Solution of the homeworks

Oneliners

One-liner First Place 2011: Animated Sky Chart (Xuan Tran Thi)

Link: http://www.wolfram.com/events/technology-conference/2011/one-liners.html

g={#,WolframAlpha["sky chart "<>#,{{"SkyMap:AstronomicalData",1},"Content"}]}&/@(DateString[DatePlus[Date[],#]]&/@Range[10]);
ListAnimate[g]

1/ Map the feedback for “sky chart” querry to WolframAlpha. 
2/ Display next ten dates and the sky chart animation on the same row.   
3/ ListAnimate function is used to animate a sequence of images g

DateString[DatePlus[Date[],#]]&/@Range[10]

WolframAlpha["sky chart "<>"Mon 17 Nov 2014 08:20:28",{{"SkyMap:AstronomicalData",1},"Content"}]

It takes time but it is worth waiting!

We are looking for the number of possibilities to represent the number 50 as a sum of squares disregrading signs and ordering

Divisors[50]

Divisors[50,GaussianIntegers->True]

Select[Divisors[50,GaussianIntegers->True],((Abs[#])^2)==50&]

Map[{Re[#],Im[#]}&,Select[Divisors[50,GaussianIntegers->True],((Abs[#])^2)==50&]]

Map[Sort[{Re[#],Im[#]}]&,Select[Divisors[50,GaussianIntegers->True],((Abs[#])^2)==50&]]

Union[Map[Sort[{Re[#],Im[#]}]&,Select[Divisors[50,GaussianIntegers->True],((Abs[#])^2)==50&]]]

squareRepresentation[n_Integer]:=Union[Map[Sort[{Re[#],Im[#]}]&,Select[Divisors[n,GaussianIntegers->True],((Abs[#])^2)==n&]]]

squareRepresentation/@{50,50.,π}

Ulam Spiral (Mezei Tamás)

The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes.It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a "long and very boring paper"[1] at a scientific meeting.

Ulam constructed the spiral by writing down a regular rectangular grid of numbers, starting with 1 at the center, and spiraling out:
Numbers from 1 to 49 placed in spiral order

He then circled all of the prime numbers and he got the following picture:
Small Ulam spiral

Remarkably, in a passage from his 1956 novel The City and the Stars, author Arthur C. Clarke describes the prime spiral seven years before it was discovered by Ulam. Apparently, Clarke did not notice the pattern revealed by the Prime Spiral because he “never actually performed this thought experiment.”

s={Re@#,Im@#}&/@Fold[Join[#1,Last@#1+I^#2 Range@#2/2]&,{0},Range@140];ListPlot[s,Joined->True,Epilog->{Point/@s[[Prime@Range@PrimePi@Length@s]]}]

Fold[Join[#1,Last@#1+I^#2 Range@#2/2]&,{0},Range@4]

t={Re@#,Im@#}&/@%

Length@t

PrimePi@%

Range@%

Prime@%

t[[%]]

Graphics@{Point/@t[[Prime@Range@PrimePi@Length@t]]}

Carsten Olm

I browsed through some of the one - liner competitions and found an awesome one - liner (1 st price in 2011, written by Zdeněk Buk) :

Hyperlink["http://blog.wolfram.com/data/uploads/2011/11/OneLinerWinners2011.nb"]

ImageAssemble[Round[Rescale[ImageData[i=Nest[Darker,ImageResize[ExampleData[{"TestImage","Elaine"}],50],3]]]9]/.n_Integer:>Nest[Lighter,i,n]]

The example is based on one of Mathematica' s example images "Elaine" :

ExampleData[{"TestImage","Elaine"}]

that is first resized to a smaller size (width 50 pixel) :

string=ImageResize[ExampleData[{"TestImage","Elaine"}],50]

The image is made darker three times:

string2=Nest[Darker,string,3]

If you modify this number if repetitions in the one-liner e.g. to 1, you see that the final image will become “too light”, vice-versa. Hence, this step was included as the averaged “light content” of the original image is more than most viewers would consider well-balanced.

The next step is to convert the resized image into a numbers representing all the pixels and rescale these numbers from 0 to 1 (the minimal value being 0, the maximal value being 1):

string3=Rescale[ImageData[string2]]

Now, the rescaled values are multiplied by 9 and rounded to the nearest integer (again, 9 is an empirically found value that adjusts the hue of the image):

Round[string3 9]

Here comes the “magic”: what we ignored so far is the index i in the ImageData function.

