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first note new shape

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  • IICSE University
First Note on the New Shape of Sconvexity
Dr. M. R. Pinheiro
October 19, 2016
Abstract
In this note we copy the work we presented in Second Note on the
Shape of S-convexity, but apply the reasoning to one of the new limit-
ing curves, limiting curves we presented in Summary and Importance
of the Results Involving the Definition of S-Convexity. This is about
Possibility 1, second part of the definition, that is, the part that deals
with negative real functions. The first part has already been dealt with
in Second Note on the Shape of S-convexity.
MSC(2010): 26A51
Key-words: Analysis, Convexity, Definition, S-convexity, geometry, shape.
I. Introduction
In [8], we have decided to keep the name S1and replace the previous class
K1
swith a new version of it, which would be one of our possible definitions,
as for [9].
So far, we have:
S1
Definition 1. A function f:X><, where |f(x)|=f(x), is told to
belong to K1
sif, for each xXwe select, and for all of them, the inequality
f((1 λ)x+λ(x+δ))
Postal address: P.O. Box 12396 A’Beckett St, Melbourne, Victoria, Australia, 8006.
Electronic address: illmrpinheiro@gmail.com.
1
(1 λ)sf(x) + λsf(x+δ)
holds λ/λ [0,1]; s=s1/0< s1<1;X/X < ∧ X= [a, b];δ/0< δ
(bx).
Definition 2. A function f:X><, where |f(x)|=f(x), is told to
belong to K1
sif, for each xXwe select, and for all of them, the inequality
f((1 λ)x+λ(x+δ))
(1 λ)
log21
12sf(x) + λ
log21
12sf(x+δ)
holds λ/λ [0,1]; s=s1/0< s1<1;X/X < ∧ X= [a, b];δ/0< δ
(bx).
Remark 1.If the inequalities are obeyed in the reverse1situation by f, then
fis said to be s1concave.
We are now going to study continuity, arc length, and maximum height of
the just-mentioned limiting line. We have discarded the original definition
of K1
s[11].
In [8], we decided to use its name and symbols to designate another class of
functions because the intentions involved in the creation of this other class
of functions are the same.
II. Continuity
We now prove that the function
f(λ) = (1 λ)
log21
12sf(x) + λ
log21
12sf(x+δ)
is continuous through a few theorems from Real Analysis.
We know, for instance, that both the sum and the product of two con-
tinuous functions are continuous functions (see, for instance, [3]). Notice
that λ
log21
12sis continuous, given that 0 λ1 and 0 < s 1.
y1=f(x) and y2=f(x+δ) are constants, therefore could be seen as
constant functions, which are continuous functions. (1 λ)
log21
12s
1Reverse here means ‘>’, not ‘’.
2
is continuous due to the allowed values for λand s. As a consequence,
(1 λ)
log21
12sf(x) + λ
log21
12sf(x+δ) is continuous.
Notice that f(λ) = (1 λ)
log21
12sf(x) + λ
log21
12sf(x+δ) is C,
that is, is smooth (see [4], for instance).
Because the coefficients that form the convexity limiting line use 100% split
between the addends and form straight lines and the coefficients that form
the Sconvexity limiting line use more than 100% or 100% split between the
addends, given that (1 λ)
log21
12s≥ −(1 λ) and λ
log21
12s
λ(we are using the negativity of the function here), we know that the
limiting line for Sconvexity lies always above or over the limiting line for
convexity, and contains two points that always belong to both the convexity
and the sconvexity limiting lines (first and last or (x1;y1) and (x2;y2)).
We now have then proved, in a definite manner, also in the shape of a paper,
that our limiting line for the Sconvexity phenomenon is smooth, contin-
uous, and located above or over the limiting line for the convexity phe-
nomenon. Our Sconvexity limiting line should also be concave when seen
from the limiting convexity line for the same points ((x1;y1) and (x2;y2))
(taking away the cases in which y1=y2= 0 or s= 1).
III. Arc Length
Arc length is defined as the length along a curve,
sZγ
|dl|,
where dl is a differential displacement vector along a curve γ(see [6]).
