 2. A move from your outer board to your home board. A move from the opponent's home board to the opponent's outer board. A move forward within the opponent's home board. A game in which both players hold advanced anchors on the opponent's side of the board in an attempt to hinder the opponent as he tries to bring his checkers home. A tournament of 1-point Nackgammon matches. [Named after Nack Ballard, who popularized the game.] A backgammon variant played using the same rules as regular backgammon except for the starting position. Players start with 2 checkers on each of the opponent's one-point and two-point, 4 checkers on the mid-point, 3 checkers on the eight-point, and 4 checkers on the six-point. With fewer checkers up front for attacking, and more checkers back for anchoring and maneuvering, games tend to be longer and more positional. [Also spelled "Nardi" and "Nardy".] A Russian game similar to Moultezim. [Devised by backgammon expert Neil Kazaross.] A mnemonic device for estimating match equity (1) based on the current match score. The leader's percent probability of winning the match is 50, plus his point (4) lead in the match multiplied by the appropriate Neil's number. Compare: Janowski's Formula and Turner's Formula. The architecture used in many of the strongest backgammon programs such as Jellyfish, Snowie, and GNU Backgammon. A neural network consists of many simple processors connected by unidirectional paths carrying numeric data. The network is "trained" by adjusting the weights of the connections until desired outputs are achieved for given inputs. 1. Pure race. 2. An easy decision. A position with a flexible game plan; a game where there is more than one reasonable strategy for winning, such as racing, priming, or blitzing (1). A consolation tournament for losers of the first round of the main tournament. Losers in later rounds of the main event do not get to enter the consolation event. Compare: Progressive Consolation A match score expressed in terms of the number of points (4) needed to win the match rather than the number of points won so far. For example, a score of 5-1 in a match to 7 would be "2-away/ 6-away". Normalized scores are used in match equity tables. The method of representing the moves of a game. The tournament division for the weakest players, particularly those who do not desire the stronger competition and higher entrance fees of the other divisions. Compare: Intermediate Division and Open Division. A player who is new to backgammon. Compare: Intermediate and Advanced level. A home board with n made points. A position in which you will bear off all of your checkers in n rolls or less. For example, having ten checkers left on your ace-point is a "5-roll position." A play which cannot be profitable for any possible sequence of future rolls. The ratio of the probability of an event happening to that of its not happening, or vice versa. Usually the higher number is given first. For example, the odds of rolling double 6's are "35 to 1 Said of checkers which have been borne off. A model for estimating winning chances in a pure race based on the players' pip counts. In this model, all of a player's pips are represented by just one checker on a infinitely-long backgammon board. The one-checker model overestimates winning chances in positions where one side has more wastage than the other. The deepest point (1) in a player's home board, the point farthest from the bar and closest to being borne off; also called the ace-point. A backgammon variant where the goal is to be the first player to bear off all of your checkers. There is no doubling cube and no bonus for gammons or backgammons (2). Since you never lose more than one point (4), back games are more of an option in this variant than in regular backgammon. A bearoff database where the arrangement of checkers on only one player's side is considered. The values in the database are calculated assuming the goal at each turn is to minimize the average number of rolls required to bear off. Compare: Two-Sided Bearoff Database. This refers to playing backgammon over the Internet. Online backgammon allows players all over the world to compete against one another. You can play for rating points or for real money. See: Backgammon Server. The player whose turn it is. You are on roll as soon as your opponent picks up his dice to end his turn, and before you throw the dice to begin your own turn. For example, the only time you may double is when you are on roll. Where a checker is placed after it is hit. When you have a checker on the bar, you may not move any of your other checkers until that checker has been entered back onto the board. The main division of a tournament; the division that any player may enter. Also called the championship division, it generally has the highest entry fee, the largest prizes, and attracts the strongest players. Compare: Novice Division and Intermediate Division. The first phase of a backgammon game where the players have yet to establish their initial game plans. Compare: Middle Game and End Game. The first roll of the game in which both players simultaneously roll one die. This roll determines both the player to go first and the numbers to be played. A position on the board not occupied by two or more of the opponent's checkers. A tournament open to any player regardless of strength or experience. See: Open Division. [Another furry rodent, by analogy to beaver and raccoon.] An immediate redouble (while retaining ownership of the cube) by the player who just accepted a raccoon. The side of the board away from where the players bear off their checkers. Each player's outer board comprises that player's points seven through twelve. Compare: Home Board. The outer board, particularly points nine, ten, and eleven. A contiguous sequence of blocked points in which the majority of those points are in the outer board. Points (4) won in excess of those needed to win a match. For example, if you win a game worth 4 points in a match in which you are 2 points away from winning, the surplus 2 points are overage. Make an unnecessarily big play. Games played face-to-face, as opposed to on the Internet or by correspondence. To have two or more checkers on a point (1) so that the opponent is blocked from landing or touching down there. The player who last accepted a double in the game. He places the cube on his side of the board to indicate that only he may make the next double. See: Cube Ownership. The player who last accepted a double is said to own the doubling cube. He places the cube on his side of the board. Only the owner of the cube may offer the next double in the same game. Compare: Centered Cube. A succession of events, each of which depends on the preceding event. The probability of the entire parlay is equal to the product of the probabilities of the individual events. A prime of fewer than six consecutive points (2). Compare: Full Prime. Chouettes with a large number of players often permit the box to take a partner. The partnership is offered in rotation, starting with the captain and moving on down the line. If no one offers to be the box's partner, a partner may be chosen by lot from among the team members other than the captain. To play safe in the current position but risk greater danger later in the game. Compare: Pay Now. See: Pay-Now-or-Pay-Later Decision. To take an immediate risk (such as leaving a shot) to avoid the prospect of a more serious risk later in the game. Compare: Pay Later. See: Pay-Now-or-Pay-Later Decision. [From a 1980's television commercial for FRAM oil filters which showed a ruined car engine and a mechanic who quipped: "You can pay me now, or you can pay me later."] The problem of whether to take a modest but definite risk on the current turn or wait and perhaps take a more serious risk in the future. The best possible roll; a joker. To hit an opposing blot and continue the same checker to safety on the same play. Hit a blot. The victim of a hustler. 1. One of the spots on a die that indicate numeric value. 2. A unit of distance on a backgammon board corresponding to the difference in point (1) numbers. For example, the 13-point and the seven-point are six pips apart. The total number of points (or pips (2)) that a player must move his checkers to bring them home and bear them off. For example, at the start of a game each player has a pip count of 167: 48 pips for 2 checkers on the 24-point, plus 65 pips for 5 checkers on the 13-point, plus 24 pips for 3 checkers on the eight-point, plus 30 pips for 5 checkers on the six-point. A Greek game in which players pin blots rather than hit them. The collection of moves a player makes in satisfying the requirements of a roll. To avoid leaving blots which might be hit. See: Safe Play. One turn by one player, a measure of how far a player (or computer program) looks ahead when selecting a play or evaluating a position. Note: There is no agreement in the backgammon community as to whether plies are counted starting at 0 (as GNU Backgammon does it) or starting at 1 (as Snowie does it). An old method of scoring in backgammon (1) that is no longer used. The winner of the game gets 1 point for each checker in the loser's home board, 2 points for each checker in the loser's outer board, 3 points for each checker in the winner's outer table, and 4 points for each checker on the bar or in the winner's home table. To hit an opposing blot with two of your checkers at the same time, thereby also making the point. Pointing on a blot in your home board is usually a very strong play. A measure of playing performance equal to the total number of points (4) won (or lost) divided by the number of games played. A backgammon variant in which you always play the lower number of a roll first. A Greek game similar to Western backgammon (1). The arrangement of checkers on a backgammon board. 1. A play that emphasizes fighting for and keeping key points over running or blitzing (1); a structural play. 2. A checker-play decision where strategy considerations dominate. Compare: Technical Play. A card with a preprinted diagram of a backgammon board designed for recording a position. The player who last accepted a double is said to own the cube. Only that player may make the next double of the game. Prior to the first double, neither player owns the cube (see centered cube) and either player may double. After the Crawford game. Analysis of a game or match after it has been completed. Acronym for "Position, Race, And Threats," a guideline for making cube decisions. According to the guideline, a player should double if he has an advantage in two of the three areas. And his opponent should pass if the player who doubled has an advantage in all three areas. Dice which have been carefully cut so their shape and balance are more accurate than regular dice and have pips (1) that are flat and not dimpled. To evacuate a high point (1) in your home board before all of your checkers are home in preparation for bearing off. You sometimes do this when the opponent holds an anchor deep in your home board and you are worried about clearing a high point safely during the bear off. By preclearing, you take advantage of opportune rolls at the time you get them. Before the Crawford game. Taking one or more checkers deep into your home board early in the game out of undue concern for short-term safety. A dice roll made by a player before the opponent has ended his turn by picking up the dice. Under U.S. rules, the premature roll is invalid and must be rethrown. Under BIBA rules, the premature roll stands but the player who did not pick up the dice may change his play in light of the new information. To advance a runner so it directly bears on an opponent's blot, forcing the opponent to cover the blot, move it, or risk it being hit. 1. Six consecutive made points. An opposing checker trapped behind a prime cannot escape until the prime is broken. 2. Several consecutive made points, such as a 4-prime or 5-prime. Trapped behind a prime. A player who accepts a double when he has one or more checkers trapped behind an opponent's prime. A game in which both players have long primes with opponent's checkers trapped behind them. The winner of these games is often the player with better timing. A type of game in which the primary strategy is to trap one or more opponent's checkers behind your prime. A consolation tournament for losers in the first several rounds of the main tournament. Progressive means that losers in later rounds of the main event get one or more byes to later rounds of the consolation event. Compare: Nonprogressive Consolation. A prearranged position played several times, usually for money, as a means of settling a dispute over which checker play (1) or cube action is best. See: Take/Drop Proposition. See Cube Proxy. 1. The German name for backgammon (1). 2. A German backgammon variant in which players enter in the same quadrant and move around the board in the same direction. ["Pure" because it focuses on one game plan.] Playing with the goal of making a prime. Pure play includes bringing builders into play quickly, slotting to make key points, keeping your checkers in front of the opponent's checkers, and recirculating checkers as necessary. A game in which the opposing forces have disengaged so there is no opportunity for further blocking or hitting by either side. In a pure race, your goal is simply to get your checkers home as quickly as possible and bear them off. Compare: Contact Position. One quarter of the playing area on a backgammon board. The first quadrant comprises a player's points 1 to 6, the second quadrant points 7 to 12, the third quadrant points 13 to 18, and the fourth quadrant points 19 to 24. The roll of 4-4 on the dice (double 4's). A quarter entry is a single elimination tournament for four players, which is held before the beginning of a greater tournament. Each of the four contestants pays an entry fee of (usually slightly more than) a quarter of the entry fee of the main event. The winner of this four person tournament is entitled to play in the main event. A technique used to reduce the element of luck in a rollout by ensuring the numbers rolled in the first few rolls of each trial are as evenly distributed as possible. For example, if you roll out a position 36 times, quasi-random dice will ensure that each trial begins with a different roll. Traditional name for the four-point. An unassuming play that does not hit, or slot, or pose an immediate threat; it just maintains the status quo. A feature of a problem that makes it interesting enough to appear on a quiz. The mere appearance on a quiz suggests that the "obvious" play may not be the correct play. Free from danger of being hit. A play that leaves no blots, or a play that leaves blots only in positions where the opponent is unlikely to hit. Compare: Bold Play. Move a checker out of danger of being hit. Cover a blot or move it out of range of being hit. 1. To conceal or misrepresent your true ability. 2. To enter a tournament division below your skill level. To leave a position in which a particular number will play comfortably next turn so you will not be forced you to destroy your position if you roll that number. Typically, you save a number to avoid having to leave a shot or break a valuable point. Compare: Kill a Number. To escape all of your checkers from the opponent's home board before he is able to bear off all of his own checkers, and thereby avoid losing a backgammon (2). To bear off one of your own checkers before the opponent has borne off all of his, and thereby avoid losing a gammon. See: Random Seed. A competitor in a tournament whose position in the draw is predetermined to ensure that he will not meet other seeded players in the early rounds of an elimination event. A checker which may or may not be available to make another point, depending on the roll. One of the four players competing in the semifinals of an elimination tournament. The second-last round of an elimination tournament; the one that determines the two players who advance to the finals. A decision to end a game early with the payment of points by one player to the other based on the agreed fair value of the position (see Equity). Settlements are generally not allowed in tournament play. A method of reducing the variance of a cubeful rollout. Any trial in which the equity of a player exceeds a given value (the settlement limit) is terminated at that point and scored as double/ See: Starting Position. To mix the dice using a dice cup prior to rolling. A good player who seeks out weaker players and persuades them to play for high stakes. (Sharks eat fish.) A Turkish game similar to Western backgammon (1). Change game plan. To give up one point in order to make an adjacent point. 1. An opportunity to hit an opposing blot. A direct shot is an opportunity to hit using a single number. An indirect shot is an opportunity to hit using both numbers of the dice played with the same checker. 2. A particular roll of the dice which could hit an enemy blot. When counting shots, you count each doubles roll once and each mixed roll twice to get a total out of 36. A separate tournament prize fund made up of additional optional entry fees which goes to the highest finishing player(s) of those who entered the side pool. The side pool allows a tournament to keep the regular entry fee low while providing players willing to pay a higher entry fee a chance to win more. [By analogy to the golden point.] A term sometimes used for the opponent's four-point, the second best point on which to anchor. A blot within range of being hit with a single number but for which there are no ways to hit using a combination of numbers on both dice. A tournament format in which a competitor continues playing until he loses. See: Elimination Format. Compare: Double Elimination. A completed game which is not a gammon or a backgammon (2); a game in which the losing player has borne off at least one checker. The winner of a single game receives the value of the doubling cube only and no bonus. One blot which can be directly hit one way. Compare: Double Shot. The sixth point (1) in a player's home board; the point adjacent to the bar. To place a single checker on a point (1) you wish to make with the intention of covering the blot on your next turn. To slot a checker in your own home board while your runners are split. A safe play when a bolder, more aggressive play is available. Compare: Big Play. A backgammon variant in which one player starts with nine checkers on the bar and his remaining six checkers in the opponent's home board. The roll of 1-1 on the dice (double 1's). The second commercial neural-net backgammon program (1998) after Jellyfish. Website: Snowie Backgammon. A prime with no gaps; a full prime. Compare: Broken Prime. An extra checker that can be used for hitting or making a point without leaving behind a blot. A bearoff position in which you expect to take at least two checkers off every roll, typically when all of your checkers are crowded onto the three lowest points of your home board. To separate two checkers which are together on a point (1) (usually the opponent's one-point) and leave them as blots. See: Major Split and Minor Split. To take advantage of the opponent's requirement make a move. You leave him a position in which the only move he can make hurts his position. Often this means he is forced to break a valuable defensive point (2) earlier than he would like. Four or more checkers piled on a point (1). See: Candlesticks. An optional rule where rolls of doubles are played like any other roll; that is, each number is played once, not twice. See: Irish. The amount wagered by the participants in a game of backgammon (1). The current stake is the initial stake multiplied by the value of the doubling cube. A measure of a rollout's variance or random error. A rollout will be within one standard deviation of its convergence value 66% of the time, within two standard deviations 95% of the time, and within three standard deviations 99.7% of the time. See also: Joint Standard Deviation. The arrangement of checkers at the start of a game. Each player has 2 checkers on the opponent's one-point, 5 checkers on the mid-point, 3 checkers on the eight-point, and 5 checkers on the six-point. See: the Rules of Backgammon. Remain in the opponent's home board. Fail to enter from the bar. Fail to enter from the bar. [What happens when a player reaches the "boiling point."] To play wildly, out of annoyance or impatience at one's bad luck. To lose one's emotional stability in a gambling context; in particular, to take bigger and bigger risks in an effort to recoup earlier losses. One who steams. The last lone checker heading for home. The overall, long range plans for a game. The reasoning behind a play. See: Game Plan. Compare: Tactics. A position barren of spare checkers or builders and thus prone to awkward numbers; too many points. To remove all but two checkers from a point (1). A position barren of spare checkers or builders and thus prone to awkward numbers. To deliberately make an illegal play or otherwise take an unfair advantage. A home board with several made points. A play which makes a strong point. To purposely leave a blot to be hit so it can be recirculated. The idea is to improve your timing or shore up your defense in the opponent's home board. Also known as a Hara-Kiri play. A backgammon variant in which players can win by arranging their checkers into specific patterns within their home board. 1. The difference in score between winning a game and losing it. 2. The difference in your equity before a roll and after it, or the difference between rolling poorly and rolling well. A position with many gaps and few adjacent made points. A method of pairing players in a tournament. Under the Swiss system, players are not eliminated, no player meets the same opponent twice, and successive rounds match players with scores as similar to each other as possible. To give up one point (2) to make another, usually in your home board. The roll of 1-6 to escape a prime, usually from the bar and often hitting a blot. Bias introduced in a rollout because of errors in checker play (2) or cube play (2). 1. An entire backgammon board. 2. One of the four quadrants of a backgammon board; for example, your inner table or outer table. 1. The English name for the Roman game Tabula. 2. A generic term for any game played on a backgammon board. A system of betting where the players' stake in a game is limited to an agreed fixed amount. The idea is to protect the players from losing more money than they have at hand. It also evens the playing field when one player has more money at his disposal than the other. A Roman game similar to backgammon in which players use three dice instead of two, and move around the board in the same direction. The game was also popular in England where it was known as Short-term, calculable aspects of the game, as opposed to strategic considerations. Tactics in backgammon include: hitting blots, making points, clearing points, and avoiding unnecessary risks. To start to throw your dice before the opponent has picked up his own dice to finish his turn; to roll prematurely. A way to settle a difference of opinion about whether a position is a take or a drop by playing a series of games starting with the position in question. The player who believes the position is a take plays the taking side owning a 2-cube and gets one point added to his score for each game played. The minimum game winning chances at which it is correct for a player to accept a double; the point at which a player is equally well off accepting a double or refusing a double; a player's drop Hit a blot. A Persian game similar to Western backgammon (1). A game popular in Bulgaria in which players pin opposing blots rather than hit them. A Turkish game similar to Western backgammon. The Greek name for games played on a backgammon board. These typically include Portes, Plakoto, and Fevga. The first strong neural-net backgammon program (1991), written by Gerald Tesauro. In a chouette, the players lead by the captain who play against the box; the captain and his crew. A checker-play decision which primarily depends on tactical considerations. Compare: Positional Play. An inadvertent clue as to whether you will be taking or dropping if offered a double. A plot showing how a position's equity is distributed among each of the 6 x 6 upcoming rolls. It provides a way to visualize aspects of a position such as volatility and duplication. See: Equity Temperature Map: Introduction. A unit of time in positional development equal to half a roll. A hit designed to forestall the opponent by depriving him of half a roll when the opponent threatens to hit a blot or make an important point, or needs to consolidate a disorganized position. To intentionally place a blot in a position where it can be hit with the idea of enticing the opponent to give up a strategic point (2). A player's two-point. A formula devised by Edward O. Thorp for making doubling decisions in pure race games. It is a modification of the basic pip count which takes into account some elements of checker distribution. Each player's Thorp count is his pip count, plus 2 for each of his checkers still on the board, minus 1 for each of his occupied home board points, plus 1 for each checker on his one-point. Then the player on roll increases his count by 10 percent if it is more than 30. Thorp advises: Double any time your count does not exceed the opponent's by more than 2; redouble any time your count does not exceed opponent's by more than 1; accept the double if your count does not exceed doubler's by more than 2. The third point (1) in a player's home board, counting from the edge of the board toward the bar. To shake a pair of dice in a dice cup and release them onto a backgammon board. If the dice are cocked, they must be rethrown. The average number of rolls or pips (2) that can be played without having to make a major concession, such as leaving a blot, breaking a key point, or burying a checker. A feature of digital chess clocks which gives each player a specified number of seconds at the start of each turn before that player's clock begins running. Typical time delays in backgammon range from 8 to 15 seconds per move. The idea is that players are charged only for "thinking time" and not for the time required to roll the dice, wait for them to settle, read the numbers and move the checkers. How long you expect to retain the desirable features of a position compared to your opponent. Good timing means your opponent will be forced to make a major concession, such as leaving a blot, breaking a key point, or burying a checker, before you. You can sometimes help preserve your timing by killing large numbers or recirculating checkers. A position which you should not double, even though your opponent has a clear drop, because your equity is higher by playing on for a gammon. An inflexible position with many made points and few spare checkers. Seven is usually "too many." To temporarily land on an intermediate open point after playing one of two numbers with the same checker. An rule rarely used today in Western backgammon, though it is common in the Middle East. The rule requires that once you touch a checker (other than to adjust it) you must move that checker, and once you remove your hand from a properly played checker, that checker must remain where it was played. A formal competition among multiple entrants in which a winner is decided. The person who organizes and oversees a tournament. A game popular in seventeenth-century France in which players have just three checkers each and play only on their own side of the board. The player who is behind in a match or behind in the race (2). Compare: Leader. Reaching the same position by different means. A deliberate attempt to squeeze the opponent off of his anchor so that the trapper can close out any blots thereby exposed and win a gammon. Traditional name for the three-point. Playing a position out to the end of the game once (or to the point of truncation). A rollout consists of multiple trials, the results of which are averaged together to yield an estimate of the equity of the position. [Named for Walter Trice.] The ideal position to aim for during bear-in, consisting of: 7 checkers on your six-point, 5 checkers on your five-point, and 3 checkers on your four-point. It has the lowest wastage of any position with all 15 checkers still on the board. 1. A game popular in French high society prior to the Revolution. Players score points for making specific plays or moving their checkers into certain configurations. 2. The French name for "backgammon." Traditional name for the three-point. A rollout which is not played to the end of the game. Instead, the position is rolled out a given number of plies (the horizon of the rollout) and estimates of the equities of the resulting positions are averaged together. A truncated rollout has more systematic error than a full rollout but is faster because each trial is shorter, and a truncated rollout has less variance so fewer trials are required to converge on a result. The sequence of actions that each player takes in alternation. One turn consists of: (a) possibly offering a double; (b) rolling the dice; (c) playing the roll; and (d) picking up the dice. A simple formula devised by Stephen Turner for estimating the match equity (1) at a given score. Expressed as a percent, the leader's match equity E = 50 + (24/T + 3) D, where T is the number of points (4) the trailer still needs and D is the difference in scores. Compare: Janowski's Formula and Neil's Numbers. Move from the opponent's outer board to your own outer board. To offer a double. To offer a double. A mode available in some backgammon-playing programs which allows the computer to evaluate your moves as you make them and alert you to any errors it thinks you made. To offer a double. The second point (1) in a player's home board, adjacent to the one-point; also called the deuce-point. A bearoff database with the correct equity for each possible combination of two opposing bearoff positions. Four separate equities are recorded for each position: three cubeful equities (one for each state of the doubling cube), and one cubeless equity. A two-sided database is more accurate than a one-sided database, but requires considerably more room.	{"url":"http://winandfun.com/backgammon_glossary.htm","timestamp":"2024-11-11T06:37:36Z","content_type":"text/html","content_length":"429396","record_id":"<urn:uuid:4494a237-c79a-4cb0-9354-808459b4a7ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00219.warc.gz"}
Enrichment, or how I learned to stop worrying and love the SWU, part 2 Okay, before we get into the techniques of enrichment, let’s spend a little time with the fun part—the numbers! How much does it take and how much does it cost? Here’s where I’ve really enjoyed learning more lately. The basic equations that describe enrichment aren’t that hard to derive. Even I can do it! So here goes: You know that you’re going to start out with some amount of input material—the “feed” as it is called. You know that after you’re done you will have split the feed into two: the “product” and the waste, which in this case is called the “tails”. Assuming that you didn’t mess up and lose lots of material along the way, you could add up the product and the tails and that would be the same amount of material that you started with. So we’ll call the mass of the feed “mf”, we’ll call the mass of the product “mp”, and we’ll call the mass of the tails “mt”. And we’ll write a really simple equation that relates all three. Next, let’s assume that we have a mixture that has only two components. They might be U-235 and U-238, or Li-6 and Li-7, or Cl-35 and Cl-37. This derivation works for any two-component mixture. Let’s say that the fraction of the less abundant component (U-235 in the case of uranium) is given by a non-dimensional variable called “x”. And let’s say that that value of x is different for the feed, product, and tails. For instance, in the case of uranium, if the feed is natural uranium, 0.0071 of it is U-235. So the value for x for the feed would be 0.0071. xf = 0.0071. And let’s say that we want to make fuel for a light-water reactor out of this, and it needs the uranium enriched up to 3%, or 0.03. So the value of x for the “product” would be 0.03. xp = 0.03. The last thing to figure out would be how much U-235 we will tolerate in our “tails”, the part that we’re going to throw away. At first blush, we might think, “hey, I don’t want to throw away any U-235, I want it to all go into my product. So I’ll set the tails to 0% and everything will be great.” Hah—that seems like a good idea but it isn’t. Because the amount of effort it will take to strip every single last atom of U-235 from the mixture is infinite. So in reality don’t do that. So then you might think, “OK, that might be too hard, maybe I should set the tails enrichment higher so I don’t have to do as much work. I’ll set it to just a little bit less than the natural level of enrichment, something like 0.006%.” Another bad idea. If you set the tails enrichment too low, you expend too much effort trying to get some small amount of stuff. If you set it too high, you end up throwing away most of the stuff you’re trying to get in the first place. So you have to be kind of careful about how you set your tails enrichment. Which will have everything to do with the economics of enrichment, as we shall see. So I’ve spent this time telling you that you know the enrichment of the feed (easy, it’s the natural enrichment) and the enrichment of the product (easy, it’s what your customer wants) but the real question is what is the enrichment of your tails. That’s going to have to be a choice based on economics, and as economics change you might find yourself revisiting that decision. OK, so let’s say we’ll set the tails enrichment at 0.003. xt = 0.003. That’s means about half of the U-235 in the original uranium is going to end up in your product and about half is going to end up in the tails. If we assume that the total amount of U-235 is constant, then we can write another equation: This is just another way of saying that all of the U-235 ends up in either the product or the tails, and if you added it all up it would be the same as the amount that you had in the feed. Now we can combine these two equations to start figuring things out. We can use some of those tricks we learned in eighth-grade algebra to solve for things when we have two equations. We can rewrite the first equation to equal the tails rather than the feed: And we can substitute that definition into the second equation: Then, again using our eighth-grade algebra skills we can expand the equation: and we can group the terms relating to the mass of the feed and the product: We pull out the common factors… …and then we can solve for the mass of the feed: Or even better, we can figure out a ratio between the mass of the feed and the mass of the product: Hooray! So why is this a big deal? Because now we have an answer that doesn’t really care how much actual mass of feed or product that we’re talking about. On the left-hand side of the equation is a ratio, and on the right-hand side of the equation are a bunch of enrichment values. Let me show you how useful this nifty little expression is. Let’s say that you want to figure out how much natural uranium you will need to fuel a nuclear reactor that uses enriched uranium. This is pretty much the situation we are in in the United States. Let’s say that you know that a typical reactor takes about 35 tonnes of enriched uranium fuel per year, and that that fuel is enriched up to 3%. You already know that the enrichment level of natural uranium is 0.0071, and let’s say that you’re comfortable with your tails enrichment being 0.003. So then you run the numbers: See how the tails enrichment shows up in both the numerator and the denominator of the expression? In this case, the ratio is 6.58, which tells you that you’ll need almost seven times more uranium as a feed material than you’ll need for fuel. So if you need 35 tonnes, you multiply 35*6.58 to get 230 tonnes. You’ll need 230 tonnes of natural uranium to make 35 tonnes of enriched uranium. Where does the rest of the uranium go? Into the tails. There’s 195 tonnes of uranium tails in this example. One of the things that you find when you do uranium calculations is that you almost always make a lot more tails than you make product. But let’s do another example, to broaden our perspective. Let’s say that we’re talking about enriching lithium now instead of uranium. We want to make lithium-7 for a LFTR, and we need it to be enriched in lithium-7 up to a point of about 0.9999 (the more nines the better). But natural lithium is only about 90% enriched in lithium-7. So how much lithium feed will we need to enrich to 0.9999? Again, it has everything to do with the enrichment we’ll tolerate in the tails. Let’s say (being foolish) that we set the tails enrichment to 0.80. Don’t want to work too hard, right? So let’s go run the numbers and find a feed-to-product ratio for our lithium enrichment. In this case, we need about twice as much lithium feed to get a particular level of product. Why is that? Because by setting the tails to 0.8 only about half of the lithium-7 ended up in the product. The other half if in the tails. So even in a case where we have something like lithium-7, where it is the dominant constituent of natural lithium, setting the tails level of enrichment has a lot to do with how much feed it will take. So let’s go back and run it again, this time with a tails enrichment of 0.5. No sense letting all that valuable lithium-7 go to waste, right? Now we only a little bit more lithium feed (25%) than we expect as product—in other words, most of the lithium-7 ended up in our product stream rather than in the tails. Seems like a better way to go, right? Well, maybe. It still depends on a lot of other things, like how hard it actually is to do the separation. And getting into that level of detail will be a subject of an upcoming post… One thought on “Enrichment, or how I learned to stop worrying and love the SWU, part 2” 1. Lithium has a number of uses in which the isotopic composition is of no importance. This suggests that if the preperation of the Lithium for seperation is free than the tails would be very little depleted in Li 7 at least until LFTRs became common. Note that seperation of isotopes of Lithium is an established procedure as purified Li 6 is used in production of Tritium for bombs. That means that the feed for the Li 7 needed in a LFTR should be the tails from the seperation of Li 6 rather than natural lithium.	{"url":"https://energyfromthorium.com/2010/08/07/loveswu2/","timestamp":"2024-11-06T05:37:50Z","content_type":"text/html","content_length":"56039","record_id":"<urn:uuid:54cb605b-fffb-4cf8-a618-2fc17e163d6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00049.warc.gz"}
Cylindrical decomposition : a way to solve 1st order logic of the real numbers Add to your list(s) Download to your calendar using vCal If you have a question about this talk, please contact Ian Orton. I will explain what is exactly 1st order logic of the real numbers and show why this is unclear whether it is even decidable. Tarski found an algorithm in 1951 but with very (very) high complexity and I will present Collins’ algorithm from 1975, which is much better. The problem is also shown to be really hard (EXPSPACE I will present the algorithm in a semi formal way (formal would require several hours) to give both the intuition and a few interesting facts, and I will also try to explain why the problem is still interesting, i.e. give some applications. This talk is part of the Logic & Semantics for Dummies series. This talk is included in these lists: Note that ex-directory lists are not shown.	{"url":"https://talks.cam.ac.uk/talk/index/69157","timestamp":"2024-11-05T00:36:38Z","content_type":"application/xhtml+xml","content_length":"11888","record_id":"<urn:uuid:e385785c-5c3d-414e-8dc5-e3546652fe0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00527.warc.gz"}
