# Experimenting with Genetic Algorithms

#### September 13, 2016

Genetic algorithm trying to find a series of four mathematical operations (e.g. -3*4/7+9) that would result in the number 42.

I’m teaching a numerical methods class that’s partly an introduction to programming, and partly a survey of numerical solutions to different types of problems students might encounter in the wild. I thought I’d look into doing a session on genetic algorithms, which are an important precursor to things like networks that have been found to be useful in a wide variety of fields including image and character recognition, stock market prediction and medical diagnostics.

The ai-junkie, bare-essentials page on genetic algorithms seemed a reasonable place to start. The site is definitely readable and I was able to put together a code to try to solve its example problem: to figure out what series of four mathematical operations using only single digits (e.g. +5*3/2-7) would give target number (42 in this example).

The procedure is as follows:

• Initialize: Generate several random sets of four operations,
• Test for fitness: Check which ones come closest to the target number,
• Select: Select the two best options (which is not quite what the ai-junkie says to do, but it worked better for me),
• Mate: Combine the two best options semi-randomly (i.e. exchange some percentage of the operations) to produce a new set of operations
• Mutate: swap out some small percentage of the operations randomly,
• Repeat: Go back to the second step (and repeat until you hit the target).

And this is the code I came up with:

genetic_algorithm2.py

```''' Write a program to combine the sequence of numbers 0123456789 and
the operators */+- to get the target value (42 (as an integer))
'''

'''
Procedure:
1. Randomly generate a few sequences (ns=10) where each sequence is 8
charaters long (ng=8).
2. Check how close the sequence's value is to the target value.
The closer the sequence the higher the weight it will get so use:
w = 1/(value - target)
3. Chose two of the sequences in a way that gives preference to higher
weights.
4. Randomly combine the successful sequences to create new sequences (ns=10)
5. Repeat until target is achieved.

'''
from visual import *
from visual.graph import *
from random import *
import operator

# MODEL PARAMETERS
ns = 100
target_val = 42 #the value the program is trying to achieve
sequence_length = 4  # the number of operators in the sequence
crossover_rate = 0.3
mutation_rate = 0.1
max_itterations = 400

class operation:
def __init__(self, operator = None, number = None, nmin = 0, nmax = 9, type="int"):
if operator == None:
n = randrange(1,5)
if n == 1:
self.operator = "+"
elif n == 2:
self.operator = "-"
elif n == 3:
self.operator = "/"
else:
self.operator = "*"
else:
self.operator = operator

if number == None:
#generate random number from 0-9
self.number = 0
if self.operator == "/":
while self.number == 0:
self.number = randrange(nmin, nmax)
else:
self.number = randrange(nmin, nmax)
else:
self.number = number
self.number = float(self.number)

def calc(self, val=0):
# perform operation given the input value
if self.operator == "+":
val += self.number
elif self.operator == "-":
val -= self.number
elif self.operator == "*":
val *= self.number
elif self.operator == "/":
val /= self.number
return val

class gene:

def __init__(self, n_operations = 5, seq = None):
#seq is a sequence of operations (see class above)
#initalize
self.n_operations = n_operations

#generate sequence
if seq == None:
#print "Generating sequence"
self.seq = []
self.seq.append(operation(operator="+"))  # the default operation is + some number
for i in range(n_operations-1):
#generate random number
self.seq.append(operation())

else:
self.seq = seq

self.calc_seq()

#print "Sequence: ", self.seq
def stringify(self):
seq = ""
for i in self.seq:
seq = seq + i.operator + str(i.number)
return seq

def calc_seq(self):
self.val = 0
for i in self.seq:
#print i.calc(self.val)
self.val = i.calc(self.val)
return self.val

def crossover(self, ingene, rate):
# combine this gene with the ingene at the given rate (between 0 and 1)
#  of mixing to create two new genes

#print "In 1: ", self.stringify()
