text: 0 ⋮ Vote. A matrix is a two-dimensional array of numbers. Create Arrays of Random Numbers. Surface plots are useful for visualizing matrices that are too large to display in numerical form and for graphing functions of two variables. Generate a 2x3 matrix of random: real numbers, each in range 0 to 1; real numbers, each in range 0 to 10; integers, each in range 5 to 20. 0 ⋮ Vote. It's right there built into MATLAB. In the following example, a 2 x 4 matrix of random integers in the range of [1, 10] is created. This example shows how to create an array of random integer values that are drawn from a discrete uniform distribution on the set of numbers –10, –9,...,9, 10. Learn more about matrix, random number generator positive semidefinite matrix random number generator. That is a different matter than constructing matrices whose entries have a uniform distribution within a given range and the matrix as a whole is PD. Save the current state of the random number generator and create a 1-by-5 vector of random numbers. Use rand, randi, randn, and randperm to create arrays of random numbers.. Random Numbers Within a Specific Range. Indexing is a key to the effectiveness of MATLAB at capturing matrix-oriented ideas in … We will create each and every kind of random matrix using NumPy library one by one with example. Such matrices appear to be relatively rare to start with; to require uniform random distribution on the entries appears to make them difficult to generate. How to generate a random matrix ?. does someone know an easy way or command to do this in matlab ? Answered: Image Analyst on 23 Dec 2014 ... And since MATLAB passes implicitly by reference anyway, it won't require a memory copy unless you … y = range(X,vecdim) returns the range over the dimensions specified in the vector vecdim.For example, if X is a matrix, then range(X,[1 2]) is the range of all elements in X because every element of a matrix is contained in the array slice defined by dimensions 1 and 2. Vote. This MATLAB function returns an n-by-n distributed matrix with random integer values in the range [1,imax]. For example, let us create a 4-by-5 matrix a − The following command creates a matrix of random integers of size m x n in a range from 1 to x. There is a help facility in MATLAB (beyond the Answers forum). 0. Random Numbers Within a Specific Range. Be careful not to confuse rand with randn, which produces Gaussian random variables. Indexing into a matrix is a means of selecting a subset of elements from the matrix. Let’s get started. Generate Random Numbers. Random Number Generator is the creation of random numbers without any decision or noticeable patterns among them. MATLAB: Generating a random matrix with range. How can normalize matrix in range [-1,1]?. This example shows how to create an array of random floating-point numbers that are drawn from a uniform distribution in the open interval (50, 100). Question: (a) Using MATLAB's Randi Function, Create A 12 X 12 Matrix Of Random Integers, Each In The Inclusive Range From - 10 To 10, And Assign The Matrix To Variable A. In matlab, one can generate a random number chosen uniformly between 0 and 1 by x = rand(1) To obtain a vector of n random numbers, type x = rand(1,n) If you type x = rand(n) you get a n-by-n matrix of random numbers, which could be way too big. Hi, I am trying to generate 12*2 matrix. Follow 90 views (last 30 days) AA on 23 Dec 2014. Vote. Answered: Image Analyst on 23 Dec 2014 ... And since MATLAB passes implicitly by reference anyway, it won't require a memory copy unless you … It is used in many programming languages for the generation of random values within the specified range. MATLAB: How to generate a random positive semi-definite matrix of certain size with real numbers in a certain range. Random Integers. So all we have to do is generate an initial random matrix with full rank and we can then easily find a positive semi-definite matrix derived from it. I only want to pick one element which is what the above does. (b) Create A Row Vector B Storing In Each Of Its Elements The Number Of Zeros Occurring In The Corresponding Column Of Matrix A. 0. The rank gives a measure of the dimension of the range or column space of the matrix, which is the collection of all linear combinations of the columns. Nearly all random matrices are full rank, so the loop I show will almost always only iterate once and is very very unlikely … Can you help me out with this. Generating a random matrix with range. This MATLAB function returns an n-by-n distributed matrix with random integer values in the range [1,imax]. Learn more about matrix normalization selecting a specific range in a matrix and cell array. Generating a random matrix with range. I'm able to generate a random integer however it always returns a negative value and a zero and not in the expected range. This example shows how to create an array of random floating-point numbers that are drawn from a … I have a matrix and i want to select a range of elements. To get normally distributed random numbers with mean and standard deviation other than the standard normal distribution ($\mu=0,\sigma=1$), you will have to use another MATLAB builtin function normrnd(),. The plot is formed by joining adjacent points with straight lines. MATLAB ® graphics defines a surface by the z-coordinates of points above a rectangular grid in the x-y plane. For things like this you can use the help documentation. The first column should contain random values between [0 5] and the second column should have random values between [5 20]. Best Answer. Than I decided to keep characters instead of the numbers, reducing the memory occupied by the matrix. selecting a specific range in a matrix and cell array. A matrix is full rank if its rank is the highest possible for a matrix of the same size, and rank deficient if it does not have full rank. for example i want to select all the elements that are lower than 182 and swap/change them. Also had problem with perms (needed to restart Matlab to be able to generate for 11 elements, I have x64 and 16GB of memory). R = normrnd(mu,sigma) generates random numbers from the normal distribution with mean parameter mu and standard deviation parameter sigma. matrix random number generator. In cases where both are important a better solution would be to expand the bounds by a very very small amount near the limit of Matlab's precision and then to include a conditional that checks if a value was drawn outside of the bounds. So you can either type "help random" "doc random" into the command window, or click the question mark icon, or (in older versions) get help from the menus. I wil apply it in the future to a matrix of zeros and 1's and apply a random movement to the element picked. I know how to make matrices with random numbers and make all the random numbers necessary, I'm just confused. I'd bet that the need for a random process would have higher importance than the need for the very small chance of hitting the bounds. Follow 104 views (last 30 days) AA on 23 Dec 2014. Learn more about random number generator, random, homework MATLAB Random Matrix with Integer values; Random Matrix with a specific range of numbers; Matrix with desired size ( User can choose the number of rows and columns of the matrix ) Create Matrix of Random Numbers in Python. Unlike rand and randn, a parameter specifying the range must be entered before the dimensions of the matrix. y = range(X,vecdim) returns the range over the dimensions specified in the vector vecdim.For example, if X is a matrix, then range(X,[1 2]) is the range of all elements in X because every element of a matrix is contained in the array slice defined by dimensions 1 and 2. I'm looking for a way to generate a *random positive semi-definite matrix* of size n with real number in the *range* from 0 to 4 for example. Thanks for your feedback Cedric. There are various ways of generating random numbers in MATLAB with different applications. s = rng; r = randn(1,5) r = 1×5 0.5377 1.8339 -2.2588 0.8622 0.3188 Learn more about matrix, function Learn more about matrix, random number generator In MATLAB, you create a matrix by entering elements in each row as comma or space delimited numbers and using semicolons to mark the end of each row. I'm trying to generate a random integer in the range of -5 and +5 using round and rand functions. Representing Data as a Surface Functions for Plotting Data Grids. MATLAB ® has several indexing styles that are not only powerful and flexible, but also readable and expressive. Learn more about positive semidefinite matrix, random number generator This MATLAB function returns an n-by-n distributed matrix with random integer values in the range [1,imax]. The simplest randi syntax returns double-precision integer values between 1 and a specified value, imax. There usually is values above and below .5. 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text:
Who wanna text on p.adlet if it is not yes or no answer i will take the points back

Question:

who wanna text on p.adlet if it is not yes or no answer i will take the points back

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text:
Relational/logic programming is quite cool. It let’s you do things that sound impossible, like run a program backwards. Using Z3py to do a lot of heavy lifting, there’s a cute, simple, and pretty powerful little microkanren you can make. You can have it answer queries like the following that run list append backwards.

q = Const("q", List)
r = Const("r", List)
runq([q,r], appendo(q,r,cons(2,cons(42,cons(17,nil)))))
'''
Response:
[{q: nil, r: cons(2, cons(42, cons(17, nil)))},
{q: cons(2, nil), r: cons(42, cons(17, nil))},
{q: cons(2, cons(42, nil)), r: cons(17, nil)},
{q: cons(2, cons(42, cons(17, nil))), r: nil}]
'''

Full (very short) implementation here https://gist.github.com/philzook58/242961b16f0e04a731760ca8d272d91c.

### A Short Spiel on Minikanren

The minikanren approach builds a embedded relational programming language out of 2 pieces

1. unification
2. Some kind of nondeterministic search

The microkanren paper is really good, if you can read a little scheme http://webyrd.net/scheme-2013/papers/HemannMuKanren2013.pdf. I talk more and have more links about microkanren here https://www.philipzucker.com/yet-another-microkanren-in-julia/

The non-deterministic search is structured around the concept of a “goal”. A goal transforms a state into a set of possible refined states based on the assertions of the goal. In other words it can be modelled as a function State -> [State].

The state of a minikanren search is described by the currently known unifications. Unification is a sort of bidirectional pattern matching. You take two patterns and traverse down them to figure out what variables have to equal, or if it is impossible for them to match. I’m going to short circuit this entire piece to just lean on Z3. This has the upside of not needing to implement it, and the downside of Z3 being rather less controlled / general in what it returns compared to using unification.

A term with variables, let’s call it a pattern, is in some sense is a representation of a set of concrete terms with no variables. For example, the pattern f(?a, c) represents f(c,c), f(bar(z),c), f(g(cons(l)),c) and so on. The patterns represents the set of terms with every possible term substitution into the variables of the pattern. Unification is a way of asking for the intersection of these term sets.

Z3 goes totally concrete though. So when minikanren returns a result that contains patterns, Z3 could return every possible instantiation (which is possibly infinite) of those patterns, depending on how we ask.

