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Tuesday, June 28, 2016

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


"The change depends on direction of the motion"

This means: "It is possible, we move but there will not be any change or even we maybe lose everything."
"The change depends on direction of the motion." is a philosophic quote in which mathematics (Differential Calculus) applied it to find out the definition of gradient vector field.
For example, in Economics, USD is still an independent variable where many curves such as x = f ($), y = f ($), z = ($) … are determining the direction of motion overall function of
w = f (x, y, z …).
In designing strategic plan, maybe whole a company's strategy – making hierarchy depends on only a person or a department where this independent variable will change the company to a star or overturned (collapsed). In physics, the internal energy of the gas (except an ideal gas) depends on the pressure, volume and temperature (K = f (p, v, T)).
Nowadays, we can see many matrices as transformations and operators which are affecting on vectors.
The purpose of this article is, to introduce some special matrices in which you can easily generate all eigenvalues without any calculation.
The question is: How can we apply these special matrices as operators in our real world?


Some special matrices

1. Let consider “A1” as set of Arithmetic Progression where:

d = 1,      a1 = 1   and    an = a1 + (n - 1) d, n = 1, 2, 3,…..

In this case, we have:


A1 = {a1, (a1+1), (a2 +1),……. (a1 + (n - 1) d)}


One of permutations of set A1 is to invert members of set A1 as follows:


A2 = {(a1 + (n - 1) d),….,(a2 +1), (a1+1), a1}      

We can generate many sets which are the periodicity of set A1 just like below cited:

A3 = {a1, (a1+1), (a2 +1),……. (a1 + (n - 1) d)}

A4 = {(a1 + (n - 1) d),….,(a2 +1), (a1+1), a1}

Finally, we will have set B:

B = {A1, A2, A3, A4, …….An    


Rule 1: If size members of set Aor AorAor An, is equal to size members of set B, we will have a square matrix (Mn*n) to be generated by sets A1, A2, A3,…….An in which Eigenvalue of this matrix will be calculated by using of Binomial Coefficient as follows:

Eigenvalue (Mn*n) = λ = C (k, 2)

Where:   k = n+1, nϵ N, λ > 0


Example:
A1 = {1, 2, 3, 4}
A2 = {4, 3, 2, 1}      


A3 = {1, 2, 3, 4}

A4 = {4, 3, 2, 1}  


Matrix (M 4*4) = 
   
1 2 3 4

4 3 2 1

1 2 3 4

4 3 2 1

λ = C ((4+1), 2) = 10



2. Let consider set “A” as follows:

A = {x | x ϵ R}

Rule 2: Each type square matrix which has been generated from set “A”  just like below matrix:

a1 0 0 0 0 0 0 0 0….

a1 a2 0 0 0 0 0 0…..

a1 a2 a3 0 0 0 0 0 ….

a1 a2 a3 a0 0 0 0……

It will show us the eigenvalues which are just equal members set A which have been included in this square matrix (λ = a1, a2, a3, a4, …)


Example:

A = {0.67, 2, 43, 5, -23, 9, -2.3}

Assume we have set B which is a subset of A:

B = {43, -23, -2.3, 9)

Matrix N will be:

43      0

43  -23    0

43 -23 -2.3 0

43 -23 -2.3 9

Eigenvalues of Matrix N are: λ = 43, -23, -2.3, 9

3. Consider square matrices 2*2 as follows:


a    -a

b    -b

Or 

 a      b
-a    -b
a , b ϵ R

Rule 3: Eigenvalues of above matrices are equal to: λ = 0 and
λ = a – b
4. Consider square matrices 3*3 as follows:
 a     -a      a
 b     -b      b
 c     -c       
Or
  a     b       c
 -a    -b     -c
  a      b       
a , b , c ϵ R
Rule 4: Eigenvalues of above matrices are equal to: λ = 0 and
λ = (a + c)-b
5. Consider square matrices 4*4 as follows:


a      -a      a     -a

b     -b      b     -b

c     -c           -c

d     -d      d     -d
Or
  a     b      c      d
 -a    -b    -c     -d
 a      b           d
-a    -b     -c     -d
 a, b , c , d ϵ R

