OpenLLM-France/wikimedia huggingface dataset:
# Thinking Tools/Hieroglyphic Hints

This page displays several hieroglyphic symbols chosen at random. These can be useful for stimulating and expanding your imagination especially in the context of solving problems. To make use of these symbols:
* Identify the problem you are working on. Write it down.
* Select a row of hieroglyphic symbols at random from this page.
* Write down what you see each symbol representing.
* Draw some connection from the problem you are working on to your interpretation of each symbol.
* Seek a clue, insight, idea, or connection to some new approach to solving your problem.
𓍠 𓍡 𓍢 𓁲 𓁖 𓁨 𓏋 𓊄 𓎎 𓍎
𓈝 𓇮 𓀕 𓁦 𓈁 𓎫 𓂻 𓂡 𓀆 𓇿 𓃈
𓊼 𓄛 𓄏 𓊓 𓉘 𓆐 𓂦 𓎠 𓀯 𓈮 𓈟
𓍁 𓇓 𓌢 𓊛 𓋇 𓊅 𓇬 𓅜 𓐎 𓁶 𓀣
𓀏 𓅙 𓂤 𓋍 𓏱 𓂵 𓐝 𓉗 𓆰 𓁴 𓀴
𓃉 𓅹 𓇺 𓄛 𓌊 𓀵 𓌶 𓀦 𓎅 𓄄 𓋖
𓄕 𓆾 𓏐 𓉿 𓇱 𓆴 𓃀 𓅢 𓉍 𓎙 𓊣
𓌂 𓋕 𓉷 𓁲 𓎌 𓅶 𓂥 𓎖 𓍂 𓍈
𓈜 𓇽 𓎘 𓈗 𓍔 𓄮 𓀯 𓉜 𓌷 𓌐 𓃬 𓃘
𓇽 𓎘 𓈗 𓍔 𓄮 𓀯 𓉜 𓌷 𓌐 𓃬 𓃘
𓃎 𓂩 𓂆 𓉙 𓂆 𓆐 𓋄 𓏲 𓆓 𓂂 𓀻
𓍠 𓉑 𓊚 𓊽 𓐃 𓎐 𓂇 𓐗 𓇎 𓁋 𓆚
𓂓 𓋴 𓇆 𓊑 𓋣 𓇒 𓄑 𓐒 𓈲 𓅌 𓉘
𓆏 𓎐 𓎯 𓅽 𓋦 𓁳 𓏕 𓄚 𓄇 𓎳 𓀥
𓍧 𓅓 𓇛 𓁞 𓌆 𓄵 𓅵 𓈨 𓄋 𓏁 𓆎
𓄃 𓁴 𓍺 𓏇 𓊹 𓁎 𓇔 𓋚 𓊔 𓍓 𓐀
𓁸 𓂠 𓈨 𓁞 𓁍 𓍧 𓉤 𓊉 𓌛 𓄬 𓋧
# Thinking Tools/Quick Reference

This quick reference lists a variety of thinking tools.

## Clarify—Strategic Thinking

* Five Ws
* Phoenix Checklist
* SMART Goals
* Webbing Tool
* Storyboarding
* Problem finding
* Stating a Problem

## Ideate—Blue Sky Thinking

* Secondary Research
* Benchmarking
* Appreciative Inquiry
* Thinking outside the box
* Metaphor—What is this problem like?
* Brainstorming
* Forced Relationships
* Hieroglyphic Hints
* Lateral Thinking
* PO – Provocative Operation
* Idea Notebooks
* Synetics
* Varying Attributes
* SCAMPER
* Six Thinking Hats
* Biomimicry
* TRIZ
* Attribute Matrix
* Reversals

## Develop—Evaluative Thinking

* Discard the Unsuitable
* POINT
* Prototyping
* Foresight Scenarios
* Affinity Diagram
* Tree Diagram
* Quality Function Deployment
* Representation

## Implement—Tactical Thinking

* Are we doing this?
* Action Planning
* Assisters and Resisters
* Stakeholder Analysis
* Project Management
* Murder board

## Navigating, Integrating, Prioritizing

* Success Zones
* Unleashing Creativity
* Additional thinking tools
# Third grade elementary math

Created: 2007 06 23 | Percent completed:
| Elementary Curriculum and Training » Elementary Math 3 | Edit template |
Lesson | 1 | 2 | 3 | 4 | 5 | 100 |

## What is Elementary Math 3

Elementary Math 3 is the teacher's guide for teaching math at the third grade level.
Each unit is numbered and can be taught in one day. Each unit also contains scripted instructions for the teacher. Many units contain workbook pages and activities.