## Generators

I rather like python generators. You can do some pretty wacko control flow with them.

My internal model of what I am doing is the state of the program is a list of Z3 assertions and goals transform into new lists of z3 assertions. This form would allow for a more normal kanren interleaving search.

But, by using the internal state of the z3 Solver with the push/pop interface we get efficiency and convenience gains but what is left behind is a rather bizarre and very stateful program.

## Implementation

So let’s write out the most basic goal for equality. The goal is parametrized by the left and right hand side of the equality. The goal itself is a generator that takes in a solver and yields with the solver in a new state with new assertions added. We add the s.check() == sat for early pruning of this branch of the search using Z3’s solving capabilities. When the generator finally dies, the solver should be returned to it’s original state. That is a contract that needs to be correctly implemented by every goal. It’s a tightwire walk for sure, but if you build your goals using combinators, you’re ok.

from z3 import *
def eq(x,y):
def goal(s):
s.push()
s.add(x == y)
if s.check() == sat:
yield
s.pop()
return goal

In fact we can abstract this to lift an arbitrary z3 assertion into a goal. via the same pattern basically. I change just s.check() == sat to s.is_good() because I’ll want the flexibility later. You can think of them as the same.

def lift(z3assert):
def res(s):
s.push()
s.add(z3assert)
if s.is_good():
yield
s.pop()
return res
def lift2(op):
def res(x,y):
return lift(op(x,y))
return res
import operator
eq = lift2(operator.__eq__)

Now we can define the kanren goal combinators disj and conj for disjunction (or) and conjunction (and). conj1 basically runs two goals in sequence, the second one being in the scope of the first. disj1runs two goals sort of in parallel, where they aren’t in thr scope of each other. disj and conj are the multiargument forms for convenience. The * are the wacky little notations for python varargs.

def conj1(f,g):
def res(s):
for _ in f(s):
if s.is_good():
#print(s.model())
yield from g(s)
return res
import functools
def conj(*args):
return functools.reduce( conj1 , args )
def disj1(f,g):
def res(s):
# itertools.chain(f(s), g(s))
yield from f(s)
yield from g(s)
return res
def disj(*args):
return functools.reduce(disj1, args)

Anther convenience function is conde which operates like a scheme cond expression and also a bit like a pattern match.

def conde(*args):
return disj(*map(lambda l: conj(*l), args))

Some other helpful routines. Zzz makes a thunked goal. This is a confusing thing, but it is necessary when you define a recursive relation or else the thing will go into an infinite loop. mempty is a failure and printmodel does not change the assertion state and can be useful for debugging.

def Zzz(thunk_goal):
def res(s):
yield from thunk_goal()(s)
return res
def printmodel(s):
if s.is_good():
print(s.model())
yield
def mempty(s):
if False:
yield # trick into making it an iterator?

Finally we need a toplevel run function. There are some choices here. This version takes in the variables of interest in qs and projects them out of the model. This does not return every possible solution to the query just one per successful branch.

def runq(qs, goal):
s = Solver()
s.is_good = lambda : s.check() == sat
for _ in goal(s):
m = s.model()
yield {q: m.eval(q) for q in qs}

Here’s a basic query. We have the full flexibility of Z3 available to us

x, y = Ints("x y")
ex_goal2 = conde( [eq(x,3) , eq(y,4) ] ,
[eq(y,5) , eq(x,6) ] )
list(runq([x,y],ex_goal2))
# [{x: 3, y: 4}, {x: 6, y: 5}]

A more complex query. Note I had to use the Zzz combinator on the recursive call to appendo to stop the thing from going off the rails.

# Declare a List of integers
List = Datatype('List')
# Constructor cons: (Int, List) -> List
List.declare('cons', ('car', IntSort()), ('cdr', List))
# Constructor nil: List
List.declare('nil')
# Create the datatype
List = List.create()
cons = List.cons
car  = List.car
cdr  = List.cdr
nil  = List.nil

def appendo(x,y,z):
h = FreshInt()
t = FreshConst(List)
res = FreshConst(List)
return conde( [ eq(x, nil), eq(y,z) ],
[ eq( x, cons(h,t)), eq(z, cons(h, res)) , Zzz( lambda : appendo(t,y,res ))]
)

q = Const("q", List)
r = Const("r", List)
x, y = Ints("x y")
list(runq([q,r], appendo(q,r,cons(2,cons(42,cons(17,nil))))))

And interesting alternative to runq is to use iterative deepening. This is a complete search method. This will return multiple times for the same path through the goals. We could avoid this by tracking known models and disallowing them as we go along.

def run_iterative_deep(goal):
s = Solver()
s.is_good = lambda : s.num_scopes() < s.max_depth and s.check() == sat
for depth in itertools.count(start=1):
s.max_depth = depth
for _ in goal(s):
yield s.model()

### Kanren’s Conj and Disj vs Z3’s And and Or

I’ve been struck by the similarity of programming Z3 and macro programming / compile time programming. Python in the use of Z3py is a macro level metalanguage for Z3. Sometimes you have a choice about whether to perform computation in the python layer or in the Z3 layer. Do you want to define a z3 Function or use a metalevel python function to define exponentiation? Do you want to use python records or Z3 records for your data types? You can use tricks of partial evaluation to do as much as possible at the python layer. The two layers are not completely interchangeable though. Python lacks the solving/inference/reversible capabilities of Z3, and Z3 notably lacks recursion and iteration in full power.

Z3 queries are inflexible in certain senses. It feels like you need to kind of define a fixed universe over which Z3 is to search. The two main sidesteppings of this restriction are the use of quantifiers and relatedly the fixedpoint engines, for both of which your mileage may vary.

It is probably desirable to use Z3 And/Or whenever possible to displace disj/conj. Z3 is going to do a much better job of searching than my junk, and it reduces the number of push/pop.

### A Quantified Z3 append for comparison

You can define append using quantifiers. This is a full definition. However, the use of quantifiers using the regular solving engine in Z3 usually means it will not be able to return a model because it can’t in general figure out if it satisfied the entire quantified definition (quantification is in a sense asserting an infinite number of formula, which is hard to check in general). The fixedpoint solvers https://rise4fun.com/z3/tutorialcontent/fixedpoints can do something with this, but that is for another post someday.

from z3 import *
# Declare a List of integers
List = Datatype('List')
# Constructor cons: (Int, List) -> List
List.declare('cons', ('car', IntSort()), ('cdr', List))
# Constructor nil: List
List.declare('nil')
# Create the datatype
List = List.create()
cons = List.cons
car  = List.car
cdr  = List.cdr
nil  = List.nil

append = Function("apppend", List, List, List)

q = Const("q", List)
r = Const("r", List)
l = Const("l", List)
h = Int("h")
t = Const("t", List)
append_def = [
ForAll([l], append(nil, l) == l),
ForAll([h,t,l], append(cons(h,t), l) == cons(h, append(t,l) ))
]
s = Solver()
s.add(append_def)
s.add( append(q,r)  == cons(2,cons(42,cons(17,nil))) )
s.check() # doesn't return

## Why the quotes around minikanren?

The Kanren family usually uses a complete search of some kind, interleaving the branches of the search tree. Since I’m abusing the push/pop mechanism of z3, it is most convenient to use a depth first search, which is incomplete. In this sense, what we have here is not really a Kanren.

And yet, I did reference the microkanren paper to write this and am building an embedded relational programming language with kanren-like surface syntax. So maybe this is still a Kanren.

### Bits and Bobbles

Use this for reachability analysis on programs (while loops, etc).

Make a version of run that exhausts a branch by asserting the model must change back to itself.

Compare and contrast with Z3 built-in fixed point solvers like BMC and Spacer.

Use unsat core to backjump more?

How to do full minikanren style interleaving search?

Combining logic programming with constraints/smt is not at all a new thing. It goes under the name CLP (constraint logic programming) and CHC (constrained horn clauses).

https://www.youtube.com/watch?v=KsC_9_-NuQg


text:
# emulsion.tools package¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Misc. tools

## emulsion.tools.calendar module¶

Classes and functions for the definition of Emulsion calendars.

class emulsion.tools.calendar.EventCalendar(calendar_name, step_duration, origin, period=None, **initial_dates)[source]

Bases: object

The EventCalendar class is intended to handle events and periods of time. Dates/times are given to the calendar in a human-readable format, and converted to integers (simulation steps).

__init__(calendar_name, step_duration, origin, period=None, **initial_dates)[source]

Initialize the calendar using the specified information.

Parameters: calendar_name (string) – the name of the calendar step_duration (datetime.timedelta) – actual duration of one simulation step origin (datetime.datetime) – date/time value of the beginning of the calendar period (datetime.timedelta) – if the calendar is periodic, actual duration of the period, None otherwise initial_dates (dict) – initial dictionary of events (event name as key, dict as value with either begin/end dates or one single date if punctual event)
add_event(name, begin_end)[source]

Add the specified event to the calendar. An event is characterized by its name and a begin_end tuple indicating the begin and end dates.

Parameters: name (str) – the name of the event begin_end (tuple) – a tuple composed of the begin date and the end date (possibly identical for handling punctual events)
date_to_step(date)[source]

Return the step corresponding the specified date.

Parameters: date (datetime.datetime) – the date to convert into time steps t (int) – the time step during which the specified date occurs (the date occurs between the returned time step t and time step t+1)
get_events()[source]

Return the list of events contained in the current calendar.

Returns: l (list) – the list of all event names
increment(steps=1)[source]

Advance the current date by the specified number of simulation steps.