After a new policy is enacted, we often want to estimate what effects the policy had. Difference-in-differences (diff-in-diff) is one way to estimate the effects of new policies. To use diff-in-diff, we need observed outcomes of people who were exposed to the intervention (treated) and people not exposed to the intervention (control), both before and after the intervention. For example, suppose California (treated) enacts a new health care law designed to lower health care spending, but neighboring Nevada (control) does not. We can estimate the effect of the new law by comparing how the health care spending in these two states changes before and after its implementation. Thanks to its apparent simplicity, diff-in-diff can be mistaken for a “quick and easy” way to answer causal questions. However, as we peer under the hood of diff-in-diff and illuminate its inner workings, we see that the reality is more complex. The table below provides a guide to the notation we use throughout this site. Symbol Meaning $Y(t)$ Observed outcome at time $t$ $A=0$ Control $A=1$ Treated $t=1,\ldots,T_0$ Pre-treatment times $t=T_0+1,\ldots,T$ Post-treatment times $Y^a(t)$ Potential outcome with treatment $A = a$ at time $t$ $X$ Observed covariates $U$ Unobserved covariates At the outset of any analysis, we first define a study question, such as “Did the new California law actually reduce health care spending?” This particular question is aimed at determining causality. That is, we want to know whether the new law caused spending to go down, not whether spending went down for other reasons. Next, we transform our question into a statistical quantity called a target estimand. The target estimand, or target parameter, is a statistical representation of our policy question. For example, the target estimand might be “the average difference in health care spending in California after the new law minus average health care spending in California if the law had not been passed.” This target estimand is written in terms of potential outcomes. In our toy scenario, California has two potential outcomes: health care spending under the new law and health care spending without the new law. Only one of these is observable (spending with the new law); the other is unobservable because it didn’t happen (spending without the new law). Third, we choose an estimator, which is an algorithm that uses data to help us learn about the target estimand. Here, we focus on the diff-in-diff estimator, which relies on some strong assumptions, including that health care spending in Nevada can help us understand what would have happened in California without the new law. That’s how we can use observed data to learn about a target estimand that is written in terms of unobservable outcomes. More on this later. With all these elements in place, now we can actually compute our estimate, a value of the estimand found by applying the estimator to the observed data. To recap, • The quantity we care about is called the estimand. We choose a target estimand that corresponds to our policy question and express it in terms of potential outcomes. • The algorithm that takes data as input and produces a value of the estimand is called the estimator. • The estimator’s output, given data input, is called the estimate. This value represents our best guess at the estimand, given the data we have. As noted above, we define the target estimand in terms of potential outcomes. In the California example, we used the average effect of treatment on the treated (ATT). This compares the potential outcomes with treatment to the potential outcomes with no treatment, in the treated group. For a diff-in-diff, the ATT is the effect of treatment on the treated group in the post-treatment period. Written mathematically, the ATT is Average effect of treatment on the treated (ATT) \[\begin{equation} ATT \equiv \mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1\right] \end{equation}\] Recall that $Y^a(t)$ is the potential outcome given treatment $a$ at time $t$. Here, $t = 2$ represents the post-treatment period, $a = 1$ represents treatment and $a = 0$ represents no treatment. Translated literally, the equation is \[\begin{equation} \mbox{Expected}\left[\mbox{Spending in CA with the new law} - \mbox{Spending in CA without the new law}\right] \end{equation}\] If we could observe the potential outcomes both with treatment and with no treatment, estimating the ATT would be easy. We would simply calculate the difference in these two potential outcomes for each treated unit, and take the average. However, we can never observe both potential outcomes at the same time. In the treated group, the potential outcomes with treatment are factual (we can observe them), but the potential outcomes with no treatment are counterfactual (we cannot observe them). So how do we estimate the ATT when the some of the potential outcomes are unobservable? In diff-in-diff, we use data from the control group to impute untreated outcomes in the treated group. This is the “secret sauce” of diff-in-diff. Using the control group helps us learn something about the unobservable counterfactual outcomes of the treated group. However, it requires us to make some strong assumptions. Next, we discuss assumptions required for diff-in-diff. For diff-in-diff, the treatment status of a unit can vary over time. However, we only permit two treatment histories: never treated (the control group) and treated in the post-intervention period only (the treated group). Thus, we will use $A=0$ and $A=1$ to represent the control and treated groups, with the understanding that the treated group only receives treatment whenever $t > T_0$ (see notation). Every unit has two potential outcomes, but we only observe one — the one corresponding to their actual treatment status. The consistency assumption links the potential outcomes $Y^a(t)$ at time $t $ with treatment $a$ to the observed outcomes $Y(t)$. Consistency Assumption \[ Y(t) = (1 - A) \cdot Y^0(t) + A \cdot Y^1(t) \] If a unit is treated $(A=1)$, then the observed outcome is the potential outcome with treatment $Y(t) = Y^1(t)$ and the potential outcome with no treatment $Y^0(t)$ is unobserved. If a unit is not treated $(A=0)$, then $Y(t) = Y^0(t)$ and $Y^1(t)$ is unobserved. However, we also assume that future treatment does not affect past outcomes. Thus, in the pre-intervention period, the potential outcome with (future) treatment and the potential outcome with no (future) treatment are the same. We write this assumption mathematically as Arrow of time \[ Y(t) = Y^0(t) = Y^1(t),\; \mbox{for}\ t \leq T_0 \] For recent work that relaxes the consistency assumption, see Wang (2023). Wang (2023) considers a panel data setting with interference over both space and time. This means that any individual unit’s potential outcome at a particular time period can depend not only on a unit’s individual assignment, but also on the treatment status of all units over all time periods. Therefore, each unit in each time period may have more than only two potential outcomes. In this setting, Wang (2023) defines the marginalized individualistic potential outcome, i.e., the average of all potential outcomes for unit $i$ at time $t$ (holding fixed the treatment history of unit $i$ up to time period $t$) corresponding to all the ways other units could be treated or untreated over all time periods. The marginalized individualistic treatment effect is simply a contrast between the marginalized individualistic potential outcomes of two treatment histories, and the overall marginalized average effect (AME) for the contrast of any two treatment histories is simply the average of the marginalized individualistic treatment effects. In practice, Wang (2023) also makes the assumption that future treatment does not affect past outcomes, which reduces the number of potential outcomes for each unit under a particular treatment history. This assumption, along with a particular “ignorability” condition facilitates estimation and inference, both of which roughly adhere to the treatments in the Estimation and Inference sections below. Counterfactual assumption (Parallel Trends) A second key assumption we make is that the change in outcomes from pre- to post-intervention in the control group is a good proxy for the counterfactual change in untreated potential outcomes in the treated group. When we observe the treated and control units only once before treatment $(t=1)$ and once after treatment $(t=2)$, we write this as: Counterfactual Assumption (1) \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 1\right] = \\ \nonumber \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 0\right] \end{align}\] This is an assumption — not something we can test — because it involves unobserved counterfactual outcomes, namely $Y^0(2)$ for $A = 1$. In the shiny app embedded below, we can see what the counterfactual assumption does and how we calculate the ATT under our assumptions. The solid black lines represent the observed data. When we click the “Impute Counterfactual from Control to Treated” button, the slope of the line of the control group is imputed to the treated group (dashed line). Finally, clicking the “Show diff-in-diff effect button” reveals how we calculate the average effect of treatment on the treated (ATT). Traditionally, this assumption is called the parallel trends assumption, but as we will soon see, that term can be ambiguous. Positivity Assumption Lastly, we make a positivity assumption. With the positivity assumption, we assume that treatment is not determinant for specific values of $X$. Thus, for any $X = x$, the probability of being treated (or untreated) lies between 0 and 1, not inclusive. Positivity Assumption \[\begin{equation} 0 < P(A = 1 \| X) < 1 \; \text{ for all } X. \end{equation}\] We will invoke the positivity assumption explicitly when we discuss semiparametric and nonparametric estimators. Using the assumptions above, we can re-write the target estimand (which involved unobserved counterfactuals) in a form that depends only on observed outcomes. This process is called “identification”. For diff-in-diff, identification begins with the ATT, applies the Counterfactual Assumption (1) and the Consistency Assumption, and ends with the familiar diff-in-diff estimator. The result is the familiar diff-in-diff estimator \[\begin{align} ATT &\equiv \mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1\right] \\ &= \lbrace \mathbb{E}\left[Y(2) \mid A = 1\right] - \mathbb{E}\left[Y(1) \mid A = 1\right] \rbrace - \\ & \ \ \ \ \ \ \lbrace \mathbb{E}\left[Y(2) \mid A = 0\right] - \mathbb{E}\left[Y(1) \mid A = 0\right] \rbrace \end{align}\] For a straightforward estimate of the ATT, we could simply plug in the sample averages for the four expectations on the right-hand side: 1. The post-intervention average of the treated group for $\mathbb{E}\left[Y(2) \mid A = 1\right]$; 2. The pre-intervention average of the treated group for $\mathbb{E}\left[Y(1) \mid A = 1\right]$; 3. The post-intervention average of the control group for $\mathbb{E}\left[Y(2) \mid A = 0\right]$; 4. The pre-intervention average of the control group for $\mathbb{E}\left[Y(1) \mid A = 0\right]$. Finding the standard error for this estimator is a little more complex, but we could estimate it by bootstrapping, for example. Sometimes the counterfactual assumption may hold only after conditioning on some observed covariates, and the identification becomes more complex. More on this in the Confounding section. When we observe the treated and control units multiple times before and after treatment, we must adapt the target estimand and identifying assumptions accordingly. Let’s start by looking at possible target estimands. Target Estimands We can calculate the ATT at any of the post-treatment time points Time-varying ATT at each time point For some $t > T_0$, \[\begin{equation} ATT(t) \equiv \mathbb{E}\left[Y^1(t) - Y^0(t) \mid A = 1\right] \end{equation}\] or we can compute the average ATT across the post-treatment time points Averaged ATT over all time points \[\begin{equation} ATT \equiv \mathbb{E}\left[\overline{Y^1}_{\{t>T_0\}} - \overline{Y^0}_{\{t>T_0\}} \mid A = 1\right] \end{equation}\] Here, the overbar $\overline{{\color{white} Y}}$ indicates averaging and the subscript $_{\{t>T_0\}}$ refers to the time points over which the outcome is averaged. The above estimands make sense when the treatment is administered at the same time for all treated groups. When treatment timing differences occur, Athey and Imbens (2022) and Goodman-Bacon (2021) discuss the weighted estimands that arise. We discuss diff-in-diff when there is variation in treatment timing briefly in the estimation section. Assumptions for Multiple Time Points What kind of assumptions do we need to estimate the ATTs above? We consider several counterfactual assumptions that may require: 1. parallel average outcomes in pre- to post-intervention periods 2. parallel outcome trends across certain time points, or 3. parallel outcome trends across all time points. First, consider an assumption that averages over the pre- and post-intervention time points, effectively collapsing back to the simple two-period case. Counterfactual Assumption (2a) Avg pre, avg post \[\begin{align} \mathbb{E} \left[\overline{Y^0}_{\{t > T_0\}} - \overline{Y^0}_{\{t \leq T_0\}} \mid A = 0\right] = \\ \mathbb{E} \left[\overline{Y^0}_{\{t > T_0\}} - \overline{Y^0}_{\{t \leq T_0\}} \mid A = 1\right] \end{align}\] Here, we assume that the difference between the average of the pre-intervention outcomes and the average of the untreated post-intervention outcomes is the same for both treated and control groups. To identify the time-averaged ATT using this assumption, we use the same identification process as in the simple case with only one observation in each of the pre- and post-intervention periods. Next, we restrict our focus to only two time points: one pre-intervention and one post-intervention. Counterfactual Assumption (2b) One pre, one post For some $t^* > T_0$, there exists a $t' \leq T_0$ such that \[\begin{align} \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 1\right] = \\ \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 0\right] \end Counterfactual Assumption (2b) essentially disregards time points other than these two. That is, the other time points need not satisfy any “parallel trends” assumption. While this assumption is perfectly valid if true, using such an assumption requires justification. For instance, why do we believe this assumption is satisfied for two time points but not the rest? To identify the ATT using this assumption, we again use the same identification process as in the simple case, since we are back to considering only one time point pre-intervention and one time point post-intervention. Counterfactual Assumption (2c) Avg pre, one post For some post-treatment time point $t^ > T_0$, \[\begin{align} \mathbb{E}\left[Y^0(t^) - \overline{Y^0}_{\{t \leq T_0\}} \mid A = 0\right] = \\ \mathbb{E}\left[Y^0(t^) - \overline{Y^0}_{\{t \ leq T_0\}} \mid A = 1\right] \end{align}\] In this version we assume that there are “parallel trends” between one post-intervention time point and the average of the pre-intervention outcomes. Counterfactual Assumption (2d) All pre, one post For some $t^* > T_0$ and each $t' \leq T_0$: \[\begin{align} \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 1\right] = \\ \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 0\right] \end{align}\] Counterfactual Assumption (2d) is a stricter version of (2c), where parallel trends holds at post-intervention time $t^$ and every possible pre-intervention time point. Note that if Counterfactual Assumption (2d) holds, then Counterfactual Assumption (2c) also must hold, but the reverse is not necessarily true. Finally, we get to the assumption we’ve been waiting for, in which the untreated potential outcomes evolve in parallel in the treatment and control groups at pre- and post-intervention time point. This is the strictest version of parallel trends and is what researchers often mean by “parallel trends”. Counterfactual Assumption (2e) All pre, all post For each $t^ > T_0$ and each $t' \leq T_0$: \[\begin{align} \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 1\right] = \\ \mathbb{E}\left[Y^0(t^) - Y^0(t') \mid A = 0\right] \end{align}\] This is the most restrictive because it requires parallel evolution of the untreated outcomes at all pre- and post-intervention time points. Many papers which use diff-in-diff methodology have a line or two stating that they assume “parallel trends” without much further elaboration. As the above assumptions illustrate, the counterfactual assumptions are both more diverse and more specific than this general statement. Sometimes authors explicitly impose parallel trends in the pre-treatment period only. This “parallel pre-trends” assumption must be paired with a second assumption called “common shocks” (see Dimick and Ryan 2014; Ryan, Burgess, and Dimick 2015): Parallel pre-trends In the pre-intervention period, time trends in the outcome are the same in treated and control units. Common shocks In the post-intervention period, exogenous forces affect treated and control groups equally. Stating the assumptions this way can be misleading for two reasons. First, not all identifying assumptions require strict parallel pre-intervention trends. For example, Counterfactual Assumption (2d) requires parallel trends in the pre-intervention period, but only Counterfactual Assumption (2e) demands parallel trends throughout the study. Second, parallel pre-intervention trends is not an assumption at all! It is a testable empirical fact about the pre-intervention outcomes, involving no counterfactuals. (See below for more discussion of parallel trends testing.) By contrast, common shocks is an untestable assumption involving exogenous forces that are likely unknown to the researcher. We prefer the counterfactual assumptions above because they are explicitly stated in terms of counterfactual outcomes, directly identify the diff-in-diff estimator, and avoid any false sense of security from tests of parallel trends. Which assumptions are reasonable in the data you see? Use the app below to explore potential outcomes that satisfy each of the above assumptions. The app randomly generates outcomes for the control group then randomly generates untreated outcomes (counterfactuals in the post-intervention period) for a treated group that satisfy each assumption above. What do you have in mind when you say that you assume “parallel trends”? Does this match what you see in the app? Testing for Parallel Trends in the Pre-Treatment Period Provided there are enough time points, researchers often test whether trends are parallel in the pre-intervention period. But the test of parallel trends is neither necessary nor sufficient to establish validity of diff-in-diff (Kahn-Lang and Lang 2020). Moreover, conditioning a test of the diff-in-diff effect on “passing” a test for parallel pre-period trends changes the performance of the whole procedure (Roth 2022). Recognizing that editors, reviewers, and readers may still want to see tests of parallel trends despite these issues, one possible work-around is to reformulate the test using a different null hypothesis, namely one designed to show “equivalence” of the pre-period trends (Hartman and Hidalgo 2018). Other possibilities include procedures that re-formulate the model to allow for non-parallel pre-period trends and focus on how this impacts treatment effect estimates (Bilinski and Hatfield 2020; Rambachan and Roth 2022). In addition, robustness checks such as pre-period placebo intervention tests can assess the sensitivity of the conclusions to pre-period trend differences. For an excellent review of parallel trends testing in diff-in-diff, see McKenzie’s World Bank blog post part1 and part2. Equivalence tests Our primary concern with (the usual) hypothesis tests of parallel trends (one in which the null hypothesis asserts parallel trends) is that we can never actually prove what we set out to prove. The only conclusions that emerge from a conventional hypothesis test are “fail to reject the null” or “reject the null.” The decision to “fail to reject” is decidedly different than accepting the null. However, there is another problem with hypotheses for testing assumptions. Let’s delve briefly into a thought experiment where the “parallelness” of trends is captured by a single parameter $\theta $ (where $\theta = 0$ denotes two lines that are perfectly parallel). Deviations from zero (either negative or positive) denote departures from “parallelness” at varying magnitudes. The hypotheses for testing parallel trends look something like: $H_0:$ $\theta = 0$ $H_1:$ $\theta \neq 0$. If we have a big enough sample size we can reject the null if the true value of $\theta$ is 5 or 3 or 1 or 0.01. But do we really care about deviations of magnitude 0.01 compared to deviations of 5? It would be better if we could insert expert knowledge into this test and incorporate some margin for deviation in our test. Equivalence tests do just this, while at the same time reversing the order of the hypotheses. Let $\tau$ denote an acceptable margin for deviations from parallel trends so that if $\|\theta\| \leq \tau$, we feel OK saying that the trends are parallel (or close enough). The hypotheses for an equivalence test could be something like: $H_0:$ $\|\theta\| > \tau$ $H_1:$ $\|\theta\| \leq \tau$. Equivalence tests are nothing new. They are sometimes used in clinical trials to determine if a new drug is no worse than a standard-of-care drug, for example. They also happen to provide an intuitive approach to testing for parallel trends in the pre-treatment periods. Unfortunately, this setup won’t solve all our (diff-in-diff) problems. Sample size considerations can be a hindrance in assumption testing, for one. However, this sort of issue arises no matter how we construct our testing framework, so we might as well set up our tests in a way that is more intuitive. The staggered adoption setting is when units enter into treatment at different time periods (but then do not ever switch out of treatment). Goodman-Bacon (2021) showed that the common TWFE estimator implicitly targets certain aggregation schemes for combining the ATTs for specific groups and time periods. Callaway and Sant’Anna (2021) expand on this argument by deriving identification assumptions for each specific group-time ATT, and then propose different aggregation schemes that researchers can use depending on the application at hand. Callaway and Sant’Anna (2021) provide two different identification assumptions, both of which are versions of parallel trends conditional on baseline covariates. The two assumptions differ in terms of the comparison group to which parallel trends applies. The first assumption compares a treated group at a particular time period to units that are never treated over all time periods. The second assumption compares a treated group at a particular time period to units who (at that time) have yet to enter treatment. For each of these two assumptions, Callaway and Sant’Anna (2021) provide three different ways of writing the treated group’s expected change in counterfactual outcomes in terms of the comparison’s groups expected change in observable outcomes. The first is in terms of a population-level outcome regression. The second is in terms of population-level outcomes weighted by the inverse of their treatment assignment probabilities. The third is the ``doubly robust’’ estimand in terms of both the population-level outcome regression and inverse probability weighted outcomes. Consider, for example, the “never treated” comparison group and the population-level outcome regression. Conditional parallel trends based on a “never-treated” group is defined as \[\begin{equation} \mathbb{E}\left[Y_{t}(0) - Y_{t - 1} \mid X, G_g = 1\right] = \mathbb{E}\left[Y_{t}(0) - Y_{t - 1} \mid X, C = 1\right], \end{equation}\] where $t \geq g - \delta$ with $\delta \geq 0$ representing some known number of periods by which units may anticipate treatment, the random variable $G_g$ is an indicator for whether a unit is treated at time $g$, and $C$ is an indicator for whether a unit belongs to the “never-treated” group. We can use this assumption to write the ATT, which is a function of the left-hand-side of the conditional parallel trends expression, a counterfactual quantity, in terms of the right-hand-side of this expression, an observable quantity. Callaway and Sant’Anna (2021) then stipulate that the population-level outcome regression is $\mathbb{E}\left[Y_t - Y_{g - \delta - 1} \mid X, C = 1\right]$. This population-level outcome regression is the same as the right-hand-side of Equation (1). Therefore, conditional parallel trends based on a “never-treated” group implies that the ATT can be identified by the population level outcome regression. Similar arguments establish identification of the ATT via the IPW and doubly-robust estimands. Related to our discussion of parallel trends is a discussion on confounding in diff-in-diff. In most settings, a confounder is a factor associated with both treatment and outcomes. This is why randomized trials are not subject to bias through confounders — no factor is associated with the randomly assigned treatment. In other words, the potential outcomes and treatment are independent. Unconditionally unconfounded \[ Y^a \perp A \] Sometimes, treatment may be randomized within levels of a covariate (conditionally randomized) and we write this relation: Conditionally unconfounded \[ Y^a \perp A \mid X \] In both of these versions, the treatment $A$ is independent of the potential outcomes $Y^a$, either unconditionally or conditional on $X$. In practice, these relations are only satisfied in randomized trials; otherwise, there is no guarantee that $X$ is sufficient to make $A$ and $Y^a$ conditionally independent. Even if we continue collecting covariates, it is likely that some unmeasured covariates $U$ are still a common cause of $A$ and $Y^a$. In diff-in-diff studies, the notion of confounding is fundamentally different. As alluded to in the previous section, confounding in diff-in-diff violates the counterfactual assumption when (1) the covariate is associated with treatment and (2) there is a time-varying relationship between the covariate and outcomes or there is differential time evolution in covariate distributions between the treatment and control populations (the covariate must have an effect on the outcome). To see more in-depth discussions of confounding for diff-in-diff, we recommend Wing, Simon, and Bello-Gomez (2018) or Zeldow and Hatfield (2021). Time-Invariant versus Time-Varying Confounding In an upcoming section, we will explicitly show the effect that a confounder has on the parallel trends assumption. We begin our discussion of confounding in diff-in-diff by highlighting an important distinction: time-invariant and time-varying. When we have a covariate that satisfies certain properties (associated with treatment group and with outcome trends), parallel trends will not hold. As the name suggests, a time-invariant confounder is unaffected by time. It is typically measured prior to administering treatment and remains unaffected by treatment and other external factors. Another, more pernicious, type of confounder is the time-varying confounder. Time-varying covariates freely change throughout the study. Examples of time-varying covariates seen in observational studies are concomitant medication use and occupational status. Time-varying covariates are particularly troublesome when they predict treatment status and then are subsequently affected by treatment, which in turn affects their treatment status at the next time point. In effect, time-varying confounders act as both a confounder and a mediator. However, recall that treatment status in diff-in-diff is monotonic: the comparison group is always untreated, and the treated group only switches once, from untreated to treated. With these treatment patterns in mind, let’s talk a bit about time-varying confounders. We need to assess whether the time-varying covariates are affected by treatment or not. In most cases, we cannot know for certain. For example, in a study assessing the effect of Medicaid expansion on hospital reimbursements, we can be fairly certain that the expansion affected insurance coverage in the population. On the other hand, factors such as the average age of the population might change from pre- to post-expansion. How much would Medicaid expansion have affected that change? If the validity of our diff-in-diff model relied on adjusting for these factors, we would have to account for these covariates in some way. Our next section will talk about estimation in diff-in-diff studies, including how to deal with confounding. Our key identifying assumptions provide a “blueprint” for estimation and inference. Having expressed the ATT in terms of observable quantities in the population, estimation often proceeds by plugging in sample analogues of these population quantities to achieve unbiased and consistent estimation under random sampling. With the key identifying assumptions for diff-in-diff freshly in mind, we now turn our attention to estimating causal effects. Recall the simple estimator we identified above: \[\begin{align} ATT &\equiv \mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1\right] \\ &= \lbrace \mathbb{E}\left[Y(2) \mid A = 1\right] - \mathbb{E}\left[Y(1) \mid A = 1\right] \rbrace - \\ & \ \ \ \ \ \ \lbrace \mathbb{E}\left[Y(2) \mid A = 0\right] - \mathbb{E}\left[Y(1) \mid A = 0\right] \rbrace . \end{align}\] Using sample means to estimate the ATT works well when there are two time periods and few covariates. In more challenging applications with many time points and many confounders, we will often specify a model that can readily be extended to more complex settings. Our discussion herein will motivate the use of regression as one way to estimate diff-in-diff parameters. A typical linear model for the untreated outcomes $Y^0_{it}$ (Athey and Imbens (2006) or Angrist and Pischke (2008) p. 228, for example) is written \[\begin{equation} Y^0_{it} = \alpha + \delta_t + \gamma I(a_i = 1) + \epsilon_{it}\;. \end{equation}\] The counterfactual untreated outcomes are presented as a sum of an intercept $\alpha$, main effects for time $\delta_t$, a main effect for the treated group with coefficient $\gamma$, and a normally distributed error term $\epsilon_{it}$. We first present a model for the untreated outcomes assuming no effect of covariates on the outcome. We then transition to the more realistic case that covariates are present and have real effects on the outcome. Now we can simply connect the untreated outcomes to the observed outcomes $Y_{it}$ using the relation \[\begin{equation} Y_{it} = Y^0_{it} + \beta D_{it}\;, \end{equation}\] where $D_{it}$ is an indicator of the treatment status of the $i^{th}$ unit at time $t$, and $\beta$ is the traditional diff-in-diff parameter. Note that $D_{it}$ often will be equivalent to an interaction between indicators for the treatment group and the post-treatment period, $D_{it} = a_i \cdot I(t > T_0)$. This will be the case when all treated units receive the intervention at the same time. When there is variation in treatment timing, $D_{it}$ cannot be interpreted as an interaction because pre- and post-treatment periods are not well defined for the control group! These models impose a constant diff-in-diff effect across units. For more about this strict assumption, please see our discussion of Athey and Imbens (2006). Let’s return to the simple scenario of two groups and two time periods $\left(t \in \{1,2\}\right)$. The model for $Y^0_{it}$ reduces to \[\begin{equation} Y^0_{it} = \alpha + \delta I(t = 2) + \gamma I(a_i = 1) + \epsilon_{it}\;. \end{equation}\] If this model is correctly specified, Counterfactual Assumption (1) holds since \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 1\right] &= (\alpha + \delta + \gamma) - (\alpha + \gamma) \\ &= \delta \end{align}\] \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 0\right] &= (\alpha + \delta ) - \alpha \\ &= \delta\;. \end{align}\] Now, let’s introduce the effect of a covariate and see how it affects our counterfactual assumption. For example, write our model for $Y^0$ including an additive effect of a covariate $X$, \[\ begin{equation} Y^0_{it} = \alpha + \delta_t + \gamma_a + \lambda_t x_i + \epsilon_{it}\;. \end{equation}\] Here, the effect of $X$ on $Y^0$ may vary across time, so $\lambda$ is indexed by \ Initially, we assume a constant effect of $X$ on $Y^0$ at $t = 1$ and $t = 2$, so $\lambda_t = \lambda$. In this case, Counterfactual Assumption (1) is still satisfied even if the distribution of $X$ differs by treatment group because these group-specific means cancel out: \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 1\right] &= (\alpha + \delta + \gamma + \lambda \mathbb{E}\left\{X \mid A = 1\right\} ) - \\ & \ \ \ \ \ \ (\alpha + \gamma + \lambda \mathbb {E}\left\{X \mid A = 1\right\}) \\ &= \delta \end{align}\] \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 0\right] &= (\alpha + \delta + \lambda \mathbb{E}\left\{X \mid A = 0\right\} ) - \\ & \ \ \ \ \ \ (\alpha + \lambda \mathbb{E}\left\{X \mid A = 0\right\}) \\ &= \delta\;. \end{align}\] Lastly, we let the effect of $X$ on $Y^0$ vary across time ($\lambda$ indexed by $t$), after which we have a different story: \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 1\right] &= (\alpha + \delta + \gamma + \lambda_2 \mathbb{E}\left\{X \mid A = 1\right\} ) - \\ & \ \ \ \ \ \ (\alpha + \gamma + \lambda_1 \ mathbb{E}\left\{X \mid A = 1\right\}) \\ &= \delta + \lambda_2 \mathbb{E}\left\{X \mid A = 1\right\} - \\ & \ \ \ \ \ \ \lambda_1 \mathbb{E}\left\{X \mid A = 1\right\} \end{align}\] \[\begin{align} \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 0\right] &= (\alpha + \delta + \lambda_2 \mathbb{E}\left\{X \mid A = 0\right\} ) - \\ & \ \ \ \ \ \ (\alpha + \lambda_1 \mathbb{E}\left\{X \ mid A = 0\right\}) \\ &= \delta + \lambda_2 \mathbb{E}\left\{X \mid A = 0\right\} - \\ & \ \ \ \ \ \ \lambda_1 \mathbb{E}\left\{X \mid A = 0\right\} \end{align}\] are not necessarily equal. They are only equal if the effect of $X$ on $Y^0$ is constant over time (i.e., $\lambda_1 = \lambda_2$) or the mean of the covariate in the two groups is the same (i.e., $\mathbb{E}\left\{X \mid A = 1\right\} = \mathbb{E}\left\{X \mid A = 0\right\}$). This illustrates an important connection between the counterfactual assumption and the regression model and introduces the notion of confounding in diff-in-diff. To better visualize this, use the app below to explore time-varying confounding in simulated data. The y-axis is the mean of the untreated potential outcomes ($Y^0$) and the x-axis is time. Remember: for Counterfactual Assumption (1) to hold, the lines connecting $Y^0$ values in the treated and control groups must be parallel. Whenever the lines are not parallel (i.e., the differential change over time is not 0), Counterfactual Assumption (1) is violated. • What happens when the covariate distributions are different in the treated and control groups? (hint: change the values of $Pr(X=1\|A=0)$ and $Pr(X=1\|A=1)$) • What happens when the covariate effect varies over time? (hint: change the effects of $X$ on $Y^0$ at $t = 1$ and $t = 2$) As you may have discovered in the app, $X$ is a confounder if two conditions hold: 1. $X$ is associated with treatment ($A$) and 2. the effect of $X$ on $Y$ varies across time. For the remaining parts of the “Estimation” section, we will give general overviews of several of the more common ways to estimate diff-in-diff parameters. We start with linear regression, then discuss matching frameworks, and conclude with semiparametric and nonparametric estimators. Probably the most commonly used estimator in diff-in-diff is a linear regression model. At the very least, the regression model will contain a treatment indicator, an indicator that equals one whenever we are in a post-treatment period, and their interaction. This interaction is typically taken to be the parameter of interest and if the usual diff-in-diff assumptions are true, will equal the ATT. When using R to perform analysis, our code will look something like: lm(y ~ a * post) Here, $a$ is a treatment indicator and $post$ is an indicator for post-treatment period. The notation a * post gives main effects for $a$ and $post$ and their interaction. In reality, most regression models will not be that sparse (only two indicators and an outcome). For example, we frequently encounter regression models such as this: This example, taken from McWilliams et al. (2014), is much more typical of the kinds of regression models we see in applied settings. Without going into unnecessary detail on this paper’s background, the covariate called “ACO_indicators” is the treatment variable. The covariate $\beta_{3k}$ represents the causal effect of interest. However, there are many other terms in the model, including time fixed effects, other fixed effects (“HRR_indicators”), and covariates. We talk about the inclusion of fixed effects below, followed by a discussion on adjusting for covariates using regression Fixed effects in diff-in-diff Let’s talk about fixed effects briefly (see Mummolo and Peterson (2018) for a more in-depth discussion of fixed effects models and their interpretation). Fixed effects, particularly unit-level fixed effects, are used in causal inference to adjust for unmeasured time-invariant confounders. Of course, there are trade-offs. The discussion from Imai and Kim (2019) explains that using unit fixed effects comes at the cost of capturing the dynamic relationship between the treatment and the outcome. Kropko and Kubinec (2020) discuss the common two-way fixed effects model, which includes unit and time fixed effects. Their main point is that estimates coming from two-way fixed effect models are difficult to interpret when we have many time periods. When we have the canonical (two-period, binary treatment) diff-in-diff setup, the $\beta$ coefficient from the two-way fixed effect model $\ left(y_{it} = \alpha_i + \delta_t + \beta D_{it} + \epsilon_{it}\right)$ equals the usual estimate. As more time periods are added within the fixed-effects framework, we implicitly add supplementary assumptions. In particular, the diff-in-diff effect is assumed homogenous across time and cases. Homogeneity across time is a stringent assumption that says the diff-in-diff effect is the same no matter how close or far apart the time periods are. We say this not to discourage use of two-way fixed effect models, but to discourage automatic use of them. True they work well for some cases (when we need to adjust for unmeasured time-invariant confounders), but we really need to examine our research goals on an application-by-application basis, consider the assumptions implicit in the models we’re thinking of using, and adjust our tools accordingly. What if treated units are treated at different times? Goodman-Bacon (2021) examines the two-way fixed effect regression model $(Y_i(t) = \alpha_i + \delta_t + \beta D_{it} + \epsilon_{it})$ as a diff-in-diffs estimator when there exists treatment variation. It turns out that the TWFE estimator is a weighted combination of all possible $2 \times 2$ diff-in-diff estimators found in the data. For example, consider three groups: an untreated group, an early-treated group and a late-treated group. The $2 \times 2$ diff-in-diff estimators consist of three $2 \times 2$ comparisons: 1. Early-treated and late-treated groups vs. Untreated group in all periods 2. Early-treated group vs. late-treated group in periods before late-group is treated 3. Late treated group and early treated group in periods after early-group is treated The weights of these $2 \times 2$ estimators are based on the sample size in each group and the variance of the treatment variable. Note that the variance of a binary random variable is its expected value multiplied by $1$ minus its expected value. Therefore, the variance of the treatment variable is decreasing in its expected value’s distance from $0.5$. The implication, then, is that $2 \times 2$ comparisons of groups that are closer in size and occur in the middle of the time window will receive the greatest weights. In this setting with variation in treatment timing, Goodman-Bacon (2021) defines the variance-weighted ATT estimand as follows: In each of the three $2 \times 2$ comparisons above, there is an ATT for the treated group in each of that comparison’s post-treatment periods. The variance-weighted ATT estimand calculates the average ATT over all post-treatment periods in each $2 \times 2$ comparison and then sums the average ATT over all $2 \times 2$ comparisons with weights equal to the probability limits of the estimators’ weights described above. So long as each timing group’s ATTs do not vary over time, Goodman-Bacon (2021) shows that a variance-weighted common trends assumption suffices for identification of the variance-weighted ATT. Moreover, when effects are constant both over time and units, the variance-weighted ATT is equal to the overall ATT, which makes the variance-weighted ATT estimand easier to interpret. Bai (2009) describes an interactive fixed effects model that incorporates time-varying dynamics. Each unit is assumed to have an $r$-vector of factor loadings, $\mathbf{\lambda}_i$, multiplies an $r$-vector of common factors at each time point $\mathbf{F}_{t}$. That is, for outcome $Y_{it}$ of unit $i$ at time $t$, the data-generating model is \[ Y_{it} = X_{it}'\beta + \lambda_{i1} F_{1t} + \ldots + \lambda_{ir}F_{rt} + \epsilon_{it}\;, \] where $X_{it}$ are observed covariates. Note that the two-way fixed effects model is a special case of this where $F_{1t} = 1$, $F_{2t} = \delta_t$ and $\lambda_{i1}= \alpha_{i}$, $\lambda_{i2} = 1$. The authors present least-squares estimators for large $N$ and large $T$. Marginal structural models can capture dynamics such as past outcomes affecting future treatments, but cannot account for time-invariant unmeasured confounders. Thus, we can either adjust for time-invariant unmeasured confounders and assume no dynamic relationship between treatment and outcome or we can assume that there are no unmeasured confounders and allow for more complicated relationships between treatment and outcome. Confounding in linear settings If we know how confounding arises, we can address it. For example, if the truth is a linear data-generating model, we can use a linear regression model to address confounding. The flowchart below outlines six linear data-generating models and the appropriate linear regression adjustment for each. Of these six scenarios, two require no adjustment at all. Of the 4 that require adjustment, only one requires the regression adjustment type nearly always found in the literature, i.e., adjusting for a time-varying covariates without any interaction with time. In the other three scenarios with confounding bias, the issue is due, in whole or in part, to time-varying covariate effects. For these cases, including an interaction of covariates with time is crucial to addressing confounding bias. See directed acyclic graphs (DAGs) (together with a brief discussion) for these scenarios by selecting an option below: In this scenario, the covariate $X$ does not vary over time. The arrow from $X$ to $A$ indicates that $X$ is a cause of $A$, satisfying the first requirement of a confounder. Additionally, are arrows from $X$ to $Y(1)$ and to $Y(2)$ as well as an arrow from $A$ to $Y(2)$. [Note: there is no arrow from $A$ to $Y(1)$ because treatment is administered after $Y(1)$.] $\ alpha$ is the effect of $X$ on $Y(1)$, and $\beta$ is the effect of $X$ on $Y(2)$. When $\alpha = \beta$, the effect of $X$ is time-invariant and we do not require covariate adjustment. When $\alpha \neq \beta$, we must adjust for the interaction of $X$ with time. In this scenario, the time-varying covariate $X$ in periods 1 and 2 is denoted $X(1)$ and $X(2)$. There is no arrow connecting $A$ to $X(2)$, indicating that treatment does not affect the evolution of $X(1)$ to $X(2)$. When $\alpha = \beta$, the effect of $X$ is time-invariant and we do not need to adjust for the covariate. When $\alpha \neq \beta$, we must adjust for the interaction of $X$ with time. In this scenario, the time-varying covariate $X$ evolves differentially by treatment group. However, most diff-in-diff analyses implicitly or explicitly assume that $X$ does not evolve based on treatment group. See our nonparametric section below. One diff-in-diff estimator that directly accounts for this phenomenon is Stuart et al. (2014), which we discuss in more detail below. When $\ alpha = \beta$, the effect of $X_t$ on $Y^0$ is time-invariant and it suffices to adjust only for $X_t$. When $\alpha \neq \beta$, we must adjust for the interaction of $X_t$ with time. Matching estimators adjust for confounding by balancing the treatment groups on measured covariates. Rather than using the entire sample population to estimate the diff-in-diff effect, units in the control group are selected by on their “closeness” to units in the treated group. We introduce this section by a series of tweets about a recent Daw and Hatfield (2018a) paper on matching and regression to the mean. Do you use diff-in-diff? Then this thread is for you. You’re no dummy. You already know diverging trends in the pre-period can bias your results. But I’m here to tell you about a TOTALLY DIFFERENT, SUPER SNEAKY kind of bias. Friends, let’s talk regression to the mean. (1/N) pic.twitter.com/M2tEEsBiyH — Laura A. Hatfield, PhD (@laura_tastic) July 27, 2018 The argument focuses on estimators that match on outcomes in the pre-treatment period. Matching on pre-treatment outcomes is attractive in diff-in-diff because it improves comparability of the groups and possibly of their outcome trends. The crux of the argument in Daw and Hatfield (2018a) is that matching estimators can be dangerous in diff-in-diff settings due to regression to the mean. Regression to the mean is a notorious phenomenon in which extreme values tend to revert to the group mean on subsequent measurements. For example, if we select the ten students who score highest on an exam, at a subsequent exam, the average score for these ten students would drop towards the class mean. For diff-in-diff, the effect is similar. By constraining the pre-treatment outcomes to be similar, we are more likely to select units of the group that are higher or lower than their respective group means. Once the matching constraint is dropped (in the post-treatment period), these units’ means can revert back to their respective group’s mean and possibly yield a spurious diff-in-diff effect. So in some cases, matching can actually introduce bias. So how can we know whether matching is useful or harmful in our diff-in-diff study? Unfortunately sometimes we can’t know. Take the paper Ryan, Burgess, and Dimick (2015) which presents a simulation study using matching estimators and shows that matching can reduce confounding bias. In their paper, they sampled the treated and control groups from the same population, but the probability of being part of the treated group increased for high pre-treatment outcomes. In contrast, Daw and Hatfield (2018a) set up similar simulations but with treated and controls groups coming from different, but overlapping, populations. For the following thought experiment, assume no diff-in-diff effect is present. If the populations are drawn as in Ryan, Burgess, and Dimick (2015), the two populations have different pre-treatment means. In a diff-in-diff study without matching, these units will regress to the mean in the post-treatment period (but they will regress to the same value since they are drawn from the same population!). This yields a non-zero diff-in-diff effect. Matching actually fixes this issue. If the populations are drawn as in Daw and Hatfield (2018a), the opposite is true. Without matching, the populations are different in the pre-intervention period and remain that way in the post-intervention period (since they are representative of their true populations, there is no regression to the mean). With matching, the populations are constrained to be the same in the pre-treatment period, and once the constraint is released in the post period, the two groups regress back to their group means. So in the Ryan, Burgess, and Dimick (2015) setup, matching is the solution to regression to the mean bias; in the Daw and Hatfield setup, matching is the cause of the regression to the mean Using real life data, there is no way to check empirically whether our groups come from the same population or from different populations. Determining this must come from expert knowledge from how the treatment assignment mechanisms work. To quote Daw and Hatfield (2018b) in the follow-up to their own paper: (R)esearchers must carefully think through the possible treatment assignment mechanisms that may be operating in the real-life situation they are investigating. For example, if researchers are aware that a pay-for-performance incentive was assigned to physicians within a state based on average per-patient spending in the past year, one may be comfortable assuming that treatment assignment mechanism is operating at the unit level (i.e., the potential treatment and control units are from the same population). In contrast, if the same incentive was assigned to all physicians within a state and a researcher chooses a control state based on geographic proximity, it may be more reasonable to assume that treatment assignment is operating at the population level (i.e., the potential treatment and control units are from separate populations). Other researchers have also noted the lurking biasedness of some matching diff-in-diff estimators. Lindner and McConnell (2019), for example, found that biasedness of the estimator was correlated in simulations with the standard deviation of the error term. As the standard error increased, so did the bias. In a pair of papers, Chabé-Ferret (Chabé-Ferret 2015, 2017) similarly concluded that matching on pre-treatment outcomes can be problematic and is dominated by symmetric diff-in-diff. Up to this point, we can think of a diff-in-diff analysis as a four-step process: 1. make assumptions about how our data were generated 2. suggest a sensible model for the untreated outcomes 3. connect the untreated outcomes to the observed outcomes 4. estimate the diff-in-diff parameter (via regression or matching or both) While this process is simple, our estimates and inference can crumble if we’re wrong at any step along the way. We’ve discussed the importance of counterfactual assumptions and inferential procedures. We now turn our attention to the modeling aspect of diff-in-diff. So far, we have discussed only parametric models. Below, we present some semiparametric and nonparametric estimators for diff-in-diff. These give us more flexibility when we don’t believe in linearity in the regression model or fixed unit effects. Semi-parametric estimation with baseline covariates Abadie (2005) addresses diff-in-diff when a pre-treatment covariate differs by treatment status and also affects the dynamics of the outcome variable. In our confounding section above, this is the “Time-invariant X with time-varying effect” scenario. Let’s return to the two-period setting. When a covariate $X$ is associated with both treatment $A$ and changes in the outcome $Y$, the Counterfactual Assumption 1 no longer holds. Thus, Abadie (2005) specifies an identifying assumption that conditions on $X$. That is, Conditional Counterfactual Assumption \[ \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 1, X\right] = \mathbb{E}\left[Y^0(2) - Y^0(1) \mid A = 0, X\right]. \] This assumption does not identify the ATT, but it can identify the CATT, that is, the conditional ATT: Conditional average effect of treatment on the treated (CATT) \[ CATT \equiv \mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1, X\right]. \] The CATT itself may be of interest, or we may want to average the CATT over the distribution of $X$ to get back the ATT. To identify the CATT, repeat the identification steps above with expectations conditional on $X$. As expected, it turns out that \[\begin{align} \mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1, X\right] &= \lbrace \mathbb{E}\left[Y(2) \mid A = 1, X\right] - \mathbb{E}\left[Y(1) \mid A = 1, X \right] \rbrace - \\ & \ \ \ \ \ \ lbrace \mathbb{E}\left[Y(2) \mid A = 0, X\right] - \mathbb{E}\left[Y(1) \mid A = 0, X \right] \rbrace. \end{align}\] Nonparametric estimators for these quantities are easy when $X$ is a single categorical variable. They are simply sample averages for groups defined by combinations of $X$ and $A$: 1. The post-treatment average of the treated group with $X=x$ for $\mathbb{E}\left[Y(2) \mid A = 1, X=x\right]$ 2. The pre-treatment average of the treated group with $X=x$ for $\mathbb{E}\left[Y(1) \mid A = 1, X=x\right]$ 3. The post-treatment average of the control group with $X=x$ for $\mathbb{E}\left[Y(2) \mid A = 0, X=x\right]$ 4. The pre-treatment average of the control group with $X=x$ for $\mathbb{E}\left[Y(1) \mid A = 0, X=x\right]$ However, if $X$ is high-dimensional or contains continuous covariates, these get tricky. Abadie proposes a semiparametric solution using propensity scores. Recall that a propensity score is the estimated probability of treatment given pre-treatment covariate $X$, $P(A = 1 \mid X)$. For this approach to work, we need to use the positivity assumption, which was introduced in the assumption section. That is, Positivity Assumption \[\begin{equation} 0 < P(A = 1 \| X) < 1 \; \text{ for all } X. \end{equation}\] This assumption ensures the estimand is defined at all $X$. If some values of $X$ lead to guaranteed treatment or control (i.e., propensity scores of $0$ or $1$) we should reconsider the study population. With positivity in hand, consider the weighted estimator of Abadie (2005): \[$$\mathbb{E}\left[Y^1(2) - Y^0(2) \mid A = 1\right] = \mathbb{E}\left[\frac{Y(2) - Y(1)}{P(A = 1)} \cdot \frac{A - P(A = 1 \mid X)}{1 - P(A = 1 \mid X)}\right].