## Lesson 1
### Math Facts

Worksheet:

### Strands

Strand 1 Strand 2

## Authors and contributors

This Wikicurriculum has been written by (please add your name here if you are a contributor in order to comply with licensing):
* Harriska2 (Contributions)
# Third Reich
## HITLER'S THIRD REICH

Course Description This course will allow the student to examine the various facets of fascism in Nazi Germany. Topics will include: post-Treaty of Versailles, fall of the Weimar Republic, Rise of the Nazis, Nazi Germany, World War II, and the Fall of the Third Reich. Students will be expected to complete readings on a nightly basis, respond to them in the forms of discussion questions or essays, and tests/examinations.
Grading Scheme This course will follow the following system of weights and measures for grading:
* 50% Tests (approximately four)
* 25% Writing assignments/essays (unlimited number)
* 25% Reading quizzes (LOWEST two scores will be dropped)
Attendance Since this is an online course, attendance records will ONLY be kept in the form of extra credit. Contributing to online discussions posted on CourseZone will allow the student to accumulate up to 5 POINTS worth of extra credit, which can be applied to tests and other examinations.

## Syllabus

* Week 1... Post-World War I Germany and the Lure of Fascism
* Week 2... Hitler and the Early Nazi Party
* Week 3... The Enabling Act and Complete Control over the Third Reich
* Week 4-5... Development of the Nazi State (1933-1939)
* Weeks 5-6... Dreams of an Aryan Empire
* Weeks 7-8... The Holocaust
* Week 9... German Involvement in World War II
* Week 10... Fall of the Third Reich and Final Examination

## FIRST ASSIGNMENT

Please read the following articles regarding German reparation payments, the Treaty of Versailles, and the weak Weimar Republic.
* w:Treaty_of_Versailles Concerning the Treaty of Versailles
* w:World_War_I_reparations Concerning German reparation payments
* w:World_War_I_reparations the Weimar Republic
1. What was the ultimate goal of the Allied powers during negotiations held at the Paris Peace Conference?
2. What were the conditions of Article 231/the War Guilt Clause contained within the Treaty of Versailles?
3. How was the map of Europe essentially redrawn following changes made by the treaty?
4. Why was the French occupation of the Ruhr Valley a disaster? What prompted the French to occupy the region in the first place? What eventually happened to the value of the German mark?
5. How did the Dawes Plan attempt to reduce reparations and stabilize Germany's payment system?
6. What were the conditions of the Treaty of Locarno?
7. How did the withdraw of American funds from German banks lead to international banking faulures?
8. Describe a few of the difficulties the Weimar Republic faced during its early years.
# Third power sum/By induction/Exercise

Prove, by induction, the following formula.
    $\sum _{i=1}^{n}i^{3}={\left({\frac {n(n+1)}{2}}\right)}^{2}\,.$
# Sigma-algebra
## Definition

Let X be some set, and let 2$^X$ represent its power set. Then a subset $\Sigma \subseteq 2^{X}$ is called a σ-algebra if it satisfies the following three properties:.
* X is in ${\mathcal {S}}$, and X is considered to be the universal set in the following context.
* Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
* ${\mathcal {S}}$ is closed under countable unions: If A$_1$, A$_2$, A$_3$, ... are in ${\mathcal {S}}$, then so is
          $A=A_{1}\cup A_{2}\cup A_{3}\cup \ldots =\bigcup _{i=1}^{\infty }A_{i}$

## Lemmas

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).
It also follows that the empty set $\emptyset$ is in ${\mathcal {S}}$, since by
* (1) X is in ${\mathcal {S}}$
* (2) asserts that its complement, the empty set, is also in ${\mathcal {S}}$. Moreover, since $\{X,\emptyset \}$ satisfies condition
* (3) as well, it follows that $\{X,\emptyset \}$ is the smallest possible σ-algebra on X.
* The largest possible σ-algebra is the power set on X, which contains is $2^{n}$ elements, if X is finite and contains n elementss.
Elements of the $\sigma$-algebra are called measurable sets. An ordered pair $(X,{\mathcal {S}})$, where X is a set and ${\mathcal {S}}$ is a $\sigma$-algebra over X, is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a $\sigma$-algebra to [0, ∞].
A $\sigma$-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see Wikipedia:Sigma algebra).

