I received a new battery for my Acer Aspire today.

One fact which I did write about, was that I own an old ‘Acer Aspire 5020′, in addition to my other computers, but that the network name of the Acer laptop was ‘Venus’, when it is running in Linux mode.

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This laptop was manufactured in 2005, and needed a new battery.

What’s remarkable about this, is the fact that the Lithium Ion battery it came with in 2005, may only have held a fraction of the charge it was supposed to, but still held a charge in 2016. This is because unlike today, in 2005, Acer produced high-quality products. But given the times, I would not buy an Acer today.

The old battery was holding a 1-hour charge in theory, although it would never actually reach that, because my settings forbade the charge from going below 25%.

Now that I received the new battery, I expect that it will hold a 2-hour charge again, and that I can safely set the laptop’s critical level to 15% again. But I do not expect, that the new, gray-market battery, will also last another 10 years.



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I needed to restrict my Vacuuming Robot today.

One fact which I had written about earlier, was that I have a “Neato XV Signature” vacuuming robot.

Neato XV _1

This robot has frequently gotten lost underneath my bed, presumably because the lower surface of my box is a canvas sheet which hangs below the sensor-line of the robot’s laser, in a way that’s curved almost parallel to the floor. The robot cannot analyze the geometry which this seems to forbid it, and can therefore sometimes end up in an endless loop. Because I want to be able to trust my robot with vacuuming 100%, when I’m not home, this error required adjustment.

Using great lengths which I have, of the magnetic boundary marker, I now also zoned off the rectangular region which is the entire underside of my bed. The robot no longer goes there.



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My Distinction Between Variables And Constants

The way I process information, applied to ‘Computer Algebra Systems’, defines the difference between constants and variables in a context-sensitive way. It’s for the purpose of solving one problem, that certain symbols in an expression become variables, others constants, and others yet, function names. The fact that a syntax has been defined to store these symbols, does not affect the fact that their status can be changed from constant to variable and vice-versa.

I’ll name an example. For most purposes a Univariate Polynomial has the single variable (x), denotes powers of (x) as its base terms, and multiplies each of the base terms by a constant coefficient. To some people this might seem immutable.

But if the purpose of the exercise is to compute a Statistical, Polynomial Regression – which is “an overdetermined system” – then we must find optimal values for prospective coefficients. We can use this as the basis to form a “Polynomial Approximation” of a system, which could be of the 8th degree for example, and yet this polynomial must fit a data-set as closely as possible, which could have a list of 20 values of (x), each associated with a real value of (y), which our optimized set of coefficients is supposed to approximate, from the powers of (x), including the power (0), which always yields the base value (1).

In order to determine our 9 coefficients, we need to decide that all the powers of (x) have become constants. The coefficients we’re trying to determine best, have now become the variables in our problem. Thus, we have a column-vector of real (y)s (still variables), and matrices which state the powers of (x) which supposedly led to those values of (y). I believe that this is a standard for doing so:

Regression Analysis Guide

Well another conclusion we can reach, is that the base values which need to be correlated with real (y), aren’t limited to powers of (x). They could just as easily be some other functions of (x). It’s just that one advantage which polynomials have, is that if there is some scaling of (x), it’s possible to define a scaled parameter (t = ux) such that a corresponding polynomial in terms of (t) can do what our polynomial in terms of (x) did. If the base value was ( sin(x) ) , then ( sin(t) ) could not simply take its place. This is important to note if we are trying to approximate orbital motions of planets for example.

But then as soon as we’ve computed our best-fitting vector of coefficients, we can treat them as constants again, so that to plug in different values of (x) which did not occur in the original data-set, will also yield the corresponding, predicted values of (y’). So now (x) and (y’) are our variables again.



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A Key Limitation to Factor Theorem

I don’t really remember my Factor Theorem from John Abbott College well. But one detail which I think I do recall, is that its use was meant for “Univariate Polynomials”, with “Invariant Coefficients”. This means, that the coefficients needed to be integers or ‘other numbers’, known in advance, but not symbolic constants. In computerized cases where the coefficients aren’t preset, there are other, narrow constraints on them. A similar problem exists with the way I was taught to invert certain matrices in Linear Algebra. The elements are well-behaved in certain cases, but just as with polynomials, if the coefficients are suddenly random, floating-point numbers, those methods no longer work. Then, we must use a brute-force approach. And in the case of polynomials, there is no sure brute-force approach that works beyond the 4th degree.

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