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Here you can see a contraciction in definition of atomic objects. Clearly \
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Also the the elemeent before the last can be picked as well  (the negative \
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Also a range can be chosen. Let\[CloseCurlyQuote]s take the elements of the \
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Let\[CloseCurlyQuote]s generate a list ofpairs of random numbers that will \
have length of 10.\
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Using function part the first element of the 4 par can be picked\
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Wolfram Mathematica supports programming by using different paradigms. \
Functional programming is one of the basics in a programming language. It can \
be defined as a way of computing, which, in its essence, behaves like a \
normal mathematical function. In this case, the data do not change, and the \
results of mathematical operations are just copies of the original data. This \
concept was taken to the program from standard mathematics, where functions \
are relations between sets. Because of that, during the computation the first \
set does not change. Of course, this is not the whole description of \
functional programming, but it summarizes its essence. Other problem with the \
definition is that computer\[CloseCurlyQuote]s memory sets limitations to \
interpreting mathematical functions. If we have two large sets, the program \
will need to work with a lot of memory. On top, we always work with a large \
amount of data and a great number of functions. \
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 "In computer science, there exists a specific branch which is interested in \
Lambda functions. Lambda function, in its nature, is a function which output \
is strictly related to its input. For two same inputs in has the same \
outputs. Generally, it can be said that it is a mathematical function written \
in a programming language. Besides that, the Lambda function can be accepted \
as a input and can also be an output of a function etc.\nIn Wolfram \
Mathematica Lambda function is called Pure Function or anonymous function (",
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If you want to use the function couple times, you can store it to a variable.\
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Programming with rules of replacement is relatively different from imperative \
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in programming templates (or patterns) and their recognition is important.
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First we have to clear Set and SetDelayed. I like this example ... Random[ ] \
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likewise, if I ask what is eq1, we will get a number that was created by the \
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Let\[CloseCurlyQuote]s look at Set Delayed, also \[OpenCurlyDoubleQuote]:=\
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but we wont get an output. The Kernel has the information, that when it \
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but when we generate 5 time eq2, we get different numbers, a new random \
number is always generated\
\>", "Text",
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Cell["\<\
so let\[CloseCurlyQuote]s get to the functions\[Ellipsis] well not quite yet. \
We need patterns first. Anything can be a pattern: Kyncl with his ID number \
and his genetical information is a very specific pattern, unique, as is the \
number 1 in Mathematica or the constant \[Pi]. On the opposite side, there \
are three underscores ___, which mean that where they are placed, there can \
be no expressions, or one expression, or even more expressions acceptable in \
Mathematica. The most common is one underscore, _, which is an equivalent to \
any one acceptable expression in Mathematica\[CloseCurlyDoubleQuote]. 
When we write:
x: _
, it means, that with the name x we are naming anything, because the \
underscore means \[OpenCurlyDoubleQuote]one anything\[CloseCurlyDoubleQuote], \
so any acceptable expression in Mathematica. Generally in Mathematica we name \
expressions with a colon. But this is used so often, that instead of using \
full syntax, we almost always use the shortened version:
x_
which has the exact same meaning.
so finally, let\[CloseCurlyQuote]s get to the functions ...
A function can be understood as a permanent replacement rule.
When I write (in full syntax it would be f [x: _]: = x + 1 + x)
f [x_]: = x + 1 + x;
, than I am telling Mathematica, that if f[\[OpenCurlyDoubleQuote]anything\
\[CloseCurlyDoubleQuote]] appears somewhere, than it should be replaced by \
\[OpenCurlyDoubleQuote]anything\[CloseCurlyDoubleQuote] +1+ \
\[OpenCurlyDoubleQuote]anything\[CloseCurlyDoubleQuote].