$$\] To estimate these quantities, we need fitted values of the propensity scores for each unit, i.e., $\hat{P}(A=1 \| X= x_i)$ and then we use sample averages in the treated and control groups. To see the math, We only need the average change in outcomes among the treated units and the weighted average change in outcomes among the control units. The weights are $\frac{\hat{P}(A=1 \| X=x_i)}{1 - \hat{P}(A=1 \| X=x_i)}$. What are these weights sensible? Well, outcome changes among control units with $x_i$ that resemble treated units (i.e., with large $\hat{P}(A=1 \| X=x_i)$) will get more weight. Outcome changes among control units with $x_i$ that resemble control units will get less weight. To model the propensity scores, we could use a parametric model like logistic regression or something more flexible like machine learning. As usual, extending the model to multiple time points in the pre- and post-treatment periods is more complicated. Nonparametric estimation with empirical distributions Athey and Imbens (2006) developed a generalization of diff-in-diff called “changes-in-changes” (of which diff-in-diff is a special case). This method drops many of the parametric assumptions of diff-in-diff and allows both time and treatment effects to vary across individuals. Again we are in a two-period, two-group setting. The Athey and Imbens (2006) model is much less restrictive than the usual parametric model. It only assumes two things: 1. $Y_i^0 = h(u_i, t)$ for unobservable characteristics $u_i$ and an unknown function $h$ (increasing in $u$) 2. Within groups, the distribution of $u_i$ does not change over time Note that the distribution of $u_i$ can differ between treatment groups so long as this difference remains constant. Below, we discuss a method for estimating diff-in-diff without this assumption. To estimate the target parameter, Athey and Imbens (2006) estimate empirical outcome distributions in a familiar list of samples: 1. The post-treatment distribution of $Y$ in the treated group, 2. The pre-treatment distribution of $Y$ in the treated group, 3. The post-treatment distribution of $Y$ in the control group, and 4. The pre-treatment distribution of $Y$ in the control group. What we are missing is 5. The post-treatment distribution of $Y^0$ in the treated group (i.e., counterfactual, untreated outcomes) Since we cannot observe this distribution, Athey and Imbens (2006) estimate it through a combination of the empirical distributions for (1), (2), and (3). After estimating (5), the effect of treatment is the difference between observed (4) and estimated (5). Let $F_{Y^0_{12}}$ be the counterfactual distribution for the untreated outcomes for the treated group at $t = 2$. We estimate this quantity through the relation: \[ F_{Y^0_{12}}(y) = F_{Y_{11}}(F^{-1}_{Y_{01}}(F_{Y_{02}}(y)))\;, \] where $F_{Y_{11}}$ is the distribution function for the (observed) outcomes for the treated group at $t = 1$; $F_{Y_{01}}$ is the distribution function for the (observed) outcomes for the untreated group at $t = 1$; and $F_{Y_{02}}$ is the distribution function for the (observed) outcomes for the untreated group at \ (t = 2\). The other distribution of note, $F_{Y^1_{12}}(y)$, is actually observed since this are the treated outcomes for the treated group at $t = 2$. Since we have estimates for both $F_{Y^1_{12}}$ and $F_{Y^0_{12}}$, we can also estimate the diff-in-diff effect. MATLAB code for this estimator is available on the author’s website. Bonhomme and Sauder (2011) extend this idea to allow the shape of the outcome distributions to differ in the pre- and post-intervention periods. The cost of this additional flexibility is that they must assume additivity. Semiparametric estimation with time-varying covariates One of the key identifying assumptions from Athey and Imbens (2006) is that the distribution of $u$ — all non-treatment and non-time factors — is invariant across time within the treated and control groups. That is, the distributions do not change with time. Stuart et al. (2014) circumvents this restriction by considering four distinct groups (control/pre-treatment, control/ post-treatment, treated/pre-treatment, treated/post-treatment) rather than just two groups (control and treated) observed in two time periods. With the four groups, the distribution of $u$ can change over time. A consequence of this setup is that it no longer makes sense to talk about the diff-in-diff parameter as the effect of treatment on the treated; instead, the estimand is defined as the effect of treatment on treated in pre-treatment period. The estimator uses propensity scores for predicting the probabilities of each observation being in each of the four groups. We can use some kind of multinomial regression. The treatment effect is then calculated as a weighted average of the observed outcomes (see section 2.4 of Stuart et al. (2014)). Whereas standard parametric techniques rely on estimation of the outcome regression, and methods such as those in Abadie (2005) leverage information in the propensity score function, some doubly robust methods use both of these components. Broadly, doubly robust methods will yield unbiased estimates if one of these regressions is estimated consistently, and they will yield efficient estimates if both are estimated consistently. Sant’Anna and Zhao (2020) propose a doubly robust estimator that allows for linear and nonlinear specifications of the outcome regression and propensity score function for panel or repeated cross-section data. Inference yielding simultaneous confidence intervals involves a bootstrapping procedure, which accommodates clusters (although the number of groups must be large). Li and Li (2019) develop a doubly robust procedure specific to diff-in-diff, with an application to data about automobile crashes before and after the implementation of rumble strips. They parameterize the treatment effect using both multiplicative and additive quantities Elsewhere, Han, Yu, and Friedberg (2017) implement a double robust weighting approach based on Lunceford and Davidian (2004) to study the impact of medical home status on children’s healthcare outcomes. General texts on double robust estimators are available (see van der Laan and Robins (2003) or van der Laan and Rose (2011)). For another type of doubly robust method that depends on the outcome regression and unit and time weights, see Arkhangelsky et al. (2021) in the Synthetic Control section. Inference and estimation are closely linked. Once we estimate the causal estimand, we want to know how uncertain our estimate is and test hypotheses about it. In this section, we highlight some common challenges and proposed solutions for inference in diff-in-diff. Whether the data arise from repeated measures or from repeated cross-sections, data used in diff-in-diff studies are usually not iid (i.e., independently and identically distributed). For example, we often have hierarchical data, in which individual observations are nested within larger units (e.g., individuals in a US state) or longitudinal data, in which repeated measures are obtained for units. In both of these cases, assuming iid data will result in standard errors that are too small. Below, we discuss three common issues in inference: serial autocorrelation, clustered data, and distributional assumptions. We draw on three papers in this area: Bertrand, Duflo, and Mullainathan ( 2004), Rokicki et al. (2018), and Schell, Griffin, and Morral (2018). Collapsing the data Collapsing the data by averaging the pre- and post-intervention observations returns us to the simple two-period setting, obviating the need to consider longitudinal correlation in the data. When treatment is administered at the same time point in all treated units, we can perform ordinary least squares on the aggregated data. On the other hand when treatment is staggered (e.g., states pass the same health care law in different years), Bertrand, Duflo, and Mullainathan (2004) suggest aggregating the residuals from a regression model and then analyzing those. See Goodman-Bacon (2021) and Athey and Imbens (2022) for more about varying treatment times. In simulation studies, Bertrand, Duflo, and Mullainathan (2004) and Rokicki et al. (2018) find that aggregation has good Type I error and coverage, but it does lose some information (and thus power). Clustered and robust standard errors The most popular way to account for clustered data in diff-in-diff is clustered standard errors (Cameron and Miller 2015; Abadie et al. 2023). This adjusts the usual variance-covariance matrix estimate by accounting for correlation in the data. In Stata, this is as simple as using the cluster option in the regress function. In R, the vcovHC package implements a huge variety of alternative variance-covariance methods. However, clustered standard error methods fail with only one treated unit (Conley and Taber 2011), for example, when a single state implements a policy of interest. There is no hard and fast rule on the number of treated units needed for clustered standard errors to be appropriate. Figures 2 (panel A) and 4 of Rokicki et al. (2018) show that when the number of groups is small, clustering standard errors results in under-coverage, which gets worse as the treated-to-control ratio becomes more unbalanced. Donald and Lang (2007) developed a two-part procedure for estimation and inference in simple models that works well even when the numbers of groups is small. Mixed models with random effects at the cluster level can account for serial correlation. This is what we used in our demonstration of confounding in a previous section. Robust standard errors estimation methods (aka sandwich estimators or Huber-White) are meant to protect against departures from the assumed distributional form of the errors. However Schell, Griffin, and Morral (2018) found that across a range of simulations, these robust standard error estimates were worse than no adjustment or other methods of adjusting standard errors. They caution (p 36), “researchers should be quite cautious about using so-called robust SEs with longitudinal data such as these.” Generalized estimating equations Generalized estimating equations (GEE) take into account covariance structure and use a robust sandwich estimator for the standard errors (see Figure 2, panel D and Figure 4 in Rokicki et al. (2018) ). Both of these methods are widely available in statistical software. In particular, GEE is powerful since it is robust to misspecification of the correlation structure. Specifying the correct covariance will increase the efficiency of the estimate. However, note that Rokicki et al. (2018) also found under-coverage in the confidence interval in the GEE estimates when the ratio of treated to control units was lopsided. Arbitrary covariance structures Throughout the diff-in-diff literature, we find simulations and inference techniques based on an autoregressive covariance AR(1) structure for the residuals within a cluster. The AR(1) covariance structure is \[ \text{Cov}(Y_i) = \sigma^2 \begin{pmatrix} 1 & \rho & \rho^2 & \cdots & \rho^{n-1} \\ \rho & 1 & \rho & \cdots & \rho^{n-2} \\ \rho^2 & \rho & 1 & \cdots & \rho^{n-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \rho^{n-1} & \rho^{n-2} & \rho^{n-3} & \cdots & 1 \end{pmatrix} \] with an unknown variance parameter $\sigma^2$ and an unknown autoregressive parameter $0 \leq \rho \leq 1$. When $\rho$ is larger, clustered values are more highly correlated; whereas when $\ rho = 0$, observations are independent. This structure assumes that correlation is positive (or zero) across all observations and that observations closer to each other are the most strongly correlated, and observations that are more distant are more weakly correlated. An assumed AR(1) correlation structure is common in diff-in-diff simulation studies and inference techniques. Bertrand, Duflo, and Mullainathan (2004) considered this correlation structure in simulations and found that “this technique [assuming AR(1)] does little to solve the serial correlation problem” due to the difficulty in estimating $\rho$. Rokicki et al. (2018) used an AR(1) in their simulations. As we illustrate in the app below, with unit and time fixed effects, AR(1) is not a plausible correlation structure. McKenzie (2012) also discusses autocorrelation, emphasizing how statistical power relates to the number of time points and to autocorrelation, ultimately concluding that ANCOVA is more powerful than The correlation structure in diff-in-diff applications may follow many different structures. For example, after de-meaning and de-trending, outcomes have a weak positive correlation in adjacent time points but a negative correlation at time points in far apart time points. In the shiny app below, we present correlation structures for simulated data and real data. The real datasets are from (a) the Dartmouth Health Atlas, (b) MarketScan claims data, and (c) Medicare claims, which are described in more detail within the app. Play around with the settings to simulate data that look like your applications. Does the correlation structure look the way you expect? Permutation tests Permutation tests are a resampling method that can be used to test statistical hypotheses. In the diff-in-diff setting, permutation tests comprise the following steps: 1. Compute the test statistic of interest on the original data. For example, calculate the interaction term between time and treatment from a regression model. Call this $\hat{\delta}$. 2. For $K$ a large positive integer, permute the treatment assignment randomly to the original data, so that the data are the same save for a new treatment assignment. Do this $K$ times. 3. For each of the $K$ new datasets, compute the same test statistic. In our example, we compute a new interaction term from a regression model. Call these $\hat{\delta}^{(k)}$ for permutation \ (k \in \{1, \dots, K\}\). 4. Compare the test statistic $\hat{\delta}$ found in the first step to the test statistics $\hat{\delta}^{(1)}, \dots, \hat{\delta}^{(K)}$ found in the third step. The fourth step is where we can get a nonparametric p-value for the parameter of interest. If, for instance, $\hat{\delta}$ is more extreme than 95% of $\hat{\delta}^{(k)}$ then the permutation test p-value is 0.05. For more on permutation inference for difference-in-differences, see Conley and Taber (2011) and MacKinnon and Webb (2020). Lagged dependent variable regression A promising technique to address serial autocorrelation is to include lagged variables. (Schell, Griffin, and Morral 2018) The form of the regression model is different from the specifications above, since it uses change coding of the treatment indicator and includes the previous time-period’s value of the outcome as a regressor: \[ Y_i(t) = \beta (D_{it} - D_{i,t-1}) + \gamma Y_i(t-1) + \alpha_i + \delta_t + \epsilon_{it} \] In simulations, these authors show dramatic impacts on Type I error rates from inclusion of lagged dependent variables with change coding. Design-based Inference We usually conceptualize Difference-in-Differences designs in the framework of random sampling of treated and control units from a target population. Yet as Manski and Pepper (2018) and others point out, it can sometime be difficult to conceive units in a study as having been sampled from a larger population of units, e.g., when the units are all provinces in a particular country. In such settings, Rambachan and Roth (2022) propose a method of inference that treats random assignment of units to treatment and control as the stochastic process that generated the data. This finite population framework has the benefit of enabling inference when units in a study were not in fact sampled from a larger target population. In this setting, the ATT (our usual causal target) is not a population quantity that is fixed over different possible random samples from this population. The ATT is a random quantity that changes over different possible random assignments depending upon which units happen to be assigned to treatment (Sekhon and Shem-Tov 2021). Hence, Rambachan and Roth (2022) define the ATT as the expected average effect among treated units, where the expectation is over the set of possible random assignments with unknown assignment probabilities. Ideally, in a finite population setting, the researcher knows the true probabilities with which units could have wound up in treatment and control conditions. In the absence of a randomized experiment, though, we do not know these probabilities. Instead, Rambachan and Roth (2022) invoke an alternative assumption: The covariance of control potential outcomes and units’ treatment assignment probabilities is $0$. This assumption will be satisfied in a randomized experiment since all units have the same treatment assignment probabilities. However, this assumption can also be satisfied when treatment assignment probabilities are not constant and control potential outcomes differ across units with different treatment assignment probabilities. The key condition is that control potential outcome do not linearly depend on treatment assignment probabilities; nonlinear dependence is okay. In a panel setting with outcomes measured over time, Rambachan and Roth (2022) show that the assumption of linear independence between control potential outcomes and assignment probabilities is a finite population analogue of the standard parallel trends assumption: The assumption implies that the after-minus-before mean of the treated group’s control potential outcomes is equal, in expectation over possible random assignments, to the after-minus-before mean in the control group’s control potential outcomes. When this condition is satisfied, the after-minus-before mean of the treated group’s treated potential outcomes minus the after-minus-before mean in the control group’s control potential outcomes is unbiased for the ATT. In addition, Rambachan and Roth (2022) derive (1) an upper bound for the estimator’s variance, (2) an unbiased estimator of this upper bound, and (3) establish the asymptotic Normality of the estimator. Taken all together, the method in Rambachan and Roth (2022) enables us to do design-based inference for DID without having to assume that our study is equivalent to a randomized experiment. In this section, we summarize robustness checks relevant to diff-in-diff. We’ve highlighted diff-in-diff’s reliance on strong, untestable causal assumptions and discussed common estimation and inference methods. Many things can go wrong — assumptions don’t hold, wrong modelling choices, etc. We discuss a few techniques here that can help us quantify the robustness of our diff-in-diff Placebo Tests Lechner (2011) perform a thought experiment that we summarize here. Imagine we are doing a data analysis using diff-in-diff, and that our data have multiple pre-treatment time points. Now, imagine we move the “treatment time” to time before the real treatment. If we conduct a diff-in-diff analysis, what would you expect the estimated treatment effect to be? Perhaps you would expect no treatment effect, since no treatment occurred at that time. This idea underlies the use of placebo tests as a sensitivity check for diff-in-diff studies. Slusky (2017) use this technique to show that significant changes in health insurance coverage among people aged 19-25 (who were affected by the ACA’s dependent coverage provision) relative to those 16-18 or 27-29 (two possible comparison groups) occurred at several time points before the ACA was implemented. Instrumental Variable Approach For Diverging Trends Freyaldenhoven, Hansen, and Shapiro (2019) propose an approach to address the issue of diverging parallel trends due to an unobserved confounder. Their approach identifies an observed covariate that serves as an instrument for the unobserved confounder. They assume there is a latent variable that triggers treatment once it passes a threshold. For example, once a person’s income falls below a particular threshold, she becomes eligible for food stamps. They show that if we can identify a covariate that is related to income but unrelated to the food stamp intervention, we can use 2-stage least squares to estimate an unbiased treatment effect. This method is appealing both because it provides a mechanism to address bias and encourages researchers to think explicitly about unobserved confounders and their role in inducing bias. However, this framework relies heavily on being able to identify and find an instrument for unobserved confounders, which may be difficult to do in Sensitivity Analysis Rambachan and Roth (2022) develop inference for the situation when parallel trends may not hold exactly, but we can impose restrictions on the extent to which parallel trends is violated. To do this, Rambachan and Roth (2022) introduce an additional parameter — the post-treatment difference in trends between treated and control groups — and inference that is valid under a restriction about the set to which the value of this parameter belongs. They provide several intuitive ways for researchers to restrict the possible values of the post-treatment difference in trends. Two such restrictions are as follows: 1. Relative Magnitude: The post-treatment difference in trends is no greater than some magnitude, $\bar{M}$, of the maximum pre-treatment difference in trends. 2. Smoothness: The change in the post-treatment difference in trends between consecutive periods is less than or equal to some $M \geq 0$. These two restrictions can be combined in useful ways, e.g., placing a bound on (a) the post-treatment change in the difference in trends between consecutive periods in terms of (b) some magnitude of the pre-treatment change in difference in trends between consecutive periods. Given such restrictions, the unknown causal target could be any of a set of values consistent with the restriction on the difference in trends. Rambachan and Roth (2022) propose a method of inference that, under reasonable assumptions, achieves the following aim: Over these possible values for the causal target, the lowest possible coverage probability for a random confidence interval is at least $1$ minus the size ($\alpha$-level) of the test. Rambachan and Roth (2022) also provide conditions that ensure the probability of rejecting causal parameter values inconsistent with the restriction tends to $1$ as the sample size grows. This inferential method also leads to a valuable form of sensitivity analysis: studying the behavior of inference under more or less lax restrictions on the post-treatment difference in trends. For example, consider increasingly lax restrictions in terms of the relative magnitude bound, $\bar{M}$. When $\bar{M} = 0$, parallel trends is true, and only one value of the causal target is consistent with this restriction. As $\bar{M}$ increases, more values of the causal target will be consistent with the restrictions on the difference in trends. Consequently, confidence intervals will become wider. Researchers can then answer questions like: How large would the post-treatment deviation from the pre-treatment difference in trends need to be in order to make my confidence interval cross the null? Much of diff-in-diff theory and its applications focus on continuous outcomes, but nonlinear outcomes are common too. Nonlinear outcomes include binary outcomes such as death status or count outcomes such as the number of hospitalizations. If we have a binary outcome, we can model the probability directly with a linear probability model; the downside to this approach is that predicted probabilities can fall outside of the $[0, 1]$ range. We can restrict predicted probabilities within $[0, 1]$ using an appropriate transformation — logit and probit transformations are perhaps the most common. However, in doing so we lose a lot of nice properties that come with the linear model. Ai and Norton (2003) first pointed out this vexing occurrence with respect to to diff-in-diff. In particular, they showed that the cross-partial effect can be nonzero even when the treatment/post-period interaction term is 0. Puhani (2012) noted that while the point from Ai and Norton (2003) is true, the true diff-in-diff estimate is still taken directly from the interaction term in the model. He shows that diff-in-diff is actually a difference of two cross-partial derivatives so the interaction term always has the same sign as the diff-in-diff effect (not necessarily the case in Ai and Norton (2003)). Thus, inference on the treatment effect can be conducted through the usual test of the interaction parameter. The Karaca-Mandic, Norton, and Dowd (2012) paper ties the previous two papers together. They show in Figures 3 and 4 how the diff-in-diff effect (on the probability scale) can change as the value of the linear predictor $X\beta$ changes, even when the model does not include an interaction term. The authors then go through an interactive example using Stata, which might be useful to researchers intending to do a diff-in-diff analysis with a nonlinear model. Another tricky aspect of nonlinear models is that the role of covariates is more complex than described above. e.g. In OLS, if you omit a variable that's correlated with your outcome but not the tx, you still have an unbiased estimate of your beta. In logistic regression, even covariates that are not correlated with treatment can act as confounders and bias your ORs! — Alyssa Bilinski (@ambilinski) February 5, 2020 In selecting a comparison group of units not impacted by the intervention, a researcher has many options. For example, to evaluate the impact of the Massachusetts health insurance expansion, a researcher could use all other states as controls. She could use nearby states as controls. She could even choose counties from around the US that look similar to Massachusetts counties. When conducting DID, researchers often use subject matter knowledge to choose comparison groups and then evaluate whether trends are parallel. The synthetic control methods is an appealing alternative because it is a data-driven way to select comparison groups. The first instance of the synthetic control method is taken from Abadie and Gardeazabal (2003), which studied effects of terrorism on economic growth in Spain. Beginning in the 1960s, the Basque Country experienced a rash of terrorism, which was broken by a 1998 cease-fire. The other regions of Spain were weighted to form a synthetic Basque Country, similar to the real Basque Country in demographics. The results showed that per capita GDP increased in Basque Country after the cease-fire relative to the synthetic Basque country with no cease-fire. In the years since that original synthetic control paper, many methodological refinements and extensions have been published. We summarize some of these below, but first, we consider potential pitfalls of synthetic controls. The assumption that a weighted set of comparison units can represent the treatment group’s counterfactual outcomes is strong, and if not met, can introduce bias. Modifications to add flexibility to SCM include allowing a level difference between the treatment and comparison groups or allowing weights that do not sum to 1 and are negative (Doudchenko and Imbens 2016). Another alternative to address the convex hull issue is to develop synthetic controls for comparison units rather than treatment units and include only those that are well-estimated in the effect estimate (Powell 2018). Another potential issue with SCM is overfitting. Several authors have observed that under a fixed penalty, SCM is a form of ridge regression. They have, therefore, proposed choosing the SCM penalty term to minimize mean-squared error to avoid overfitting (Doudchenko and Imbens 2016; Kinn 2018). In an alternative approach, Powell (2018) modeled unit-specific trends to reduce noise. A third concern is the ‘curse of dimensionality’ (Ferman and Pinto 2021). Traditional SCM are consistent as the number of time periods goes to infinity. However, an increasing number of time periods decrease the likelihood that appropriate weights exist; with more time periods, the treatment unit may not fall inside the convex hull of the comparison units. Ben-Michael, Feller, and Rothstein ( 2021) propose adjusting the SCM estimate by the outcome regression-estimated average difference between treatment and comparison units (including lagged outcomes and possibly covariates). We discuss their augmented synthetic control method in more detail below. Finally, because synthetic controls weight on pre-period outcomes (and since weighting and matching are close cousins), one might wonder about regression to the mean bias (as discussed above). A working paper by Illenberger, Small, and Shaw (2020) uses simulation to show that the same issues that plague diff-in-diff with matching on pre-period outcomes also affect synthetic controls. As in diff-in-diff, the problem worsens with larger baseline differences between treated and control group means and decreases with the auto-correlation in the outcomes. Those authors propose a correction that subtracts off the portion of the treatment effect due to regression to the mean. By assuming various values of baseline means, auto-correlation, and residual variance parameters, researchers can understand what values would be sufficient to materially change their treatment effect estimates. Ding and Li (2019) study the relationship between diff-in-diff and related methods that assume treatment assignment is ignorable conditional on past outcomes (which includes synthetic controls and lagged dependent variables regression). These authors extend results from linear models to nonparametric settings. Their results apply when two conditions hold: stationarity (i.e., the outcomes would not get infinitely large over time) and stochastic monotonicity (i.e., either treated or control has smaller lagged outcomes), both of which can be empirically checked. When stationarity holds and the treated group has smaller lagged outcomes, the diff-in-diff treatement effect will be greater than or equal to the lagged dependent variable effect, and vice versa. Synthetic Difference-in-Differences Arkhangelsky et al. (2021) propose an extension to SCM called Synthetic Difference-in-Differences (SDID), which combines elements of both DID and SCM. Most DIDs incorporate both unit and time fixed effects, which weights each comparison unit and each time period equally. SCM incorporate unit-level weights and time fixed effects. SDID adds additional flexibility by incorporating all of these elements: 1) unit and time fixed effects and 2) unit- and time-level weights. \[ \hat{\omega}^{sc} = \underset{\omega \in W}{\text{arg min}} \sum^{T-1}_{t = 1} \left(\sum^{N-1}_{i=1} \omega_iY_i(t) - Y_N(t)\right)^2 \] \[ \hat{\lambda}^{sc} = \underset{\lambda \in L}{\text{arg min}} \sum^{N-1}_{i = 1} \left(\sum^{T-1}_{t=1} \lambda_iY_i(t) - Y_i(T)\right)^2 \] \[ \left(\hat{\theta}^{sdid}, \hat{\tau}^{sdid}\right) = \underset{\theta}{\text{arg min}} \sum^N_{i=1}\sum^T_{t=1}\left( Y_i(t) - g(\theta)_{it} - \tau W_{it}\right)^2\hat{\omega}_i\hat{\lambda}_t The authors demonstrate that this approach is doubly-robust: as $N$ and $T$ go to infinity, the SDID estimators is consistent if either 1) the unit and time weights or 2) the outcome model are correctly specified. (This roughly corresponds to either SCM with additional weight flexibility or DID being consistently estimated.) Augmented Synthetic Control Augmented SCM (ASCM) is another extension of synthetic controls. Ben-Michael, Feller, and Rothstein (2021) select weights to minimize the difference in pre-intervention level then subtract an estimate of the remaining level difference from the post-intervention difference. This addresses the “curse of dimensionality”: while SCM is unbiased as the number of pre-intervention time periods grows, researchers are less likely to identify a good fit as the number of time periods grows, even when one exists, unless they have a very large number of control units. Both ASCM and SDID methods can weight up recent periods relative to distant periods when estimating this level difference. Further, both ASCM and SDID involve a correction to the SCM estimator that is 0 if there is exact pre-intervention balance. These methods only apply when there is imperfect pre-intervention balance and matter most when there is substantial pre-intervention imbalance. Generalized Synthetic Control Xu (2017) proposed the generalized synthetic control method, which combines an interactive fixed effects model with the framework of synthetic control. The basic idea is to use a parametric factor model for the outcomes to predict the “missing” untreated potential outcomes for the comparison group. The factor model assumes that a small set of time-varying factors interact with unit-specific “factor loadings” to generate the outcome. Note that the popular two-way fixed effects model (with unit and time fixed effect) is a special case of a factor model. Within the generalized synthetic control framework, Xu (2017) writes the outcome for unit $i$ at time $t$ as \[ Y_i(t) = \delta_{it}D_{it} + x_{it}\beta + \lambda_i f_t + \epsilon_{it}, \] where $D_{it}$ is a treatment indicator which equals 1 whenever the unit $i$ receives treatment at time $t$, $\delta_{it}$ are heterogeneous effects, $x_{it}$ and $\beta$ are covariates and their coefficients, and $\lambda_i$ and $f_t$ are factor loadings and factors, respectively. To use the generalized synthetic control method, we first estimate the latent factors, $f_t$, and the coefficients on the covariates, $\beta$, using only the data from the control units (with least squares and some additional restrictions like orthogonality of the factors). Then we estimate the factor loadings of each treated unit, $\lambda_i$, by minimizing a least squares equation for the treated units’ outcomes in the pre-treatment period, conditional on the factors and coefficients estimated in the first step. Finally, we use the estimated coefficients, factors, and factor loadings in the parametric outcome model to predict counterfactual untreated outcomes in the post-treatment period for the treated units: \[ \hat{Y}^0_i(t) = x_{it}\hat{\beta} + \hat{\lambda}\hat{f_t}. \] Following this step, the ATT is simply the average (within the treated group only) of the observed post-treatment outcomes minus the predicted untreated outcomes: \[ ATT(t) = \frac{1}{N_t} \sum_i Y_i(t) - \hat{Y}^0_i(t), \] where the summation iterates over all treated units and $N_t$ denotes the number of treated units. The strengths of the method are that the factor structure can address time-varying confounding (unlike diff-in-diff); it incorporates heterogeneous treatment effects (unlike interactive fixed effects); and it can accommodate multiple treated units, observed covariates, and treated units outside the convex hull of the controls (unlike the synthetic control formulation of Abadie, Diamond, and Hainmueller (2010)). The limitations of this method include the requirement for a reasonably long pre-treatment period, the reliance on a parametric model for the outcome, and the lack of obvious safeguards against inappropriate controls (i.e., those that lack common support with the treated units). A related technique (sometimes described as equivalent to diff-in-diff with multiple time points) is comparative interrupted time series (CITS). The causal assumptions of the two methods are different, however. In CITS, the counterfactual is constructed by 1) fitting linear models to the comparison group’s outcomes in the pre- and post-intervention periods, 2) computing the pre- to post-period changes in the intercepts and slopes, 3) fitting a linear model to the treated group’s outcomes in the pre-intervention period, and 4) assuming the comparison group’s intercept and slope changes computed in step 2) would have held in the treated group in the absence of intervention. We highlight some important differences between DID and CITS: 1. CITS does not require parallel outcome evolution in the treated and comparison groups in the pre-intervention period 2. CITS does require a linear model to capture the pre- to post-intervention change in the outcome process of the comparison group (which is then assumed to also hold for the treated group’s counterfactual untreated outcomes) Notice that these are not merely differences in modeling. They are differences in the construction of the counterfactual: • DID assumes the pre-to-post change in the average outcomes of the comparison group would also have been observed in the treated group, absent the intervention. • DID with a pre-period slope difference assumes the pre-to-post change in the average outcomes of the comparison group plus the linearly growing difference observed in the pre period would combine to produce the treated group’s counterfactual outcomes, absent the intervention. • CITS assumes the pre-to-post change in the intercept and slope of the comparison group would have been observed in the treated group, absent the intervention. Up to this point, identifying the diff-in-diff estimator required parallel trends or some type of counterfactual assumption. Mora and Reggio (2012) and Mora and Reggio (2019) developed a diff-in-diff estimator using an alternative assumption called the parallel growth assumption. The parallel growth assumption essentially requires that the derivatives of the paths are parallel. For example, imagine we have two linear functions — $f(x) = 2x$ and $g(x) = 3x$. These functions are not parallel, but their derivatives, $f'(x) = 2$ and $g'(x) = 3$, are. Imagine the following scenario with five time points in which treatment is administered after time 3. The trajectory of the untreated outcomes for the control group (all observed by the consistency assumption) are shown with the orange dotted-dashed line. The trajectory of the untreated outcomes for the treated (observed up to the 3rd time point) is shown in the blue solid line. Clearly, this scenario violates the parallel trends assumption. In fact using parallel trends, the counterfactual untreated outcomes for the treated would deviate from their true trajectory shown by the dashed line below. In this example any inference using diff-in-diff methods based on parallel trends will be biased. However, in this particular case, Mora and Reggio (2012) showed that the treatment effect is identified under an alternative assumption – parallel growth assumption. This assumption is similar to parallel trends except that we require the derivatives of the trajectories to be parallel. In the above graph, the two trajectories are not parallel, but their derivatives are! Both are straight lines with constant derivatives. Since the derivatives are parallel, the authors show that the estimator is identified. While interesting in theory, it is difficult enough justifying parallel trends using our original data (i.e., not derivatives or lagged differences), and it’s hard to imagine when we could be confident in parallel trends on the derivatives but not on the original scale. However, this paper is useful in understanding the role of the underlying diff-in-diff assumptions and which other assumptions may be possible to obtain similar quantities, especially in the case where parallel trends fails. Ben-Michael et al. (2023) propose the use of multitask Gaussian processes (MTGPs) in a panel data setting. This approach models a unit’s untreated potential outcomes in time period $t$ in terms of a model component plus a random error term with mean equal to $0$. As the term Gaussian in the name of the method suggests, researchers conventionally suppose that untreated potential outcomes follow a multivariate Gaussian distribution. The model component is indexed by unit, $i$, and time period, $t$. Defined on the model components is a Gaussian process prior characterized by time and unit kernels. The role of the time kernel is to determine how the parameter values of the model in time $t$ correlate with the model’s parameter values in later time periods. That is, the kernel captures how much information the model in one time period provides for the model in later time periods. At a high level, the unit kernel captures the correlation between untreated potential outcomes of any two distinct units, $i$ and $i^{\prime}$. Common kernels, such as the usual squared exponential kernel, are such that the correlation between model parameters is decreasing in the difference between any two time periods. This property is intuitively sensible in that information about potential outcomes in the year 1996 will probably contain more information about potential outcomes in 1997 than in 2023. Likewise, given a suitable (but perhaps less immediately apparent) definition of distance between two units, the untreated potential outcome of one unit provides less information about that of another unit the “further away” it is. Estimation and inference proceed by updating the Gaussian process priors on the model parameters via a model-based likelihood function. With this prior distribution on the model parameters, researchers can then draw from the posterior distribution of the model parameters; input each draw as the parameters of the model of untreated potential outcomes; draw each of the treated units’ untreated potential outcomes; and then directly calculate the ATT from the treated units’ observed treated potential outcomes and their imputed control potential outcomes. Repeating this procedure many times will yield a simulation-based approximation to the posterior distribution of the ATT. Two of the key benefits of the MTGP model are as follows. First, the functional form of the model component is nonparametric. Along these lines, Ben-Michael et al. (2023) show that the MTGP model’s posterior predictive mean can be represented as a weighting estimator (with weights for both time and units). While the mean squared error of this weighting estimator depends on the true, unknown functional form of the model component, Ben-Michael et al. (2023) show that the weights are effectively chosen to minimize the estimator’s worst-case, mean squared error over all functions in a large class of possible functions. Second, because of the time kernels, MTGP model has intuitive appeal in that uncertainty about ATTs will be greater for periods further away from the onset of treatment, which is a property that many existing methods of uncertainty quantification lack. The bulk of this website was written and edited by Bret Zeldow, Thomas Leavitt and Laura Hatfield. Funding for this website was provided by the Laura and John Arnold Foundation. Additional content contributors are Alyssa Bilinski, Carrie Fry, and Sherri Rose. Many thanks to Savannah Bergquist, Austin Denteh, Alex McDowell, Arman Oganisian, Toyya Pujol-Mitchell, and Kathy Swartz for their helpful comments. This website is built using the R package blogdown and hosted on Netlify. The design is based on the Kraiklyn Hugo theme. Abadie, A. (2005). Semiparametric difference-in-differences estimators. The Review of Economic Studies (1), 1–19. Abadie, A., Athey, S., Imbens, G. W., & Wooldridge, J. M. (2023). When should you adjust standard errors for clustering? The Quarterly Journal of Economics (1), 1–35. Abadie, A., & Cattaneo, M. D. (2018). Econometric methods for program evaluation. Annual Review of Economics (1), 465–503. Abadie, A., Diamond, A., & Hainmueller, J. (2010). Synthetic control methods for comparative case studies: Estimating the effect of alifornia’s tobacco control program. Journal of the American Statistical Association (490), 493–505. Abadie, A., Diamond, A., & Hainmueller, J. (2011). Synth: An r package for synthetic control methods in comparative case. Journal of Statistical Software (13), 1–17. Abadie, A., Diamond, A., & Hainmueller, J. (2012). Comparative politics and the synthetic control method. American Journal of Political Science (2), 495–510. Abadie, A., & Gardeazabal, J. (2003). The economic costs of conflict: A case study of the The American Economic Review (1), 113–132. Ai, C., & Norton, E. C. (2003). Interaction terms in logit and probit models. Economics Letters (1), 123–129. Altman, D. G., & Bland, J. M. (1995). Statistics notes: bsence of evidence is not evidence of absence. British Medical Journal (7003), 485. Angrist, J. D. (2001). Estimation of limited dependent variable models with dummy endogenous regressors: Simple strategies for empirical practice. Journal of Business & Economic Statistics (1), 2–28. Angrist, J. D., & Pischke, J.