Fundamental Theorem of Algebra - A Rough Sketch of its Derivation

The fundamental theorem of algebra connects the number of roots a polynomial has in the complex plane to the degree of the highest exponent.  Like if the highest exponent you have in the polynomial is four, then you can expect there to be four roots of the polynomial in the complex plane.  Here's a rough argument for why this happens to be true.  I definitely do not promise that it is error-free.

First of all, since derivatives are preserved under addition and multiplication, polynomials are always complex differentiable.  They have a derivative everywhere.  And because they have a derivative everywhere, the Cauchy-Riemann equations are applicable.  

These equations cancel out nicely when you do a contour integral on a complex surface.  So this makes the value at the center of a circle integral equal the average of the values of the circumference.

Now, unless the function is unbounded, this mean value contour property is exactly what leads to Liouville's theorem in complex analysis which says that every bounded function with a derivative everywhere must be constant.  

We can use this theorem to show by contradiction that every non-constant polynomial contains at least one root.  

Assume it's non-constant, and arrive at a contradiction.

Suppose we have a non-constant polynomial with no roots.  These polynomials are unbounded like all polynomials.  The reciprocal of this polynomial will be bounded though.  When the original polynomial goes to infinity, the reciprocal will go to zero.  So it will be bounded instead.  So this reciprocal is bounded, non-constant, and have a derivative everywhere.  But by Liouville's theorem, since it's bounded and has a derivative everywhere, it should be constant.  Now we have a contradiction from assuming the polynomial was non-constant with zero roots.  Therefore it must be non-constant and possess at least one root.

So we cannot claim a non-constant polynomial contains no roots.  It must contain at least one.

So for whatever non-constant polynomial we have, we can divide it by x minus its root, (x-r), since we know it has at least one of them.  We can repeat this process until we have no roots left.  We will end up with some product of (x-r)'s, and the number of them will equal the degree of the polynomial.  That's the fundamental theorem of algebra.

The most interesting thing to me about this is the restrictive properties of fully differentiable functions on the complex plane. They seem to be much more restrictive than differentiable functions on the reals.

Conviction

How sad it is when the biggest conviction someone has is the conviction to avoid effort.  (Don't tell my intimidating but well-meaning friend that though.  She'll accuse me of possessing such a conviction.)

Do Republicans Ignore Obligations the Rich Possess?

Biblical doctrine does not state that possessing wealth is a sin, but it does state that possession of such wealth comes with more duties.  Those that have means are Biblically obligated to assist those that don't.  Now if you remove Biblical doctrine altogether and focus on the secular alone, then no such obligation exists, unless it's a legal requirement in the form of indirect taxation and distribution.

I can definitely understand people voting against the Republican party for not doing enough in their eyes to pressure the wealthy into alleviating the stresses of the working and handicapped poor.  The poor that are in their condition as a result of their own vices like laziness and drunkenness I do not believe are entitled to be pulled up in an attempt to close the wealth gap, unless they can change their ways.  Trying to close the wealth gap with them is fruitless because they will waste whatever they get, and it's just as bad as sending a reimbursement check to the rich that engage in the same type of wasteful behaviors.  Just because someone is poor doesn't automatically make them virtuous.  Some poor people are just as greedy as the rich and would act exactly in the same way if they had the opportunity.

So how much do Republican politicians ignore the plight of the deserving poor by their methods of cutting taxes for the rich and cutting back on government social programs they deem to be wasteful?  I have no idea.  I do believe a well-behaved full time worker who is homeless long term is a travesty, and things like that should definitely be addressed and corrected in some way.  Is that an obligation of the rich?  I would say possibly so.  The poor sure as heck can't fix a problem like that.  When stuff like that is going on, the rich should probably voluntarily open their pocketbooks a bit more either in the form of wages, charity, or smaller markups on what they sell, so poor people don't vote for higher taxes on the rich to fund more social programs.  Even if done more out of compulsion more than compassion, it seems like the wise thing to do.  I really don't care how wide the gap between the rich and poor is, so long as the working and handicapped poor can live a life in clean surroundings with decent food and medical security.

Rigid vs Flexible - In Physical Structures and Civilizations

A building can't be too rigid or too flexible in order to stand for a long period of time.  If it's too rigid without any give whatsoever, a traumatic event like an earthquake will take it down.  But even though it needs some flexibility, it needs MUCH more solidarity or it won't be able to stand at all.

I think the same thing applies to civilizations.  They need a little give to be able to adapt to traumatic events, but they need far more of a rigid structure than a flexible one to stand for long periods of time.  Rigid with some flexibility rather than flexible with some rigidity.  

A Small Tribute to Yitang Zhang

Got to have respect for a guy who makes a huge breakthrough in prime gaps even though at one point he had to leave academia to work in a Subway sandwich shop.  Talk about being overqualified.  I guess you never know who is behind that counter when you order.

Mr. Zhang, let this post be a small acknowledgement of your amazing accomplishments.  I wish you well going forward in your career and life.

A Conundrum

 Moral absolutism is unverifiable, moral relativism is inconsistent, and moral nihilism is worthless.

Perpendicular Motion - The Key to Maxwell's Equations?

I remember I had somewhat of a difficult time trying to get the last two formulas in Maxwell's equations to make sense to me.  Then one day I stumbled upon some physics textbook I found in the UHCL library that made something in my brain click.  I reviewed the book on Amazon, but unfortunately I can't remember the name of the book at the moment. 

I believe the book first mentioned how magnetic poles on a pair of magnets either attract or repel each other.  But both do so more or less linearly.  If you put an electric charge and a magnetic charge close to each other, the outcome is quite different.  Instead of attracting or repelling each other linearly, the motion of one is almost perpendicular as the other approaches.  

This clarified the whole circular motion in Maxwell's equations for me.  Immediately the curl operation made sense, because before I was exposed to this idea it just seemed like an arbitrary operation in vector calculus.