Tag: list

Suppose we have a function $f:Atomathbb{R}$ where $Asubseteq(a,b)$, $a,binmathbb{R}$ and $Ssubseteq A$.

Now suppose $f$ is

$$f(x)=begin{cases}
x^2+1 & xin S_1\
x & xin S_2\
end{cases}$$

Where $S_1$ and $S_2$ are pairwise disjoint subsets of $A=S_1cup S_2$ where $S_1=left{frac{1}{c}+frac{sqrt{2}}{2}:cinmathbb{Z}right}cap(0,1)$, $S_2=left{frac{1}{2^{(d_1)}}+frac{1}{2^{(d_2)}}:d_1,d_2inmathbb{Z}right}cap(0,1)$ and $(a,b)=(0,1)$ where $a=0$ and $b=1$.

Suppose we divide $(0,1)$ into $r$ sub-intervals $left{(x_{i-1},x_i)right}_{i=1}^{r}$, where $a=x_0le x_1 le …le x_i le …le x_{r-1}le x_r=b$; and, if $f$ is defined in that sub-interval we take a point from that sub-interval; (however, if $f$ is not defined, we set $f=0$) and multiply it by measures $mu(S_1cap(x_{i-1},x_i),A)$, for $f(S_1)$, and $mu(S_2cap(x_{i-1},x_i),A)$, for $f(S_2)$,

We define $mu(S_1,A)$ and $mu(S_2,A)$ as the following:

In each sub-interval $(x_{i-1},x_i)$, we cover $S_1cap (x_{i-1},x_i)$ by $m_1$ sub-intervals, of each of the sub-intervals in $left(I_{i}right)_{i=1}^{r}=left{(x_{i-1},x_i)right}_{i=1}^{r}$. We define this as $left(I_{i_1,k}right)_{k=1}^{m_1}$, where $1 le m_1 le maxleft{|S_1|,1right}$. Then cover $S_2cap(x_{i-1},x_i)$ by $m_2$ sub-intervals of each of the sub-intervals in $left(I_{i}right)_{i=1}^{r}=left{(x_{i-1},x_i)right}_{i=1}^{r}$, which we define as $left(I_{i_2,k}right)_{k=1}^{m_2}$, where $1 le m_2 le maxleft{|S_2|,Aright}$. Last, cover $A=S_1cup S_2$ by $n$ sub-intervals of $J=(a,b)$, which we define as $left(J_{k}right)_{k=1}^{n}$ where $1 le n le maxleft{|A|,1right}$. All these sub-intervals have the same length which we define as $cinmathbb{R}^{+}$.

We wish to minimize (or find the infimum) of the total sum of the lengths of the $m_1$ sub-intervals covering $S_1cap(x_{i-1},x_i)$, where the infimum depends on $c$. This means we must set $c$ to specific values before taking the infimum. This would give us $m_1$ divided by $n$. The same is applied with $m_2$ sub-intervals covering $S_2cap (x_{i-1},x_i)$ and the $n$ sub-intervals covering $Acap (x_{i-1},x_i)$. This would give us $m_2$ divided by $n$.

Next we want want to find the infimum of $m_1$ divided by $n$ over $c$, (both $m_1$ and $n$ depends on $c$). We want the same with the infimum of $m_2$ divided by $n$ over $c$. This gives us $mu(S_1cap(x_{i-1},x_i),A)$ and $mu(S_2cap(x_{i-1},x_i),A)$.

To get our average one must take

$$sum_{i=1}^{r}(x_i^2+1)mu(S_1cap(x_{i-1},x_i),A)+sum_{i=1}^{r}(x_i)mu(S_2cap(x_{i-1},x_i),A)$$

Questions

1. How would we do this on Mathematica?
2. How would we generalize this to any piece-wise function defined on a countable set?

My Attempt

Subscript(S, 1)(c_Integer, r_) := 
 Select(Union@
   Flatten(Table(
     1/2^(Subscript(d, 1)) + 1/2^(Subscript(d, 2)), {Subscript(d, 1), 
      1, c}, {Subscript(d, 2), Subscript(d, 1), c})), 
  Between(#, 
    r) &) 

This lists all elements in $S_1$ as c$toinfty$ and groups them into $r$ sub-intervals of $(0,1)$

Subscript(S, 2)(c_Integer, q_) := 
 Select(Union@
   Flatten(Table(
     1/Subscript(u, 1) + Sqrt(2)/2, {Subscript(u, 1), 1, c})), 
  Between(#, 
    q) &) 

This lists all elements of Set $S_2$ as and groups them into $r$ sub-intervals of $(0,1)$

Subscript(f, 1)(x_) := 
 x^2 

This is the function $f=f_1$ on $xin S_1$

Subscript(f, 1, 1)(c_Integer, r_) := 
 Subscript(f, 
  1) /@ (Subscript(S, 1)(c, #) & /@ Partition(Subdivide(r), 2, 1)) 

