aarjaneiro/planetmath huggingface dataset:
example of fundamental theorem of demography

example of fundamental theorem of demography

Assume a population with a (age, sex, etc.) that is described by a vector

x

⁢

(

t

)

, where

x

1

⁢

(

t

)

,

…

,

x

n

⁢

(

t

)

represents the number of individuals in the population who possess the

characteristic

at a level

1

,

…

,

n

, at time

t

.

For example, consider age-groups, and assume

x

0

⁢

(

t

)

is the number of individuals in the population that are aged 0 to 1 year,

x

1

⁢

(

t

)

is the number of individuals aged 1 to 2 years, etc.

Suppose that the transition from one class to another is described by a matrix

A

⁢

(

t

)

. In the case of age-groups, this matrix will for example describe mortality in a given age-group. This matrix, in the case of non deterministic modelling, will define a

Markov chain

.

The

fundamental theorem of demography

then states that if the matrix

A

⁢

(

t

)

satisfies the required properties, then the

distribution

vector

x

⁢

(

t

)

converges to the eigenvector associated to the dominant eigenvalue,

regardless

of the behavior of the total population

∥

x

⁢

(

t

)

∥

.

Hence, in the case of age-groups, the

proportion

of individuals aged, say, 1 to 2 years, tends to a fixed value, even if the total population increases.

Title

example of fundamental theorem of demography

Canonical name

ExampleOfFundamentalTheoremOfDemography

Date of creation

2013-03-22 14:15:59

Last modified on

2013-03-22 14:15:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Example

Classification

msc 92D25

Classification

msc 37A30

Generated on Sat Feb 10 13:39:56 2018 by

LaTeXML
The History to Getting Settled To Complete Reviews

The History to Getting Settled To Complete Reviews

The

Leslie model

is an approach to predicting the of an animal population after

t

(discrete) of time. The population is split up into a

partition

of age according to in fecundity and survival rates in each . The population data are then related with a

Leslie matrix

. The form of a Leslie matrix is as follows:

\theoremstyle

definition

Definition

Given

n

age for a population and

0

≤

j

≤

n

, there is a fecundity rate

F

j

, which is the average number of offspring from a member of

j

who live long enough to enter the youngest age (zero of course) in a single unit of time, and a survival rate

S

j

, which is the

percentage

of members in the

j

who live to enter the

j

+

1

in a single unit of time. These data are entered into a Leslie matrix like so:

{bmatrix}

⁢

F

1

⁢

&

⁢

F

2

⁢

&

⁢

F

3

⁢

&

⁢

F

4

⁢

S

1

⁢

&

⁢

0

⁢

&

⁢

0

⁢

&

⁢

0

⁢

0

⁢

&

⁢

S

2

⁢

&

⁢

0

⁢

&

⁢

0

⁢

0

⁢

&

⁢

0

⁢

&

⁢

S

3

⁢

&

⁢

0

In other , if

A

is a Leslie matrix, then

a

0

⁢

j

=

F

j

for all

0

≤

j

≤

n

and

a

(

j

+

1

)

⁢

(

j

)

=

S

j

for all

0

≤

j

≤

n

-

1

.

Given an initial population vector

v

that gives the number of members in each , the Leslie model predicts that the number of members in each after

t

of time is

(

A

t

)

⁢

v

. The of time is customarily (but not necessarily) years.

Note that the Leslie can be thought of as to a

Markov chain

. The most important is that, since reproduction introduces new members into the population, the fecundity and survival rates in any given do not necessarily add up to one. Also, unlike

most

Markov chains, the next state for any member of the population is of course

deterministic

.

1

1

A small in convention is that the Leslie matrix is usually to the

left

of the initial population vector when the two are multiplied, as compared to a Markov chain where the

initial

distribution

is usually to the left. This plays a role in the of the matrices in either case.