-S. (2008). Mostly harmless econometrics: An empiricist’s companion . Princeton, NJ: Princeton University Press. Angrist, J. D., & Pischke, J.-S. (2010). The credibility revolution in empirical economics: How better research design is taking the con out of econometrics. The Journal of Economic Perspectives (2), 3–30. Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). Synthetic difference in differences. American Economic Review (12), 4088–4118. Athey, S., Bayati, M., Doudchenko, N., Imbens, G. W., & Khosravi, K. (2021). Matrix completion methods for causal panel data models. Journal of the American Statistical Association (536), 1716–1730. Athey, S., & Imbens, G. W. (2006). Identification and inference in nonlinear difference-in-differences models. (2), 431–497. Athey, S., & Imbens, G. W. (2017). The state of applied econometrics: Causality and policy evaluation. The Journal of Economic Perspectives (2), 3–32. Athey, S., & Imbens, G. W. (2022). Design-based analysis in difference-in-differences settings with staggered adoption. Journal of Econometrics (1), 62–79. Bai, J. (2009). Panel data models with interactive fixed effects. (4), 1229–1279. Basu, S., Meghani, A., & Siddiqi, A. (2017). Evaluating the health impact of large-scale public policy changes: lassical and novel approaches. Annual Review of Public Health (1), 351–370. Bauhoff, S. (2014). The effect of school district nutrition policies on dietary intake and overweight: synthetic control approach. Economics & Human Biology , 45–55. Ben-Michael, E., Feller, A., Franks, A., & Raphael, S. (2023). Estimating the effects of a alifornia gun control program with multitask aussian processes. The Annals of Applied Statistics (2), 985–1016. Ben-Michael, E., Feller, A., & Rothstein, J. (2021). The augmented synthetic control method. Journal of the American Statistical Association (536), 1789–1803. Bertrand, M., Duflo, E., & Mullainathan, S. (2004). How much should we trust differences-in-differences estimates? The Quarterly Journal of Economics (1), 249–275. Bilinski, A., & Hatfield, L. A. (2020). Nothing to see here? Non-inferiority approaches to parallel trends and other model assumptions. Blundell, R., & Costa Dias, M. (2009). Alternative approaches to evaluation in empirical microeconomics. The Journal of Human Resources (3), 565–640. Bonhomme, S., & Sauder, U. (2011). Recovering distributions in difference-in-differences models: A comparison of selective and comprehensive schooling. The Review of Economics and Statistics (2), 479–494. Brown, T. T., & Atal, J. P. (2019). How robust are reference pricing studies on outpatient medical procedures? hree different preprocessing techniques applied to difference-in differences. Health Economics (2), 280–298. Callaway, B., & Sant’Anna, P. H. C. (2021). Difference-in-differences with multiple time periods. Journal of Econometrics, 225(2), 200–230. Cameron, A. C., & Miller, D. L. (2015). A practitioner’s guide to cluster-robust inference. The Journal of Human Resources (2), 317–372. Chabé-Ferret, S. (2015). Analysis of the bias of matching and difference-in-difference under alternative earnings and selection processes. Journal of Econometrics (1), 110–123. Chabé-Ferret, S. (2017). Should we combine difference in differences with conditioning on pre-treatment outcomes? (IDEAS Working Paper Series from RePEc No. 824). St. Louis, MO: Federal Reserve Bank of St. Louis. Chernozhukov, V., Wüthrich, K., & Zhu, Y. (2021). An exact and robust conformal inference method for counterfactual and synthetic controls. Journal of the American Statistical Association (536), 1849–1864. Conley, T. G., & Taber, C. R. (2011). Inference with “difference in differences” with a small number of policy changes. The Review of Economics and Statistics (1), 113–125. Daw, J. R., & Hatfield, L. A. (2018a). Matching and regression to the mean in difference-in-differences analysis. Health Services Research (6), 4138–4156. Daw, J. R., & Hatfield, L. A. (2018b). Matching in difference-in-differences: Between a rock and a hard place. Health Services Research (6), 4111–4117. Dimick, J. B., & Ryan, A. M. (2014). Methods for evaluating changes in health care policy: ifferences approach. JAMA: The Journal of the American Medical Association (22), 2401–2402. Ding, P., & Li, F. (2019). A bracketing relationship between difference-in-differences and lagged-dependent-variable adjustment. Political Analysis (4), 605–615. Donald, S. G., & Lang, K. (2007). Inference with difference-in-differences and other panel data. The Review of Economics and Statistics (2), 221–233. Doudchenko, N., & Imbens, G. W. (2016). Balancing, regression, difference-in-differences and synthetic control methods: A synthesis (No. NBER Working Paper No. 22791). National Bureau of Economic Research. Dube, A., & Zipperer, B. (2015). Pooling multiple case studies using synthetic controls: An application to minimum wage policies (IZA Discussion Paper No. 8944). Bonn, Germany: IZA — Institute of Labor Economics. Ferman, B., & Pinto, C. (2021). Synthetic controls with imperfect pretreatment fit. Quantitative Economics (4), 1197–1221. Ferman, B., Pinto, C., & Possebom, V. (2020). Cherry picking with synthetic controls. Journal of Policy Analysis and Management (2), 510–532. Fretheim, A., Zhang, F., Ross-Degnan, D., Oxman, A. D., Cheyne, H., Foy, R., … Soumerai, S. B. (2015). A reanalysis of cluster randomized trials showed interrupted time-series studies were valuable in health system evaluation. Journal of Clinical Epidemiology (3), 324–333. Freyaldenhoven, S., Hansen, C., & Shapiro, J. M. (2019). Pre-event trends in the panel tudy design. American Economic Review (9), 3307–3338. Gaibulloev, K., Sandler, T., & Sul, D. (2014). Dynamic panel analysis under cross-sectional dependence. Political Analysis (2), 258–273. Glymour, M. M., Weuve, J., Berkman, L. F., Kawachi, I., & Robins, J. M. (2005). When is baseline adjustment useful in analyses of change? n example with education and cognitive change. American Journal of Epidemiology (3), 267–278. Gobillon, L., & Magnac, T. (2016). Regional policy evaluation: Interactive fixed effects and synthetic controls. The Review of Economics and Statistics (3), 535–551. Goodman-Bacon, A. (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics (2), 254–277. Greenaway-McGrevy, R., Han, C., & Sul, D. (2012). Asymptotic distribution of factor augmented estimators for panel regression. Journal of Econometrics (1), 48–53. Greene, W. (2004). The behaviour of the maximum likelihood estimator of limited dependent variable models in the presence of fixed effects. The Econometrics Journal (1), 98–119. Greene, W. (2010). Testing hypotheses about interaction terms in nonlinear models. Economics Letters (2), 291–296. Hahn, J., & Shi, R. (2017). Synthetic control and inference. (4), 52–63. Han, B., Yu, H., & Friedberg, M. W. (2017). Evaluating the impact of parent‐reported medical home status on children’s health care utilization, expenditures, and quality: A D ifferences analysis with causal inference methods. Health Services Research (2), 786–806. Hartman, E., & Hidalgo, F. D. (2018). An equivalence approach to balance and placebo tests. American Journal of Political Science (4), 1000–1013. Illenberger, N. A., Small, D. S., & Shaw, P. A. (2020). Impact of regression to the mean on the synthetic control method: ias and sensitivity analysis. (6), 815–822. Imai, K., & Kim, I. S. (2019). When should we use unit fixed effects regression models for causal inference with longitudinal data? American Journal of Political Science (2), 467–490. Imbens, G. W., & Angrist, J. D. (1994). Identification and estimation of local average treatment effects. (2), 467–475. Kahn-Lang, A., & Lang, K. (2020). The promise and pitfalls of differences-in-differences: Reflections on 16 and pregnant and other applications. Journal of Business & Economic Statistics (3), 613–620. Karaca-Mandic, P., Norton, E. C., & Dowd, B. (2012). Interaction terms in nonlinear models. Health Services Research ), 255–274. Kaul, A., Klößner, S., Pfeifer, G., & Schieler, M. (2022). Standard synthetic control methods: he case of using all preintervention outcomes together with covariates. Journal of Business & Economic Statistics (3), 1362–1376. King, G., & Zeng, L. (2006). The dangers of extreme counterfactuals. Political Analysis (2), 131–159. Kinn, D. (2018). Synthetic control methods and big data Kreif, N., Grieve, R., Hangartner, D., Turner, A. J., Nikolova, S., & Sutton, M. (2016). Examination of the synthetic control method for evaluating health policies with multiple treated units. Health Economics (12), 1514–1528. Kropko, J., & Kubinec, R. (2020). Interpretation and identification of within-unit and cross-sectional variation in panel data models. PLoS ONE (4), e0231349. Lechner, M. (2011). The estimation of causal effects by difference-in-difference methods. Foundations and Trends in Econometrics (3), 165–224. Li, F., & Li, F. (2019). obust estimation in ifferences with an application to traffic safety evaluation. Observational Studies (1), 1–23. Lindner, S., & McConnell, K. J. (2019). Difference-in-differences and matching on outcomes: tale of two unobservables. Health Services and Outcomes Research Methodology (2–3), 127–144. Lipsitch, M., Tchetgen, E. T., & Cohen, T. (2010). Negative controls: tool for detecting confounding and bias in observational studies. (3), 383–388. Lopez Bernal, J., Soumerai, S., & Gasparrini, A. (2018). A methodological framework for model selection in interrupted time series studies. Journal of Clinical Epidemiology , 82–91. Lunceford, J. K., & Davidian, M. (2004). Stratification and weighting via the propensity score in estimation of causal treatment effects: comparative study. Statistics in Medicine (19), 2937–2960. MacKinnon, J. G., & Webb, M. D. (2020). Randomization inference for difference-in-differences with few treated clusters. Journal of Econometrics (2), 435–450. Manski, C. F., & Pepper, J. V. (2018). How do right-to-carry laws affect crime rates? oping with ambiguity using bounded-variation assumptions. The Review of Economics and Statistics (2), 232–244. McKenzie, D. (2012). Beyond baseline and follow-up: he case for more in experiments. Journal of Development Economics (2), 210–221. McWilliams, J. M., Landon, B. E., Chernew, M. E., & Zaslavsky, A. M. (2014). Changes in patients’ experiences in edicare accountable care organizations. The New England Journal of Medicine (18), 1715–1724. Meyer, B. D. (1995). Natural and quasi-experiments in economics. Journal of Business & Economic Statistics (2), 151–161. Mood, C. (2010). Logistic regression: Why we cannot do what we think we can do, and what we can do about it. European Sociological Review (1), 67–82. Moon, H. R., & Weidner, M. (2015). Linear regression for panel with unknown number of factors as interactive fixed effects. (4), 1543–1579. Mora, R., & Reggio, I. (2012). Treatment effect identification using alternative parallel assumptions (Working Paper, Economic Series (48) No. 12–33). Getafe, Spain: Universidad Carlos III. Mora, R., & Reggio, I. (2019). Alternative diff-in-diffs estimators with several pretreatment periods. Econometric Reviews (5), 465–486. Mummolo, J., & Peterson, E. (2018). Improving the interpretation of fixed effects regression results. Political Science Research and Methods (4), 829–835. O’Neill, S., Kreif, N., Grieve, R., Sutton, M., & Sekhon, J. S. (2016). Estimating causal effects: Considering three alternatives to difference-in-differences estimation. Health Services and Outcomes Research Methodology (1), 1–21. Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error structure. (4), 967–1012. Powell, D. (2018). Imperfect synthetic controls: Did the Massachusetts Health Care Reform save lives? (Working Papers No. WR-1246). Santa Monica, CA: RAND Corporation Puhani, P. A. (2012). The treatment effect, the cross difference, and the interaction term in nonlinear Economics Letters (1), 85–87. Pustejovsky, J. E., & Tipton, E. (2018). Small-sample methods for cluster-robust variance estimation and hypothesis testing in fixed effects models. Journal of Business & Economic Statistics (4), 672–683. Rambachan, A., & Roth, J. (2022a). A more credible approach to parallel trends. Rambachan, A., & Roth, J. (2022b). Design-based uncertainty for quasi-experiments. Reese, S., & Westerlund, J. (2018). Estimation of factor-augmented panel regressions with weakly influential factors. Econometric Reviews (5), 401–465. Robbins, M. W., Saunders, J., & Kilmer, B. (2017). A framework for synthetic control methods with high-dimensional, micro-level data: valuating a neighborhood-specific crime intervention. Journal of the American Statistical Association (517), 109–126. Rokicki, S., Cohen, J., Fink, G., Salomon, J. A., & Landrum, M. B. (2018). Inference with difference-in-differences with a small number of groups: A review, simulation study and empirical application Medical Care (1), 97–105. Roth, J. (2022). Pretest with caution: Event-study estimates after testing for parallel trends. American Economic Review: Insights (3), 305–322. Ryan, A. M. (2018). Well-balanced or too matchy-matchy? he controversy over matching in difference-in-differences. Health Services Research (6), 4106–4110. Ryan, A. M., Burgess, J. F., & Dimick, J. B. (2015). Why we should not be indifferent to specification choices for difference-in-differences. Health Services Research (4), 1211–1235. Samartsidis, P., Seaman, S. R., Presanis, A. M., Hickman, M., & De Angelis, D. (2019). Assessing the causal effect of binary interventions from observational panel data with few treated units. Statistical Science (3), 486–503. Sant’Anna, P. H. C., & Zhao, J. B. (2020). Doubly robust difference-in-differences estimators. Journal of Econometrics (1), 101–122. Schell, T. L., Griffin, B. A., & Morral, A. R. (2018). Evaluating methods to estimate the effect of state laws on firearm deaths: A simulation study . Santa Monica, CA: Sekhon, J. S., & Shem-Tov, Y. (2021). Inference on a new class of sample average treatment effects. Journal of the American Statistical Association (534), 798–804. Slusky, D. J. G. (2017). Significant placebo results in difference-in-differences analysis: The case of the ’s parental mandate. Eastern Economic Journal (4), 580–603. Sofer, T., Richardson, D. B., Colicino, E., Schwartz, J., & Tchetgen Tchetgen, E. J. (2016). On negative outcome control of unobserved confounding as a generalization of difference-in-differences. Statistical Science (3), 348–361. Stuart, E. A., Huskamp, H. A., Duckworth, K., Simmons, J., Song, Z., Chernew, M. E., & Barry, C. L. (2014). Using propensity scores in difference-in-differences models to estimate the effects of a policy change. Health Services and Outcomes Research Methodology (4), 166–182. van der Laan, M. J., & Robins, J. M. (2003). Unified methods for censored longitudinal data and causality . New York, NY: Springer. van der Laan, M. J., & Rose, S. (2011). Targeted learning: Causal inference for observational and experimental data . New York, NY: Springer. VanderWeele, T. J., & Shpitser, I. (2013). On the definition of a confounder. The Annals of Statistics (1), 196–220. Wang, Y. (2023). Causal inference under temporal and spatial interference. Wing, C., Simon, K., & Bello-Gomez, R. A. (2018). Designing difference in difference studies: Best practices for public health policy research. Annual Review of Public Health (1), 453–469. Xu, Y. (2017). Generalized synthetic control method: Causal inference with interactive fixed effects models. Political Analysis (1), 57–76. Zeldow, B., & Hatfield, L. A. (2021). Confounding and regression adjustment in difference-in-differences studies. Health Services Research (5), 932–941.	{"url":"https://diff.healthpolicydatascience.org/","timestamp":"2024-11-15T03:43:55Z","content_type":"application/xhtml+xml","content_length":"187808","record_id":"<urn:uuid:c1494bfe-6018-4fc6-b865-093631e547cf>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00298.warc.gz"}
Molecular Modeling Basics Let’s start by considering a simple example with one coordinate (for example a distance), $R$, and where the energy is given by $$ E= \frac{1}{2}(R-R_e)^2$$ The corresponding potential energy surface, known as a quadratic PES , is shown in Figure 1. Figure 1. Plot of the (quadratic) potential energy surface given by Equation 1. Here $R_e$ is the value of $R$ at which the energy is lowest (this is known as the equilibrium geometry ) and this is what we’d like to find. We start by taking a guess at $R$, $R_g$. We already know how to check whether this is an energy minimum: we need to evaluate the gradient, which is $$\left(\frac{\partial E}{\partial R}\right)_{R_g}=k(R_g-R_e)$$ It’s clear that the gradient is non-zero for $R_e \ne R_g$. However, by rearranging the equation the gradient also tells us how to change $R$ to get an energy minimum $$R_e=R_g-\frac{1}{k}\left(\frac{\partial E}{\partial R}\right)_{R_g}=R_g+\frac{1}{k}F_g$$ where $F$ is the force, which is the negative gradient. If we know $k$ we can find $R_e$ in one step starting from $R_g$ (Figure 2). Figure 2. The equilibrium geometry can be found in one step on a quadratic PES If we don’t know $k$ then it is safest to take many small steps, i.e. scale the gradient by some small constant ($c$) and repeat until the gradient falls below some threshold (Figure 3) $$R_{n+1}=R_n-c\left(\frac{\partial E}{\partial R}\right)_{R_n}$$ Figure 3. Energy minimization by steepest descent. This general approach is known as steepest descent , since the gradient tells you in which direction where the energy is decreasing (descending) the most. Notice that this means that minimizing the energy will find the closest minimum to the starting geometry . When steepest descent is used to find equilibrium geometries it is often combined with a so-called “line search”, which tries to find the lowest energy in the direction of the current gradient, but the general idea is still the same. Another use of steepest descent is to connect a transitions state with its two closest minima, since this tells you what two minima the transition state connects. Here the guess structure is a transition state structure displaced along the normal mode corresponding to the imaginary frequency (the transition state structure itself cannot be used because its gradient is zero). This path is minimum energy path (MEP) between reactions and products and the resulting collection of structures is known as the intrinsic reaction coordinate (IRC). An IRC is usually depicted as a plot of the potential energy vs the mass-weighted root-mean-square-displacement of the Cartesian coordinates relative to some reference geometry (usually the transition state), usually denoted $s$ (Figure 4). When we draw a potential energy surface for a reaction we usually mean an IRC. Figure 4. Depiction of an IRC or MEP: two steepest descent paths starting from the transition state Clearly, the more steps we take, the higher the computational cost, and in general we want to take as few steps as possible. As I mentioned above, if we knew $k$ we could find the energy minimum in one step. For a quadratic surface $k$ is the second derivative of $E$ with respect to $R$ (or the first derivative of the gradient) so Eq 3 becomes $$ R_e=R_g-\left(\frac{\partial^2 E}{\partial R^2}\right)^{-1}_{R_g}\left(\frac{\partial E}{\partial R}\right)_{R_g}$$ Such a step is known as a Newton-Raphson step quadratic step . This works fine if the surface is quadratic, which is a good approximation near the minimum but not far away. For really bad guesses, where the PES tends to be flat, the quadratic step can be too large (Figure 5). Figure 5. Quadratic steps (a) close to and (b) far from the minimum. Most algorithms will scale back quadratic steps that are considered unreasonably large, and even using quadratic energy minimizers many steps are needed: $$R_{n+1}=R_n-\left(\frac{\partial^2 E}{\partial R^2}\right)^{-1}_{R_n}\left(\frac{\partial E}{\partial R}\right)_{R_n}$$ $$ \mathbf{q}_{n+1}=\mathbf{q}_{n}-c\mathbf{H}^{-1}_n\mathbf{g}_n$$ Eq. 7 is the same equation as 6 but in many dimensions: qn are the current coordinates (for example Cartesian coordinates) for step number $n$ ($n = 0$ for the guess coordinates), $\mathbf{q}_{n+1}$ are the new coordinates, $\mathbf{g}_n$ is the gradient, and $\mathbf{H}_n$ is the Hessian both evaluated at the current coordinates. Note that the Hessian should be computed at each step. The April issue of Computational Chemistry Highlights is out. CCH is an overlay journal that identifies the most important papers in computational and theoretical chemistry published in the last 1-2 years. CCH is not affiliated with any publisher: it is a free resource run by scientists for scientists. You can read more about it here. Table of content for this issue features contributions from CCH editors Mario Barbatti, Steven Bachrach, and Jan Jensen: Big Data Meets Quantum Chemistry Approximations: The ∆-Machine Learning Approach The Nucleoside Uridine Isolated in the Gas Phase Interested in more? There are many ways to subscribe to CCH updates. Also, for your daily computational chemistry fix subscribe to Computational Chemistry Daily This work is licensed under a Creative Commons Attribution 4.0	{"url":"https://molecularmodelingbasics.blogspot.com/2015/05/","timestamp":"2024-11-03T12:04:20Z","content_type":"application/xhtml+xml","content_length":"120913","record_id":"<urn:uuid:d80d89ff-fd20-4f0c-a1b7-f6269e7c7ea1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00465.warc.gz"}
Revolution to Second Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like angle finds its use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the conversion of different units of measurement like r to ″ through multiplicative conversion factors. When you are converting angle, you need a Revolutions to Seconds converter that is elaborate and still easy to use. Converting Revolution to Second is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in Revolution to Second conversion along with a table representing the entire conversion.	{"url":"https://www.unitsconverters.com/en/Revolution-To-Second/Unittounit-3672-3669","timestamp":"2024-11-05T11:52:02Z","content_type":"application/xhtml+xml","content_length":"114133","record_id":"<urn:uuid:4bf4f3ae-e654-424b-a68f-44b379321705>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00119.warc.gz"}
Speed of train going through a tunnel There is a tunnel with a train track and two people are at 1/4th distance from one end of the tunnel. They heard the train sound, and both started running in the opposite directions immediately. Both barely missed the train from hitting. Both run at the same speed. What is the speed of the train? It is not possible to find the speed of the train with just this information. But, if this kind of question is asked in the interviews, it means, we need to find the speed of the train relative to others like the speed of the person, or the distance of the tunnel etc. In the above diagram, AB is the tunnel, and AC is 1/4th of the tunnel and BC is 3/4th of the tunnel. The train is at D when they heard the train sound. The train enters the tunnel at point A. If the train enters the tunnel at point B, then both will not miss the train barely. If the train enters the tunnel at point B, By the time the person travels from C to B, the other person can comfortably cross the tunnel. So, the train enters the at point A only. Since both barely missed the train, it means, both the train and the first person reached point A at the same time. At that time, the second person would be at the middle of the tunnel. The second person also misses the train barely at point B. It means, by the time, the second person reaches from the middle of the tunnel to B, the train reached from point A to B. It means, the train travels twice faster than the person. By the time, the first person reached from C to A, the train reached from D to A. From the relative speeds of the person and the train, we can conclude that, DA is double to CA, and half of AB. No comments:	{"url":"http://algos.tnsatish.com/2010/08/speed-of-train-going-through-tunnel.html","timestamp":"2024-11-02T08:22:05Z","content_type":"application/xhtml+xml","content_length":"41398","record_id":"<urn:uuid:aea9c0eb-64a3-4c10-9034-d0675b6870af>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00405.warc.gz"}
Equations - WBPREP The nature of roots of quadratic equations is determined by whether the roots are real, complex, they are equal, or unequal. The nature of the roots of a quadratic equation can be determined by the discriminant of the equation. Let’s learn about that in this post. What are Roots of a Quadratic Equation An equation … Read more Solve x^6-1=0 and Find the Roots The solutions of x6=1 (x to the power 6 equals 1) are given by x= 1, -1, (-1+√3i)/2, (-1-√3i)/2, (1+√3i)/2, and (1-√3i)/2, where i = √-1 is an imaginary complex number. In this post, we will learn how to find the roots of x^6=1. Solutions of x6=1 To solve the given equation x6=1, we need … Read more Solution of x^4=1 \| Roots of x^4=1 The solutions of x4=1 (x to the power 4 equals 1) are given by x= 1, -1, i, and -i where i=√-1 is the imaginary complex number. In this post, we will learn how to find the roots of x4=1. Solutions of x4=1 To solve the given equation x4=1, we need to follow the below … Read more x^3=1 Solution \| x^3=1 Roots The solutions of x3=1 (x cubed equals 1) are given by x= 1, ω, and ω2 where ω = (-1+√3i)/2 is the complex number. In this post, we will learn how to find the root of x3=1. Solutions of x3=1 The given equation is x3=1. Step 1: Apply the formula of a3-b3. ⇒ x3 -1 … Read more How to Solve Quadratic Equation There are many techniques for how to solve quadratic equations. For example, we can solve a quadratic equation by factoring or directly using the quadratic formula. In this post, we will learn about the same. Solving Quadratic Equations using Factoring To solve a quadratic equation using the factoring method, we will follow the below steps … Read more Factorise x^4+4x^2+3 \| How to Solve x^4+4x^2+3=0 The equation x4+4×2+3=0 is a bi-quadratic equation. In this post, we will first factorise x4+4×2+3 and then solve the equation x4+4×2+3=0. How to Factorise x4+4×2+3 Question: Factorize x4+4×2+3. Solution: To factorize x4+4×2+3, our aim is to express it in the form of a2-b2. For that, we need to add and subtract 1 to both sides. … Read more How to Solve Linear Equations A linear equation in one variable is always of the form ax+b=c, where a, b, and c are constants. Linear equations are also known as simple equations. In this post, we will learn how to solve linear equations with examples. ax+b=c Put c=0. Then ax+b = 0 ⇒ ax = -b This is an equation … Read more How to Factorise x^2+25 \| Solve x^2+25=0 In this post, we will learn how to factorise the quadratic algebraic expression x2+25, then solve the quadratic equation x2+25=0. How to Factorise x2+25? Answer: The factorization is given by x2+25 = (x-5i)(x+5i), where i = √-1 is an imaginary complex number. Solution: To factorise the expression x2+25, we first write the expression in the … Read more How to Factorise and Solve x^4+x^2+1=0 In today’s article, we will first factorize the expression x4+x2+1, and then we will solve the bi-quadratic equation x4+x2+1=0. How to Factorise x4+x2+1 Answer: The factorisation of x4+x2+1 is given by x4+x2+1 = (x2-x+1)(x2+x+1). Solution: At first, we will add and subtract x2 to the given expression. x4+x2+1 = (x4+x2+1) + x2 – x2 = … Read more	{"url":"https://www.wbprep.com/category/equations/","timestamp":"2024-11-03T11:58:50Z","content_type":"text/html","content_length":"156372","record_id":"<urn:uuid:0d6b5a9d-e335-48ac-be53-6f584fd0827f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00516.warc.gz"}
Jobs in math and stat General information There is a strong demand for people with good quantitative skills. Various career rating sites rated math related jobs as the best available. For example, CareerCast lists the top four jobs in the country as mathematician, university professor, statistician, and actuary (see below). Forbes: Best jobs for 2014 says: Most jobs for mathematicians used to be confined to the cloistered halls of academia. But in the age of big data, an increasing number of companies are hiring mathematicians to crunch numbers for all sorts of projects, from energy firms that need to figure out the most efficient ways to get products to distributors, to the U.S. Department of Transportation, which must calculate how best to spend agency money. The Bureau of Labor Statistics projects that job opportunities for mathematicians will grow 23% between 2012 and 2022 and it pegs the median 2012 mathematicianâ€™s salary at $110,000. Those statistics land mathematician at the top of career advice and job listing website CareerCastâ€™s 2014 list of the 10 best jobs in the U.S. Here are some other websites that describe the range of jobs you can have with a mathematics or statistics degree. NOTE: This information is provided as a resource; I do not endorse any of the organizations and cannot vouch for their reputations. Mathematics Association of America career info lists statistician, actuary, computer science, operations research, biomathematics, cryptography, teaching, and finance among math careers. US government jobs in mathematics and statistics Private (not official) listing of government jobs in math and stat American Statistical Association career info discusses careers using statistics. ThisIsStatistics discusses careers using statistics. Panel on jobs in statistics in Washington DC on 17 February 2016, 3:30-6:45 pm, see pages 11-12 Data Science The Math/Stat Department at American University has a few courses in Data Science; in Fall 2017 there will be more courses and the ability to get a certificate, a minor, or a major. Information about Data Science describes this new field that combines statistics and computation. Actuarial Science Actuarial Science is a great career that involves using mathematics and statistics to analyze data and predict trends. The traditional job for an actuary was with an insurance company, where one would model costs from auto accidents, fire, natural disasters, life and health insurance, etc. These models are used to set premium prices. Now there is a need for people with these skills in government agencies, large private companies and unions. Various job rating surveys have rated actuary as one of the best jobs in the future. This is because there is a long term demand for these positions, they are secure long term careers, opportunities to advance, strong salaries (above those of economists, attorneys, accountants or professors!), safe working conditions, etc. If you are interested in actuarial science, you should take the following courses: Math 221 (Calculus I), 222 (Calculus II), 223 (Calculus III), 501 (Probability), 565 (Math. Applications of Interest and Derivatives); Stat 202 (Basic Stat), 302 (Intermediate Stat), 502 (Introduction to Mathematical Statistics); and CSC 280 (Introduction to Computer Science). Some of these, e.g. Math 501 and Math 565 prepare you directly for two of the actuarial exams. Below are some links to websites that can give you more information. Internships, jobs, and summer programs for students Look at the links above for Statistics and Actuarial Science for internships in those special fields. Below are some links to websites that can give you more information. Teaching Math • US Department of Education Information on teaching, in particular geographical areas where there are teacher shortages. • MathTeaching.org Resources for potential teachers. • Southern Teachers Agency Teaching jobs at private/independent high schools in the southern U.S. Other information	{"url":"https://edspace.american.edu/jpnolan/information-on-jobs-in-mathematics-and-statistics/","timestamp":"2024-11-02T05:43:48Z","content_type":"text/html","content_length":"37930","record_id":"<urn:uuid:45970bff-b0ae-4e5a-96af-04a265f24257>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00170.warc.gz"}
Marksheet In Excel What Is Marksheet Format In Excel? Marksheet in Excel is created in an Excel workbook to maintain a cumulative data for present and future analysis. It helps us keep track of the performances of the employees in a company, students in an educational organization, etc. An Excel Marksheet format is an automated Marksheet that gives us the required results when we modify or update the existing data, or add new data. Key Takeaways • The Marksheet in Excel is created to calculate the result of the students. • Excel functions such as IF, SUM, AVERAGE, COUNTIF, ROUND, etc., are used to calculate the results, like finding the total, average, and count of students falling under specific criteria, etc., using Marksheet format and formulas. • We can provide an alternative output instead of the default “True” or “False” using the conditional functions such as IF() and COUNTIF(), as we saw in the examples like output as Pass or Fail, Promoted or Distinction, etc. How To Make Marksheet In Excel Format? We can Make Marksheet In Excel Format using the following methods, namely, 1. SUM function. □ Comma Method. □ Colon Method (Shift Method). 2. AVERAGE function. 3. ROUND function. 4. IF function. 5. COUNTIF function Excel VBA – All in One Courses Bundle (35+ Hours of Video Tutorials) If you want to learn Excel and VBA professionally, then Excel VBA All in One Courses Bundle (35+ hours) is the perfect solution. Whether you’re a beginner or an experienced user, this bundle covers it all – from Basic Excel to Advanced Excel, Macros, Power Query, and VBA. #1 – SUM Function It adds the selected numeric cell values and returns the total value. Its syntax is, a. SUM function – Comma Method – In this method, we enter the cell values separately in the formula using commas. The following example depicts the marks of the subjects of the students of a class, and we will calculate the total marks using the Excel SUM Function Comma Method. The steps to evaluate the values using the Excel SUM Function Comma Method are as follows: • Step 2: Next, we will enter the Excel SUM Formula in cell F2. □ Enter the value of ‘number1’ as B2, i.e., the sum values in cell B2. □ Enter the value of ‘number2’ as C2, i.e., the sum values in cell C2. □ Enter the value of ‘number3’ as D2, i.e., the sum values in cell D2. □ Enter the value of ‘number4’ as E2, i.e., the sum values in cell E2. Therefore, the complete formula is =SUM(B2,C2,D2,E2) in cell F2. • Step 3: Press the “Enter” key. The result is “353” in the image below. • Step 4: Drag the formula from cell F2 to F6 using the fill handle in excel. The output is shown below. 2. SUM function – Colon Method (Shift Method) – In this method, the cell values are entered by selecting the cell range or dragging the cursor on the required cells. Using the same example as above, we will calculate the marks of the subjects of the students of a class and the total marks using the Excel SUM Function Colon Method. The steps to evaluate the values using the Excel SUM Function Colon Method are as follows: • Step 2: Next, we will enter the Excel SUM Formula =SUM(B2:E2) in cell F2. • Step 3: Press the “Enter” key. The result is “353” in the image below. • Step 4: Drag the formula from cell F2 to F6 using the fill handle. The output is shown below. Output Observation: Hence, both the methods of the SUM function, i.e., the Comma Method and the Colon Method, gives the same output. #2 – AVERAGE Function It returns the average of the selected numerical values. The values can be numbers, percentages, or time. The function first calculates the sum of the values, and divides it by the count of the values of the list. For instance, the data below depicts the marks of the subjects of the students of a class, and we will calculate the average marks using the Excel AVERAGE Function. The steps to evaluate the values using the AVERAGE Function are as follows: • Step 1: Select cell H2, and enter the formula =AVERAGE(D2:G2). • Step 2: Press the “Enter” key. The result is “82.5” in the image below. • Step 3: Drag the formula from cell H2 to H6 using the fill handle. The output is shown below. #3 – ROUND Function It is used to round up the numerical data according to the number of digits assigned to round the values. ROUND functions Arguments Explanation • number: It is the numerical value we want to round up or down. It is a mandatory argument. • num_digits: It is the number of digits we want to round up. It is a mandatory argument. The following example depicts the marks and average marks of the subjects of the students of a class, and we will calculate the round value of the average marks using the Excel ROUND Function. The steps to evaluate the values using the Excel ROUND Function are as follows: • Step 1: Select cell I2, and enter the formula =ROUND(H2,0). • Step 2: Press the “Enter” key. The result is “83” in the image below. • Step 3: Drag the formula from cell I2 to I6 using the fill handle. The output is shown below. #4 – IF Function It tests the logical conditions and gives the output as true or false. In addition, it performs analytical tests for the given data using mathematical operators. The argument can have desired output other than true and false. The following example depicts the scores of two subjects of the students of a class, and we will evaluate whether they are passing or failing using the Excel IF Function. The steps to evaluate the values using the Excel IF Function are as follows: • Step 1: Select cell D2, and enter the formula =IF(AND(B2>=60, C2>=90), “Pass,” “Fail”) [Note: The ‘logical test’ value is B2>=60 AND C2>=90, i.e., the condition values of cells B2 & C2. ] • Step 2: Press the “Enter” key. The result is “Pass” in the image below. • Step 3: Drag the formula from cell D2 to D6 using the fill handle. The output is shown below. #5 – COUNTIF Function It counts all the cells with cell values in the range as specified by the conditions. COUNTIF functions Arguments Explanation • range: It is a range on which the criteria argument is applied. It is a mandatory argument. • criteria: It is a condition applied to the range of values argument. It is a mandatory argument. The following example depicts the annual scores of the subjects of the students of a class, and we will whose score is greater than 80 using the Excel COUNTIF Function. The steps to evaluate the values using the Excel COUNTIF Function are as follows: • Step 1: Select cell B7, and enter the formula =COUNTIF(B2:B6,“>80”), i.e., the ‘range’ value is B2:B6, and the ‘criteria’ value is ‘>80’. • Step 2: Press the “Enter” key. The result is “2”, i.e., in the cell range B2 to B6, there are 2 cells greater than 80 [cell B2 and B3]. Important Things To Note • All the formulas used should be written in the correct syntax, or else we will get the “#NAME?” error. • For entering any text in the function, we should use double quotes (“ ”) as we have used while writing “Pass,” or “Fail,” etc, in the IF and COUNTIF function examples. • If the dataset consists of non-numeric values in the Marksheet, the formulas ignore them and finds the total of the other numeric values, irrespective of the functions used. Frequently Asked Questions (FAQs) 1. How to create Marksheet in Excel? Following are the steps to create a Marksheet in Excel. 1) Insert Personal Details. The first part of our mark sheet contains the student details. 2) Insert the Subject Names as column Headers. 3) Insert respective Marks of the subjects of individual students. 4) Insert Subject wise Grades 5) Calculate Total Marks using the formulas. 6) Calculate the Result and display the same. 2. How to calculate grades in the Marksheet in Excel? To calculate the grade, we need to, • Add up all scores, • Divide by the total number of exam points, and • Multiply by the weighted percentage. • Then calculate the percentage for the project, and add the two totals together to get the final grade. 3. Example of calculating Result from Marksheet in Excel? The following example depicts the annual scores of the students of a class, and we will evaluate whether they have got Distinction using the Excel IF Function. The procedure to evaluate the values using the Excel IF Function is, • Select cell C2. • Enter the formula =IF(B2>=80), “Distinction,” “Promoted”) • Press the “Enter” key. • Drag the formula from cell C1 to C6 using the fill handle. [Note: The ‘logical test’ value is B2>=80 AND C2>=90, the ‘value if true’ value is ‘Distinction,’, and the ‘value if false’ value is ‘Promoted,’] The output is shown above, i.e., student details in cells C2, C4, and C5, are promoted, and the student details in cells C3 and C6, are promoted with distinction. Download Template This article must help understand the Marksheet in Excel’s formula and examples. You can download the template here to use it instantly. Recommended Articles This has been a guide to Marksheet In Excel. Here we create marksheet in excel using Top 5 Methods along with examples & a downloadable template. You can learn more from the following articles –	{"url":"https://www.excelmojo.com/marksheet-in-excel/","timestamp":"2024-11-10T11:47:52Z","content_type":"text/html","content_length":"230284","record_id":"<urn:uuid:89264a66-f160-44f1-a224-7b25ef975c20>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00648.warc.gz"}
The short and long hands of a clock are 4cm and 6cm long respectively. Find the Total distance travelled will be equal to distance travelled in one revolution that will it’s circumference multiplied by total revolution. You have to add distance travelled by short hand and distance travelled by long hand. Complete step-by-step answer: In two days total distance covered will be equal to distance covered by shorthand + distance covered by long hand of clock. Distance covered by small hand or hour hand tip = total revolution \[ \times 2\pi r\] =$\frac{{2 \times 24}}{{12}} \times 2 \times 3.14 \times 4 = 100.48{\text{cm}}$(in two days total hours is 48 and hour hand completes one revolution in 12 hours so you have to divide by 12 to find total revolution) Distance travelled by large hand or minute hand tip =total revolution $ \times 2\pi {\text{r}}$ $ \Rightarrow \frac{{2 \times 24}}{1} \times 2 \times 3.14 \times 6 = 1808.64{\text{cm}}$ In two days total hours is 48 and a minute hand completes one revolution in 1 hour so to find the total number of revolutions you have to divide by 1. So total distance covered= 100.48cm + 1808.64cm=1909.12cm Note: - Whenever you get this type of question the key concept of solving is you have to find total distance covered by short hand and total distance covered by long hand and add them to find total distance travelled. And to find the total number of revolutions you have to count the total number of hours and divide by as much number of hours in which that hand completes one revolution.	{"url":"https://www.vedantu.com/question-answer/the-short-and-long-hands-of-a-clock-are-4cm-and-class-7-maths-cbse-5ee49b0a5cbfd47b4697762b","timestamp":"2024-11-07T13:03:52Z","content_type":"text/html","content_length":"148412","record_id":"<urn:uuid:5072f8e0-a4ad-4aaa-a8a1-4e8fd5a99375>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00210.warc.gz"}