Groups $S_1$ into $r$ sub-intervals

Subscript(f, 1, 1, 1)(c_Integer, r_) := 
 Subscript(f, 1, 1)(c, r) /. {} -> {0}

Converts null sets to zero

Subscript(f, 1, 1, 1)(30, 100);

Subscript(f, 2)(
  x_) := x 

This is the function $f=f_2$ on $xin S_2$

Subscript(f, 2, 2)(c_Integer, r_) := 
 Subscript(f, 
  2) /@ (Subscript(S, 2)(c, #) & /@ 
    Partition(Subdivide(r), 2, 
     1)) (* Groups Subscript(S, 2) into r sub-intervals *)
Subscript(f, 2, 2, 2)(c_Integer, r_) := 
 Subscript(f, 2, 2)(c, r) /. {} -> {0} 

Converts null sets to zero

In(205):= 
Subscript(I, 1)(r_) := 
 Block({m = 10^16}, 
   With({k = Ceiling@Log2@N@m + 2}, 
    Total(#) + 1 - Last(#) &@
     Unitize@BinCounts(
       Total(Tuples(2^Range(-k, -1), 2), {2}), {Min(#1) &, Max(#1) &, 
        1/m}))) /@ Partition(Subdivide(r), 2, 1) 

This is supposed to give $m_1$ from where the total sum of the length of all $m_1$ sub-intervals, of each of the $r$ sub-intervals, covering the intersection of $S_1$ and the $r$ sub-intervals are as small as possible.

In(206):= Subscript(I, 1)(10)

During evaluation of In(206):= BinCounts::bins: The bin specification {Min(#1)&,Max(#1)&,1/10000000000000000} is not a list of 2 or 3 real values.

Out(206)= If(29155653097335068495 === $SessionID, 
Out(206), Message(
MessageName(Syntax, "noinfoker")); Missing("NotAvailable"); Null)

Unfortunately, there is something wrong with this function.

Subscript(I, 2)(r_) := 
 Block({m = 10^16}, 
   With({k = Ceiling@N@m + 2}, 
    Total(#) + 1 - Last(#) &@
     Unitize@BinCounts(
       Total(Tuples(Range(-k, -1), 1), {2}), {Min(#1) &, Max(#1) &, 
        1/m}))) /@ 
  Partition(Subdivide(r), 2, 
   1) 

This is supposed to give $m_2$ from where the total sum of the length of all $m_2$ sub-intervals, of each of the r sub-intervals, covering the intersection of $S_2$ and the $r$ sub-intervals are as small as possible

Subscript(I, 2)(10)
J(r_) := Sum(
  Subscript(I, 1)(i) + Subscript(I, 2)(i), {i, 1, 
   r}) 

This is supposed to give $n$ from where the total sum of the length of all $n$ sub-intervals, of each of the $r$ sub-intervals, covering the intersection of $A$ and the $r$ sub-intervals are as small as possible

Subscript(Avg, 1)(r_) := 
 Sum(Subscript(f, 1, 1, 1)(30, i)((i, 1))*
   Subscript(I, 1)(r)((i))/J(r), {i, 1, 
   r}) (* Subscript(I, 1)(r)((i))/A(r) is mu(Subscript(S, 1),A) as m
(Rule) Infinity, the whole sum is supposed to give the average over 
Subscript(f, 1) *)
Subscript(Avg, 2)(r_) := 
 Sum(Subscript(f, 2, 2, 2)(30, i)((i, 1))*
   Subscript(I, 2)(r)((i))/J(r), {i, 1, 
   r}) (* Subscript(I, 1)(r)((i))/A(r) is mu(Subscript(S, 2),A) as m
(Rule)Infinity, the whole sum is supposed to give the average over 
Subscript(f, 2) *)
Subscript(Avg, 1)(10000) + 
 Subscript(Avg, 1)(10000) (* This is the combined total sum *)

During evaluation of In(217):= General::nomem: The current computation was aborted because there was insufficient memory available to complete the computation.

During evaluation of In(217):= Throw::sysexc: Uncaught SystemException returned to top level. Can be caught with Catch((Ellipsis), _SystemException).

Out(218)= SystemException("MemoryAllocationFailure")

During evaluation of In(217):= BinCounts::bins: The bin specification {Min(#1)&,Max(#1)&,1/10000000000000000} is not a list of 2 or 3 real values.

During evaluation of In(217):= BinCounts::bins: The bin specification {Min(#1)&,Max(#1)&,1/10000000000000000} is not a list of 2 or 3 real values.

During evaluation of In(217):= General::nomem: The current computation was aborted because there was insufficient memory available to complete the computation.

During evaluation of In(217):= Throw::sysexc: Uncaught SystemException returned to top level. Can be caught with Catch((Ellipsis), _SystemException).

Out(222)= SystemException("MemoryAllocationFailure")

Unfortunately, as you see here I am not getting any results.

How do we fix the errors in my code? Is my attempt correct?