## Learning Task

* Explore the measurement problem and explain why a $\sigma$-algebra is necessary for the definition of probability distributions!
* Explain why a discrete probability distribution can use the
# Sigma - pi separability
## Sigma - pi separability

There is nothing magical about the separation of σ and π electrons. It is a simple consequence of symmetry. If we have a planar compound, then reflection in the plane that contains all the atoms is a symmetry operation in the point group that the molecule belongs to. For example, ethene (ethylene) has D$_{2h}$ symmetry and reflection in the plane of the 6 atoms is one of the 3 reflection operations of the D$_{2h}$ point group.
All molecular orbitals must transform like one of the irreducible representations of the group. This jargon simply indicates that for these planar compounds the molecular orbitals must be symmetric or antisymmetric with respect to reflection in the plane. Molecular orbitals which are symmetric, i.e the same above and below the plane, are called σ orbitals. Those that are antisymmetric, i.e. they change sign going through the plane, are called π orbitals. σ molecular orbitals must be formed from symmetric atomic orbitals. This means that only s orbitals and the p$_x$ and p$_y$ orbitals (we specify the xy axis as being in the plane and the z axis being perpendicular to the plane) can contribute to the σ orbitals. Similarly only antisymmetric atomic orbitals can contribute to the π molecular orbitals.
This means that only the 2p$_z$ orbital contributes to the π molecular orbitals. Note that in these empirical methods we are only using atomic orbitals that are occupied in the free atom. This is called a minimal basis set. We are not using d orbitals.
The simple conclusion from this is that the π molecular orbitals are built from just one atomic orbital (the 2p$_z$ orbital) on each heavy atom (carbon for hydrocarbons). The hydrogen atomic orbitals only contribute to the σ orbitals.
In a proper treatment, the form of the π orbitals is determined by the nature of the σ orbitals and vice versa because of the repulsion between the π and σ electrons. However, in an empirical method, we simply fudge in the repulsion from the σ electrons into the terms we include in the calculation of the π orbitals.
# Siepinski conjectures
## Definition

For the original Sierpinski problem, it is finding and proving the smallest k such that k×b$^n$+1 is not prime for all integers n ≥ 1 and GCD(k+1, b-1)=1.

### Extended definiton

Finding and proving the smallest k such that (k×b$^n$+1)/GCD(k+1, b-1) is not prime for all integers n ≥ 1.
# Sierpinski conjectures
## Definition

For the original Sierpinski problem, it is finding and proving the smallest k such that k×b$^n$+1 is not prime for all integers n ≥ 1 and GCD(k+1, b-1)=1.

### Extended definiton

Finding and proving the smallest k such that (k×b$^n$+1)/GCD(k+1, b-1) is not prime for all integers n ≥ 1.
# Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 2
## Problem 1: Part 2: Energy equation

For an adiabatic process, the rate of change of temperature can be written as
    ${\dot {T}}={\cfrac {\chi }{\rho C_{p}}}{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}^{p}$
where $\chi$ is the Taylor-Quinney coefficient, $\rho$ is the density, and $C_{p}$ is the specific heat. Express ${\dot {T}}$ in terms of ${\dot {\gamma }}$ and $\partial f/\partial {\boldsymbol {\sigma }}$. This is the evolution law for $T$.
Plugging in the expression for ${\dot {\boldsymbol {\varepsilon }}}^{p}$, we get
    ${{\dot {T}}={\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}~.}$
# Nonlinear finite elements/Homework 1
## Problem 1: Mathematical Model Development

Consider the motion of a pendulum (see Figure 1).
* Figure 1. The motion of a pendulum.
Assume that:
* The rod and the mass at the end are rigid.
* The mass of the rod is negligible compared to the mass of the bob.
* There is no friction at the pivot.
Find the following:
* The equation of motion of the bob (angular displacement as a function of time). State all other assumptions.
* Why is the equation of motion "nonlinear"?
* Assume that $\theta$ is "small". Derive the equation of motion for this situation.
* Why is the small $\theta$ equation of motion "linear"?
* Derive a finite element formulation for the linear equation of motion.
* Use ANSYS or some other tool of your choice (including your own code) to solve the linear equation of motion via finite elements. Compare with the exact solution. You can use any reasonable values for time ($t$), mass ($m$), and length ($l$).
* What happens when you try to solve the nonlinear equation of motion with finite elements? (You will need to formulate the FE formulation to see the difference.)