Parameters: steps (int) – number of steps to ‘add’ to the current date
step_to_date(step)[source]

Return the date when the specified step begins.

Parameters: step (int) – the time step to convert to a date d (datetime.datetime) – the corresponding date
exception emulsion.tools.calendar.InvalidIntervalException(begin, end)[source]

Bases: Exception

Exception raised when trying to insert an inconsistent event in the calendar. An event is considered inconsistent if the begin date is posterior to the send date in a non-periodic calendar.

__init__(begin, end)[source]

Create the exception with the incorrect begin and end dates.

emulsion.tools.calendar.date_in(begin, end, period=None)[source]

Build a function which tests if a date belongs to the specified interval (from begin to end). If a period is specified, the test relies upon the periodicity, otherwise dates are considered absolute.

Parameters: begin (datetime.datetime) – the date when the interval begins end (datetime.datetime) – the date when the interval ends period (datetime.timedelta) – the duration of the period after which the calendar cycles lambda – a function which maps a date (datetime.datetime) to a bool to indicate whether or not the date belongs to the begin-end interval. InvalidIntervalexception – if begin is posterior to end in a non-periodic calendar

## emulsion.tools.functions module¶

Additional functions for symbolic computing in YAML model definition files.

All functions in this module can be used in EMULSION models.

emulsion.tools.functions.AND(*values)[source]

Return a logical AND (conjunction) between all the values.

Example

To define symptomatic individuals as infected for at least 5 days:

is_symptomatic:
desc: 'test if individuals has symptoms'
value: 'AND(is_I, duration_in_health_state >= 5)'

Parameters: *values (list) – boolean values separated by commas bool – True if all values are True, False otherwise.
emulsion.tools.functions.IfThenElse(condition, val_if_true, val_if_false)[source]

Ternary conditional function: return either val_if_true or val_if_false depending on condition.

Example

Here in a parameter definition (assuming that summer_period is defined e.g. in the calendars section of the model):

average_temperature:
desc: 'average temperature for the season'
value: 'IfThenElse(summer_period, 25, 8)'

Parameters: condition (bool) – a boolean expression val_if_true (number) – value to return if the expression is True val_if_false (number) – value to return if the expression is False number – One of the two values, depending on condition.
emulsion.tools.functions.MAX(*values)[source]

Return a the maximum of all values.

Parameters: *values (list) – boolean values separated by commas float – the highest value
emulsion.tools.functions.MIN(*values)[source]

Return a the minimum of all values.

Parameters: *values (list) – boolean values separated by commas float – the lowest value
emulsion.tools.functions.OR(*values)[source]

Return a logical OR (disjunction) between all the values.

Example

To sell animals depending on either their weight or their age:

transitions:
...
- from: Fattening
to: Sold
proba: 1
cond: 'OR(weight >= weight_thr, age >= age_thr)'

Parameters: *values (list) – boolean values separated by commas bool – True if at least one of the values is True, False otherwise.
emulsion.tools.functions.random_beta(a: float, b: float) → float[source]

Return a random value drawn from a beta distribution of parameters a and b.

Example

In a prototype definition:

age: 'random_beta(2, 5) * age_max'

Parameters: a (float, positive (>0)) – first shape parameter of the beta distribution b (float, positive (>0)) – second shape parameter of the beta distribution float – a random value sampled according to a beta distribution of parameters a and b

numpy.random.beta()

emulsion.tools.functions.random_bool(proba_success: float) → int[source]

Return a random boolean value (actually, 0 or 1) depending on proba_success.

Example

Set the value of a state variable has_symptoms when entering in the infectious state:

states:
...
- I:
...
on_enter:
- set_var: has_symptoms
value: 'random_bool(proba_symptomatic)'

Parameters: proba_success (float in [0,1]) – probability of returning 1 (True) int – either 1 with probability proba_success, or 0 with probability 1-proba_success
emulsion.tools.functions.random_choice(*values)[source]

Return a value chosen randomly among those provided (equiprobable sampling).

Example

To init age among three typical values:

prototypes:
individuals:
init_individual:
age: 'random_choice(10, 50, 200)'

Parameters: *values (list) – the possible values, separated by commas val – one of the values (equiprobable choice)
emulsion.tools.functions.random_choice_weighted(*values)[source]

Return a value chosen randomly among those provided (but not equiprobably).

Example

To init age among three typical values which respectively represent 10%, 20% and 70% of the population:

prototypes:
individuals:
init_individual:
age: 'random_choice_weighted(10, 0.1, 50, 0.2, 200, 0.7)'

Parameters: values (list) – possibles choices and their weight, alternatively, separated by commas. Weights are normalized to be used as probabilities. One of the choices
emulsion.tools.functions.random_exponential(scale: float) → float[source]

Return a random value drawn from an exponential distribution of rate 1/scale (thus of mean scale).

Example

In a prototype definition:

time_to_live: random_exponential(mean_duration)

Parameters: scale (float) – the scale parameter of the distribution, i.e. the inverse of the rate float – a random value sampled according to an exponential distribution of rate 1/scale

numpy.random.exponential()

emulsion.tools.functions.random_gamma(shape: float, scale: float) → float[source]

Return a random value drawn from a gamma distribution of parameters shape and scale.

Example

In a prototype definition:

age: 'random_gamma(3, 2)'

Parameters: shape (float, positive (>0)) – the shape of the gamma distribution scale (float, positive (>0)) – the scale of the gamma distribution float – a random value sampled according to a gamma distribution of parameters shape and scale

numpy.random.gamma()

emulsion.tools.functions.random_integers(low: int, high: int) → int[source]

Return a random integer value drawn from a discrete uniform distribution between low and high (both inclusive).

Example

In a prototype definition:

age: random_integers(min_age, max_age)

Parameters: low (int) – lower boundary of the sample interval (inclusive) high (int) – upper boundary of the sample interval (inclusive) int – a random integer value sampled according to a discrete uniform distribution between low and high

numpy.random.random_integers()

emulsion.tools.functions.random_multinomial(number_of_samples, *probas)[source]

Return a multinomial sample based on the specified probabilities.

Parameters: number_of_samples (int) – number of experiments probas (list) – list of probabilities list – the drawn samples
emulsion.tools.functions.random_normal(mn: float, sd: float) → float[source]

Return a random value drawn from a normal distribution of mean mn and standard deviation sd.

Example

In a prototype definition:

age: 'random_normal(100, 5)'

Parameters: mn (float) – the mean of the normal distribution sd (float, positive (>=0)) – the standard deviation of the normal distribution float – a random value sampled according to a normal distribution of mean mn and standard deviation sd

numpy.random.normal()

emulsion.tools.functions.random_poisson(lam: float) → int[source]

Return an integer random value drawn from a Poisson distribution of mean lam.

Example

In an action, e.g. here when computing how many newborn individuals will be produced:

- from: Gestating
to: NonGestating
on_cross:
- produce_offspring: newborn
amount: 'random_poisson(1.05)'

Parameters: lam (float) – the mean of the Poisson distribution int – a random integer value sampled according to a Poisson distribution of mean lam

numpy.random.poisson()

emulsion.tools.functions.random_uniform(low: float, high: float) → float[source]

Return a random value drawn from a uniform distribution between low (inclusive) and high (exclusive).

Example

In a prototype definition:

age: random_uniform(min_age, max_age)

Parameters: low (float) – lower boundary of the sample interval (inclusive) high (float) – upper boundary of the sample interval (exclusive) float – a random value sampled according to a uniform distribution between low and high

numpy.random.uniform()

## emulsion.tools.graph module¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Tools aimed at handling graphs for the state machines

class emulsion.tools.graph.EdgeTypes

An enumeration.

PRODUCTION = 2
TRANSITION = 1
linestyle = 'solid'
class emulsion.tools.graph.MultiDiGraph(**attributes)[source]

Bases: object

An oriented multigraph (two nodes can be linked by several edges). Replacement for MultiDiGraph in networkx with a reproducible order in accessing edges and nodes.

__init__(**attributes)[source]

Create an instance of MultiDiGraph. If keywords arguments are specified, they are considered attributes of the whole graph.

add_edge(from_id, to_id, type_id=<EdgeTypes.TRANSITION>, key=None, **attributes)[source]

Add the specified edge to the graph. If nodes are not already created, they are automatically added. Edge attributes can be specified. Since this graph allows multiple edges between pairs of nodes, two calls to this method lead to two distinct edges (even if the attributes are the same), unless a key is specified. The key is used to identify edges between a given pair of nodes. By default, keys are consecutive integers.

add_node(node_id, **attributes)[source]

Add the specified node to the graph. Node attributes can be specified. If the node is already present, updated the attributes.

edges()[source]

Return a list of edge tuples.

edges_from(from_id, type_id=<EdgeTypes.TRANSITION>)[source]

Return a list of tuples (to_id, attributes) corresponding to all edges going out of the from_id node.

## emulsion.tools.misc module¶

Collection of various useful functions used in EMULSION framework, especially regarding introspection.

emulsion.tools.misc.AGENTS = 1

Constant value for the index where list of agents are stored in ‘populations’ tuples.

emulsion.tools.misc.POPULATION = 1

Constant value for the index where population amounts are stored in ‘populations’ tuples.

emulsion.tools.misc.add_all_test_properties(agent)[source]
emulsion.tools.misc.add_new_property(agent, property_name, getter_function)[source]

Add a new property to an agent.

Actually, the property is added to the class of the agent, as Python properties are descriptors. Yet, the dynamic attribution of properties must me done through instances rather than classes, since agents must add the name of the property to their _mbr_cache attribute.