This snippet is left:     /.n_Integer:>Nest[Lighter,i,n]
i.e. a delayed rule assignment for n (what does “_Integer” mean? That n is always an integer?). Again, Nest is used and Lighter is applied to i for n times., i.e. it’s some kind of recursion.

This somehow does the job, although I don’t know what “_Integer” and “/.” (is that some kind of execution command? I didn’t find anything about that symbol) stand for.

f[x_Integer]:=(x^2)

f/@{1,2,5,3.,E}

/.  is the shorthand for ReplaceAll, this can be found if you press F1.

Nevertheless, it's a nice tool and I will play around with it in future a bit more. Here are two tests with photos  taken from my social network profile:

SetDirectory[NotebookDirectory[]];
testcol=Import["Olm07_test_color.jpg","Image"];
testbw=Import["Olm07_test_bw.jpg","Image"];

ImageAssemble[Round[Rescale[ImageData[i=Nest[Darker,ImageResize[testcol,50],3]]]9]/.n_Integer:>Nest[Lighter,i,n]]

Unfortunately, this doesn’t work. But why? Of course, because it’s not a black-and-white image just like the one of Elaine. So let’s try another example.

ImageAssemble[Round[Rescale[ImageData[i=Nest[Darker,ImageResize[testbw,50],3]]]9]/.n_Integer:>Nest[Lighter,i,n]]

Still not. That’s annoying! Let us try to find the differences between the images. You can show the images by removing the semicolons above, right-click and click on “Get image information”. Here we go! The main difference is:

ImageColorSpace[ExampleData[{"TestImage","Elaine"}]]
ImageColorSpace[testbw]
ImageColorSpace[testcol]

This means that I was lazy when saving testbw. I kept the color space “RGB” instead of converting it into “Grayscale”. Let’s do this now:

testbw=ColorConvert[testbw,"Grayscale"];
testcol=ColorConvert[testcol,"Grayscale"];
ImageColorSpace[testbw]
ImageColorSpace[testcol]

ImageAssemble[Round[Rescale[ImageData[i=Nest[Darker,ImageResize[testcol,80],3]]]9]/.n_Integer:>Nest[Lighter,i,n]]
ImageAssemble[Round[Rescale[ImageData[i=Nest[Darker,ImageResize[testbw,50],3]]]9]/.n_Integer:>Nest[Lighter,i,n]]

Tadaaa!

One-liner problem (Tipary Bence)

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &, Reverse@ImageData@AstronomicalData["Earth", "Image"]ᵀ, {2}]

But this code needs some installation, because it works with a built in picture

AstronomicalData["Earth","Image"]=

The result would be something like this:

So let us consider an even smaller picture to have faster solution

picture=

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &, Reverse@ImageData@picture\[Transpose], {2}]

At first let us check the Reverse@ImageData@pictureᵀ part

FullForm[picture]

This is the second pixel row without the RawArray[“Byte”,..] function which inverts the colors

Image[List[List[List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[159,223,159,8],List[135,195,135,34],List[77,164,77,59],List[102,178,102,70],List[125,195,125,81],List[108,181,108,73],List[80,165,80,63],List[103,182,103,42],List[163,218,163,14],List[0,255,0,1],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0],List[0,0,0,0]]]]

From the FullForm we can see that the picture’s main part is the pixels which are represented in a form ..{{row1},{row2},..}.. where row1 is {pixel1,pixel2,..} and pixel 1 is {redvalue,greenvalue,bluevalue,transparency} all from 0-255 (it is solid if transparency is 255)

ImageData function will give us similar - inverted colors

{(ImageData@picture)[[2]]}
Image[%]

ᵀ  stands for the Transpose

Reverse and Transpose are only needed for reset the orientation of the picture

The second row after reset

Image@ImageData@picture

Image@(Reverse@ImageData@picture\[Transpose])

The next part will be Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &

This will create  a solid circle with the RGB color representation of the {pixel1} list and with the center position of the pixel and some random value, and with a random radius

RGBcolor will give us back the original colors

RGBColor @@ {0.596078431372549`,0.9490196078431372`,0.596078431372549`,1.`}

Graphics[{RGBColor @@{0.596078431372549`,0.9490196078431372`,0.596078431372549`,1.`}, Disk[{1,1} + RandomReal[1, 2], RandomReal[1.5]]}]