#print "In 2: ", ingene.stringify()
new_seq_a = []
new_seq_b = []
for i in range(len(self.seq)):
if (random() < rate): # swap
new_seq_a.append(ingene.seq[i])
new_seq_b.append(self.seq[i])
else:
new_seq_b.append(ingene.seq[i])
new_seq_a.append(self.seq[i])

new_gene_a = gene(seq = new_seq_a)
new_gene_b = gene(seq = new_seq_b)

#print "Out 1:", new_gene_a.stringify()
#print "Out 2:", new_gene_b.stringify()

return (new_gene_a, new_gene_b)

def mutate(self, mutation_rate):
for i in range(1, len(self.seq)):
if random() < mutation_rate:
self.seq[i] = operation()

def weight(target, val):
if val <> None:
#print abs(target - val)
if abs(target - val) <> 0:
w = (1. / abs(target - val))
else:
w = "Bingo"
print "Bingo: target, val = ", target, val
else:
w = 0.
return w

def pick_value(weights):
#given a series of weights randomly pick one of the sequence accounting for
# the values of the weights

# sum all the weights (for normalization)
total = 0
for i in weights:
total += i

# make an array of the normalized cumulative totals of the weights.
cum_wts = []
ctot = 0.0
cum_wts.append(ctot)
for i in range(len(weights)):
ctot += weights[i]/total
cum_wts.append(ctot)
#print cum_wts

# get random number and find where it occurs in array
n = random()
index = randrange(0, len(weights)-1)
for i in range(len(cum_wts)-1):
#print i, cum_wts[i], n, cum_wts[i+1]
if n >= cum_wts[i] and n < cum_wts[i+1]:

index = i
#print "Picked", i
break
return index

def pick_best(weights):
# pick the top two values from the sequences
i1 = -1
i2 = -1
max1 = 0.
max2 = 0.
for i in range(len(weights)):
if weights[i] > max1:
max2 = max1
max1 = weights[i]
i2 = i1
i1 = i
elif weights[i] > max2:
max2 = weights[i]
i2 = i

return (i1, i2)

# Main loop
l_loop = True
loop_num = 0
best_gene = None

##test = gene()
##test.print_seq()
##print test.calc_seq()

# initialize
genes = []
for i in range(ns):
genes.append(gene(n_operations=sequence_length))
#print genes[-1].stringify(), genes[-1].val

f1 = gcurve(color=color.cyan)

while (l_loop and loop_num < max_itterations):
loop_num += 1
if (loop_num%10 == 0):
print "Loop: ", loop_num

# Calculate weights
weights = []
for i in range(ns):
weights.append(weight(target_val, genes[i].val))
# check for hit on target
if weights[-1] == "Bingo":
print "Bingo", genes[i].stringify(), genes[i].val
l_loop = False
best_gene = genes[i]
break
#print weights

if l_loop:

# indicate which was the best fit option (highest weight)
max_w = 0.0
max_i = -1
for i in range(len(weights)):
#print max_w, weights[i]
if weights[i] > max_w:
max_w = weights[i]
max_i = i
best_gene = genes[max_i]
##        print "Best operation:", max_i, genes[max_i].stringify(), \
##              genes[max_i].val, max_w
f1.plot(pos=(loop_num, best_gene.val))

# Pick parent gene sequences for next generation
# pick first of the genes using weigths for preference
##        index = pick_value(weights)
##        print "Picked operation:  ", index, genes[index].stringify(), \
##              genes[index].val, weights[index]
##
##        # pick second gene
##        index2 = index
##        while index2 == index:
##            index2 = pick_value(weights)
##        print "Picked operation 2:", index2, genes[index2].stringify(), \
##              genes[index2].val, weights[index2]
##

(index, index2) = pick_best(weights)

# Crossover: combine genes to get the new population
new_genes = []
for i in range(ns/2):
(a,b) = genes[index].crossover(genes[index2], crossover_rate)
new_genes.append(a)
new_genes.append(b)

# Mutate
for i in new_genes:
i.mutate(mutation_rate)

# update genes array
genes = []
for i in new_genes:
genes.append(i)

print
print "Best Gene:", best_gene.stringify(), best_gene.val
print "Number of iterations:", loop_num
##

```

When run, the code usually gets a valid answer, but does not always converge: The figure at the top of this post shows it finding a solution after 142 iterations (the solution it found was: +8.0 +8.0 *3.0 -6.0). The code is rough, but is all I have time for at the moment. However, it should be a reasonable starting point if I should decide to discuss these in class.