Parameters: agent (AbstractAgent) – the agent to which the property must be added property_name (str) – the name of the property getter_function (lambda) – the function upon which the property is built
emulsion.tools.misc.aggregate_probabilities(probability_values: Iterable[float], delta_t: float) → Iterable[float][source]

From the specified probability_values, intended to represent probabilities of events duting one time unit, compute the probabilities for the specified time step (delta_t).

Parameters: probability_values (list) – a list of probability values for several events (during 1 time unit) delta_t (float) – the value of the time step (expressed in time units) list – the probabilities of the same events during delta_t time units.
emulsion.tools.misc.aggregate_probability(probability: float, delta_t: float) → float[source]

Transform the specified probability value, intended to represent a probability for events tested each time unit, into the probability for the specified time step (delta_t).

Parameters: probability (float) – the probability value of an event (during 1 time unit) delta_t (float) – the value of the time step (expressed in time units) float – the probability of the event during delta_t time units.
emulsion.tools.misc.count_population()[source]

Return the amount of atoms represented in agents_or_pop.

Parameters: agents_or_pop (tuple) – either (‘population’, qty) or (‘agents’, list of agents) int or float – the amount corresponding to the population: generally, an int value, but deterministic models produce float values.
emulsion.tools.misc.create_aggregator(sourcevar: str, operator: str)[source]

Create an aggregator function to be used as property getter. This aggregator function has to collect all values of sourcevar for agents contained in a given host, and reduce them to one avalue using the specified operator.

Parameters: sourcevar (str) – the name of the variable to collect in the sublevel operator (str) – the name of the operator to apply to the collected values lambda – A function which can be applied to a MultiProcessManager agent (i.e. with explicit sublevel agents) which returns the aggregated values for the whole population.
emulsion.tools.misc.create_atoms_aggregator(sourcevar: str, operator: str, machine_name: str, state_name: str)[source]

Build a getter for property of the form newvar_X where X if the value of state_name (a state of state_machine), newvar is an aggregate variable based on collecting all values of sourcevar for the specific group_name and aggregating the values using operator. This function is intended to work on IBMProcessManager agents, which do not benefit from groupings.

Parameters: sourcevar (str) – the name of the variable to collect in the sublevel operator (str) – the name of the operator to apply to the collected values machine_name (str) – the name of the state machine for which the getter is created state_name (str) – the state name lambda – A function which can be applied to an IBMProcessManager agent (i.e. with explicit but ungrouped sublevel agents) which returns the aggregated value for the specified group.
emulsion.tools.misc.create_counter_getter(machine_name, state_name)[source]
emulsion.tools.misc.create_duration_getter(machine_name)[source]
emulsion.tools.misc.create_group_aggregator()[source]

Build a getter for property of the form newvar_X_Y where X and Y are states of two different states machines used in a grouping, newvar is an aggregate variable based on collecting all values of sourcevar for the specific group_name and aggregating the values using operator. The functions can handle groupings with an arbitrary number of states.

Parameters: sourcevar (str) – the name of the variable to collect in the sublevel operator (str) – the name of the operator to apply to the collected values process_name (str) – the name of the process associated with the grouping for which the population getter is created group_name (str or tuple) – either a string representing the state name (if only one), or a tuple of several state names lambda – A function which can be applied to an MultiProcessManager agent (i.e. with explicit sublevel agents) which returns the aggregated value for the specified group.
emulsion.tools.misc.create_new_serial(end=None, model=None)[source]

Create the serial number generator associated to the specified variable.

emulsion.tools.misc.create_population_getter()[source]

Build a getter for property of the form total_X_Y where X and Y are states of two different states machines used in a grouping.

The functions can handle groupings with an arbitrary number of states.

Parameters: process_name (str) – the name of the process associated with the grouping for which the population getter is created group_name (str or tuple) – either a string representing the state name (if only one), or a tuple of several state names callable – A function which can be applied to an AbstractProcessManager agent which returns the population size for the specified group.
emulsion.tools.misc.create_state_tester(state_name)[source]
emulsion.tools.misc.create_weighted_random(machine_name, weights, model=None)[source]

Create a random choice function which returns a random state from the given state machine (among non-autoremove states), according to the weights. Weights are interpreted either directly as probabilities (if the number of weights is stricly one below the number of available states, the last state getting the complement to 1), or as true weights which are then normalized to be used as probabilities.

machine_name: str
the name of the state machine where the states must be chosen among the N non-autoremove states.
weights: list
a list of N or N-1 model expressions assumed to produce positive numbers
model: EmulsionModel
the model where this function is defined
Returns: lambda (a function that returns a random state according to the) – values of the weights list, interpreted either as probabilities (if size N-1) or as weights (if size N) which are then normalized to provide probabilities
emulsion.tools.misc.find_operator(operator: str)[source]

Return an aggregation function named operator.

Search starts with emulsion functions (module emulsion.tools.functions), which includes Python built-ins, then in numpy.

A special shortcut is provided for percentiles: percentileXX is interpreted as the partial function numpy.percentile(q=int(XX)).

Parameters: operator (str) – the name of the aggregation operator lambda – a function that takes a list (or array-like) as input and returns the application of the operator to the values.
emulsion.tools.misc.load_class(module=None, class_name: str = None, options: dict = {})[source]

Dynamically load the class with the specified class_name from the given module.

Parameters: module – a Python module, where the class is expected to be located. class_name (str) – the name of the class to load options – some options tuple – a tuple composed of the class (type) and the options.

Todo

• clarify the role of options
emulsion.tools.misc.load_module(module_name: str)[source]

Dynamically load the module with the specified module_name and return it.

Parameters: module_name (str) – the name of a valid Python module (accessible in the PYTHONPATH environment variable). ref – A reference to the Python module.
emulsion.tools.misc.moving_average(values, window_size, mode='same')[source]

Compute a moving average of the specified values with respect to the window_size on which the average is calculated. The return moving average has the same size as the original values. To avoid boundary effects, use mode='valid', which produce a result of size len(values) - window_size + 1.

Parameters: values (array-like) – contains the values for moving average computation window_size (int) – width of the moving average window mode (str) – a parameter for numpy.convolve nd_array – a numpy nd_array containing the values of the moving average.
emulsion.tools.misc.probabilities_to_rates(probability_values: Iterable[float]) → List[float][source]

Transform a list of probabilities into a list of rates. The last value is expected to represent the probability of staying in the current state.

Parameters: probability_values (list) – a list of probabilities, the last one representing the probability to stay in the current state list – a list of rates corresponding to those probabilities.
emulsion.tools.misc.rates_to_probabilities(total_rate: float, rate_values: List[float], delta_t: float = 1) → List[float][source]

Transform the specified list of rate_values, interpreted as outgoing rates, into probabilities, according to the specified time step (delta_t) and normalized by total_rate.

For exit rates $$\rho_i$$ (one of the rate_values), the probability to stay in current state is given by:

$p_0 = e^{-\delta t.\sum_i \rho_i}$

Thus, each rate $$\rho_i$$ corresponds to a probability

$p_i = \frac{\rho_i}{\sum_i \rho_i} (1 - p_0)$
Parameters: total_rate (float) – the total exit rate, used for normalization purposes. If rate_values represent all possible exit rates, total_rate is their sum. rate_values (list) – the list of rates to transform delta_t (float) – the value of the time step (expressed in time units) list – the list of probabilities corresponding to the rate_values.
emulsion.tools.misc.read_from_file(filename: str)[source]

Read the specified YAML filename and return the corresponding Python document.

Parameters: filename (str) – the name of the YAML file to load object – a Python object built by parsing of the YAML file, i.e. either a dict, list, or even str/int/etc… (Most YAML document will produce dictionaries.)
emulsion.tools.misc.retrieve_value(value_or_function, agent)[source]

Return a value either directly given by parameter value_or_function if it is a true value, or computed from this parameter seen as a function, with the specified agent as argument.

Parameters: value_or_function – either a callable (function that applies to an agent to retrieve an individual value), or the value itself agent – the agent to use as parameter of the callable if necessary value – the expected value (agent-based or not).
emulsion.tools.misc.rewrite_keys(name, position, change_list)[source]
emulsion.tools.misc.select_random(origin: Iterable, quantity: int, exclude: sortedcontainers.sortedset.SortedSet = SortedSet([], key=None, load=1000)) → List[source]

Return a random selection of quantity agents from the origin group, avoiding those explicitly in the exclude set. If the origin population proves too small, all available agents are taken, irrespective to the quantity.

Parameters: origin (iterable) – the population where agents must be selected quantity (int) – the number of agents to select in the population exclude (set) – agents which are not available for the selection list – a list of randomly selected agents according to the above constraints.
emulsion.tools.misc.serial(start=0, end=None, model=None)[source]

A very simple serial number generator.

## emulsion.tools.parallel module¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Tools for parallel computing.

emulsion.tools.parallel.job(target_simulation_class, proc, **others)[source]

Simple job for a simple processes

emulsion.tools.parallel.job_dist(total_task, workers)[source]

Return a distribution of each worker need to do.

emulsion.tools.parallel.parallel_multi(target_simulation_class=<class 'emulsion.tools.simulation.MultiSimulation'>, nb_simu=None, nb_proc=1, **others)[source]

Parallel loop for distributing tasks in different processes

emulsion.tools.parallel.parallel_sensi(target_simulation_class=<class 'emulsion.tools.simulation.SensitivitySimulation'>, nb_proc=1, **others)[source]

Parallel loop for distributing sensitivity tasks in different processes

## emulsion.tools.plot module¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Plotting tools… to be improved!

emulsion.tools.plot.build_machine_plot(machine_name, model, params)[source]

Return a bokeh figure based on the representation of the state machine, with the associated colors.

Parameters: machine – the state machine to which the states are related A bokeh gridplot.
emulsion.tools.plot.build_state_plot(counts, cols, machine, model, y='quantity', group='state', ylab='Number of individuals')[source]

Return a bokeh figure based on the representation of the states in the counts dataframe, with the associated colors.

Parameters: counts – a Pandas dataframe containing the values to plot cols (dict) – dictionary which maps state/variable names to colors machine – the state machine to which the states are related model – the Emulsion model related to this plot y (str) – the name of the field containing y values in the dataframe group (str) – either ‘state’ (default) or ‘variables’, name of the field containing each legend items. A bokeh gridplot (reduced to one figure a the herd level, one figure per herd at metapopulation level).
emulsion.tools.plot.plot_outputs(params)[source]

Read outputs from previous runs and plot the corresponding figures. In the params dictionary, output_dir is expected to contain a counts.csv file; figure_dir is where the plot is saved.

## emulsion.tools.simulation module¶

Tools for providing generic simulation classes.

class emulsion.tools.simulation.AbstractSimulation(start_id: int = 0, model=None, model_path: str = '', stock_agent: bool = True, output_dir: str = 'outputs/', target_agent_class=None, save_results: bool = True, input_dir: str = None, load_from_file: str = None, save_to_file: str = None, **_)[source]

Bases: object

Abstract class from which any simulation class inherits.

__init__(start_id: int = 0, model=None, model_path: str = '', stock_agent: bool = True, output_dir: str = 'outputs/', target_agent_class=None, save_results: bool = True, input_dir: str = None, load_from_file: str = None, save_to_file: str = None, **_)[source]

Initialize the simulation.

Parameters: start_id – ID of the (first) simulation model – instance of the model to run model_path – path to the filename holding the description of the model, used if model is None stock_agent – TODO output_dir – name of the directory for simulation outputs target_agent_class – agent class representing the top level in the simulation save_results – True if simulation outputs have to be saved, False otherwise. TODO: should be removed, and replaced by a set of OutputManagers dedicated to the specific expected outputs. load_from_file – a filename from which the initial state of the simulation (agents corresponding to levels with their state) is read (instead of running the initialize_level method) save_to_file – a filename in which the final state of the simulation (agents corresponding to levels with their state) is written (after running the finalize_level method)
evolve(steps: int = 1)[source]

Operations to perform at each time step. Should be defined in subclasses.

Parameters: steps –
run()[source]

Entry point to simulation execution. Should be defined in subclasses.

update_csv_counts(df, dparams: dict = {})[source]

Update the CSV recording of populations in each state.

class emulsion.tools.simulation.MultiSimulation(multi_id: int = 0, nb_simu: int = 100, set_seed: bool = False, silent: bool = False, quiet: bool = False, dparams: dict = {}, **others)[source]

MultiSimulation can handle multiple repetitions of a given model. For sensibility study (same model with different values of variables), please check out SensitivitySimulation.

__init__(multi_id: int = 0, nb_simu: int = 100, set_seed: bool = False, silent: bool = False, quiet: bool = False, dparams: dict = {}, **others)[source]

Initialize a simulation with multiple repetitions of the same model.

counts
evolve(steps=1)[source]

Operations to perform at each time step. Should be defined in subclasses.

Parameters: steps –
run(update=True)[source]

Run all repetitions one by one.

write_dot()[source]
class emulsion.tools.simulation.OutputManager(model=None, output_dir='', output_file='counts.csv', log_file='log.txt')[source]

Bases: object

Manager to handle different outputs (csv, database,… etc)

__init__(model=None, output_dir='', output_file='counts.csv', log_file='log.txt')[source]

Initialize the output manager, specifying the model and the directory where outputs will be stored.

update_output_information()[source]

Update csv file path or database connection engine

update_output_type()[source]

Update output type if specified in model

update_outputs(df=None)[source]

Update outputs: writing in csv file or in database

class emulsion.tools.simulation.SensitivitySimulation(scenario_path=None, df=None, nb_multi=None, **others)[source]

SensitivitySimulation can handle sensibility study with a given pandas DataFrame of parameters or a path linked with file which contains scenarios of parameters. Then it will be transformed to a dictionary of scenario in the d_scenario attribute.

For instance, d_scenario could be the form (QFever model example) :
{0: {‘m’: 0.7, ‘q’: 0.02 …},
1: {‘m’: 0.5, ‘q’: 0.02 …}, 2: …, … }
__init__(scenario_path=None, df=None, nb_multi=None, **others)[source]

Initialize the simulation.

Parameters: start_id – ID of the (first) simulation model – instance of the model to run model_path – path to the filename holding the description of the model, used if model is None stock_agent – TODO output_dir – name of the directory for simulation outputs target_agent_class – agent class representing the top level in the simulation save_results – True if simulation outputs have to be saved, False otherwise. TODO: should be removed, and replaced by a set of OutputManagers dedicated to the specific expected outputs. load_from_file – a filename from which the initial state of the simulation (agents corresponding to levels with their state) is read (instead of running the initialize_level method) save_to_file – a filename in which the final state of the simulation (agents corresponding to levels with their state) is written (after running the finalize_level method)
counts
run()[source]

write_dot()[source]
class emulsion.tools.simulation.Simulation(steps: int = 100, simu_id: int = 0, silent: bool = False, quiet: bool = False, **others)[source]

Simulation class is aimed at running one repetition of a given model (for several repetitions, use MultiSimulation).

__init__(steps: int = 100, simu_id: int = 0, silent: bool = False, quiet: bool = False, **others)[source]

Create an instance of simulation.

Parameters: steps – number of time steps to run simu_id – ID of the simulation silent – if False, show a progress bar during simulation execution quiet – if True, show no progressbar at all
counts

Return a pandas DataFrame contains counts of each process if existing. TODO: column steps need to be with one of process and NO column steps for inter herd

evolve(steps: int = 1)[source]

Make the target agent evolve.

Parameters: steps – the number of time steps to run
init_agent(**others)[source]

Create an agent from the target class.

log_path()[source]

Return the log path used by current simulation

run(dparams: dict = {})[source]

Make the simulation progress.

## emulsion.tools.state module¶

This module defines:

• a class StateVarDict which is a special auto-referential dictionary to handle state variables
• a class EmulsionEnum which is a special kind of enumeration intended to handle states from state machines
class emulsion.tools.state.EmulsionEnum[source]

Bases: enum.Enum

This class represents enumerations for states of state machines in EMULSION. They are endowed with some special features:

1. They provide total ordering between items (based on __lt__ and __eq__ methods).
2. A comparison with None is provided (any state is always greater than None).
3. Other features will be developed soon.
class emulsion.tools.state.StateVarDict(*args, **kwargs)[source]

Bases: dict

A special dictionary aimed at handling the state variables of agents in EMULSION models. In addition to the classical dict key-based access, it provides an attribute-like access syntax.

This class is used in EMULSION to store agent properties. Such properties include those defined automatically and used by the EMULSION engine, such as the values of the states for each state machine, the current time step, or “hidden” values such as the time spent in the current state for each state machine. They are also used to include user-defined attributes (e.g. age, weight…).

When searching for a model statevar, the engine tries first to find a classical instance variable (which can be mimicked by a Python @property-decored function), then looks inside the agent’s statevar attribute; finally, the search continues in the agent’s host (if any).

Example

s = StateVarDict(age=10, sick=True)
s['age'] += 1
s.sick = False
s.new_property = 'Wow !'

__init__(*args, **kwargs)[source]

Initialize self. See help(type(self)) for accurate signature.

## emulsion.tools.timing module¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Tools for performance assessment.

emulsion.tools.timing.timethis(times=None)[source]

A decorator function for printing execution time.

## emulsion.tools.view module¶

A Python implementation of the EMuLSion framework (Epidemiologic MUlti-Level SImulatiONs).

Tools for data / map visualization.

emulsion.tools.view.build_animation(values, title, unit=None, filename=None, cmap='coolwarm', framerate=10, resolution=100, writer='imagemagick', **kwargs)[source]

Create an animation based on the values list, with the specified title. A special color map name can be specified, as well as the presence of a colorbar and additional keyword arguments passed to imshow. If a specified unit is given, each frame is marked with its number and the corresponding unit. If a filename is provided in the save parameter, the animation is stored in that file.

emulsion.tools.view.show_contour(value, cmap='hot', save=None, colbar=False, **kwargs)[source]

Display the specified value as a contour map with values. A special color map name can be specified, as well as the presence of a colorbar and additional keyword arguments passed to contourf. If a filename is provided in the save parameter, the image is stored in that file.

emulsion.tools.view.show_histo(value, xlabel, facecol='green', save=None)[source]

Display the distribution of the specified value array using xlabel for the legend. A specific face color can be used. If a filename is provided in the save parameter, the image is stored in that file.

emulsion.tools.view.show_img(value, cmap='hot', save=None, colbar=True, **kwargs)[source]

Display the specified value as an image. A special color map name can be specified, as well as the presence of a colorbar and additional keyword arguments passed to imshow. If a filename is provided in the save parameter, the image is stored in that file.


text:
PDF (letter size)

## A very simple introduction to using Python in Latex

May 26, 2022   Compiled on May 26, 2022 at 5:35pm

### Contents

This note describes all the steps to use Python inside Latex. The directions are based on using Linux, since this is the system I tried this on.

Using texlive 2020, this package (pythontex) is already there. There is not additional installation needed. Once you install texlive itself, then you have this package.