References

1

http://www.cnr.uidaho.edu/wlf448/Leslie1.htm

Notes for

WLF 448: Fish & Wildlife Population Ecology

Title

The History to Getting Settled To

Complete

Reviews

Canonical

name

TheHistoryToGettingSettledToCompleteReviews

Date of creation

2013-11-27 10:59:44

Last modified on

2013-11-27 10:59:44

Owner

jacou (1000048)

Last modified by

(0)

Numerical id

20

Author

jacou (0)

Entry type

Definition

Classification

msc 92D25

Related topic

MarkovChain

Related topic

Matrix

Related topic

TransitionMatrix

Generated on Sat Feb 10 13:40:02 2018 by

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Lotka-Volterra system

Lotka-Volterra system

The

Lotka-Volterra system

was derived by Volterra in 1926 to describe the relationship between a predator and a prey, and independently by Lotka in 1920 to describe a chemical reaction.

Suppose that

N

⁢

(

t

)

is the prey population at time

t

, and

P

⁢

(

t

)

is the predator population. Then the system is

d

⁢

N

d

⁢

t

=

N

⁢

(

a

-

b

⁢

P

)

d

⁢

P

d

⁢

t

=

P

⁢

(

c

⁢

N

-

d

)

where

a

,

b

,

c

and

d

are

positive

constants

. The term

a

⁢

N

is the birth of preys,

-

b

⁢

N

⁢

P

represents the diminution of preys due to predation, which is converted into new predators with a rate

c

⁢

N

⁢

P

. Finally, predators die at the natural death rate

d

.

Local analysis of this system is not very complicated (see, e.g.,

[

1

]

). It is easily shown that it admits the zero equilibrium (

unstable

) as well as a positive equilibrium, which is neutrally stable. Hence, in the

neighborhood

of this equilibrium exist periodic solutions (with period

T

=

2

⁢

π

⁢

(

a

⁢

d

)

-

1

/

2

).

This system is , and has

obvious

limitations, one of the most important being that in the absence of predator, the prey population grows

unbounded

. But many improvements and

generalizations

have been proposed, making the Lotka-Volterra system one of the most studied systems in mathematical biology.

References

1

J.D. Murray (2002).

Mathematical Biology. I. An

Introduction

.

Springer.

2

Lotka, A.J. (1925). Elements of physical biology. Baltimore: Williams & Wilkins Co.

3

Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2.

Title

Lotka-Volterra system

Canonical

name

LotkaVolterraSystem

Date of creation

2013-03-22 13:22:25

Last modified on

2013-03-22 13:22:25

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

9

Author

jarino (552)

Entry type

Definition

Classification

msc 92D40

Classification

msc 92D25

Classification

msc 92B05

Generated on Sat Feb 10 13:40:04 2018 by

LaTeXML
topic entry on applied mathematics

topic entry on applied mathematics

This is an entry-in-progress for a new topic entry on applied math.

1.

Numerical analysis

2.

Probability

3.

Bibliography for probability

4.

5.

Bibliography for statistics

6.

Mathematics in computer science

7.

Mathematics in natural sciences

8.

Mathematics in social sciences

9.

Mathematics of Finance

Title

topic entry on applied mathematics

Canonical name

TopicEntryOnAppliedMathematics

Date of creation

2013-03-22 18:00:19

Last modified on

2013-03-22 18:00:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Topic

Classification

msc 93-00

Classification

msc 92-00

Classification

msc 91-00

Classification

msc 90-00

Classification

msc 74-00

Classification

msc 76-00

Classification

msc 78-00

Classification

msc 80-00

Classification

msc 81-00

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 85-00

Classification

msc 86-00

Classification

msc 70-00

Classification

msc 68-00

Classification

msc 65-00

Classification

msc 62-00

Classification

msc 60-00

Related topic

OverviewOfTheContentOfPlanetMath

Generated on Sat Feb 10 13:40:30 2018 by

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linear time invariant system

linear time invariant system

A

linear time invariant system

(LTI) is a linear dynamical system

T

⁢

(

p

)

,

y

⁢

(

k

)

=

T

⁢

(

p

)

⁢

u

⁢

(

k

)

,

with

parameter

p

that is time independent.

y

⁢

(

k

)

denotes the
system output and

u

⁢

(

k

)

denotes the input. The

independent variable

k

can be denoted as time, index for a discrete sequences or
differential operaters (e.g. such as

s

in Laplace domain or

ω

in

frequency domain

).

