Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Addition-Subtraction - The PjProblemStrings

*Addition* and *Subtraction* are premier concepts in the physical universe. The creation of the universe would have been impossible without *addition* and it will be impossible to end the universe without *subtraction*. The sequence of PjProblemstrings that represents *addition* and *subtraction* reveals a deeper understanding of the concepts beyond the symbols humans have created to represent them.

(a) Describe the sequence of PjProblemStrings that *add* *matter* to an arbitrary space with a *containership property*.

(b) How is the sequence of PjProblemStrings for *addition* different from the sequence of PjProblemStrings for *subtraction*?

(c) True or false? whenever there is an *addition* of *matter* into a given *space*, there is a *subtraction* of matter from another *space*.

(d) Indicate the *addition* and *substraction* in the following scenarios:

(i) A fisherman catches a beautiful fish with his fishing line and quickly returns the fish into the river.

(ii) Putting the golf ball into a hole.

(iii) An airplane in flight.

(e) Consider a game of chess. Which piece on the board is most able to visit different spaces on the board?

**The strings**:
S_{7}P_{3}A_{32} (Force - Push).
**The math**:

Several Pj Problems are at play. However, Pj Problems of Interest is of type *force* (pull and push).

**(a)** Given an arbitrary space with a containership property, the sequence of PjProblemStrings that *add* matter to the space is *force-motion-force*. The force may be a pull or a push and the *motions* associated with the force may be any of the four types of motion (linear, rotary, reciprocal, oscillator). The first force *fetches* matter and places it in space of interest . The second force keeps the matter in the space even if just transiently. *Addition* and *subtraction* results in *change*.

So,the sequence of PjProblemStrings = S_{i}P_{3}A_{k} S_{i}P_{4}A_{k'}S_{i}P_{3}A_{k}.

Where S_{i} is an arbitrary space (assume to be the same space in the sequence. it may vary).

Where, for the force, k = 1, 2 (pull, push)

and for the motion, k' = 1, 2, 3, 4 (linear, rotary, reciprocal, oscillatory).
**(b)** The sequence for *subtraction* is also *force-motion-force*. The first force *removes* matter from space of interst and places it in another space. There is no difference in sequence but there is difference in force objective. So, the types of force and motions may vary.
**(c)** True. This is the *add-substract duality*. An *addition* to a space implies a *subtraction* from another space (also conservation of mass).
**(d)** (i) Fisherman *subtracts* when he pulls fish out of river and *adds* to the space the fish is pulled into. Fisherman *adds* when he pushes fish back into the river and *subtracts* from the space from which the fish is pushed.

(ii) The golfer *subtracts* from where he putts the golf ball and *adds* to the hole into which the golf ball enters.

(iii) The airplane is serially *subtracted* from the coordinates that constitute its flight path. In essence, *motion* is a series of spatial *addition* and *subtraction*.
**(e)** The *queen* because she has the *greatest degree of motion freedom*. In general those with the greatest degree of motion freedom are most able to visit different spaces.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings