# Laplace Transform vs Fourier transform

Started by October 27, 2009
```On 10/31/2009 9:43 AM, Jerry Avins wrote:
> Randy Yates wrote:
>> Jerry Avins <jya@ieee.org> writes:
>>> [...]
>>> Complex numbers are a clever and useful way to represent those
>>> quantities in cases where phase shift has meaning.
>>>
>>> Compact notations expand out ability to comprehend (same root as in
>>> prehensile) complicated things. Vector analysis, quaternions, complex
>>> numbers, matrix algebra .. without them, we'd be hard pressed to
>>> express, let alone understand, some phenomena as
>>> concepts. Nevertheless, being built up from simpler stuff (Shall we go
>>> back to Peano's axioms?) thay constitute the HLLs of math.
>>
>> Jerry,
>>
>> First of all, Peano's axioms have nothing to do with complex, or even
>> real, numbers. They are another way to get, eventually, to the ring of
>> integers (other than the standard group-based development) and integral
>> domains.
>>
>> And once again I must point out that your intimations are, at a minimum,
>> at odds with the field of mathematics (namely, abstract algebra) that
>> has proven extremely useful (re: the realization that no formulas exist
>> for roots of equations greater than order 4). Under this system of
>> mathematics, the complex numbers are NOT just a notational
>> convenience. Rather, they are a field without which the solution of
>> arbitrary algebraic equations over the reals cannot be determined.
>>
>> You have argued many times that they are just a "notational
>> convenience." I posit that the extremely rich utility in viewing
>> algebraic systems through the lens of two operations (addition and
>> multiplication), i.e., as in the basic ring definition, makes this
>> assertion patently false.
>
> I wouldn't argue even about promoting "notational convenience" to
> "notational necessity". I argue that the notation and the phenomenon
> described by it are not the same thing. As Korzybski wrote, "The map is
> not the territory it represents". Granted, he went on, "but if correct,
> it has a similar structure to the territory, which accounts for its
> usefulness". Still, I think it is important to distinguish between
> "similar" and "same".
>
> Jerry

That's appropriate.  Maps can represent three dimensions, and therefore
topology, using different means.  Is the notation in any one more or
less similar, or have increased "sameness" than the others?  What's the
level of dissimilarity or unsameness that would make one fail your test
of having connection to the real world?

I think complex quantities exist if things exist that have magnitude and
phase.  I think such things do "exist", but perhaps that just moves the
discussion to what "exist" means.

--
Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.abineau.com
```
```Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes:
>> [...]
>> I wouldn't argue even about promoting "notational convenience" to
>> "notational necessity". I argue that the notation and the phenomenon
>> described by it are not the same thing. As Korzybski wrote, "The map
>> is not the territory it represents". Granted, he went on, "but if
>> correct, it has a similar structure to the territory, which accounts
>> for its usefulness". Still, I think it is important to distinguish
>> between "similar" and "same".
>
> Jerry,
>
> This is not a matter of mistaking the map for reality, but rather of
> arguing whether or not the map accurately portrays reality.
>
> Since neither one of us knows with absolute certainty and complete
> characterization what the "phenomenom" (i.e., some aspect of reality)
> is, neither one of us can argue that it "is" or "is not" correctly
> portrayed by a certain "map."

That goes only one way. It is certainly possible in some instances that
a particular map misrepresents the area it purports to describe.

> However, the longer, more extensively, and more repeatedly a "map"
> demonstrates itself by experience and experiment to accurately
> represent, and even to predict, that phenomenom - that aspect of
> reality - the less likely it is that the map is wrong.
>
> I believe that is the case with complex numbers. Arguing against their
> existence is, in my opinion, similar to arguing against the statement
> that you have four grandchildren (e.g.) on the basis that the concept of
> "4" is a map and not necessarily representative of reality.

If course complex numbers exist .. on paper. That doesn't mean that the
physical phenomena that we use them to describe are themselves complex
or can be understood only as complex numbers. Remember, there is not one
complex number in Maxwell's Treatise.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Jerry Avins <jya@ieee.org> writes:

> Randy Yates wrote:
>> Jerry Avins <jya@ieee.org> writes:
>>> [...]
>>> I wouldn't argue even about promoting "notational convenience" to
>>> "notational necessity". I argue that the notation and the phenomenon
>>> described by it are not the same thing. As Korzybski wrote, "The map
>>> is not the territory it represents". Granted, he went on, "but if
>>> correct, it has a similar structure to the territory, which accounts
>>> for its usefulness". Still, I think it is important to distinguish
>>> between "similar" and "same".
>>
>> Jerry,
>>
>> This is not a matter of mistaking the map for reality, but rather of
>> arguing whether or not the map accurately portrays reality.
>>
>> Since neither one of us knows with absolute certainty and complete
>> characterization what the "phenomenom" (i.e., some aspect of reality)
>> is, neither one of us can argue that it "is" or "is not" correctly
>> portrayed by a certain "map."
