## Sunday, April 25, 2010

## Friday, April 16, 2010

### Where the Graphs Come From

Where the Graphs Come From

Curve, as a word for graph, in math, is misleading, as a label for count pairs, on many levels. First, it is because it refers to a visual impression, since many of mathematical functions represented have the form of curvy line, of the concept. What it fails to capture and signify is that the “curve” represents pairs of numbers, i.e. pairs of counts. Function, in mathematics, is a set of pairs. It represents counts you have paired for this or that reason. Function is not a formula. Formula is only a rule. Functions is more of a map, or the most precisely, it is a set of paired numbers.

Then, how we ended up with the “curve” word? Some genius came up with the idea to consider counts of length, i.e. magnitude of length, number that is obtained by measuring length, length of line. The count obtained by measuring the length of a line will be the same as the count obtained say, by counting cars, measuring mass, temperature, stating the price of goods, or speed, or how many trains go through the station during the day, or any other counts that you can think of. Thus, showing the line on the graph, and knowing that the next step is measuring its length of that line, and matching that count (obtained from length) with the count of another object will give you the representation of the quantity you are interested in, as depicted in Fig 1.

Fig 1. Matching the counts.

Each point on the curve represents a pair of lengths, namely x and y. This is the most important property of the curve, that its points represent the pairs of lengths. The shape of curve is not there, in most of the cases, to be considered aesthetically. The curve shape tells you the relation between the paired lengths, and that’s the most important information you can get by visually inspecting the curve of the graph in mathematics.

Fig 2. Another view of matching the counts and using a graph for respresentation.

So, the major conclusions follow. Function in math is a defined pair of counts. Functions is not a formula. The other good word for function is mapping, map, between two or more numbers. Pairing numbers is another better word for function. Curve in math graph is a visual representation of paired lengths. Lengths of the curve, line are equivalent to the numbers, counts obtained from other sources, for instance, by measurements, agreements, counting, picking the number.

More links on math:

Curve, as a word for graph, in math, is misleading, as a label for count pairs, on many levels. First, it is because it refers to a visual impression, since many of mathematical functions represented have the form of curvy line, of the concept. What it fails to capture and signify is that the “curve” represents pairs of numbers, i.e. pairs of counts. Function, in mathematics, is a set of pairs. It represents counts you have paired for this or that reason. Function is not a formula. Formula is only a rule. Functions is more of a map, or the most precisely, it is a set of paired numbers.

Then, how we ended up with the “curve” word? Some genius came up with the idea to consider counts of length, i.e. magnitude of length, number that is obtained by measuring length, length of line. The count obtained by measuring the length of a line will be the same as the count obtained say, by counting cars, measuring mass, temperature, stating the price of goods, or speed, or how many trains go through the station during the day, or any other counts that you can think of. Thus, showing the line on the graph, and knowing that the next step is measuring its length of that line, and matching that count (obtained from length) with the count of another object will give you the representation of the quantity you are interested in, as depicted in Fig 1.

Fig 1. Matching the counts.

Each point on the curve represents a pair of lengths, namely x and y. This is the most important property of the curve, that its points represent the pairs of lengths. The shape of curve is not there, in most of the cases, to be considered aesthetically. The curve shape tells you the relation between the paired lengths, and that’s the most important information you can get by visually inspecting the curve of the graph in mathematics.

Fig 2. Another view of matching the counts and using a graph for respresentation.

So, the major conclusions follow. Function in math is a defined pair of counts. Functions is not a formula. The other good word for function is mapping, map, between two or more numbers. Pairing numbers is another better word for function. Curve in math graph is a visual representation of paired lengths. Lengths of the curve, line are equivalent to the numbers, counts obtained from other sources, for instance, by measurements, agreements, counting, picking the number.

More links on math:

- Real world examples for rational numbers, for kids
- Math and its relationship with real world
- How math can be applied to so many different fields?
- Where the graphs in mathematics and physics come from?
- Tweets about math, physics, and how to approach math calculations
- One insight about mathematical axioms, logic and their relation to other disciplines
- How the ideas are born and notes on creative thinking
- Mathematics Axiomatic Frontier
- Why math can be an independent discipline?
- More on creative thinking, math, innovations, physics, emotions
- Comparison between making movies and math and physics
- More tweets about math
- Domains of math applications and math development
- Where all those number series in math are coming from?
- How to understand the role of math in economics, physics, engineering, and in other fields

## Thursday, April 8, 2010

### Mapping between intercellular communication and brain interpretation of the signals

The trick in biochemistry and in cellular biochemistry is to understand where our understanding stands. We know about receptors embedded in the cell membrane and we know about inter-cellular communication via ion concentration changing and channel, gate opening closing. But, what we don't know is how the INTERPRETATION is done in the brain. The interpretation is present, we know it, and we can eventually make a MAP between the receptor(-signal-ion concentration-channel dynamics) and interpretation in brain, but we don't know how the interpretation in brain is realized. That's the state of molecular biology and cellular biochemistry now. Still, even this approach allows to design drugs and understand mechanisms in brain and of signals.

