SOME PROPERTIES OF MY CENTRAL
MATHEMATICS
I started my first investigation in
the year 2007. This investigation is considered as one of the
greatest single investigation in the history of Mathematical Sciences.
This single investigation was later confirmed and proved by me in
the year 2008 for many mathematical scientists in Ghana
witness. Even though, I first named it “Balance Point Concept”, I
currently proffered the general hypothetical name Central Point
Law which has many properties in the establishment of modern sciences
and technology: is it already used in running computer programs(see Adongo’s
Point Values Interval Law) and practical engineering designed.(see Adongo’s
parallelnear theorem)?
This Law which most people think, it
should later be named as Central Mathematics has profound blessing in the life
of research scientists and even behavioral and social sciences. Below are the
properties of the Central Mathematics.
1)MAIN LAW
a)Adongo's Central Point Law:
For all integers x, and y
and all real numbers A ,B and C in equation Ax+By=C; then at
central point;
i)the value of x and y
are x=C/2A and y=C/2B respectively.
ii) the axial term Ax is
equal to the axial term Bx
iii) the slope s=-y/x of the
line is the central point pairs (x, y)
iv) the intercept of a line is (x-x,
y+y) or (x+x, y-y)
v) the central point line which
makes an angle of θ with positive axis has gradient –y/x=tanθ
vi) the two lines l1 and
l2 are perpendicular if the gradient of line l1 and l2
are m1=-y/x and m2=x/y respectively
b)Adongo's Linear Point Values Interval
Law:
At central point (x, y), the point
variable x is adding itself to infinity when the point variable y
is subtracting itself to infinity and vice-versa.
[x ; y ]=[(x+x;
x+x+x, ….); (y-y; y-y-y, ….)]
or
[x ; y]=[(x-x;
x-x-x, ….); (y+y; y+y+y, ….)]
[x ; y ]=[ℓxx;
ℓyy]
For
ℓx=1,2,3,4,………, is correspond to ℓy=0,-1,-2,-3,……..,
IMPORTANCE OF CENTRAL POINT LAW
1)
Used to determine Centre of mass, moment of initial,
kinetic energy, area and volume of a lamina.
2)
Used in mechanical drawing, navigation, survey,
and topology.
3)
Used to determine orbit of satellites, planets,
and comets; shapes of galaxies, gears and cams.
4)
Very important in economics, business and
finances.
5)
Very important in computer science.
Very important in engineering sciences
EXAMPLE
If 3x + y = 6 , find
i). the central pairs, x and y of the equation.
ii). the slope of its graph using Adongo’s central slope formula approach.
iii).
the intercept of the line.
iv). the
possible values of x and y
to fourth interval.
SOLUTION
i)The gravitation pairs x and y
is calculated as;
x = C / 2A
x = 6 / [ 2 (3) ]
x = 1
y = C / 2B
y = 6 / [ 2 ( 1)]
y = 3
ii)Slope of the line equation or graph is
S = - ( y / x )
S = - ( 3 / 1 )
S = - 3
iii)
The intercept of the line is calculated as
( xG ; 0 ) = (2x , 0 )
= [ 2 (1) , 0 ]
= ( 2 , 0 )
And:
( 0 ; y )= ( 0 , 2y
)
= (0 , 2 * 3 )
= (0 , 6 )
iv)
Applying the central point values interval law, we have,
( x ; y ) = [ ( 1 , 0 , -1 , 2 , -3
) ; ( 3 , 6 , 9 , 12 , 15 ) ]
or
or
=[ ( 1 ,
2 , 3 , 4 , 5 ) ; ( 3 , 0 , -3 , -6 , -y )]
KEY PROPERTIES OF THE CENTRAL POINT
LAW
PROPERTY ONE:
For all integers x and y;
and all real numbers a, b, and c.
