performance – sum of multiple piecewise functions python

suppose I have a few lists

b_wi=((1,2,3,4),(6,7,8,9,10,11)) #b_wi is a subset of x
f_wi=((5,4,2,7,9),(5,4,3,7,2,3,4))
x=((1,2,3,4,5,6,7,8,9,2,5,3),(1,24,36,42,35,6,7,8,91,2,5,3))
#the following two are step functions formed by the above lists.

'''
F1 = f_wi(0)(0) if x< b_wi(0)(0) ;
     f_wi(0)(1) if  x< b_wi(0)(0) <=x< b_wi(0)(1);
      ...;
     f_wi(0)(-1) if x>= b_wi(1)(-1)


F2 = f_wi(1)(0) if x< b_wi(1)(0) ;
     f_wi(1)(1) if  x< b_wi(1)(0) <=x< b_wi(1)(1);
      ...;
     f_wi(1)(-1) if x>= b_wi(1)(-1)
'''

Now I want to get max (F1+F2) and the corresponding interval. I did some searching and found this : Evaluate sum of step functions

However, since the length of intervals is not the same for these step functions, I cannot apply the solution in the link directly. Instead, I did this:

import numpy as np
from pandas.core.common import flatten
def func(x,b,f): 
    return f(np.searchsorted(b,x,side='right'))

intval= np.unique(list(flatten(b_wi)))
x=np.concatenate(((-10000),(intval(:-1)+intval(1:))/2,(10000)))  #b_wi is a subset of x. That is why I can use this.
a=np.zeros((len(x)))
for b, f in zip(b_wi,f_wi):
    a=a+ func(x,b,np.asarray(f))
print(a/2)

Now I get can get the maximum of (F1+F2) using

np.amax(a)

and I can get the interval as well.

This is just a simple example I used to illustrate my question. My actual lists are longer than these and there are 100000 step functions. Since I ‘flatten’ ‘b_wi’ in order to find the corresponding interval, the length of ‘intval’ becomes too large. Hence, my method is too slow. Does anyone know how I could speed it up? I feel like I am using the wrong method. Thank you very much.

differential equations – Piecewise functions and ODEs

I had a ODE involving Piecewise functions, which mathematica solved to give (in input form)

{{Bph -> Function({r}, -((2*((-j0)*Pi*r^2 - a^2*j0*Pi*UnitStep(-a + r) + j0*Pi*r^2*UnitStep(-a + r)))/r))}}.

Now I want to use this in solving another equation, say mhseqn2, which doesn’t involve any other Piecewise stuff, in which this function Bph appears as a coefficient. I had named the above out put as sol1, hence ran mhseqn2 //. sol1((1)) // Simplify. The output did not involve something Piecewise or the UnitStep function, but was

(Derivative(1)(p)(r) == 0 && a < r) || (2*j0^2*Pi*r == Derivative(1)(p)(r) && a > r),

where p is the function I have to solve the differential equation for.

My question is how can this be converted to something DSolve understands, like a piecewise function. In this case the answer can be eyeballed, but what about something more complicated?

ds.dynamical systems – On topological entropy of continuous piecewise linear maps

Let $f:(0,1) to (0,1)$ be a continuous piecewise linear function with $k$ pieces (say the turning points are $c_1,…c_{k}$ and slopes $s_1,…,s_k$). I am curious to see if $c_1,…,c_k$ and $s_1,…,s_k$ are chosen uniformly at random from (0,1), if $f$ will have topological entropy positive with probability $p>0.$ Can you suggest maybe some related literature on this? I have read papers on the field but none of them deals with “random” functions.

How to rigorously prove from set theory that piecewise functions exist?

Suppose we have two functions $f$ and $g$ from the set $S$ into the set $T$. Suppose further that we have subsets $S_1$ and $S_2$ of $S$ which are disjoint and whose union is $S$. How does one rigorously prove, from ZFC set theory, that there is a unique function $h$ which agrees with $f$ on $S_1$ and agrees with $g$ on $S_2$?

reference request – Embedding theorems for fractional Sobolev spaces $W^{s,p}(Gamma)$ where $Gamma$ is closed piecewise $C^1$ curve in $Bbb R^2$

I am interested in embedding theorems for the fractional Sobolev space $W^{s,p}(Gamma)$ where $Gamma$ is closed piecewise $C^1$ curve in $Bbb R^2$ such as the boundary of a triangle or rectangle. What are basic results for this? Also, is there some $p$ for which $W_{s,p}(Gamma)subset C(Gamma)?$

Let’s use $Gamma=partial Omega$ where $Omega$ is a square as our basic example.
This is supposedly a smooth manifold as it is homemorphic to a circle, is that right? So we can define $W^{s,p}(Gamma)$ as Brezis mentions in the comments of Chapter 9 in his functional analysis books.

piecewise – Expressing $f(x)=-x ;;; -2leq x leq 0\f(x)=x ;;; 0

Can we express functions such as:

$$f(x)=-x quad -2leq x leq 0\f(x)=x quad 0< x leq 2\f(x+4)=f(x)$$

On Mathematica? I tried something with piecewise functions but it didn’t work. Namely, I couldn’t make the last condition. I tried:

f(x_) := Piecewise({{-x, -2 <= x <= 0 }, {x, 0 < x <= 2 }})
f(x_ + 4) := f(x)

plotting – Different kind of PlotStyle via Piecewise

I have a Piecewise. For vertical line I have to add Exclusions -> None.

    a1 = 1; 
    b1 = 2; 
    c1 = 2; 
    L(x_) := (x - a1)/(b1 - a1); 
    R(x_) := (c1 - x)/(c1 - b1); 
    A(x_) = Simplify(Piecewise({{0, x < a1}, {L(x), a1 <= x <= b1}, {R(x), Inequality(b1, LessEqual, x, Less, c1)}, {0, x >= c1}}, 0)); 
    p1 = Plot(A(x), {x, a1, c1 + 0.1}, PlotStyle -> {Black, Dashed}, Exclusions -> None)

What are the other options available? Can I make dashed to * or o.

Finding probabilities in a piece-wise distribution function

Suppose I have a cumulative distribution function,
enter image description here

How do I find P(0.25<X<1) since the given distribution function is not differentiable at points like x=0.5 ?

Modeling a piece-wise objective function for a linear program?

I am attempting to design a linear program where I optimize the amount of money I make by selling good $A$. Selling units of $A$ below a threshold $t$ results in an income of $a$ dollars per unit of $A$ and selling quantities greater than that threshold $t$ results in an income of $b$ dollars per unit.

How might I model this as a linear objective function?

how to plot the amplitude and phase spectrum of Fourier Transform of periodic function Abs[Sin[t]] and corresponding non-periodic piecewise function

the two function

x(t_) := Piecewise({{Sin(t) , 0 <= t <= Pi}}) # will get frequency function X(w)
xhat(t_) := RealAbs(Sin(t)) # will get fourier coeff ck

I want to show the amplitude-frequency relations graph plotted as in Fig.D.2 between periodic function xhat(t_) and corresponding non-periodic function x(t_)
enter image description here

Amplitude and Phase of a Periodic Signal (Fourier Series)

how to plot the amplitude and phase spectrum of a Fourier Transform in this specific pattern?

similar answers does not work for me.