Yeppp! is a high-performance SIMD-optimized mathematical library. This Julia package makes it possible to call Yeppp from Julia. Install this package by For common 64-bit platforms, this will download dependencies automatically. For some other platforms such as the PowerPC 64 architecture, you may still be able to use this package by downloading Yeppp! and extracting from the binaries folder the file(s) specific to your OS. Check the platforms supported by Yeppp! here. Make sure the extracted files are available on the system library search path or in the current directory. For example, in Julia's bin folder. See example usage below. Yeppp's vectorized log is 3x faster than the one in Base, which is written in Julia. using Yeppp x = rand(10^7) ty = @elapsed Yeppp.log!(similar(x), x) t = @elapsed log(x) The following functions are available for Array{Float64}. Inputs are in x, and outputs are in y. dot(x1, x2) max!(y, x1, x2) min!(y, x1, x2) add!(y, x1, x2) subtract!(y, x1, x2) multiply!(y, x1, x2) log!(y, x) exp!(y, x) evalpoly!(y, x_coeff, x) sin!(y, x) cos!(y, x) tan!(y, x) See the Yeppp! documentation for the full set of functions available in Yeppp!.	{"url":"https://juliapackages.com/p/yeppp","timestamp":"2024-11-04T00:48:51Z","content_type":"text/html","content_length":"41317","record_id":"<urn:uuid:dd4cd9d1-11f9-4cb5-bfdd-f5c278cf1c81>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00171.warc.gz"}
Excel Conditional Formatting Formula In this tutorial, we will learn how to use a Python formula to apply conditional formatting in Excel. Specifically, we will focus on highlighting text in one tab if it is not present in another tab. This technique can be useful when you want to quickly identify text that is missing or not matching between two tabs. To achieve this, we will use the COUNTIF function in Excel, along with a comparison operator. The COUNTIF function allows us to count the number of cells in a range that meet a specific condition. In our case, we want to count the number of cells in the 'Legend' tab that match any of the values in the 'Offer' tab. The formula we will use is as follows: =COUNTIF(Legend!I1:I100, Offer!D1:D100)=0 Let's break down the formula step-by-step: 1. We specify the range in the 'Legend' tab that we want to check for matches using the notation 'Legend!I1:I100'. This range represents the cells from I1 to I100 in the 'Legend' tab. 2. We specify the range in the 'Offer' tab that we want to compare against using the notation 'Offer!D1:D100'. This range represents the cells from D1 to D100 in the 'Offer' tab. 3. We use the COUNTIF function to count the number of cells in the 'Legend' tab that match any of the values in the 'Offer' tab. 4. We compare the count to 0 using the equal operator (=). If the count is 0, it means that there are no matches, and the formula returns TRUE. If the count is greater than 0, it means that there are matches, and the formula returns FALSE. To apply this formula as a condition for conditional formatting, you can follow these steps: 1. Select the range of cells in the 'Offer' tab that you want to apply the conditional formatting to. 2. Go to the 'Home' tab in Excel and click on 'Conditional Formatting' in the 'Styles' group. 3. Choose 'New Rule' from the drop-down menu. 4. In the 'New Formatting Rule' dialog box, select 'Use a formula to determine which cells to format'. 5. Enter the formula '=COUNTIF(Legend!I1:I100, Offer!D1:D100)=0' in the 'Format values where this formula is true' field. 6. Choose the formatting style you want to apply to the cells that meet the condition. 7. Click 'OK' to apply the conditional formatting. By following these steps, the text in the specified range of cells in the 'Offer' tab will be highlighted if it is not present in the specified range of cells in the 'Legend' tab. In conclusion, using a Python formula in Excel, we can easily apply conditional formatting to highlight text based on a condition. This technique can be helpful in various scenarios where you need to compare and identify missing or non-matching text between different tabs in Excel. A Google Sheets formula =COUNTIF(Legend!I1:I100, Offer!D1:D100)=0 Formula Explanation This formula uses the COUNTIF function to check if the values in range I1:I100 in the "Legend" tab match any of the values in range D1:D100 in the "Offer" tab. If there are no matches, it means that the text in range D1:D100 in the "Offer" tab is not present in range I1:I100 in the "Legend" tab. Step-by-step explanation 1. The COUNTIF function is used to count the number of cells in range I1:I100 in the "Legend" tab that match any of the values in range D1:D100 in the "Offer" tab. 2. The formula compares the count to 0. If the count is 0, it means that there are no matches, and the formula returns TRUE. If the count is greater than 0, it means that there are matches, and the formula returns FALSE. 3. The result of the formula can be used as a condition for conditional formatting. If the formula returns TRUE, the text in range D1:D100 in the "Offer" tab will be highlighted. For example, let's say we have the following data in the "Offer" tab: \| D \| \| \| \| Cat \| \| Dog \| \| Cow \| \| Pig \| And the following data in the "Legend" tab: \| I \| \| \| \| Dog \| \| Pig \| In this case, the formula =COUNTIF(Legend!I1:I100, Offer!D1:D100)=0 would return TRUE, because none of the values in range D1:D100 in the "Offer" tab (Cat, Dog, Cow, Pig) are present in range I1:I100 in the "Legend" tab (Dog, Pig). This means that the text in range D1:D100 in the "Offer" tab would be highlighted using conditional formatting.	{"url":"https://codepal.ai/excel-formula-generator/query/2KkSiciN/excel-formula-conditional-formatting-highlight-text","timestamp":"2024-11-08T11:01:30Z","content_type":"text/html","content_length":"99507","record_id":"<urn:uuid:9dce5780-e034-41e1-b53a-bf86ce9af94f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00386.warc.gz"}
Nonparametric and Empirical Bayes Estimation Methods You are here Nonparametric and Empirical Bayes Estimation Methods Date Issued: In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski's method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an $(r+1)$-dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic $(r+1)$-dimensional function, we show that by choosing Besov balls of mixed smoothness, we can avoid the ''curse of dimensionality'' and, hence, obtain higher than usual convergence rates when $r$ is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels. Title: Nonparametric and Empirical Bayes Estimation Methods. Benhaddou, Rida, Author Pensky, Marianna, Committee Chair Name(s): Han, Deguang, Committee Member Swanson, Jason, Committee Member Ni, Liqiang, Committee Member University of Central Florida, Degree Grantor Type of text Date Issued: 2013 Publisher: University of Central Florida Language(s): English In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski's method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an $ (r+1)$-dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive Abstract/ wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic $(r+1)$-dimensional function, we show that by choosing Besov balls Description: of mixed smoothness, we can avoid the ''curse of dimensionality'' and, hence, obtain higher than usual convergence rates when $r$ is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels. Identifier: CFE0004814 (IID), ucf:49737 (fedora) Note(s): Sciences, Mathematics This record was generated from author submitted information. Subject(s): Empirical Bayes -- functional deconvolution -- minimax convergence rate -- wavelets. Link to This http://purl.flvc.org/ucf/fd/CFE0004814 Restrictions public 2013-08-15 on Access: Host UCF In Collections	{"url":"https://ucf.digital.flvc.org/islandora/object/ucf%3A49737","timestamp":"2024-11-04T11:36:46Z","content_type":"text/html","content_length":"39241","record_id":"<urn:uuid:b786e57a-88f6-4fd9-9906-a295cfbaf2b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00295.warc.gz"}
Eclipse Qrisp Theoretical Overview# In this tutoral we will take a peek into the inner workings of the Qrisp QAOA module. From learning the main concepts of the algorithm we’ll take a deep dive into the QAOAProblem class, exploring its various components and how they come together to implement the algorithm. From defining quantum parameters to optimizing and measuring the final state, we’ll really get to see how this class functions under the hood. QAOA in a nutshell# The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed for solving combinatorial optimization problems. The quantum and classical part work together iteratively: the quantum computer prepares a quantum state and measures it, producing a classical output; this output is then fed into a classical optimization routine which produces new parameters for the quantum part. This process is repeated until the algorithm converges to an optimal or near-optimal solution. Before even running the algorithm there is a need to define some initial state $\|\psi_0\rangle$, often chosen to be equal superposition state $$\|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z\|z\rangle, $$ where $n$ is the number of qubits. The QAOA operates in a sequence of layers, each consisting of a problem-specific operator and a mixing operator. To be a little more exact, the state $\|\psi_0\rangle$ is then evolved under the action of $p$ layers of QAOA, where one layer consists of applying the unitary phase separating operator $$U_P(C,\gamma)=e^{-i\gamma C}=\prod_{\alpha=1}^me^{-i\gamma C _{\alpha}}, $$ which applies phase to each computational basis state based on its cost function value; and the unitary mixer operator $$U_M(B,\beta)=e^{-i\beta B}, $$ where $B$ represents a specific mixer operator that drives the transitions between different states. Because of properties of the unitaries’ eigenvalues, the QAOA parameters are bound to hold values $\gamma\in\{0, 2\pi\},$ and $\beta\in\{0,\pi\}$ which are then optimized classically. After $p$ layers of QAOA, we can define the angle dependent quantum state $$\|\psi_p\rangle=\|\boldsymbol\gamma,\boldsymbol\beta\rangle=U_M(B,\beta_p)U_P(C,\gamma_p)\cdots U_M(B,\beta_1)U_P(C,\ The end goal in QAOA is now to optimize the variational parameters $\gamma_p$ and $\beta_p$ in order to minimize the expectation value of the cost function with respect to the final state $\ket {\psi_p}$. This is done using classical optimization techniques. It’s important to remember that QAOA provides an approximate solution, and its performance depends on factors like problem size and structure, choice of initial state, and number of layers $p$. Increasing $p$ generally leads to better solutions but also increases computational cost. Alternating Operator Ansatz# QAOA on its own is an efficient tool for exploring complex combinatorial optimization problems. However, as with any tool, there is always room for improvement and expansion. Its potential can be further expanded by introducing a broader range of operators, not just those derived from the time evolution under a fixed local Hamiltonian proposed in the original paper. Exactly this concept is well illustrated in the paper “From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz” by Stuart Hadfield and his team. They have taken the foundational principles of the QAOA and extended them, creating an upgraded version that is both more adaptable and easier to implement. The main ideas we use in our implementations are inspired by this work. Similarly to the original QAOA, we have several key components when implementing the Quantum Alternating Operator Ansatz: • COST FUNCTION: the cost function is problem-specific and defines the optimization landscape. In the Quantum Alternating Operator Ansatz, cost functions can be represented by more general families of operators. • INITIAL STATE: initial state can be any state over all computational basis states. It is in most cases chosen to be a uniform superposition. • PHASE SEPARATOR: this applies a phase to each computational basis state based on its cost function value. In the Quantum Alternating Operator Ansatz, we can use a wider range of operators tailored to each problem instance. • MIXER: drives transitions between different states. In the Quantum Alternating Operator Ansatz, mixers can be chosen from a broader set of unitaries, which allows for more efficient implementation and potentially better exploration of the solution space. In their paper, Hadfield and his colleagues give us some really useful examples of how to formulate different problem instances using these building blocks. The appendix section, in particular, stands out as it provides detailed problem formulations, most of which are now implemented in Qrisp using the QAOAProblem class. The following table provides a detailed overview of the problem instances already implemented within our framework, along with their corresponding mixer type: Our QAOA journey doesn’t stop here. In the next tutorials we’re going to tackle two fascinating problems: MaxCut and Max-$\kappa$-Colorable Subgraph, showcasing multiple unique features of Qrisp, including the functionality of creating custom QuantumVariable types - get ready to add a splash of QuantumColor to your code.	{"url":"https://www.qrisp.eu/general/tutorial/Quantum%20Alternating%20Operator%20Ansatz/Theoretical.html","timestamp":"2024-11-05T00:03:29Z","content_type":"text/html","content_length":"32043","record_id":"<urn:uuid:09a0dd55-afb9-421a-9f45-66ed2ff13a83>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00259.warc.gz"}
CS 182/282A: Deep Neural Networks - Fall 2022 Deep Networks have revolutionized computer vision, language technology, robotics and control. They have a growing impact in many other areas of science and engineering, and increasingly, on commerce and society. They do not however, follow any currently known compact set of theoretical principles. In Yann Lecun's words they require "an interplay between intuitive insights, theoretical modeling, practical implementations, empirical studies, and scientific analyses." This is a fancy way of saying “we don’t understand this stuff nearly well enough, but we have no choice but to muddle through anyway.” This course attempts to cover that ground and show you how to muddle through even as we aspire to do more. This is a graduate-level/advanced undergraduate course about a particular approach to information processing using (simulated) analog circuits where the desired circuit behavior is tuned via optimization involving data since we have no idea how to do hand-tuning at scale. Probabilistic frames are useful to understand what is going on, as well as how we navigate certain design choices. Overall, we expect students to have a strong mathematical background in calculus, linear algebra, probability, optimization, and statistical learning. Berkeley undergraduate courses that can help build maturity include: • Calculus: Math 53 (note: Math 1B or AP Math is not enough) • Linear Algebra and Optimization: EECS 16B and EECS 127/227A is ideal, but EECS 16B alone might be enough if students have complete mastery of that material. Math 110 is also helpful. (note: Math 54 or EECS 16A is required as a minimum, but are not nearly enough.) • Probability: EECS 126, Stat 134, or Stat 140 (note: CS 70 is required at a minimum, but might not be enough for everyone) • Statistical Learning: CS 189/289A or Stat 154 (note: Data 102 is insufficient, even when combined with Data 100.) Math 53 and EECS 126 and EECS 127 and CS 189 is the recommended background. Prerequisites are not enforced for enrollment, but we encourage you to consider taking some of the classes listed above and save this course for a future semester if you feel shaky on the The course assumes familiarity with programming in a high-level language with data structures. Homeworks and projects will typically use Python. We encourage you to check out this tutorial if you haven’t used it before. Students who have taken Berkeley courses like CS 61A and CS 61B are well-prepared for the programming components of the class. We do not have the staff bandwidth to help students with material that they should have understood before taking this course. If you choose to proceed with this course, you are accepting full responsibility to teach yourself anything in your background that you are missing. We will not be slowing down to accommodate you, and questions pertaining to background material will always have the lowest priority in all course forums. The goal is to teach a principled course in Deep Learning that serves the diverse needs of our students while also codifying the present understanding of the field. Topics covered may include, but are not limited to: • Underlying themes of deep learning, including building beyond underlying machine learning concepts like supervised vs unsupervised learning, regression and classification, training/validation/ testing, distribution shifts, regularization, the fundamental underlying tradeoffs; • Defining and training neural networks: features, computation graphs, backpropagation, iterative optimization (SGD, Newton’s Method, Momentum, RMSProp, AdaGrad, Adam), strategies for training (explicit and implicit regularization, batch and layer normalization, weight initialization, gradient clipping, ensembles, dropout), hyperparameter tuning • Families of contemporary models: fully connected networks, convolutional nets, graph neural nets, recurrent neural nets, transformers • Problems that utilize neural networks: computer vision, natural language processing, generative models, and others. • Conducting experiments in a systematic, repeatable way, leveraging and presenting data from experiments to reason about network behavior.	{"url":"https://cogak.com/course/241","timestamp":"2024-11-02T19:03:26Z","content_type":"text/html","content_length":"143406","record_id":"<urn:uuid:e7eb3ef6-8425-45a4-a486-8e524df3c6d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00146.warc.gz"}
Homotopy category Short description: Concept in math In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. The naive homotopy category The category of topological spaces Top has objects the topological spaces and morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category^[1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f: X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other. There is a functor from Top to hTop that sends spaces to themselves and morphisms to their homotopy classes. A map f: X → Y is called a homotopy equivalence if it becomes an isomorphism in the naive homotopy category.^[2] Example: The circle S^1, the plane R^2 minus the origin, and the Möbius strip are all homotopy equivalent, although these topological spaces are not homeomorphic. The notation [X,Y] is often used for the set of morphisms from a space X to a space Y in the naive homotopy category (but it is also used for the related categories discussed below). The homotopy category, following Quillen Quillen (1967) emphasized another category which further simplifies the category of topological spaces. Homotopy theorists have to work with both categories from time to time, but the consensus is that Quillen's version is more important, and so it is often called simply the "homotopy category".^[3] One first defines a weak homotopy equivalence: a continuous map is called a weak homotopy equivalence if it induces a bijection on sets of path components and a bijection on homotopy groups with arbitrary base points. Then the (true) homotopy category is defined by localizing the category of topological spaces with respect to the weak homotopy equivalences. That is, the objects are still the topological spaces, but an inverse morphism is added for each weak homotopy equivalence. This has the effect that a continuous map becomes an isomorphism in the homotopy category if and only if it is a weak homotopy equivalence. There are obvious functors from the category of topological spaces to the naive homotopy category (as defined above), and from there to the homotopy category. Results of J.H.C. Whitehead, in particular Whitehead's theorem and the existence of CW approximations,^[4] give a more explicit description of the homotopy category. Namely, the homotopy category is equivalent to the full subcategory of the naive homotopy category that consists of CW complexes. In this respect, the homotopy category strips away much of the complexity of the category of topological spaces. Example: Let X be the set of natural numbers {0, 1, 2, ...} and let Y be the set {0} ∪ {1, 1/2, 1/3, ...}, both with the subspace topology from the real line. Define f: X → Y by mapping 0 to 0 and n to 1/n for positive integers n. Then f is continuous, and in fact a weak homotopy equivalence, but it is not a homotopy equivalence. Thus the naive homotopy category distinguishes spaces such as X and Y, whereas they become isomorphic in the homotopy category. For topological spaces X and Y, the notation [X,Y] may be used for the set of morphisms from X to Y in either the naive homotopy category or the true homotopy category, depending on the context. Eilenberg–MacLane spaces One motivation for these categories is that many invariants of topological spaces are defined on the naive homotopy category or even on the true homotopy category. For example, for a weak homotopy equivalence of topological spaces f: X → Y, the associated homomorphism f[]: H[i](X,Z) → H[i](Y,Z) of singular homology groups is an isomorphism for all natural numbers i.^[5] It follows that, for each natural number i, singular homology H[i] can be viewed as a functor from the homotopy category to the category of abelian groups. In particular, two homotopic maps from X to Y induce the same homomorphism on singular homology groups. Singular cohomology has an even better property: it is a representable functor on the homotopy category. That is, for each abelian group A and natural number i, there is a CW complex K(A,i) called an Eilenberg–MacLane space and a cohomology class u in H^i(K(A,i),A) such that the resulting function [math]\displaystyle{ [X,K(A,i)]\to H^i(X,A) }[/math] (giving by pulling u back to X) is bijective for all topological spaces X.^[6] Here [X,Y] must be understood to mean the set of maps in the true homotopy category, if one wants this statement to hold for all topological spaces X. It holds in the naive homotopy category if X is a CW complex. Pointed version One useful variant is the homotopy category of pointed spaces. A pointed space means a pair (X,x) with X a topological space and x a point in X, called the base point. The category Top[] of pointed spaces has objects the pointed spaces, and a morphism f: X → Y is a continuous map that takes the base point of X to the base point of Y. The naive homotopy category of pointed spaces has the same objects, and morphisms are homotopy classes of pointed maps (meaning that the base point remains fixed throughout the homotopy). Finally, the "true" homotopy category of pointed spaces is obtained from the category Top[*] by inverting the pointed maps that are weak homotopy equivalences. For pointed spaces X and Y, [X,Y] may denote the set of morphisms from X to Y in either version of the homotopy category of pointed spaces, depending on the context. Several basic constructions in homotopy theory are naturally defined on the category of pointed spaces (or on the associated homotopy category), not on the category of spaces. For example, the suspension ΣX and the loop space ΩX are defined for a pointed space X and produce another pointed space. Also, the smash product X ∧ Y is an important functor of pointed spaces X and Y. For example, the suspension can be defined as [math]\displaystyle{ \Sigma X=S^1\wedge X. }[/math] The suspension and loop space functors form an adjoint pair of functors, in the sense that there is a natural isomorphism [math]\displaystyle{ [\Sigma X, Y]\cong [X,\Omega Y] }[/math] for all spaces X and Y. Concrete categories While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions (in the naive homotopy category) or "zigzags" of functions (in the homotopy category). Indeed, Freyd showed that neither the naive homotopy category of pointed spaces nor the homotopy category of pointed spaces is a concrete category. That is, there is no faithful functor from these categories to the category of sets.^[7] Model categories There is a more general concept: the homotopy category of a model category. A model category is a category C with three distinguished types of morphisms called fibrations, cofibrations and weak equivalences, satisfying several axioms. The associated homotopy category is defined by localizing C with respect to the weak equivalences. This construction, applied to the model category of topological spaces with its standard model structure (sometimes called the Quillen model structure), gives the homotopy category defined above. Many other model structures have been considered on the category of topological spaces, depending on how much one wants to simplify the category. For example, in the Hurewicz model structure on topological spaces, the associated homotopy category is the naive homotopy category defined above.^[8] The same homotopy category can arise from many different model categories. An important example is the standard model structure on simplicial sets: the associated homotopy category is equivalent to the homotopy category of topological spaces, even though simplicial sets are combinatorially defined objects that lack any topology. Some topologists prefer instead to work with compactly generated weak Hausdorff spaces; again, with the standard model structure, the associated homotopy category is equivalent to the homotopy category of all topological spaces.^[9] For a more algebraic example of a model category, let A be a Grothendieck abelian category, for example the category of modules over a ring or the category of sheaves of abelian groups on a topological space. Then there is a model structure on the category of chain complexes of objects in A, with the weak equivalences being the quasi-isomorphisms.^[10] The resulting homotopy category is called the derived category D(A). Finally, the stable homotopy category is defined as the homotopy category associated to a model structure on the category of spectra. Various different categories of spectra have been considered, but all the accepted definitions yield the same homotopy category. Original source: https://en.wikipedia.org/wiki/Homotopy category. Read more	{"url":"https://handwiki.org/wiki/Homotopy_category","timestamp":"2024-11-08T20:17:11Z","content_type":"text/html","content_length":"51722","record_id":"<urn:uuid:15460261-e1dd-4419-a8be-87655564a294>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00719.warc.gz"}
Category:Zkl - Rosetta CodeCategory:Zkl programming language may be used to instruct a computer to perform a task. Listed below are all of the tasks on Rosetta Code which have been solved using Zkl. zkl is a general purpose object oriented programming language. It is imperative but borrows concepts from many programming paradigms, including functional and prototype based. It is curly-bracketed, dynamic, reflective, and threaded. It has built in garbage collection, strings, lists, dictionaries, threads, fibers (continuations and co-routines) and more. The syntax strongly resembles the C programming language while the data model is closer to that of Python and Smalltalk. The goals of the language are rapid prototyping and programmer productivity. Program speed and the the constraints of production oriented programming are not high on the "needs" list. It is open source and free. Download, compile and play with it. The VM is written in C, it compiles with clang, GCC or VisualStudio (makefiles, project files included) on Linux/FreeBSD/MS Windows. Running it is old school: command line or REPL, no IDE or GUI. Will work with emacs or vi! I use c mode. External Links and References Web Pages This category has only the following subcategory. Pages in category "Zkl" The following 200 pages are in this category, out of 1,011 total. previous page ) ( next page previous page ) ( next page	{"url":"https://rosettacode.org/wiki/Category:Zkl?pagefrom=Identity+matrix","timestamp":"2024-11-06T20:56:59Z","content_type":"text/html","content_length":"70917","record_id":"<urn:uuid:6aad05aa-0eca-472c-8d37-584fcf5d3d75>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00733.warc.gz"}
Bellman Ford Prerequisites: Shortest Path Bellman Ford is an algorithm that finds the shortest path from one source node to every other node in the graph. The running time is O(n^2) and is able to find negative cycles. Bellman Ford can be done using backtracking to find the shortest path in a graph. We first start at the starting node with a starting cost of 0 and 0 edges used. For each node thats connected to that node, we repeat and add to the cost of the node. We will do an example of the Bellman Ford algorithm on the above graph. At each node we have the node index and the current weight to reach that node. We start at node 0 with a weight of 0. From node 0, we can reach node 1 and node 3. At node 1, we have an accumulative weight of 3. At node 3, we have an accumulative weight of 5. From node 1, we can reach node 2 and node 4 with respective accumulative weights of 10 and 5. From node 3, we can reach node 4 with an accumulative weight of 9. From node 2, we can reach node 5 with an accumulative weight of 19. From node 4, we can reach node 5 with an accumulative weight of 11. From node 4, we can reach node 5 with an accumulative weight of 15. Let N be the number of nodes in the graph Let edges an adjacency list of the graph where: edges[source] contains all edges of the graph where source is the source edge An edge is represented as an object where: edge.weight is the weight of the edge edge.target is the target node of the edge edge.source is the source node of the edge Let start be the starting node Let shortestPath[target] be the shortest path from the source node to the target node Let bellmanFord(target,n,w) be the shortest path from the source node to the target node using n edges and cost of w. bellmanFord(target, n , w) Base Case: bellmanFord(target, N , w): bellmanFord(source, n, w): shortestPath[source] = min(shortestPath[source], w) bellmanFord(edge.dest, n + 1, w + edge.weight) for edge in edges[source] shortestPath = [0] * N We can rewrite this solution using dynamic programming to be more efficient. class edge { int weight, source, dest; public edge(int source, int dest, int weight) { this.source = source; this.dest = dest; this.weight = weight; public static int UNDEFINED = Integer.MIN_VALUE; public static int BellmanFord(Vector<Vector<edge>> adjList, int startNode, int endNode) { int n = adjList.size(); // Let dist[i] be minimum distance from start to i. int[] dist = new int[n]; // initialize dist[i]=0 and used[i]=false for (int i = 0; i < n; i++) { dist[i] = UNDEFINED; dist[startNode] = 0; // Maximum path to take is n-1 steps. for (int i = 0; i < n - 1; i++) { // Iterate through nodes. for (int j = 0; j < n; j++) { // Iterate through neighbors of the node. for (int k = 0; k < adjList.get(j).size(); k++) { // Only visit node if path is defined. if (dist[j] == UNDEFINED) { edge e = adjList.get(j).get(k); // If dist[e.source] has been used if (dist[e.source] != UNDEFINED) { // If new dist < cur dist or not used, then update node. int newDist = dist[e.source] + e.weight; if (newDist < dist[e.dest] \|\| dist[e.dest] == UNDEFINED) { dist[e.dest] = newDist; // Check if negative cycle exists. for (int j = 0; j < n; j++) { for (int k = 0; k < adjList.get(j).size(); k++) { edge e = adjList.get(j).get(k); // Check if edge can create negative cycle. if (dist[e.source] + e.weight < dist[e.dest]) { System.out.println("Negative cycle exists."); // Check if no path exists. if (dist[endNode] == UNDEFINED) { System.out.println("No path from start to end"); // Return distance from start to end return dist[endNode]; 1. Prove Bellman-Ford works. 2. Arbitrage occurs when you can exchange currencies for another and make a profit. For example given a currency exchange table: USD CAD EURO USD / 1.12 0.72 CAD 0.90 / 0.64 EURO 1.38 1.56 / Notice that 1 USD -> 1.12 CAD -> 1.008 USD. Bellman Ford can be used to find methods of arbitrage by using the vertex as currency and edges as transactions, and the weight as the exchange rate. All that is needed is to find a path that maximizes product of weights and finding a negative cycle. Write a program that detects a path for arbitrage to occur.	{"url":"https://www.thecshandbook.com/Bellman_Ford","timestamp":"2024-11-12T15:14:15Z","content_type":"text/html","content_length":"12990","record_id":"<urn:uuid:3fe48b66-8f04-40c1-9e19-81e54f792857>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00385.warc.gz"}
2,428 research outputs found Different electron acceleration regimes in the evanescent field of a surface plasma wave are studied by considering the interaction of a test electron with the high-frequency electromagnetic field of a surface wave. The non-relativistic and relativistic limits are investigated. Simple scalings are found demonstrating the possibility to achieve an efficient conversion of the surface wave field energy into electron kinetic energy. This mechanism of electron acceleration can provide a high-frequency pulsed source of relativistic electrons with a well defined energy. In the relativistic limit, the most energetic electrons are obtained in the so-called electromagnetic regime for surface waves. In this regime the particles are accelerated to velocities larger than the wave phase velocity, mainly in the direction parallel to the plasma-vacuum interface Numerical modelling of convection driven dynamos in the Boussinesq approximation revealed fundamental characteristics of the dynamo-generated magnetic fields and the fluid flow. Because these results were obtained for an incompressible fluid, their validity for gas planets and stars remains to be assessed. A common approach is to take some density stratification into account with the so-called anelastic approximation. The validity of previous results obtained in the Boussinesq approximation is tested for anelastic models. We point out and explain specific differences between both types of models, in particular with respect to the field geometry and the field strength, but we also compare scaling laws for the velocity amplitude, the magnetic dissipation time, and the convective heat flux. Our investigation is based on a systematic parameter study of spherical dynamo models in the anelastic approximation. We make use of a recently developed numerical solver and provide results for the test cases of the anelastic dynamo benchmark. The dichotomy of dipolar and multipolar dynamos identified in Boussinesq simulations is also present in our sample of anelastic models. Dipolar models require that the typical length scale of convection is an order of magnitude larger than the Rossby radius. However, the distinction between both classes of models is somewhat less explicit than in previous studies. This is mainly due to two reasons: we found a number of models with a considerable equatorial dipole contribution and an intermediate overall dipole field strength. Furthermore, a large density stratification may hamper the generation of dipole dominated magnetic fields. Previously proposed scaling laws, such as those for the field strength, are similarly applicable to anelastic models. It is not clear, however, if this consistency necessarily implies similar dynamo processes in both settings.Comment: 14 pages, 11 figure International audienceIn a multiphoton photoelectric process, an electron needs to absorb a given number of photons to escape the surface of a metal. It is shown for the first time that this number is not a constant depending only on the characteristics of the metal and light, but varies with the interaction duration in ultrashort time scales. The phenomenon occurs when electromagnetic energy is transferred, via ultrafast excitation of electron collective modes, to conduction electrons in a duration less than the electron energy damping time. It manifests itself through a dramatic increase of electron production. A basic hypothesis of the photoelectric process is that the photoemissive properties of matter remain unaltered during the interaction with light. Light-metal coupling is tacitly assumed as a perturbation of the electron population that remains in equilibrium during the interaction. Now, it has recently been shown that transient nonequilibrium electron states can exist in ultrashort time scales, in particular , when electromagnetic energy is transferred from a laser pulse to conduction electrons in a lapse of time shorter than the electron-phonon energy transfer duration [1– 4]. In this Letter, we address the basic question of whether the photoemissive properties of a metal can be modified through ultrafast energy transfer and nonequilib-rium electron heating. In a metallic electron gas, transient density disturbances can result in electron collective oscillation modes in the volume and near the surface. Under certain conditions, these so-called surface plasmon (polariton) modes can be excited by light [5,6]. In the case of thin metal films, the surface plasmon modes on the two surfaces can be coupled [7–9] and energy can be transferred from one surface plasmon mode to the other [10]. Collective electron oscillations can exist as well at the interface [11] between two perfect metals due to symmetry breaking at the metal-metal interface. Furthermore, interface and surface plas-mon modes can be coupled [12] in a bilayer metal system made of a metal M 1 (of electron density n 1) covered by a thin metallic layer M 2 (of electron density n 2 < n 1). If the overlayer metal M 2 is thin enough, the field of the surface plasmon can tunnel through the M 2 bulk and excite electron density fluctuations at the interface between the two metals (see Fig. 1). If the metal overlayer is too thick, the field of the surface plasmon must tunnel through too large a distance to excite the density fluctuations between the two metals. Conversely, if it is too thin, the surface plasmon amplitude is damped because of increasing coupling between the two opposite faces of the overlayer. There exists therefore an optimum thickness of the overlayer for which the amplitude of the induced interface plasmon is maximum. An interesting consequence of the interface or surface plasmon coupling effect is that the electron population in the metal overlayer can be in transient nonequilibrium energy states through ultrafast energy transfer from the coupled interface and surface plasmons. Actually, the conduction electrons near the surface and the metal-metal interface experience an effective nonlinear low-frequency force, the so-called ponderomotive force [13,14], resulting from the strongly inhomogeneous high-frequency field of the plasmons, and are accelerated toward regions of decreasing field amplitude. The ponderomotive force plays the role of an applied electrostatic force that transfers electromagnetic energy in a coherent way to an electron population, in contrast with stochastic energy transfer via thermal heating. The maximum energy that can be transferred to a free electron with initial energy E 0 through ponderomotive acceleration in a strong oscillating electri Given a discrete group $\Gamma$, a finite factor $\mathcal N$ and a real number $p\in [1, +\infty)$ with $peq 2,$ we are concerned with the rigidity of actions of $\Gamma$ by linear isometries on the $L_p$-spaces $L_p(\mathcal N)$ associated to $\mathcal N$. More precisely, we show that, when $\Gamma$ and $\mathcal N$ have both Property (T) and under some natural ergodicity condition, such an action $\pi$ is locally rigid in the group $G$ of linear isometries of $L_p(\mathcal N)$, that is, every sufficiently small perturbation of $\pi$ is conjugate to $\pi$ under $G$. As a consequence, when $\Gamma$ is an ICC Kazhdan group, the action of $\Gamma$ on its von Neumann algebra ${\mathcal N}(\Gamma)$, given by conjugation, is locally rigid in the isometry group of $L_p({\mathcal N}(\ Gamma)).$Comment: 20 page We determine the quotient category which is the representation category of the kernel of the homomorphism from Nori's fundamental group scheme to its \'etale and local parts. Pierre Deligne pointed out an error in the first version of this article. We profoundly thank him, in particular for sending us his enlightning example reproduced in Remark 2.4 2).Comment: 29 page International audiencePhotoelectrons emitted from a gold target via a surface-plasmon-assisted multiphoton photoelectric process under a femtosecond laser pulse of moderate intensity are much more energetic than in an ordinary photoeffect without electron collective excitation. The phenomenon is interpreted in terms of time-dependent ponderomotive acceleration of the particles by the resonant field localized at the metal surface. The amplitude of the plasmon resonance may be directly estimated by means of the electron energy spectra. The development of powerful lasers more than three decades ago has allowed the investigation of the generalization of the classical photoelectric emission from metals to processes involving the absorption of several photons [1]. In recent years, the advent of laser pulses of ultra-short duration has favored studies in the femtosecond time regime [2]. These investigations can lead to the creation of new high-current ultrafast electron sources. Experimental studies have revealed that the electron emission rate can be greatly enhanced by the excitation of collective electron modes of the metal, the so-called surface plasmons [3,4]. The increase of the photoelectric signal can be qualitatively explained in terms of an assisted photoelectric effect where the energy of femtosecond light pulses is stored by the surface plasmon, creating a hot-electron population that does not have enough time to transfer its energy to the crystal lattice. While the presence of a surface-plasmon excitation is efficient in increasing the production of photoelectrons, an important open question is how the energy of the emitted electrons in such a " surface-plasmon-assisted " photoelectric process may differ from the energy predicted by the familiar photoelectric equation generalized to multiphoton processes. In this Letter, we show that the photoelectron energy is strongly affected by the surface-plasmon field, the modification from the classical values depending on the characteristics of the plasmon resonance. This fact may be easily understood by considering a simple analysis of the photo-electron behavior in the inhomogeneous high-frequency electric field surrounding the metal surface. The analysis involves simple classical concepts such as the notion of time-dependent ponderomotive effects, which have been successfully used in the context of multiphoton ionization of atoms in high-intensity lasers [5]. Consider an electron released from the metal surface after having absorbed a required number n of photons from the laser beam to overcome the work function W of the metal. While traveling in the vacuum dressed by the high-frequency field E sp of the surface plasmon, the total energy of the electron consists of the sum of its kinetic energy § n (given by the Einstein multiphoton photoelectric equation § n ෇ n ¯ hv 2 W) and its quiver energy U sp ෇ e 2 International audienceThe relativistic acceleration of electrons by the field of surface plasma waves created in the interaction between ultrashort high-intensity laser pulses with sharp-edged overdense plasmas has been investigated. It is shown that the initial phase of the wave experienced by the electrons play a leading part by yielding a well-defined peaked structure in the energy distribution function. This study suggests that resonant excitation of surface plasma waves could result in quasi-monokinetic energetic electron bunches. When the space charge field becomes too strong, this mechanism can evolve toward a true absorption process of the surface wave energy via an enhanced ''vacuum heating'' mechanism generalized to the case of surface plasma waves International audienceThe possibility of inducing a magnetic field via surface plasma-wave excitation is investigated with a simple nonrelativistic hydrodynamic model. A static magnetic field is predicted at the plasma surface, scaling with the square of the surface-wave field amplitude, and the influence of the electron plasma density is studied. In the case of resonant surface-wave excitation by laser this result can be applied to low intensities such that the electron quiver velocity in the field of the surface wave is less than its thermal velocity	{"url":"https://core.ac.uk/search/?q=authors%3A(M.%20Raynaud)","timestamp":"2024-11-02T19:14:56Z","content_type":"text/html","content_length":"171770","record_id":"<urn:uuid:cc7ca455-3bf1-4a90-a400-3c6fa20af037>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00262.warc.gz"}