Citing this post: Urbano, L., 2016. Experimenting with Genetic Algorithms, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Vpython on the Web with Glowscript

#### February 25, 2016

Using glowscript.org, we can now put vpython programs online. Here’s a first shot of the coil program:

http://www.glowscript.org/#/user/lurbano/folder/Private/program/magnet-coil.py

Moving the magnet through a wire coil creates an electric current in the wire.

Some changes to the standard vpython language must be used and there are some limitations, but it seems to work.

Citing this post: Urbano, L., 2016. Vpython on the Web with Glowscript, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Dilation

#### December 14, 2015

Dilation (scaling) of a quadrilateral by 2x.

A quick program that animates scaling (dilation) of shapes by scaling the coordinates. You type in the dilation factor.

dilation.py

```from visual import *

#axes
xmin = -10.
xmax = 10.
ymin = -10.
ymax = 10.
xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

#tick marks
tic_dx = 1.0
tic_h = .5
for i in arange(xmin,xmax+tic_dx,tic_dx):
tic = curve(pos=[(i,-0.5*tic_h),(i,0.5*tic_h)])
for i in arange(ymin,ymax+tic_dx,tic_dx):
tic = curve(pos=[(-0.5*tic_h,i),(0.5*tic_h,i)])

#stop scene from zooming out too far when the curve is drawn
scene.autoscale = False

# define curve here
shape = curve(pos=[(-1,2), (5,3), (4,-1), (-1,-1)])

shape.append(pos=shape.pos[0])
shape.color = color.yellow
shape.visible = True

#dilated shape
for i in shape.pos:
dshape.append(pos=i)

#label
note = label(pos=(5,-8),text="Dilation: 1.0", box=False)
intext = label(pos=(5,-9),text="> x", box=False)

#scaling lines
l_scaling = False
slines = []
for i in range(len(shape.pos)):

#animation parameters
animation_time = 1. #seconds
animation_smootheness = 30
animation_rate = animation_smootheness / animation_time

x = ""
while 1:
#x = raw_input("Enter Dilation: ")
if scene.kb.keys: # event waiting to be processed?
s = scene.kb.getkey() # get keyboard info

#print s
if s <> '\n':
x += s
intext.text = "> x "+x
else:
try:
xfloat = float(x)
note.text = "Dilation: " + x

endpoints = []
dp = []
for i in shape.pos:
endpoints.append(float(x) * i)
dp.append((endpoints[-1]-i)/animation_smootheness)
#print "endpoints: ", endpoints
#print "dp:        ", dp
for i in range(animation_smootheness):
for j in range(len(dshape.pos)):
dshape.pos[j] = i*dp[j]+shape.pos[j]
rate(animation_smootheness)
if slines:
for i in range(len(shape.pos)):
slines[i].pos[1] = vector(0,0)
slines[i].pos[-1] = dshape.pos[i]

for i in range(len(shape.pos)):
dshape.pos[i] = endpoints[i]
slines[i].pos[-1] = dshape.pos[i]

for i in range(len(shape.pos)-1):
print shape.pos[i], "--->", dshape.pos[i]

except:
#print "FAIL"
failed = True
intext.text = "> x "
x = ""

```

Citing this post: Urbano, L., 2015. Dilation, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# A Visual Introduction to Differentiation (using vpython)

#### October 15, 2015

Screen capture: Enter an x value and the program calculates the slope for the function and draws the tangent line.