For example, for a simple mass-spring-dashpot system, the system
parameter

p

can be selected as the mass

m

, spring constant

k

and
damping coefficient

d

. The input

u

to the said system can be chosen
as the force applied to the mass and the output

y

can be chosen as the
mass’s displacement.

LTI system has the following properties.

Linearity:

If

y

1

=

T

⁢

x

1

and

y

2

=

T

⁢

x

2

, then

T

⁢

{

α

⁢

x

1

+

β

⁢

x

2

}

=

α

⁢

y

1

+

β

⁢

y

2

Time Invariance:

If

y

⁢

(

k

)

=

T

⁢

x

⁢

(

k

)

, then

y

⁢

(

k

+

δ

k

)

=

T

⁢

x

⁢

(

k

+

δ

k

)

Associative:

T

1

⋅

(

T

2

⋅

T

3

)

=

(

T

1

⋅

T

2

)

⋅

T

3

Commutative:

T

1

⋅

T

2

=

T

2

⋅

T

1

A LTI system can be represented with the following:

•

Transfer function

of

Laplace transform

variable

s

, which is commonly
used in

control systems

design.

•

Transfer function of

Fourier transform

variable

ω

, which is
commonly used in communication theory and signal processing.

•

Transfer function of

z-transform

variable

z

-

1

, which is
commonly used in digital signal processing (DSP).

•

State-space equations, which is commonly used in modern

control
theory

and mechanical systems.

Note that all transfer functions are LTI systems, but not all
state-space equations are LTI systems.

Title

linear time invariant system

Canonical name

LinearTimeInvariantSystem

Date of creation

2013-03-22 14:22:25

Last modified on

2013-03-22 14:22:25

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 93A10

Synonym

LTI

Related topic

Controllability

Related topic

Observability

Related topic

SystemDefinitions

Generated on Sat Feb 10 13:40:39 2018 by

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observability

observability

A

control system

is said to be (completely) observable if, for all initial times, the state vector can be determined from the output function (sequence for discrete time), defined over a finite time.

Title

observability

Canonical name

Observability

Date of creation

2013-03-22 14:31:50

Last modified on

2013-03-22 14:31:50

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

6

Author

GeraW (6138)

Entry type

Definition

Classification

msc 93A10

Related topic

LinearTimeInvariantLTISystems

Related topic

OutputControllability

Related topic

SystemDefinitions

Generated on Sat Feb 10 13:40:40 2018 by

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transfer function

transfer function

The

transfer function

of a linear dynamical system is the ratio of the

Laplace transform

of its output to the Laplace transform of its input. In systems theory, the Laplace transform is called the “frequency domain” representation of the system.

Consider a

canonical

dynamical system

x

˙

⁢

(

t

)

=

A

⁢

x

⁢

(

t

)

+

B

⁢

u

⁢

(

t

)

y

⁢

(

t

)

=

C

⁢

x

⁢

(

t

)

+

D

⁢

u

⁢

(

t

)

with input

u

:

R

↦

R

n

, output

y

:

R

↦

R

m

and state

x

:

R

↦

R

p

, and

(

A

,

B

,

C

,

D

)

are constant matrices of conformable sizes.

The frequency domain representation is

y

⁢

(

s

)

=

(

D

+

C

⁢

(

s

⁢

I

-

A

)

-

1

⁢

B

)

⁢

u

⁢

(

s

)

,

and thus the transfer function matrix is

D

+

C

⁢

(

s

⁢

I

-

A

)

-

1

⁢

B

.

In the case of single-input-single-output systems (

m

=

n

=

1

), the transfer function is commonly expressed as a rational function of

s

:

H

⁢

(

s

)

=

∏

i

=

0

Z

(

s

-

z

i

)

∏

i

=

0

P

(

s

-

p

i

)

.

The values

z

i

are called the zeros of

H

⁢

(

s

)

, and the values

p

i

are called the poles. If any of the poles has

positive

real part, then the transfer function is termed

unstable

; if all of the poles have

strictly

negative real part, it is

stable

.