## Wednesday, April 7, 2010

### Hockey, Physics, Axioms and Where Innovations Come From

Physics has strict laws. But, hockey, also, has strict rules, yet many different games can be developed within those rules. Same thing in Physics. You are free to choose, arbitrary, by your feeling, by your human experience, the initial conditions for your Physics problem. Or, invention. That's completely arbitrary, can be called art, skills, talent, intuition, genius. Like, how high you want to put ball before you leave it to the gravitational forces. That initial height IS NOT dictated by laws of Physics. It is determined by you. But, once you leave the ball, the Physics law takes over, and their strictness has to be followed and be aware of. Hence, the creativity in Engineering is in determining the initial conditions. They are up to you. Then, and only then you use laws of Physics to obtain the next result. For instance, Physics Laws, or even Laws of Electromagnetics, will not tell you, nor will prevent you, to determine the shape, amplitude of voltages for your generators. If sinusoidal voltages suit you, chose them. If other voltage shape suits you, chose that shape. And only then, you use the laws of electrical circuits to find the system states, voltages, currents, energy within the circuit. Creativity in Engineering is in choosing correct, useful, desired initial conditions.

Similar thing is in hockey. The rules of hockey will not tell you how to win. The same way language grammar will not and can not tell you how to write a winning novel or screenplay. The rules of hockey are hockey axioms, and you use them to create instances of the game, within the rules framework. Each game is a set of theorem obtained from the initial hockey axioms, and proved by the way of arbitrary not interfering with the game.

In Physics, differential equations can not tell you how to construct a new machine or make an invention. Why? Because differential and integral equations are waiting for YOU to tell them their initial and boundary conditions, i.e. STARTING CONDITIONS, they are waiting for you to tell them what and when to calculate, in the form of integrand, integral limits, initial and boundary conditions.

Similar thing is in hockey. The rules of hockey will not tell you how to win. The same way language grammar will not and can not tell you how to write a winning novel or screenplay. The rules of hockey are hockey axioms, and you use them to create instances of the game, within the rules framework. Each game is a set of theorem obtained from the initial hockey axioms, and proved by the way of arbitrary not interfering with the game.

In Physics, differential equations can not tell you how to construct a new machine or make an invention. Why? Because differential and integral equations are waiting for YOU to tell them their initial and boundary conditions, i.e. STARTING CONDITIONS, they are waiting for you to tell them what and when to calculate, in the form of integrand, integral limits, initial and boundary conditions.

Labels:
electrical engineering,
engineering,
hockey,
physics

### Three phase circuits - a comment

Conservation of Energy Law in an electric circuit, say 3PH, does not limit you what kind of voltages you are going to apply or produce at the generator points. And this particular freedom, in choosing the shape of voltages, is the root of creativity and inventions in 3PH circuits. Any voltage can be there. We choose, almost axiomatically, a priori, that there will be sinusoidal, periodic voltages.

Pic 1. Three phase systems with generators and loads.

As in any axiomatic system, there is a World #1 and the World #2. The World # 1 determines the reason, motivation why we have the voltages we have. The World #2 accepts those voltages as given, as axioms, as starting assumptions, premises. Finding the reason what voltage shapes we will have usually has nothing to do with the World #2. For instance, we need 3PH voltages to create rotational magnetic field, which in turn, we need to rotate the shaft, which, in turn, we need to move a car. Moving a car is the World # 1, while chosen voltages are the World # 2.

2010 Copyright by Nesha. All Rights Reserved.

Pic 1. Three phase systems with generators and loads.

As in any axiomatic system, there is a World #1 and the World #2. The World # 1 determines the reason, motivation why we have the voltages we have. The World #2 accepts those voltages as given, as axioms, as starting assumptions, premises. Finding the reason what voltage shapes we will have usually has nothing to do with the World #2. For instance, we need 3PH voltages to create rotational magnetic field, which in turn, we need to rotate the shaft, which, in turn, we need to move a car. Moving a car is the World # 1, while chosen voltages are the World # 2.

2010 Copyright by Nesha. All Rights Reserved.

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