### 1 Example using pyconsole

Here is the most simple tex ﬁle that uses python

\documentclass[11pt]{article}%
\usepackage{pythontex}
\begin{document}
This document uses Python
\end{document}

Now comes the tricky part. To compile it, we have to do the following commands

pdflatex foo.tex
/usr/local/texlive/2022/texmf-dist/scripts/pythontex/pythontex.py  foo.tex

Which prints on the screen this

>/usr/local/texlive/2022/texmf-dist/scripts/pythontex/pythontex.py  python_file.tex
This is PythonTeX 0.17

--------------------------------------------------
PythonTeX:  python_file - 0 error(s), 0 warning(s)

If you get an error ModuleNotFoundError: No module named 'pygments' then try the following to install pygments: pip install pygments ﬁrst.

Now need to compile the latex ﬁle on more time

pdflatex foo.tex

We have to run the above 3 commands each time. On my installation of texlive 2022, the script pythontex.py was not already in the path, so instead of keep typing the above long command each time, I added an alias to my bashrc

vi $HOME/.bashrc #add this line below to the file alias PYX=/usr/local/texlive/2022/texmf-dist/scripts/pythontex/pythontex.py #after exiting vi, typed source$HOME/.bashrc

So now I can write (less typing) the following

pdflatex foo.tex
PYX  foo.tex
pdflatex foo.tex

Or you can add it to the path. You must have python itself installed. Version 2.7 or better, else none of the above will work.

In the above example, we did not do anything interesting. Let us now make a python variable, raise it to the power 2, and show the result in Latex

\documentclass[11pt]{article}%
\usepackage{pythontex}
\usepackage{nopageno}
\begin{document}
\begin{pyconsole}
x = 987.27
x = x**2
\end{pyconsole}

The variable is $x=\pycon{x}$
\end{document}

Which produces the following PDF ﬁle when compiled

Here is screen shot of the above

### 2 Example using pycode

In this example we will call Python to do symbolic computation, which is integrating a function, then obtain the latex back of the result. For this sympy is used inside Python.

Create Latex ﬁle like this

\documentclass[11pt]{article}%
\usepackage{pythontex}
\usepackage{nopageno}
\begin{document}
\begin{pycode}
from sympy import *
x=symbols('x')
result = latex(integrate("(1+x)**(1/2)",x))
\end{pycode}

The result of integrating $\int \sqrt{ 1+x } \, dx$ is given by $\py{result}$

\end{document}

Now compile using the same commands as ﬁrst example above

pdflatex foo.tex
/usr/local/texlive/2022/texmf-dist/scripts/pythontex/pythontex.py  foo.tex
pdflatex foo.tex

Which produces the following PDF ﬁle when compiled

Here is screen shot of the above

### 3 Calling python passing it argument to process and getting the Latex back

In this example, we make a small function inside pycode which takes one argument which is string that represents the integrand to integrate and second argument which is the integration variable.

Create Latex ﬁle like this

\documentclass[11pt]{article}%
\usepackage{amsmath}
\usepackage{pythontex}
\usepackage{nopageno}
\begin{document}

\begin{pycode}

from sympy import *

def int(theIntegrand,var):
var  = symbols(var)
anti = integrate(theIntegrand,var)
return latex(anti)

\end{pycode}

The result of integrating $\int \frac{1}{\sqrt{ 1+x }} \, dx$ is given by $\py{int("1/(1+x)**(1/2)","x")}$

Here is some list of integrations to do

\begin{align*}
\int \frac{1}{\sqrt{ 1+x }} \, dx &=  \py{int("1/(1+x)**(1/2)","x")} \\
\int \sin x \, dx &=  \py{int("sin(x)","x")} \\
\int x \sin x \, dx &=  \py{int("x*sin(x)","x")} \\
\int x^2 \sin x \, dx &=  \py{int("x**2 * sin(x)","x")} \\
\int x e^{2 x} \, dx &=  \py{int("x*exp(2*x)","x")} \\
\int \frac{1}{1+u} \, du &=  \py{int("1/(1+u)","u")} \\
\end{align*}

\end{document}

Compiling again using the same commands as the above examples

pdflatex foo.tex
/usr/local/texlive/2022/texmf-dist/scripts/pythontex/pythontex.py  foo.tex
pdflatex foo.tex

Produces the following PDF ﬁle

Here is screen shot of the above

### 4 Calling python from latex to solve diﬀerential equations and show its solution

In this example, we make a small function inside pycode which takes and ode to solve, using diﬀerent initial conditions.

Create Latex ﬁle like this

\documentclass[11pt]{article}%
\usepackage{amsmath}
\usepackage{pythontex}
\usepackage{nopageno}
\begin{document}

\begin{pycode}

from sympy import *

def solveODE(ode,y,x):
x = Symbol(x)
y = Function(y)
lhs,rhs = ode.split('=')
ode = Eq(S(lhs),S(rhs))
sol = dsolve(ode,y(x))
return latex(sol)

def solveODEwithIC(ode,y,x,ic):
x   = Symbol(x)
y   = Function(y)
lhs,rhs = ode.split('=')
ode = Eq(S(lhs),S(rhs))
sol = dsolve(ode,y(x),ics= S(ic) )
return latex(sol)

\end{pycode}

Given the first order differential equation $y'(x)=1+2 x$, it has the solution
$\py{solveODE("Derivative(y(x),x)=1+2*x","y","x")}$
%
When the initial conditions are $y(0)=3$, then the solution becomes
$\py{solveODEwithIC("Derivative(y(x),x)=1+2*x","y","x","{y(0):3}")}$
We can also solve second order ODE in Latex. For example, given the ode $y''(x)+3 y(x)=1+2 x$
then its solution is
$\py{solveODE("Derivative(y(x),x,x)+y(x)=1+2*x","y","x")}$
The above solution with initial conditions $y(0)=1$ and $y'(0)=0$ is given by
$\py{solveODEwithIC("Derivative(y(x),x,x)+y(x)=1+2*x","y","x","{y(0):1,y(x).diff(x).subs(x, 0): 0}")}$

\end{document}

Compiling again using the same commands as the above examples produces the following PDF ﬁle

Here is screen shot of the above

References:

1. Pythontex main documenation http://www.ctan.org/pkg/pythontex
2. pythontex quickstart PDF
3. Pythontex GitHub
4. A Gentle Introduction to PythonTeX