This quick program is intended to introduce differentiation as a way of finding the slope of a line. Students know how to find the slope of a tangent line at least conceptually (by drawing). We pick a curve: in this case:

$f(x) = x^2$

then enter values of x in the program to see how x, the function value and the differential compare to each other.

x f(x) f'(x)
0.5 0.25 1
1 1 2
2 2 4
3 9 6

Because it’s quick you have to change the function in the code, and enter the values for x in the python shell.

With a sin curve.

differentiation_intro_numeric.py

```from visual import *

class tangent_line:
def __init__(self):
self.dx = 0.1
self.line = curve()
self.tangent_line = curve()
self.point.visible = False
self.label = label(pos=(-5,-8))

'''CHANGE FUNCTION (y) HERE'''
# the original function
def f(self, x):
#y = sin(x)
y = x**2
return y
'''END CHANGE FUNCTION HERE'''

def find_slope(self, x):
sdx = .00001
m = (self.f(x+sdx)-self.f(x))/sdx
return round(m,3)

def draw(self):
for x in arange(xmin, xmax+self.dx, self.dx):
self.line.append(pos=(x, self.f(x)))

def draw_tangent(self, x):
m = self.find_slope(x)
y = self.f(x)
b = y - m * x
print "When x = ", x, " slope = ", m
self.label.text = "point: (%1.2f, %1.2f)\nSlope: %1.2f" % (x,y,m)
self.plot_point(x)

#draw tangent
self.tangent_line.visible = False
self.tangent_line = curve(pos=[(xmin,m*xmin+b),(xmax,m*xmax+b)], color=color.yellow)

def plot_point(self, x):
self.point.visible = True
self.point.pos = (x, self.f(x))

#axes
xmin = -10.
xmax = 10.
ymin = -10.
ymax = 10.
xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

#tick marks
tic_dx = 1.0
tic_h = .5
for i in arange(xmin,xmax+tic_dx,tic_dx):
tic = curve(pos=[(i,-0.5*tic_h),(i,0.5*tic_h)])
for i in arange(ymin,ymax+tic_dx,tic_dx):
tic = curve(pos=[(-0.5*tic_h,i),(0.5*tic_h,i)])

#stop scene from zooming out too far when the curve is drawn
scene.autoscale = False

# draw curve
func = tangent_line()
func.draw()

# get input
while 1:
xin = raw_input("Enter x value: ")
func.draw_tangent(float(xin))

```

Citing this post: Urbano, L., 2015. A Visual Introduction to Differentiation (using vpython), Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Coding Online on the Coding Ground

#### August 18, 2015

For some of my students with devices like Chromebooks, it has been a little challenging finding ways for them to do coding without a simple, built-in interpreter app. One interim option that I’ve found, and like quite a bit is the TutorialsPoint Coding Ground, which has online interfaces for quite a number of languages that are great for testing small programs, including Python.

Screen capture of Python coding at Tutorial Point’s Coding Ground.

Citing this post: Urbano, L., 2015. Coding Online on the Coding Ground, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# How to Write a Web Page from Scratch

#### March 14, 2015

```<html>

<body>

</body>
</html>

```

I gave a quick introduction to writing HTML web pages over the last interim. The basic outline of a web page can be seen above (but there’s no content in there, so if you tried this file in a web browser you’d get a blank page). Everything is set between tags. The open html tag (<html>) starts the document, and the close html tag (</html>) ends it.

The head section (<head>) holds the information needed to set up the page, while the body section (<body>) holds the stuff that’s actually on the page.

For example, if I wanted to create a heading and a paragraph of text, I would just put it in the body, like so:

```<html>

<body>

This is my text.

</body>
</html>

```

Note that the heading is placed between h1 tags. And the result should look like this:

Basic webpage.

Now save this file as an html file, which means save it as a plain text file using a .html extension (e.g. test.html) and open the file in a browser.

Note on Editors: I’ve found that the best way to do this is by using a simple coding editor. I recommend my students use GEdit on Linux, Smultron (free version here) or GEdit on Mac, and Notepad++ on Windows. You don’t want to use something like Microsoft Word, because complex word processing software like it, LiberOffice and Pages need to save a lot of other information about the files as well (like the font style, who created the file etc.).

