text:
Talk:Cellular Automata

Schelling
One good example of CA from the social sciences is Schelling, T. C. (1971) ‘Dynamic models of segregation’, Journal of Mathematical Sociology, 1, pp. 143–86. Shown at the end of this page:. MShonle 17:40, 8 August 2005 (UTC)

Format
How about if the Introduction page was merged into this main page, making the end result look something like the Computer Science:Data Structures book? I could also create a nav bar to help chapter browsing. MShonle 19:50, 12 August 2005 (UTC)

Some pages for deletion
Some experiments and some renamed images.

Do you want me to delete these for you? MShonle 01:44, 17 August 2005 (UTC)

Yes, thank you. I thought you were watching this talk page. I will ask you again when I will need some administration. --IzI 10:38, 17 August 2005 (UTC)

 * Done. :-) MShonle 13:54, 17 August 2005 (UTC)

Another one IzI 14:33, 17 August 2005 (UTC)

Another one, misspelled name. --IzI 12:31, 25 August 2005 (UTC)

 * Done. MShonle 15:05, 25 August 2005 (UTC)

 * Done. MShonle 21:06, 2 October 2005 (UTC)

I am sorry for all this pictures that need deletion, but I use this page for an exam and I am late (as usually). --IzI 16:27, 3 October 2005 (UTC)

title: Talk:Cellular Automata


text:
SPM/VBM

Voxel-Based Morphometry

Criticisms of VBM
There are many criticisms that can be made of VBM. In particular, the accuracy of the spatial normalisation used by SPM is an issue that upsets many people. Spatial normalisation in SPM uses only about 1000 parameters, and only fits the overall brain shapes. It is unable to warp one brain so that it exactly matches another. In order to compensate for this inaccuracy, the data are smoothed, where the amount of smoothing should partially depend on the accuracy of the inter-subject registration. There are many ways of assessing the "accuracy" of spatial normalisation. One approach is to look at the amount of overlap among the different tissue classes. What this shows is that e.g. grey matter is matched with grey matter. It does not indicate that e.g. the various sulci of the different subjects are in alignment. For example, with a registration model that has too much freedom, a sulcus of one subject could be warped so that it appears to becomes a different sulcus of another subject. For the purpose of comparing brain images with mass-univariate statistics, the objective of spatial normalisation should not be about making the brains of different subjects appear identical, but should be about bringing homologous regions as closely into alignment as possible. There is indication that models with too much freedom can be worse than models with less freedom. For example, the evaluations of Hellier et al., whereby registration accuracy of different approaches are compared indicated that the spatial normalisation of SPM2 was as accurate as that of any of the other methods. The average distances between registered sulci were 9.9, 10.3, 11.5, 10.7 and 10.8mm for other methods. The average distance obtained by the simplistic approach in SPM2 was 8.7mm. Obviously, there is still plenty of scope for improvement, but for an almost fully automated approach, it does not do too badly. There are many inter-subject registration, and tissue segmentation programs out there. These can also be used for preprocessing if the functions within SPM are found wanting. In particular, surface based approaches may prove to be superior.

Another criticism is that shape is multivariate, so any kind of morphometry should be based on multivariate statistics, rather than mass-univariate tests. I (John Ashburner) largely agree with this. Unfortunately, the results of multivariate analyses can not be communicated quite so easily among the brain imaging community - especially if the results are limited to the 2D pictures required by papers. The community wants to localise a limited number of discrete volumetric differences (rather than have a full characterisation of the differences).

Another one is that different ways of doing an analysis give different results. The reason for this is that a different analysis is asking a different question, to which the answer is likely to be different. In particular, the choice of "global normalisation" method will dramatically change the resulting pattern of blobs. Note also that different preprocessing will also give different results. The most "accurate" preprocessing will give the most accurate interpretations of the results and the most sensitivity. Less accurate preprocessing will give less accurate interpretations. The fact that different registration or segmentation algorithms produce different results is exactly analogous to the fact that the use of different landmarks give different results for other kinds of morphometric analyses. Registration can never be perfect, so some spurious volumetric differences will be found. If one averages over larger regions (i.e. smoothes using a larger FWHM), then the effect of misregistration is reduced. It is therefore possible to be confident that any differences are really volumetric. If the arguments against VBM were taken seriously, then all manual volumetric analyses are wrong because structures can never be outlined with 100% accuracy.

The non-stationary residual variance from the t-tests is also a problem for interpreting results. This causes blobs to move towards brain regions with low residual variance, and away from regions with high residual variance; MASCOI may help with this problem. Many people see significant differences outside the brain because the residual variance approaches zero as one moves towards regions without any grey matter; this is one motivation for using some kind of mask (the other is to reduce the severity of the multiple-testing problem). Ask your local statistician for further explanations.

Interpreting Results
The most general interpretation is that there is a significant difference in intensities in the preprocessed images in this region. The hard bit is to determine what may have caused these differences. The two possible explanations are:

1) The structure is smaller in one group than the other.

2) There is some other significant difference between the anatomies of the groups that shows up in the analysis in this region. There are loads of possibilities here.

Generally, we hope that the pre-processing has sensitised the tests to the first explanation, but other causes can never be discounted. e.g. one group has bigger ventricles, which will systematically influence spatial normalisation. Another possibility is that there is a contrast difference, and maybe the structure shows up better (and is therefore better segmented as grey matter) in one group rather than the other.

The t statistic is proportional to the difference between the groups, divided by the square root of the residual variability. If there is the same volumetric difference, but more variability in the grey matter, then the statstics in the grey matter will be less significant. There are also likely to be issues relating to estimated residual smoothness. If the residuals are more smooth, then the correction is less severe so the corrected stats are more significant.

The statistics are more sensitive further away from regions of greater variability. In VBM, this means a tendency for the most significant difference to be shifted away from the centre of a structure.

The contrast images (con_*.img)  that are estimated with VBM are the same as if you took the unsmoothed preprocessed data, made a difference image, and then smoothed this difference image. The initial difference is mostly at the edges, but is blurred (by the smoothing) so that it covers a larger region.

"Modulation"
Spatial normalisation expands and contracts some brain regions. Modulation involves scaling by the amount of contraction, so that the total amount of grey matter in the modulated GM remains the same as it would be in the original images.

If you take the limiting case of extremely precise (not necessarily extremely accurate) registration and segmentation, the pre-processed concentration images are likely to be all identical. An analysis of these data would not show you anything. I therefore tend to interpret unmodulated data as a representation of registration error, rather than an attempt to preprocess the data in order to sensitise the t tests to more meaningful volumetric differences.

Shapes are multivariate. The volume of one structure is likely to be related to that of another. For example, smaller brains may have smaller putamen. If you include "globals" in the model by proportional scaling, then this could be accounted for. Similarly, the size of a putamen may, or may not, correlate with the size of surrounding structures. A modulated analysis attempts to correct the volumes for regional expansion/shrinkage during spatial normalisation. An unmodulated analysis would really be comparing volumes after scaling out the volume changes in neighbouring regions. The definition of a neighbouring region is vague, and is dependent on the precision of the spatial normalisation.

For example, consider that spatial normalisation is able to register only to the resolution of the temporal lobe, and everything in this brain region expands or contracts by roughly the same amount. If one group has hippocampi that are large w.r.t. the volume of the temporal lobe, then this procedure would be more sensitive to such differences. Unfortunately, it is not straightforward to say exactly what the scale of the spatial normalisation is, so the results of such an analysis are a little bit arbitrary.

VBM in Practice
Basics

Start MATLAB, make sure SPM is in your path (e.g. run pathtool and add the SPM directory), and type spm pet to get the GUI up.

Unified Segmentation in SPM5
Preliminaries

(EDITORS: add stuff about DICOM import?)

First ensure that your images are reasonably well aligned with the template: click Check Reg, go to the SPM5/templates/ directory and choose T1.nii then go to the directory with your dataset and select the images. If they are not approximately in registration, then you will need to use the Display button and then adjust the values (up, pitch, etc.) then click Reorient and reselect the image(s).

Segmentation

Clicking the Segment button will bring up a page of options; you must select your files (under Data), and you will probably want to change the Output Files options. Unless you have reason to do otherwise (these notes are meant for beginners!) select modulated normalised for the tissues that you are interested in, don't save bias corrected, and don't do cleanup. Leave all the Custom options as they are. You can now save the job (as a .mat file) if you want, and/or click Run to segment the images. As a very rough estimate, allow 30 minutes per image.

Smoothing

Click Smooth, select the images (Modulated normalised/Warped tissue Class images are prefixed with mwc), leave the data type option as same, and choose the smoothing width. This is a difficult decision... commonly used widths are 8 to 12 mm, you may have to experiment to decide what works best for your scans/subjects, though there is an argument that the kernel width should match the expected size of the difference for which you wish to search. Unless you have very good reasons to do otherwise enter the same number three times, since you normally wish to smooth equally in all directions (i.e. with an isotropic kernel).

Statistics

This is hard to give a short introduction to, since it very much depends on your experiment, but to give a very simple example of an unpaired two-group t-test: click Basic Models near the top-left of the GUI, then choose the two-sample t-test factorial design. Enter the scans, and enter any covariates that you want (e.g. age, Total Intracranial Volume, etc. See below). Now for masking... it's probably best, if this is your first attempt at VBM, to select absolute threshold masking and enter a value of 0.05. Rather like smoothing though, exactly what masking to use is a somewhat difficult decision. Set the output Directory where you want the results to be written. It's probably best to leave the other options alone on a first attempt, though maybe check the section below...

Click Run, check the design matrix that appears, and then click Estimate on the left, and choose the SPM.mat file in the directory you just chose for output. Then click Results, and choose this file again.

Now you need to worry about the general linear model contrasts, which can be quite a complicated area. For a simple unpaired two-group t-test with no covariates, a t-contrast of [-1 1] will give a right-tailed (group2 > group1) t-test; [1 -1] will give a left-tailed (group1 > group2) t-test, and an F-contrast of either [-1 1] or [1 -1] will give an F-test (group2 $$\ne$$ group1) equivalent to a two-tailed t-test.

After defining and choosing your contrast, you'll then be asked about masking it with others (say no, unless you know you want this kind of analysis). Next you'll be asked about how to correct/adjust for multiple comparisons. If you have no idea, I'd select FDR, and leave the default of 0.05; better still, read the Genovese and Nichols articles on FWE and FDR. You should then be presented with the standard Maximum Intensity Projection (MIP) plot of the significant regions. Clicking one of the p-values buttons on the left (probably want volume) will produce a table of results, while selecting overlays... sections and choosing e.g. the T1 template will allow you to move around in the brain viewing the significant areas.

Optimised VBM in SPM2 and SPM99
This is not necessary for SPM5 because the Segment button should do everything in a single unified model. "Optimised VBM" was a half-baked procedure invented in about 10 minutes in order to provide a quick short term solution for improving the spatial normalisation in SPM. It does improve the spatial normalisation though, and should also give better inter-subject registration (and therefore better results) for fMRI data. If you want to do optimised VBM in SPM2 or SPM99, then the procedure is:

Segment the original images

This involves an implicit registration of a template to the image. The transformation that is estimated here is used to overlay the prior probability maps, which assist the segmentation. Before doing this step, you may need to reorient the images via Display. This is so that the affine registration has better starting estimates. A search for the keywords reorient Display at will give you all the hints about this that you need.