We will show how it works (even the \[OpenCurlyDoubleQuote]SET Delayed\
\[CloseCurlyDoubleQuote], also \[OpenCurlyDoubleQuote]: =\
\[CloseCurlyDoubleQuote]): \
\>", "Text",
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we can see that
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But if it by chance had a value, than the function wouldn\[CloseCurlyQuote]t \
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\>", "Text",
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And indeed, even according to \[OpenCurlyDoubleQuote]help\
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On the contrary, there is no danger in using \[OpenCurlyDoubleQuote]Set \
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It might seem that using \[OpenCurlyDoubleQuote]Set\[CloseCurlyDoubleQuote], \
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Let\[CloseCurlyQuote]s see how \[OpenCurlyDoubleQuote]SetDelayed\
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at the moment of sending the expression f [x _]: = Integrate [2x, x] to the \
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dx... which is nonsense. Let\[CloseCurlyQuote]s also look how \
\[OpenCurlyDoubleQuote]help\[OpenCurlyDoubleQuote] see\[CloseCurlyQuote]s \
this function:\
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when using \[OpenCurlyDoubleQuote]Set\[CloseCurlyDoubleQuote],also \
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but if x had some specific value at the moment of sending f [x _] = Integrate \
[2x, x] to the Kernel, it wouldn\[CloseCurlyQuote]t work, the global x would \
be integrated \
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Cell["\<\
To put it in two sentences: we usually use \[OpenCurlyDoubleQuote]: =\
\[CloseCurlyDoubleQuote], if that doesn\[CloseCurlyQuote]t work, we try \
\[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote]. It is not a short circuit \
in the nuclear alternator, we can solve it by trial and error:
Let\[CloseCurlyQuote]s look at the \[OpenCurlyDoubleQuote]overusing\
\[CloseCurlyDoubleQuote] of functions. Now that we understand patterns a \
little bit better, it makes more sense. When i write:
ClearAll [f];
f [x_]: = 1 + x;
f [x_, y_]: = x + y ^ 2;
I am actually saying, what should happen if f[\[OpenCurlyDoubleQuote]anything\
\[CloseCurlyDoubleQuote]] occurs (this expression is replaced by \
\[OpenCurlyDoubleQuote]anything\[CloseCurlyDoubleQuote]+ \
\[OpenCurlyDoubleQuote]1\[CloseCurlyDoubleQuote] and when f [\
\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote], \
\[OpenCurlyDoubleQuote]anything2\[CloseCurlyDoubleQuote]] appears, this \
expression is replaced by \[OpenCurlyDoubleQuote]\\\[CloseCurlyDoubleQuote] \
\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote] + \\! \\ (\\ n \\ * \
SuperscriptBox [\\ (\[OpenCurlyDoubleQuote]anything2\[CloseCurlyDoubleQuote] \
\\), \\ (2 \\)] \\) \[OpenCurlyDoubleQuote]\
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Cell["\<\
Honestly: overusing of functions is used mostly in emergencies, when we \
forget about something and it is easier to solve past mistakes by overusing, \
to tell the function  what to do in the next situation. However what we use \
more often, is that the parameter is a List, but that is obvious, if I write:
ClearAll [f];
f [{x_, y_}]: = {x + y ^ 2, y-x};
than it will return:\
\>", "Text",
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Cell["Rules of replacement", "Subsection",
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Cell["\<\
This certainly isn\[CloseCurlyQuote]t a standard treatise about the rules in \
Mathematica:
Functions, as we wrote in aboutFuknctionsVYA.nb, are actually permanent rules \
of replacement.
For example: f [x _]: = x ^ 2 means that when f \
[\[OpenCurlyDoubleQuote]anything\[CloseCurlyDoubleQuote]] occurs, this \
expression is replaced by the expression anything ^ 2. However, it is \
sometimes unnecessary to create a permanent global rule, when we want to \
replace an expression by some rule just in one row of our program. For this \
we can use the commands \[OpenCurlyDoubleQuote]Rule\[CloseCurlyDoubleQuote] \
(also \[OpenCurlyDoubleQuote]->\[CloseCurlyDoubleQuote]), \
\[OpenCurlyDoubleQuote]RuleDelayed\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]:->\[CloseCurlyDoubleQuote]) and \
\[OpenCurlyDoubleQuote]ReplaceAll\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]/.\[CloseCurlyDoubleQuote]). The \
\[OpenCurlyDoubleQuote]All\[CloseCurlyDoubleQuote] means that the rules apply \
to everything, that fulfills the exchange rule, for example:  expr / .x-> 1 \
searches the entire expression expr and replaces all x in it  by ones.
