The Products of Hyperreal Series and the Limitations of Cauchy Products

Cauchy products are used to take the products of convergent series. Here, we show the limitations of this approach in divergent series, including those that can be analyzed through the BGN method. Alternative approaches and formulas for divergent series are suggested, as well as their benefits and drawbacks.

Following the Science

More and more frequently we are hearing the words "follow the science" spoken by those who believe that they are right and are frustrated by those who disagree with them. To what degree is this a legitimate effort to avoid rehashing incorrect ideas compared to a way to stifle questions about weakly-supported concepts?

2D Puzzle Visualizations of Boolean Formulae

In the comparison between human and computational intelligence, often times the comparison is not straightforward because humans can possess domain knowledge inaccessible to the program they are competing with. To provide a level playing field, it is helpful to have humans and computers compete in a domain where both start with equal domain knowledge, and the domain is well understood.

Solution of the Grazing Goat Problem: A Conflict between Beauty and Pragmatism

What is the ideal solution of a problem in mathematics? It depends on your nerd ideology. Pure mathematicians worship the beauty of a mathematics result. Closed form solutions are particularly beautiful. Engineers and applied mathematicians, on the other hand, focus on the result independent of its beauty. If a solution exists and can be calculated, that's enough. The job is done. An example is solution of the grazing goat problem. A recent closed form solution in the form of a ratio of two contour integrals has been found for the grazing goat problem and its beauty has been admired by pure mathematicians. For the engineer and applied mathematician, numerical solution of the grazing goat problem comes from an easily derived transcendental equation. The transcendental equation, known for some time, was not considered a beautiful enough solution for the pure mathematician so they kept on looking until they found a closed form solution. The numerical evaluation of the transcendental equation is not as beautiful. It is not in closed form. But the accuracy of the solution can straightforwardly be evaluated to within any accuracy desired. To illustrate, we derive and solve the transcendental equation for a generalization of the grazing goat problem.

A Corollary of the Conant-Ashby Theorem Applied to Abiogenesis

From the Conant-Ashby theorem about the "good regulator" is possible to derive a corollary about the origin of life (OOL). This corollary introduces the concept of "good constructor." Thenit is shown as nature, seen as a material system ruled by the laws of physics, cannot be a "good constructor" of the basic machinery necessary for a living cell. As a consequence OOL needs intelligent design.

Comets, Water, and Big Bang Nucleosynthesis

We argue that the cosmological origin-of-life problem is tightly connected to the origin-of-water problem, because life is not possible without abundant water. Since comets are astronomically dark and composed of water, as well as possessing microfossils, they are an underestimated candidate for the origin of life. If in addition dark matter is composed of comets, then water outweighs the visible stars, possibly solving several cosmological mysteries simultaneously. This motivates us to consider how it is possible to build a cosmological model in which water is formed in the Big Bang and then hidden from modern astronomy. In the process, we discover that magnetic fields play an important role in making water, as well as addressing several well-known deficiencies of the standard lambda-CDM cosmological model of the Big Bang. We do not see this paper as a demonstration but as an outline of how to address the origin of life problem with dark comets.

Tiling Efflorescence of Expanding Kernels in a Fixed Periodic Array

Continually expanding periodically translated kernels on the two dimensional grid can yield interesting, beau- tiful and even familiar patterns. For example, expand- ing circular pillbox shaped kernels on a hexagonal grid, adding when there is overlap, yields patterns includ- ing maximally packed circles and a triquetra-type three petal structure used to represent the trinity in Chris- tianity. Continued expansion yields the flower-of-life used extensively in art and architecture. Additional expansion yields an even more interesting emerging ef- florescence of periodic functions. Example images are given for the case of circular pillbox and circular cone shaped kernels. Using Fourier analysis, fundamental properties of these patterns are analyzed. As a func- tion of expansion, some effloresced functions asymp- totically approach fixed points or limit cycles. Most interesting is the case where the efflorescence never repeats. Video links are provided for viewing efflores- cence in real time.

Is Information Content a Single, Static Quantity?

Before information may be measured it must first manifest as a specific kind of information, and that manifestation always occurs within a fixed context. If any critical element of the context is changed, the information that is manifested also changes. The implication of that is significant: information is \emph{not} a single, static entity but instead is a variable, dynamic entity that acquires fixed definition only within a context.

Deciding a Bitstring of 1s is Non-Random is Impossible in General

Without domain knowledge, an algorithm given an extremely long sequence of 1s would be unsure whether the sequence is completely random. When asked to predict the next digit, the algorithm can only give an equal weighting to 0 and 1.

Proving the Derivative of sin(x) Using the Pythagorean Theorem and the Unit Circle

This provides a more straightforward and student-friendly proof of the derivative of sin(x) which doesn't explicitly utilize the limit (x->0) of sin(x)/x