Rule 5: Eigenvalues of above matrices are equal to: λ = 0 and
λ = (a + c -d)-b
6. Consider square matrix “A” as follows:

            a       0      0

A =      a       b      0

            a       b       



If we turn this matrix around first column (as axis) and then we turn it around first row (as axis), we will have matrix “B”:

          c      b     a

B =    0      b     a

               0      


a , b , c ϵ R

Rule 6: Eigenvalue of A + B is equal to λ = c
and eigenvalue of A – B or B - A is equal to zero (λ = 0)
Example:
If matrix A is:

            1.67     0        0

A =      1.67     4        0

            1.67     4       -3  


            -3     4      1.67
B =       0      4     1.67
             0      0     1.67  
                    -1.33     4        1.67
A + B =        1.67      8       1.67
                     1.67      4      -1.33  
λ =  -3
                4.67     -4       -1.67
A - B =    1.67      0       -1.67
                1.67      4       -4.67  
λ = 0




7. Here is another special matrix. Consider matrix A is a square matrix n*n just like below forms in which (a1, a3, a5, a7 a9…an) are diagonal of matrix as follows:


A =

a1   a2    0         0
   a3    a4    0     0
0    0      a5   a6    0
0    0      0     a7   a8
0    0      0      0   a9
…………………....an

Or

A =
a1        0       0
a2    a3       0    0
0     a4   a5       0
0     0    a6   a7    0
0     0    0    a8   a9
…………………....an
a1, a2, a3,….an ϵ R  

Rule 7: Eigenvalues of matrix A are all members on diagonal of matrix A. In fact, property of matrix A to generate eigenvalues is just like Diagonal Matrix.

Example:
A =
-1.56    2     0
  0     -12     3
  0        0     4  

λ = -1.56, -12 and 4

Consequently, we can find out that inverse of previous special matrix (Rule 7) will have the same eigenvalues (members of diagonal). According to Rule 7, we had below matrix M:
M =

a1 0 0 0 0 0 0 0 0….

a1 a2 0 0 0 0 0 0…..

a1 a2 a3 0 0 0 0 0 ….

a1 a2 a3 a0 0 0 0……

Therefore, eigenvalues of M^-1 are all members on diagonal of M^-1.

8. Consider "T" as set of vectors as follows:

T = {V1, V2, V3, ….Vn}

Where:

V1 = a1i

V2 = a2 i + a3 j

V3 = a4 i + a5 j + a6 k

Vn = a7 i + a8 j + a9 k + a10 t …….. amtn   and     a1, a2, a3, a4, a5, …. am are members of R (real 
numbers)

Theorem 8:

If "A" is a square matrix n*n inferred from the set of above vectors just like below cited:

A = {(a1, 0, 0, 0,…), (a2, a3, 0, 0, 0, …), (a4, a5, a6, 0, 0, 0, …..), …....Vn}

In fact, Scalar amounts of V1, V2, V3, …Vn are respectively rows of matrix A.

Then:   Eigenvalues of (A^n) = all members on diameter of matrix A^n (n member of N).

A =

4
0
0
0
0
-6.5
1
0
0
0
5
2
7
0
0
3
9
-5
12
0
-3.45
9
43
15
72

A^4 =
256
0
0
0
0
-552.5
1
0
0
0
2210
800
2401
0
0
-13147.5
13995
-18335
20736
0
-249381
4740402
17264326
6713280
26873856

Eigenvalues are = 256, 1, 2401, 20736, 26873856

Wednesday, June 1, 2016

The Particles with velocity more than the speed of light


 Around 27 months ago, I wrote an article of "Analysis and Design Open Oscillatory Systems with Forced Harmonic Motion" posted on linkhttp://www.emfps.org/2014/02/analysis-and-design-open-oscillatory.html

According to the conclusion of above article and using of Newtonian mechanics, it can be proved that there is some systems with velocity more than the speed of light.

Let me tell you another example about the kinetic theory of gases.