We can see that using \[OpenCurlyDoubleQuote]Rule\[CloseCurlyDoubleQuote] \
(also \[OpenCurlyDoubleQuote]->\[CloseCurlyDoubleQuote]) is similar to using \
\[OpenCurlyDoubleQuote]Set\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote]) and using \
\[OpenCurlyDoubleQuote]RuleDelayed\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]:->\[CloseCurlyDoubleQuote]) is similar to using \
\[OpenCurlyDoubleQuote]SetDelayed\[CloseCurlyDoubleQuote] (also, \
\[OpenCurlyDoubleQuote]: =\[CloseCurlyDoubleQuote]).
If we want to make changes to an expression, we do it by typing the \
expression /. rule, in the case of more rules, by typing /. {rule1, rule2, \
...}.
In the alternate syntax: ReplaceAll [expression, rule], or ReplaceAll \
[expression, {rule1, rule2, ...}].
Note: we can put even one rule in the list,it wont harm us, so the expression \
/. {rule} works the same as expression /. rule.
Before looking in more detail on the differences between \
\[OpenCurlyDoubleQuote]Rule\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]\\ [->]\[CloseCurlyDoubleQuote]) and \
\[OpenCurlyDoubleQuote]RuleDelayed\[CloseCurlyDoubleQuote] 
(also, \[OpenCurlyDoubleQuote]:->]\[CloseCurlyDoubleQuote]), we will observe \
that they maintain a form of a \[OpenCurlyDoubleQuote]list smaller by one\
\[CloseCurlyDoubleQuote]:\
\>", "Text",
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Cell["\<\
When there will be multiple rules of change in the list, the first relevant \
one will be used, if the rules in the rule list are only represented as \
elements of this list:\
\>", "Text",
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Cell["\<\
If there are rules in the rule list, that are in separate lists, we get a \
list of answers:\
\>", "Text",
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Cell["\<\
Such lists of assignment lists , return for example: commands, that can have \
more \[OpenCurlyDoubleQuote]solutions\[CloseCurlyDoubleQuote], for example: \
Solve, if we want a list of results , we can get to the list by: x / .Solve \
[x ^ 3 == - 1, x]\
\>", "Text",
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If we want to input into a specific expression, it doesn\[CloseCurlyQuote]t \
matter if we use \[OpenCurlyDoubleQuote]Rule\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]->\[CloseCurlyDoubleQuote]) or \
\[OpenCurlyDoubleQuote]RuleDelayed\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]:->\[CloseCurlyDoubleQuote])\
\>", "Text",
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Firstly we have to realize, that the important part is what the Kernel is \
\[OpenCurlyDoubleQuote]seeing\[CloseCurlyDoubleQuote], not what we see. For \
example, we see in FrontEnd (that is how we read) a == b and the Kernel \
\[OpenCurlyDoubleQuote]sees\[CloseCurlyDoubleQuote] an expression as its \
FullForm (in simplified terms), it see\[CloseCurlyQuote]s the expression as \
Equal [a, b], and that is how it executes (/.) required exchanges (-> or:->), \
thus executing it in the internal representation, not in what we see: Well \
let\[CloseCurlyQuote]s try it ourselves by exchanging \
\[OpenCurlyDoubleQuote]equals\[CloseCurlyDoubleQuote] with \
\[OpenCurlyDoubleQuote]multiplication\[CloseCurlyDoubleQuote]:\
\>", "Text",
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Cell["\<\
using \[OpenCurlyDoubleQuote]{FullForm [replacedExpression], FullForm \
[expression]} /. FullForm-> TreeForm\[CloseCurlyDoubleQuote]in the previous \
is kind of a joke, but it fits. We just used the exchange rule again. We can \
also see, that we can use \[OpenCurlyDoubleQuote]preservation of  the list\
\[CloseCurlyDoubleQuote] in this example: (compare with x /. {{X-> Pi}, {x-> \
E}}}, only the \[OpenCurlyDoubleQuote]form\[CloseCurlyDoubleQuote] is \
exchanged (a variable without  a value, the \[OpenCurlyDoubleQuote]form\
\[CloseCurlyDoubleQuote] itself will actually not be used, because it will be \
changed as FullForm or TreeForm and only these existing functions will be \
evaluated.))\
\>", "Text",
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Cell["\<\
both of the previous exercises are good to try to code by yourself , but here \
we care about the rules, so I\[CloseCurlyQuote]ll show it to you. 
