Powered By Blogger

Sunday, December 4, 2016

The change depends on direction of the motion: Generating Eigenvalues from special matrices (CON)

In the reference with the article of “Can We Solve a Nonlinear Equation With Many Variables? (Con)” posted on link: : http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html and following to article of "The change depends on direction of the motion: Generating Eigenvalues from special matrices" posted on link: https://emfps.blogspot.com/2016/06/the-change-depends-on-direction-of.html, here is another new theorem as follows:


Theorem: “If and only if there are three points on a sphere which are members of natural number, they make a matrix 3*3 where the eigenvalue of this matrix is easily calculated by using below 
formulas:

A =
 a11    a12    a13
a21     a22    a23
a31     a32    a33

λ1 = a11 - a12  
 
λ2 = a11+ a12 + a13

Example (1):

The points P1 (1, 2, 2) and P2 (2, 1, 2) and P3 (2, 2, 1) are on a sphere with radius of 3 (r =3), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

1
2
2
2
1
2
2
2
1

λ1 = 1 – 2 = -1  

λ2 = 1+ 2 +2 = 5

Example (2):
The points P1 (8, 16, 16) and P2 (16, 8, 16) and P3 (16, 16, 8) are on a sphere with radius of 24 (r =24), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

8
16
16
16
8
16
16
16
8

λ1 = 8 – 16 = -8  

λ2 = 8+ 16 +16 = 40

Example (3):
The points P1 (4, 8, 8) and P2 (8, 4, 8) and P3 (8, 8, 4) are on a sphere with radius of 12 (r =12), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

4
8
8
8
4
8
8
8
4
λ1 = 4 – 8 = -4 

λ2 = 4+ 8 +8 = 20

Saturday, November 12, 2016

Energy Saving by Using the Solution of a Nonlinear Equation With Many Variables

Following to article of "Can We Solve a Nonlinear Equation with Many Variables? (Con)" posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.htmlthe purpose of this article is to solve a nonlinear equation with five variables in limited domain and range. I have started it with a sample of thermal conduction in curtain wall system in buildings in which this example pursues the energy saving and cost management in shopfront glass in malls, shopping centers and mini-markets. This is a real example because the people can directly use from the data and results of this article to save energy.
Before starting of example, let me tell you about a problem and a restriction when we are applying 
this method to solve the equations as follows:

1. As you understood, this method gives us the same domain and range for all variables while we 
need to different domains and ranges to solve the equations in the fields of engineering. For instance, domains for variables can be between 1 - 100 or 1 – 1000 but sometimes we have to have real numbers for the variables including: x (0.0025, 0.48) or
y (-23.5, 0.78) or z (4, 563.75) and so on. I can say to you that we are able to solve easily this problem by using of some theorems in mathematics and change the domains and ranges of variables to ideal and desired ones. I have shown you this changes in below example.

2. But, there is a restrictive factor something like discrete functions instead of continuous functions so that we have to follow this limitation forever. It means that we should consider sub-intervals between domains and ranges for our variables. In fact, the numbers which belong to variables between a domain and rage are discrete not continuous. Of course, we can decrease and decrease sub-intervals step by step. The best way is, to apply Arithmetic Progression for sub-intervals.


"Please be informed that we can use many methods to reach our targets which are new data and results. Therefore, the most important thing is, to extract new data and results by any method. In fact, the analysis of these new data and results lead us to reach the gates of new worlds. The methods are only tools to reach our targets."

A real example of energy saving: Thermal conduction in shopfront glass

The law of thermal conduction (Fourier's Law) gives us the great opportunity to calculate the rate of energy transfer by heat for a slab of infinitesimal thickness (dx) and temperature difference (dT) in which the rate of│dT/dx│is named the temperature gradient. Simple conductive heat transfer (in watts) through a uniform body can be calculated from below equation: 
P = k. A.│∆T/ x      and       T = Th – Tc   and     Th > Tc

Where:

- A is the area of the body (m2)
- k is the body's thermal conductivity (W/m°C)
- T is the temperature difference across the body (°C)
- x is the body's thickness (m)
- Th is one face of the slab with high temperature (°C)
- Tc is other face of the slab with low temperature (°C)
- P is the rate of heat transfer (Watt)