text:
0% average accuracy. An updated version of this instructional video is available. - e-eduanswers.com i) Describe the Domain: i) Describe the Domain: ii) Describe the Range: ii) Describe the Range: Correct answer to the question Decide if the function is an exponential growth function or exponential decay function, and describe its end behavior using limits. Find the End Behavior f(x)=-(x-1)(x+2)(x+1)^2. End Behavior of Exponential Functions Assignment Identify the end behavior for the following functions. 1. Read our Privacy Policy and Terms of Use. This function, also denoted as  (), is called the "natural exponential function", or simply "the exponential function". Logarithmic Functions . Graphs of Exponential Functions. what that tells us about the base of the exponential function. c For very high x-values, f(x) moves toward the horizontal asymptote. Please ask your teacher to reset your password for you. This is the currently selected item. Answer: An asymptote is a line that the graph of a function approaches, as $$x$$ either increases or decreases without bound. This calculator will determine the end behavior of the given polynomial function, with steps shown. cazielke1_32927. Your email address is safe with us. It turns out exponential functions also … • The end behavior of the parent function is consistent. End Behavior: Exponential Functions. (a) y 1 = 2 x (b) y 2 = 3 x (i) Which point is common to all four graphs ? Exponential Functions, Functions Change the a and b values for the function and then test an x value to see what the end behavior would look like. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Finish Editing . As we can see from the graph above, unless the exponential function is shifted up or down, the asymptote is at y = 0. We will shortly turn our attention to graphs of polynomial functions, but we have one more topic to discuss End Behavior.Basically, we want to know what happens to our function as our input variable gets really, really large in either the positive or negative direction. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Question 1075973: Compare the number of intercepts and end behavior of an exponential function in the form of y=A(b)^x, where A > 0 and 0 b 1 to the polynomial where the highest degree tern is -2x^3, and the constant term is 4 Answer by stanbon(75887) (Show Source): True False, Value: 2Which of the following describes the end behavior of the graph of the function above? 15 minutes ago. 2. b) Graph the exponential decay function from part a. c) What are the domain & range of the function? 1) What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? Practice. As x approaches 0 from the right (denoted as … Share practice link. End behavior of polynomials. In the exponential function f(x) = 3 + 3, what is the end behavior of f(x) as x goes to co? This quiz is incomplete! Give an overview of the instructional video, including vocabulary and any special materials needed for the instructional video. This is how we are often taught in school, but there is seldom any further investigation as to why this is true. Intro to end behavior of polynomials. End Behavior in Math: Definition & Rules Exponential Growth vs. Value: 2Which of the following describes the end behavior of the graph of the function that gives the number of pennies as a function of days? We recommend keeping it to 1-2 paragraphs. Let's take a look at the end behavior of our exponential functions. This end behavior of graph is determined by the degree and the leading co-efficient of the polynomial function. Estimate the end behaviour of a function as $$x$$ increases or decreases without bound. We’ll use the function $f\left(x\right)={2}^{x}$. Mathematics. 15 minutes ago. The behavior of the graph of a function as the input values get very small ( x → − ∞ x → − ∞) and get very large ( x → ∞ x → ∞) is referred to as the end behavior of the function. Our teacher grew mold on a piece of bread for around twenty days. What are the key features of each graph (include domain, range, intercepts, asymptotes, and end behavior)? Louis bought a new car for \$20,000. In this section we would like to explore $$\displaystyle a$$ to be $$\displaystyle\infty$$ or $$\displaystyle -\infty$$. Answer: An asymptote is a line that the graph of a function approaches, as $$x$$ either increases or decreases without bound. Since any exponential function can be written in terms of the natural exponential as = , it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one.The natural exponential is hence denoted by Let's take a look at the end behavior of our exponential functions. Question 1075973: Compare the number of intercepts and end behavior of an exponential function in the form of y=A(b)^x, where A > 0 and 0 b 1 to the polynomial where the highest degree tern is -2x^3, and the constant term is 4 Answer by stanbon(75887) (Show Source): Subsection Graphs of Exponential Functions The graphs of exponential functions have two characteristic shapes, depending on whether the base, $$b\text{,}$$ is … Apply the distributive property. Its population over the last few years is shown below. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Edit. This is often called the Leading Coefficient Test. Date last modified: March 20, 2012. 0 times. Tap for more steps... Simplify by multiplying through. Take a look at the graph of our exponential function from the pennies problem and determine its end behavior. He estimates that the value of the car will decrease by 12% each year. The pattern of getting close to zero (very close), without ever reaching zero, is observable in the table. The End Behavior of Exponentials • The end behavior of the parent function is consistent. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. The basic result is: in a race to inﬁnity, exponential functions come in ﬁrst, followed by power functions, and logarithmic functions are in last place. End Behavior: Exponential Functions DRAFT. Answer to 7. The function $$f(x)→∞$$ or $$f(x)→−∞.$$ What is the End behavior on the right side. Since the leading coefficient of this odd-degree polynomial is positive, then its end-behavior is going to mimic that of a positive cubic. Assign HW. After every day he counted the number of square centimeters covered in mold. This topic covers: - Radicals & rational exponents - Graphs & end behavior of exponential functions - Manipulating exponential expressions using exponent properties - Exponential growth & decay - Modeling with exponential functions - Solving exponential equations - Logarithm properties - Solving logarithmic equations - Graphing logarithmic functions - Logarithmic scale NC.M1.F-LE.3 Compare the end behavior of linear, exponential, and quadratic functions … We write as x → ∞, f ( x) → ∞ x → ∞, f ( x) → ∞ . In this lesson you will learn how to determine the end behavior of a polynomial or exponential expression. It helps many students once they see the numbers. 1) What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? Exponential functions are defined and continuous for all real numbers. Apply the distributive property. 0. When the base of an exponential function is greater than 1, a large positive exponent will cause the entire expression to be very large, and a large negative exponent will cause the entire expression to be very small (although still positive). by cazielke1_32927. Mathematics. To answer this question, the important things for me to consider are the sign and the degree of the leading term. Find the End Behavior f (x)=- (x-1) (x+2) (x+1)^2 f(x) = - (x - 1)(x + 2)(x + 1)2 Identify the degree of the function. Thus graphs of exponential functions have horizontal asymptotes. We will shortly turn our attention to graphs of polynomial functions, but we have one more topic to discuss End Behavior.Basically, we want to know what happens to our function as our input variable gets really, really large in either the positive or negative direction. These unique features make Virtual Nerd a viable alternative to private tutoring. Describe the end behavior of a power function given its equation or graph. As x approaches infinity, the y-values slowly get larger, approaching infinity. Describe the end behavior of the graph of f(x)= x 8 … Learn what the end behavior of a polynomial is, and how we can find it from the polynomial's equation. Matching end behaviors to specific functions Skills Practiced . Since mold grows exponentially our job is to find a function to model the data. The only problem is that the function has to have a horizontal asymptote where the maximum number of squares has been reached, since … 11th - 12th grade. Edit. DRAFT. A positive cubic enters the graph at the bottom, down on the left, and exits the graph at the top, up on the right. B. We can use words or symbols to describe end behavior. Graph exponential functions. The behavior of a function as $$x→±∞$$ is called the function’s end behavior. 0. HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. We can use words or symbols to describe end behavior. a) Write an exponential decay function that represents the value of the car after t years. Logarithmic functions are essentially just inverses of exponential functions. If students continue struggling with the End Behavior, I will pull up the table for the exponential function on the calculator. What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Typescript Class Vs Interface, Yaara Release Date Zee5, Belmont Law Courses, St Moriz Tanning Mousse Medium, Elko Building Permits, Aurora University Sign In, Bert For Text Classification,


text:
# So many P's!

Suppose $$a,b,c$$ are three numbers in G.P. If the equations $$ax^2+2bx+c=0$$ and $$dx^2+2ex+f$$ have a common root, then $$\frac{d}{a} , \frac{e}{b} , \frac{f}{c}$$ are in :

• A.P.

• G.P.

• H.P.

• none of the above.

Note: A.P.,G.P. and H.P. above indicate the arithmetic, geometric and harmonic progressions.

Note by Paramjit Singh
4 years, 4 months ago

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paragraph 1paragraph 2

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Consider two equations,
$$E_{1} : a_{1}x^2 + b_{1}x + c_{1}$$
$$E_{2} : a_{2}x^2 + b_{2}x + c_{2}$$
If they have a common root say $$\alpha$$, then we can say that,
$$a_{1}\alpha^2 + b_{1}\alpha + c_{1} = 0$$
$$a_{2}\alpha^2 + b_{2}\alpha + c_{2} = 0$$
Solving them simultaneously for $$\alpha$$ and $$\alpha^2$$ and eliminating $$\alpha$$ , we get an equation of the form,
$$(a_{1}c_{2} - a_{2}c_{1})^{2} = (b_{1}c_{2} - b_{2}c_{1})(a_{1}b_{2} - a_{2}b_{1})$$
Here, $$a_{1} = a, b_{1} = 2b, c_{1} = c, a_{2} = d, b_{2} = 2e, c_{2} = f$$
Substituting the values, we get,
$$(af - dc)^2 = 4(bf - ce)(ae - bd)$$
$$(ac)^2(\frac{f}{c} - \frac{d}{a})^2 = 4ab^2c(\frac{f}{c} - \frac{e}{b})(\frac{e}{b} - \frac{d}{a})$$

$$(\frac{f}{c} - \frac{d}{a})^2 = 4(\frac{f}{c} - \frac{e}{b})(\frac{e}{b} - \frac{d}{a})$$ (As $$a,b,c$$ form a geometric progression)

Let $$\frac{d}{a} = l, \frac{e}{b} = m, \frac{f}{c} = n$$. Therefore the above equation can be written as,

$$(l - n)^2 + 4nl = 4mn + 4lm - 4m^2$$
$$(l + n)^2 - 2\times2m(l+n) + (2m)^2 = 0$$
$$(l + n -2m)^2 = 0$$ which implies $$2m = l + n.$$
Therefore, they form an arithmetic progression.

- 4 years, 4 months ago

Great!

- 4 years, 4 months ago

a p

- 4 years, 3 months ago


text:
# Probabilities of Four Points Being in Convex/Deltoid Configurations

Question:

what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum Spanning Tree (MST)?

Put differently, what are the probabilities for the following situations:

• the points are in convex configuration and
• the MST has two leaf nodes?
• the MST has three leaf nodes?
• the points are in deltoid configuration
• the MST has two leaf nodes?
• the MST has three leaf nodes?

I assume that a quadruplet of points has a higher probability of being in convex configuration if its MST has two leaf nodes than if it has three, but that that probability is less than 1.

Denote by $$P_1,P_2,P_3,P_4$$ the $$4$$ random points and by $$p_{12}$$ the probability that the points $$P_3,P_4$$ lie on the same side of the line determined by $$P_1,P_2$$. Then, as shown by Renyi and Sulanke we have $$\bar{v}=\binom{4}{2}p_{12}= 6p_{12}.$$ Given two points $$P,Q$$ we denote by $$L(P,Q)$$ the line determined by $$P,Q$$ and by $$a(P,Q)$$ the area of the part of the disk that lies on the same side of $$L(P,Q)$$ as the center of the disk. Set $$\bar{a}(P,Q)=\frac{1}{\pi} a(P,Q).$$ Then $$p_{12}= \frac{1}{\pi^2}\int_{\bD^2}\Big(\; \bar{a}(P,Q)^2+(1-\bar{a}(P,Q))^2\;\Big)dPdQ$$ $$=\int_{\bD^2}\Big(\; 2\bar{a}(P,Q)^2-2\bar{a}(P,Q)+1\;\Big) dPdQ$$ $$=1-2\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ.$$ The line $$L(P,Q)$$ has a unique equation of the form $$x\cos \theta+y\sin\theta =p,\;\;p\in[0,1], \;\;\theta\in[0,2\pi].$$ We denote by $$L_{\theta,p}$$ the line described by such an equation. The chord cout-ut by the disk on this line has length $$\ell(p)=2\sqrt{1-p^2}.$$ For such a line we have $$a(P,Q)=\frac{\pi}{2}+2\int_0^p \sqrt{1-x^2} dx =\frac{\pi}{2} +p\sqrt{1-p^2} +\arcsin(p).$$ We set $$\alpha(p):=\frac{1}{2}+ \frac{1}{\pi} \big(\; p\sqrt{1-p^2} +\arcsin(p)\;\big)=\bar{a}(P,Q).$$ As explained in the equality (5) in here we have $$\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ=\frac{1}{3}\int_0^{2\pi} \int_0^1\alpha(p)(1-\alpha(p)) \ell(p)^3 dp d\theta,$$ $$=\frac{2\pi}{3} \int_0^1 \alpha(p)(1-\alpha(p)) \ell(p)^3 dp.$$ This integral can be computed/approximated using a computer algebra system such as MAPLE or MATHEMATICA.


text:
#### Term 3 Important Questions

9th Standard

Reg.No. :
•
•
•
•
•
•

Maths

Time : 02:30:00 Hrs
Total Marks : 100
10 x 1 = 10
1. Find the value of m from the equation 2x + 3y = m. If its one solution is x = 2 and y = −2 .