For other examples, we will use the previous example and \
let\[CloseCurlyQuote]s try to see what we would do, if we didn\
\[CloseCurlyQuote]t\[OpenCurlyQuote]t know, that \
\[OpenCurlyDoubleQuote]Equal[a, b]\[CloseCurlyDoubleQuote] is an internal \
representation of \[OpenCurlyDoubleQuote]a == b\[CloseCurlyDoubleQuote], but \
we understand patterns and knew \[OpenCurlyDoubleQuote]SetDelayed\
\[CloseCurlyDoubleQuote] (also \[OpenCurlyDoubleQuote]:-> \
\[OpenCurlyDoubleQuote]), brackets on the left hand side of the expression \
(lhs_==rhs_):->lhs*rhs are useless (just for clarification lhs = left hand \
side, rhs = right hand side).\
\>", "Text",
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Cell["\<\
notice that bland green color, which was already at  \
\[OpenCurlyDoubleQuote]Set\[CloseCurlyDoubleQuote] (also \
\[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote]) and \
\[OpenCurlyDoubleQuote]SetDelayed\[CloseCurlyDoubleQuote] ( also \
\[OpenCurlyDoubleQuote]: =\[CloseCurlyDoubleQuote]). This color actually \
tells us the same about the order of evaluation and \
\[OpenCurlyDoubleQuote]locality\[CloseCurlyDoubleQuote] of  what the \
\[OpenCurlyDoubleQuote]lhs\[CloseCurlyDoubleQuote] and the 
 \[OpenCurlyDoubleQuote]rhs \[OpenCurlyDoubleQuote] are.  Even further on \
behaves in the same way, because the function differs from the use (/.) and \
the rules {-> or:>} only by the fact, that a local exchange is in place and \
not a globally introduced replacement rule. Actually, if you applyed the rule \
f [x _]:> x ^ 2 than it would act as if you defined a global rule (a \
function) f [x _]: = x ^ 2, and if you applied the function on every \
expression during the evaluation f[ x _] -> x ^ 2, it would be like \
introducing a global rule (function):f [x _] = x ^ 2.
, with all the properties, that we observed at \[OpenCurlyDoubleQuote]Set\
\[CloseCurlyDoubleQuote] and \[OpenCurlyDoubleQuote]SetDelayed\
\[CloseCurlyDoubleQuote].