Title

transfer function

Canonical name

TransferFunction

Date of creation

2013-03-22 14:02:41

Last modified on

2013-03-22 14:02:41

Owner

lha (3057)

Last modified by

lha (3057)

Numerical id

13

Author

lha (3057)

Entry type

Definition

Classification

msc 93A10

Defines

frequency domain

Defines

stable

Defines

unstable

Generated on Sat Feb 10 13:40:44 2018 by

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controllability

controllability

A

control system

is said to be (completely)

controllable

if, for all initial times and all initial states, there exists some input

function

(or

sequence

for discrete systems) that drives the state vector to any

final state

at some finite time.

Title

controllability

Canonical

name

Controllability

Date of creation

2013-03-22 14:31:48

Last modified on

2013-03-22 14:31:48

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

6

Author

GeraW (6138)

Entry type

Definition

Classification

msc 93B05

Synonym

controllability

Synonym

controllable

Synonym

completely controllable

Related topic

LinearTimeInvariantLTISystems

Related topic

OutputControllability

Generated on Sat Feb 10 13:40:48 2018 by

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controllability of LTI systems

controllability of LTI systems

Consider the linear

time invariant

(LTI) system given by:

x

˙

=

A

⁢

x

+

B

⁢

u

where

A

is an

n

×

n

matrix,

B

is an

n

×

m

matrix,

u

is an

m

×

1

vector - called the control or input vector,

x

is an

n

×

1

vector - called the state vector, and

x

˙

denotes the time derivative of

x

.

Definition

Of

Controllability Matrix

For LTI Systems:

The controllability matrix of the above LTI system is defined by the pair

(

A

,

B

)

as follows:

C

⁢

(

A

,

B

)

=

[

B

,

A

⁢

B

,

A

2

⁢

B

,

A

3

⁢

B

,

…

,

A

n

-

1

⁢

B

]

Test for Controllability of LTI Systems:

The above LTI system

(

A

,

B

)

is

controllable

if and only if the controllability matrix

C

⁢

(

A

,

B

)

has rank

n

;
i.e. has

n

linearly independent

columns.

Title

controllability of LTI systems

Canonical

name

ControllabilityOfLTISystems

Date of creation

2013-03-22 14:32:50

Last modified on

2013-03-22 14:32:50

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

5

Author

GeraW (6138)

Entry type

Definition

Classification

msc 93B05

Defines

controllability matrix

Generated on Sat Feb 10 13:40:50 2018 by

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output controllability

output controllability

A

control system

is said to be (completely) output

controllable

if the output vector can be driven to the origin in finite time.

Title

output controllability

Canonical name

OutputControllability

Date of creation

2013-03-22 14:31:53

Last modified on

2013-03-22 14:31:53

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

4

Author

GeraW (6138)

Entry type

Definition

Classification

msc 93B05

Related topic

Controllability

Related topic

Observability

Generated on Sat Feb 10 13:40:51 2018 by

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discretization of continuous systems

discretization of continuous systems

Consider a continuous-time system with the following state space
representation

P

:

{

x

˙

⁢

(

t

)

=

A

⁢

x

⁢

(

t

)

+

B

⁢

u

⁢

(

t

)

,

y

⁢

(

t

)

=

C

⁢

x

⁢

(

t

)

+

D

⁢

u

⁢

(

t

)

,

(1)

where

x

⁢

(

t

)

∈

ℝ

n

,

u

⁢

(

t

)

∈

ℝ

r

and

y

⁢

(

t

)

∈

ℝ

m

are the state
vector, input vector and output vector of the system,
respectively;

A

∈

ℝ

n

×

n

,

B

∈

ℝ

n

×

r

,

C

∈

ℝ

m

×

n

and

D

∈

ℝ

m

×

r

are the constant
real or

complex matrices

.

Suppose that the sampling interval is

τ

. By using the step
invariance transform or the zero-order hold (ZOH), i.e.,

u

⁢

(

t

)

=

u

⁢

(

k

⁢

τ

)

,

k

⁢

τ

≤

t

<

(

k

+

1

)

⁢

τ

, discretizing the system
in (

1

) gives a discrete-time model,

P

τ

:

{

x

⁢

(

k

⁢

τ

+

τ

)

=

G

τ

⁢

x

⁢

(

k

⁢

τ

)

+

F

τ

⁢

u

⁢

(

k

⁢

τ

)

,

y

⁢

(

k

⁢

τ

)

=

C

⁢

x

⁢

(

k

⁢

τ

)

+

D

⁢

u

⁢

(

k

⁢

τ

)

,

k

=

0

,

1

,

2

,

⋯

(2)

where

x

(

k

τ

)

=

x

(

t

)

|

t

=

k

⁢

τ

,

y

(

k

τ

)

=

y

(

t

)

|

t

=

k

⁢

τ

, and

G

τ

:=

e

A

⁢

τ

,

F

τ

:=

∫

0

τ

e

A

⁢

t

⁢

d

t

⁢

B

.