Clean up the grey matter.

This is only for SPM99. In SPM2, this procedure is combined with the segmentation. The seg1 and seg2 images are entered into the program, and the result is a brain_*.img, which has values of 1 for brain, and 0 for non-brain. The resulting brain_*.img is used to remove a few misclassified voxels from the *_seg1.img file. This is done using ImCalc, selecting the seg1_, seg2_, seg3_, abd brain_ images, entering an output filename, and the following expression:

i1.*i4./(i1+i2+i3+eps)

Estimate warps from the GM.

The reason for the first segmentation and cleanup is so that a set of spatial normalisation parameters can be estimated by matching GM with GM. In SPM99, you would disable 'Mask brain when registering?', and set the resolution for the spatially normalised images to be higher than the current defaults. The template would be an image of grey mater, which could be the one in the apriori directory, or it could be a "custom" made template.

Apply these warps to the original T1 image. This gives you a spatially normalised T1 image, which should have a resolution of about 1mm isotropic. The original *_seg* and brain_* images could be deleted at this stage.

Segment the spatially normalised T1.

Tell the segmentation routine that the image is spatially normalised, so no affine registration is done.

Clean up the grey matter.

The same as above (only for SPM99).

Modulate

Spatial normalisation expands and contracts some brain regions. Modulation involves scaling by the amount of contraction, so that the total amount of grey matter in the modulated GM remains the same as it would be in the original images.

Smooth.

Best if you do this by about 12mm.

Stats.

The whole idea behind the pre-processing is that the t-tests are sensitised to volumetric differences in the GM. Remember that your results could reflect other differences among the images.

Custom Templates
Spatial normalisation in versions of SPM prior to SPM5 required templates that had similar contrasts to the image to be matched with them. The reason for this is that the objective function is based on the mean squared difference between the images. This only makes sense if the intensities of the different tissue types correspond between the images (with an additional scaling factor that is estimated, which rescales the overall intensities). In SPM5, the segment button allows spatial normalisation to be achieved using a different objective function, which does not rely on a simple relationship between the intensities of a pair of images. For full details about the mechanism, I would suggest taking a look at Ashburner & Friston. Unified segmentation. NeuroImage 26(3):839-851 (2005).

The idea is that the segmentation of SPM5 warps a set of tissue probability maps so that they overlay on to the image to segment. After warping, these will represent (part of) the prior probability of each voxel belonging to a particular class.

Ideally, these tissue probability maps should represent the prior probabilities for the population under investigation, and are made by averaging a large number of spatially normalised tissue class images of different subjects (wc*.img). I suspect that if you don't have a large number of subjects, then the tissue probability maps released with SPM5 would be more representative than an average made up of only about 20 subjects.

Globals
The mass univariate testing approach assumes that all voxels in an image are independent (apart from the GRF correction part). The use of "globals" is a compromise in order to model dependencies between each voxel and some global measure.

Most people who study shape use a multi-variate framework. Shape is based on the relationship between the positions of corresponding features, after accounting for size, position and orientation. Unfortunately, we do not have enough subjects to do a full multi-variate analysis, so we restrict our analyses to being mass-univariate, and possibly use a "global" as a confound, or use some kind of proportional scaling. We attempt to answer a more clinically interesting question by limiting our investigation to identifying regions of increased or decreased GM.

"Globals" are a compromise (fudge) that is needed to introduce some kind of multi-variateness into the analysis. People generally want to see a map of significant blobs, so a mass-univariate approach (SPM) is usually adopted. In reality, shape (and hence relative volumes) is really multivariate. The volume of one structure is related to the volume of another. Bigger brains may (on average) have bigger hippocampi. Brains with globally thicker grey matter are more likely to have grey matter that is thicker at a particular sulcus. Also, brains where the segmentation (for some reason) overestimates the amount of grey matter, are also likely to appear to have more grey at a particular region.

Unfortunately, this multi-variateness is not a simple linear relationship. For example, bigger brains are likely to have proportionally more white matter than grey. For brains that are globally different, it becomes very difficult to interpret exactly what the differences mean in a mass univariate way. Basically, it is not clear what to use as globals for any particular experiment. This will depend on what theory you have about your subjects.

Proportional scaling converts volumes into values that are a proportion of total brain/GM volume. For example, you may be able to say that a region around some point contains 3% of the total GM volume. If you are interested in differences in such measures, then a proportional scaling model may be preferable. Alternatively, if you want to localise regions where the trends in GM volume differ from the total GM volume, then an ANCOVA model may be preferable.

Using no "globals" and a "modulated" analysis is intended to show regions of absolute volumetric difference in grey matter. Using total grey matter as a confound will show regions of absolute difference that can not be explained by the total grey matter differences.

You can include additional columns in your design matrix that model various effects of no interest (e.g. an ANCOVA correction for some global measure). This allows you to localise differences that can not be explained by these uninteresting effects.

For example, in a group comparison, one group's brains may be bigger than those of the other group. In a situation like this, you may only be interested in patterns of volumetric differences that differ from those of the total brain size.

Using proportional scaling to total grey matter will show differences where a region contains a disproportionately large or small region of total grey matter. For example, a structure may normally represent 2% of the total brain grey matter, but in patients it may represent 1.5%.

The males versus females test is a tricky one. Human males are generally bigger than females, with bigger heads. If you were doing an analysis of nose sizes, then it is likely that males would have bigger noses. Maybe you would want to know if this extra size was still significant if total head size was taken into account, either by modelling head size as a covariate, or by proportionally scaling the noses such that they were rendered proportional to the total head size (giving us a measure of nose as a percentage of head volume).

Normally, a brain with e.g. twice the volume of another brain will not have twice as much grey matter. The relationship between grey and white-matter volume approximately obeys a power law, with a 4/3 exponent (Zhang & Sejnowski, PNAS 97(10):5621-5626, 2000). If we use a proportional scaling model based on whole brain volume, our null hypothesis is that grey matter volume varies linearly with brain volume. This is simply not true, although it may serve as a useful first approximation.

The choice of model is up to the investigator, and it really does depend what you want to test. I would really rather not be give any firm recommendations. When a VBM experiment is written up, the model should be accurately described. Different models will give different results, which may appear to conflict with each other.

In the case of dementia, the total intra-cranial volume is often used to proportionally scale the data. Thus, each structure/region is treated as a proportion of intra-cranial volume. Head sizes vary, so this normalisation should reduce the residual variance from the model.

Obtaining Covariates
Globals (total segmented tissue volumes)

spm_global is not a good way of computing "globals" for VBM, as it was intended to compute a kind of average brain intensity by excluding "non-brain" voxels (with the assumption that these are below a certain fraction of the pre-exclusion mean), and computing the mean of what is left.

The function spm_get_volumes can be used to integrate over tissue segmentations, returning volumes in litres. The differences between native space (c), DARTEL-Imported (rc), modulated normalised (mwc) and smoothed modulated normalised (smwc) segmentation volumes are usually very small. The main difference is the amount of the spinal cord included in the white matter segment, which depends on the field of view of the images. Manual segmentation protocols (e.g. as in Whitwell et al., 2001) often set an arbitrary cut-off, such as the most inferior slice (in MNI space) that includes cerebellum; this is perhaps closest to the results of mwc or smwc images.

Total Intracranial Volume

To get TIV, one option is to sum the volumes of GM, WM and CSF, for example using spm_get_volumes on each tissue class and then adding these results. Use of this approach with the mwc segments from the New Segmentation Toolbox in SPM8 performs very well compared to manual or other automatic measurements (Ridgway et al., 2011).

Other covariates e.g. from spreadsheets/textfiles

You can specify any variables in the MATLAB workspace as covariates. If you have covariates in a text-file you can read this into MATLAB first, e.g. using textscan or textread. One approach for importing from spreadsheets (e.g. Microsoft Excel) is to use the program to save as .csv (comma-separated values) and then to use MATLAB's csvread to get variables from the CSV file.

Data from different scanners
You can generally model out the effect of using different scanners or scan sequences by modelling confounding effects in the design matrix. Note that if you are doing a comparison of one group versus another, where one group is collected on one scanner, and the other group is collected on another, then modelling out the scanner effect will also model out the difference between your groups. See Stonnington et al. 2008

Papers, tutorials, background, etc.

 * Wright et al. 1995 --- The original VBM paper
 * Ashburner and Friston 2000 --- Highly cited authoritative paper on the methods
 * Ashburner and Friston 2005 --- Recent paper on SPM5's Unified Segmentation model
 * Human Brain Function (2nd ed.) --- Very useful tutorial-style material, especially Ch.6
 * John Ashburner's PhD Thesis --- More thorough than HBF but less recent
 * Brain Warping --- edited by A. Toga, contains several useful chapters
 * JHU-PNI VBM Methods page --- Tutorial info, references, and links to discussion
 * CBGM at Northwestern University --- Many short notes, tutorials, discussion

Toolboxes, helper scripts, and other software

 * Christian Gaser's VBM Toolboxes including Optimised VBM for SPM2 and SPM5
 * CBGM at Northwestern University contains many useful scripts
 * John's Gems compiled by Tom Nichols
 * Ged Ridgway's VBM scripts includes resize_img, get_totals, make_diffs
 * Masking toolbox

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: SPM/VBM


text:
User talk:Stumples

Welcome, Stumples!

Come introduce yourself in the general reading room or your project in the project reading room. If you have any questions, you can ask in the assistance reading room or contact me personally. Kellen T 19:22, 8 August 2005 (UTC)
 * Please also check out the Cookbook policy page and the Cookbook talk pages.

title: User talk:Stumples


text:
User:Sandpiper

Hi people. Just arrived here from 'pedia looking to see what had happened to all the content from Harry Potter, And I discovered it was struggling to survive here too. Perhaps we should help it.

title: User:Sandpiper


text:
Ordinary Differential Equations/Graphing 1

First Order Differential Equations

First-order differential equations basically relate an x value, a y value and a gradient, y' at the point (x,y). As there are 3 variables, it is impossible to represent the solution to a DE in a 2D form. However, we can draw diagrams in 2 dimensions to represent the solutions by eliminating one of the variables.

There are two main ways of doing this, slope fields, described below, and isoclines, which are related to slope fields.

Slope Fields
A slope field is a set of sections of tangents to solutions of the DE arranged over the xy plane. It attempts to show the solutions of a DE not by showing one or more solutions, but by showing a representation of all solutions across the xy plane.

The gradient of a (straight) line is defined as the change in y divided by the change in x. By setting the change in x to 1, we have the change in y equal to the gradient, which is y', the subject of the DE. Some rearrangement may be required to get the DE in to the required form, i.e.