If the variables \[OpenCurlyDoubleQuote]lhs\[CloseCurlyDoubleQuote] and \
\[OpenCurlyDoubleQuote]rhs\[CloseCurlyDoubleQuote] are clean as a whistle, \
than the use \[OpenCurlyDoubleQuote]->\[CloseCurlyDoubleQuote] instead of \
\[OpenCurlyDoubleQuote]:->\[CloseCurlyDoubleQuote] doesn\[CloseCurlyQuote]t \
give as a problem. The right hand side of the expression (lhs _ == rhs _) -> \
lhs * rhs, ei a * b, but if \
\[OpenCurlyDoubleQuote]lhs\[CloseCurlyDoubleQuote] and \
\[OpenCurlyDoubleQuote]rhs\[CloseCurlyDoubleQuote] are clean as a whistle, \
than only \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] and \
\[OpenCurlyDoubleQuote]b\[CloseCurlyDoubleQuote] remain and the rule is \
applied in the most logical way, in this case \
\[OpenCurlyDoubleQuote]correctly\[CloseCurlyDoubleQuote], which is what we \
wanted in this case\
\>", "Text",
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similarly with functions, I will write the same case on purpose and you will \
see that understanding functions like (global) replacement rules quite fits :\
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let\[CloseCurlyQuote]s see what happens, when for example \
\[OpenCurlyDoubleQuote]rhs\[CloseCurlyDoubleQuote] isn\[CloseCurlyQuote]t \
clean as a whistle\
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let\[CloseCurlyQuote]s see again what happens, when for example \
\[OpenCurlyDoubleQuote]rhs\[CloseCurlyDoubleQuote] isn\[CloseCurlyQuote]t \
clean as a whistle.
It can be ok, if we want this result, but normally it isn\[CloseCurlyQuote]t\
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Let\[CloseCurlyQuote]s show, what happen\[OpenCurlyQuote]s during  \
\[OpenCurlyDoubleQuote]RuleDelayed\[CloseCurlyDoubleQuote] (also  \
\[OpenCurlyDoubleQuote] :->\[CloseCurlyDoubleQuote]) and  \
\[OpenCurlyDoubleQuote]SetDalyed\[CloseCurlyDoubleQuote](also  \
\[OpenCurlyDoubleQuote] :=\[CloseCurlyDoubleQuote])
\[OpenCurlyDoubleQuote]Delayed\[CloseCurlyDoubleQuote] can be recognized by \
the extra colon in the beginning \
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Lastly let\[CloseCurlyQuote]s show the link between  \
\[OpenCurlyDoubleQuote]functions \[OpenCurlyDoubleQuote] and  \
\[OpenCurlyDoubleQuote]/. \[OpenCurlyDoubleQuote]  and  \
\[OpenCurlyDoubleQuote]:-> \[OpenCurlyDoubleQuote] or  \
\[OpenCurlyDoubleQuote] ->\[CloseCurlyDoubleQuote]\
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Cell["Patterns", "Subsection",
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Cell["\<\
We will be talking about the use of patterns, not patterns in general . Who \
wants, can look up the keyword Pattern in  Help and can go on through Help \
using SeeAlso. Of course, it still stands, that ANYTHING can still be a \
pattern. Me, you, he ... but also an electron and event the word electron \
itself, all this can be a pattern. 
In the next part it is important to show how to write patterns with a certain \
degree of generality, so they will contain an underline (\
\[OpenCurlyDoubleQuote]_\[CloseCurlyDoubleQuote]).
The basic syntax is:
\[OpenCurlyDoubleQuote]PatternName\[CloseCurlyDoubleQuote]:\
\[CloseCurlyDoubleQuote]pattern\[CloseCurlyDoubleQuote]
(original syntax, we will use it for a while)
As a revision we will show:
(\[OpenCurlyDoubleQuote]pattern\[CloseCurlyDoubleQuote] is {\
\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote],\
\[CloseCurlyDoubleQuote] anything2\[CloseCurlyDoubleQuote]}, in the internal \
syntax List [\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote],\
\[CloseCurlyDoubleQuote]anything2\[CloseCurlyDoubleQuote]]).
(if I don\[CloseCurlyQuote]t put it in the right form, it becomes a different \
pattern a function that \[OpenCurlyDoubleQuote]doesn\[CloseCurlyQuote]t \
function\[CloseCurlyDoubleQuote], it should have one parameter, a pair, so f \
[{\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote], \
\[OpenCurlyDoubleQuote]anything2\[CloseCurlyDoubleQuote]}], for f [\
\[OpenCurlyDoubleQuote]anything1\[CloseCurlyDoubleQuote], \
\[OpenCurlyDoubleQuote]anything2\[CloseCurlyDoubleQuote]]. The Kernel has no \
rule, so the evaluation will not change anything.)\
\>", "Text",
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Equivalent with
 f[{x_,y_}]:=x^y\
\>", "Text",
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Cell["\<\
Usually we write patterns by starting with the \[OpenCurlyDoubleQuote]general\
\[CloseCurlyDoubleQuote] and then we in \[OpenCurlyDoubleQuote]general\
\[CloseCurlyDoubleQuote] add some conditions. 