## CSS Styling

But we typically don’t want something so plain, so we’ll style our heading using CSS. Usually, we’ll want all of our headings to be the same so we set up a style to make all of our headings blue. So we’ll add a styling section to our header like so:

```<html>

<style type="text/css">
h1 { color: blue; }
</style>

<body>

This is my text.

</body>
</html>

```

Which gives:

There are a few named colors like blue, but if you want more freedom, you can use the hexadecimal color codes like “#A3319F” to give a kind of purple color (hexadecimals are numbers in base 16).

## DIV’s to Divide

Finally, we’ll often want pages with separate blocks of information, so let’s make a little column for our heading and paragraph. We’ll make the background yellowish, put a border around it, and give it a set width.

To do this we’ll place our heading and paragraph into a DIV tag, which does not do anything but encapsulate the part of the web page we want into sections.

```<html>

<style type="text/css">
h1 { color: blue; }
</style>

<body>
<div id="section1">

This is my text.
</div>
</body>
</html>
```

Note that I’ve given the div tag an id (“section1), this is so that I can refer to that section only in the styling section using “#section”:

```<html>

<style type="text/css">
h1 { color: blue; }

#section1 {
background-color: #A2A82D;
border: solid black 2px;
width: 400px;
}
</style>

<body>
<div id="section1">

This is my text.
</div>
</body>
</html>
```

to give:

More styling.

## Unleashed

With that introduction, I set students loose to make their own web pages. They had to make two page, both linking to the other, and one of them being an “About Me” page.

There are lots of places on line to find out what HTML tags and CSS styles are available and how to use them, so my goal was to introduce students to the language they needed to know.

One issue that came up was the use of copyrighted images. Current adolescents see everything online as part of a sharing culture–most of my students for this lesson had Pintrest accounts–and it took some explanation for them to understand why they should not use that cute gif of a bunny eating a slice of pizza without getting permission from the author (or at least finding out if the artwork was free to use).

Finally, I did do a quick intro on how to using JavaScript (with Jquery) to make their pages more interactive, but given that we only had two days for this project, that was pushing things a little too far.

Citing this post: Urbano, L., 2015. How to Write a Web Page from Scratch, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Limiting Chemical Reactions

#### January 1, 2015

Figuring out the limiting reactant in a chemical reaction integrates many of the basic chemistry concepts including: unit conversions from moles to mass and vice versa; the meaning of chemical formulas; and understanding the stoichiometry of chemical reactions. So, since we’ll need a number of these, I wrote a python program to help me design the questions (and figure out the answers).

Program examples come zipped because they require the program file and the elements_database.py library:

# Baking Powder and Vinegar (Common Molecules)

Limiting_component-Common.py: This has the baking powder and vinegar reaction limited by 5 g of baking soda. It’s nice because it uses a few pre-defined “common molecules” (which are defined in the elements_database.py library.

You enter the reactants and products and the program checks if the reaction is balanced, then calculates the moles and masses based on the limiting component, and finally double checks to make sure the reaction is mass balanced.

Limiting_component-Common.py

```from elements_database import *
import sys

print "LIMITING REACTANT PROGRAM"
print
print "  Determines the needed mass and moles of reactants and products if reaction is limited by one of the components"

c = common_molecules()

'''Create Reaction'''
rxn = reaction()
#       molecule: from molecule class in elements_database
#       stoichiometry: integer number

'''Print out the reaction'''
print
print "Chemical Formula"
print "  " + rxn.print_reaction()
print

'''Check if reaction is balanced'''
balanced = rxn.check_for_balance(printout=True)

'''Calculate limits of reaction'''
if balanced:
rxn.limited_by_mass(c.baking_soda, 5, printout=True)