i)the equation ax-by=c is
equal to:
c/x+by/x=a
PROPERTY TWO:
For all integers x and y; and
all real numbers a, b, and c
ii)the equation ax+by=c has
centre point if:
ax/[c-by]=by/[c-ax]
PROPERTY THREE:
For all integers x and y; and all
real numbers a, b, and c
iii) the equation ax+by=c has
centre point if:
ax(by)=by(ax) ; ax=by
where;
ax=c-by
by=c-ax
PROPERTY FOUR:
For all integers x and y;
and all real numbers a, b, and c
ii)the equation ax+by=c has
centre point if:
ax/c+by/c=1
c=[2by+2ax]/2
PROPERTY FIVE:
For all integers x and y;
and all real numbers a, b, and c where x=y
ii)the equation ax+by=c has
centre point if:
x=(x+y)/2=[c/2a+c/2b]/2
PROPERTY SIX:
For all integers x,y
and z; and all real numbers a, b, c and d
ii)the equation ax+by+cz=d has
centre point if:
z=[(d-2ax)+(d-2by)]/2
x=(d-cz)/2a
y=(d-cz)/2b
PROPERTY SEVEN
For all integers x and y;
and all real numbers a, b, c and q
ii)the equation [ax/q+by/q]m=c
has centre point if:
x=(qc1/m)/2a
y=(qc1/m)/2b
More also, the properties of the Central
Point Law is too worthy, as it cannot be measured in the sense of all
observers. The Central Point Properties is not only essential for creating new
models and theories by continuous transformations, but some variables can be
predicted in the absent of others. It is the first contribution to use one single
observed datum for statistical purposes. I hope this worthy contribution
(that cannot be compared to others for now); will be carefully transformed by
all Academic institutions in the world for future academic curriculum. All good
practical projects holders and consultants are invited. FIA/FSA is
invited.
Below are formulas that can be newly
transformed by using the properties of the Central Point Mathematics.
FORMULAS THAT ARE NEEDED FOR ECONOMICAL
TRANSFORMATION
1)
PV=Σ[(R-C)/(1+i)t]
2)
PV=Σ[PtQt-(VtQ+F)/(1+i)t]
3) QT=(TFC+π)/(P-AVC)
4)
NPV=Σ[NCFt/(1+i)t] - C > 0
5)
PI=Σ[R/(1+i)t]/Co
6)
P=(a-c)/(b+c)
7)
Y=1/(1-b)[a-bT+I +G]
APPLYING FORMULA NPV=Σ[NCFt/(1+i)t] - C > 0
NPV CRITERION
Theorem:
Given the secondary time data which
net present value plus the initial cost is the present value and is
PV=NPV+C=∑NCFt /(1+i)t will have independent primary
function NCFt=1/2(NPV+C)(1+i)t for each t=1,2,3,……, if the
capital project accept NPV>0.
EXAMPLE
Used the net present value criterion
to evaluate the project of Central Bank at requires project that has cost of
25000000.
year(t)
|
Net present flow(NCF)
|
1
|
600000
|
2
|
800000
|
3
|
800000
|
4
|
600000
|
5
|
250000
|
SOLUTION
1
|
600000
|
0.86957
|
521742
|
|
2
|
800000
|
0.75614
|
604912
|
|
3
|
800000
|
0.65752
|
526016
|
|
4
|
600000
|
0.59175
|
343050
|
|
5
|
250000
|
0.49718
|
124295
|
|
6
|
NCFt
|
[1/(1+0.15)t]
|
Total=2120015
|
NCFt=1/2(NPV+C)/(1+i)t
Computing the average net cash flow
for year 1,2,3, 4 and 5.
NCF1=1/2(2120015)/(10.15)1=921745.6522
NCF2=1/2(2120015)/(10.15)2=801517.9584
NCF3=1/2(2120015)/(10.15)3=696972.1377
NCF4=1/2(2120015)/(10.15)4=606062.7285
NCF5=1/2(2120015)/(10.15)5=527011.0682
PROOF!