Together with Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &, {...} , {2}] we will get the Graphics function mapped onto the pixels which will look like:

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}]&,{{{1,0,0,0.5},{0,1,0,0.5}},{{0,0,1,0.5},{0,0,0,0.5}}},{2}]

Without the random functions it will be clearer

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 , 0.5]}]&,{{{1,0,0,0.5},{0,1,0,0.5},{0,1,1,0.5}},{{0,0,1,0.5},{0,0,0,0.5},{1,0,1,0.5}}},{2}]

We can see the operation of the MapIndexed function

MapIndexed[f,{{{1,0,0,0.5},{0,1,0,0.5}},{{0,0,1,0.5},{0,0,0,0.5}}},{2}]

We will have two arguments of f, the first will represent the RGBcolor and the second is the position of the pixel within the picture, but here our coordinates are in the I plane quarter {y,x} instead of the IV plane quarter {x,-y} therefore we need a transformation here (Reverse, Transpose)

Putting the whole stuff together we get some kind of dotted representation of the picture with some random parameters which finally looks pretty attractive

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &, Reverse@ImageData@picture\[Transpose], {2}]

Which is identical with the following

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[{#2[[2]],-#2[[1]]}+ RandomReal[1, 2], RandomReal[1.5]]}] &, ImageData@picture, {2}]

Without the Reverse and Transpose function we would get a transformed form of the picture and this is because the MapIndexed function process on the lists of list

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 + RandomReal[1, 2], RandomReal[1.5]]}] &, ImageData@picture, {2}]

Without the random values

Show@MapIndexed[Graphics[{RGBColor @@ #1, Disk[#2 , 0.5]}] &, Reverse@ImageData@picture\[Transpose], {2}]

This One-liner was a pretty good example for picture manipulation with the help of a dense code

With some small modification we can create some unique picture manipulation effect

Show@MapIndexed[Graphics[{RGBColor @@ {#1*{1,0,1,1}}, Circle[#2 + RandomReal[1,2], RandomReal[0.5]]}] &, Reverse@ImageData@picture\[Transpose], {2}]

Data Analysis

Tipary Bence

Data

?Data*

At first let us import some data - practice dataset from a subject called Analysis of Technical and Economic Data

ClearAll[importdata]
importdata=Import["http://www.hds.bme.hu/letoltesek/targyak/BMEGEVGAG14/feladatsorok2013/2013_f1_boxplot_hisztogram.xlsm","Data"];

Let us gather the important data from the sheet

ClearAll[data]
data0=Table[importdata[[2,i,2]],{i,4,Length@importdata[[1]]}]

Then apply a random sampling

data=RandomSample[data0,500];

Run this, if the data cannot be imported!

ClearAll[data0];
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Statistics

The probability density and cumulative distribution functions

DiscretePlot[PDF[EmpiricalDistribution@data,x],{x,data}]
DiscretePlot[CDF[EmpiricalDistribution@data,x],{x,data}]

The histogram

Histogram[data]

The mean

Mean@data

The Median

Median@data

The mode

Commonest[data]
Length@%

We can see that there are no coincidences between the elements, therefore there is no uniqe mode

The varience

Variance@data

The deviation

(Sqrt[Variance@data])

StandardDeviation@data

The range

sampleRange[l_List]:=Max[l]-Min[l]
sampleRange[data]

The lower and upper quartile

qlow[l_List]:=Quantile[l,1/4]
qhigh[l_List]:=Quantile[l,3/4]

qlow[data]

qhigh[data]

InterquartileRange[data]

Xuan Tran Thi

SetDirectory[NotebookDirectory[]]

data= Import["Xuan09.xlsx"]

data= Import["Xuan09.xlsx","Elements"]

data= Import["Xuan09.xlsx","FormattedData"]

splitdata = Delete[data,{1,1}][[1]]

split1 = splitdata[[All,1]] 
split2 = splitdata[[All,2]]

GraphicsRow[{Histogram[split1],Histogram[split2]}]

ordereddata[x_] := Sort[x]
Show[{ListPlot[ordereddata[split1]],ListPlot[ordereddata[split2]]}]

Mean[split1]
Variance[split1]

Calculate mean and variance as definition

length1 = Length[split1]
k=0;
avg = 0;
For[k =1,k≤length1,k++,
avg = avg + ordereddata[split1][[k]]
]

Average = avg/length1

var = 0;
For[k =1,k≤length1,k++,
var = var + (ordereddata[split1][[k]] - Average)^2
]
var = var/length1

Although I do not like For, it is good to see a recursive cacluation of a statistics.

