## #Foundationals: “Policies are many, whilst principles are few; and as Principles are timeless, so are Policies ephemeral” (#Discrete #Math – on Boolean Algebra Practical Applications)

3 JunYup… at the core network?

01010101010!

I am, I am NOT

False/True – or empty, according to a recent post that I plucked from my current customer

Meaning, when asked:

“…two founders of Boolean algebra and its applications (Boole and Shannon). After performing some research, share with the class some practical applications of Boolean algebra. Please cite your source…”

In other words, seems like at the ROOT of all these decision points in our flowcharts lies this Boolean extrapolations.

As recalcitrant stalwarts that they are, mainstay stewards of the ‘process’ as in the end, this “ON/OFF” paradigm appears to cover a LOT of practicalities.

Such as:

**– Circuits (yeah, electrical ones!)
– Databases (Ellison’s got a foundation, of course!)
**

Ready?

(On Manufacturing Efficiency)

Boolean algebra finds its most practical use in the simplification of logic circuits. If we translate a logic circuit’s function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or arithmetic operations, **the simplified equation may be translated back into circuit form for a logic circuit performing the same function with fewer components. **If equivalent function may be achieved with fewer components, the result will be increased reliability and decreased cost of manufacture.

(Then they go “Checkmate” on un-simplifying the first equation?… neat!)

Note how the last rule (A + AB = A) is used to “un-simplify” the first “A” term in the expression, changing the “A” into an “A + AB”. While this may seem like a backward step, it certainly helped to reduce the expression to something simpler! Sometimes in mathematics we must take “backward” steps to achieve the most elegant solution. **Knowing when to take such a step and when not to is part of the art-form of algebra, just as a victory in a game of chess almost always requires calculated sacrifices. **

*“Live Long and Prosper” *I guess it’s what I need to say now?

Cheat Sheet!

o summarize, here are the three new rules of Boolean simplification expounded in this section:

Then, just to extrapolate to the other end of the Continuum…

“As most people are well aware, mathematical expressions are used to make computations and other math problems simpler and smaller. Boolean Algebra is used to do the same thing, except it does it with logic circuits instead of other mathematical problems.

The first thing that needs to be known for Boolean algebra is the meaning of the various signs. There are three of these, all deriving from the three basic gates. They are AND, OR and NOT.

The AND operation is also known as conjunction. It gives the product of two binary bits. Using A and B as inputs, it would be written as AB or sometimes A·B. This of course means that A is multiplied by B, which is exactly how an AND gate functions.

The OR operation is also known as disjunction. This operation gives the sum of two binary bits. Again, using A and B as inputs, it would be written as A + B, which is how an OR gate functions. NOTE, the “+” does necessarily mean the same thing that it does in normal mathematics we are all used to. In Boolean algebra, it stands for OR. For example, 1 + 1 would equal to 1, not 10, the binary equivalent of two. If either of the inputs is 1, the output is 1. The NOT operation is also known as negation. This operation gives the opposite of a single term. For example, the negation of A is written as . While the negation of AB would be . We can relate the AND, OR and NOT operations to the corresponding gates…”

(Sounds familiar by now? GOOD!)

How does that tie into coding?

Check out the code samples HERE

https://forums.oracle.com/forums/thread.jspa?threadID=1005678

Again, all ‘sponsored’ by those we hope to eventually hire us!

So yes, as I’ve oft-repeated: **“Policies are many, whilst principles are few; and as Principles are timeless, so are Policies ephemeral”**

(Betcha there’s an Equation out there for that!)

**“Tank!… I need an EXIT!… NOW!”**

@Kankuchito

References

AllAboutCircuits.Com. Boolean Rules for Simplification. Retrieved from: http://www.allaboutcircuits.com/vol_4/chpt_7/5.html

ThinkQuest.Com – Boolean Algebra (2013) Retrieved from: http://library.thinkquest.org/C0116405/Basics/booleanalgebra.php

## Six Degrees of Ramification – on Fractals, Models, Randomization (#Discrete #Math Class)

27 May… of Fractal-ization?

I mean, with a question like:

“Random graphs are a fascinating subject of applied and theoretical research. These can be generated with a fixed vertex set … and edges added to the edge set … **based on some probability model, such as a coin flip.**

[First] Speculate (did you just say introduce ‘fuzzyness’ on the answer?) on how many connected components a random graph might have if the likelihood of an edge (v1,v2) being in the set E is 50%.