 * $$\frac{dy}{dx}=f(x,y).$$

Now, if we draw a line at a point P(xy), with the x component equal to 1, and the y component equal to y' at that point, we have part of the tangent to the solution passing through that point. The solutions to the DE will follow these lines' slopes.

By drawing these lines at intervals across the xy plane, we can get a good idea of what the solutions to the DE look like. Notice that at no point did we have to actually solve the DE, we just plug in the values of x and y, at the points and we have slope field. This is especially useful when the DE is not solvable, for example

 * $$\frac{dy}{dx}=x^x.$$

Despite being theoretically easy to evaluate, slope fields are difficult to do without a computer, as even a small slope field has hundreds of lines, and the plotting of these is not practical by hand (even if possible). Often, the tangent sections are scaled for clarity so that they do not interfere with each other.

Example 1: y'=1
Consider the DE

 * $$\frac{dy}{dx}=1.$$

This is probably the simplest DE there is (other than y'=0). It simply states that the gradient of the solution is always 1. This clearly leads to the general solution

 * $$y=x+C, \,$$

where C is our constant of integration.

The slope field of this DE (right) is composed of lines with an x component of 1 and a y component of y', i.e. 1, spaced across the xy plane. The solutions to this DE, such as

 * $$y=x\,$$

and
 * $$y=x+1\,$$

clearly follow the slope field. Some solutions are shown in Fig 2, and the similarities can readily be seen.

Notice that the solutions do not necessarily touch any of the lines in the slope field. These lines are only a subset of the infinite number of possible tangents that could be drawn to the infinite number of solutions.

If the DE was

 * $$\frac{dy}{dx}=k,$$

then the gradient of the slope field would be k, as would the gradient of the solutions.

Example 2: y'=x
This slope field will be more complex, as the gradient of the tangent sections will vary, depending on where the section is along the x-axis. The x component of the lines is still 1, but the y component is now x, making the gradient more positive the higher x gets, and more negative the lower x gets.

The solution to the DE

 * $$\frac{dy}{dx}=x$$

is

 * $$y=\frac{1}{2}x^2+c$$

A set of these solutions is drawn along with the slope field in Fig. 4 to show how the two relate.

Example 3: y'=sin x
This is the last example of a DE where the derivative is equal to a function of x only. The DE is

 * $$\frac{dy}{dx}=\sin x$$

The slope field is shown in Fig. 5 below.

The solution to the DE is

 * $$y=-\cos x + C \,$$

and a set of solutions overlaid on the slope field looks like this:

Example 4: y'=y
Now we come to the trickier problem of DE where the derivative is related to the dependent function, y. The easiest one is when y(x) is constant, but this has already been covered, as it is the same as x being constant (see Example 1).

In the slope field, the x component of the lines remains 1, but the y component is equal to y. A common mistake is to assume that because the gradient is a function of y, the y component is set to 1. However, this will mean that the lines no longer represent the gradient as defined by Δy/Δx.

The inclination of the lines will increase as y becomes greater, but will not change if x changes. The slope field looks like Fig. 7.

The solution is as follows:

 * $$\frac{dy}{dx}=y$$
 * $$\int \frac{1}{y}dy=\int 1 dx$$
 * $$\ln y=x +C \, $$
 * $$y=Ce^x \,$$

This makes sense as the exponential function is the only function not to change when differentiated or integrated. Fig. 8 shows some solutions superimposed on the slope field.

This is a good example of where a slope field can be slightly misleading. The even spacing of the field (an artificial property, as the points with lines are chosen on an evenly spaced grid) does not convey the fact that as C increases linearly (as they do in Fig. 8), the solutions get further and further apart.

Example 5: y' = sin y
This is a difficult integration to do (I used a computer), but the slope field is still easy to draw: the y component of each line depends on the sine of the yvalue. This repeats every 2π units up the y axis, but between π and -π, the slope field looks like Fig. 9.

The general solution to

 * $$\frac{dy}{dx}=\sin y$$

is

 * $$y=\pm 2 \arccot e^{-x-C}. \,$$

Now, even though I didn't actually do the DE myself, and I have no idea if it is correct or not, I can compare it to the slope field, as in Fig. 10. This high degree of agreement between the two strongly suggests that the solution to the DE is correct. This is a very useful way of checking a result if you have access to a computer or a good graphing calculator.

Example 6: y'=xy
This is the last example of an slope field that I will do, and it involves a derivative related to both x and y:

 * $$\frac{dy}{dx}=xy.$$

The slope field looks like Fig. 11, and the x components of the lines are still all 1, despite, the presence of both x and y in the DE. This will never change.

The slope field along with its solutions is shown in Fig. 12.

The next method of graphing First-Order DEs is using isoclines, which have a significant relationship with slope fields.

 * Next: Isoclines


text:
Ordinary Differential Equations/First Order Linear 2

Return to Exponential Growth
Remember the population growth problem, where $$\frac{dP}{dt}=(B-D)P$$? Now that we can solve linear equations, we can also solve variations where a factor $$f(t)$$ is added in. The new equation is $$\frac{dP}{dt}=(B-D)P+f(t)$$, and can be solved by the linear methods taught in the last section.

Immigration
Lets say that 1000 people move into a city, in addition to the normal population growth. This can be interpreted by making $$f(x)=1000$$. This gives us a linear differential equation to solve

$$\frac{dP}{dt}=kP+1000$$

$$\frac{dP}{dt}-kP=1000$$

Step 1: Find $$e^{\int P(t)dt}$$

$$\int kdt=kt+C$$

$$e^{\int P(t)dt}=Ce^{kt}$$

Letting C=1, we get $$e^{kt}$$

Step 2: Multiply through

$$e^{kt}P'+e^{kt}P=1000e^{kt}$$

Step 3: Recognize that the left hand is $$\frac{d}{dt} e^{\int P(t)dt}y$$

$$\frac{d}{dt} e^{kt}P=1000e^{kt}$$

Step 4: Integrate

$$\int (\frac{d}{dt} e^{kt}P)dt=\int 1000e^{kt}dt$$

$$e^{kt}P=\frac{1000}{k}e^{mt}+C$$

Step 5: Solve for y

$$P=\frac{1000}{k}+\frac{C}{e^{kt}}$$

See how the answer is a constant addition to the normal solution, as expected.

Hunting
Lets say the government allows 10 animals to be killed a year. This makes $$f(t)=-10t$$. How does this effect the solution?

$$\frac{dP}{dt}=kP-10t$$

$$\frac{dP}{dt}-kP=-10t$$

Step 1: Find $$e^{\int P(t)dt}$$

$$\int kdt=kt+C$$

$$e^{\int P(t)dt}=Ce^{kt}$$

Letting C=1, we get $$e^{kt}$$

Step 2: Multiply through

$$e^{kt}P'+e^{kt}P=-10te^{kt}$$

Step 3: Recognize that the left hand is $$\frac{d}{dt} e^{\int P(t)dt}y$$

$$\frac{d}{dt} e^{kt}P=-10te^{kt}$$

Step 4: Integrate

$$\int (\frac{d}{dt} e^{kt}P)dt=\int -10te^{kt}dt$$

$$e^{kt}P=\frac{-10(kx-1)e^{kx}}{k^2}+C$$

Step 5: Solve for y

$$P=\frac{-10(kx-1)}{k^2}+\frac{C}{e^{kx}}$$

Mixture problems
Imagine we have a tank containing a solution of water and some other substance (say salt). We have water coming into the tank with a concentration $$C_i$$, at a rate of $$R_i$$. We also have water leaving the tank at a concentration $$C_o$$ and rate $$R_o$$. We therefore have a change in concentration in the tank of

$$\frac{dx}{dt}=R_iC_i-R_oC_o$$

Thinking this through, $$R_i$$, $$C_i$$, and $$R_o$$ are constants, but $$C_o$$ depends on the current concentration of the tank, which is not constant. The current concentration is $$\frac{x}{V}$$ where V is the volume of water in the tank. Unfortunately, the volume is changing based on how much water is in the tank. If the tank initially has $$V_0$$ volume, the volume at time t is $$V(t)=V_0+t(r_i-r_o)$$. This makes the final equation

$$x'=R_iC_i-\frac{R_ox}{V_0+t(R_i-R_o)}$$

which is an obvious linear equation. Lets solve it.

$$x'+\frac{R_ox}{V_0+t(R_i-R_o)}=R_iC_i$$

Step 1: Find $$e^{\int P(t)dt}$$

$$\int \frac{R_o}{V_0+t(R_i-R_o)}=\frac{R_oln((R_i-R_o)t+V_0)}{R_i-R_o}+C$$

$$e^{\int P(t)dt}=Ce^{\frac{R_oln((R_i-R_o)t+V_0)}{R_i-R_o}}=C((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}$$

Letting C=1, we get $$((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}$$

Step 2: Multiply through

$$((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}x'+((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}\frac{R_ox}{V_0+t(R_i-R_o)}=((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}R_iC_i$$

Step 3: Recognize that the left hand is $$\frac{d}{dt} e^{\int P(t)dt}x$$

$$\frac{d}{dt} ((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}x=((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}R_iC_i$$

Step 4: Integrate

$$\int (((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}x)dt=\int (R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}R_iC_idt$$

$$((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}x=\frac{C_iV_0((R_i-R_o)t+V_0)^{\frac{R_i}{V_0}}}{(R_i-R_o)}$$

Step 5: Solve for y

$$x=\frac{C_iV_0((R_i-R_o)t+V_0)^{\frac{R_i}{V_0}}}{(R_i-R_o)((R_i-R_o)t+V_0)^{\frac{R_o}{R_i-R_o}}}$$

Ugly, isn't it. Most of the time when dealing with real world mixture problems, you'll plug in much earlier and use numbers, which makes it easier to deal with.


text:
Talk:Playful Organizing

I changed the naming structure from ChapterNameX to Playful Organizing/ChapterNameX --Azertus 21:21, 8 August 2005 (UTC)

title: Talk:Playful Organizing


text:
Topology/Order

Recall that a set $$X$$ is said to be totally ordered if there exists a relation $$\leq$$ satisfying for all $$x,y,z\in X$$

 * 1) $$(x\leq y)\land (y\leq x)\implies x=y$$ (antisymmetry)
 * 2) $$(x\leq y)\land (y\leq z)\implies x\leq z$$ (transitivity)
 * 3) $$(x\leq y)\lor (y\leq x)$$ (totality)

The usual topology $$\mathcal{U}$$ on $$\mathbb{R}$$ is defined so that the open intervals $$(a,b)$$ for $$a,b\in\mathbb{R}$$ form a base for $$\mathcal{U}$$. It turns out that this construction can be generalized to any totally ordered set $$(X,\leq)$$.