NCFt=1/2(NPV+C)/(1+i)t
NPV=2X921745.6584(1.15)-2500000=-379985.00
NPV=2X921745.6584(1.15)2-2500000=-379985.00
NPV=2X921745.6584(1.15)3-2500000=-379985.00
NPV=2X921745.6584(1.15)4-2500000=-379985.00
NPV=2X921745.6584(1.15)5-2500000=-379985.00
SECOND APPLICATION
Introduction:
The availability of Central Point
Theory open the way for the Central Prediction Theory as long as applied
statistics is concerned. The earliest form of more accurate Credibility Theory
(or Model) is the Buhlmann and Bultmann-Straub Theory (or Model) which I
revolutionized as Central Credibility Theory. The availability of the Central
Credibility Theory (or Model) in turn open way for the establishment my Central
Rating Theory(or Model) that can applied practical in most Insurance and
Financial Industries.
The Central Point Theory is not only
the cause of my Theories Mention above but also applied by me to produce
practical theories(or Models) like Central Cost and Pricing Model,
Inventory Theory(or Model) of Uncertainty, Strategic option Model, Minimum
Uncertainty Theory(or Method), Growth Rate, Growth Theory, Billian Capital
Stock Valuation Theory(or Model), Base Rate Model and Billian Interaction
Theory(or Model).
I cannot talk of Point Values
Interval Law without Central Point Theory. One of the useful areas in which I
can apply my Point Values Interval Law is assessment of Systematic Risk and
Beta Coefficients.
PROBLEM OF STATEMENT
Existing Theory in this field of
analysis is limited because; people could not apply the existing concept to
perfectly predict infinite expected returns of portfolio in order differences
or infinite betas of portfolio in order of differences that proved important
here. It could not predict some infinite different possible portfolio expected
returns and betas by varying the percentages invested in two assets. The
existing Theory on assessment of systematic risk and beta could not also use to
perfectly predict corresponding relationship between expected returns and betas
in order of differences to infinity.
SOLUTION TO THE PROBLEM
This theory of mine is to perfectly
predict infinite expected returns of portfolio in order differences or infinite
betas of portfolio in order of differences. This theory is also possible for
predicting some infinite different possible portfolio expected returns and
betas by varying the percentages invested in two assets. My theory also
perfectly predicts corresponding relationship between expected returns and
betas in order of differences to infinity.
ASSESSMENT OF SYSTEMATIC RISK AND
BETA
There is no type of risk that
affects a large number of assets than systematic risk. A systematic risk that
affects a large number of assets is also known as market risk. The systematic
risk principles states that the mean expected return of securities asset are in
linear relation and the reward for bearing risk depends only on the systematic
risk of an investment in order of differences, the underlying rationale for
this principle straightforward: Since unsystematic risk can be eliminated at
virtually no linear relation and reward for bearing it. Put another way, the
market does not reward risk that are borne unnecessarily.
MEASUREMENT
Since systematic risk is crucial
determinant of an asset’s expected return, in their order of differences, we
need some way of measuring the level of systematic risk for different
investments. The specific measure we will use is called Point Values Interval
Law of betas coefficient I have propounded. Beta coefficient is amount of
systematic risk present in a particular risky asset relation to an average
risky asset. The most important thing to remember is that the expected return
in order of differences respectively, on an asset depends only on its
systematic risk in order differences. Since assets with betas have expected
returns in order of differences. Let us considered security A and security B
which expected return, and beta coefficient of portfolio are giving
E(RP)= {2WAE[E(RA)]+2WBE[E(RB)]}/2
E(RP)= {2WAE[E(RA)]+2WBE[E(RB)]}/2
βP= {2WAE(βA)
+ 2WBE(βB)}/2
The mean amount of systematic risk
present in security A asset and security B asset relative to an average security
A asset and security B asset respectively are:
E(βA) =2WAE(βA)
E(βB) = 2WBE(βB)
And the mean expected return on
asset and security A and B are:
E[E(RP)] = 2WAE(RA)
E[E(RP)] = 2WBE(RB)
POINT VALUES INTERVAL LAWS
LAW ONE:
The mean expected return of security
A asset and mean expected return of security B asset are in linear relation.