(a)

2

(b)

-2

(c)

10

(d)

0

2. Which of the following is a solution of the equation 2x − y = 6

(a)

(2,4)

(b)

(4,2)

(c)

(3, −1)

(d)

(0,6)

3. A pair of linear equations has no solution then the graphical representation is

(a)

(b)

(c)

(d)

4. If $P(\frac{a}{3},\frac{b}{2})$is the mid-point of the line segment joining A(−4,3) and B(−2,4) then (a,b) is

(a)

(-9, 7)

(b)

$(-3, \frac{7}{2})$

(c)

(9, -7)

(d)

$(3, -\frac{7}{2})$

5. The ratio in which the x-axis divides the line segment joining the points A(a1,b1) and B(a2 ,b2 ) is

(a)

b1 : b2

(b)

−b1 : b2

(c)

a1 : a2

(d)

−a1 : a2

6. The value of $\frac { tan15° }{ cot75° }$ is

(a)

cos900

(b)

sin300

(c)

tan450

(d)

cos300

7. The value of $\frac { 1-{ tan }^{ 2 }{ 45 }^{ 0 } }{ 1+{ tan }^{ 2 }{ 45 }^{ 0 } }$

(a)

2

(b)

1

(c)

0

(d)

$\frac { 1 }{ 2 }$

8. The perimeter of an equilateral triangle is 30 cm. The area is

(a)

$10\sqrt { 3 }$ cm2

(b)

$12\sqrt { 3 }$ cm2

(c)

$15\sqrt { 3 }$ cm2

(d)

$25\sqrt { 3 }$ cm 2

9. The probability based on the concept of relative frequency theory is called

(a)

Empirical probability

(b)

Classical probability

(c)

Both (1) and (2)

(d)

Neither (1) nor (2)

10. The six faces of the dice are called equally likely if the dice is

(a)

Small

(b)

Fair

(c)

Six-faced

(d)

Round

11. II. Answer the questions shortly
15 x 2 = 30
12. Check whether $\frac { 1 }{ 4 }$ is a solution of the equation 3(x + 1) = 3( 5–x) – 2( 5 + x).

13. Solve the following linear equations
(i) $\frac { 2(x+1) }{ 3 } =\frac { 3(x-2) }{ 5 }$
(ii) $\frac { 2 }{ x+1 } =4-\frac { x }{ x+1 } ,(x\neq -)$

14. It takes 24 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 8 hours and the pipe of the smaller diameter is used for 18 hours. Only half of the pool is filled. How long would each pipe take to fill the swimming pool.

15. The age of Arjun is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father

16. If the mid-point (x,y) of the line joining (3,4) and (p,7) lies on 2x + 2y +1 = 0 , then what will be the value of p?

17. If the centroid of a triangle is at (4,−2) and two of its vertices are (3,−2) and (5,2) then find the third vertex of the triangle.

18. If  $\left( \frac { 3 }{ 2 } ,5 \right) ,\left( 7,\frac { -9 }{ 2 } \right)$and$(\frac{13}{2},\frac{-13}{2})$ are mid-points of the sides of a triangle, then find the centroid of the triangle.

19. From the given figure find all the trigonometric ratios of angle B.

20. If cos A = $\frac { 3 }{ 5 }$, then find the value of $\frac { sinA-cosA }{ 2tanA }$

21. A boy standing at a point O finds his kite flying at a point P with distance OP=25m. It is at a height of 5m from the ground. When the thread is extended by 10m from P, it reaches a point Q. What will be the height QN of the kite from the ground? (use trigonometric ratios)

22. Find the area of an equilateral triangle whose perimeter is 180 cm.

23. A park is in the shape of a quadrilateral. The sides of the park are 15 m, 20 m, 26 m and 17 m and the angle between the first two sides is a right angle. Find the area of the park.

24. Find the TSA and LSA of the cube whose side is (i) 8 m (ii) 21 cm (iii) 7.5 cm

25. The dimensions of a brick are 24 cm × 12 cm × 8 cm. How many such bricks will be required to build a wall of 20 m length, 48 cm breadth and 6 m height?

26. 1500 families were surveyed and following data was recorded about their maids at homes

 Type of maids Only part time Only full time Both Number of families 860 370 250

A family is selected at random. Find the probability that the family selected has
(i) Both types of maids
(ii) Part time maids
(iii) No maids

27. III. Answer all the questions:
5 x 3 = 15
28. Draw the graph for the following
(i) y = 2x
(ii) y = 4x - 1
(iii) $y=\left( \frac { 3 }{ 2 } \right) x+3$
(iv) 3x + 2y = 14

29. Solve by cross multiplication method : 3x + 5y = 21; −7x − 6y = −49

30. If (x,3), (6,y), (8,2) and (9,4) are the vertices of a parallelogram taken in order, then find the value of x and y.

31. Find the six trigonometric ratios of the angle $\theta$ using the given diagram.

32. Find the Total Surface Area and Lateral Surface Area of the cube, whose side is 5 cm.

33. IV. Answer all in detail
9 x 5 = 45
34. (Graphing made easier!) Draw the graph of the line given by the equation y = 4x – 3.

35. Use graphical method to solve the following system of equations x + y = 5; 2x – y = 4.

36. The sum of the digits of a given two digit number is 5. If the digits are reversed, the new number is reduced by 27. Find the given number.

37. Solve for x and y: 8x − 3y = 5xy, 6x − 5y = −2xy by the method of elimination.

38. Solve 2x = −7y + 5; −3x = −8y −11 by cross multiplication method.

39. Find the points of trisection of the line segment joining (−2,−1) and (4, 8)

40. In what ratio does the point P(–2, 4) divide the line segment joining the points A(–3, 6) and B(1, –2) internally?

41. Find the values of the following:
(i) (cos00 + sin450 + sin300)(sin900 - cos450 + cos600)
(ii) tan2600 - 2tan2450 - cot2300 +2sin2300$\frac { 3 }{ 4 }$ cosec2 450

42. Find the area of a quadrilateral ABCD whose sides are AB = 8cm, BC = 15 cm, CD = 12 cm, AD = 25 cm and = 90°.


text:
# Find a basis for the null space of the transpose operator

1. Mar 19, 2014

### nateHI

1. The problem statement, all variables and given/known data

Let $n$ be a positive integer and let $V = P_n$ be the space of polynomials over $R$. Let D be the differentiation operator on $V$ . Find a basis for the null space of the transpose operator $D^t: V^*\to V^*$.

2. Relevant equations

Let $T:V\to W$ be a linear transformation. Then $(im T)^0=kerT^t$
Let let $\{f_0,f_1,...f_n\}$ be a dual basis for $\{v_0,...,v_n\}$ then $f_i(v_j)=\delta_{ij}$

3. The attempt at a solution
Let $\{1,x,...x^n\}$ be a basis for $V$ and let $\{f_0,f_1,...f_n\}$ be the dual basis. The image of $D$ is the set of polynomials with degree less than n. We know from $f_i(v_j)=\delta_{ij}$ that $f_n$ will annihilate any polynomial without an $x^n$ term. But that is the entire image of $D$ so $\{f_n\}$ is the basis we seek, where I used $(im T)^0=kerT^t$.

I think this is correct, I'm just looking for tips to make it cleaner.

Last edited: Mar 19, 2014
2. Mar 19, 2014

### maajdl

What does that mean:

Find a basis for the null space of the transpose operator Dt:V∗→V∗.

What is the relation between D and DT?
What is the definition of V* ?

3. Mar 19, 2014

### nateHI

Sorry for my lack of understanding, are you asking or prompting me to think harder because I missed something?

4. Mar 19, 2014

### maajdl

No, I just ask because I think I might answer if I understood the question a bit more.
By the way, informal discussion is often helpful.
That's my own experience.

5. Mar 19, 2014

### maajdl

I just understood that your question involved the dual space and Dt is probably the equivalent of D in this space.
On wikipedia, the dual space is defined on the basis of "coordinates" , which is not very encouraging for me.
How is the norm defined, how can I construct the dual space?

6. Mar 19, 2014

### nateHI

Hmm, very good questions. I'll take a crack at answering tomorrow night. I just finished my homework and need some sleep. Unless of course someone smarter than me (of which there are many to choose from) gets to it first.

7. Mar 19, 2014

### pasmith

If $V$ is a vector space over $\mathbb{R}$, then the dual space $V^{*}$ is the set of linear functions from $V$ to $\mathbb{R}$. It is not necessary for $V$ to be an inner product space or normed space for the dual space to be defined.

A linear map $f: V \to W$ has a corresponding dual map $f^{t} : W^{*} \to V^{*}$ given by
$$f^t : \phi \mapsto (\phi \circ f)$$

Here we are asked for a basis of $\ker D^t = \{\phi \in P_n^{*}: D^t(\phi) = 0\}$, where $D: v \mapsto v'$ is the differentation operator on $V$.

You need to define $v_j = x^j$.

I think you have actually to say what $f_n$ is, other than "the dual vector such that $f_n(x^k) = \delta_{kn}$".

The short proof is to write down
$$D^t(\phi)\left(\sum_{k=0}^n a_kx^k\right) = \sum_{k=1}^n ka_k\phi(x^{k-1})$$
and see that $D^t(\phi) = 0$ requires $\phi(1) = \phi(x) = \dots = \phi(x^{n-1})= 0$, so that $\phi(v) = cv^{(n)}(0)$ for some scalar $c$.