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

Types Of PjProblemStrings Sequences

PjProblemStrings Sequences are the work-horses of dynamic spaces. The primary types are as follows:

(1) **Stand-alone** PjProblemStrings Sequences

(2) **Queueing** PjProblemStrings Sequences

(3) **Colliding** PjProblemStrings Sequences

(4) **Displacement** PjProblemStrings Sequences

(5) **Overlapping** PjProblemStrings Sequences

(6) **Hovering** PjProblemStrings Sequences

(7) **Linked** PjProblemStrings Sequences

(a) Give at least one example for each of the seven types

of PjProblemStrings Sequences.

(b) Interprete Newton's first and third laws of motion in the context of PjProblemStrings Sequences.

**The strings**:
S_{7}P_{3}A_{3k} (force -k =1, 2), S_{7}P_{4}A_{4k} (motion- k =1, 2, 3,4).
**The math**:

All Pj Problems are at play. However, Pj Problems of Interest are of types *force* and *motion*.

**(a)**Scenarios:

(1) **Stand-alone** PjProblemStrings Sequences: examples are a crane offloading containers one at a time; walking; running; driving a car; flying an airplane. In essence all solo motions are of this type.

The remaining six types are different groupings/interactions of *stand-alone* PjProblemStrings Sequences.

(2) **Queueing** PjProblemStrings Sequences: examples are, customers queueing in front of the service counter of a service company; cars queueing at a traffic light, marching in single file, etc.

(3) **Colliding** PjProblemStrings Sequences: examples are, two lovers kissing; a sperm fertilizing and ovum; two or more cars colliding; a missile hitting an aircraft; a batter hitting the baseball thrown by a pitcher; kinetic collisions when heat is applied to matter, etc.

(4) **Displacement** (or Replacement) PjProblemStrings Sequences: examples are, drinking with a straw (suction - air moves, liquid replaces); convection flow (warm air rises, cold air sinks).

(5) **Overlapping** PjProblemStrings Sequences: covalent bonding (atomic orbitals overlap to form molecular orbitals).

(6) **Hovering** PjProblemStrings Sequences: examples are, a cop helicopter trailing a suspect, a skateboarder's jump; a stuntman's jump. In this type, the hovering PjProblemStrings Sequence hovers above another PjProblemStrings Sequence sequence.

(7) **Linked** PjProblemStrings Sequences: examples are, a moving train; a tow truck towing a vehicle.
**(bi)** Newton's First Law: *an object at rest will remain at rest and an object in motion will remain in motion at constant velocity unless acted upon by an unbalanced force*.

In the context of PjProblemStrings Sequences, colliding PjProblemStrings Sequences are needed to overcome the inertia of an object at rest and an object in motion at constant velocity. A stand-alone PjProblemString Sequence is created in the former scenario and the stand-alone PjProblemString Sequence in the latter scenario is change to an accelerating sequence after collision. In the case of an automobile, the collision occurs between the foot and the accelerator pedal.
**(bii)** Newton's Third Law: *for every action, there is an equal and opposite reaction*. In other words, all forces come in pairs.

Hence PjProblemStrings Sequences are motions sandwiched between two forces. The *first force* is a resultant force (action - reaction) causing motion. The second force opposes motion. Motion continues until the resultant force equals zero.

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