[Blindmath] Basic mathematical concept question

Sat Mar 24 20:17:15 UTC 2012

Now for the practical.

Many real-life "curves" cannot be analytically defined in precise
mathematical terms without using an infinite series. Even for those that
can be analytically defined, you might find this information useful.

Consider, for example, the set of values that define the red color values
in a single row of pixels in a digital image. It is highly unlikely that
you can, with any reasonable number of terms, fit a mathematical expression
to to that shape. However, using a computer, you can approximate the first
derivative of that shape by successively subtracting the red value of one
pixel from the red value of the next pixel from the left end of the row to
the right end of the row and saving those difference values along the way.

If you plot those values, you will be plotting an estimate of the first
derivative of the curve.

Viewed from a completely different viewpoint, you will have applied a
high-pass frequency filter to the curve. This filter will suppress the low
frequency values and accentuates the high frequency values. In general, you
can expect the new function to appear to be more "ragged" than the first.

If the set of values to which the process is applied happen to be samples
of audio data, and you listen to the "before" and "after" versions, you
might be able to audibly detect the change in the frequency spectrum.

Perform that process for all three color values for every row of pixels in
the image, normalize the results to guarantee that all of the new values
fall in the range from 0 to 255 inclusive, display those new values as an
image, and you will have produced a new image in which the features tend to
be sharper than in the original image on a horizontal basis. Since you
didn't do the same thing vertically, you won't experience the increased
sharpness along the vertical axis of the image.

Math is a wonderful thing. I continue to be amazed as to how mathematical
concepts in one area tie into mathematical concepts in an apparently
different area if you keep an open mind for such connections.

Now back to the practical. When I took differential calculus about 100
years ago, there was no such thing as a personal computer and barely such a
thing as a mainframe computer. However, with the availability of personal
computers, you now have the opportunity to estimate and graph the
derivative of a given function and to compare that graph with a graph of
the derivative function that you produce analytically. If they don't match
-- it's back to the analytic drawing board.

Here is the procedure:

1. Evaluate the original function at a set of equally and closely spaced
points and save those values as a sequence of sampled data values.

2. Perform the difference operation that I described above on those sampled
data values, save, and graph those results. (Be aware that you cannot
perform the difference operation on the last data value.)

3. Graph your analytically derived differential function and compare the
two.

Hope this helps.
Dick Baldwin

On Sat, Mar 24, 2012 at 2:30 PM, David Tseng <davidct1209 at gmail.com> wrote:

> With regard to accessible materials I would start with seeking a
> non-lossy medium for your textbook. When I went through my
> undergraduate work in mathematics, I employed a variety of techniques
> to accomplish this. If you can find it, audio recordings of Calculus
> books (Stewart comes to mind) are available in the U.S. Despite the
> very sequential (tape) nature of this method, humans reading the book
> render the text precisely and insert appropriate descriptions for
> figures/graphs when appropriate. This becomes invaluable as you move
> through Calculus in its differential, integral, and multi variable
> variants.
>
> There are also now Daisy equivalents of the above. So, you can
> download an audio book that has chapter/section indecies.
>
> Alternatively, you can try to have relevant materials hand-annotated
> by whatever resources your college may offer.
>
> Finally, wikipedia, though it reads more like a reference than say a
> tutorial, is quite accessible. Images have alt tags that expose the
> raw LaTeX of the expression under discussion.
>
> Hth,
> David
>
> On 3/24/12, Andrew Stacey <andrew.stacey at math.ntnu.no> wrote:
> > That was a nearly perfect explanation.
> >
> > Just one small technicality.  The properties that Richard refers to as
> > "continuous" and "discontinuous" are actually "differentiable" and
> > "non-differentiable".  The rough-and-ready description of a function
> being
> > continuous is that its graph can be drawn without taking the pencil off
> the
> > paper.  The actual definition is a little more complicated, but that's
> the
> > basic idea.  A continuous function has no jumps, no gaps.  It is
> > a differentiable function that has a well-defined slope everywhere.
> >
> > A differentiable function is automatically continuous, but not vice
> versa.
> >
> > Some examples:
> >
> > 1. A discontinuous function: define f(t) = 0 if t < 0 and f(t) = 1
> > otherwise.
> > This has a jump at 0 so is discontinuous.
> >
> > 2. A continuous function that isn't differentiable: f(t) = 0 if t < 0 and
> > f(t)
> > = t otherwise.  This is continuous, but it has no well-defined slope at
> > 0 since as we approach 0 from the left the slope appears to be 0 (flat)
> > whereas if we approach it from the right then the slope appears to be 1.
> >
> > Most functions that you encounter are differentiable, in fact probably
> even
> > better than that.  But properly defined, most functions that exist are
> not
> > differentiable anywhere.  It's just that since differentiable functions
> are
> > so
> > easy to study (compared to arbitrary functions), we use them if at all
> > possible.
> >
> > Andrew Stacey
> >
> > _______________________________________________
> > Blindmath mailing list
> > Blindmath at nfbnet.org
> > http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> > To unsubscribe, change your list options or get your account info for
> > Blindmath:
> >
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/davidct1209%40gmail.com
> >
>
> _______________________________________________
> Blindmath mailing list
> Blindmath at nfbnet.org
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> To unsubscribe, change your list options or get your account info for
> Blindmath:
>
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/baldwin%40dickbaldwin.com
>

-- 
Richard G. Baldwin (Dick Baldwin)
Home of Baldwin's on-line Java Tutorials
http://www.DickBaldwin.com

Professor of Computer Information Technology
Austin Community College
(512) 223-4758
mailto:Baldwin at DickBaldwin.com
http://www.austincc.edu/baldwin/