[Then] Do you think the number of components would depend on the size of the vertex set V? Explain why or why not.”

Where does one end up – BUT in a random corner of the cloud

** – Enter Fractals
**

Again, from the realm of Aesthetics, it’s now commonplace (some may argue, even BORING!) to get to see these CGI-facilitated illustrations all over the place; as with most anything out there, I can bet you that someone already set their Ruby-on-Rails atop these equations, and well, the

*slogan could yield a couple of interesting iterations..*

**“there’s an app for that”**So as to the math of fractals, since they’re graphs to begin with, let’s jump in, shall we? (and careful where one ends up at!)

“… One of the most amazing facets of mathematics is the experience of **starting with a problem in one area of mathematics and then following the trail through several other areas to the solution (or several versions of the solution).**

We *** illustrate *** [See? Gotta satisfy the Right Hemisphere with these most complex ones!] this with a problem that starts out as a problem in rendering the attractor of an Iterated Function System (IFS), which leads to a solution that involves finding an *** **Eulerian** *** cycle in a certain graph and then to finding generators for the multiplicative group of a finite field.

We start with an introduction to IFS fractals and the problem of generating an image of the attractor of an IFS.

[…and what is an] ITERATED FUNCTION SYSTEMS. The basic idea behind Iterated Function System fractals is that we wish to formalize the concept of self-similarity. That is, given an image like… we [then] wish to formalize our notion that the set is made up of three smaller copies of itself…” (Mendivil, 2003)

– Enter Instantiation – CGI-Style

– Enter Instantiation – CGI-Style

As I’ve mentioned elsewhere, the Computer Graphics Industry is one of the most complex, yet gratifyingly simply iterations of what Computing has become – and as the excerpt above includes, the need to draw up an illustration, based off of very precisely and accurately calculated principles?Timeless – and profitable, for those venturing into the realms of the unknown, coming back after spending a quarter billion dollars to see their ROI (Return on Investment) quadruple – and all off of a fable of runaway mining and organic blue giants? on a place with 60% of our gravity field?

So EBSCO Host, when queried “acm siggraph fractal instantiation” spat out a SINGLE answer – yet, how lucky does one feel when it states…

Parts of the tree are transformed only when required, thus our approach is much faster than common algorithms such as Open L-Systems or space colonization methods.

**Input is a skeleton-based tree geometry that can be computed from common tree production systems or from reconstructed laser scanning models.
**

Our approach enables content creators to directly interact with trees and to create visually convincing ecosystems interactively. We present different interaction types and evaluate our method by comparing our transformations to biologically based growth simulation techniques.”

(.pdf is taking its time downloading… so)

(whoa!… so many equations!… guess there’s a CONNECTION?)

“”…focus on phototropism and gravitropism. Phototropism is the tendency of a given branch to grow towards the light direction. We estimate the effects of phototropism for each branch at the time the branch was growing using our temporal light model described above. Gravitropism controls bending of the branches either away from or towards gravity. While we compute the strength of the tropisms directly from the input tree, we later expose it as a parameter that the user can modify to control the

transformation behavior of the trees…” (ibid)

Well, let’s just say that yes, we’re pretty good at deciphering and explaining what’s ALREADY there – and yes, all of the work above, again, serves to build near-perfect backgrounds to a story? one that only happens within arrays and CPU’s, rendering farms, mice and tablets – and in out heads, as everything’s put together within our synapses?

“Real landscapes also have varying statistical behaviour from place to place, so for example sandy beaches don’t exhibit the same fractal properties as mountain ranges. A fractal function, however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behaviour spatially. Additionally real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model” (Wiki, 2013)

**connection or no connection?**

References

Mendivil, F. (2003). Fractals, graphs, and fields. The American Mathematical Monthly, 110(6), 503-508,510-511,513-514. Retrieved from http://search.proquest.com/docview/203742447?accountid=35812

WIkipedia (2013) Computer Generated Imagery Retrieved from: http://en.wikipedia.org/wiki/Computer-generated_imagery

Wikipedia (2013) Fractals Retrieved from: http://en.wikipedia.org/wiki/Fractal_landscape

## WBS – per PMBOK – and how it ties to #Recursion – Discrete Math Class

11 MayQuestion was…

*“Describe a situation in your professional or personal life when recursion, or at least the principle of recursion, played a role in accomplishing a task, such as a large chore that could be decomposed into smaller chunks that were easier to handle separately, but still had the semblance of the overall task. *

*Did you track the completion of this task in any way to ensure that no pieces were left undone, much like an algorithm keeps placeholders to trace a way back from a recursive trajectory? *

*If so, how did you do it? If not, why did you not?”*

And my answer, was…

Work

Breakdown

Structure

(Finally!)