This is simply telling us that, at any mean return of security A asset which
are linear related to mean return of security B asset in order of correspondence;
the mean return of security A asset is adding itself to infinity when the mean
return of security B asset is subtracting itself to infinity and vice-versa.
LAW TWO:
Mean amount of systematic risk
present in security A asset and security B asset relative to an average
security A asset and security B asset respectively are in linear relation. This
is telling us that at any mean amount of systematic risk present in security A
asset and security B asset relative to an average security A asset and security
B asset is adding itself to infinity when the mean amount of systematic risk
present in security A asset and security B asset relative to an average
security A asset and security B asset is subtracting itself to infinity and
vice-versa.
Suppose we had the following
investments.
SECURITY
|
AMOUNT INVESTED
|
EXPECTED RETURN
|
BETA
|
STOCK A
|
$1000
|
8%
|
0.80
|
STOCK B
|
$2000
|
12%
|
0.95
|
What is the expected return on this portfolio?
What is the beta of this Portfolio? Does this portfolio have more or less
systematic risk than an average asset in order of differences?
SOLUTION
To answer, we first have to
calculate the portfolio weights. Notice that the total amount is $3000;
of this WA=1000/3000=0.333333 is invested in stock A.
similar WB=2000/3000=0.666667 is invested in stock B.
E(RP)= [2x0.333333x8%
+2x0.66667x12%]/2=10.666668%
βp = [2x0.333333x0.080
+2x0.666667x0.95]/2=0.90
The expected beta of asset A is:
E(βP) = 2WAE(βA)
= 0.533333
The expected beta of asset B is:
E(βP) = 2WBE(βB)
= 1.266667
Applying the point values interval
law, we have
E(βP) = 2WA)E(βA)=
0.53, 1.06, 1.6, 2.13
E(βP) = 2WBE(βB)
= 1.27, 0.00, -1.27, -2.53
The beta of portfolio in order of
differences is:
E(βP) = (2WBE(βB)
+ 2WAE(βA))/2
=( 1.27+0.53)/2, (1.06+0.00)/2, (1.6-1.27)/2, (2.13-2.53)/2
This portfolio has an expected
return 10.67 percent and a betas of ( 1.27+0.53)/2, (1.06+0.00)/2,
(1.6-1.27)/2, (2.13-2.53)/2 that is less than 1.00, less
than 1.00, less than 1.00, less than 1.00, meaning that
this portfolio has less risk in order of differences.
REWARD OF MARKET LINE
Let us now tune our minds to see how
risk is rewarded in the marketplace. To begin, suppose that asset has an
expected return and beta of portfolio, the risk-free rate is Rf. The
risk-free asset, by definition, has no systematic risk (or unsystematic risk),
so risk-free asset has a beta of o. The reward market slope of security asset
is;
Slop=-Rf/2βi
THE RESULT
The situation have described for
asset A and B cannot persist in a well-organized, active market, because
investors would be attract to asset A and away from asset B. As a result, asset
A’s price would fall. Since price and returns move in opposite directions. The
result is that A’s expected return would decline and B’s would rise.
This buying and selling would
continue until the two assets plotted on exactly the same line.
To finish up, if we let E(Ri)
and βi stand for the expected return and beta,
respectively on any asset in the market, then we know that it must plot on
Reward Line(RL). As a result, we know that its reward-to-risk ratio is the same
as the overall markets;
-Rf/2βi=E(Rm)-Rf
E(Ri)= Rf/2
CONCLUSION
Existing Theory in this field of analysis is limited
because; people could not apply the existing concept to perfectly predict infinite
expected returns of portfolio in order differences or infinite betas of
portfolio in order of differences that proved important here. It could not
predict some infinite different possible portfolio expected returns and betas
by varying the percentages invested in two assets. The existing Theory on
assessment of systematic risk and beta cannot also use to perfectly predict
corresponding relationship between expected returns and betas in order of
differences to infinity.