Lam Van Thanh Binh

SetDirectory[NotebookDirectory[]];

col1=Import["Book1Lam09.xlsx",{"Data",1,{2,3,4,5,6,7},2}]

Import["Book1Lam09.xlsx"]

col2=Import["Book1Lam09.xlsx",{"Data",1,{2,3,4,5,6,7},3}]
col3=Import["Book1Lam09.xlsx",{"Data",1,{2,3,4,5,6,7},4}]
Histogram[{col1,col2,col3}]
ListPlot[Sort[col1]]
Mean[{col1,col2,col3}]
Variance[{col1,col2,col3}]
MeanDeviation[{col1,col2,col3}]
Median[{col1,col2,col3}]

Carsten Olm

How lucky am I that I have a beer diary that I can use to solve this homework! :-)

Sampling

SetDirectory[NotebookDirectory[]];
beer=Import["Olm09_beer.xlsx","XLSX"];dim=Dimensions[beer[[1]]];
len=dim[[1]];
countrycountlist=beer[[1,All,2]];

beer//Short

Let' s calculate some statistics of this list of beers I drank per country. I partially will deal with some actual sampling in “Hypothesis testing”.

TableForm[{Length[countrycountlist],Max[countrycountlist],Min[countrycountlist],Mean[countrycountlist],HarmonicMean[countrycountlist],Commonest[countrycountlist],StandardDeviation[countrycountlist]},TableHeadings->{{"Count","Max","Min","Mean","Harmonic Mean", "Most common","Standard deviation"},{"Value"}}]
BoxWhiskerChart[countrycountlist,"Outliers",PlotLabel->"Box plot without far outliers removed"]
Histogram[countrycountlist,10]

Let us remove some far outliers :

countrycountlistMod=Select[countrycountlist,#<100&];
TableForm[{Length[countrycountlistMod],Max[countrycountlistMod],Min[countrycountlistMod],Mean[countrycountlistMod],HarmonicMean[countrycountlistMod],Commonest[countrycountlistMod],StandardDeviation[countrycountlistMod]},TableHeadings->{{"Count","Max","Min","Mean","Harmonic Mean", "Most common","Standard deviation"},{"Value"}}]
BoxWhiskerChart[countrycountlistMod,"Outliers",PlotLabel->"Box plot with far outliers removed"]
Histogram[countrycountlistMod,10]

Note that the functions BoxWhiskerPlot and BoxOutliers seem to be replaced by another syntax. Unfortunately, they still appear in the help function.

Hypothesis testing

Instead of replicating what we discussed during the last lecture, let me see if my hypothesis “I rate beer according to a normal distribution” holds. Note that I use the German grading system (1 = very good, 6 = insufficient)

ClearAll[len,dim];
dim=Dimensions[beer[[2]]];
len=dim[[1]];
gradelist=beer[[2,All,2]];
?? DistributionFitTest

μ=Mean[gradelist];
σ=StandardDeviation[gradelist];
DistributionFitTest[gradelist,NormalDistribution[μ,σ]]

Really?

Manipulate[DistributionFitTest[SeedRandom[];RandomSample[gradelist,length],NormalDistribution[μ,σ]],{{length,2,"Sample size:"},2,len,1,Appearance->{"Open","Labeled"}}]

Again, if I did know more about math, I wouldn’t have made such a stupid hypothesis. A list of integers cannot be normally distributed (by definition)! The values have to be continuous.

Anyway, I like the example, so let’s modify the list of grades a bit.