a stem-and-leaf plot How do you make a stem-and-leaf plot of data? How to Make a Stem-and-Leaf Plot 1. Step 1: Determine the smallest and largest number in the data. The game stats: 2. Step 2: Identify the stems. 3. Step 3: Draw a vertical line and list the stem numbers to the left of the line. 4. Step 4: Fill in the leaves. 5. Step 5: Sort the leaf data. What is a stem-and-leaf plot mode? A stem-and-leaf plot is a type of plot that displays data by splitting up each value in a dataset into a stem and a leaf. What can a stem-and-leaf plot detect? Stem-and-leaf plots are a method for showing the frequency with which certain classes of values occur. You could make a frequency distribution table or a histogram for the values, or you can use a stem-and-leaf plot and let the numbers themselves to show pretty much the same information. How do you make a key for a stem-and-leaf plot? Lastly, create a key for the stem-and-left plot. The key tells the reader what values are represented in the display. To do this, select one term in the plot and separate the digits by a vertical line, then use the equal sign to show what it is equivalent to. It will look like this, 6\|8=68. What is median in stem-and-leaf plot? Using Stem-and-Leaf Plots to Find The Mean, Median, and Mode of a Data Set. The mean is the average of a set of data. The median is the middle number of a set of data. The mode is the number that occurs the most in a set of data. What is the count of a stem-and-leaf plot? The value for a row below the median represents the total count for that row and all the rows below it. For each row, the number in the “stem” (the middle column) represents the first digit (or digits) of the sample values. The “leaf unit” at the top of the plot indicates which decimal place the leaf values represent. How do you read a stem plot? Steps to Interpreting a Stem Plot The stems are on the left of the vertical line and the leaves are on the right. The stems are usually the first digit of a number. So if you have a value of 25, 2 is the stem that goes on the left of the vertical line and 5 is the leaf that goes on the right.	{"url":"https://cowetaamerican.com/2022/06/29/how-do-you-make-a-stem-and-leaf-plot-of-data/","timestamp":"2024-11-13T05:33:09Z","content_type":"text/html","content_length":"57678","record_id":"<urn:uuid:82a0d50f-cb45-4944-a915-84aae9eaa3b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00367.warc.gz"}
Monday Morning Mulling: July 2023 Challenge On the final Friday of each month, we set an Excel / Power Pivot / Power Query / Power BI problem for you to puzzle over for the weekend. On the Monday, we publish a solution. If you think there is an alternative answer, feel free to email us. We’ll feel free to ignore you. The Challenge Imagine that you need to filter rows in a table that have specific keywords within the text strings contained therein. Manually filtering every single keyword and copying / pasting them to a new location can be a tedious and time-consuming process. To address this, we challenged you to develop a user-friendly solution that allows users to select the desired keywords and return a list having all the text strings associated with those keywords. You can download the original question file here. Your aim was to create a filter using the keywords "Awesome", "Amazing", and "Perfect" as filter criteria, as shown in the picture below: As always, there were some requirements: • no Power Query / Get & Transform or VBA was allowed • the formula(e) should be dynamic so that they should update when a new entry was added. Suggested Solution 1 You can find our Excel file here, which shows our suggested solution. The steps are detailed below. Find the keywords To begin, we create four [4] new columns in the Data_v1 table. One is named Included and the other three [3] represent the heading names based upon the keywords provided: We accept this is a manual interaction as Table headers may not contain formulae. To check whether the text strings in the Data_v1[Teams] column have specific keywords, we need to create a formula that uses the ISNUMBER and SEARCH functions. These functions work together to confirm the keywords are within the text strings (note that the FIND function could also be used, but beware that it's case-sensitive and requires an exact match between the capitalisation of the keywords and text strings). The SEARCH function will return a number if the keywords is found within the text string. If the text string did not contain the keyword, then it will return #VALUE! Then, the ISNUMBER checks whether the output of the SEARCH function is a number or not and will return TRUE or FALSE accordingly. At this point, we can choose to keep it as logical value or turn it into number. We will change all these logical values into numbers, so we multiply the numbers by one [1]. Therefore, all TRUE values will be restated as one [1] and all FALSE values will turn to zero [0]. Similarly, we will do the same for the other two [2] columns, viz. Hidden power of SUBTOTAL Let’s move on the Output table. In this table, we first enter the keywords of this challenge in the Teams column and we will leave the Name column blank: Then, we added one column to this Table and we call this column ‘Included in Filter’. This column will specify which keywords to include in the filter. The formula for this new column will be: The SUBTOTAL function has the advantage of being able to exclude hidden cells from calculations. When the function_num is greater than 100, the SUBTOTAL function will not include any hidden cells in the calculation. Using function_num 103 of SUBTOTAL, which is COUNTA, will return one [1] if a cell is not hidden and zero [0] if the cell is hidden. For instance, if we completely hide the row 23, the result will be as follows: As we can see cell F23 has a value of zero [0] while another unhidden cell has a value of one [1]. We did not format this table or add any colouring as we will need to set the pixel height of these rows to one [1] later, so not putting formatting here will prevent any colours condensing together later on. Matching keywords We will now add another column to the Data_v1 Table called ‘Included’. This column will determine whether to include a particular row in our Output table based upon a formula. The formula is as =MIN(MMULT(Data_v1[@[Awesome]:[Amazing]],Output[Included in Filter]),1) The first part of the formula, MMULT(Data_v1[@[Awesome]:[Amazing]],Output[Included in Filter]) multiplies two [2] selected vectors to obtain the dot product. You can read more about MMULT here. The resulting dot product shows how many matches the text strings have with the filter. We then wrap this dot product in the MIN function to ensure that our output is either one [1] or zero [0], like a TRUE or FALSE value. Below the Output table we will use the FILTER function to filter out the list that contains the keywords: This will use the Included column in the Data_v1 table to filter out all of the matches we have on the Data_v1 table and output it out in the form of an array. Therefore, we have the following output It is important to note that anything typed below a Table will automatically add a new row to the Table. Therefore, you may need to resize the Output table to exclude the new formula that was just Hide, but not completely The project is currently 80% complete, but we need to hide rows 23 to 25 from our output. However, we need to be careful when hiding these rows. We do not want to completely hide them, as this could cause the SUBTOTAL to become zero [0] or the filter to stop working. The lowest pixel height for a row that will still allow the SUBTOTAL function to work is 1/3 pixel (0.25 points). To achieve this, we can select the rows we want to hide and go to Home -> Cells -> Format -> Row Height (or use the shortcut key Alt + H + O + H) and enter 0.25 points in the box. However, the filter of Excel will not work if cell is under one [1] pixel (0.75 points). Therefore, we need to adjust the row height to one [1] pixel to ensure that our filter works properly. After setting the row height to one [1] pixel, our filter is now complete: Now, we may use the drop-down menu from the Table to filter out Data_v1[Teams] that contains the keywords we need. The drop-down filter will automatically set the row height to zero [0] pixels for unselected criteria and maintain one [1] pixel for selected criteria. Suggested Solution 2 The first solution is effective, but is constrained in terms of scaling ability. Therefore, we have another solution to offer to users that can use the LAMBDA function and dynamic arrays. To do this, we will alter few formulae and steps here to make our solution more dynamic and able to absorb more inputs such as a list of keywords that we want to be able to filter by. First of all, we create a Look Up table on the Lookup sheet for Teams, we will name this table LU_Teams: Next, we create our dynamic heading from these inputs by using the following formula: As our inputs are in rows but we want our headings across the conditionally formatted columns we will need to transpose our inputs: Next, we need to construct a matrix that will tell us which of the keywords Teams contains using this formula: This formula uses the dynamic headings (in the range G10#) as the find_text argument and the Teams column in the Data_v2 table as the within_text argument of the SEARCH function. Since G10# is a column vector and the Teams column is a row vector, the formula will search for the first keyword, which is ‘Awesome’, in every item in the Teams column and output it to the first column of the matrix then it proceeds to the second keyword and outputs this to the second column of the matrix and the same for the third keyword and third column of the matrix. After the search, we will have a matrix of integer numbers, wrapping this within the ISNUMBER function will convert these to logical values and then multiplying by one [1] will create a matrix of one [1] and 0 [zero] values, as with solution 1. Next to the output, instead of using the Table here we will use Dynamic Arrays to create the filter. In cells D22:F22 we create the same headings as we did in solution 1: In cell E23 we enter the following formula: This creates the keywords lookup in form of a row vector. In cell F23 we will need a formula that is able to spill down if we have more keywords added in the LU_Teams table. However, it will be hard without using a recursive function like LAMBDA here as SUBTOTAL doesn’t create arrays. Thus, we will use the following formula to create a Dynamic Array that uses SUBTOTAL: As we know from solution 1, this part of the formula is the main engine of our filter: This portion of the formula will use an input (called row here) and apply that input as the second argument of our SUBTOTAL function. The final function, which is BYROW, will help us dissect the matrix into multiple rows and feed these into the row parameter of the LAMBDA function. The LAMBDA function will treat each of those row array inputs individually and apply the SUBTOTAL function to each, outputting an array of SUBTOTAL. After typing the formula in cell F23 we will have a dynamic array of SUBTOTAL results that behave similarly to our formulae in solution 1. The next step here would be using the FILTER function, similarly to our approach in solution 1, in cell D29 we enter the following formula: The MMULT function here will multiply the matrix we created within the range G11# by the row vector we created in the range F23#. This would result in a row vector outlining which rows contain the applicable keywords under the current filter context. Our Table is then filtered based on this result. The last step here is to apply the filter, we simply select D22:F27 then click Home -> Editing -> Sort & Filter -> Filter (or Alt + H + F + S for short). Then we hide column F and set row 23 to row 28 to one [1] pixel. Voila, we have a scalable filter for this challenge! Word to the Wise We appreciate there are many, many ways this could have been achieved. If you have come up with an alternative, radically different approach, congratulations – that’s half the fun of Excel! The Final Friday Fix will return on Friday 25 August 2023 with a new Excel Challenge. In the meantime, please look out for the Daily Excel Tip on our home page and watch out for a new blog every business working day.	{"url":"https://www.sumproduct.com/blog/article/fff-mmm/monday-morning-mulling-july-2023-challenge","timestamp":"2024-11-08T16:00:49Z","content_type":"text/html","content_length":"60086","record_id":"<urn:uuid:8298564d-10d7-4c22-8cba-1c8322268faa>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00679.warc.gz"}
8.3 Impact of short tests on predicting IQ \| Child Development with the D-score 8.3 Impact of short tests on predicting IQ 8.3.1 Measurement and prediction In Section 8.2, we saw that a short test can measure the middle or one tail of the distribution, but cannot be precise for both at the same time. If we want to identify children at risk for delayed development, we are interested in the lower tail of the distribution, so in that case, the standard set is suitable. But what set should we use if we want to predict a later outcome? This section explores that effect of taking different milestone sets on the quality of prediction. 8.3.2 UKKI Hafkamp-de Groen et al. (2009) studied the effect of the D-score on later intelligence, using a subset of 557 SMOCC children that were followed up at the age of five years. The Utrechtse Korte Kleuter Intelligentietest (UKKI) (Baarda 1978) is a short test to measure intelligence. The UKKI is a simple test with just three components: • Redraw five figures (square, triangle, cross, trapezoid, rhomboid); • Draw human figure, with 28 characteristics, like legs, eyes, and so on; • Give meaning to 13 words like knife, banana, umbrella, and so on. Administration time is about 15-20 minutes. The UKKI has a reasonable test-retest reliability for group use (Pearson $r = 0.74$, 3-month interval). 8.3.3 Exploratory analysis Figure 8.5 shows the empirical IQ distribution of 557 children. The mean IQ score is 108, and the standard deviation is 15, so the IQ-scores of children in the sample is about a half standard deviation above the 1978 reference sample. Figure 8.6 shows that the relation between the D-score 0-2 years and IQ at five years is positive for all milestone sets and all ages. The strength of the association increases with age. At the age of 2 years, the regression coefficient for D-score is equal to $\beta(D) = 1.4$ (SE: $0.21, p < 0.0001$), so on average an increase of 1.0 unit in the D-score at the age of 2 years corresponds to a 1.4 IQ-score points increase at the age five years. Table 8.2: Pearson correlation between D-score (0-2 years) and IQ at 5 years. Visit Standard set Additional set All milestones 1m 0.059 0.005 0.027 2m 0.051 0.056 0.048 3m 0.036 0.100 0.102 6m 0.040 0.038 0.036 9m 0.094 0.143 0.132 12m 0.046 0.162 0.137 15m 0.180 0.153 0.187 18m 0.129 0.153 0.146 24m 0.245 0.255 0.267 Table 8.2 summarizes the Pearson correlations between the D-score and later IQ. The association between D-score and IQ is weak during the first year of life but gets stronger during the second year. In general, having more (and more informative) milestones helps to increase the correlation, but the effects are relatively small. So even from the standard set of the seven easy milestones at 24m, we obtain a reasonable correlation of 0.245. All in all, these results suggest that neither the amount nor the difficulty level of the milestones is critical in determining the strength of the relation between the D-score and IQ.	{"url":"https://d-score.org/dbook1/sec-predictiq.html","timestamp":"2024-11-10T06:21:47Z","content_type":"text/html","content_length":"46576","record_id":"<urn:uuid:1806d98b-caf9-417a-a6c6-b513e1d55c16>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00135.warc.gz"}
Samonte, Joseph Angelo \| Institute of Mathematics Samonte, Joseph Angelo Professorial Lecturer Research Interests Actuarial science and mathematical finance. Statistical modeling and applied statistics. Data science and predictive analytics. Risk measurement and management. Academic Groups • Education □ Master of Statistics (Risk Assessment Methods concentration) University of the Philippines Diliman Thesis: Modeling and Predicting Fire Insurance Claims Using Hurdle Models □ Professional Masters in Applied Mathematics major in Actuarial Science (2015) University of the Philippines Diliman Thesis: Pricing Equity-Index Linked Single Pay Variable Universal Life Insurance Product □ Bachelor of Science in Mathematics (2010), cum laude University of the Philippines Diliman Thesis: Sustainability Analysis using the Collective Risk Model • Teaching 1st Semester AY 2024-25 □ Math 164 (Mathematics of Life Contingencies). This course introduces students with the field of actuarial science. It discusses the mathematics behind basic pricing and establishing reserves for traditional life insurance products or any long-term financial product that involves life contingent risks. □ Math 262.1 (Actuarial Science I: Product Management Aspects in Actuarial Science). This course tackles the technical and practical actuarial concepts involving product pricing and development, and valuation of liabilities for life insurance. In addition, it introduces the concept of the actuarial control cycle framework as a problem solving algorithm in dealing with actuarial tasks. Previous Courses □ Math 162 (Theory of Interest). This course provides an in-depth knowledge of the different measures of interest, how they are derived from a cashflow or series of cashflows, and how these interest concepts may be applied in amortization, sinking funds, and pricing investment securities such as bonds and stocks. □ Math 260 (Actuarial Theory and Practice). This is the second course in actuarial mathematics covering life contingencies. This covers analysis of reserves, policy values, cash values, nonforfeiture options, asset shares and profit analysis; and multiple life, multiple decrement and multiple state models. □ Math 261 (Survival and Loss Models). This course is an overview of various survival and loss models as applied on risk theory and short-term insurances. This covers frequency, severity and aggregate models; insurance, exposure and policy modifications; tail weight of distributions and risk measures. □ Math 262.2 (Actuarial Science II: Financial Management Aspects in Actuarial Science). This course discusses the continuation of the actuarial control cycle with emphasis on professionalism and ethics. In addition, it introduces basic actuarial accounting concepts and financial reporting framework for life insurance. □ Math 297 (Actuarial Models). This course walks through the actuarial modeling process as applied on formulating premium rates for short-term insurances. It covers empirical and parametric models, and credibility theory. □ Math 297 (Advanced Statistical Modeling and Predictive Analytics). This course covers statistical modeling with focus on data-centric models that can carry out predictive analytics. In particular, this is an overview of various supervised and unsupervised statistical learning methods that are prominently used in insurance and finance industry practice. □ Math 297 (Group Insurance). This course surveys various aspects of group insurance. Topics include benefit provisions, pricing, underwriting, experience rating, funding methods and reserving □ Math 297 (Mathematics of Investment and Finance). This course presents the quantitative methods and analysis applicable to fundamental concepts, theory and practice in investments and finance. It covers financial reporting and analysis, project valuation, portfolio theory, asset pricing models, and capital structure and financing. □ Math 297 (Statistical Modeling and Data Analysis). This course introduces statistical modeling as applied on actuarial science and finance with the aid of R software. Topics include an introduction to data analysis, basic and multiple linear regression, and classical methods in time series analysis and forecasting. □ Math 297 (Survey on Areas of Actuarial Practice). This course presents introductory actuarial methods for various areas of actuarial practice. Topics include pension and retirement benefit valuation; profit measures and asset share pricing for traditional life insurance; introduction to participating and universal life insurances; short-term insurance ratemaking and loss reserving for health, group life and non-life insurance. □ Math 2 (Mathematics in Everyday Life). This general education course covers basic mathematical concepts and skills encountered in everyday life and gain appreciation to its practical and disciplinary values. □ Math 295 (Special Project). Supervised facilitation and directed research course for actuarial science graduate students finishing their project and orally presenting a defensible result of their research work.	{"url":"https://math.upd.edu.ph/faculty/jlosam","timestamp":"2024-11-11T21:20:54Z","content_type":"text/html","content_length":"59865","record_id":"<urn:uuid:22666ab8-c69f-4ff7-8b64-481433759c73>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00669.warc.gz"}
Halves and Quarters Class 4 Math JKBOSE Solutions - Learn JKBOSE “Halves and Quarters” is Chapter 9 from Merry Math IV for students of Class 4th of JKBOSE. This particular post is about Halves and Quarters Class 4 Math JKBOSE Solutions. In a previous post, you read about Carts and Wheels Class 4 Math JKBOSE Solutions. Let’s get started: Halves and Quarters Class 4 Math JKBOSE Solutions Page No. 117 Solutions If the cats ask you to divide the chapati equally, how will you divide it? Ans. Take a chapati and fold it in such a way that the edges of both the parts touch each other. Then press the chapati on its fold and cut it out in two equal parts. Half of half If two more cats come for food, how will you divide one chapati equally for four cats? Ans. Take a chapati and fold it in such a way that the edges of both the parts touch each other. Then press the chapati on its fold and cut it out in two equal parts. After that, take each part one by one and fold it in such a way that the circular edges touch each other at every point, press the fold and divide it. Similarly, the other half should also be divided in the same way. Half of Many Pieces Meena got a chocolate. She divided it equally and gave half to her friend Veena. Circle the portion that Veena got. Ans. Veena got half a portion of the chocolate. Page No. 118 Solutions How many pieces of chocolate are there? Ans. There are six pieces of chocolate. How many pieces were left with Meena? Ans. Three pieces were left with Meena. Many Shapes from a Half Sheet Shade the two triangles with different colours. Draw different shapes using these triangles. One such shape is shown here. Ans. These are some different shapes using these triangles. Page No. 118-119 Solutions Many Ways to Cut Into Half In how many different ways can you cut a rectangle into half? Draw 5 different ways. Ans. These are the 5 different ways to cut a rectangle in half. Many Ways to Make Quarters In how many different ways can you cut a rectangle into four equal parts? Draw 5 different ways. Ans. A rectangle can be cut into four equal parts as shown: Can you check if they are equal? Ans. Yes, I can check that they are equal. Page No. 120 Solutions Cutting the Cake Ameena’s father bought a cake. She divided the cake into 4 equal parts herself, her brother Babloo, her father and her mother. Colour each share with different colours. How much does each get? Ans. Each gets one-fourth (1/4) of the cake. Mother gave her share of the cake to Ameena. Now colour the total part that Ameena will get. Ans. Ameena will get half (1/2) of the cake. Out of 4 parts Ameena will get 2 parts, which is equal to half of the cake. So, she can write it as 2/4 or ½ Before Ameena’s mother gave her share to Ameena, she had only ½ of ‘half of the cake’, which was ¼ of the total cake. Colour the share Babloo got How much of the cake do Ameena and Babloo together get? Colour their total share. Ans. Ameena and Babloo together get 3 parts out of 4. So they get ¾ part of the cake. Page No. 121 Solutions Greedy Guddu First pumpkin-seller- ¼ of this pumpkin is for ₹10. This full pumpkin will cost ___________. Ans. This full pumpkin will cost = ₹ 40 Ans. If the cost of ¼ pumpkin is = 10 then the cost of 1 pumpkin is = 10 × 4 = ₹ 40 Guddu walks to the next seller and looks for a pumpkin of the same size. Guddu- How much of this pumpkin will I get for 10? Second pumpkin-seller – Half This full pumpkin will cost ___________. Ans. This full pumpkin will cost = ₹ 20. If the cost of ½ pumpkin = 10 Then the cost of full pumpkin = 2 ×10 = ₹ 20 Page No. 122 Solutions Using a Price List (A) How much does ½ kg of tomatoes cost? Sol. Cost of 1 kg of tomatoes = 8 Cost of ½ kg of tomatoes = ½ x 8 = ₹ 4 Ans. (B) Which costs more – ½ kg of onions or ¼ kg of carrots? Sol. In the First Case. Cost of 1 kg of onions = 10 Cost of ½ kg of onions = ₹10 x ½ = ₹ 5 In the Second Case. Cost of 1 kg of carrots = 16 Cost of ¼ kg of carrots =₹16 x ¼ = ₹ 4 So, Onions cost more than carrots. (C) Aruna is going for shopping. She has only 20 with her. Can she buy all the things on her shopping list? Sol. Cost of 1 kg of potatoes = ₹ 12 Cost of ½ kg of potato = ₹ 12 × ½ = ₹ 6 Cost of 1 kg of pumpkin = ₹ 4 Cost of 2 kg of pumpkin = ₹ 4 × 2 = ₹ 8 Cost of 1 kg of carrots = 16 Cost of ¼ kg of carrots = 16 x ¼ = ₹ 4 Total cost of ½ kg potatoes + 2 kg pumpkin and ¼ kg of carrots = ₹ (6+8+4) = ₹ 18 Yes, she can buy all the things on her shopping list. (D) What is the Price of ¾ kg of Potatoes? Sol. Cost of 1 kg of Potatoes = ₹ 12 Cost of ¾ kg of Potatoes = ₹ 12 × ¾ = 9 Ans. (E) Make two questions yourself from the price list. 1. What is the price of ¾ kg of pumpkin? 2. Which costs more, 2 kg of tomatoes or ¾ kg of carrots? Page No. 123 Solutions Practice Time (a) What part of the whole is coloured? Write below in each shape. (b) Colour that part of the shape which is written below. (c) Cut in half Draw a line which divides these shapes into half. Page No. 124 Solutions (d) Colour half the number of shapes as shown here. (e) Colour ¼ of these shapes (f) Match the coloured part as shown. Page No. 125 Solutions Make the other half (g) ½ of the picture is drawn here. Can you complete the picture by drawing the other half? Ans. Yes, I can complete it by drawing the other half. (h) This is a quarter of a picture. Can you complete it? How many more quarters will you draw to complete it? Ans. Yes, I can complete it. I have to draw three more quarters to complete it. Half and Quarter of a Metre Using your metre scale, cut a string of one metre. On this string mark the length ½ metre, ¼ metre and ¾ metre. Using your string, draw a line of length ½ metre on the floor. How many centimetres long is the line? 1 metre = 100 cm Then ½ metre = 100/2 cm = 50 cm. Ans Page No. 126 Solutions ½ metre = ……… cm ¼ metre = ……… cm ¾ metre = ……… cm Ans. ½ metre = 50 cm ¼ metre = 25 cm (¼ x 100 = 25) ¾ metre = 75 cm (¾ x 100 = 75) Sharing Milk This bottle is full of milk and it holds one litre. The milk is put into 4 other bottles so that each bottle has ¼ litre of milk. Shade the bottles to show the level of milk in each. The level of milk is shown as below: – How many millilitres of milk does each bottle have? _________ Ans. Each bottle has ¼ litre of milk. So, 1000 ml ÷ 4 = 250 millilitre milk. Aahan poured 1 litre of milk into two bottles so that the first bottle holds ¾ litre and the other holds ¼ litre. Shade the level of milk in each bottle. How many millilitres of milk does each bottle hold? Ans. The first bottle holds ¾ litre of milk So, 1000 × ¾ ml = 750 ml The second bottle holds ¼ litre of milk So, 1000 × ¼ ml = 250 ml Page No. 127 Solutions Balance the Weight Choose from the weights above to make the two pans equal. In how many ways can you do it? Ans. There are so many ways to make the two pans equal. (i) 1kg + 250g + 500g + 250g (ii) 500g + 500g + 1kg (iii) 1kg + 250g + 250g + 250g + 200g + 50g (iv) 500kg + 250g + 200g + 50g + 1kg (v) 500g + 500g + 250g + 250g + 200g + 50g + 250g (vi) 1kg + 100g + 200g + 500g + 200g (vii) 500g+200g + 100g + 200g + 1kg a) Draw the weights in the empty pan. b) In how many different ways can you balance this weight of ¾ kg? 1) …………………… 2) …………………… 3) …………………… Ans. (1) 500g + 250g (2) 200g + 100g + 200g + 250g (3) 500g + 200g + 50g Page No. 128 Solutions Why Is It Wrong? Kamraan shaded some parts as shown. But his friend Maria says that it is wrong. Explain why it is wrong. Ans. In this rectangle, the four divisions are not of equal size. So, the shaded part is not equal to ¼ of the rectangle. It is ⅖ of the rectangle. In the case of the triangle, the shaded part is not equal to ½ of the triangle. Practice Time There are 60 mangoes. ½ of them are ripe. How many mangoes are ripe? Sol. The number of ripe mangoes ½ of 60. = 60 × ½ = 30 Ans. There are 32 children. ½ of them are girls. How many children are boys? Sol. Total number of children = 32 Number of girls = ½ of 32 = ½ x 32 = 16 So, the number of boys = Total number of children – number of girls = 32 – 16 =16 Ans. There are 20 stars. A quarter of them are red. How many stars are red? Sol. Total number of stars = 20 So, the number of red stars = ¼ of 20 = ¼ × 20 = 5 Ans. How many are not red? Sol. Total number of stars number of Red stars = Number of non-red stars = 20 – 5 = 15 Ans. Aslam wants a pencil. It costs ₹ 2. He gives a one-rupee coin, one half-rupee coin and one quarter-rupee coin. Is it enough? Sol. Cost of a pencil = 2 The money he gives = one rupee + one-half rupee + one quarter rupee = (1.00 +0.50 +0.25 = 1.75) Total money he had given = 1.75 But the cost of a pencil is 2.00 and he gives only 1.75. So, it is not enough. Page No. 129 Solutions Let’s Try These-(Activity) 1. For each drawing, answer the following questions: (i) How many equal parts are there in the shape? (ii) What fraction is each part of the whole shape? (iii) What fraction is shaded? Sol. (a) (i) Three (ii) Each part is ⅓ of the whole. (iii) ⅓ part is shaded. (b) (i) Four (ii) Each part is ½ of the whole (iii) ½ part is shaded. (c) (i) Four (ii) Each part is ¼ of the whole. (iii) ¾ part is shaded. (d) (i) Four. (ii) Each part is ¼ of the whole. (iii) ¾ part is shaded. (e) (i) 8 parts (ii) Each part is ⅛ of the whole. (iii) ⅝ part is shaded. (f) (i) 6 parts (ii) Each part is ⅙ of the whole (iii) ⅚ part is shaded. (g) (i) 5 parts (ii) Each part is ⅕ of the whole (iii) ⅗ part is shaded. (h) (i) 12 parts. (ii) Each part is 1/12 of the whole (iii) The 6/12 parts are shaded. 2. Write the fraction for the shaded part of each shape. Also, write the fraction for the white part of each shape. Ans. Fraction for the shaded portion: (a) ½ (b) ¾ (c) ½ (d) 3/6 or ½ (e) ½ (f) 4/8 or ½ Fraction for the white portion: (a) ½ (b) ¼ (c) ½ (d) 3/6 or ½ (e) ½ (f) 4/8 or ½ 3. Find if the two fractions are equal [or equivalent]. Write equal or not equal. Sol. (a) ½ ≠ ⅔ So, A ≠ B (b) 3/6 = ½ So, A = B (c) 2/6 = ⅓ So, A = B (d) 2/8 = ¼ So, A = B. (e) ⅜ = ½ So, A ≠ B (f) 2/10 = ⅕ So, A = B That’s it about Halves and Quarters Class 4 Math JKBOSE Solutions. Hope you found it useful. Do share your views about this post in the comment section below.	{"url":"https://learnjkbose.com/halves-and-quarters-class-4-math-jkbose-solutions/","timestamp":"2024-11-12T13:31:06Z","content_type":"text/html","content_length":"136397","record_id":"<urn:uuid:b64cbee2-0d71-47b8-972b-d35fa7154be1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00747.warc.gz"}
columns of data 1. In Input column(s), enter one or more columns to standardize. 2. In Store results in, enter a column number (for example, C1) or a column name for each input column. If the name contains spaces, enclose the name in single quotation marks. 3. Select the method to standardize the data: □ Subtract mean and divide by standard deviation: Center the data and change the units to standard deviations. For a regression analysis, select this method to standardize predictors in order to reduce multicollinearity and to compare the size of the coefficients on a comparable scale. □ Subtract mean: Center the data. For a regression analysis, select this method to standardize predictors in order to reduce multicollinearity. This method is helpful when your model contains highly correlated predictors, higher-order terms, and interaction terms. □ Divide by standard deviation: Standardize the scale for each variable that you specify, so that you can compare them on a similar scale. For a regression analysis, select this method to standardize predictor variables in order to determine which predictors have a larger effect, while controlling for differences in scale. □ Subtract first value, then divide by second: Enter your own values (such as known values for the mean and the standard deviation) to subtract from and divide by. □ Make range from start to end: Transform the data linearly so that the resulting data have the first value that you specify as a minimum, and the second value that you specify as a maximum.	{"url":"https://support.minitab.com/en-us/minitab/help-and-how-to/calculations-data-generation-and-matrices/standardize/standardize-columns-of-data/","timestamp":"2024-11-03T07:25:12Z","content_type":"text/html","content_length":"11735","record_id":"<urn:uuid:a09f91ba-be71-466e-9088-31d77d7738e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00039.warc.gz"}
How the Random number is generated in Java using Random class ? \| Sololearn: Learn to code for FREE! How the Random number is generated in Java using Random class ? Logic behind the process of generating random numbers import java.util.Random; public class Program { public static void main(String[] args) { Random rand = new Random(); int value = rand.nextInt(50); System.out.println(value) ; } } java.util.Random class đ is the built-in method in Java to generate random number... not truly random but uniformly distributed based on mathematical functions... good enough in most cases the argument 50 passed provide the upper bound limit... check this out for more đ	{"url":"https://www.sololearn.com/en/discuss/1234492/how-the-random-number-is-generated-in-java-using-random-class","timestamp":"2024-11-14T11:03:35Z","content_type":"text/html","content_length":"917417","record_id":"<urn:uuid:01d8b954-9008-4f9d-a137-36030bce0446>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00193.warc.gz"}
Box Plots Introduced in 1977 by mathematician John Tukey, the box plot is a two-part depiction of a given data set, using a central "box" and two lines, often called "whiskers." The box shows the set's median, as well as first and third quartile values, typically distributed over a vertical axis. The whiskers, which protrude on the top and bottom of the box, can be used in a variety of ways – most commonly to show the minimum/maximum of a set, or one standard deviation above and below the "box." Box plots have the ability to represent a variety of characteristics of a given set in a succinct manner, as well as compare multiple sets to one another (as seen in the image below). There are at least five relevant pieces of information per set (six if there is a signifier of a set's mean), making them extremely thorough compared to one-dimensional data visualizations such as pie charts or histograms. That said, given the amount of data points per set, box plots can become difficult to follow if many sets are being compared on the same plot. It appears that box plots are used primarily by scientists/statisticians as a pragmatic way to interpret and display data, and less so by popular news outlets. As a result, it is difficult to find a "bad" box plot - perhaps because the comprehensiveness of box plots makes them borderline foolproof to malicious manipulation. Below is a box plot that displays data on airline departure delays over the course of twelve days. While the median barely deviates over the course of the study, the obvious change in maximum value could imply a negative pattern that isn't particularly relevant to travelers. Above is an example of a box plot being used as a complement to a histogram. Because box plots address many different data points, it can be useful to focus on one particular attribute and use a box plot as a separate device to visualize the data set as a whole. Below is, in my opinion, an example of a box plot containing too many data sets, making it difficult to intuit/digest ––	{"url":"https://ldld.samizdat.cc/2017/box-plots/","timestamp":"2024-11-08T21:30:41Z","content_type":"text/html","content_length":"11640","record_id":"<urn:uuid:14a767ec-193f-4881-9f97-bf84e8653816>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00324.warc.gz"}
How to Measure Residual Noise In evoked potential testing, there are various ways to measure residual noise. But which one should you use? The fully objective and perhaps easiest method is to use the automated residual noise calculator. This feature analyses the sweep-to-sweep variability of the measured and provides a numerical and graphical display of the residual noise. The lower the variance, the more “stable” the averaged waveform becomes. Here is an example of the graphical and numerical display. The y-axis shows residual noise (0-200 nV), and the x-axis shows number of sweeps (0-8000). A target value of 40 nV is shown by the arrow. As the number of sweeps increases the variability decreases (more noise is averaged away), showing a curve that decreases rapidly, but then begins to asymptote. This is perfectly normal and the reason for the asymptote is because the remaining noise in an average trace decreases as a function of the square root of the number of trials. This popular statistical method to evaluate noise is available for the most commonly used evoked potential application – threshold estimation in infants using the Auditory Brainstem Response. There are other methods also in common use, and could be used during ABR measures but would be relied upon in applications where the automated residual noise calculator is not available, such as the Auditory Middle Latency Response, the Auditory Late Response and Neuro Latency/Rate exams (i.e ABR for differential diagnosis of retrocochlear pathology). One approach is the “average gap” method, demonstrated below for an N1-P2 complex of the Auditory Late Response. The lower (blue) trace is an average response of 50 presentations of a 2 kHz toneburst signal; 25 condensation sweeps and 25 rarefaction. These two sub-averages are shown above (A+B curves), with their baselines overlaid. If the noise was zero and these two sub-averages contained the evoked potential only (a hypothetical scenario) then they would overlay perfectly. Any differences between them therefore represents the residual noise. The “average gap” method involves estimating (“by eye”) the difference between the trace by observing the gaps between the trace. The average gap of these A+B curves is shaded for illustration. It is a semi-objective approach since the results are dependent on the interpretation of the observer, although they are based on objectively obtained A related method is via the differential (A-B) curve. This subtraction curve will be smaller amplitude (a “flat” trace) when the noise is low and greater in amplitude (a “wavy” trace) when the noise is high. The amplitude of the waveform (e.g. how much it deviates from the baseline) thus gives an objective indication of the residual noise. For example, below is the same data as the previous example but instead of the A and B curves displayed overlaid, the differential curve (A-B) is displayed overlaid onto the average trace. It may be noticed that where the “gap” is large in the previous example, the differential curve deviates far from the baseline whereas when the “gap” is small, the differential curve remains close to the baseline. In the final example, below, we see an ABR measured in an infant at 40 dB nHL using the CE-Chirp stimulus in the left ear. The child has a mild hearing loss and the response here is close to threshold. However, a clear wave V is nonetheless present, as demonstrated by the waveform markers showing a faint but clear wave V; the data have an Fmp value of 11.05. Crucially the residual noise is low with an automated indication of 7nV, very little gap between A+B curves and a “flat” differential curve. Related course	{"url":"https://www.interacoustics.com/academy/evoked-potentials/abr-training/how-to-measure-residual-noise","timestamp":"2024-11-13T01:02:38Z","content_type":"text/html","content_length":"97795","record_id":"<urn:uuid:e707fdea-9fef-4b7c-9c0f-3f8e589a0497>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00762.warc.gz"}
Burgers-Mohr/Power-Mohr Model: Loading/Unloading Compression Test To view this project in 3DEC, use the menu command . The project’s main data files are shown at the end of this example. This example shows the influence of loading rate on the axial stress response of a viscoplastic sample in an unconfined compression test. The viscoelastic behavior of the sample obeys a generalized Kelvin law. Yielding is characterized by a Mohr-Coulomb failure criterion in both cases. The viscoplastic behavior is compared to that of a second sample, made of elasto-plastic Mohr-Coulomb material, and undergoing the same velocity-controlled compression test. All values quoted in this section may be interpreted in any consistent system of units, but are probably not representative, and are given only for purposes of illustration. The viscoplastic and viscoelastic samples are represented by one brick each, using the Burgers-Mohr model, and the Mohr-Coulomb model. To represent the generalized Kelvin viscous behavior, the viscous component of the Maxwell cell in viscosity-maxwell is not activated. In the first part of the test, a vertical compressive velocity of magnitude 10^-4 (in units of distance per unit time) is applied on both sides of the samples for a total of 1500 steps. The timestep is set to 10^-3, a value small compared to the ratio $\eta ^K/G^K$ of 10. For the unconfined compression test considered here, the Mohr-Coulomb failure criterion predicts that shear yielding will take place when the axial stress reaches the value of $-2C\sqrt{N_\phi}$ (\ (\simeq -1.28\) × 10^6). On the other hand, the axial stress in the elasto-plastic sample is given, up to incipient failure, by the elastic relation (1)\[\sigma_{xx} = [\alpha_1 - 2 {\alpha_2}^2 / ({\alpha_1} + {\alpha_2})] \epsilon_{xx}\] where $\alpha_1 = K + 4/3 G$, $\alpha_2 = K - 2/3 G$, $\epsilon_{xx}=-2vt/L$, $v$ is the applied velocity magnitude, $t$ is the simulation time elapsed to incipient failure, and $L$ is the horizontal length of the sample. The numerical results are presented in Figure Figure #compression-viscoplastic-slow-3dec. Note that the Burgers-Mohr sample fails at the same stress level, but later in time, thus reflecting the effect of creep (at incipient failure, for the model parameter used in the simulation, the time is about 1.00 for the Burgers-Mohr sample). When the loading rate is increased, and the simulation is repeated for the same final amount of deformation (the applied velocity is increased, the same number of steps is used but the timestep is reduced), the responses of the two models become more similar. For a velocity of 10^-2, the effect of creep cannot be detected on the plot—see Figure 2 (at incipient failure, the creep time is now about 0.75 × 10^-2). In the second part of the test, the compressive velocity is set to zero and the models are cycled for 500 steps. While the Mohr-Coulomb sample stays at yield, the Burgers-Mohr sample unloads as creep develops (see Figure 2). The interaction between creep and plastic flow in the Burgers-Mohr sample may be appreciated by comparing the viscoplastic behavior in Figure 2 and Figure 3; in the latest plot, more plastic flow (measured by strain-shear-plastic) is allowed to take place before the compressive velocity is set to zero and, subsequently, the magnitude of maximum creep unloading is In the third part of the test, the samples are “reloaded” by application of a 3DEC velocity of 5 × 10^-5, for a total of 2000 steps. At this stage, both samples are yielding at the same stress level (see Figure 4). To complete the test, the compressive velocity is set to zero again and the models are cycled for another 1500 steps. The evolution of axial stress during this final stage may be observed in Figure 4. Data Files model new ;File: CompressionBurgers.dat (formerly example 1.6.7) ;Title:Compression test on Burger-creep viscoplastic and Mohr material model configure creep model large-strain off block create brick 0 3 0 1 0 1 block create brick 6 9 0 1 0 1 block zone generate edgelength 10 block zone group 'mo' block zone group 'cv' range position-x 0 3 block zone cmodel assign mohr-coulomb range group 'mo' block zone property density 2.5E3 bulk 1.19E10 shear 1.1E10 friction 44 ... cohesion 2.72E5 tension 2E5 range group 'mo' block zone cmodel assign burgers-mohr range group 'cv' block zone property density 2.5E3 bulk 1.19E10 shear-kelvin 1.1E10 ... shear-maxwell 1.1E10 viscosity-kelvin 1.1E10 ... cohesion 2.72E5 friction 44 tension 2E5 range group 'cv' ; --- fish functions --- program call 'compression.fis' block history displacement-x position 0.0 0.0 0.0 fish history CVisc fish history Mohr model history creep time-total model save 'compression_test_ini' model restore 'compression_test_ini' model creep timestep starting 0.0010 model creep timestep fix 0.0010 model cycle 1500 model title 'Slow Compression Test' model save 'compression_test_slow' model restore 'compression_test_ini' model creep timestep starting 1e-5 model creep timestep fix 1e-5 model cycle 1500 model title 'Rapid Compression Test' model save 'compression_test_fast' model restore 'compression_test_ini' model creep timestep starting 1e-3 model creep timestep fix 1e-3 model cycle 1500 model cycle 1500 model title 'Less Plastic Flow' model save 'compression_test_lessflow' model restore 'compression_test_ini' model creep timestep starting 1e-3 model creep timestep fix 1e-3 model cycle 3000 model cycle 1500 model title 'More Plastic Flow' model save 'compression_test_moreflow' model restore 'compression_test_ini' model creep timestep starting 1e-3 model creep timestep fix 1e-3 model cycle 1500 model cycle 1500 model cycle 2000 model cycle 1500 model title 'Several Load Cycles' model save 'compression_test_cycles' program return Was this helpful? ... Itasca Software © 2024, Itasca Updated: Aug 13, 2024	{"url":"https://docs.itascacg.com/itasca910/3dec/block/test3d/Creep_Material_Models/CompressionBurgers/compressionviscoplastic.html","timestamp":"2024-11-13T19:15:29Z","content_type":"application/xhtml+xml","content_length":"25797","record_id":"<urn:uuid:5a80c243-023e-4643-b66e-4a2244e7b0ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00677.warc.gz"}