```

Outputs results in the Results table (using scientific notation):

```LIMITING REACTANT PROGRAM

Determines the needed mass and moles of reactants and products if reaction is limited by one of the components

Chemical Formula
NaHCO3 + HCl  --> CO2 + H2O + NaCl

Check for balance
---------------------------------------------
| Element | Reactants | Products | Difference |
---------------------------------------------
|   Na    |    1      |    -1    |     0      |
|   H     |    2      |    -2    |     0      |
|   C     |    1      |    -1    |     0      |
|   O     |    3      |    -3    |     0      |
|   Cl    |    1      |    -1    |     0      |
---------------------------------------------
Balance is:  True

Given: Limiting component is 5 g of NaHCO3.
Molar mass = 84.00676928
Moles of NaHCO3 = 0.0595190130849

Results
------------------------------------------------------------------
| Molecule | Stoich.*| Molar Mass (g) | Moles req. |    Mass (g)   |
------------------------------------------------------------------
|NaHCO3    |    1    |    84.0068     |  5.952e-02 |   5.000e+00   |
|HCl       |    1    |    36.4602     |  5.952e-02 |   2.170e+00   |
|CO2       |    -1   |    44.0096     |  5.952e-02 |   2.619e+00   |
|H2O       |    -1   |    18.0156     |  5.952e-02 |   1.072e+00   |
|NaCl      |    -1   |    58.4418     |  5.952e-02 |   3.478e+00   |
------------------------------------------------------------------
* negative stoichiometry means the component is a product

Final Check: Confirm Mass balance:
Reactants:    7.1701 g
Products:    -7.1701 g
--------------------------
=      0.0000 g
--------------------------
```

# General Example

If not using the common molecules database, you need to define the components in the reaction as molecules yourself. This example reacts magnesium sulfate and sodium hydroxide, and limits the reaction with 20 g of magnesium sulfate.

Limiting_component-General.zip

The main file is:

Chem_Exam-limiting.py

```from elements_database import *
import sys

print "LIMITING REACTANT PROGRAM"
print
print "  Determines the needed mass and moles of reactants and products if reaction is limited by one of the components"

# create reaction

rxn2 = reaction()
#       molecule: from molecule class in elements_database
#       stoichiometry: integer number

'''Print out the reaction'''
print
print "Chemical Formula"
print "  " + rxn2.print_reaction()
print

'''Check if reaction is balanced'''
balanced = rxn2.check_for_balance(printout=True)

'''Calculate limits of reaction'''
if balanced:
rxn2.limited_by_mass(molecule("Mg:1,S:1,O:4"), 20, True)

# print out masses
print "Masses Involved in Reaction"
print "  Reactants:"
for i in rxn2.reactants:
#print "rxn", i
print "    {m}: {g} g".format(m=i.molecule.print_formula().ljust(10), g=i.mass)
print "  Products:"
for i in rxn2.products:
#print "rxn", i
print "    {m}: {g} g".format(m=i.molecule.print_formula().ljust(10), g=-i.mass)

```

This program is slightly different from the common molecules example in that, at the end, it prints out masses calculated in a more easily readable format in addition to the other data.

```Masses Involved in Reaction
Reactants:
MgSO4     : 20.0 g
NaOH      : 13.2919931774 g
Products:
MgO2H2    : 9.69066512026 g
Na2SO4    : 23.6013280571 g
```

When I have some time I’ll convert this to JavaScript like the molecular mass example.

Citing this post: Urbano, L., 2015. Limiting Chemical Reactions, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Molar Mass of Molecules

#### December 5, 2014

Calculate the molar mass of a molecule:

The notation for the chemical formula is a little funky: you put the element symbol and then the number of atoms separated by a colon; each element/number of atoms pair are separated by commas, so sodium chloride (NaCl) would be “Na:1,Cl:1“.

This will have to do until I can write something to parse the regular chemical formula notation.

On the plus side, you can link to a specific molecular mass calculation by adding the formula to the url. So magnesium chloride (MgCl2) can be found with this url:

http://soriki.com/js/chem/chem_db/molecular_mass.html?formula=Mg:1,Cl:2

Citing this post: Urbano, L., 2014. Molar Mass of Molecules, Retrieved May 29th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.