Let us assume I was not 100% sober when I made decisions about grades. Which means that the grades will have some “uncertainty”, which we can reconstruct/randomly assign to each value:

uncertainty=RandomReal[{-0.5,+0.5},2577];
gradelistMod=gradelist+uncertainty;
Min[gradelistMod]
Max[gradelistMod]

Values <1 and >6 cannot even be explained with the influence of alcohol, so let’s restore the minimal and maximal possible grades:

Do[
If[
gradelist[[n]]==1,gradelistMod[[n]]=gradelist[[n]],
If[
gradelist[[n]]==6,gradelistMod[[n]]=gradelist[[n]]
]
],{n,1,len}]
Min[gradelistMod]
Max[gradelistMod]

Manipulate[DistributionFitTest[SeedRandom[];RandomSample[gradelistMod,length],NormalDistribution[μ,σ]],{{length,2,"Sample size:"},2,len,1,Appearance->{"Open","Labeled"}}]

A bit better, but still not “satisfactory” as far as the fulfilment of my hypothesis is concerned...


I also liked the use of HypothesisTestData in the examples given by the help function:

ℋ=DistributionFitTest[RandomSample[gradelistMod,20],Automatic,"HypothesisTestData"];
ℋ["TestDataTable",All]

Estimation

Let' s see how well a non-random sample (taken from the n first elements of the above list) represents the full dataset:

norm=NormalDistribution[Mean[gradelistMod],StandardDeviation[gradelistMod]];
Clear[sample,s1,s]
sample[s1_]:=gradelistMod[[1;;s1]];
Manipulate[Plot[{PDF[estsample[n],x],PDF[norm,x]},{x,1,6},PlotRange->{{1,6},{0,0.7}},PlotLegends->{"Estimation via non-random sample","Normally-distributed full dataset"}],{{n,1000,"Sample Size"},2,len,1,Appearance->{"Open","Labeled"}}]

This is another very interesting plot, although I’m not 100% sure about its correct interpretation (we might discuss this!):

Clear[sample2,s1,s]
sample2[s1_]:=RandomSample[gradelistMod,s1];
estsample2[s_]:=EstimatedDistribution[sample2[s],NormalDistribution[Mean@sample2[s],StandardDeviation@sample2[s]]]
Manipulate[Plot[{PDF[estsample2[n],x],PDF[norm,x]},{x,1,6},PlotRange->{{1,6},{0,0.7}},PlotLegends->{"Estimation via random sample","Normally-distributed full dataset"}],{{n,1000,"Sample Size"},2,len,1,Appearance->{"Open","Labeled"}}]

Programming remarks

Polynomials, tricks

Expand,Factor (FactorTerms,FactorList,FactorTermsList),Collect,Exponent,Coefficient
CoefficientList,PolynomialQ,PolynomialGCD,

Less multiplications

Horner form

HornerForm[11 (x^3)-4 (x^2)+7 x+2]

Rational functions

HornerForm[(11 (x^3)-4 (x^2)+7 x+2)/((x^2)-3x+1)]

Multivariate functions

HornerForm[(11 (x^3)y-4 (x^2)+7 x+2(y^2))(y (x^2)-3x+1)]

Parameters as coefficients

HornerForm[(11 (x^3)y-4 p (x^2)+7 q x+2(y^2))(y (x^2)-3(p^2)x+r),{x,y}]

Karatsuba algorithm

ClearAll[a,b,c,d];
p1=(a x+b)(c x+d)//Expand

Collect[%112,x]

4 multiplications

Expand[u=(a+b)(c+d)]

p2=a c (x^2)+(u-a c-b d)x+b d

p1==p2

Simplify@%115

p1-p2//Simplify

Karatsuba, 3 multiplications

Matrix multiplications

Hyperlink["Strassen","http://en.wikipedia.org/wiki/Strassen_algorithm"]

Not only polynomials

ClearAll[f];
D[f[(Sqrt[(x^2)+1])],{x,3}]

Together[%]

FullForm[f'[(Sqrt[1+(x^2)])]]

Collect[%122,Derivative[_][f][_]]

Collect[%,Derivative[_][f][_],Together]

Approximate solution

SolveAlways[a x+b==0,x]

f[x_]:=x^3+a x^2+b x+c;
SolveAlways[Implies[f[x]==0&&f[y]==0,x==y],{x,y}]

y''[x]+2 y[x]==0/.y->Function[x,Sin[a x]]

Series[%,{x,0,3}]

SolveAlways[%,x]

A control problem

Studying the problem solved in the help

nds=NDSolve[{x''[t] + x[t]==0,x'[0]==π,x[0]==0, WhenEvent[{x[t]==2,x[t]==-3},x'[t]->-x'[t]]},x,{t,0,20}];
Plot[{x[t]/.nds,2,-3},{t,0,20}]

let us describe the heating of a room. If the temperature reaches 22 C, then the heating is turned off, cooling takes places following Newton´s law. If it reaches 18 C, an oven of 25 C is turned on. Let us show 4 periods.