## Problem 2: Numerical Exercise

In the learning resources section, you have read about the process of formulating a FE model for an axial bar under a distributed body load. Ansys 9 shows you how to solve the same problem using ANSYS. Work the example out on your own.
Next, consider a mini centrifuge of the type manufactured by Kisker Biotech (see Figure 2).
* Figure 2(b). Plan view of mini centrifuge.
This type of mini centrifuge is widely used in the fields of genetic research and microbiology. The centrifuge is rate to run at 15,000 RPM. The allowable temperature range is -20 C to 40 C. The nominal radius is 140 mm. Assume other geometric quantities on the basis of the figure.
We want to extend the temperature range to -80 C - 100 C and the allowable RPM to 25,000.
Assume that the current material is a light metal (any other material of your choice will also do). The only requirement is that it should be cheap and stiff. Ignore bending and the weight of the sample tubes.
Find the thinnest beam of the material of your choice that can stand the axial load that is generated due to rotation at the new speed. Use ANSYS or your own code to solve the problem. State all simplifying assumptions.
We just need a ballpark value. A 1-D simulation will do.
# Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 4
## Problem 1: Part 4: Consistency condition

The consistency condition during plastic flow requires that
    ${\dot {f}}({\boldsymbol {\sigma }},\alpha ,T)=0~.$
Write down an expression for the time derivative of $f({\boldsymbol {\sigma }},\alpha ,T)$ using the chain rule.
The time derivative is
    ${\dot {f}}({\boldsymbol {\sigma }},\alpha ,T)={\frac {\partial f}{\partial t}}={\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}:{\frac {\partial {\boldsymbol {\sigma }}}{\partial t}}+{\frac {\partial f}{\partial \alpha }}{\frac {\partial \alpha }{\partial t}}+{\frac {\partial f}{\partial T}}{\frac {\partial T}{\partial t}}~.$
In shorter form
    ${\dot {f}}=f_{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\sigma }}}+f_{\alpha }~{\dot {\alpha }}+f_{T}~{\dot {T}}~.$
# Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 8
## Problem 1: Part 8: Consistency condition - 2

The yield stress $\sigma _{y}$ is given by the Johnson-Cook model
    $\sigma _{y}(\alpha ,T)=\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]$
where $\sigma _{0}$ is the initial yield stress, $B,n$ are constants, $T_{0}$ is a reference temperature, and $T_{m}$ is the melt temperature.
Derive expressions for $\partial f/\partial \alpha$, and $\partial f/\partial T$ for the von Mises yield condition with the Johnson-Cook flow stress model.
The yield function is
    $f={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\sigma _{y}={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]$
Therefore,
    ${{\frac {\partial f}{\partial \alpha }}=f_{\alpha }=-n~B~\alpha ^{n-1}\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}$
and
    ${{\frac {\partial f}{\partial T}}=f_{T}=\left({\cfrac {1}{T_{m}-T_{0}}}\right)\left[\sigma _{0}+B\alpha ^{n}\right]~.}$
# Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 5
## Problem 1: Part 5: Consistency condition - 1

Use the consistency condition and the expressions you have derived in the previous parts to derive an expression for ${\dot {\gamma }}$ in terms of $\partial f/\partial {\boldsymbol {\sigma }}$, $\partial f/\partial \alpha$, $\partial f/\partial T$, ${\boldsymbol {\mathsf {C}}}$, and ${\dot {\boldsymbol {\varepsilon }}}$.
From the consistency condition
    ${\dot {f}}=f_{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\sigma }}}+f_{\alpha }~{\dot {\alpha }}+f_{T}~{\dot {T}}=0$
From the previous parts
    ${\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}:\left({\dot {\boldsymbol {\varepsilon }}}-{\dot {\gamma }}f_{\boldsymbol {\sigma }}\right)~;\qquad {\dot {\alpha }}={\sqrt {\cfrac {2}{3}}}~{\dot {\gamma }}~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _}}~;\qquad {\dot {T}}={\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}~.$
Plug into the consistency condition to get
    $f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:\left({\dot {\boldsymbol {\varepsilon }}}-{\dot {\gamma }}f_{\boldsymbol {\sigma }}\right)+{\sqrt {\cfrac {2}{3}}}~{\dot {\gamma }}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _}}+{\cfrac {\chi }{\rho ~C_{p}}}~{\dot {\gamma }}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}=0~.$
Collect terms containing ${\dot {\gamma }}$:
    ${\dot {\gamma }}\left[-f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}+{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _}}+{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}\right]+f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}=0$
Therefore,
    ${\dot {\gamma }}={\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}~.$
# PlanetPhysics/N Groupoid