Learning the Simplicity of Scattering A SciPost Submission Page Learning the Simplicity of Scattering Amplitudes by Clifford Cheung, Aurélien Dersy, Matthew D. Schwartz Submission summary Authors (as registered SciPost users): Aurélien Dersy Submission information Preprint Link: https://arxiv.org/abs/2408.04720v1 (pdf) Code repository: https://github.com/aureliendersy/spinorhelicity Date submitted: 2024-09-06 21:26 Submitted by: Dersy, Aurélien Submitted to: SciPost Physics Ontological classification Academic field: Physics Specialties: • High-Energy Physics - Theory Approach: Computational The simplification and reorganization of complex expressions lies at the core of scientific progress, particularly in theoretical high-energy physics. This work explores the application of machine learning to a particular facet of this challenge: the task of simplifying scattering amplitudes expressed in terms of spinor-helicity variables. We demonstrate that an encoder-decoder transformer architecture achieves impressive simplification capabilities for expressions composed of handfuls of terms. Lengthier expressions are implemented in an additional embedding network, trained using contrastive learning, which isolates subexpressions that are more likely to simplify. The resulting framework is capable of reducing expressions with hundreds of terms - a regular occurrence in quantum field theory calculations - to vastly simpler equivalent expressions. Starting from lengthy input expressions, our networks can generate the Parke-Taylor formula for five-point gluon scattering, as well as new compact expressions for five-point amplitudes involving scalars and gravitons. An interactive demonstration can be found at https://spinorhelicity.streamlit.app . Author indications on fulfilling journal expectations • Provide a novel and synergetic link between different research areas. • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work • Detail a groundbreaking theoretical/experimental/computational discovery • Present a breakthrough on a previously-identified and long-standing research stumbling block Current status: Awaiting resubmission Reports on this Submission The article presented by the authors provides a novel and compelling approach to simplifying expressions in spinor-helicity variables, addressing challenges posed by redundancies from momentum conservation and Schouten identities. By leveraging machine learning (ML) techniques, the authors provide a fresh perspective on this complex problem. The presentation is clear and detailed. First, the authors present a one-shot simplification technique for expressions of moderate size. Then, they expand on this by investigating a sequential simplification approach where sub-expressions are simplified after being grouped together based on their cosine similarity. This allows the simplification of larger expressions. Their open-source code available on GitHub adds further value to their contribution. In summary, I believe this article will be a valuable contribution to both the machine learning and the scattering amplitude literature. I appreciate the effort made by the authors to connect these fields while making the article accessible to both communities. I therefore recommend its publication. Beforehand, I suggest the following minor revisions to enhance clarity and impact: 1. Towards the end of page 2, in the introduction, the authors review ML applications to high-energy physics. It may be worthwhile to include in this discussion prior studies aimed at reproducing the numerical output of the amplitudes, rather than achieving exact analytic simplifications. For example, consider referencing arXiv:2002.07516 and arXiv:2107.06625, (and perhaps check references therein). Mentioning these works would strengthen the connection to existing literature. 2. Towards the end of page 5, the authors cite references [40-42] in the context of little-group and dimensional analysis constraints. While these are indeed relevant, the main simplification in those works arises from analysing singularities. Perhaps the wording could be rephrased as "These constraints, together with information from singular kinematic limits, can [...]" to more accurately reflect that work. Additionally, arXiv:2203.04269 is a recent advancement in this approach, which can simplify spinor-helicity expressions in the roundabout way (complex analytic -> numeric -> simpler 3. On page 9, in section 2.4, projective coordinates and twistors are mentioned, but without any reference. In the context of momentum twistors arXiv:0905.1473 comes to mind, and additional references could help guide readers unfamiliar with these topics. 4. On page 11, the authors mention that a numerical check is performed on candidate expressions generated by the one-shot simplification approach to verify their validity. Looking in the code at add_ons/numerical_evaluations.py, it appears they are using double-precision (16-digit) phase space points, requiring 9-digit precision to declare a value as zero (ZERO_ERROR_POW_LOCAL = 9). It might be beneficial stating this in the paper. In principle, one may be concerned that numerical similar, but not analytically identical, simplified expressions could be erroneously accepted, or, on the contrary, that valid simplifications could be discarded due to precision loss. While this is probably unlikely until expressions have hundreds or thousands of terms, it might be worth commenting upon. Higher-precision and/or finite-field evaluations would greatly reduce room for errors, if needed. The authors may also wish to consider a native python implementation of spinor-helicity evaluations, rather than using a Mathematica link to S@M; the python package "lips" could be an 5. The particular redundancy of four-point amplitudes is referred to on multiple occasions. A more mathematically sound statement is that at four point, factorization is not unique (see "unique factorization domain", arXiv:2203.17170). While at n-point, n>4, factorization is (conjecturally) unique. This implies that there exists a unique least common denominator (LCD) for n>4, but not This is evident in the first two amplitudes in appendix G, which admit representations with different denominators. The first Parke-Taylor formula is more commonly written as (⟨1\|2⟩^3)/(⟨1\|4⟩⟨2\|3⟩⟨3\|4⟩), while the second expression could be written as (⟨1\|2⟩^6[3\|4])/(⟨1\|3⟩⟨1\|4⟩⟨2\|3⟩⟨2\|4⟩⟨3\|4⟩). The authors could comment on how this choice is made: does the ML model return multiple candidate representations, and is one picked at random among the valid ones? Or perhaps, are the denominator factors in the simplified expression restricted to be a subset of those in the original expression? Similarly, for n>4, the authors could comment on the ability of the model to identify the least common denominator. For instance, in the last amplitude before the bibliography, the denominators contain manifestly spurious factors [25] and [45]. I imagine this is an artefact of the compact form chosen to write this expression in the paper, even if the ML algorithm may return an expression without those factors in the denominator. It is worth noting that a clear and efficient algorithmic way to determine the LCD exists, through univariate interpolation and factor matching. 6. On a related note to 5, it has been widely observed that since rational functions in scattering amplitudes are sparse, multivariate partial fraction decompositions are an important tool to tame their complexity. The authors could comment on whether this already affects their ML approach or how it could be included. 7. While the authors consider up to six-point amplitudes, it appears that only maximally helicity violating trees are considered. It might be worthwhile to comment on what changes would be required to handle NMHV trees, which may include three-particle irreducible denominator factors s_ijk, and potentially spurious spinor chains. Similarly, I would imagine that a more compelling application of this method would be to loop-integral rational coefficients, rather than tree amplitudes. Like for NMHV trees, these may include several more denominator factors, other than 2-particle spinor 8. In loop amplitudes, the numerical constants in the rational coefficients can be rational numbers with fairly large denominators and numerators. In their work, the authors encounter mostly simple numerical coefficients (like ±1 or ±2), and by default choose to blind all constants (see page 22). They could comment on how their method could be reconciled with the numbers observed in loop coefficients. Perhaps a similarity could be defined among the constants, on top of that among the spinor monomials? Publish (easily meets expectations and criteria for this Journal; among top 50%) The manuscript presents a machine learning approach for the simplification of scattering amplitudes, more precisely for the algebraic simplification of rational functions in spinor helicity variables. Such a simplification is indeed a challenge and one that many scientists in this field have stumbled upon. The authors start by discussing the type of input and output amplitudes/expressions and the generation of the training data set via a backward generation. Two approaches, a one-shot simplification and a sequential version, are then discussed in depth. Both setups -- including transformer models, beam searches, and contrastive learning -- consist of state-of-the-art ML methods. The network architectures are carefully chosen for the problem at hand and are reasonably motivated by the authors. For both approaches the manuscript includes a plethora of results for the performance that show high efficacy and success rates for many applications, but also point to limitations of these methods. I agree with the authors' conclusion that their approach has the power and flexibility to tackle real problems in theoretical high-energy physics (and beyond!). The article is written in a clear and concise language and understandable, in most parts, also for non-experts in machine learning. I highly appreciate the authors' efforts in that respect, as the target group -- mostly theoretical physicists -- might not be very familiar with many of ML concepts. Furthermore, the methods described here are a beautiful showcase of the application of ML to obtain exact results. I believe that these and similar methods are also applicable to other areas of theoretical physics where one seeks exact analytical data. Finally, the submission is accompanied by a git repository containing the code and data set used in this project, which allows for reproductability of the results and which is a valuable resource for the community. I, therefore, recommend this article for publication after a few minor things have been answered/corrected in an updated version: 1 - In Sec 2.2: I would like to see some justification for the choice of the target data set and/or how well it will work for more complicated terms and/or how is there any bias by doing a backward generation compared to a forward generation (see e.g. https://arxiv.org/pdf/1912.01412). It seems biased to me in the sense that it might work great for amplitudes where we expect such very simple final forms, but I wouldn't expect it to capture well more complicated final expressions. I am thinking here of amplitudes with a "simplest" form of 10-20 numerator terms or even an application of these ideas to intermediate expressions where one might be interested to simplify very complicated rational functions to slightly less complicated rational functions. Or formulated from a different perspective: Is the target data set optimal for the sequential simplification or is there some potential for improvement? 2 - In the end of section 2.2 the authors mention that their setup is restricted to unitary relative coefficients of terms. This seems quite a bit restrictive for "real-world" applications (e.g. not N=4 SYM). I have seen amplitudes where the (probably) "simplest" form contains relative coefficients like 3/7. There seems to be a partial resolution to this restriction in footnote 14, but I'd suggest to extend that discussion as it wasn't clear to me how this additional difficulty could be handled most efficiently. 3 - Similar to the point above, the restriction to Nterms <= 3 in section 2.2 might not be optimal for more complicated problems. In summary, some more care should be taken to the choice of data sets and a discussion of their validity/bias/potential for improvement would be useful to the reader. 4 - How does the approach described in this paper compare to existing software, as e.g. cited in [13,14]? Further ideas for improvement: 5 - Section 2.4 seems out of place and most of its content might better fit into the introduction. 6 - The same holds for the last few sentences of section 4.1. These comments might fit better in thes introductory part of section 4 or even in the introduction section. 7 - Figure 12: It might be useful to also somehow mark the distinction of circle and triangle markers within the plots. 8 - It would be interesting to see a comparison of the approach described in this paper with existing software, as e.g. cited in [13,14]. 9 - The conclusions might benefit from an extended discussion related to future developments in this research area. In particular it might be worth emphasizing that these methods produce exact analytical results and may be applicable to other problems where fast numerical checks of the answers are available. I can easily think of a handful of applications that fulfill the latter criterion. Publish (easily meets expectations and criteria for this Journal; among top 50%) In the present article, the authors consider the use of machine-learning techniques to simplify tree-level scattering amplitudes written in terms of spinor-helicity variables, a task of relevance to analytic calculations in theoretical high-energy physics. While machine-learning techniques have in the past years revolutionized many different fields related to numeric and noisy data, comparably little work has been done on applying machine-learning techniques to analytic and exact subjects, such as theoretical physics and mathematics. The present paper is a very interesting specimen of such work. As such, it indeed provides a novel link between different research areas and opens a new pathway in an existing research field, with clear potential for multi-pronged follow-up work. In an introduction, the authors summaries the challenge and prospective reward behind simplifying complicated expression in theoretical physics, and in particular those written in terms on spinor-helicity variables. They sketch their machine-learning approach to this challenge and give an overview of related machine-learning approaches in the literature. In the second section, the authors introduce the spinor-helicity formalism and describe how they construct their training data by applying a number a scrambling steps to randomly generated simple expressions to produce expressions that should be simplified in a supervised-learning approach. In the third section, the authors describe a transformer model that is able to simplify an expression if the expression is related to the simple expression by three or less scrambling steps. However, the authors find that the accuracy drops with the number of scrambling steps and does not generalize beyond the number of steps seen at training. In a fourth section, the authors address the problem in generalization by training a second transformer to identify subsets of terms that are likely to be simplified by the first transformer. This allows the combined model to reliably simplify expressions of up to 50 terms. They also demonstrate that their model is able to simplify complicated expressions arising from tree-level scattering The authors conclude with a summary of their results and perspective on future work. In particular, the authors argue that a similar machine-learning approach could be used for a range of different simplification problems. The authors give further more technical details on their approach in seven appendices. They have made their code available via a github repository, further facilitating the reproduction of their Moreover, the authors provide a link to an interactive online demonstration of their model, allowing interested readers to apply the model and gauge its strengths and limitations themselves. The paper is well written and presents very interesting results. I thus recommend it for publication provided the following minor comments are addressed in a satisfactory fashion: 1. The authors mention their interactive online demonstration only in the abstract. It might improve the impact of the paper to refer to this interactive demonstration also in other places, as well as to elaborate on its capabilities. 2. One case for machine-learning in this context is that there exists no clear algorithmic way to simplify expressions in spinor-helicity variables analytically. The authors make this case in section 2.4, but this seems to be so fundamental to their work that it might already be mentioned in the introduction. 3. A second case for machine-learning in this context is that the simplification is hard to achieve but that its correctness is easy to verify via numerics. The authors mention this numeric verification on page 11. But again, this aspect seems to be so fundamental to their work that it would benefit from being mentioned already in the introduction. (It is widely known that transformers are prone to hallucinations. The possibility to numerically verify their output is the reason that hallucinations are not a problem for the authors' approach.) 4. One of the motivations that the authors bring up for their work is the simplicity of the Parke-Taylor amplitude (1.1). The authors mention that their model successfully simplifies the corresponding expressions for four and five gluons. I could not find a corresponding statement for six gluons though. In contrast, Parke and Taylor successfully simplified their earlier results for six gluons in 1986, using slightly different but related variables. (Parke and Taylor came up with an educated guess that they checked numerically, but that is also how the author's model works.) Could the authors place their impressive(!) achievements using a machine-learning approach into perspective of what is currently happening in the field of scattering amplitudes using a more traditional approach? 5. On page 11, the authors give many relevant technical details on the training of their model. Could they also mention how long training took on the A100 GPUs that they used? 6. From figure 3, it seems like five-point amplitudes are harder to learn for the model than four- and six-point amplitudes. Do the authors have an explanation as to why? 7. In figure 8, the authors give the averages of cosine similarities. Would it be useful to give also standard deviations? 8. Below (4.6), the authors write ``even as c(t) increases''. Since $0<c_0<1$ and $\alpha>0$, doesn't c(t) decrease with t? 9. While the authors consider massless amplitudes, many interesting processes involve also massive particles. Could the authors comment on whether it is possible to extend their approach to the massive case, for which a variant of spinor-helicity variables exists as well? 10. As previously written, the paper is in general very well written. I have spotted only two typos that the authors might wish to correct. On page 25, ``structure words'' should likely read ``structure of words''. On page 31, ``away [...] form completely dissimilar terms'' should likely read ``away [...] from completely dissimilar terms''.	{"url":"https://scipost.org/submissions/2408.04720v1/","timestamp":"2024-11-04T04:18:00Z","content_type":"text/html","content_length":"56272","record_id":"<urn:uuid:778e9b13-6e37-4d08-9454-fb6ee0b6b124>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00223.warc.gz"}
Tan 3pi/4 - Find Value of Tan 3pi/4 \| Tan 3π/4 A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. Tan 3pi/4 The value of tan 3pi/4 is -1. Tan 3pi/4 radians in degrees is written as tan ((3π/4) × 180°/π), i.e., tan (135°). In this article, we will discuss the methods to find the value of tan 3pi/4 with • Tan 3pi/4: -1 • Tan (-3pi/4): 1 • Tan 3pi/4 in degrees: tan (135°) What is the Value of Tan 3pi/4? The value of tan 3pi/4 is -1. Tan 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°). We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi) ⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees ∴ tan 3pi/4 = tan 3π/4 = tan(135°) = -1 For tan 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 3pi/4 value = -1 Since the tangent function is a periodic function, we can represent tan 3pi/4 as, tan 3pi/4 = tan(3pi/4 + n × pi), n ∈ Z. ⇒ tan 3pi/4 = tan 7pi/4 = tan 11pi/4 , and so on. Note: Since, tangent is an odd function, the value of tan(-3pi/4) = -tan(3pi/4). Methods to Find Value of Tan 3pi/4 The tangent function is negative in the 2nd quadrant. The value of tan 3pi/4 is given as -1. We can find the value of tan 3pi/4 by: • Using Unit Circle • Using Trigonometric Functions Tan 3pi/4 Using Unit Circle To find the value of tan 3π/4 using the unit circle: • Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive x-axis. • The tan of 3pi/4 equals the y-coordinate(0.7071) divided by the x-coordinate(-0.7071) of the point of intersection (-0.7071, 0.7071) of unit circle and r. Hence the value of tan 3pi/4 = y/x = -1 Tan 3pi/4 in Terms of Trigonometric Functions Using trigonometry formulas, we can represent the tan 3pi/4 as: • sin(3pi/4)/cos(3pi/4) • ± sin(3pi/4)/√(1 - sin²(3pi/4)) • ± √(1 - cos²(3pi/4))/cos(3pi/4) • ± 1/√(cosec²(3pi/4) - 1) • ± √(sec²(3pi/4) - 1) • 1/cot(3pi/4) Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of tan 3pi/4 will be negative. We can use trigonometric identities to represent tan 3pi/4 as, • cot(pi/2 - 3pi/4) = cot(-pi/4) • -cot(pi/2 + 3pi/4) = -cot 5pi/4 • -tan (pi - 3pi/4) = -tan pi/4 ☛ Also Check: 1. Example 1: Find the value of (2 sin (3pi/8) cos (3pi/8) sec (3pi/4)). [Hint: Use tan 3pi/4 = -1] Using sin 2a formula, 2 sin (3pi/8) cos (3pi/8) = sin (2 × 3pi/8) = sin 3pi/4 ⇒ 2 sin (3pi/8) cos (3pi/8) sec(3pi/4) = sin(3pi/4) sec(3pi/4) = sin(3pi/4)/cos(3pi/4) = tan 3pi/4 ⇒ (2 sin (3pi/8) cos (3pi/8) sec(3pi/4)) = -1 2. Example 2: Find the value of tan 3pi/4 if cot 3pi/4 is -1. Since, tan 3pi/4 = 1/cot(3pi/4) ⇒ tan 3pi/4 = 1/(-1) = -1 3. Example 3: Using the value of tan 3pi/4, solve: (sec²(3pi/4) - 1). We know, (sec²(3pi/4) - 1) = (tan²(3pi/4)) = 1 ⇒ (sec²(3pi/4) - 1) = 1 Show Solution > Ready to see the world through math’s eyes? Math is at the core of everything we do. Enjoy solving real-world math problems in live classes and become an expert at everything. FAQs on Tan 3pi/4 What is Tan 3pi/4? Tan 3pi/4 is the value of tangent trigonometric function for an angle equal to 3π/4 radians. The value of tan 3pi/4 is -1. How to Find Tan 3pi/4 in Terms of Other Trigonometric Functions? Using trigonometry formula, the value of tan 3pi/4 can be given in terms of other trigonometric functions as: • sin(3pi/4)/cos(3pi/4) • ± sin(3pi/4)/√(1 - sin²(3pi/4)) • ± √(1 - cos²(3pi/4))/cos(3pi/4) • ± 1/√(cosec²(3pi/4) - 1) • ± √(sec²(3pi/4) - 1) • 1/cot(3pi/4) ☛ Also check: trigonometric table What is the Value of Tan 3pi/4 in Terms of Cosec 3pi/4? Since the tangent function can be represented using the cosecant function, we can write tan 3pi/4 as -1/√(cosec²(3pi/4) - 1). The value of cosec 3pi/4 is equal to 1.41421. What is the Value of Tan 3pi/4 in Terms of Sin 3pi/4? Using trigonometric identities, we can write tan 3pi/4 in terms of sin 3pi/4 as, tan(3pi/4) = -sin(3pi/4)/√(1 - sin²(3pi/4)) . Here, the value of sin 3pi/4 is equal to 1/√2. How to Find the Value of Tan 3pi/4? The value of tan 3pi/4 can be calculated by constructing an angle of 3π/4 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.7071, 0.7071) on the unit circle. The value of tan 3pi/4 is equal to the y-coordinate(0.7071) divided by the x-coordinate (-0.7071). ∴ tan 3pi/4 = -1 Download FREE Study Materials Math worksheets and visual curriculum	{"url":"https://www.cuemath.com/trigonometry/tan-3pi-4/","timestamp":"2024-11-02T17:57:28Z","content_type":"text/html","content_length":"221726","record_id":"<urn:uuid:7f7580f4-3225-4786-b981-f113ee027483>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00008.warc.gz"}
PiezoLoadCases \| Piezotechnics Dr. Jaenker GmbH Main Load Cases for Piezo Actuators The linear constitutive equation of piezoelectricity describes a mechanically linear-elastic material with the superimposed effect of an electrically induced force. The piezoelectric force, in turn, is directly linearly related to the applied electric field. A piezo actuator is also a capacitor with the superimposed effect of an induced charge, which is linearly related to an applied force. The displacement of the piezo actuator is the electrically controlled deformation of a solid body. Once a voltage is applied to a piezoelectric actuator, piezoelectric forces act instantaneously within the solid body. As a result, it begins to deform until the load and the linear deformation forces of the piezoelectric solid body reach mechanical equilibrium. No Load - Free Stroke The displacement X X of a piezo actuator is equal to the piezoelectric (charge) constant d d multiplied by the voltage U U. X = d U Piezo actuators utilize multiple thin layers, and the achievable displacement is the value of a single layer multiplied by the number of layers. The thickness of each layer is 100 µm, the voltage is 150 V, and the electric field strength ranges from 1 to 2 kV/mm. Weight Loading - Constant Load Force T is constant. The displacement X of the actuator remains constant under a constant load force and is the product of the piezoelectric (charge) constant d and the voltage U. The gravitational force statically deforms the piezo actuator and overlays the electronically controlled deformation path of static deformation. This applies even when the external load forces are significantly lower than the blocking force. Linearelastische Last - Federtyp The displacement of the actuator is reduced by the ratio of the stiffness of the load spring k load k load to the stiffness of the actuator ka . X = X0 ka / (ka + kload) (free displacement X 0 = d ⋅ U X 0 =d⋅U) Blocked Actuator The generated force is equal to the area multiplied by the piezoelectric constant d d multiplied by the electric field strength divided by the compliance s. Fb = A d E / s Abrupt Increase of Voltage When the voltage source is abruptly switched on, the actuator experiences an impulse excitation. The electrical behavior of the actuator is that of a capacitor, drawing large currents. The amplifier provides maximum current until the voltage reaches the commanded final value after a rise time d T dT. I = C dU / dt If the amplifier is capable of delivering high currents, the actuator may overshoot in this situation, posing a risk of internal tensile stresses. To prevent the risk of damaging an actuator, the following measures must be taken: • Current limiting is a reliable method to reduce the rise time and prevent damage due to overshoots. • A force biasing mechanism can be installed to compensate for the transient tensile stress in the piezo material during rapid acceleration.. Dynamische Ansteuerung von Piezoaktuatoren In dynamic control, periodically varying voltages (sine or square waves) or non-periodic signals are applied to the actuator. An example is high-pressure fuel injectors in modern automobiles. They spray fuel very precisely into the combustion chamber. Piezo actuation is technically superior to electromagnets. A fuel injector driven by this technology can generate multiple fast and precise impulses during an engine's combustion cycle. The dynamic behavior of a piezo actuator with coupled mechanical load is determined by masses, stiffnesses, and damping rates. The actuator itself represents a low-damping spring-mass system. The low-frequency displacement behavior of such a system is limited or given by the free displacement capability of the actuator. At higher frequencies, displacement is limited by the inertia of the effective actuator mass. The achievable displacement of an actuator in sinusoidal operation is determined by the balance between the piezoelectric force and the inertia force of the accelerated effective mass. The following equations illustrate the mechanical response far below and above the mechanical resonance frequency f r f r : a) f <fr X = d U b) f> fr X = Fb / (MLast (2 pi f) 2) Highly dynamic operation of a piezoelectric actuator leads to outstanding mechanical (force x velocity) and electrical (voltage x current) performance values. However, piezoelectric materials are subject to damping effects, which result in losses and corresponding heating of the material during dynamic operation. Piezo stacks heat up quickly during continuous dynamic operation. It is essential to take sufficient measures to avoid reaching excessively high temperatures. • Limiting the amplitude (voltage, displacement), • The cutoff frequency, • Limiting operating time and • Adequate Cooling PIEZOTECHNICS is the first choice when it comes to high performance and dynamics!	{"url":"https://www.piezotechnics.com/loads","timestamp":"2024-11-03T06:21:28Z","content_type":"text/html","content_length":"30417","record_id":"<urn:uuid:9e0d31d9-075c-47f7-8237-4482de23e109>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00284.warc.gz"}
The STL Format - GuLab.cn Standard Data Format of STL Reprinted from http://www.fabbers.com/tech/STL_Format Reprinted from Section 6.5 of Automated Fabrication by Marshall Burns, Ph.D. Technical source: StereoLithography Interface Specification, 3D Systems, Inc., October 1989 An STL (“StereoLithography”) file is a triangular representation of a 3-dimensional surface geometry. The surface is tessellated or broken down logically into a series of small triangles (facets). Each facet is described by a perpendicular direction and three points representing the vertices (corners) of the triangle. These data are used by a slicing algorithm to determine the cross sections of the 3-dimensional shape to be built by the fabber. Format Specifications An STL file consists of a list of facet data. Each facet is uniquely identified by a unit normal (a line perpendicular to the triangle and with a length of 1.0) and by three vertices (corners). The normal and each vertex are specified by three coordinates each, so there is a total of 12 numbers stored for each facet. Facet orientation. The facets define the surface of a 3-dimensional object. As such, each facet is part of the boundary between the interior and the exterior of the object. The orientation of the facets (which way is “out” and which way is “in”) is specified redundantly in two ways which must be consistent. First, the direction of the normal is outward. Second, the vertices are listed in counterclockwise order when looking at the object from the outside (right-hand rule). These rules are illustrated in Figure 1. Figure 1. Orientation of a facet is determined by the direction of the unit normal and the order in which the vertices are listed. Vertex-to-vertex rule. Each triangle must share two vertices with each of its adjacent triangles. In other words, a vertex of one triangle cannot lie on the side of another. This is illustrated in Figure 2. Figure 2. The vertex-to-vertex rule. The left figure shows a violation of the rule. A correct configuration is shown on the right. The object represented must be located in the all-positive octant. In other words, all vertex coordinates must be positive-definite (nonnegative and nonzero) numbers. The STL file does not contain any scale information; the coordinates are in arbitrary units. The official 3D Systems STL specification document states that there is a provision for inclusion of “special attributes for building parameters,” but does not give the format for including such attributes. Also, the document specifies data for the “minimum length of triangle side” and “maximum triangle size,” but these numbers are of dubious meaning. Sorting the triangles in ascending z-value order is recommended, but not required, in order to optimize performance of the slice program. The STL standard includes two data formats, ASCII and binary. We will introduce ASCII formart only. STL ASCII Format The ASCII format is primarily intended for testing new CAD interfaces. The large size of its files makes it impractical for general use. The syntax for an ASCII STL file is as follows: 1 solid name 2 % facet syntax loop 3 endsolid name The syntax for facet unit is: 1 % facet unit syntax 2 facet normal ni nj nk 3 outer loop 4 vertex v1x v1y v1z 5 vertex v2x v2y v2z 6 vertex v3x v3y v3z 7 endloop 8 endfacet More about STL format https://all3dp.com/what-is-stl-file-format-extension-3d-printing/ https://en.wikipedia.org/wiki/STL_(file_format)	{"url":"http://gulab.cn/post/STL-formart/","timestamp":"2024-11-14T22:14:05Z","content_type":"text/html","content_length":"20074","record_id":"<urn:uuid:3d49e553-7c62-47af-ba36-642ffb089deb>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00289.warc.gz"}
Theory of Computation Assignment Help, TOC Homework Help, Computer Science Tutorial Are you often getting anxious over accomplishing Theory of Computation assignments within strict deadline? Well, in this situation, it’s quite natural to feel stressed-out. It’s time to sweep your all sorts of academic worries with Theory of Computation Assignment Help service offered by TutorsGlobe. With us, you will always get well-researched, correctly structured, and top-notch quality papers from industry best subject matter experts. We ensure that we will never fail to score the highest grade. Theory of Computation/Automata: In theoretical computer science & mathematics, the theory of computation is the branch which deals along with whether and how efficiently problems can be resolved on a model of computation, by using an algorithm. The field is divided into three main branches:, computability theory, automata theory and computational complexity theory. To perform a rigorous study of computation, computer scientist’s work along with a mathematical abstraction of computers called as model of computation. There are various models in use, but the most commonly analysis is the Turing machine. Computer scientists examine the Turing machine because it is easy to formulate, can be analyzed and utilized to prove results, and because this represents what various consider the most powerful possible "reasonable" model of computation (see Church–Turing thesis). It might appear that the potentially infinite memory capacity is an unrealizable attribute; however any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in principle, any difficulty that can be resolved (decided) by a Turing machine can be solved out by a computer that contains a bounded amount of memory. Automata theory: This is the study of abstract machines (or more suitably, abstract 'mathematical' machines or systems) and the computational problems that can be solved out by using these machines. These abstract machines are called as automata. Automata come through the Greek word that means that something is doing something by itself. Automata theory is also nearly associated to formal language theory, as the automata are frequently classified by the class of formal languages they are able to recognize. An automaton may be a finite representation of a formal language that can be an infinite set. Computability theory: This deals primarily with the question of the extent to which a difficulty is solvable on a computer. The statement that the halting problem can’t be solved out by a Turing machine is one of the most significant results in computability theory, as it is an instance of a concrete problem that is both simple to formulate and impossible to solve by using a Turing machine. Much of computability theory prepares on the halting problem result. Another significant step in computability theory was Rice's theorem, which states that for all of the non-trivial properties of partial functions; it is undecidable whether a Turing machine computes a partial function with that property. It is closely associated to the branch of mathematical logic called recursion theory, which eliminates the limitation of studying only models of computation which are reducible to the Turing model. Various mathematicians and computational theorists who study recursion theory will refer to it like computability theory. Computational complexity theory: Complexity theory assumed not only whether a problem can be solved out at all on a computer, but also how efficiently the problem can be solved out. Two main aspects are considered: time complexity and space complexity, which are respectively how several steps does it take to perform a computation, and how much memory is needed to perform that computation. To analyze how much time and space a given algorithm needed, computer scientists express the time or space needed to solve the problem as a function of the size of the input problem. For instance, searching a particular number in a long list of numbers becomes harder as the list of numbers grow up larger. If we say there are n numbers in the list, then if the list is not indexed or sorted in any way we might have to look at each number in order to discover the number we're seeking. Thus we say that to solve this problem, the computer require to perform a number of steps that linearly grows in the size of the problem. Pick our top-rated Theory of Computation Assignment Help service and grab the chance to score impeccable grades. We at TutorsGlobe have assisted thousands of students in fulfilling their dream grades by offering them top-notch Theory of Computation Homework Help service at the price that suits their pocket. You could be next. Having TutorsGlobe by your side, you can score the best possible marks, without shedding your blood, sweat, and tears while finishing off the assigned academic tasks. Now you will have someone, highly qualified and experienced, to take care of your all academic problems and will help you in acing your academic grades and performance. So, don't delay your academic growth; be ready to pass your academic curriculum with flying colors. Latest technology based Theory of Computation Assignment Help service online: Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in your Theory of Computation based Assignments and Homework tasks via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Theory of Computation, project ideas and tutorials. We provide email based Theory of Computation homework help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Theory of Computation. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Computer Science Homework Help and assignment help services. They use their experience, as they have solved thousands of the Computer assignments, which may help you to solve your complex issues of Theory of Computation. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay. So far, our Assignment Help experts have worked on more than a few topics of Computer Science, and some of them are as illustrated below:	{"url":"https://www.tutorsglobe.com/homework-help/theory-of-computation-assignment-help-73718.aspx","timestamp":"2024-11-02T04:42:14Z","content_type":"text/html","content_length":"52644","record_id":"<urn:uuid:91a8b5c2-7066-400a-bf25-dd057115c997>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00286.warc.gz"}
Class ContractionHierarchyPrecomputation<V,E> Type Parameters: V - the graph vertex type E - the graph edge type public class ContractionHierarchyPrecomputation<V,E> extends Object Parallel implementation of the contraction hierarchy route planning precomputation technique The original algorithm is described the article: Robert Geisberger, Peter Sanders, Dominik Schultes, and Daniel Delling. 2008. Contraction hierarchies: faster and simpler hierarchical routing in road networks. In Proceedings of the 7th international conference on Experimental algorithms (WEA'08), Catherine C. McGeoch (Ed.). Springer-Verlag, Berlin, Heidelberg, 319-333. Parallel version of the algorithm is described in the article: Vetter, Christian. "Parallel Time-Dependent Contraction Hierarchies." (2009). This algorithm speeds up shortest paths computation by contracting graph vertices. To contract a vertex means to remove it from the graph in such a way that shortest paths in the remaining overlay graph are preserved. This property is achieved by replacing paths of the form $\langle u, v, w\rangle$ by a shortcut edge $(u, w)$. Note that the shortcut $(u, w)$ is only required if $\langle u, v, w\rangle$ is the only shortest path from $u$ to $w$. Contraction is performed as follows. First a priority is computed for each vertex in the graph. This implementation uses edge quotient, complexity quotient and hierarchical depth metrics for computing priority. A hierarchy is then generated by iteratively contracting independent sets of vertices. A vertex is independent iff it`s priority is less than the priority of every vertex in its 2-neighbourhood. A 2-neighbourhood of a vertex $v$ is defined as a set of vertices that are reachable from $v$ using at most 2 hops. After contraction each vertex gets its unique contraction level - its position in the computed hierarchy. Finally, after all vertices are contracted each edge is set to be either upward if its source has lower level that its target, or downward if vice versa. Computing initial priorities, independent sets and shortcuts, updating neighbours priorities and marking upward edges is performed in parallel what gives this implementation performance speedup comparing to the sequential approach. For parallelization, this implementation relies on the ThreadPoolExecutor which is supplied to this algorithm from outside. Semen Chudakov • Nested Class Summary Modifier and Type static class Edge for building the contraction hierarchy. static class Return type of this algorithm. static class Vertex for building the contraction hierarchy, which contains an original vertex from graph. • Constructor Summary Constructs a new instance of the algorithm for a given graph and executor. Constructs a new instance of the algorithm for a given graph, randomSupplier and executor. Constructs a new instance of the algorithm for a given graph, parallelism, randomSupplier, shortcutsSearchHeapSupplier and executor. • Method Summary Modifier and Type Computes contraction hierarchy for graph. Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait • Constructor Details □ ContractionHierarchyPrecomputation Constructs a new instance of the algorithm for a given . It is up to a user of this algorithm to handle the creation and termination of the provided . For utility methods to manage a graph - graph executor - executor which will be used for parallelization □ ContractionHierarchyPrecomputation Constructs a new instance of the algorithm for a given . Provided should return different random generators instances, because they are used by different threads. It is up to a user of this algorithm to handle the creation and termination of the provided . Utility methods to manage a graph - graph randomSupplier - supplier for preferable instances of Random executor - executor which will be used for parallelization □ ContractionHierarchyPrecomputation Constructs a new instance of the algorithm for a given . Provided should return different random generators instances, because they are used by different threads. It is up to a user of this algorithm to handle the creation and termination of the provided . For utility methods to manage a graph - graph randomSupplier - supplier for preferable instances of Random shortcutsSearchHeapSupplier - supplier for the preferable heap implementation. executor - executor which will be used for parallelization	{"url":"https://jgrapht.org/javadoc/org.jgrapht.core/org/jgrapht/alg/shortestpath/ContractionHierarchyPrecomputation.html","timestamp":"2024-11-07T17:12:17Z","content_type":"text/html","content_length":"29159","record_id":"<urn:uuid:c8a2a2ff-9e7c-4ca4-bbb5-613e5b6de775>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00449.warc.gz"}