nd1=NDSolve[{T'[t]==-(T[t]-18),T[0]==21,WhenEvent[{T[t]==18.01},T'[t]->25-T[t]],WhenEvent[{T[t]==21.99},T'[t]->-(T[t]-18)]},T,{t,0,30}]

Let us take the derivative of the equation (then, one has to supplement the initial conditions).

nd=First@NDSolve[{D[T'[t]==-(T[t]-18),t],T[0]==21,T'[0]==18-T[0],WhenEvent[{T[t]==18.01},T'[t]->25-T[t]],WhenEvent[{T[t]==21.99},T'[t]->18-T[t]]},T,{t,0,30}]

Plot[T[t]/.nd,{t,0,30}]

Presentation: GUI (Csikja Rudolf)

If you are able to correct in such a way that also the last part works then tell us, please.

Sketch

Panel[
Grid[{{"Panel1 - Initial Values","Graphics"},{"Panel2 - Simulation Control",SpanFromAbove},{"Panel3 - Results",SpanFromAbove}},Frame->All,Alignment->{Left},Spacings->{1.5, 1.5}]
,"MainPanel",ImageSize->{300,150}, Alignment->{Center, Center}]

Panel1

Sketch

LeftPanelSize[height_]:={300,height}

Panel[Grid[{{"StaticText","InputField1", "RadioButtons1"},{"StaticText","InputField2", "RadioButtons2"},{"StaticText","InputField3","CheckBox3"}},Frame->All,Alignment->{Left},Spacings->{1.5, 1.5}],"Panel1",ImageSize->{300,150}, Alignment->{Center, Center}]

Realization

Panel1[{inputField1_,radioButtons1_},{inputField2_,radioButtons2_},{inputField3_,checkBox3_}]:=Deploy[
Panel[
Grid[{{"Attitude: ",InputField[inputField1,Number,FieldSize->{3,1}], RadioButtonBar[radioButtons1,{# π/180&->" deg     ",#&->" rad"}]},
{"Speed: ",InputField[inputField2,Number,FieldSize->{3,1}], RadioButtonBar[radioButtons2,{# 6/10&->" km/h  ",#&->" m/s"}]},
{"Drag: ",InputField[inputField3,Number,FieldSize->{3,1}, Enabled->checkBox3], Checkbox[checkBox3]}}, Alignment->{Left},Spacings->{1.5, 1.5}]
,Style["Initial Values",Bold],ImageSize->{300,130}, Alignment->{Center, Center}]
]

DynamicModule[{initialAngle=45,convertAngle=# π/180&,initialSpeed=100,convertSpeed=# 6/10&,initialDrag=1,enableDrag=False},
{Panel1[{Dynamic[initialAngle],Dynamic[convertAngle]},{Dynamic[initialSpeed],Dynamic[convertSpeed]},{Dynamic[initialDrag],Dynamic[enableDrag]}],{Dynamic[convertAngle[initialAngle]],Dynamic[enableDrag],Dynamic[initialSpeed]}}]

ú

Panel2

Sketch

Panel[Grid[{{"Button1"},{"Button2"}},Frame->All,Alignment->{Left},Spacings->{1.5, 1.5}],"Panel2",ImageSize->{300,100}, Alignment->{Center, Center}]

Realization

Panel2[{action1_,action2_}]:=Deploy[
Panel[Grid[{{Button["Run Simulation",pt=action1]},{Button["Clear Plot", action2]}},Alignment->{Left},Spacings->{1.5, 1.5}],Style["Simulation Control",Bold],ImageSize->{300,120}, Alignment->{Center, Center}]
]

Panel2[{a,b}]