The basic use is:
PatternName:_typeOfWhatItShouldBeApplyedToo, for example
n: _Integer
Let\[CloseCurlyQuote]s show this on an example, when we look for divisors of \
a function parameter.  Because it only makes sense to do this for integer \
parameters and we don\[CloseCurlyQuote]t want to get any errors if someone \
entered a non-integer parameter, we restrict the pattern validity to only \
integer numbers:\
\>", "Text",
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Cell["\<\
Only we have relatively a small amount of types (Real, Integer, Complex, \
Subscript, List ..). We will help ourselves by constraining the pattern with \
a question mark and a Boolean function (which returns True or False)
Let\[CloseCurlyQuote]s show it on an example (the function \
\[OpenCurlyDoubleQuote]test\[CloseCurlyDoubleQuote]  has a parameter \
\[OpenCurlyDoubleQuote]y\[CloseCurlyDoubleQuote] on purpose, to show it more \
clearly, that with \[OpenCurlyDoubleQuote]SetDelayed\[CloseCurlyDoubleQuote] \
(also \[OpenCurlyDoubleQuote]: =\[CloseCurlyDoubleQuote]), it does not matter \
how we name our variable, because it will never be used. During the calling \
of the function, the parameter generally already has a different value.\
\>", "Text",
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Just to remind, we rewrite the functions with the help of \
\[OpenCurlyDoubleQuote]pure\[CloseCurlyDoubleQuote] functions:\
\>", "Text",
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Cell["\<\
The character \[OpenCurlyDoubleQuote]&\[CloseCurlyDoubleQuote] has a big \
priority, and that is why it doesn\[CloseCurlyQuote]t work without bracket:\
\>", "Text",
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Cell["\<\
Let\[CloseCurlyQuote]s show a typical situation, where it is use full to \
shorten some pattern, let\[CloseCurlyQuote]s show it on an example of a list \
of pairs:
{{voltage1, current1}, {voltage2, current2}, {voltage3, current4} ...} and we \
want to make a resistance list out of it, than it would be: {voltage1 / \
current1, voltage2 / current2, voltage3 / current3 ...}\
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Cell["\<\
In the previous example it works well. What if our listOfPairs would contain \
only two pairs.\
\>", "Text",
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Cell["\<\
..that wasn\[CloseCurlyQuote]t what we wanted it to do. Well what happened \
is, that the pattern {u_, i_} fits both {2,1} and {4,2}, but also {{2,1}, \
{4,2}}, those \[OpenCurlyDoubleQuote]any two elements\[CloseCurlyDoubleQuote] \
can be lists {2,1} and {4,2}. If we\[CloseCurlyQuote]re not sure if the \
pattern fits something, we have the command MatchQ. If the internal function \
ends with a Q (shortened from the word \[OpenCurlyDoubleQuote]Question\
\[CloseCurlyDoubleQuote]), then it returns True or False. There are quite a \
lot of these kind of functions and you can you learn more about them, if you \
send this to the Kernel: \[OpenCurlyDoubleQuote]?*Q\[CloseCurlyDoubleQuote]\
\>", "Text",
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Cell["\<\
We can solve it in many different ways, but out of the two shown, the second \
way is better. The currents and voltages will generally not be integers, but \
they should have a whole numerical value.\
\>", "Text",
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Cell["\<\
patterns are useful to us, because we have powerful commands in Mathematica \
that use them, such as Cases.
Cases are used mostly in the syntax Cases[inWhatIAmLookingFor, pattern, {0, \
\\ [Infinity]}], where {0, \\ [Infinity]} tells us, that we  should search \
through the whole expression inWhatIAmLookingFor. Bassically, the main \
problem often is to come up with the right pattern.