Learn the talents that will set you up for fulfillment in numbers and operations; fixing equations and systems of equations; linear equations and capabilities; and geometry. Coursera presents a broad range of programs in math and logic, all of which are delivered by instructors at top-quality establishments similar to Stanford University and Imperial College London. Online courses are a preferred way to learn about many various matters in laptop science, and this format additionally lends itself nicely to constructing your math and logic expertise. In reality, many students use online courses to satisfy arithmetic prerequisites for superior computer science degrees. If you are in search of online math courses, there are a lot of assets at your disposal. • Fear not – our complete listing of the highest virtual math classes is bound to help. • Learn third grade math—fractions, space, arithmetic, and a lot more. • The finest on-line math programs will provide you with a private tutor to deal with your individual needs and regular assessments to repeatedly track your progress. • ” Using AI has been an effective software for helping college students be higher writers, more knowledgeable thinkers, and efficient researchers. But, a math schooling is helpful for individuals who aspire to careers in many different fields, from science to art. Build your mathematical abilities and explore how edX courses can help you get started on your studying journey right now. A math training may help you perceive the ideas that information the world you reside in. Studying this discipline not only develops your mathematical pondering abilities, but also your problem-solving abilities. I Did perhaps not know that!: Top 10 tynker.com of the decade No matter the mathematics course you select, you’ll stroll away with new abilities and both a certificate or diploma to showcase your learnings. Alison’s math certificates and diplomas show that you simply had been able to effectively be taught and complete the work. This will help you stand out when applying to schools or a new job. Enroll in one of our math programs today to earn your certificates or diploma. Khan Academy is a learning useful resource for the arts, sciences, and arithmetic. Here, you will develop an understanding of the underlying mathematical construction of everyday mathematics. You will study why mathematical methods work and find the connections between totally different areas of maths so as to recognize the interconnected nature of various maths matters. Pre-algebra is the primary step in high school math, forming the constructing blocks that result in geometry, trigonometry, and calculus. To reach faculty math, one should be taught to think outdoors the box. A key function of mathematical thinking is the flexibility to suppose exterior the box, which is extremely priceless in at present’s world. The objective of this course is to develop that essential mind-set. Just as teachers look via textbooks or do on-line searches, they can save time by utilizing ChatGPT. We hope you’ve found this Online Math courses with certificates listing useful and intriguing. Since you have made it this far then definitely you’re prepared to be taught more and right here at Coursesity, it’s our responsibility to enlighten people with knowledge on matters they are prepared to be taught. This course consists of assorted modules which can help you in understanding how numbers are organized, stored, displayed, and utilized in mathematics. Learn Precalculus aligned to the Eureka Math/EngageNY curriculum —complex numbers, vectors, matrices, and extra. Learn multivariable calculus—derivatives and integrals of multivariable functions, utility problems, and extra. It also covers a quantity of college-level programs, corresponding to calculus and statistics. It’s worth noting, nevertheless, that Khan Academy doesn’t provide stay classes or any tutor guidance. This is an actual draw back, as there’s nowhere to turn immediately if you’re stuck. Khan Academy goals https://www.topschoolreviews.com/tynker-review to provide free, standards-aligned math lessons for anybody. Their courses cover math topics for kindergarteners up to college students. The study of mathematics and logic as a discipline provides as a lot as a lot more than what you discovered in highschool algebra. sylvan tynker Recommendations This course will help you velocity up your ideas of Geometry and finding out Divisibility guidelines for every quantity in the number system. Develop and follow differential calculus methods with functions. My name is Lewis Keegan and I am the writer and editor of SkillScouter.com. I’m extremely enthusiastic about on-line education and what it could possibly do for those to raised their lives. The course is 6 weeks in length, eight to 10 hours per week, and has four programs within the larger program. tynker Could Be Fun For All So, the perfect math course for you is one that may meet the priorities you’ve. With this in mind, custom-made 1-on-1 classes are your best wager for finding a math class that’s perfectly tailor-made to your particular needs. That said, math is greatest taught via 1-on-1 tutoring in order that instructors can offer you customized steerage and clear explanations to your questions. For this cause, if you want to examine math on-line, your greatest option is working with a personal tutor. With this in thoughts, Preply’s expert tutors can really speed up your learning for any math matter. They’ll also regularly assess your progress and provide useful materials that keep you motivated and develop your confidence.	{"url":"https://ecosolutions.gl/2022/11/13/picking-math-websites-is-simple/","timestamp":"2024-11-09T03:56:10Z","content_type":"text/html","content_length":"54077","record_id":"<urn:uuid:e2819f0b-618c-4f3e-adf0-2e9fa9eb5dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00804.warc.gz"}
Computer Science 210 Data Structures Fall 2019, Siena College Ratios BlueJ Project Click here to download a BlueJ project for Ratios. Ratios Source Code The Java source code for Ratios is below. Click on a file name to download it. * Example Ratios -- the Ratio class * This class encapsulates a numerator and denominator and * includes the capability to set the numerator or denominator, * retrieve the numerator or denominator, retrieve the decimal * equivalent of the ratio, and return a "pretty" String * representation of the ratio * @author Jim Teresco, The College of Saint Rose, CSC 202, Fall 2012 * Siena College, CSIS 210, Fall 2016 * @version Fall 2019 public class Ratio { // We first define the fields -- also called instance variables // that define the state of a ratio. // In this case, we need only two integers, one each for the numerator // and denominator. // Note that these look like local variables in a method, except we // add the qualifier "private" to indicate that no one except the // methods of this class can access these variables. This is the // most common qualifier for instance variables. // Instance variables have a class scope -- they are visible inside all // methods we write within this class. private int numerator; private int denominator; // We next provide the methods that operate on a Ratio. The first // is usually a special method that gets called when we create a // Ratio, called the constructor. It looks like the method calls // we have seen earlier in the semester, except we do not need // "static" or "void". The name of the constructor must match the // name of the class, and it should take any parameters needed to // give the class its initial value. Construct a new Ratio object with the given numerator and denominator. @param num the numerator @param den the denominator public Ratio(int num, int den) { // Inside the constructor, we initialize our instance variables. In // this case, we initialize them based entirely on the parameters. // In some other cases, the initialization might be to constant values // or may use more complex expressions. Note that "num" and "den" are // parameters whose scope is only this constructor -- they will not // exist when we get to other methods later on. While "numerator" // and "denominator" are instance variables that will remain in scope // and retain their values in other methods we call later. numerator = num; denominator = den; // We next provide two "mutator" methods. These methods are capable of // modifying the "private" instance variables based on the values in the // parameters. Mutator methods modify the state of an object, but do // not usually return any information, so they are specified with a "void" // qualifier in addition to "public". Note that we do not specify "static" // on any of our methods in the class now that we are defining our own // classes that represent objects. Change the numerator. @param num the new numerator public void setNumerator(int num) { numerator = num; Change the denominator. @param den the new denominator public void setDenominator(int den) { denominator = den; // Objects aren't useful if we cannot get information back out from them. // This is done with "accessor" methods -- methods that allow us to access // the private instance variables within the class describing our object. // We start with 2 simple accessors -- ones that allow us to retrieve the // numerator and denominator. Since those are both integer values, we specify // "int" return values on our methods. Get the numerator of the Ratio @return the numerator public int getNumerator() { return numerator; Get the denominator of the Ratio @return the denominator public int getDenominator() { return denominator; // Methods can also compute and return something interesting. In this case, // we add the capability to compute and return the "decimal" equivalent // of the ratio. This will be a double. Get the decimal value represented by the Ratio. @return the decimal value represented by the Ratio public double getDecimalValue() { return 1.0 * numerator / denominator; // All Java classes provide a mechanism for printing a meaningful // representation of each instance of the class. It is always in a // method named toString that returns a String. If we do not provide // such a method, Java will use a builtin one (which will work but // which probably does not do quite what we'd like). So we provide one // here to print out our Ratio in a "nice" format. Return string representation of the Ratio. @return the string representation of the Ratio public String toString() { return numerator + "/" + denominator; * Example Ratios -- a main method that demonstrates the use of * the Ratio class (also provided in this project) * @author Jim Teresco, The College of Saint Rose, CSC 202, Fall 2012 * Siena College, CSIS 210, Fall 2016/Fall 2017/Fall 2019 * @version Fall 2019 public class Ratios { Testing the Ratio class. @param args not used public static void main(String[] args) { // Now to represent ratios, we create an instance of Ratio // for each. The "new" results in a call to the constructor // of the class. Ratio a = new Ratio(4, 6); Ratio b = new Ratio(2, 4); // print some information about these. If we try to "print out" // an instance of a class, Java will implicity call the class's // toString method. We can also call it explicitly, as in the // second example. System.out.println("Ratio a is " + a); System.out.println("Ratio b is " + b.toString()); // we can print their decimal equivalents, which is now done by // the Ratio class by calling its getDecimalValue method System.out.println("a as a decimal is " + a.getDecimalValue()); System.out.println("b as a decimal is " + b.getDecimalValue()); // let's change the ratios a bit, this time by calling mutator // methods of our Ratio instances, then do the printouts again System.out.println("Ratio a is " + a); System.out.println("Ratio b is " + b); System.out.println("a as a decimal is " + a.getDecimalValue()); System.out.println("b as a decimal is " + b.getDecimalValue());	{"url":"https://courses.teresco.org/cs210_f19/examples/Ratios/","timestamp":"2024-11-07T12:47:30Z","content_type":"application/xhtml+xml","content_length":"8691","record_id":"<urn:uuid:4b131890-c02a-4b5c-b837-ddeeb2e76815>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00152.warc.gz"}
Elias Jääsaari I am a PhD student in Machine Learning at Carnegie Mellon University, where I am advised by Tianqi Chen and Ameet Talwalkar. My research interests include automated and scalable machine learning systems and algorithms. Before starting my PhD, I lived in Cambridge, UK, where I was an early employee at a University of Cambridge spin-out company. I am originally from Finland and received my BSc and MSc degrees in Computer Science from the University of Helsinki, where I was advised by Teemu Roos and affiliated with the Information, Complexity and Learning research group. E. Jääsaari, M. Ma, A. Talwalkar, T. Chen. SONAR: Joint Architecture and System Optimization Search. arXiv preprint. [pdf] V. Hyvönen, E. Jääsaari, T. Roos. A Multilabel Classification Framework for Approximate Nearest Neighbor Search. Proceedings of the 35th Conference on Neural Information Processing Systems (NeurIPS 2022). [pdf] E. Jääsaari, V. Hyvönen, T. Roos. Efficient autotuning of hyperparameters in approximate nearest neighbor search. Proceedings of the 23rd Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD 2019). [pdf] T. Silander, J. Leppä-aho, E. Jääsaari, T. Roos. Quotient normalized maximum likelihood criterion for learning Bayesian network structures. Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS 2018). [pdf] E. Jääsaari, J. Leppä-aho, T. Silander, T. Roos. Minimax optimal Bayes mixtures for memoryless sources over large alphabets. Proceedings of the 29th International Conference on Algorithmic Learning Theory (ALT 2018). [pdf] V. Hyvönen, T. Pitkänen, S. Tasoulis, E. Jääsaari, R. Tuomainen, L. Wang, J. Corander, T. Roos. Fast nearest neighbor search through sparse random projections and voting. Proceedings of the 4th IEEE International Conference on Big Data (IEEE Big Data 2016). [pdf] [code]	{"url":"https://eliasjaasaari.com/","timestamp":"2024-11-12T02:35:02Z","content_type":"text/html","content_length":"4448","record_id":"<urn:uuid:4601da48-634c-4392-878a-b7e86c49197e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00876.warc.gz"}
Information Ratio - What Is It, Explained, Example, Vs Sharpe Ratio Information Ratio Last Updated : 21 Aug, 2024 What Is An Information Ratio? The information ratio (IR) measures an investment manager's skill in generating returns above a benchmark or an index while considering the volatility of the returns. It compares the return on investment or portfolio to the returns of a benchmark or an index. The primary purpose of the information ratio (IR) parameter is to evaluate the investment manager's effectiveness in producing returns that surpass the benchmark while considering the level of risk taken to achieve those returns. It also demonstrates the portfolio returns' stability, reflecting its performance over time. In its calculation, another parameter called "tracking error" is used to indicate the risk-adjusted returns of an investment. • The information ratio measures the risk-adjusted performance of a portfolio relative to a benchmark. • This ratio is significant because it can help investors understand the skill level of an asset or investment manager and show the consistency of returns over a specific period. • Typically, investors prefer a higher positive information ratio of 0.4 to 0.6. A value above 0.6 is considered excellent. • It is considered reliable because it accounts for the volatility of returns. Information Ratio Explained The information ratio is an important aspect of investing, wherein the investment manager uses their expertise to generate more returns on investment above the benchmark. Typically, a benchmark such as S&P 500, Nasdaq Composite, etc., is used as a reference point. Before investing, the investor would be aware of the expected return, known as the benchmark return. It is based on the performance of the index over the years, as well as other factors that are considered. After investing, asset or portfolio returns usually differ from benchmark returns. Ideally, investors expect the portfolio returns to be higher, indicating that they have earned above-average or more-than-expected returns. However, if the benchmark returns are higher, the investment has underperformed, and the investment efficiency is unsatisfactory. This is where the information ratio comes into play as it measures the portfolio's performance relative to the benchmark and considers the amount of risk taken to generate those returns. Furthermore, a higher value of the information ratio is generally considered better, indicating better performance and investment efficiency. Now, let's look at the information ratio formula: Tracking error means the standard deviation of the difference in value between the portfolio and the benchmark returns. This plays a major role in identifying the consistency of the returns and the index performance. The formula shows that a positive information ratio (IR) occurs when the portfolio returns are higher than the benchmark returns. Conversely, a negative IR indicates underperformance. A good IR typically falls between 0.4 and 0.6, while a value below 0.4 suggests an unfavorable investment or lack of skill for the manager. Conversely, although rare, an IR exceeding 0.6 suggests a remarkable So, what is the importance of the information ratio? Firstly, it evaluates the skills and efficiency of the investment manager. This helps the investor decide whether to trust the manager or look for other options. Secondly, it helps evaluate the consistency of returns. Thirdly, it helps assess the benchmark index's performance against which the portfolio is being evaluated. Finally, the information ratio considers the risk volatility of returns, making it a more comprehensive performance measure. Here are a few essential points to remember: • A negative information ratio does not indicate that there are no returns or that the investor makes a loss. The investor can make extremely high returns and still have a negative ratio if the returns exceed the benchmark. • Different investors have different risk tolerance. Similarly, different securities have a different risk-return values associated with them. Therefore, a one-size-fits-all approach is not recommended in this case. Refer to the examples below – a simple and real-life calculation. Example #1 Here's a calculation example. • The benchmark returns from index X in 2022 = 9.3% • The annualized return from portfolios = 11.2% • Tracking error = 6.7% Though the ratio is positive, it is below the 0.4 mark. Therefore, it is neither a good investment nor a bad one. Example #2 Recent research shows that the information ratio of 23 equity sectors under the Investment Association has reached its lowest in at least 16 years. The study was conducted from 2007 to 2022, and the IRs of each of these sectors were monitored. Surprisingly, in 2022, the funds showed an average IR of -0.68. This is attributed to the decrease in the efficiency and skills of investment managers. The best performers are the IA Commodity/ Natural Resources sector with 1.13 and the IA Global Equity Income sector with 0.4. Unfortunately, these are the only two sectors with a positive IR. The worst-performing sector is IA Financials and Financial Innovation, with -1.48. The second lowest in the study period was -0.42 in 2016. However, the decline was due to the Brexit Referendum and the U.S. Presidential election. Information Ratio vs Sharpe Ratio The following table outlines the main differences between the information ratio and the Sharpe ratio: Basis Information Ratio Sharpe Ratio Purpose Measures the consistency of returns from an investment manager Measures the excess returns relative to the risk of an investment Calculation (Portfolio return - Benchmark return) / Tracking error (Portfolio return - Risk-free rate) / Standard deviation of returns Secondary Factor Benchmark return Risk-free rate Focus Manager's skill in outperforming a benchmark Investment's return per unit of risk Use Evaluating the performance of an investment manager Comparing the performance of different investments The information ratio and Sharpe ratios measure risk-adjusted returns, but the information ratio focuses on evaluating an investment manager's ability to outperform a benchmark. At the same time, the Sharpe ratio looks at an investment's excess return relative to the amount of risk taken. The secondary factor in the information ratio is the benchmark return, while the Sharpe ratio uses the risk-free rate. Investors can use both ratios to evaluate investment performance, but they are typically used for different purposes. Frequently Asked Questions (FAQs) What is the Treynor vs. information ratio? Treynor and information ratios are performance metrics used to evaluate the risk-adjusted return of an investment portfolio. The main difference is that the Treynor ratio measures the excess return earned per unit of systematic risk. In contrast, the information ratio measures the excess return earned per unit of tracking error. What are the drawbacks of the information ratio? One potential issue is that the method assumes a normal distribution of returns, which may not always be the case in real-world scenarios. In cases where returns are not normally distributed, the information ratio may not accurately reflect the risk-adjusted performance of the strategy. Another issue is that the information ratio only considers the strategy's relative performance compared to a benchmark and does not consider the strategy's absolute performance. Who commonly uses the information ratio? The information ratio is a widely used performance measure in the investment management industry. It is commonly used by portfolio managers, investment analysts, and other financial professionals to evaluate the risk-adjusted performance of investment strategies or portfolios. In addition, it is used by both active and passive managers and individual investors to assess the performance of their investments relative to a benchmark. Recommended Articles This has been a guide to what is Information Ratio. Here, we explain it with its comparison with Sharpe Ratio and its examples. You can learn more about accounting from the following articles –	{"url":"https://www.wallstreetmojo.com/information-ratio/","timestamp":"2024-11-08T07:26:39Z","content_type":"text/html","content_length":"330168","record_id":"<urn:uuid:75e11f06-7489-4824-b8c2-c2f0be145f42>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00214.warc.gz"}
Calculating interest rate given present and future value Present value: The current worth of a future sum of money or stream the payments are invested at a given rate of interest. 6 Jun 2019 Calculation Formulas. Simple Interest Rate. Given a present value and a future value based on simple interest, interest rate can be found out by PV is the present value and INT is the interest rate. You can read the formula, "the future value (FVi) Free online finance calculator to find any of the following: future value (FV), periods (N), interest rate (I/Y), periodic payment (PMT), present value (PV), or starting that $100 today is worth $110 in one year, given that the interest rate is 10%. When you are considering an investment, you want to know what rate of return an investment will give you. Some investments promise a fixed cost and a fixed Calculating the Interest Rate (i) Now we will show how to find the interest rate (i) for discounting the future amount in a present value (PV) calculation. To do this, we need to know the three other components in the PV calculation: present value amount (PV), future amount (FV), and the length of time before the future amount is received (n). the relevant time future. If interest is compounded n times a year at an annual rate r for t years, then the relationship between FV and PV is given by the formula. d Describe how time and discount rate affect present and future values; money originally borrowed, which interest is calculated on, is called the principal. have been given up by the lender, including lending to others, investing elsewhere,. Calculate the PV of a future amount Enter the calculated present value, the discount rate as the annual interest rate, and set the other options to match how When we study interest problems, we always go into A) Future Value of Simple Interest and Given some initial amount that we call the principal (P), the number of years you will use this It is the easiest type of interest to calculate and understand because its value I = Prt (Simple Interest = Principal x Interest Rate x Time). 23 Feb 2018 This is called calculating the future value of your goal. r= annual rate of inflation mutual fund · excel · financial goals · Future Value · Inflation · present value Coronavirus India LIVE news · Coronavirus death toll worldwide · Coronavirus impact on mutual funds · EPF interest rate · Dow Jones Trading This is because a dollar in the present will grow to be more than a dollar at a future date due to inflation and investment returns. This total growth rate is the interest rate of an investment. The unknown interest rate of an investment can be calculated if its initial present value, expected future value and years of investment are given. The formula for calculating the present value of a future amount using a simple interest rate is: P = A/(1 + nr) Present value (also known as discounting) determines the current worth of cash to Compound interest calculations can be used to compute the amount to which an investment will grow in the future. Compound interest is also called future value. For instance, a 12% annual interest rate, with monthly compounding for two future value (FV) considering compound interest, and an annual (or monthly or quarterly) value Annual Value (AV) is PV amortized or annualized to express a given amount as equal Definitions and Mechanics of Time Value Calculations. Time – The end of a year or period. MARR – Minimum Attractive Rate of Return. In other words, this formula is used to calculate the length of time a present value would need to reach the future value, given a certain interest rate. The formula Compute the present value. Given: a future value, fv; an interest rate compounded once per period, of Issuers calculate the future value of annuities to help them decide how to the discount rate (I) by the number of payments per year to find the rate of interest paid Anything But Ordinary: Calculating the Present and Future Value of Annuities the relevant time future. If interest is compounded n times a year at an annual rate r for t years, then the relationship between FV and PV is given by the To determine the period interest rate, simply take the annual rate of interest, and divide it by the number of compounding frequencies in a year. If 12% interest is compounded quarterly (4 times a year), then the period interest rate is 3% (12% 4). Comparing the interest costs with simple interest is very easy, i = interest rate Simple compound interest with one-time investments This is the formula that will present the future value (FV) of an investment after n years if Excel (and other spreadsheet programs) is the greatest financial calculator ever made. Solve for periodic interest rate, I/Yr, Rate(nper,pmt,pv,fv,type,guess) In this problem, the $100 is the present value (PV), NPer is 5, and Rate is 10%. To find the future value of this lump sum investment we will use the FV function, If we are given the future value of a series of payments, then we can calculate the given, then use the following formula to convert it to an effective interest rate:. future value (FV) considering compound interest, and an annual (or monthly or quarterly) value Annual Value (AV) is PV amortized or annualized to express a given amount as equal Definitions and Mechanics of Time Value Calculations. Time – The end of a year or period. MARR – Minimum Attractive Rate of Return. This is because a dollar in the present will grow to be more than a dollar at a future date due to inflation and investment returns. This total growth rate is the interest rate of an investment. The unknown interest rate of an investment can be calculated if its initial present value, expected future value and years of investment are given. The formula for calculating the present value of a future amount using a simple interest rate is: P = A/(1 + nr) Simple interest rate can also be calculated using Excel INTRATE function.. Compound Interest Rate. Given a present value, a series of equal values that occur after equal intervals in future and/or a single value at some future date that are subject to compound interest, the interest rate can be worked out using either of the following equations:	{"url":"https://bestftxediries.netlify.app/kromm3221vib/calculating-interest-rate-given-present-and-future-value-feny.html","timestamp":"2024-11-03T07:02:03Z","content_type":"text/html","content_length":"31875","record_id":"<urn:uuid:6f77c85f-b585-4297-a934-8475e6ed8da7>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00047.warc.gz"}
String-Math 2015search Item Successfully Added to Cart An error was encountered while trying to add the item to the cart. Please try again. Please make all selections above before adding to cart A co-publication of the AMS and International Press of Boston Hardcover ISBN: 978-1-4704-2951-5 Product Code: PSPUM/96 List Price: $139.00 MAA Member Price: $125.10 AMS Member Price: $111.20 eBook ISBN: 978-1-4704-4276-7 Product Code: PSPUM/96.E List Price: $135.00 MAA Member Price: $121.50 AMS Member Price: $108.00 Hardcover ISBN: 978-1-4704-2951-5 eBook: ISBN: 978-1-4704-4276-7 Product Code: PSPUM/96.B List Price: $274.00 $206.50 MAA Member Price: $246.60 $185.85 AMS Member Price: $219.20 $165.20 Click above image for expanded view String-Math 2015 A co-publication of the AMS and International Press of Boston Hardcover ISBN: 978-1-4704-2951-5 Product Code: PSPUM/96 List Price: $139.00 MAA Member Price: $125.10 AMS Member Price: $111.20 eBook ISBN: 978-1-4704-4276-7 Product Code: PSPUM/96.E List Price: $135.00 MAA Member Price: $121.50 AMS Member Price: $108.00 Hardcover ISBN: 978-1-4704-2951-5 eBook ISBN: 978-1-4704-4276-7 Product Code: PSPUM/96.B List Price: $274.00 $206.50 MAA Member Price: $246.60 $185.85 AMS Member Price: $219.20 $165.20 • Proceedings of Symposia in Pure Mathematics Volume: 96; 2017; 297 pp MSC: Primary 14; 51; 53; 81 This volume contains the proceedings of the conference String-Math 2015, which was held from December 31, 2015–January 4, 2016, at Tsinghua Sanya International Mathematics Forum in Sanya, China. Two of the main themes of this volume are frontier research on Calabi-Yau manifolds and mirror symmetry and the development of non-perturbative methods in supersymmetric gauge theories. The articles present state-of-the-art developments in these topics. String theory is a broad subject, which has profound connections with broad branches of modern mathematics. In the last decades, the prosperous interaction built upon the joint efforts from both mathematicians and physicists has given rise to marvelous deep results in supersymmetric gauge theory, topological string, M-theory and duality on the physics side, as well as in algebraic geometry, differential geometry, algebraic topology, representation theory and number theory on the mathematics side. This book is co-published with International Press of Boston. Advanced graduate students, post-docs, and post Ph.D. mathematicians and mathematical physicists interested in string theory. □ Articles □ Katrin Becker and Melanie Becker — Superstring compactifications to all orders in $\alpha ’$ □ Francesco Benini and Alberto Zaffaroni — Supersymmetric partition functions on Riemann surfaces □ Huai-Liang Chang, Jun Li, Wei-Ping Li and Chiu-Chu Melissa Liu — On the mathematics and physics of Mixed Spin P-fields □ Cheol-Hyun Cho — Homological mirror functors via Maurer-Cartan formalism □ Charles F. Doran, Andrew Harder and Alan Thompson — Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds □ Muxin Han — SL(2,$\mathbb {C}$) Chern-Simons theory and four-dimensional quantum geometry □ Yuan-Pin Lee, Hui-Wen Lin and Chin-Lung Wang — Quantum cohomology under birational maps and transitions □ Gregory W. Moore, Andrew B. Royston and Dieter Van den Bleeken — $L^2$-kernels of Dirac-type operators on monopole moduli spaces □ Nikita Nekrasov — $\mathfrak {BPS/CFT}$ correspondence: Instantons at crossroads and gauge origami □ Xiaowei Wang and Yuguang Zhang — Balanced embedding of degenerating Abelian varieties □ Noriko Yui — The modularity/automorphy of Calabi–Yau varieties of CM type • Book Details • Table of Contents • Additional Material • Requests Volume: 96; 2017; 297 pp MSC: Primary 14; 51; 53; 81 This volume contains the proceedings of the conference String-Math 2015, which was held from December 31, 2015–January 4, 2016, at Tsinghua Sanya International Mathematics Forum in Sanya, China. Two of the main themes of this volume are frontier research on Calabi-Yau manifolds and mirror symmetry and the development of non-perturbative methods in supersymmetric gauge theories. The articles present state-of-the-art developments in these topics. String theory is a broad subject, which has profound connections with broad branches of modern mathematics. In the last decades, the prosperous interaction built upon the joint efforts from both mathematicians and physicists has given rise to marvelous deep results in supersymmetric gauge theory, topological string, M-theory and duality on the physics side, as well as in algebraic geometry, differential geometry, algebraic topology, representation theory and number theory on the mathematics side. This book is co-published with International Press of Boston. Advanced graduate students, post-docs, and post Ph.D. mathematicians and mathematical physicists interested in string theory. • Articles • Katrin Becker and Melanie Becker — Superstring compactifications to all orders in $\alpha ’$ • Francesco Benini and Alberto Zaffaroni — Supersymmetric partition functions on Riemann surfaces • Huai-Liang Chang, Jun Li, Wei-Ping Li and Chiu-Chu Melissa Liu — On the mathematics and physics of Mixed Spin P-fields • Cheol-Hyun Cho — Homological mirror functors via Maurer-Cartan formalism • Charles F. Doran, Andrew Harder and Alan Thompson — Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds • Muxin Han — SL(2,$\mathbb {C}$) Chern-Simons theory and four-dimensional quantum geometry • Yuan-Pin Lee, Hui-Wen Lin and Chin-Lung Wang — Quantum cohomology under birational maps and transitions • Gregory W. Moore, Andrew B. Royston and Dieter Van den Bleeken — $L^2$-kernels of Dirac-type operators on monopole moduli spaces • Nikita Nekrasov — $\mathfrak {BPS/CFT}$ correspondence: Instantons at crossroads and gauge origami • Xiaowei Wang and Yuguang Zhang — Balanced embedding of degenerating Abelian varieties • Noriko Yui — The modularity/automorphy of Calabi–Yau varieties of CM type Please select which format for which you are requesting permissions.	{"url":"https://bookstore.ams.org/view?ProductCode=PSPUM/96","timestamp":"2024-11-10T19:16:57Z","content_type":"text/html","content_length":"100655","record_id":"<urn:uuid:faf90dfb-2bd2-425e-9ffc-4475218149ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00458.warc.gz"}
Road Repair Cost Prediction This project aims to predict the total cost of road repairs using a Linear Regression model. The process involves data preprocessing, training a machine learning model, evaluating its performance, and visualizing the results. This structured approach demonstrates how data preprocessing, machine learning, and visualization techniques can be integrated to develop a predictive model for road repair costs. The results offer valuable insights for city planners and engineers in budget forecasting and resource allocation for road maintenance projects. Import Necessary Libraries These libraries are 'pandas' for data manipulation and analysis; 'numpy' for numerical operations; 'matplotlib' for plotting graphs; 'seaborn' for statistical data visualization; 'sklearn.model_selection.train_test_split' to split data into training and testing sets; 'sklearn.preprocessing.StandardScaler' to standardize features; 'sklearn.preprocessing.LabelEncoder' to encode categorical variables; 'sklearn.linear_model.LinearRegression' to perform linear regression; 'sklearn.metrics' for evaluating the model's performance. import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns from sklearn.model_selection import train_test_split from sklearn.preprocessing import StandardScaler, LabelEncoder from sklearn.linear_model import LinearRegression from sklearn.metrics import mean_squared_error, r2_score Load the Data Load the dataset containing road repair data from a CSV file into a pandas DataFrame. data = pd.read_csv("file_location") Encode the Categorical Variables Convert categorical variables (Type and Condition) into numerical format, as machine learning models require numerical inputs. le = LabelEncoder() data["Type"] = le.fit_transform(data["Type"]) data["Condition"] = le.fit_transform(data["Condition"]) Separate the Target Variable and Features Target Variable: 'Total_Cost', which we want to predict. Features: All other columns used as inputs to the model. y = data["Total_Cost"] X = data.drop("Total_Cost", axis=1) Scale the Features Scale the features to have zero mean and unit variance, which helps in improving the performance of the machine learning model. scaler = StandardScaler() X_scaled = scaler.fit_transform(X) Split the Data into Training and Testing Sets Training Set: 70% of the data, used to train the model. Testing Set: 30% of the data, used to evaluate the model. Random State: ensures reproducibility of the results. X_train, X_test, y_train, y_test = train_test_split(X_scaled, y, test_size=0.3, random_state=42) Train and evaluate the Model Model Initialization: Create an instance of the Linear Regression model. Model Training: Fit the model to the training data. Predictions: Use the trained model to predict the Total_Cost on the testing set. Mean Squared Error (MSE): Measure of the average squared difference between actual and predicted values. Lower values indicate better performance. R-squared Score: Indicates the proportion of variance in the dependent variable that is predictable from the independent variables. Values closer to 1 indicate better performance. model = LinearRegression() model.fit(X_train, y_train) y_pred = model.predict(X_test) mse = mean_squared_error(y_test, y_pred) r2 = r2_score(y_test, y_pred) Visualize the Model's Predictions Scatter Plot: Plot actual vs. predicted values to visualize the model's performance. Diagonal Line: The dashed line represents a perfect prediction. The closer the points are to this line, the better the model's predictions. plt.scatter(y_test, y_pred, alpha=0.5) plt.xlabel("Actual Total Cost") plt.ylabel("Predicted Total Cost") plt.title("Actual vs. Predicted Total Cost of Road Repairs") plt.plot([min(y_test), max(y_test)], [min(y_test), max(y_test)], "k--", linewidth=2) Below is the full code with additional comments embedded. # Import the necessary libraries import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns from sklearn.model_selection import train_test_split from sklearn.preprocessing import StandardScaler, LabelEncoder from sklearn.linear_model import LinearRegression from sklearn.metrics import mean_squared_error, r2_score # Load the data data = pd.read_csv("file_location") # Encode the categorical variables (Type and Condition) le = LabelEncoder() data["Type"] = le.fit_transform(data["Type"]) data["Condition"] = le.fit_transform(data["Condition"]) # Separate the target variable and features y = data["Total_Cost"] X = data.drop("Total_Cost", axis=1) # Scale the features scaler = StandardScaler() X_scaled = scaler.fit_transform(X) # Split the data into training and testing sets # Split the data (70% for training and 30% for testing) X_train, X_test, y_train, y_test = train_test_split(X_scaled, y, test_size=0.3, random_state=42) # Train the model # Initialize the model model = LinearRegression() # Train the model model.fit(X_train, y_train) # Evaluate the model # Make predictions on the test set y_pred = model.predict(X_test) # Calculate the mean squared error mse = mean_squared_error(y_test, y_pred) print("Mean Squared Error:", mse) # Calculate the R-squared score r2 = r2_score(y_test, y_pred) print("R-squared Score:", r2) # Visualize the model's predictions # Create a plot for actual vs. predicted values plt.scatter(y_test, y_pred, alpha=0.5) plt.xlabel("Actual Total Cost") plt.ylabel("Predicted Total Cost") plt.title("Actual vs. Predicted Total Cost of Road Repairs") plt.plot([min(y_test), max(y_test)], [min(y_test), max(y_test)], "k--", linewidth=2)	{"url":"https://ml-nn.eu/p/project2.html","timestamp":"2024-11-09T22:31:54Z","content_type":"text/html","content_length":"23334","record_id":"<urn:uuid:32a642bc-06a8-4259-955c-d35b8a9549a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00858.warc.gz"}
Class in San Diego Excel Advanced Class in San Diego CA Basic Functions: • Average (average of a range of cells) • Max (highest number in a range cells) • Min (lowest number in a range of cells) • Sum (sum or total of a range of cells) Count Functions: • Count (count cells containing a number) • CountA (count non-blank cells containing text or a number) • CountBlank (count blank cells) • CountIf (count only those cells meeting a single criteria - a word, a number, etc.) • CountIfs (count only those cells meeting multiple criteria - word, number, etc.) Sum Functions: • SumIf (sum or total only those cells meeting a single criteria - a word, a number, etc.) • SumIfs (sum or total only those cells meeting multiple criteria - words, numbers, etc.) • SumProduct (sum the product of multiple rows) If Functions: • If (check to see if cells meet a single criteria and assign a result - we also incorporate AND & OR conditions) • And (assign a True or False value to rows that meet multiple criteria) • Or (assign a True or False value to rows meet one or another criteria) • IfError (substitute something else if forumal results in an error message) HLookup & VLookup Functions: • HLookup (have Excel look up data in a table) • VLookup (have Excel look up data in a table) Financial Functions: • Pmt (calculate payments and interest rates on car and home loans) Text Functions: • Concatenate (combine data from multiple fields into one) • Trim (remove extra blank spaces between Concatenated text) • Upper (convert all text to uppercase) • Lower (convert all text to lowercase) • Proper (convert text to capitalize first letter of each word) • Left (extract a number of characters from the left side of a text string) • Right (extract a number of characters from the right side of a text string) • Mid (extract a number of characters from the middle of a text string) • Text to Columns (separate text into columns) Date /Time Functions: • Date (display current date) • Now (display current date/time) • Calculate number of days/months/years between two different dates • Month • Year • AverageIf (average a range cells if they meet a single criteria) • AverageIfs (average a range cells if they meet multiple criteria) • "Nest" functions (place one function within another) • Workbook Controls (Sliders, Spin Button, Option Button) to create an interactive spreadsheet • Dashboard (create one sheet that combines charts, sheets, workbook controls, etc) Goal Seek allows you to calculate an unknown value in a given formula, but is only useful for problems that involve finding a single variable. When the desired result of a calculated cell is known, but not the input value that calculation needs to reach that result, you can use Goal Seek. Solver is a tool that helps you find solutions involving multiple variables. Scenarios are part of a group of commands that can be called what-if analysis tools. A scenario is a set of values that a user can create and save and then substitute at any time in the worksheet. The user can then switch to any of these new scenarios to view different results in the worksheet. To compare several scenarios, you can create a report that summarizes them on the same page. "WHAT-IF" TABLES: What-If Tables allow you to analyze data and produce a table to show the results. When a single variable like the unit price changes, use a one-input What-If Table. If the unit price AND number of units sold changes, use the two-input What-If Table. Create Absolute and/or Relative Macros to automate common, repetitive tasks. Add buttons and sliders to a spreadsheet that run macros.	{"url":"https://computersetcsoftwaretrainingcenter.com/computers-etc-software-training-center-classes/excel_adv_2007_fd.htm","timestamp":"2024-11-06T15:51:16Z","content_type":"text/html","content_length":"16879","record_id":"<urn:uuid:e510e7d4-dcb2-4d91-a22f-7a1af22a57a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00519.warc.gz"}