>
> That goes only one way. It is certainly possible in some instances
> that a particular map misrepresents the area it purports to describe.
>
>> However, the longer, more extensively, and more repeatedly a "map"
>> demonstrates itself by experience and experiment to accurately
>> represent, and even to predict, that phenomenom - that aspect of
>> reality - the less likely it is that the map is wrong.
>>
>> I believe that is the case with complex numbers. Arguing against their
>> existence is, in my opinion, similar to arguing against the statement
>> that you have four grandchildren (e.g.) on the basis that the concept of
>> "4" is a map and not necessarily representative of reality.
>
> If course complex numbers exist .. on paper.

You know that's not what I meant. At the point we fail to communicate,
or refuse to hear one another, we kill the discussion.

> That doesn't mean that the physical phenomena that we use them to
> describe are themselves complex

My point is, it doesn't mean they aren't, either. And interpreting them
as such gives us so much more understanding than any other method that
it's utter foolishness to abandon that understanding, all to hold the
door open for some mythical alternate reality that no one has yet seen.

> or can be understood only as complex numbers.

Of course this is true.
--
Randy Yates                      % "She's sweet on Wagner-I think she'd die for Beethoven.
Digital Signal Labs              %  She love the way Puccini lays down a tune, and
mailto://yates@ieee.org          %  Verdi's always creepin' from her room."
http://www.digitalsignallabs.com % "Rockaria", *A New World Record*, ELO
```
```Randy Yates wrote:

...

>> If course complex numbers exist .. on paper.
>
> You know that's not what I meant. At the point we fail to communicate,
> or refuse to hear one another, we kill the discussion.

Sorry. I might have guessed that would push your buttons. I really did
mean something non-trivial.

>> That doesn't mean that the physical phenomena that we use them to
>> describe are themselves complex
>
> My point is, it doesn't mean they aren't, either. And interpreting them
> as such gives us so much more understanding than any other method that
> it's utter foolishness to abandon that understanding, all to hold the
> door open for some mythical alternate reality that no one has yet seen.

Interpreting them as complex is a powerful analysis. Rejecting other
interpretations narrows our view. We don't need to choose one. We can

>> or can be understood only as complex numbers.
>
> Of course this is true.

We seem to agree then.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Jerry Avins <jya@ieee.org> writes:

> Randy Yates wrote:
> [...]
>> My point is, it doesn't mean they aren't, either. And interpreting them
>> as such gives us so much more understanding than any other method that
>> it's utter foolishness to abandon that understanding, all to hold the
>> door open for some mythical alternate reality that no one has yet seen.
>
> Interpreting them as complex is a powerful analysis. Rejecting other
> interpretations narrows our view. We don't need to choose one. We can
> profit from the broader view.

Name one (profit) from doing so.
--
Randy Yates                      % "I met someone who looks alot like you,
Digital Signal Labs              %             she does the things you do,
mailto://yates@ieee.org          %                     but she is an IBM."
http://www.digitalsignallabs.com %        'Yours Truly, 2095', *Time*, ELO
```
```Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes:
>
>> Randy Yates wrote:
>> [...]
>>> My point is, it doesn't mean they aren't, either. And interpreting them
>>> as such gives us so much more understanding than any other method that
>>> it's utter foolishness to abandon that understanding, all to hold the
>>> door open for some mythical alternate reality that no one has yet seen.
>> Interpreting them as complex is a powerful analysis. Rejecting other
>> interpretations narrows our view. We don't need to choose one. We can
>> profit from the broader view.
>
> Name one (profit) from doing so.

Deeper understanding. Let's illustrate by calculating the harmonic
content of a waveform for which we have a picture, but no equation.

This is a real example. Using a tube's or transistor's operating
characteristic, presented as a set of curves on a graph, it is
straightforward to derive a graph of the transfer characteristic.
Without an mathematical representation of that curve, a Fourier analysis
based on complex exponentials seems out of reach. If you think that
complex exponentials are the be-all and end-all of the analysis, you're
stuck. The various harmonic analyzers of the early 20th and late 19th
centuries handled this problem easily. It can be done with dividers and
graph paper, with a little slide-rule calculation.

Multiply the curve by sine and cosine basis functions at the harmonics
needed. Usually, the first 10 or so are considered enough. The curve
needs to be sampled in enough places to accommodate the highest harmonic
of interest, and only the values at the sample points are needed. these
can be tabulated and the whole thing worked out in a spreadsheet, except
when I needed to do this, there were neither spreadsheets nor computers
that would fit in anything smaller than a ballroom.

I suggest that one has a much better understanding of what a Fourier
transform is having done this by hand with sines and cosines than by
turning the crank on pairs of complex exponentials. I don't suggest that
it is more efficient.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```On Nov 1, 12:22 am, Jerry Avins <j...@ieee.org> wrote:
> Randy Yates wrote:
> > Jerry Avins <j...@ieee.org> writes:
>
> >> Randy Yates wrote:
> >> [...]