The heat transfer occurs only if there is a difference in temperature between two parts of a slab.
For this example, I used from manual of "CODE OF PRACTICE FOR USE OF GLASS IN BUILDINGS" an Indian Standard which was adopted by the Bureau of Indian Standards. You can find this paper posted on below link:


The page 27 of above Standard shows us the boundaries for maximum area of designed normal glass in compare with its thickness in shop fronts. (See Table 5.3)  



According to Table 5.3, I chose below domains for A (area) and x (thickness):

A = (6, 15) and x = (15, 25)

And so, I consider domains for the temperature of outside and inside as follows:

Th = (15, 32) and Tc = (-20, 4)

In this case, we will have a maximum of heat transfer equal to 41600 W and a minimum heat transfer equal to 2112 W. (kglass = 0.8 W/m°C)  

P max = 41600 W,       and Pmin = 2112 W

Therefore, the range for heat transfer is:  P = (2112, 41600)

Now, I apply previous method and solve above equation for five domains and range. The rate of heat transfer (P) to the Number of Results has been presented on below graph:




I only apply one sub-interval and get the total sum of results equal to 195 in which we will have 30 results for P = 12800 W. If you spend more time and apply many sub-intervals, it is possible, you will find much more results.

Analysis of findings and results

The most important part of this article is, to analyze the results to reach our target which is the energy saving in the same conditions accompanied by cost management.
According to the results, here is many outcomes which lead us to handle the cost management and the energy saving. Let me present only several states as follows:

1. For instance, if we compare maximum transfer rate with another results, we will have below analysis: (Please see below figure)




According to above table, if you decrease the temperature of your shop from 32 °C to 15°C and also increase the thickness of glass from 15 mm to 16.935 mm, then you will save about 40% of your energy consumption. What is your cost management?
Usually shop front glasses have around 8 years guarantee. If you open your shop 12 hour per day, total hours for energy consumption is: Total hours = 8*365*12 = 35040 hour
Then, you have around 40% (0.403846) energy saving which is equal to 16800 W. Therefore, you will be able to save the energy totally around 588772 KWh in the period of 8 years (16800*35040). The cost of electricity power is about 12 cent per KWh. In the result, you will save totally amount of USD70, 640 for 8 years and USD8, 830 per year. If we add future value of each annual saving with an average return rate (Cost of capital) equal to 11% per year, it means that you are saving amount of money around USD 104,718. What is your costs? You should purchase normal glass with thickness 17 mm instead of 15mm. Therefore, you should pay more USD 40 per square meter (Price of normal glass with 15mm thickness is around USD 130 and 17mm is around USD170). Consequently, your additional investment is USD 600. It means that you have still saved the amount of money around USD 104,000. 

2. If we compare some results together, sometimes we can see very exiting outcomes as follows:




According to above results, if we increase the thickness from 15mm to 25mm, we are saving around 42% energy consumption and if we increase the thickness from 15mm to 23 mm, we are saving around 35% energy consumption. For calculation of cost management, you can apply the same method mentioned in Item (1).

3. In the reference with below results, if we are able to change the design for area of 7.74 m2 to 6 m2, we have already saved around 23% energy consumption.




As I told you, we can obtain 30 results for P = 12800 W. You can find these results on below figure for your better analysis and design:



Wednesday, October 19, 2016

Can We Solve a Nonlinear Equation With Many Variables? (Con)

Following to article of "Can We Solve a Nonlinear Equation with Many Variables?" posted on below link:


Let me start second example for an equation with four variables as follows:

Example (2): Solving of Sphere Equation

As you know, the sphere equation has many applications in all fields of engineering and physics. When we talk about a sphere equation, our discussion can be expanded not only macroscopic systems but also microscopic particles such as the quantum model of the Hydrogen atom. Therefore, let me start by solving of a sphere equation for a limited domain and range as follows:

Consider the sphere equation with below domain:    

If   x^2 + y^2 + z^2 = r^2     x, y,z ϵ N,            x, y,z ≤ 100         

Then the range for the radius of sphere will be:    r ϵ N,            r ≤ 173


Now, I apply previous method and get all results of "x, y, z" for "r" in given range. The number of 
results related to "r" has been presented on below graph:





In this case, total sum of possible answers is only and only equal to 4935.
Above graph shows us that there is a maximum number of answers equal to 165 for r = 99. For instance, I have brought some results on below figure:





As we can see, for r = 15 and  r = 150, the number of results are the same equal to 15 while for r = 31, we have 24 answers

We can apply this domain and range as a template for all macroscopic and microscopic numbers.