DynamicModule[{initialAngle=45,convertAngle=# π/180&,initialSpeed=100,convertSpeed=# 6/10&,initialDrag=1,enableDrag=False,pt={Graphics[]}, tend=0,tbl={},f=ParametricNDSolveValue[{x''[t]==-((10 )^(-3))drag Sign[x'[t]](((x'[t])^2)+((y'[t])^2)),y''[t]==-9.84-((10 )^(-3))drag Sign[y'[t]](((x'[t])^2)+((y'[t])^2)),x[0]==0,y[0]==0,x'[0]==v Cos[α],y'[0]==v Sin[α], WhenEvent[y[t]==-0.001,{"StopIntegration"}]},{x,y},{t,0,∞},{α,v,drag}]},
Deploy@Panel[
Grid[{{Panel1[{Dynamic[initialAngle],Dynamic[convertAngle]},{Dynamic[initialSpeed],Dynamic[convertSpeed]},{Dynamic[initialDrag],Dynamic[enableDrag]}],Dynamic[Show[pt, Frame->True,ImageSize->{500,500},GridLines->Automatic,GridLinesStyle->Dotted, AspectRatio->Automatic,PlotRange->{{0,1000},{0,1000}}, PlotRangeClipping->True, FrameLabel->{"Distance [m]","Height [m]"}, FrameStyle->Directive[12]]]},
{Deploy[
Panel[
Grid[{
{Button["Run Simulation",xx=First[f[convertAngle[initialAngle],convertSpeed[initialSpeed],If[enableDrag,initialDrag,0]]];
yy=Last[f[convertAngle[initialAngle],convertSpeed[initialSpeed],If[enableDrag,initialDrag,0]]];tend=Nest[First,xx,2]//Last;tbl=Append[tbl,Style[Round[#,0.1],ColorData[81][Length[tbl]+1]]&/@{xx[tend],FindMaximum[yy[t],{t,5}, AccuracyGoal->2]//First,tend}];pt=Append[pt,ParametricPlot[{xx[s],yy[s]},{s,0,tend},PlotRange->All,PlotStyle->ColorData[81][Length[pt]]]];]},
{Button["Clear Plot",pt={Graphics[]};tbl={};]}},Alignment->{Left},Spacings->{1.5, 1.5}],Style["Simulation Control",Bold],ImageSize->{300,120}, Alignment->{Center, Center}]
],SpanFromAbove},
{Panel[
Dynamic@Text@Grid[Prepend[tbl,{TraditionalForm[(Subscript[x, max]) ],TraditionalForm[(Subscript[y, max])],TraditionalForm[(Subscript[t, end])]}],Background->{None,{Lighter[Black,.98],{White,Lighter[Black,.95]}}},Dividers->{{Lighter[Gray,0.6],{False},Lighter[Gray,.6]},{Lighter[Black,.2],Lighter[Black,.2],{False},Darker[Gray,.6]}},Alignment->{Left},ItemSize->Automatic,Frame->Darker[Gray,.6],ItemStyle->13,Spacings->{Automatic,.8}
],Style["Results",Bold],ImageSize->{300,Automatic}, Alignment->{Center, Center}],SpanFromAbove}},Alignment->{Left, Center},Spacings->{1.5, 1.5}]

,Style["Projectile Simulation",24,Bold]]]

Homeworks

1. Investigate the error of the parameters when fitting the sum of two exponentials as an effect of measurement error and also as the effect of the distance between the exponents. Let us see some details.

pl=Plot[3 (E^(-0.1t))-2(E^(-0.2t)),{t,0,12},PlotStyle->Directive[Thickness[.02],Red]]

,

data=Table[{t,(3(E^(-0.1t))-2(E^(-0.2t)))(1+0.02RandomReal[{-1,1}])},{t,Range[0,12,0.05]}];
ListPlot[data,Epilog->First@pl]

nlm=NonlinearModelFit[data,A (E^(-a t))-B (E^(-b t)),{A,a,B,b},t]

nlm["ParameterTable"]

ListPlot[data,Epilog->First@Plot[nlm[t],{t,0,12},PlotStyle->Directive[Red,Thickness[0.02]]]]

ListPlot[nlm["FitResiduals"],PlotLabel->Style["It is symmetric, OK.",Bold,Italic,24]]

No initial estimates were needed. It is possible and most often it is desirable to provide initial estimates. Study the effect of giving different initial estimates of the parameters.

The parameters are quite good.

Study the dependence of the estimated parameters on the error distribution. (Type - normal, uniform etc. Dispersion - small, large.)

The conjecture is (actually, the experience of many people) that the estimates will be bad if the difference between a and b is small. Study this effect also.

2. Find necessary and/or sufficient conditions for the stability of a stationary point and also for the existence of a periodic solution of an ordinary differential equation.