This theory of mine is to perfectly
predict infinite expected returns of portfolio in order differences or infinite
betas of portfolio in order of differences. This theory is also possible for
predicting some infinite different possible portfolio expected returns and
betas by varying the percentages invested in two assets in order of their
differences. My theory also perfectly predicts corresponding relationship
between expected returns and betas in order of differences to infinity.
THIRD APPLICATION
CENTER OF MASS:
The Central(or Gravitational)
Point Theory has application to many practical situations in Physics.
Some of its practical areas are Center of Mass and Moment
of Initial. In practical situation it is convenient to regard thin
sheets of material, such as copper stripping, as two-dimensional. The area of
plane that represents a two-dimensional distribution of matter is called
lamina.
In general, however, substances are
not homogeneous and so the mass density is variable, suppose a lamina is
represented by a region R of the xy-plane and its mass
density ϱ= ϱ(x, y), varies continuously over the region R.
The center of mass (or center of
gravity) of a lamina, is defined as;
mxg = My …(1)
myg = Mx …(2)
The later is a purely
geometric property of the lamina and coincides with the center of mass in the
case of a variable density will almost always have its center of mass
off-center. The variable mass density of a lamina which is directly
proportional to the distance from the vertex opposite any side due to symmetric
character of the region R is given by
ϱ(x, y) =kxn + kyn ……..(3)
Also, the mass density of lamina which is directly proportional to the distance from the vertex opposite any side to non-symmetric character is given by
ϱ(x, y) = kxxn + kyyn ……(4)
Further analysis of the formulae above, we have
ϱ(x, y)*m = k(Mxn + Myn) …..(5)
ϱ(x, y)*m=kxMxn
+ kyMyn……..(6)
Given degree n, if
the degree is equal to 1, then for symmetric character, we have;
Mx = [ϱ(x, y)*m]/2k
……..(7)
My = [ϱ(x,
y)*m]/2k………(8)
Also, if the degree is equal to 1,
then for non-symmetric character, we have
Mx = [ϱ(x, y)*m]/2kx …….. .(9)
My = [ ϱ(x, y)*m]/2ky………(10)
If the mass density ϱ is directly proportional to the square (second degree) of the distance from the vertex opposite any side, then for symmetric character, we have
Mx = [√ϱ(x, y)*√m]/√2k
……..(11)
My = [√ϱ(x,
y)*√m]/√2k………(12)
For non-symmetric character, we have
Mx = [√ϱ(x, y)*√m]/√2kx
……..(13)
My = [√ϱ(x, y)*√m]/√2kx………(14)
EXAMPLE
Find the moment of the mass
with respect to x-axis and y-axis if the mass density and entire mass a lamina
is 3 and 8 in the shape of an isosceles right triangle to the square of
the distance from the vertex opposite the hypotenuse.
SOLUTION
The mass density of the lamina is
given as;
ϱ(x, y) = kxn + kyn
but ϱ(x, y)=3
Due to symmetric character of the
region R and the mass density we apply equation (7) and (8).
Mx=√(3*8)/√2k = 3.464/k
My=√(3*8)/√2k = 3.464/k
The center of mass (xg,
yg) must lie on the line y=x. Consequently, in equation (1) and (2),
we have
xg = Mx/m=0.43/k
yg = My/m=0.43/k
NOTE: The symbols kx, ky and k is denoted as non-symmetry constant in x-axis, non-symmetry constant in y-axis and symmetry constant respectively.
REFERENCE
*Adongo Ayine William(Me). Transcript(2008), posted(EMS-Bolgatanga Branch) to Mathematical Association of Ghana( Tittle: New Mathematical Concept..........,) in the year 2008.
*Rene Descartes.(1637). "The Geometry".
NOTE: The symbols kx, ky and k is denoted as non-symmetry constant in x-axis, non-symmetry constant in y-axis and symmetry constant respectively.
REFERENCE
*Adongo Ayine William(Me). Transcript(2008), posted(EMS-Bolgatanga Branch) to Mathematical Association of Ghana( Tittle: New Mathematical Concept..........,) in the year 2008.
*Rene Descartes.(1637). "The Geometry".
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