Sunday, October 16, 2016

Solving a Nonlinear Equation with Many Independent Variables By Using Microsoft Excel Plus VBA

Typically, we are able to solve system of linear or nonlinear equations which are a set of simultaneous equations (SE). Definitely, solving of a linear SE is very easy while we have to use Newton's method to solve nonlinear SEs. The common case for both of them is, to generate an equation for each variable. It means, we can solve an equation with three variables, if we have three simultaneous equations or solving of four variables needs to find a system of four simultaneous equations and so on.

Can we solve a nonlinear equation with many variables? Yes. In the special conditions, the answer is 
positive.

The purpose of this article is to present some examples which show us possibility to solve a nonlinear equation with many variables where we have a good estimation for limited domain and range of variables. The method applied is the same method stated in article of "The Generating New Probability Theorems" posted on link:


The experienced physicists and engineers have usually the true speculation of domain and range for the variables while they need to obtain precise amounts for the variables. Therefore, this method can be useful for them.
At the first, I start by a nonlinear equation with three variables then four variables and finally five 
variables.

Example (1):

One of professional people asked me a question in math and statistics group of social media as follows:


"I need some help to interpret a stock regression:

SZ is high or low size of the company

BM is high or low book to market size

R is the return of the stock

R = 5% + 6%Bm +2%SZ – 2%*BM*SZ

My question is whether I should short high size and high book to market stocks (stocks that have both characteristics)?

I analyzed his problem by using above method as follows:

Here is my analysis:

1. To reach the maximum R, you should stay SZ the constant in low size and then if you increase BM, you will reach the maximum R.

2. If you increase both of them (SZ and BM), you will significantly decrease R.

3. The most important thing is about R = 0.11 because in this case, it does not take any difference. In fact, above analysis does not work for R = 0.11

4. If 0< SZ and BM <1, then maximum amount of R will be always equal to 0.11 (Rmax = 0.11)


I think that this is really a magic formula.

Example (2)

This is an example about financial and risk management.

As you know, the basic theory which links risk and return rate for all assets is, the Capital Asset Pricing Model. The equation of CAPM is as follows:











Now, suppose you want to invest on an asset in which your expected return rate (required return) is equal 12%. The question is: What are the alternatives or scenarios for three independent variables of risk – free, beta and market return?
Here, by applying the method stated in this article, I have obtained 21 answers for three independent variables as follows:






















Example (3): The equation of State for an Ideal Gas



Let me tell you an example about the equation of state for an ideal gas.

We have:

P.V = N.KB.T

Suppose we have a constant volume (V) equal to 0.03 m3

 If Boltzmann’s constant (KB) is equal to 1.38E-23 J/K, what are the answers for P and N and T?


According to my method, I found 9 answers which are as follows:



:Where

T = temperature (K) and P = pressure (Pa) and N = number of molecules


Example (4): Solve Circle Equation

  When we open a calculus book, we can see the signs and footprints of Pythagoras (582 B.C – 496 B.C) everywhere. Therefore, let me start by solving of circle equation for a limited domain and range as follows:

Consider the circle equation with below domain:     

If 

           x^2 + y^2 = r^2

x, y ϵ N,            x, y ≤ 100

Then the range will be

r ϵ N,            r ≤ 141


 Now, I apply above method and get all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph







In this case, total sum of possible answers is equal to 126. 
For instance, above graph shows us, if r = 25, 50, 75, 100 then the number of results for "x, y" are equal to 4 and if r = 65, 85 then the number of results for "x, y" are equal to 8. The results are as follows 






 Now, consider the circle equation with below domain

    

If 

           x^2 + y^2 = r^2

    0 0 0 x, y ϵ N,            x, y ≤ 1

Then the range will be

 4 r ϵ N,            r ≤ 14 1

 If I apply above method, I will generate all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph:






In this case, total sum of possible answers is equal to 2068. 

For instance, above graph shows us, if r = 325, 425, 650,725,850,925,975 then the number of results for "x, y" are equal to 14. The results are as follows: 




Note: All researchers and individual people, who are interested in having this model, don’t hesitate to send their request to below addresses:




WhatsApp: +98 9109250225