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$0<m\leq n,$ each $m$-arrow is invertible with respect to the $(m-1)$--composition; in the case of an infinite groupoid, the notation $\infty$-groupoid is used in the literature (rather than $\omega$-groupoid that has a distinct meaning from that of $\omega$-category).
An important reason for studying $n$--categories, and especially $n$-groupoids, is to use them as coefficient objects for non-Abelian cohomology theories. Thus, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) are relevant to non-Abelian algebraic topology (NAAT) and NAQAT (or NA-QAT).
In particular, a 2-groupoid is a 2-category whose morphisms are all invertible ones.
One needs to distinguish between a 2-groupoid and a double-groupoid as the two concepts are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true two-dimensional geometric representations with special properties that allow generalizations of important theorems in algebraic topology and higher dimensional algebra, such as the generalized Van Kampen theorem with significant consequences that cannot be obtained through Abelian means.
Furthermore, whereas the definition of an $n$-groupoid is a straightforward generalization of a 2-groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extension of the concept of double groupoid.
# PlanetPhysics/N Groupoids

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
$0<m\leq n,$ each $m$-arrow is invertible with respect to the $(m-1)$--composition; in the case of an infinite groupoid, the notation $\infty$-groupoid is used in the literature (rather than $\omega$-groupoid that has a distinct meaning from that of $\omega$-category).
An important reason for studying $n$--categories, and especially $n$-groupoids, is to use them as coefficient objects for non-Abelian cohomology theories. Thus, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) are relevant to non-Abelian algebraic topology (NAAT) and NAQAT (or NA-QAT).
In particular, a 2-groupoid is a 2-category whose morphisms are all invertible ones.
One needs to distinguish between a 2-groupoid and a double-groupoid as the two concepts are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true two-dimensional geometric representations with special properties that allow generalizations of important theorems in algebraic topology and higher dimensional algebra, such as the generalized Van Kampen theorem with significant consequences that cannot be obtained through Abelian means.
Furthermore, whereas the definition of an $n$-groupoid is a straightforward generalization of a 2-groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extension of the concept of double groupoid.
# PlanetPhysics/Non Abelian Algebraic Topology 5
## Non--Abelian algebraic topology

A relatively recent development in Algebraic Topology that began in 1960s which considers algebraic structures in dimensions greater than 1 which develop the non-Abelian character of the fundamental group of a topological space, or a novel approach to higher dimensional, non-Abelian algebraic treatments of topological invariants in Algebraic Topology.

### Recent reference:

Ronald Brown, Bangor University, UK, Philip J. Higgins, Durham University, UK Rafael Sivera, University of Valencia, Spain.2010. Nonabelian Algebraic Topology: Filtered Spaces, \htmladdnormallink{crossed complexes {http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html}, Cubical Homotopy groupoids.} EMS Tracts in Mathematics, Vol.15, an EMS publication: September 2010, approx. 670 pages. ISBN 978-3-03719-083-8.
# PlanetPhysics/Natural Transformations of Organismic Structures
### Natural Transformations of Organismic Structures

Biological systems, or living organisms are characterized by relational structures and their dynamic transformations which can be represented as natural transformations of heterofunctors in organismic supercategories(OS). Such OS-structures can be specified mathematically either by using the Yoneda-Grothendieck Lemma and construction, or they can be directly derived by a mathematical interpretation of the first ten axioms of ETAS, plus two additional axioms defining both `self-repair' of metabolic components and complete reproduction in terms of genetic coding, translational genomics and epigenetic meta-processes.
Natural transformations of organismic structures allow for the extension of both Rashevsky's organismic set theory and Robert Rosen's Metabolic-Replication Systems Set-Categorical approach to relational biology based on the concepts of molecular set variables and variable categories; the latter concepts are generalizations of Anthony Bartholomay's definition of molecular sets, and the consideration of variable categories with structure instead of only the discrete topology of sets. Further details concerning mathematical, logical and complex modeling are provided in the following list of publications and related web (html) links.
# PlanetPhysics/Neutrino

Neutrinos as elementary particles known as leptons. Currently there is 3 known neutrinos; electron, mu and tau neutrinos. They also have 3 corresponding anti-neutrinos. Neutrinos react via the weak and gravitational force. Their mass is of yet unknown, but there are very good estimates on the relative differences in mass between the 3 types of neutrinos. Some of the more famous neutrino experiments include SNO (Canada), Super-Kamiokande (Japan), MiniBooNE (USA), etc.
# PlanetPhysics/N Tuple Algebroids

This is a contributed topic on important generalizations of algebroids to higher dimensional algebra relevant to quantul operator algebras, QFT and quantum gravity.

## N-tuple Algebroids

See also the paper entry on "N-tuple Algebroids"