Definition
Let $$(X,\leq)$$ be a totally ordered set. The topology $$\mathcal{T}$$ on $$X$$ generated by sets of the form $$(-\infty, a)$$ or $$(a, \infty)$$ is called the order topology on $$X$$

title: Topology/Order


text:
Cookbook:Chicken and Broccoli Casserole

Ingredients

 * 1 lb chicken breast, cubed
 * 14 oz frozen broccoli
 * 10.5 oz canned cream of celery soup
 * 4 oz mayonnaise
 * 6 oz skim milk
 * 1 teaspoon curry powder

Procedure

 * 1) Sear the cubed chicken in saucepan.
 * 2) Place cooked chicken and remaining ingredients into a casserole dish.
 * 3) Cover with foil and bake in oven at 375°F for 30 minutes.
 * 4) Serve with rice.

title: Cookbook:Chicken and Broccoli Casserole


text:
Category:Broccoli recipes

Recipes involving broccoli.

title: Category:Broccoli recipes


text:
User talk:Pennbradly~enwikibooks

Welcome, Pennbradly!

Come introduce yourself in the general reading room or your project in the project reading room. If you have any questions, you can ask in the assistance reading room or contact me personally. Kellen T 01:49, 9 August 2005 (UTC)
 * Please also check out the Cookbook policy page and the Cookbook talk pages.

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Pennbradly. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Pennbradly~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:34, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:22, 19 April 2015 (UTC)

title: User talk:Pennbradly~enwikibooks


text:
Cookbook talk:Pasta

I suggest breaking this up into Italian/Euro pastas and Asian pastas (at least within the page). While they are fundamentally similar, they quite different in usage, tastes, and textures. Kellen T 01:52, 9 August 2005 (UTC)
 * I agree that Asian noodles shouldn't be here. I suggest putting the Asian noodles on a page called Noodles, and keeping the pasta page for Italian pasta. The distinction isn't perfect (spaghetti are technically noodles, and ramen are technically a pasta), but that would make it the easiest for people to find what they want. People expect Italian stuff under Pasta and Asian stuff under Noodles, I think. risk 14:05, 11 August 2005 (UTC)

 * My definitions: Pasta is Italian-style material, no matter what the shape. Noodles are string-like, and may be made of any starchy material. Spaghetti, udon, soba, and ramen all fall into both categories. Elbow macaroni is only pasta. Rice noodles are not pasta. The main Cookbook page should probably list both. AlbertCahalan 02:57, 13 August 2005 (UTC)

 * I don't really think udon and ramen should fall under pasta (maybe a cultural difference, the Dutch word "pasta" has no connotations of Asian cuisine). I agree with Kellen that the two should be broken up because they are used in very different ways. I'd personally prefer one page for Italian pasta exclusively, since all Italian pasta's have so much in common (the recipe for pasta dough, for instance, is the same for any pasta). I think for the sake of simplicity it would be good to keep the Pasta page for Italian Pasta, with a link at the top of the page to whatever page would hold the Asian pasta's. Although, as I said, my understanding of the word Pasta may not agree with the universal one. risk 14:51, 13 August 2005 (UTC)

 * My understanding is that the word "pasta" has come to mean the vast category of shaped foods made from flour and water. (Wikipedia agrees; see Pasta.) Noodles are a sub-group of pasta, as are non-noodles like couscous. So, I would like to see one main pasta page, with links to region (Italian, Asian, etc.) I agree that they should be seperated in some way; there are just too many kinds of pasta to keep in one list. Wikipedia splits them up by shape, but it would probably make more sense for a cookbook to split them up by country. --- PurplePieman 21:22, 13 August 2005 (UTC)

 * I'm beginning to see that using the Pasta page for Italian Pasta exclusively isn't going to work. I suggest moving this page to Italian pasta and starting the main Pasta page afresh. I think the Asian Pasta's can be divided into Chinese Noodles and Japanese Noodles ("Japanese Pasta" doesn't really sound right, I don't think the Asian cuisines have any non-noodle pasta). Those are the only categories of Pasta I can think of. Miscelaneous stuff, like couscous, can just a link on the main Pasta Page. Would a Noodle page with cross references be necessary? Or should that just be a redirect? risk 00:12, 14 August 2005 (UTC)

 * I'm certainly OK with using this page just for Italian pasta. That might not be perfectly accurate, but it's probably what people expect. While I admit that ramen and udon might be pasta, I sure don't imagine them when someone says pasta. Noodles are definitely not a sub-group of pasta, mainly because of rice noodles. In any case, Noodles and Pasta should be distinct and should link to each other. AlbertCahalan 01:08, 14 August 2005 (UTC)

 * So, why are rice noodles not pasta? They seem to fit the definition to me. --- PurplePieman 05:41, 15 August 2005 (UTC)

 * I agree that, technically, many kinds of noodles are pasta. Pasta is a term usually reserved for products with wheat flour as their base (and sometimes potatoes), but not those based on rice, mung beans, etc. So we can't draw a tidy partition that encompasses either all pasta or all noodles. We could simply redirect Cookbook:Pasta and Cookbook:Noodle to to a new page called Cookbook:Pasta and noodles. Having two separate pages would invite unneeded duplication, because of the overlap. Also, where would wonton wrappers and rice-based spring roll wrappers go, otherwise? Geo.T 06:14, 15 August 2005 (UTC)

Handmade Pasta/Noodles
There will (probably) be people (like me) who will want to look up such information as pertaining to the making of pasta/noodles as opposed to the prep of ready-made pasta/noodles, and among those people there would be those who do not have or do not desire the use of a pasta-making machine for that purpose.

Merged
Cookbook:Italian Pasta and Cookbook:Pasta of Italy have been merged here.

Talk merged from Cookbook talk:Italian Pasta

It doesn't exists such a pasta named veggeroni, not as far as Italy is concerned :) visual reference to different pasta shapes

-- balubino (10:49, 18 Aug 2004)

 * I'm residing in Singapore and here we have these colourful fusili-like pasta thats made of vegetable fibres and the label on the packaging says "Veggeroni". I suppose its a modern type of pasta? -- Voed

title: Cookbook talk:Pasta


text:
Talk:Music Theory

Feedback from Hutchies and Musical May
1.Non-tonal, post-tonal, modern and contemporary music, 20th century music would be more correct terms for what is called as atonal music. Atonal implies in a negation of tonal music that is not true.

2.Is it really neccessary to say that some people think that it is ugly and other people don't?

-- Hutchies, (30 December 2005)

 * Hutchies: I've edited this page extensively to make it more complete as an overview of alternatives to tonality in 20th-century music; I've left the title as-is though, because I do think that although (like most generic titles) it leaves a lot to be desired in terms of accuracy, it's still the most commonly used term for this period (and with the revisions to the text it's more appropriate now).
 * The title of the Wikibook is Tonal Music Theory. I've looked around for how to rename the title of a book, and it seems like we should have consensus here before such a request is sent to the bot who deals with this. It sounds like you folks are all about adding sections about atonality/non-tonality/serialism/post-tonality/modernism/post-modernism or whatever you care to call it. If that's the consensus, then the title of the book should be changed. I don't have a problem with that at all. Also, it seems that people want to add information about non-Western music. This is, in my mind, another indication that the title of the book should be changed to simply Music Theory. Or maybe just Music. Now that I think about it, wouldn't it be better if we simply limit the scope of this book to common practice styles? And create other books for other styles? Something to think about. I don't care either way, since I've got this burnin' feelin' to start the counterpoint section. --A.J. 08:32, 28 May 2007 (UTC)

1.Hutchies is right; atonal is an erroneous term for some great music, but it is the term that is still used in most schools of music, so I believe that it is fine for this book as long as the other terms are presented alongside it.

2.I believe that this textbook could go much beyond what it is now, and include sections on musical form and analysis (including various forms of pop, jazz, rock, etc. as well as classical forms) as well as information on counterpoint and voice leading in various situations. Orchestration and Arranging may be too far outside the scope of this book, but might grow as its own section here until someone transfers it to its own textbook. Of course, some of this would be very difficult to do without the aid of graphics and media files.

3.I'd like to see media files being included of simple things such as scales, chords, and examples of each musical genre.

-- MusicalMay (27 May 2006)

 * http://meta.wikimedia.org/wiki/Music_markup for easy finding, if something like this was enabled then I would use it. Tuxide 18:52, 9 August 2006

 * The awesome Score extension generates a musical score from music notation inline with the page text, and turns it into a .MIDI file you can download or play if your browser supports MIDI, and converts that into an audio file you can listen to. User:Skierpage/Music intervals/Minor practice is a simple sample I made using the extension and http://lilybin.com/ for quicker feedback on my notation. The Score extension should be used throughout this book and in all Wikipedia musical examples! -- Skierpage (discuss • contribs) 11:54, 19 November 2017 (UTC)

Cleanup-NC
Now that this book has been moved to Music Theory, I am marking it as  because its naming conventions doesn't abide to WB:NP. Links to individual pages should be  as opposed to , etc.  Tuxide 18:35, 9 August 2006 (UTC)
 * Yeah, I know. I've got it on my todo list. Over then next week or so I'll rename these pages correctly. Derek Parnell 00:17, 10 August 2006 (UTC)
 * I could've done it myself, but there was one conflicting thing I noticed. Some link to the pages in the Western Music History book instead of this one such as Baroque, but others do not such as Classical.  I suppose all can be moved except Baroque, but I really don't know what should be done in this case. Tuxide 06:09, 10 August 2006 (UTC)
 * I went ahead and renamed the red links, since those don't require moves. Tuxide 06:15, 10 August 2006 (UTC)

This whole section is dreadful! I will edit when I have time, but the inaccuracies are legion. For instance, the modes are defined against a C major scale without the 12-tone equal-tempered system - much less the major, or C major scale - ever specified. The twelve-tone equal-tempered system is never mentioned as the foundation of Western music theory. Stylistic peculiarities are listed as "theory," when actually music theory fundamentals are objective descriptions of the tonal system, and are only style-specific as one moves from fundamentals to more advanced concepts.

Anything labeled "music theory" needs to be defined in terms of a specific pitch universe; i.e., the fundamentals of the Western, equal-tempered system vs. Javanese slendro or the system of Hindusthani thats. Music theory describes the fundamental materials and procedures of the pitch (and rhythmic) system in question. Most of the genre divisions listed in this section differ in terms of form, style, and musical function, but not in terms of how they employ basic theoretical idioms.