In the reference with below links, Physicists at CERN generated ions with temperatures of more than 1.6 trillion degrees Celsius. At the Brookhaven National Laboratory in Upton, have set a new record for the highest temperature ever measured: 4 trillion degrees Celsius.



Assume, we have a 0.500 mole sample of hydrogen gas at 1.6 - 4 trillion degrees Celsius.
By using of Maxwell–Boltzmann speed distribution function, we can calculate number of molecules with velocity between 300000 km/s to 400000 km/s at 1.6 trillion degrees Celsius which is equal 1.01326*10^21. It means that about 0.34 % of total molecules of hydrogen have velocity more than 300000 km/s. At 4 trillion degrees Celsius, number of molecules which have velocity between 300000 km/s to 700000 km/s, is equal 4.23143*10^22. It means that about 14.05% of total molecules of hydrogen have velocity more than 300000 km/s.

In record of CERN, it has been stated: "… ions together at close to the speed of light…"

The question is: Had all (100%) ions the velocity close to the speed of light?

Therefore, we have only two alternatives:

1. If the answer to above question is positive, then all reference books should apply the limited velocity of 300000 km/s for Maxwell–Boltzmann speed distribution function.

2. If the answer to above question is negative, then we can use from special theory of relativity only as a simulation model in which limited speed of light is an assumption of this model.

Of course, previous experiments showed that Newtonian mechanics is contrary to modern experimental results and is clearly a limited theory in which velocity of the particles in the Universe always remains less than the speed of light.
But, here there is a strange case and the interesting point. Because this temperature which has been generated by Physicists at CERN (1.6 trillion degrees Celsius), is approximately the boundary between using of the Maxwell-Boltzmann distribution and Maxwell–Jüttner distribution. It means that all particles (100%) in temperature less than 1.6 trillion degrees Celsius have velocity between 0 to less than 300000 km/s and we can still use from Maxwell-Boltzmann distribution instead of Maxwell–Jüttner distribution.

Thursday, January 21, 2016

Nice Quotes of Pythagoras and Immanuel Kant


Let us see some nice quotes of Pythagoras (582 B.C – 496 B.C) and Immanuel Kant (1724–1804).

Quotes of Pythagoras (582 B.C – 496 B.C)

A Greek Philosopher about Number and Control
 
 

- “All is number”

- “Number rules the Universe”

- “Number is the ruler of forms and ideas, and the cause of gods and daemons”

- “Number is the within of all thing”

- “No man is free who cannot control himself”

Quotes of Immanuel Kant (1724–1804)

A German Philosopher about Science and Universal Laws
 
 

- All human knowledge begins with intuitions, proceeds from thence to concepts, and ends with ideas.

- Give me matter, and I will construct a world out of it!

- In scientific matters ... the greatest discoverer differs from the most arduous imitator and apprentice only in degree, whereas he differs in kind from someone whom nature has endowed for fine art. But saying this does not disparage those great men to whom the human race owes so much in contrast to those whom nature has endowed for fine art. For the scientists' talent lies in continuing to increase the perfection of our cognitions and on all the dependent benefits, as well as in imparting that same knowledge to others; and in these respects they are far superior to those who merit the honor of being called geniuses. For the latter's art stops at some point, because a boundary is set for it beyond which it cannot go and which has probably long since been reached and cannot be extended further.

- The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience

- Act only in accordance with that maxim through which you can at the same will that it become a universal law.

 

 

Saturday, December 5, 2015

A Model to Derive Strategies in Strategic Management


One of the most important and also hardest step of a strategic analysis or designing strategic plan is to derive and to develop strategies by using of SWOT matrix. Because we have to compare and to match SO (strengths – opportunities) Strategies, WO (weakness – opportunities) Strategies, ST (strengths – threats) Strategies and WT (weaknesses – threats) Strategies to make new ones by thinking and using of our own ideas. Consequently, we need to take high level of focus and concentration on our job. Unfortunately, by using only SWOT matrix as a tool, we are not able to focus at ease on all strengths, opportunities, weaknesses and threats simultaneously. The purpose of this article is to increase students and strategic analysts’ convenience to have better concentration on content of SWOT matrix to make reasonable strategies. In this case, a new simple rule in set theory has been utilized to solve a case in excel and then by using of the result of excel case and also Monte Carlo method, a template on excel spreadsheet to derive strategies has been made and presented.