We also sometimes want to transform the expressions ... \
let\[CloseCurlyQuote]s show this on a serial RLC circuit, where the input \
voltage is uin[t].\
\>", "Text",
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for a solution in the time domain, we  need unknowns and we have to create \
initial conditions:\.08\
\>", "Text",
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I used {uin[t] -> 0} because uin[t] also fits the pattern s_Symbol [t], but I \
don\[CloseCurlyQuote]t want it among the unknowns. How could it be done \
differently?\
\>", "Text",
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I used {uin[t] -> 0} because uin[t] also fits the pattern s_Symbol [t], but I \
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We can see, that this works, but care full, let\[CloseCurlyQuote]s try an \
exact value of fi\
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    DisplayFunction -> Identity, Ticks -> {Automatic, Automatic}, 
     AxesOrigin -> {0, 0}, FrameTicks -> {{Automatic, 
        Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, 
        Charting`ScaledFrameTicks[{Identity, Identity}]}}, 
     GridLines -> {None, None}, DisplayFunction -> Identity, 
     PlotRangePadding -> {{
        Scaled[0.02], 
        Scaled[0.02]}, {
        Scaled[0.05], 
        Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, 
     DisplayFunction -> Identity, AspectRatio -> 
     NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, 
     AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, 
     BaseStyle -> {FontFamily -> "Arial"}, DisplayFunction :> Identity, 
     Frame -> {{False, False}, {False, False}}, 
     FrameLabel -> {{None, None}, {None, None}}, 
     FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, 
     GridLines -> {None, None}, GridLinesStyle -> Directive[
       GrayLevel[0.5, 0.4]], 
     Method -> {
      "DefaultBoundaryStyle" -> Automatic, 
       "DefaultGraphicsInteraction" -> {
        "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
         "Effects" -> {
          "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
           "Droplines" -> {
            "freeformCursorMode" -> True, 
             "placement" -> {"x" -> "All", "y" -> "None"}}}}, 
       "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None,
        "CoordinatesToolOptions" -> {"DisplayFunction" -> ({
           (Identity[#]& )[
            Part[#, 1]], 
           (Identity[#]& )[
            Part[#, 2]]}& ), "CopiedValueFunction" -> ({
           (Identity[#]& )[
            Part[#, 1]], 
           (Identity[#]& )[
            Part[#, 2]]}& )}}, 
     PlotRange -> {{-1, 1}, {-0.8414709627547404, 0.8414709627547404}}, 
     PlotRangeClipping -> True, PlotRangePadding -> {{
        Scaled[0.02], 
        Scaled[0.02]}, {
        Scaled[0.02], 
        Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}],FormBox[
    FormBox[
     TemplateBox[{
       RowBox[{"sin", "(", 
         TagBox["x", HoldForm], ")"}], 
       RowBox[{"sin", "(", 
         SuperscriptBox[
          TagBox["x", HoldForm], "2"], ")"}], 
       RowBox[{"sin", "(", 
         SuperscriptBox[
          TagBox["x", HoldForm], "3"], ")"}], 
       RowBox[{"sin", "(", 
         SuperscriptBox[
          TagBox["x", HoldForm], "4"], ")"}], 
       RowBox[{"sin", "(", 
         SuperscriptBox[
          TagBox["x", HoldForm], "5"], ")"}]}, "LineLegend", 
      DisplayFunction -> (FormBox[
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         StyleBox[
          PaneBox[
           TagBox[
            GridBox[{{
               TagBox[
                GridBox[{{
                   GraphicsBox[{{
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.368417, 0.506779, 0.709798], 
                    AbsoluteThickness[1.6]], {
                    LineBox[{{0, 10}, {20, 10}}]}}, {
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.368417, 0.506779, 0.709798], 
                    AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, 
                    ImageSize -> {20, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, {
                   GraphicsBox[{{
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.880722, 0.611041, 0.142051], 
                    AbsoluteThickness[1.6]], {
                    LineBox[{{0, 10}, {20, 10}}]}}, {
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.880722, 0.611041, 0.142051], 
                    AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, 
                    ImageSize -> {20, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}, {
                   GraphicsBox[{{
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.560181, 0.691569, 0.194885], 
                    AbsoluteThickness[1.6]], {
                    LineBox[{{0, 10}, {20, 10}}]}}, {