Wiki Notation
I wonder if there is a way to notate music as a wiki, instead of using pictures. In this way somebody could edit any mistakes in the music and nobody would have to install music notation software in their computer ( and learn to use it ). Gus 2007-02-01 12:21

 * Hey Gus, I'm both a computer professional and an amature percussionist with an interest in composition on my PC; so I'll see what I can dig up on this topic. But my gut feel is that most browsers won't support this so it'll need to happen as a generated image, server side. Which probably means Mediawiki will need to update their system. (Perhaps not... I'll download and install Mediawiki tonight and see what they let you customize) If they let you execute external programs to generate images I'll look into the myriad of open source notation generators and give a brief summary of the pros and cons of each, unless this isn't the place for that discussion...  I'll keep you posted. -- Jason C Daniels 00:55, 3 April 2007 (UTC)

 * There is a mediawiki extension for the GNU Lilypond rendering engine, for rendering music notations. However, that extension isn't installed here (or on any wikimedia project, for that matter). If there was enough support, we might be able to convince the developers to activate the extension for en.wikibooks, but we would probably need several votes from community members. If you want to get this issue started, you can post a note about it on Staff lounge, and we can start voting on the idea. --Whiteknight (talk) (projects) 01:01, 3 April 2007 (UTC)
 * Thanks for the heads up. I believe I found that extension -- WikiTeX -- on Wikisophias site. The extension appears to support many "LaTex" object/drawing extensions, not just Lilypond. Whiteknight, is this the item you were referencing? -- Jason C Daniels 01:36, 3 April 2007 (UTC)
 * Probably, and thanks for fixing my typo. I remember reading something about it on meta the other day, and sort of forgot about it until i saw this discussion today. If the extension supports more types of media besides music notation, it might be a benefit to the entire wikibooks project, not just the music books. Bring it up at the staff lounge. --Whiteknight (talk) (projects) 01:38, 3 April 2007 (UTC)
 * Whiteknight, what's the status on this? I'd like to be able to write up/edit a music notation page. But I'm not familiar or comfortable enough with the image insertion proceedures to do this yet. It strikes me that using a "built-in" wiki tool for music notation rendering would be easier than creating examples in an engraving program, and exporting them as images to be uploaded and embedded into a page... --A.J. 08:10, 28 May 2007 (UTC)

 * There is an extension for this, called mw:Extension:Score, it's completely awesome. Within a Score tag, you enter Lilypond notation, and it renders as a musical score, with an optional link to an audio file you can listen to! Skierpage (discuss • contribs) 11:12, 19 November 2017 (UTC)

Musical Styles
Is it okay for me to add another style under there (Electronica, including ambient, IDM, trance, etc.) although there is a Miscellaneous Styles section? I feel that electronica deserves its own section because of its uniqueness. I also plan on writing the section. --MagiNinjA 06:41, 15 February 2007 (UTC)
 * Dude, I say go for it. I know nothing about that style of music, but if you're willing to write it, knock yourself out! --A.J. 08:32, 28 May 2007 (UTC)

Page Formatting
I noticed some areas on this TOC page where the formatting seems a little off. I will go through it and fix up some things if people want, but i dont want to step on any toes. Let me know if you are interested and I will see what I can do. --Whiteknight (talk) (projects) 22:46, 4 April 2007 (UTC)

It seemed like someone came in and copy/pasted the page code onto itself, so all the TOC entries were listed twice... I fixed this and removed the "Music Theory/" part of each section title. I am brand spanking new to the Wiki scene, so I figured I'd act first and ask for forgiveness if necessary - if I stepped on toes, or if this is against Wikiquette, please don't hesitate to let me know. This book has been languishing in "disrepair" for so long that it finally got to me, so I decided to dust out the cobwebs. Countless other issues with this book which are driving me batty! I'll work on them as diligently as I can. --A.J. 08:06, 28 May 2007 (UTC)

Content Organization
Schoenberg (In Theory of Harmony ) makes three distinctions in the pedagogy of music: Harmony, Counterpoint, and Form. I think this would be the most beneficial chapter/unit breakdown for this book. For those without any knowledge of music whatsoever would have an introductory chapter/unit from which to gain knowledge of the basics of musical notation and so on. What say you all, if there are still authors about? Again, the idea is:
 * Fundamentals (of Western Music)
 * Harmony
 * Counterpoint
 * Form --A.J. 08:32, 28 May 2007 (UTC)

 * I just wrote up a potential "contents" section for the fundamentals chapter/unit. Here
 * Fundamentals of Common Practice Style
 * Notation
 * Staves
 * Clefs
 * Note-length values
 * Pitch
 * Overtone series
 * Harmonic/Melodic Intervals
 * Consonance/Dissonance
 * Tertian Harmony
 * Tonality
 * Keys
 * Intervals appearing in a Key
 * Modes
 * Diatonicism
 * Chords
 * Chord Construction
 * Diatonic Triads/Seventh Chords
 * Primary and Secondary Diatonic Chords
 * Chord Connection (Progressions)
 * Techniques for Writing and Analysis
 * Basic Four-Part Writing/Voice-Leading
 * Roman Numeral Harmonic Analysis
 * Figured Bass (and Realization of Figured Bass)
 * Chord Symbols --A.J. 18:40, 28 May 2007 (UTC)

 * I agree; there's to much overlap throughout this book. We need to have clear sections about each subject. Herr Beethoven (talk) 14:31, 25 July 2010 (UTC)

 * I think the content in "Notation" and "Pitch" is not exactly supposed to be in the right spot. I think the problem stems from how "Common Practice Styles" is before the "Basics" when I think "Basics" should be before "Common Practice Styles." Perhaps place the content found in "Notation" and "Pitch somewhere within [The Physics of Music] and/or [Introduction]?

Organization
This book is not organized. It starts off talking about pitch and rythym and harmony. It first needs to start with basic notes and rests and then scales then chords etc. I'll help out with the basics organizing it when I can. - Portlandiaman

 * yeah, its really disorganized and confusing. please fix, thx

Reggaeton is not western music, please create a section just like western music for latin music so we can put all music of latin origin thank you

Semitones are not in western music
The reference to semitones as intervals in western music is patently wrong and should be replaced with half tones. In western music there are whole tones and half tones based on frequencies of the diatonic scale. Semitones are in other music cultures such as in India etc..This is confusing and misleading information. 2601:204:C600:C65:55F1:596E:E718:CBA9 (discuss) 03:08, 15 September 2017 (UTC)

 * You are correct that semitones are the same as half steps. However, "semitones" is also used in Western Music, as well as half steps. Here in the United States, "half steps" is more commonly used than "semitones"; however, in Germany, for instance, "semitones" is more commonly used than "half steps." This is all dependent on context. Musical Inquisit (discuss • contribs) 08:50, 9 August 2019 (UTC)

Some rewrites
Hello to anyone still here, figured I should make myself known.

I've done some big rewrites and additions and whatnot, especially with stuff like tuning and temperament as well as some 20th century theories, expect more in the future. Still kinda new to MediaWiki stuff but I'm figuring it out. Obviously I would like to change the organization of all of the pages but I want to bother only with pages since actually editing them seems to be getting the least attention as of right now. Sorry if I terribly mess things up. TromboneBoi9 (discuss • contribs) 07:24, 29 June 2024 (UTC)

title: Talk:Music Theory


text:
User:Aurochs~enwikibooks

{insert stuff here}

title: User:Aurochs~enwikibooks


text:
Radiation Oncology/Head & Neck/Sinonasal/Nasal cavity

Epidemiology

 * M>F 2:1
 * 3% of upper respiratory tract cancers

Anatomy

 * For the purpose of staging, is divided into four subsites: septum, floor, lateral wall, and vestible.

 * Nasal vestibule is the anterior nasal cavity extending from the nostrils to the internal nasal valve.

 * Lateral wall of nasal cavity includes the superior, medial and inferior turbinates, the nasolacrimal duct, and the osteomeatus.

 * 5-12% incidence of nodal mets at diagnosis.

 * Draining lymphatics of nasal vestibule include Level I, facial and preauricular.

Staging: see Staging

Histology
Nasal Cavity
 * SCC 50%
 * Esthesioneuroblastoma 13%.
 * Melanoma 10-15%

Nasal Vestibule
 * SCC 80%

Treatment of Carcinoma of Nasal Vestibule

 * Tumors of the nasal vestibule are typically considered separate from nasal cavity tumors, because they are essentially skin cancers and as such have a different natural history
 * Definitive radiation therapy preferred over primary surgical approach for T1-2N0 SCC of nasal vestibule because of good local control of xrt and major cosmetic sequellae of surgery.

Definitive Radiation for Nasal Vestibule

 * The Netherlands, 2004 PMID 15275715 -- Langendijk JA et al. "Radiotherapy of squamous cell carcinoma of the nasal vestibule."  Int J Radiat Oncol Biol Phys 2004 Aug 1;59(5):1319-25.
 * 56 pts w/ Wang Classification T1 and T2 tumors tx'd w/ external beam xrt +/- endocavitary brachytherapy boost.
 * Local control rate at 2 yrs was 80%. 12% developed nodal metastases.

 * Gainesville, 1999 (1970-1995) PMID 10402517 -- Mendenhall WM et al. "Squamous cell carcinoma of the nasal vestibule."   Head Neck. 1999 Aug;21(5):385-93.
 * 56 pts tx'd w/ xrt alone and 4 w/ xrt followed by surg
 * 5yr local control 94%, T4 71%
 * 5yr CSS 94%, T4 86%

 * Denmark, 1984 PMID 6331090 -- Johansen LV et al. "Squamous cell carcinoma of the nasal vestibule. Treatment results."  Acta Radiol Oncol. 1984;23(2-3):189-92.
 * 66 pts w/ SCC of nasal vestibule, mostly tx'd w/ xrt alone.
 * Cure rate for Wang's T1 (81%) and Wang's T2 (52%). 12/22 local recuurence salvaged successfully w/ surgery.
 * Local recurrence w/ T3 was 88%

Prognostic Factors for Nasal Vestibule

 * Princess Margaret, 2007 (1979-2000) PMID 17298305 -- Jeannon J-P et al. "Prognostic indicators in carcinoma of the nasal vestibule."  Clin Otolaryngol. 2007 Feb;32(1):19-23.
 * Largest published series of SCC of nasal vestibule.
 * 84 pts w/ SCC of nasal vestibule w/ 77% managed w/ primary xrt
 * 5yr OS 58%, 5yr DFS 52%
 * Wang classification only prognostic indicator significant on multi-variate analysis.

 * MGH, 1976 PMID 820421 -- Wang CC. "Treatment of carcinoma of the nasal vestibule by irradiation."  Cancer. 1976 Jul;38(1):100-6.
 * 36 pts w/ SCC of nasal vestibule
 * 3yr NED rates were 83% for Wang's T1, 71% for Wang's T2, and 50% for Wang's T3
 * Pts w/ bone destruction best tx'd w/ primary surgical approach +/- xrt.