WBS’s are what many of you have already put out there; i.e. at a large Corporate Customer of mine, I participated in large rollout, slated for about two years – yet, as they were somewhat similar in nature, given that the ‘target’ audience was always the same (Branches, Regional Offices, and Corporate) they were all ‘tagged’ under the exact same project number.

That it’d later mushroom into filling out a whole wing/building with a call center, trainers, engineers, contractors for the phone system, etc. did NOT matter – it was all ONE project

**Enter WBS’s**

That’s the PMBOK (Project Management Book of Knowledge) way to ‘describe’ not only the breaking-down of the actual tasks (and the actuarials that ensue, as for example, we had a whole team dealing their way of naming an Exception Process, where a 600+ step technical hardware installation manual had to stop due to technical incompatibilities that required a VERY human interface to negotiate with multiple stakeholders, work on assessing impact and even ROI (Return on Investment) as the overall relationship between Branches and Corporate somehow echoed that of Franchisors and Franchisees, giving Franchisees a lot of latitude in Capital Investitures like these, etc…)

Anyway…. that’s the anecdotal: Theoretically, then…

https://www.workbreakdownstructure.com/work-breakdown-structure-according-to-pmbok.php

(Does that NOT look like a fractal?… I mean, a graph from the textbook?)

Relevant excerpt?

“According to the Project Management Body of Knowledge (PMBOK®), and the Practice Standard for Work Breakdown Structures – Second Edition from the Project Management Institute (PMI), the work breakdown structure can be used to effectively decompose the project scope, to improve estimating, to better control the project execution and to more accurately verify project completion. In addition, using a work breakdown structure approach summarizes project information to improve the opportunity for use of historical information, which, can aid in both speed and accuracy of future projects. The work breakdown structure is a repeatable process that can be used as a template for future projects.” (PMBOK, 2013)

Hence, the ‘recursiveness’ of these projects – which as mirrored by my Customer’s experience, where they were able to ‘boilerplate’ these projects/rollouts, onto something that’s somewhat repeatable – yet mold it to fit the scope and constraints, and above, all, the budget of each – which as I mentioned earlier, had a lot do to with the way the entire structure was legally incorporated, as those were the boundaries, that much like a ‘Demarc’** (anyone who’s been in telco – telecommunications – knows that’s the lingo for “rat’s-nest-filled-with-a-gajillion-bits-of-twisted-pair-and-orange-tags-that-only-a-tech-can-decipher”) = ‘Demarcation Point’**

So yes, how does ‘recursiveness’ extend to the real world?… look around!… any Franchise/Branch of a nationwide or regional entity, most likely, there are processes that a Project Manager oversees – and tickles, tackles and otherwise shapes to fit the needs and budgets allowed to move ‘forward’ – in a remarkable fashion, when one considers that the endpoints at said Customer’s project? about a quarter million, ONLY at the Branch/Franchise levels!

Now, since it was asked as to ‘how’ it was tracked, a bit more on the ‘what’ is being tracked?

“A work breakdown structure is deliverable-oriented. So what is a deliverable? In a word, it can best be described as a noun. What is the difference between “write xyz specifications” and “xyz specifications”? One describes the end product and the other describes a single step to produce it. The end product is described as a noun without a verb. A deliverable can be delegated to a team or leader who can then be responsible for the work product and complete work should be returned when complete. A work breakdown structure is complete when all of the deliverables necessary to obtain the project goals are identified.

A work breakdown structure is a hierarchy. That means that deliverables can be further decomposed into parent and child relationships. In this case “xyz specification” may be further decomposed into “xyz functional specifications” and “xyz performance specifications”. It is important that if decomposing the deliverable that the lower child decomposition represents 100% of the parent. This is similar to how we learned to outline our writing in our freshman English class. We can continue decomposing the deliverables until we feel comfortable that we have defined in a way that we can effectively manage and control the project.

