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

Periodic Signals

Signaling is a ubiquitious phenomenon in the societies of cognitive beings.

(a) What is a signal?

(b) What is a periodic signal?

(c) Identify the following periodic waveforms:

(i)

(ii)

(iii)

**The string**:

S_{7}P_{1}A_{17} (Containership -Location)

S_{7}P_{2}A_{21} (Identity - Physical Properties).
**The math**:

Pj Problem of interest is of type *identity*.

(a) *Signals* are essentially indicators of *presence* (the *being* in a location). Thereafter, one is interested in the *identity*, analyses and possible uses of the *presence*. In electric circuits, the voltage and current signals indicate the *presence* of voltage and current sources. The identities of voltage and current signals are expressed as *waveforms* (graphical representations of functions).

(b) *Periodic Signals* are *time-dependent cyclic signals*. In other words, there is an interval called the *period* of the signal during which the signal repeats the same waveform pattern.

Mathematically, an arbitrary signal x(t) is periodic if it can be expressed as follows:

x(t) = x(t + nT)-------(1)

Where n = 1, 2, 3 ...; and T is the *period* (the time it takes to complete 1 cycle) of x(t).

The *frequency* (f) of *x(t)* in cycles/sec (Hertz) is:

*f = 1/T*-------(2)

(ci) The waveforms in c(i) are cosine and sine periodic signals (sinusoids).

The red waveform is a cosine function. Its general form is:

v(t) = V_{max}cosω----(3)

Where the radian frequency (angular velocity), ω = 2πf.

The signal expressed in (3) is often the reference sinusoid.

A sinusiodal signal has the general form:

v(t) = V_{max}cos(ωt + θ)-----(4)

The sinusiodal signal expressed in equation (4) is said to *lead* the sinusiodal signal expressed in equation (3) by *phase angle, θ*.

The *phase angle θ* = *ωτ*

Where τ is the *time shift*

The *Blue* waveform is a sine function. Its general form is:
*v(t) = V _{max}sinωt*----(5)

The relationship between the cosine signal and the sine signal is:

V

Since the cosine signal

The relationship can also be expressed as follows:

V

Since the sine signal

(cii) The waveforms in c(ii) is a periodic pulse signal.

(ciii) The waveform in c(iii) is a periodic sawtooth signal.

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