Treatment of Carcinoma of Nasal Cavity

 * Definitive radiation therapy for nasal cavity is preferred over surgery, and may include a brachytherapy boost.
 * SEER Database analysis, 2002 PMID 12220216 -- Bhattacharyya N. "Cancer of the nasal cavity: survival and factors influencing prognosis."  Arch Otolaryngol Head Neck Surg 2002 Sep;128(9):1079-83.
 * 981 pts, 97% non-metastatic, SCC 50%, esthesioneuroblastoma 13%, xrt delivered in 50% cases.
 * Overall 5 yr survival 57%
 * MD Anderson, 1992 (1969-1985) PMID 1410570 -- Ang KK. "Carcinomas of the nasal cavity."  Radiother Oncol. 1992 Jul;24(3):163-8.
 * 45 pts w/ carcinoma of nasal cavity tx'd curatively (18 xrt alone, 27 surg + xrt)
 * All pts w/ dz originating from septum controlled. 9/31 w/ dz from floor or lateral wall of nasal cavity died of dz
 * OS at 5 yrs 75%, at 10 yrs 60%
 * 4 pts had blindness (2 from xrt, 2 from exenteration).
 * Mallinkrodt, 1988 (1969-84) PMID 2846481 -- Hawkins RB. "Carcinoma of the nasal cavity--results of primary and adjuvant radiotherapy." Int J Radiat Oncol Biol Phys. 1988 Nov;15(5):1129-33.
 * 62 pts w/ CA of nasal cavity tx'd w/ either definitive xrt or surg + xrt. No elective nodal tx given.
 * 52% 5yr OS, 32% 10 yr OS
 * 49% tx'd w/ xrt alone controlled, 82% tx'd w/ surg + xrt controlled
 * 19% failed in nodes in pts who were initially N0.


text:
Climate Change/Mitigation Strategies/Fossil Fuel Economics

 * Considering Fossil Fuel's Full Cost: Looking at the long term impact of our present use of fossil fuel may help considerably in properly valuing it, and thereby encouraging the healthy development of sustainable energy sources.  If one simply notes that our posterity will not have the source to use in emergencies, it would proabably be a big step in that direction. Even our present ability to imagine can bring up significant and persisting concerns, given natures ways, humanity included.  That being done, one tends to include the cost of replacing it in some substantive sense so our posterity may address such concerns.  On the whole, fossil fuels are probably most validly used as an emergency material.


text:
Diagnostic Radiology/Musculoskeletal Imaging/Infection/Necrotizing fasciitis

Necrotizing fasciitis is a rare, often fulminant, rapidly progressive infectious process primarily involving the fascia and subcutaneous tissue. The pathophysiology involves the rapid spread of infection along the fascial planes, fascial necrosis and thrombosis of the subcutaneous blood vessels, leading to cutaneous gangrene.

Approximately 500 to 1500 cases reported annually in the US. The most often associated comorbities are diabetes mellitus and peripheral vascular disease. It is most often peripheral, involving the lower limb.

Delay to diagnosis is one of the most prominent predictors of mortality (along with diabetes). Prognosis for necrotizing fasciitis depends heavily on early recongnition and determination of the extent of necrosis in the preoperative planning.

Clinical studies are lacking that compare different imaging modalities to the gold standard of fasciotomy. These series are typically small and only look at a handful of cases.

Clinical Findings
Patients usually present with the triad of exquisite pain (often out of proportion to physical findings), swelling and fever. Tenderness, erythema, and warm skin are commonly the only signs of early disease. Distinction between uncomplicated infectious cellulitis or fasciitis, and necrotizing fasciitis is important. The former entities are treated with high-dose antibiotics without the need for surgical debridement. Necrotizing fasciitis is a surgical emergency.

Classification schemes describe the syndromes associated with necrotizing fasciitis and have overlapping features

 * Type I: mixed infection of anaerobes plus facultative species such as streptococci or Enterobacteriaceae. Acute, rapidly developing infection of the deep fascia, marked pain, tenderness, swelling, and often crepitus.
 * Type II: infection with group A streptococci. Acute infection, often accompanied by toxic shock syndrome, rapid progression of marked edema to violaceous bullae and necrosis of subcutaneous tissue, absence of crepitus.

Plain Films
Plain films: Gas on plain films is thought to be an inconsistent sign, seen is well less than half of cases.

Fig. 1 56 year old man with acute leukemia and necrotizing fasciitis.

Fig. 2 51 year old woman with history of IV drug abuse, now presenting with necrotizing fasciitis.

Image above looks like postmortem of neurofibromatosis not necrotizing fasciitis. Might want to check your data saved under "NF"

MRI
MRI: In identifying necrotizing fasciitis on MRI, findings include involvement of the deep fascia and demonstration of fascial necrosis.
 * 1) Fascial inflammation or fluid accumulation is demonstrated by low intensity on T1 and high intensity on T2.
 * 2) Fascial necrosis is evidenced by the lack of contrast enhancement of the involved area upon the administration of gadolinium.

MRI also plays a role in determining the extent of fascial involvement to aid in the preoperative assessment. However, Arlan et al. point out that the findings in MRI can be nonspecific and most preoperative decisions should be based on clinical status.

Fig. 3 T1W and T2W MR of the lower extremity showing fascial thickening and fluid accumulation between the subcutaneous tissues and fascial layer (respectively) in this patient with necrotizing fasciitis.

CT
Contrast enhanced CT can help define the extent of disease and possible complications, especially in sites that are difficult to study. However, large clinical studies correlating CT findings with pathologic findings are lacking.

Ultrasound
Some believe there is limited use for ultrasound in the diagnosis of necrotizing fasciitis. Its main limitation being when the anatomic sites are difficult to ascertain (too deep) by US. However, in their series, Yen et al. found ultrasound to have a sensitivity of 88% and a specificity of 93% (positive predictive value of 83%). Their criteria included diffuse thickening of the subcutaneous tissue accompanied by a fluid accumulation more than 4 mm in depth along the fascial layer, when compared to the normal limb.

Summary
Although one can see findings of necrotizing fasciitis on practically any imaging modality, none of these findings are absolutely specific. The optimal diagnostic approach consists of pooling all of the sometimes contradictory clinical and imaging data and making the best estimate from that. When in doubt, one should approach these challenging patients with a high index of suspicion and err on the side of patient safety. In other words, if there is sufficient likelihood that a patient has necrotizing fasciitis (your mileage may vary), surgical intervention should proceed as soon as possible.

Treatment
Fasciotomy: Obviously the gold standard. Timely intervention in those cases where clinical suspicion for necrotizing fasciitis is high is imperative. Fasciotomy allows both for confirmation of the diagnosis with histology, but also rapid iniation of treatment. Typical findings include gray discoloration of the fascia and resistance to blunt dissection.


text:
Diagnostic Radiology/Musculoskeletal Imaging/Tumors Basic/Fibrous dyplasia

Fibrous dysplasia is a benign, developmental anomaly of the bone-forming mesenchyme in which the medullary cavity is replaced with fibrous material, woven bone, and spindle cells. Virtually any bone in the body can be affected. It is a nonhereditary disorder of unknown cause. Fibrous dysplasia does not spread or proliferate, and malignant transformation is rare (0.5%).

Pathophysiology
Fibrous dysplasia is caused by abnormal proliferation and maturation of fibroblasts; it has been likened to a localized developmental arrest because all the pregenitors of normal bone are present, but they do not differentiate into their mature structures. Benign fibrous tissue with abnormally arranged, dysplastic trabeculae of immature woven bone replaces normal bone. The lesions are medullary but may replace both cancellous and cortical bone.

Epidemiology

 * Age:The peak age of diagnosis is 5 to 20 years
 * Two thirds of patients with polyostostic disease are asymptomatic before the age of 10
 * With monostotic disease, patients as old as 20 – 30 years are asymptomatic
 * Race: No specific racial predilection
 * Sex: The incidence is equal in males and females.

Clinical Findings
The followling four disease patterns are recognized:
 * 1) Monostotic form
 * 2) Polyostostic form
 * 3) Craniofacial form
 * 4) Cherubism

Monostotic form
Approximately 80% of cases of fibrous dysplasia are monostotic. The typical sites of involvement are the proximal femur, tibia, ribs, and facial bones. This form most frequently occurs in those aged 10–30 years. The degree of bone deformity is relatively less severe compared with that of the polyostotic form.

Polyostotic form
About 20% of cases of fibrous dysplasias are polyostotic. In polyostotic fibrous dysplasia, the distribution of lesions is unilateral in up to 90% of cases; monomelic and widespread distribution are less frequent. Most bony lesions occur in the lower extremities, including the pelvis, legs, and feet. Rib, spine, or skull lesions are also often seen. Common deformities include leg-length discrepancy (lesions are in the longer leg) and bowing of the weight-bearing bones due to the weakened structural integrity of the bone. This bowing phenomenon can lead to a varus angulation of the proximal femur, the shephard's crook deformity.

The polyosttotic form is associated with McCune-Albright syndrome - sexual precocity in girls, polyostotic fibrous dysplasia, and cafe-au-lait spots with serpiginous borders ("rocky coast of Maine"). Up to 30% of women with polyostotic fibrous dysplasia may have the full spectrum of the McCune-Albright syndrome.

The clinical course ranges from asymptomatic to multiple pathologic fractures, rapidly progressive deformities, severe pain, and debilitation. Malignant transformation has been documented but is rare.

Craniofacial form
This pattern of the disease occurs in 10-25% of patients with the monostotic form and in 50% with the polyostotic form. It also occurs in an isolated craniofacial form. Sites of involvement most commonly include the frontal, sphenoid, maxillary, and ethmoidal bones. The clinical manifestations include frontal bossing, facial asymmetry, inferolateral orbital displacement, and blindness.

Cherubism
This special variant of fibrous dysplasia is an autosomal-dominant disorder of variable penetrance. It occurs in children and is more severe in boys. Regression may occur after adolescence. The jaw is broad and protruding. Involvement of maxilla and mandible is symmetric.

Other features
Fibrous dysplasia may be associated with endocrinopathies in 2-3% of cases - hyperthyoridism, hyperparathyroidism, and precocious puberty. The polyostotic fibrous dysplasia can be associated with intramuscular myxomas (usually adjacent to the affected bone). This association is known as Mazabraud syndrome.

Radiologic Findings
Plain radiography is the first line study. CT may be required to assess complex regions such as the spine, pelvis, chest, and facial skeleton. Bone scintigraphy has a limited role in the detection of subtle pathologic fractures. MRI may be necessary to assess cord compression when the spine is involved.

 * Radiolucent expansile medullary lesions. The lesions are medullary but may replace both cancellous and cortical bone.
 * The abnormal area may be sharply circumscribed and marginated by a thick layer of reactive bone, or it may blend gradually with the adjacent normal bone.
 * Degree of lucency depends on the amount of osteoid
 * "Ground-glass" appearance is due to dysplastic microtrabeculae, which are no thicker than 0.1 mm.
 * Bowing deformities: biomechanically insufficient bone - shepherd's crook deformity of the femur (coxa vara).
 * Base-of-skull lesions tend to be sclerotic (in contrast to lucent lesions elsewhere)

Fig. 1 Fibrous dyplasia of the skull.

Fig. 2 Fibrous dyplasia is the most common etiology of benign rib lesions (left first rib).

Fig. 3 Fibrous dyplasia of the left iliac, radiograph and CT.

Fig. 4 Fibrous dysplasia of the right femur with "shepherd's crook" (coxa vara) deformity.

Fig. 5 Fibrous dysplasia of the tibia ("long lesion in a long bone").

Fig. 6 Fibrous dysplasia of the humerus.