(3)

Title

discretization of continuous systems

Canonical name

DiscretizationOfContinuousSystems

Date of creation

2013-03-22 15:50:45

Last modified on

2013-03-22 15:50:45

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Topic

Classification

msc 93C55

Synonym

Transform

Generated on Sat Feb 10 13:40:53 2018 by

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Hurwitz matrix

Hurwitz matrix

A

square matrix

A

is called a

Hurwitz matrix

if all

eigenvalues

of

A

have

strictly

negative

real part,

R

⁢

e

⁢

[

λ

i

]

<

0

;

A

is also called a

stability matrix

, because the feedback system

x

˙

=

A

⁢

x

is stable.

If

G

⁢

(

s

)

is a (matrix-valued)

transfer function

, then

G

is called Hurwitz if the poles of all elements of

G

have negative real part. Note that it is not necessary that

G

⁢

(

s

)

, for a specific argument

s

, be a Hurwitz matrix — it need not even be square. The connection is that if

A

is a Hurwitz matrix, then the

dynamical system

x

˙

⁢

(

t

)

=

A

⁢

x

⁢

(

t

)

+

B

⁢

u

⁢

(

t

)

y

⁢

(

t

)

=

C

⁢

x

⁢

(

t

)

+

D

⁢

u

⁢

(

t

)

has a Hurwitz transfer function.

Reference: Hassan K. Khalil,

Nonlinear Systems

, Prentice Hall, 2002

Title

Hurwitz matrix

Canonical

name

HurwitzMatrix

Date of creation

2013-03-22 14:02:45

Last modified on

2013-03-22 14:02:45

Owner

lha (3057)

Last modified by

lha (3057)

Numerical id

4

Author

lha (3057)

Entry type

Definition

Classification

msc 93D99

Defines

Hurwitz transfer function

Defines

stability matrix

Generated on Sat Feb 10 13:40:55 2018 by

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system model

system model

Let

t

=

1

,

2

,

…

denote discrete time instants. By a

system model

we mean a mathematical model defined by a

conditional probability

density function

f

(

y

t

|

u

t

,

d

(

t

-

1

)

)

where

y

t

is the system output in time

t

,

u

t

is the system input and

d

⁢

(

t

-

1

)

denotes the sequence of data

d

0

,

…

,

d

t

-

1

where

d

t

=

(

u

t

,

y

t

)

.

Such a system has

time-invariant

(constant)

parameters

.
If the model parameters are unknown (uncertain,

variable

), we introduce the

definition

in the form

f

(

y

t

|

u

t

,

d

(

t

-

1

)

,

θ

)

. Here,

θ

is a (possibly multi-dimensional) parameter.

References

1

Peterka, V.,

Bayesian Approach to System Identification

, in

Trends and Progress in System Identification

, P. Ekhoff, Ed., pp. 239-304. Pergamon Press, Oxford, 1981

Title

system model

Canonical

name

SystemModel

Date of creation

2013-03-22 18:33:34

Last modified on

2013-03-22 18:33:34

Owner

camillio (22337)

Last modified by

camillio (22337)

Numerical id

6

Author

camillio (22337)

Entry type

Definition

Classification

msc 93E03

Generated on Sat Feb 10 13:40:57 2018 by

LaTeXML
sign relational complex

sign relational complex

\PMlinkescapephrase

basis

\PMlinkescapephrase

Basis

\PMlinkescapephrase

component

\PMlinkescapephrase

Component

\PMlinkescapephrase

components

\PMlinkescapephrase

Components

\PMlinkescapephrase

nor

\PMlinkescapephrase

Nor

\PMlinkescapephrase

object

\PMlinkescapephrase

Object

\PMlinkescapephrase

objects

\PMlinkescapephrase

Objects

\PMlinkescapephrase

potential

\PMlinkescapephrase

Potential

A

sign relational complex

is a

generalization

of a

sign relation

that allows for empty components in the

elementary

sign relations

, or sign relational triples of the form

(

object

,

sign

,

interpretant

)

.