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.560181, 0.691569, 0.194885], 
                    AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, 
                    ImageSize -> {20, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #3}, {
                   GraphicsBox[{{
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.922526, 0.385626, 0.209179], 
                    AbsoluteThickness[1.6]], {
                    LineBox[{{0, 10}, {20, 10}}]}}, {
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.922526, 0.385626, 0.209179], 
                    AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, 
                    ImageSize -> {20, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, {
                   GraphicsBox[{{
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.528488, 0.470624, 0.701351], 
                    AbsoluteThickness[1.6]], {
                    LineBox[{{0, 10}, {20, 10}}]}}, {
                    Directive[
                    EdgeForm[
                    Directive[
                    Opacity[0.3], 
                    GrayLevel[0]]], 
                    PointSize[0.5], FontFamily -> "Arial", 
                    Opacity[1.], 
                    RGBColor[0.528488, 0.470624, 0.701351], 
                    AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, 
                    ImageSize -> {20, 10}, PlotRangePadding -> None, 
                    ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}}, 
                 GridBoxAlignment -> {
                  "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, 
                 AutoDelete -> False, 
                 GridBoxDividers -> {
                  "Columns" -> {{False}}, "Rows" -> {{False}}}, 
                 GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
                  GridBoxSpacings -> {
                  "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, 
             GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, 
             AutoDelete -> False, 
             GridBoxItemSize -> {
              "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
             GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], 
            "Grid"], Alignment -> Left, AppearanceElements -> None, 
           ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> 
           "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], {
         FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> 
         False], TraditionalForm]& ), 
      InterpretationFunction :> (RowBox[{"LineLegend", "[", 
         RowBox[{
           RowBox[{"{", 
             RowBox[{
               RowBox[{"Directive", "[", 
                 RowBox[{
                   RowBox[{"FontFamily", "\[Rule]", "\"Arial\""}], ",", 
                   RowBox[{"Opacity", "[", "1.`", "]"}], ",", 
                   InterpretationBox[
                    ButtonBox[
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                    GrayLevel[0], 
                    RectangleBox[{0, 0}]}, {
                    GrayLevel[0], 
                    RectangleBox[{1, -1}]}, {
                    RGBColor[0.368417, 0.506779, 0.709798], 
                    RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 
                    "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, 
                    FrameStyle -> 
                    RGBColor[
                    0.24561133333333335`, 0.3378526666666667, 
                    0.4731986666666667], FrameTicks -> None, PlotRangePadding -> 
                    None, ImageSize -> 
                    Dynamic[{
                    Automatic, 1.35 CurrentValue["FontCapHeight"]/
                    AbsoluteCurrentValue[Magnification]}]], 
                    StyleBox[
                    RowBox[{"RGBColor", "[", 
                    RowBox[{"0.368417`", ",", "0.506779`", ",", "0.709798`"}],
                     "]"}], NumberMarks -> False]], Appearance -> None, 
                    BaseStyle -> {}, BaselinePosition -> Baseline, 
                    DefaultBaseStyle -> {}, ButtonFunction :> 
                    With[{Typeset`box$ = EvaluationBox[]}, 
                    If[
                    Not[
                    AbsoluteCurrentValue["Deployed"]], 
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Mathematical optimizationa is a selection best element, with regard to some \
criterion, from some set of available alternatives. In engineering practice \
it can be a specific parameter of an apparatus.  Also, a common problem in \
engineering is model identification. For example, a computation of a \
parameter in the model from measured data using regression. Fundametally, in \
both cases engineeres work with mathematical models and optimization.\
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