Treatment
Therapy is often restricted to the orthopedic management of complications.


text:
Diagnostic Radiology/Musculoskeletal Imaging/Joint Disorders/Scleroderma

Radiologic Images
Radiographic findings in the extremities include acroosteolysis, soft tissue calcifications (calcinosis), and atrophy of the distal finger tips. Compared with hand involvement in scleroderma, foot involvement has a later onset and is relatively less frequent but can be disabling. Soft tissue changes included flexion deformities, generalized or localized atrophy, and dystrophic calcifications. In a study by Bassett et al., resorption of distal phalanges was the most common bony change in patient's with scleroderma, though osteolysis in other sites (feet, ribs, and mandibles) was also frequent. Nearly 20% showed radiographic evidence of inflammatory arthritis and joint destruction that could not be attributed to overlap with rheumatoid arthritis or mixed connective tissue disease.

Radiographic findings of other organ systems include fibrosis in the lungs and abnormal esophogram related to dysmotility.

Differential Diagnosis
The differential diagnosis for acroosteolysis includes diseases that affect peripheral nerve sensation such as diabetes, leprosy, syrinx, myelomeningocele and congenital insensitivity to pain; injury such as frostbite, thermal or electrical burns, or snake bite; trauma; psoriasis; congenital abnormality such as with Hadju-Cheney; metabolic causes such as hyperparathyroidism. Soft tissue calcifications in the extremities can also be seen with dermatomyositis, heterotopic ossification, and metastatic calcification (hyperparathyroidism, chronic renal disease).

Treatment
No treatment for acroosteolysis. The treatment for scleroderma varies depending on organ involvement and is beyond the scope of this discussion, but includes anti-inflammatory medication, both non steroidal and steroids.

Prognosis
Depends on the type and degree of systemic sclerosis. There is no cure for scleroderma and most people die from involvement of the heart, lungs and/or kidney. Involvement of visceral organs is not uncommon such as GI tract (esphageal and small bowel dismotility).


text:
Wikibooks:Title pages

As each book has a front cover, so to, does a Wikibook have a title page. You can create a title page separate from table of contents for your book to provide a short description of scope of your book.

Abstract
A book's cover should describe the purpose of the book, and thus, there should be a short paragraph explaining what the book is trying to teach, and the topics that it may cover. In short, the title page should summarize the entire book to the best of your ability. You can also a cover image for decorative purposes.

Example

 * How To Build A Computer/Cover

Naming conventions
As per naming conventions, if your book is named Book, then title page should be named Book/Cover and book table of contents should be named just Book.

What should go on the title page?
As a rule, link to the table of contents should be placed on your title page. Title pages should also link to a book's specific manual of style. There should be also a link to list containing the names of all the users actively working on the project - it can be placed in manual of style, foreword or separate "Authors" page.

It's good to point a central discussion page for the whole book so that contributors have one place for discussing book-related issues.

If a print version or PDF version of your book is available, you can announce it using print version and PDF version templates or just add a normal link to them.

Cover images should be kept to a minimum, and should not dominate your title page. This is as if your book is nominated for book of the month or collaboration of the month, the cover image is used to represent your book on the main page.

If wikibook is listed in categories and has interwiki links, all of them should be placed at cover page, not table of contents.

Cover page and table of contents
If a book has a cover page, its table of contents should not contain any interwiki links or categories. You should provide link to the cover at the top of book TOC page like:

<< Cover and interwiki

How to link to title page
If you use only default backlinks for pages using slash-convention, all chapters of your book will automatically link to table of contents, not cover page. If you use custom navigation templates, you should also not link to cover page.

External links, like links from bookshelves, should point to the book cover, not table of contents.

Cataloging
The main page of your book is where the important information regarding how the book is in relation to other books, or Wikibooks in general, resides. If your book needs to be transwikied or is put up for deletion, then that information will be on the book's title page. Our category system will also only categorize title pages of the books.

Like a library, each Wikibook must be categorized. Currently, the card catalog office organizes books by their bookshelf, by the Dewey Decimal Classification system, as well as the Library of Congress system. If you are unaware of how your book is to be categorized, please place a link to your book at the CCO's main page and other Wikibookians will help you categorize your book.

Title page and print version
If you are preparing a print version of your book, include the title page in it.

Some things from title page should not be included in print version - like interwiki, categories and some templates (for example, print version). Hence, you should put them inside ... tag. You are also encouraged to add some information visible only in print version (using ... tag), similar to:

title: Wikibooks:Title pages


text:
User:Monedula

w:en:User:Monedula

w:ru:User:Monedula

title: User:Monedula


text:
User:Wikinaut

 * please proceed to my user page on Wikipedia, where you can e-mail me.

title: User:Wikinaut


text:
Cellular Automata/Neighborhood

1D neighborhood
Since in 1D there are no shapes, the definition of the neighborhood is usually very simple.

Radial neighborhood

Usually the neighborhood in 1D is described by its radius $$r$$, meaning the number of cell left and right from the central cell that are used for the neighborhood. The output cell is positioned at the center.

 * Formal definition

Formally the radial neighborhood is the set of neighbors
 * $$ N = \{-r,-(r-1),\dots,-1,0,1,\dots,r-1,r\}$$

or simply the neighborhood size $$k=2r+1$$ with the output cell at the center $$k_0=r$$.

 * Symmetries
 * reflection symmetry

Stephen Wolfram's notation
In Wolframs's texts and many others the number of available cell states $$|S|$$ and the radius $$r$$ are combined into a pair
 * $$ (|S|,r) \,$$

 * See also
 * Stephen Wolfram, Statistical Mechanics of Cellular Automata (1983)

Brickwall neighborhood

An unaligned neighborhood, usually the smallest possible $$k=2$$. The output cell is positioned at $$k_0=0.5$$ between the two cells of the neighborhood. It is usually processed by alternatively shifting the output cell between $$k_0=0$$ and $$k_0=1$$.

von Neumann neighborhood

It is the smallest symmetric 2D aligned neighborhood usually described by directions on the compass $$N=\{N,W,C,E,S\}$$ sometimes the central cell is omitted.

 * Formal definition

Formally the von Neumann neighborhood is the set of neighbors
 * $$ N = \lbrace \{0,-1\}, \{-1,0\}, \{0,0\}, \{+1,0\}, \{0,+1\} \rbrace $$

or a subset of the rectangular neighborhood size $$k_x=k_y=3$$ with the output cell at the center $$k_0x=k_0y=1$$.

 * Symmetries
 * reflection symmetry
 * rotation symmetry 4-fold

 * See also
 * [mathworld] - [von Neumann Neighborhood]

Moore neighborhood

Is a simple square (usually 3×3 cells) with the output cell in the center. Usually cells in the neighborhood are described by directions on the compass $$N=\{NW,N,NE,W,C,E,SW,S,SE\}$$ sometimes the central cell is omitted.

 * Formal definition

Formally the Moore neighborhood is the set of neighbors
 * $$ N = \lbrace

\{-1,-1\}, \{0,-1\}, \{1,-1\}, \{-1,0\}, \{0,0\},  \{+1,0\}, \{-1,+1\}, \{0,+1\},  \{1,+1\} \rbrace $$ or simply a square size $$k_x=k_y=3$$ with the output cell at the center $$k_0x=k_0y=1$$.

 * Symmetries
 * reflection symmetry
 * rotation symmetry 4-fold

 * See also
 * [mathworld] - [Moore Neighborhood]

Margolus neighborhood
reversible

see also

Unaligned rectangular neighborhood

An unaligned (brickwall) rectangular neighborhood, usually the smallest possible $$k_x=k_y=2$$. The output cell is positioned at $$k_{0x}=k_{0y}=0.5$$ between the four cells of the neighborhood. It is usually processed by alternatively shifting the output cell to $$k_{0x}=k_{0y}=0$$ and $$k_{0x}=k_{0y}=1$$.

Hexagonal neighborhood

 * Symmetries
 * reflection symmetry
 * rotation symmetry 6-fold

Small unaligned hexagonal neighborhood

Formally the small (3-cell) unaligned hexagonal neighborhood represented on a rectangular lattice is the set of neighbors
 * Formal definition
 * $$ N = \lbrace \{0,0\}, \{1,0\}, \{0,1\} \rbrace $$

 * Symmetries
 * reflection symmetry
 * rotation symmetry 3-fold

title: Cellular Automata/Neighborhood


text: Template:Basicnutritionsummary

title: Template:Basicnutritionsummary


text:
Cellular Automata/Information Dynamics

 * Information speed

 * $$ v = \frac{ \Delta x }{ \Delta t } $$

It is defined as the number

 * Speed of light

Is the maximum speed information can travel inside CA.It depends on the size and shape of the neighborhood and it may depend on the direction.

For asymmetric 1D CA (left and right direction)

 * $$ c_L = k_0 $$ $$ c_R = k-k_0-1 $$

For symmetric 1D CA (radious $$r$$)

 * $$ c = r $$

title: Cellular Automata/Information Dynamics


text:
Latin/Chapter 3 Verse

The following poem is written in Hendecasyllabic. It is an introductory, dedication poem written by the poet Gaius Valerius Catullus.

It is commonly referred to as "Catullus 1" or by its first line.
 * Cui dono lepidum novum libellum
 * arida modo pumice expolitum?
 * Corneli, tibi: namque tu solebas
 * meas esse aliquid putare nugas.
 * Iam tum, cum ausus es unus Italorum
 * omne aevum tribus explicare cartis...
 * Doctis, Iuppiter, et laboriosis!
 * Quare habe tibi quidquid hoc libelli—
 * qualecumque, quod, o patrona virgo,
 * plus uno maneat perenne saeclo!

title: Latin/Chapter 3 Verse


text:
DataPerfect

This book is about the DataPerfect product as described at http://dataperfect.nl/CMS/index.php

Subpages

 * /Functions/
 * /User Formulas/
 * /User Web Enabling/

title: DataPerfect


text:
Category:Wikibooks policies

This category contains accepted Wikibooks policies. Policies are designated by placing policy at the top of the policy page, which adds the page automatically to this category.

title: Category:Wikibooks policies


text:
Talk:Topology/The fundamental group

I do not understand the sentence "This section is dedicated to the calculation of the fundamental group of S1 that we can consider contained in the complex topological space". Is "complex topological space".the space of complex numbers? Why not use RxR? Would the topology be any different?

Is it true that the specialty of S1 (compared with S2 or higher dimensions) is that there is not enogh "room" for a "film" between loops with different numbers of turns?

Artie01: Hi! I'm not the author of the above sentence, but I think I can explain it a bit. By the "complex topological space" I presume he/she means the complex numbers C, equipped with its standard topology. (Remember that stricly speaking, you can't just specify the underlying set --- you have to specify the topology on that set too.) As you suggest, there's no practical difference between C and RxR as topological spaces (when both have their usual topologies). I'd guess that the author discusses things in terms of C since the proof of the main theorem --- \pi_1 S^1 is isomorphic to Z --- views S^1 as the unit circle in C.

As for your second question: there are lots of intuitive ways to view the difference between the fundamental group of S1 and those of all other spheres. I like to think about a loop of stretchy string that's constrained to lie on the surface of S^n. For n > 1, it's clear that, no matter how crazy the loop is, we can always contract it down to a single point. But on S^1, no matter how we twist or stretch the string we can never unwind it.

Hope this helps...