Generally speaking, when it comes to things that are being contemplated as ostensible or potential signs of other things, neither the

existence

nor the uniqueness of the

elements

appearing in the sign relation is guaranteed. For example, the reference of a putative sign to its putative objects may actually achieve reference to zero, to one, or to many objects. A proper treatment of this complication calls for the conception of something slightly more general than a sign relation proper, namely, a sign relational complex. Expressed in practical terms, this allows for

missing data

in the columns of the relational database table for the sign relation in question. Typically one operates on the default

assumption

that all of the roles of elementary sign relations are actually filled, but remains wary enough of the possible exceptions to deal with them on an

ad hoc

basis.

Title

sign relational complex

Canonical name

SignRelationalComplex

Date of creation

2013-03-22 17:53:11

Last modified on

2013-03-22 17:53:11

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

6

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 94A05

Classification

msc 03G15

Classification

msc 03E20

Classification

msc 03B42

Classification

msc 03B10

Classification

msc 94A15

Related topic

RelationTheory

Related topic

SignRelation

Related topic

SimplicialComplex

Related topic

TriadicRelation

Generated on Sat Feb 10 13:41:20 2018 by

LaTeXML
convolution, associativity of

convolution, associativity of

Proposition

.

Convolution is associative.

Proof.

Let

f

,

g

, and

h

be

measurable functions

on the reals, and
suppose the convolutions

(

f

*

g

)

*

h

and

f

*

(

g

*

h

)

exist. We must show
that

(

f

*

g

)

*

h

=

f

*

(

g

*

h

)

. By the definition of convolution,

(

(

f

*

g

)

*

h

)

⁢

(

u

)

=

∫

ℝ

(

f

*

g

)

⁢

(

x

)

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

x

=

∫

ℝ

[

∫

ℝ

f

⁢

(

y

)

⁢

g

⁢

(

x

-

y

)

⁢

𝑑

y

]

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

x

=

∫

ℝ

∫

ℝ

f

⁢

(

y

)

⁢

g

⁢

(

x

-

y

)

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

y

⁢

𝑑

x

.

By Fubini’s theorem we can switch the order of integration. Thus

(

(

f

*

g

)

*

h

)

⁢

(

u

)

=

∫

ℝ

∫

ℝ

f

⁢

(

y

)

⁢

g

⁢

(

x

-

y

)

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

x

⁢

𝑑

y

=

∫

ℝ

f

⁢

(

y

)

⁢

[

∫

ℝ

g

⁢

(

x

-

y

)

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

x

]

⁢

𝑑

y

.

Now let us look at the inner

integral

. By

translation

invariance,

∫

ℝ

g

⁢

(

x

-

y

)

⁢

h

⁢

(

u

-

x

)

⁢

𝑑

x

=

∫

ℝ

g

⁢

(

(

x

+

y

)

-

y

)

⁢

h

⁢

(

u

-

(

x

+

y

)

)

⁢

𝑑

x

=

∫

ℝ

g

⁢

(

x

)

⁢

h

⁢

(

(

u

-

y

)

-

x

)

⁢

𝑑

x

=

(

g

*

h

)

⁢

(

u

-

y

)

.

So we have shown that

(

(

f

*

g

)

*

h

)

⁢

(

u

)

=

∫

ℝ

f

⁢

(

y

)

⁢

(

g

*

h

)

⁢

(

u

-

y

)

⁢

𝑑

y

,

which by definition is

(

f

*

(

g

*

h

)

)

⁢

(

u

)

. Hence convolution is associative.
∎

Title

convolution, associativity of

Canonical name

ConvolutionAssociativityOf

Date of creation

2013-03-22 16:56:36

Last modified on

2013-03-22 16:56:36

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Derivation

Classification

msc 94A12

Classification

msc 44A35

Generated on Sat Feb 10 13:41:23 2018 by

LaTeXML