Simply put the work breakdown structure technique divides projects into smaller more manageable chunks that can be more easily estimated and controlled. It gives a black and white version of the work effort needed and almost as important if the work is not in the work breakdown structure it is not a part of the project. The work should be decomposed until it is clear to teams performing the work. This provides a clear line of sight between the work and the goals for the project.” (ibid)

**Enter MS Project**

I’ll wrap up this paste up with a word on an app many cannot live without (per above, as you can see, PMBOK applies to MANY disciplines – not only I.T.)

Relevant excerpt:

“Outline numbers

Outline numbers are the simplest type of WBS coding. Microsoft Office Project automatically calculates an outline number for each task, basing the numbering on the outline structure of the task list. For example, the first task in your task list is numbered 1. If that task has three subtasks, the subtasks are numbered 1.1, 1.2, and 1.3.

Outline numbers consist of numbers only (no letters), and you cannot edit them. They do, however, change automatically when you move a task up or down in the task list and when you indent or outdent tasks. For example, if a subtask currently has an outline number of 3.5.4, and if you move it up one row in the list, the outline number is automatically updated to 3.5.3. If you then outdent that same subtask, the outline number is automatically updated to 3.6.” (Microsoft.Com, 2013)

So I guess Indices and other typographical artifacts, again, straddle that boundary of logic and usefulness?

@Kankuchito on Twitter/Instagram/Tumblr & Pinterest

## “Off with their Hats!” – on Clear Thinking, Hemispheres and Predominances (#Discrete #Math Class – the Wraparounds)

5 MaySomeone remarked about a post of mine… as we were about to wrap the ‘discrete’ math class…

————-

**“…Good post. Your posts are fun to read. I got the Ellison reference too…”**

————-

“One is glad to be of service” M.!

Yes – I’m not 100% clear, as for example, within the “Cloud” of some SaaS? we got **True/False… and Null?**

(Per earlier post, on the Programming Tools for a certain “no-software” product line)

So yes – how much of it can be reduced to an equation?

Dunno, BUT the class has somehow encouraged me to stop… breathe… think… and DO THE MATH!

Meaning, rather than JUST clicking or toggling (as one may argue that’s ALL one can do to a Boolean, right?… or a Boolean NOT in the Best Coast!) sometimes the consequences, well, that’s where all this logic/math/philosophy falls in…

… and yes, I must admit, wished I had paid MORE attention – it’s definitely a skill worth thousands, as I’m married to someone that swims in math all day long (Taxes) and others in my surroundings, I can definitely tell they took advantage of a “clear thinking” angle, that is definitely a major asset.

Good news? sometimes THEY wonder at what us** “Right Hemisphere Predominants” **can come up with (specially as we toy and juggle with their logic… and make them smile!)… which brings me to another interesting ‘interpolation’

– anyone heard the once popular comparison between a **“Black Hat” **and a **“Red Hat” **thinker?

(Yeah, later ‘transmogrified’ into a certain Distro’s ‘Brand’, aha!)

## Mathematics of Belief? #TED “Daniel Wolpert: The real reason for brains” on @YouTube

3 MayObjective Reality & Subjective Wisdom; Bayesian Decision Math & Feedback Processing Theorems.

Science catching up with Realities, hopefully shared in ways that not only validate the immense Fortune represented by being Human, but also, the immense Responsibility of tweaking our Movements, our Actions in a more Compassionate (Noise-free, according to Prof. Wolpert) Direction.

## Bending reality? or how reality manages to bend US? #Mathematics #Discrete Math Class

20 AprSo I’m doing actual research at the actual library… and I found THIS!

Mathematics and Reality : Mathematics and Reality Oxford Scholarship Online.

# Mathematics and Reality

## Mary Leng

### Abstract

This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects.

Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories).

According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true.

But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true.

Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories.

This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.

Neat!

*Keywords: * mathematics, philosophy, realism, fictionalism, naturalism, indispensability, science, ontology, objects, truth

### Bibliographic Information

Print publication date: 2010 | Print ISBN-13: 9780199280797 |

Published to Oxford Scholarship Online: May 2010 | DOI:10.1093/acprof:oso/9780199280797.001.0001 |