Missing check of "independence of representant"
When we define the fundamental group (or actually any group really) using a quotient, we also need to show the operation does not depend on the choice of representant, i.e. is well-defined. I currently lack the time to edit the article, so if someone want, feel free and go ahead. I'll put it into my todo list but no guarantees 94.112.136.34 (discuss) 14:50, 15 June 2014 (UTC)

title: Talk:Topology/The fundamental group


text:
User talk:Hayford Peirce

rhubarb pie pic
I actually would have preferred the image above the box, but the html & css there was being uncooperative. Did you take that pic, btw? Kellen T 17:48, 9 August 2005 (UTC)

title: User talk:Hayford Peirce


text:
Category:Pineapple recipes

Recipes involving pineapple.

title: Category:Pineapple recipes


text:
Category:Cilantro recipes

Recipes involving cilantro.

title: Category:Cilantro recipes


text:
User talk:Kellen/Categories

Categories
Do you intend to scope them all as per WB:NP (i.e. Category:Cookbook:Recipes)?

Also, I'm fairly certain an olive is a fruit. Generally a fruit is the part of the plant that contains the seed, although olives are generally pitted first. - Aya T C 19:34, 9 August 2005 (UTC)

title: User talk:Kellen/Categories


text:
Blender 3D Reference Guide/Windows Reference/Outliner

Outliner
The Outliner is a hierarchical diagram displaying a list of data in Blender and its dependencies. The SHIFT+F4 'databrowse' doesn't really show it, and 'Oops Schematic' is too chaotic still. And most of all, the former two don't offer much functionality.


text:
User:Iztok.jeras

My wikipedia page iztok.jeras

Homepage: [www.rattus.info]

Studies:
 * 1) Faculty of Electrical Engineering
 * 2) Faculty of Computer and Information Science

My books:
 * Cellular Automata
 * UML with Umbrello

title: User:Iztok.jeras


text:
Jet Propulsion/Thermal efficiency

For heat engines the thermal efficiency is the mechanical work output for each unit of heat input. For the Carnot cycle it is

$$\eta_c = 1 - \frac {T_0}{T_{max}}$$

where

$$T_0 = $$ ambient absolute temperature

$$T_{max} = $$ maximum cycle absolute temperature

Actual engines using other cycles will have lower efficiencies than the Carnot cycle since they have irreversible processes such as heat flow across finite temperature differences and non-adiabatic compression and expansions, as well as friction.

Heat engines use as high a temperature as possible to maximize the efficiency and are limited by temperature capabilities of real materials used to make the engines.

title: Jet Propulsion/Thermal efficiency


text:
Jet Propulsion/Propulsive efficiency

The propulsive efficiency is defined as:

$$\eta_p = \frac{PropulsiveWorkDeliveredToVehicle} {MechanicalWorkGeneratedByEngine}$$

The mechanical work generated by the engine is crudely split between the work difference between the input and exit mass flows from the engine as well as to the work imparted to the vehicle. For a jet engine in the air frame of reference the input mass flow has zero kinetic energy. Thus for a jet engine with a constant relative exit velocity (a close approximation of a turbojet) the propulsive efficiency increases with aircraft velocity.

Mathematically the propulsive efficiency :

$$\eta_p = \frac{2 u_0}{u_e + u_0}$$

where

$$u_0 =$$ velocity of the vehicle

$$u_e =$$ velocity of the exhaust

Thus reducing the exhaust velocity will give a higher propulsive efficiency. The thrust of the engine is given by

$$F = \dot{m} ( u_e - u_0 )$$

where $$ \dot{m} $$ is the mass flow through the engine. Thus to for a given thrust the exhaust velocity can be decreased if the mass flow is increased. This is usually done by adding a fan or a propeller to the turbojet, resulting in a turbofan or turboprop engine. The reduction in exhaust velocity also reduces the exhaust noise, but has the disadvantage that the thrust reduces more markedly with speed than for a turbojet.

title: Jet Propulsion/Propulsive efficiency


text:
Jet Propulsion/Specific impulse

Specific impulse is defined as the thrust (N) divided by the fuel weight flow rate (N/s). The resulting measure is usually quoted in seconds and defines the weight fraction that is necessary to give a particular delta-V for a rocket or range for an aircraft with a given lift to drag ratio.

For a jet engine, the specific impulse can be determined from the specific fuel consumption. For example, an engine that has an SFC of 15mg/N-s will have a specific impulse of 6800s. Current turbofan engines have cruise SFC below 15.

For a chemical rocket, the specific impulse ranges to about 500s.

title: Jet Propulsion/Specific impulse


text:
Category:Copyright violations

Works in this category are believed to have been copied or uploaded in violation of their copyright or licensing terms. Textual contributions must be compatible with the GFDL and media files must follow the media policy. Should 1 week pass without clarification of their status, these works may be deleted.

To add pages or media files to this category, place at the top of the page or file description page.

You may ask the copyright holder(s) for permission to use their work under a compatible license.

it:Categoria:Sospette violazioni di copyright

title: Category:Copyright violations


text:
Jet Propulsion/Thermodynamic Cycles

Gas turbines are based on the Brayton cycle.

All jet engines and gas turbines are heat engines that convert thermal energy into useful work. The useful work may be in the form of mechanical power, as from a shaft which may be used to drive a propeller, a vehicle, a pump, an electric generator, or any other mechanical device. In Jet engine applications the work is in producing compressed air and combustion products which are then accelerated to provide reaction propulsion.

Brayton Cycle
The Brayton cycle is the fundamental constant pressure gas heating cycle used by gas turbines. It consists of

0-2: isentropic compression

2-3: constant pressure heating

3-4: isentropic expansion

4-0: constant pressure cooling (absent in open cycle gas turbines)

Ramjet cycle
A ramjet uses the open Brayton cycle. In the diagram below a 2D supersonic intake is shown, downstream of which is a divergent subsonic diffuser. Fuel is then injected into the compressed air and evaporates producing a mixture that is ignited when it reaches the flame front. The flameholders provide the turbulent circulation necessary to stabilize the flame, since deflagration velocities are usually much smaller (<10m/s) than the average velocity of air in the combustor. The combustion products are then exhausted through the nozzle.

To understand how thrust is produced if we assume that the flow of fuel is negligible compared to the air mass then the exhaust flow will be at approximately the same Mach number as the input flow. However the total temperature of the exhaust is much higher and the exit velocity will be correspondingly higher than the input velocity. This difference in velocity (and momentum) produces thrust.

The temperature rise in the intake-diffuser is related to the freestream Mach number $$M_0$$:

 * $$\frac{T_2}{T_0}= 1 + \frac {\gamma-1}{2} M_0^2 $$

Maximum efficiency is reached if temperature rise in combustor is small.

$$\eta = \frac {[(\gamma-1)/2] M_0^2}{1+ [(\gamma-1)/2] M_0^2}$$

 * where $$\gamma = c_p/c_v \approx 1.4$$ is the ratio of specific heats of air.

Ramjets are inefficient at subsonic speeds and their efficiency improves at supersonic speeds.

At hypersonic speeds the compression and dissociation processes make full diffusion unattractive and supersonic combustion is being researched. A Scramjet slows the air down to low supersonic speeds and then burns high flame velocity fuels such as hydrogen or methane.

Turbojet cycle
Adding a compressor to a ramjet powered by a turbine in the exhaust allows increased combustor inlet temperature, and a consequent increase in possible thermal efficiency. The turbine however is limited in the temperature it can handle, so maximum power is also limited.

In the T-S diagram below the presence of the compressor allows us to raise the combustor inlet temperature (3). The raising of the combustor segment increases the cycle area and the thermal efficiency.

Addition of an afterburner (5-6) allows thrust augmentation as can be seen from the increased area of the diagram shown below. The afterburner operates in the higher entropy range and has lower efficiency than the base turbojet.

Turbofan cycle
A turbofan diverts some of the pressure energy of the core flow to power a fan which moves a larger mass flow, providing an increase in thrust and propulsive efficiency.

Turbofans normally have two or three shafts. Since the diameter of the fan is larger the same tip speed can be achieved at a lower rpm than the smaller diameter compressor and two shafts become necessary. The alternate method is to employ a gearbox to step down the shaft speed which is used in some smaller turbofans. In most turbofans however a multistage LP turbine is used to extract the same energy with smaller stage loadings and lower tangential velocity. The smaller diameter HP compressor is run with one or two turbine stages with higher tangential velocity than the LP turbine.

Cycle Losses
The compression incurs several loss mechanisms:

 * Shock:

 * Separation

 * Tip clearance

 * Seal clearance

 * Blockage

 * Whirling

title: Jet Propulsion/Thermodynamic Cycles


text:
Cookbook:Oatmeal Cookies II

| Vegan Cuisine | Dessert

This recipe originates with the New York chapter of Food Not Bombs. These delicious cookies are very simple to make, and they require no unusual ingredients.

Ingredients

 * ¾ cup (160 g) vegetable oil
 * 1½ cup (300 g) brown sugar
 * ⅓ cup (80 ml) water
 * 1 tsp vanilla extract
 * 3 cups (275 g) rolled oats
 * 1 cup (145 g) white flour
 * 1 tsp salt
 * ½ tsp baking soda

Procedure

 * 1) Put the oil, sugar, water, and vanilla together in a bowl. Mix well.
 * 2) Add the oats, flour, salt, and baking soda.
 * 3) Place on a greased baking sheet in lumps about 1 inch across and about 3 inches apart.
 * 4) Bake at 191 °C (375 °F) for 10–12 minutes, and serve when the cookies have begun to brown.

Notes, tips, and variations

 * You can add additions such as ½ cup nuts, dried fruits and/or chocolate chips. Be aware that mix will be looser with additional dry ingredients.
 * Leftover dough can be spread in a thin layer on a cookie sheet and baked, turning every 4–5 minutes, to make a tasty and crunchy granola.
 * This recipe can very easily be cut in half or doubled.

title: Cookbook:Oatmeal Cookies II


text:
Game Maker Programming/Installing

Version 6.1
Older versions of Game Maker (6.1A and below) can be downloaded from the old Game Maker Pages. It is recommended that you download the zipped version as some tutorials on this site will assume you have the sprites included in it.

Once you have downloaded Game Maker open the zipped folder and double click on gmaker.exe. This will bring up the setup. Click through it, being careful to read the Licensing Agreement. That's it! You've Installed Game Maker in your computer!

Version 7
Version 7 is the latest version of Game Maker and can be found at the YoYo Games Download Page. It features several improvements over older versions including enhanced security and a new extension mechanism. As of version 7.0 Game Maker is managed by YoYo Games.

CTW Version (5.3)
The Children's Technology Workshop Version of Game Maker is a non-violent version made for the CTW camps. Though still powerful it is a few versions behind the standard edition, is only available as a 15 day trial and lacks any 3D functionality. Most Tutorials should be compatible with the CTW version and sections that are not backward compatible will be clearly marked. The CTW version of Game Maker can be downloaded from Children's Technology Workshop.

title: Game Maker Programming/Installing