A new simple rule in set theory

In the reference with item 5 and rule 5 included in article of “Are These Rules new Conjectures in Set Theory?” posted on link:  http://emfps.blogspot.com/2015/11/are-these-rules-new-conjectures-in-set_17.html, we have below formula:

N' = n - r +1

One of the properties this rule is, to invert members of a set for instance:

 A = {1, 2, 3, 4}

 B = {(4-1+1), (4-2+1), (4-3+1), (4-4+1)}

 I used this property to solve a case in excel.

Case: How can we turn a matrix to 180 degree in excel?

We can easily turn a matrix to 90 degree by using of TRANSPOSE’ formula in excel. I am not a specialist in excel maybe there is another formula to turn a matrix to 180 degree. Here, I am willing to show you my solution to solve this case step by step by an example as follows:

- Assume we have a matrix of 16*8 including members of a, b, c…

- Copy matrix between cells N1:U1 and N16:U16

- On cell M1 to M16 fill numbers 1 to 16

- On cell A1 fill formula = 16 – M1 + 1 which is the same n - r +1 and copy to A16

- On cell B1 copy below formula and copy to K1 and A16 to K16


 =IFERROR(INDEX($M:$W,MATCH($A1,$M:$M,0),COLUMN(B1)),"")

I have enclosed new matrix below cited:



A template on spreadsheet to derive strategies in Strategic Management

Assume, we have made SWOT matrix and we will derive SO strategies. I used an example extracted from book of “Strategic Management Concepts and Cases” Thirteenth Edition by Fred R. David, Chapter 6 and Figure 6-3 as follows:

                 Strengths

S1. Inventory turnover up 5.8 to 6.7

S2. Average customer purchase up $97 to $128

S3. Employee morale is excellent

S4. In – store promotions = 20% increase in sales

S5. Newspaper advertising expenditures down 10%

S6. Revenues from repair/service in-store up 16%

S7. In-store technical support persons have MIS degrees

S8. Store’s debt-to-total assets ratio down 34%

                 Opportunities
O1. Population of city growing 10%
O2. Rival computer store opening 1mile away
O3. Vehicle traffic passing store up 12%
O4. Vendors average six new products/yr
O5. Senior citizen use of computers up 8%
O6. Small business growth in area up 10%
O7. Desire for Web sites up 18% by Realtors
O8. Desire for Web sites up 12% by small firms
I utilized from Monte Carlo method in which I had already explained this method in many articles such as below ones:

-A Monte Carlo Analysis on Case of Nike, Inc.: Cost of Capital (link: http://emfps.blogspot.com/2012/01/monte-carlo-analysis-on-case-of-nike.html
-Application of Pascal’s Triangular plus Monte Carlo Analysis to Find the Least Squares Fitting for a Limited Area (link: http://emfps.blogspot.com/2012/05/application-of-pascals-triangular-plus_23.html)
- Application of Pascal’s Triangular plus Monte Carlo Analysis to Design a Strategic Plan (link: http://emfps.blogspot.co.uk/2012/07/application-of-pascals-triangular-plus_10.html)
How should we make Cut Off tables?
First of all, we should divide the probability distribution in accordance with the number of Strengths (S) and Opportunities (O).
The most crucial thing to bear in mind is, to make two Cut Off tables for Strengths (S) in which the probability distribution is constant but Strengths (S) are inverse and also this is for Opportunities (O). In this case, we will cover all Strengths (S) and Opportunities (O) on template.
In this step, we have to use to turn a matrix to 180 degree in excel.
I have attached the example for Cut Off tables as follows:



Now, we have four cell which we can change Strengths (S) and Opportunities (O) by each click on ENTER or F9 and also it cover all Strengths (S) and Opportunities (O) just like below cited:


Of course, for having a complete template, we should continue all above steps for WO (weakness – opportunities), ST (strengths – threats) and WT (weaknesses – threats).

Below video recorded shows you the components of this model:



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