How the chain rule applies to integral equations.

In Calculus, one of the most basic things that can be solved for, is that a principal function receives a parameter, multiplies it by a multiplier, and then passes the product to a nested function, of which either the derivative or the integral can subsequently be found. But what needs to be done over the multiplier, is opposite for integration, from what it was for differentiation. The following two work-sheets illustrate:

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EPUB File for Mobile Devices

Please pardon the poor typesetting of the EPUB File. It’s the result of some compatibility issues (with EPUB readers which do not support EPUB3 that uses MathML.)

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Exploring the Discrete Sine Transform…

I can sometimes describe a way of using certain tools – such as in this case, one of the Discrete Cosine Transforms – which is correct in principle, but which has an underlying flaw, that needs to be corrected, from my first approximation of how it can be applied.

One of the things which I had said was possible was, to take a series of frequency-domain ‘equalizer settings’, which be at one per unit of frequency, not, at so many per octave, compute whichever DCT was relevant, such that the result had the lowest frequency as its first element, and then to apply that result as a convolution, in order finally to apply the computed equalizer to a signal.

One of the facts which I’m only realizing recently is, that if the DCT is computed in a one-sided way, the results are ‘completely non-ideal’, because it gives no control over what the phase-shifts will be, at any frequency! Similarly, such a one-sided convolution can also not be applied as the sinc function, because the amount of sine-wave output, in response to a cosine-wave input, will approach infinity, when the frequency is actually at the cutoff frequency.

What I have found instead is, that if such a cosine transform is mirrored around a centre-point, the amount of sine response, to an input cosine-wave, will cancel out and become zero, thus giving phase-shifts of zero.

But a result which some people might like is, to be able to apply controlled phase-shifts, differently for each frequency, such that those people specify a cosine as well as a sine component, for an assumed input cosine-wave.

The way to accomplish that is, to add-in the corresponding (normalized) sine-transform, of the series of phase-shifted response values, and to observe that the sine-transform will actually be zero at the centre-point. Then, the thing to do is, to apply the results negatively on the other side of the centre-point, which were to be applied positively on one side.


 

 

I have carried out a certain experiment with the Computer Algebra System named “wxMaxima”, in order first to observe what happens if a set of equal, discrete frequency-coefficients belonging to a series is summed. And then, I plotted the result of the definite integral, of the sine function, over a short interval. Just as with the sinc function, The integral of the cosine function was (sin(x) – sin(0)) / x, the definite integral of the sine function will be (1 – cos(x)) / x, and, Because the derivative of cos(x) is zero at (x = 0), the limit equation based on the divide by zero, will actually approach zero, and be well-behaved.


 

 

(Update 1/31/2021, 13h35: )

There is an underlying truth about Integral Equations in general, which people who studied Calculus 2 generally know, but, I have no right just to assume that any reader of my blog did so. There exist certain standard Integrals, which behave in the reverse way of how the standard Derivatives behave, just because ‘Integrals’ are ‘Antiderivatives’…

When one solves the Derivatives of certain trig functions repeatedly, one obtains the sequence:

sin(x) -> cos(x) -> -sin(x) -> -cos(x) -> sin(x)

Solving the Indefinite Integrals of the same trig functions yields the result:

sin(x) -> -cos(x) -> -sin(x) -> cos(x) -> sin(x)

Hence, the Indefinite Integral of sin(x) is in fact -cos(x), and:

( -(-cos(0)) = +1 )

(End of Update, 1/31/2021, 13h35.)

 

(Updated 2/04/2021, 17h10…)

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Power Failure Today, Downtime

I take the unusual approach of hosting this blog and site, on a server, that is running on my personal computer at home. I don’t recommend that everybody do it this way; this is only how I do it. That makes the availability of my blog and site no better, than the reliability with which I can keep my PC running, as well as that, of my Internet connection.

Unfortunately, I experienced a brief power failure this morning, between 8h45 and 9h00. As a result, this site was down until about 9h40. I apologize for any inconvenience to my readers.

BTW, There have been remarkably few failures in the recent 3 months or so.

Dirk

 

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Samsung’s Auto Hot-Spot Feature

I own a Samsung Galaxy S9 smart-phone, and have discovered that, in its tethering settings, there is a new setting, which is named “Auto Hotspot”. What this setting aims to do if activated is, on other Samsung devices, which normally only have WiFi, when the user is roaming along with his phone, there should appear an additional access point for them to connect to. The following screen-shots show, where this can be enabled on the phone

Screenshot_20201220-072343_Settings_e

Screenshot_20201220-072354_Settings_e

Screenshot_20201220-072404_Settings_e

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I believe that this explains a fact which I’ve already commented on elsewhere, which is, that when I try to set up Google Instant Tethering, the negotiation between my ‘Asus Flip C213 Chromebook’ and this phone, no longer adds Instant Tethering to the list of features which are enabled. My Samsung S9 phone will now only unlock the Chromebook. What I am guessing is that, because the feature I’m showing in this posting is a Samsung feature, with which Samsung wants to compete with the other companies, Samsung probably removed to offer Instant Tethering from their phone.

Obviously, this is only a feature which I will now be able to use, between my S9 phone, and my Samsung Galaxy TAB S6 tablet.


 

 

The reader may ask what the advantages of this feature might be, over ‘regular WiFi tethering’, or ‘a WiFi hotspot’. The advantage could be, that even though it remains an option compatible with all clients, to have the phone constantly offer a WiFi hot-spot could drain the battery more. Supposedly, if Samsung’s Auto Hotspot is being used, it can be kept enabled on the phone, yet not drain the battery overly, as long as client devices do not connect. The decision could then be made directly from the client device, whether to connect or not… This is similar, to what Google’s system offers.

Also, the Samsung phones with Android 10 have as feature, that their ‘regular hotspots’ will time out, say after 20 minutes of inactivity, again, to save battery drain. Yet, if the user is carrying a tablet with him that has been configured to connect to the mobile hotspot Automatically, the phone which is serving out this hotspot will never detect inactivity.


 

Further, I’ve been able to confirm that, as long as I have Auto Hotspot turned on on my phone, indeed it does not show up as an available WiFi connection, on devices that are not joined to my Samsung account. This is as expected. But it also adds hope that, as long as I don’t connect to the phone’s Auto Hotspot from another device, the battery drain due to my leaving this feature enabled on my phone constantly, may not be very high. I will comment by the end of this day, after having left my phone with its own WiFi Off, which means that my phone will be using its Mobile Data, but, not connecting my Samsung TAB S6, whether doing this seems to incur any unusually high amount of battery drain, on the phone…

 

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