>
< snip >
>
> I suggest that one has a much better understanding of what a Fourier
> transform is having done this by hand with sines and cosines than by
> turning the crank on pairs of complex exponentials. I don't suggest that
> it is more efficient.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF

OK, back to the ~original question, why is the LT defined for
postitive values of "t" ?
At one point I thought this had something to do w/ turning on the
signal at time "0"

```
```"Jerry Avins" <jya@ieee.org> wrote in message
> glen herrmannsfeldt wrote:
>> Jerry Avins <jya@ieee.org> wrote:
>> (snip, someone wrote)
>>
>>>> This is because the magnitude of the imaginary component of a complex
>>>> signal is a real signal.
>>
>>> Of course. All things measurable are real. "Complex" numbers are merely
>>> a clever and useful bookkeeping scheme for manipulating related pairs of
>>> real quantities.
>>
>> Or pairs of real numbers are a convenient way of measuring complex
>> quantities.  Impedance and index of refraction are both complex, and for
>> similar
>> reasons.  We can separately measure resistance and reactance, or
>> the real and imaginary parts of the index of refraction.  (The imaginary
>> part comes from absorption.)  Both are due to
>> the interaction of electrons with atoms, and with each other.
>>
>> Often the available materials are fairly close to ideal, such
>> that we can separate the quantities.  Resistors do have inductance,
>> inductors (except superconductors) do have resistance.
>
> Complex numbers are a clever and useful way to represent those quantities
> in cases where phase shift has meaning.
>
> Compact notations expand out ability to comprehend (same root as in
> prehensile) complicated things. Vector analysis, quaternions, complex
> numbers, matrix algebra .. without them, we'd be hard pressed to express,
> let alone understand, some phenomena as concepts. Nevertheless, being
> built up from simpler stuff (Shall we go back to Peano's axioms?) thay
> constitute the HLLs of math. Ultimately, all running code executes
> assembly language.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
>
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Well, I'd say machine code rather than assembly, though there's usually a
1:1 mapping.

--Phil

```
```stevem1 wrote:
> On Nov 1, 12:22 am, Jerry Avins <j...@ieee.org> wrote:
>> Randy Yates wrote:
>>> Jerry Avins <j...@ieee.org> writes:
>>>> Randy Yates wrote:
>>>> [...]
> < snip >
>> I suggest that one has a much better understanding of what a Fourier
>> transform is having done this by hand with sines and cosines than by
>> turning the crank on pairs of complex exponentials. I don't suggest that
>> it is more efficient.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
>
> OK, back to the ~original question, why is the LT defined for
> postitive values of "t" ?
> At one point I thought this had something to do w/ turning on the
> signal at time "0"

I don't know what M. Laplace had in mind, but I do know that LTs are a
formalization of Oliver Heavyside's operational calculus (D operators
and all that) which he developed to simplify the solution of homogenous
linear differential equations with constant coefficients. These are
usually accompanied by initial conditions, which account for any past
history that might exist. (Heavyside also invented the step function, an
integral of an impulse.)

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Phil Martel wrote:
> "Jerry Avins" <jya@ieee.org> wrote in message
>> glen herrmannsfeldt wrote:
>>> Jerry Avins <jya@ieee.org> wrote:
>>> (snip, someone wrote)
>>>
>>>>> This is because the magnitude of the imaginary component of a complex
>>>>> signal is a real signal.
>>>> Of course. All things measurable are real. "Complex" numbers are merely
>>>> a clever and useful bookkeeping scheme for manipulating related pairs of
>>>> real quantities.
>>> Or pairs of real numbers are a convenient way of measuring complex
>>> quantities.  Impedance and index of refraction are both complex, and for
>>> similar
>>> reasons.  We can separately measure resistance and reactance, or
>>> the real and imaginary parts of the index of refraction.  (The imaginary
>>> part comes from absorption.)  Both are due to
>>> the interaction of electrons with atoms, and with each other.
>>>
>>> Often the available materials are fairly close to ideal, such
>>> that we can separate the quantities.  Resistors do have inductance,
>>> inductors (except superconductors) do have resistance.
>> Complex numbers are a clever and useful way to represent those quantities
>> in cases where phase shift has meaning.
>>
>> Compact notations expand out ability to comprehend (same root as in
>> prehensile) complicated things. Vector analysis, quaternions, complex
>> numbers, matrix algebra .. without them, we'd be hard pressed to express,
>> let alone understand, some phenomena as concepts. Nevertheless, being
>> built up from simpler stuff (Shall we go back to Peano's axioms?) thay
>> constitute the HLLs of math. Ultimately, all running code executes
>> assembly language.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>>
> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
> Well, I'd say machine code rather than assembly, though there's usually a
> 1:1 mapping.
>
>    --Phil

Yes. I used to program my IMSAI Nova from the front panel toggles.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```