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... Geometry.1
For centuries the parallel postulate which says parallel lines never meet was thought to be a self evident logical necessity. Eventually geometries were developed such as that on the surface of a sphere in which this postulate is false. 
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... halt3.1
Many computers have a specific instruction to stop processing instructions or halt. Today programmers never use such instructions unless they are writing operating systems but in the early days of computing there were no operating systems and programmers had to halt the computer when the program completed. The Halting Problem need have nothing to do with halting. It is rather the question will a program ever do some specific action. 
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... induction3.2
To use induction to prove a property is true for all integers requires two steps. First you prove the property holds for 0. Then you prove that if the property is true for any number $n$ it is also true for $n+1$. Having established these two results the principle of induction allows you to conclude the property is true for all integers. 
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... function3.3
A function has a domain or set of inputs and a range or set of outputs. For each possible input there is a unique output. For example $f(x)=x+1$ is a function that adds one to its input $x$. We could limit its domain to the integers greater than 0 and then its range would be the integers greater than 1. A more complex example is the function that gives the payments on a $100,000 mortgage from the interest rate. Such a function might have a domain of interest rates between 3% and 10% and a corresponding limited range of payments. Many functions like these two examples are computable so we could write a computer program to go from the input to the output. Mathematical functions need not be computable as long as they are well defined. 
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... Paradox4.1
The barber paradox concerns a barber who shaves everyone in the town except those who shave themselves. If the barber shaves himself then he must be among the exceptions and cannot shave himself. If he does shave himself that he does not shave himself. Such a barber cannot exist. 
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... distance5.1
The Planck distance is $\sqrt{G \hbar/c^3}$ or approximately $10^{-33}$ meters. Where $G$ is the gravitational constant, $\hbar$ is Planck's constant divided by $2\pi$ and $c$ is the speed of light. 
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... digital.5.2
The most prominent attempt to reconcile relativity and quantum mechanics is string theory. This theory establishes minimum particle sizes to avoid the domain where the two fundamental theories of physics are incompatible. One cannot know if string theory is valid because its predictions are impossible to test with existing or foreseeable technology. String theory is not a branch of physics. It is mathematical philosophy. Science requires experiments. 
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... equation5.3
A differential equation describes how a single variable (such as the level of a lake) changes as a function of all the variables that affect it such as time and location. A partial differential equation specifies change relative to other variables such as time or location. The wave equation relates change relative to time to change relative to location. 
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...accelerating5.4
If you are moving 60 miles an hour and travel for 2 hours you will go 120 miles. If you are accelerating at 20 miles per hour per second and go for three seconds from a standing start you will be going 60 miles an hour. Your car almost certainly cannot accelerate that fast but if you have a hot motorcycle it might. 
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... point5.5
We understand acceleration in time from driving. Acceleration is zero when speed is constant neither increasing or decreasing. Acceleration across distance is similar. A flat plane or a uniform slope has zero acceleration. It is only when the steepness of the hill is changing that there is acceleration in space. The hill that keeps getting steeper or that bottoms out as you approach level terrain has acceleration. 
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... mass5.6
Any particle has some energy and thus mass. But some particles like photons that make up light travel at the speed of light and are said to have no rest mass. No amount of energy is sufficient to make a particle that has rest mass move at the velocity of light. In contrast a particle with zero rest mass must always move at the speed of light. 
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... limit5.7
Consider a sequence $a_n = \frac{n+1}{2n}$. The limit as $n$ approaches $\infty$ is $\frac{1}{2}$. No value in the sequence ever equals $\frac{1}{2}$ but each $a_n$ differs from $\frac{1}{2}$ by $\frac{1}{2n}$ which gets arbitrarily close to zero as $n$ goes to $\infty$. 
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... differences.5.8
The second order difference in time is an acceleration. To get an average velocity we divide distance by time. If you go 100 miles in two hours your average velocity is 50 miles per hour. To get an acceleration we divide the change in velocity by time. If you go from 30 miles per hour to 60 miles per hour in 10 seconds the acceleration is 3 miles per hour per second. 
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... behavior.5.9
Chaos theory studies the very complex behavior that can be exhibited by continuous nonlinear systems. These are usually far more complex than linear systems. Discretized linear finite difference equations can be made nonlinear by forcing them to assume only integer values as we did using the truncation function $T$. This can make the behavior of the discretized difference equation for more complex than the linear differential equation from which it was derived although it is not chaotic because it is a discrete and not continuous system. 
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...thermodynamics5.10
Thermodynamics is the study of heat. Initially heat was thought of as a liquid that flows. Eventually it was discovered that heat is a measure of the average random motion of molecules. Thermodynamics studies the macroscopic aspects of heat as if it were a fluid. It ignores the motion of individual molecules. Thus it is a statistical theory. 
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... laws5.11
We have all heard for every action there is an equal an opposite reaction. This is an informal statement of the law of conservation of momentum. Momentum is the product of velocity and mass. Assume a 1000 pound object traveling at 10 miles an hour smashes head on to a 100 pound object traveling at 100 miles an hour. The two objects will have equal and opposite momentum. They will both come to a dead stop. This is required by the conservation of momentum. If a large truck smashes head on into a massive concrete building the earth itself (or at least a portion of it connected to the buildings foundation) will move to conserve the momentum of the truck. There are many other conservation laws. 
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... second6.1
In contemporary physics the speed of light is assumed to define locality. In general locality is satisfied if there is any speed that limits the rate at which effects can propagate. 
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... reference6.2
In special relativity two events are said to be space-like separated if their separation in space exceeds the distance light can travel in the time between the two events. The order that such events seem to occur depends on the inertial frame of reference. Thus two events like the measurements in tests of locality in quantum mechanics will occur in a different order in different frames of reference. 
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... explanation6.3
The observation that the photons in a pair, as used by us, are always found to have different polarization can not as easily be understood as the fact that the socks in a pair, as worn by Bertlmann, are always found to have different color[6]. 
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... symmetric6.4
An equation is symmetric in time if the solution for $f_{t-1}$ is the same as the solution for $f_{t+1}$. The fundamental laws of physics are symmetric in time with some exceptions. Time symmetric models are reversible. Reverse the order in time of the initial conditions and the sequence of states goes in the opposite direction in time. 
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... radians6.5
Radian is a measurement of angle like degrees. There are $2\pi$ radians in $360^\circ$. 
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... time6.6
The Planck time is $(G\hbar/c^5)^{(1/2)} \cong 10^{-43} seconds$ where $G$ is the gravitational constant, $\hbar$ is Plancks constant divided by $2\pi$ and $c$ is the speed of light. 
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... identical6.7
The substitution of $\psi$ for $f$ between the two equations has no effect. 
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... Joule-seconds.6.8
A Joule is unit of energy. One Joule is 0.2388 calories. 
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Mountain Math Software
home consulting videos book QM FAQ contact
Email comments to: webmaster@mtnmath.com