In Cartesian coordinates, that means that the Arc Length of a curve is given
by
pZb
ap1 + f02(x)dx
whenever the curve is written in the shape r(x) = xˆx+f(x)ˆy.
Our limiting curve for Sconvexity could be expressed as a function of λin
the following way:
f(λ) = (1 λ)
log21
12sy1+λ
log21
12sy2.
3
In deriving the above function in terms of λ, we get:
f0(λ) = log21
12s(1 λ)
log21
2(12s)y1
+ log21
12sλ
log21
2(12s)y2.
With this, our arc length formula will return:
pZ1
0
v
u
u
u
t1+[log21
12s(1 λ)
log21
2(12s)y1+ log21
12sλ
log21
2(12s)y2]2dλ.
We will make use of a constant function, and we know that every constant
function is convex, therefore also S-convex (for every allowed value of s), to
study the limiting line for Sconvexity better.
We choose f(x) = 1 to work with (this function is suitable because |f(x)|=
| − 1|=1=f(x)).
We then have:
pZ1
0
v
u
u
u
t1 + [log21
12s(1 λ)
log21
2(12s)log21
12sλ
log21
2(12s)]2dλ.
Notice that s0pindeterminate and s>1p>1.
See:
Figure 1: Maple Plot, s=0.9
4
Notice that 0.25 will become 0.09 when raised to log21
120.5and its
supplement through the formula (1 λ), 0.75, will become 0.6.
In convexity, our results would have been 0.25 and 0.75 instead, that is, 64%
and 20% less in negativity is gotten with Sconvexity, respectively.
We now calculate the area under the curve by hand because Maple could not
compute it inside of an acceptable time interval: We notice that the vertex
of the graph that represents the function we are interested in is located on
(0.5; 1) in both cases. We can then draw a triangle on both sides of the
space we are interested in, and find an approximation to our target area.
After that, we can subtract an approximation to the piece of the triangle we
cannot consider. From eye observation, we can tell that the second leaf is
about half of the first, so that whatever we put for the first, we just halve it
for the second.
f(λ) = v
u
u
u
t1 + [log21
12s(1 λ)
log21
2(12s)log21
12sλ
log21
2(12s)]2
Figure 2: Maple Plot, s=0.5 Figure 3: Maple Plot, s=0.25
5
The top triangle of the first graph has base of size 0.5 and height of size 1.
The top triangle of the second graph has base of size 0.5 and height of size
of about 1.8. Base times height will then be equal to 0.5 and 0.9.
The area of the just-mentioned triangles is then 0.25 and 0.45, but we have
two of each. We then have 0.5 and 0.9, respectively, as areas for the top part
of each one of the graphs, but we still have to take away the leaves.
Upon using the grid option from Maple, we concluded that it would be fair
saying that approximately 5% of each superior triangle from the first graph
is above the curve, so that we have 95% of the original area left. That would
mean 95% of 0.25 twice or 0.475.
Similar reasoning leads to 0.6 for the second set, that is, 6x(0.5x0.1)x2 (ap-
proximation).
Now we have to add the rectangle we skipped when doing that, which is
what comes before the base line we drew: A rectangle of one unit of length
and one unit of height, and therefore a rectangle of area that is equal to 1.
To get the right approximation, we need to now put it all together: For
the figure 1, it will then be 0.475 + 1 = 1.475, and, for Figure 2, it will be
0.6 + 1 = 1.6.
We finish this section with a table containing three of the possible values
for sand their respective arc lengths (good approximations. Only the first
value is precise) for the situation in which f(x) is replaced with 1 in the
arc length formula:
sArc Length
1 1.3
0.5 1.475
0.25 1.6
IV. Maximum height
The maximum height of the sconvexity limiting curve is reached when
λ= 0.5 if fis constant and |f(x)|=f(x) because the first derivative of
the function describing the limiting line gives us zero for λ= 0.5 and changes
sign from positive to negative there.
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V. Conclusion
We here seem to have a more uniform limiting curve than the one we pre-
sented in Second Note because of the figures we get for the length. Our
studies on length are based on rough approximations, however.
We got 1,1.57, and 2 for the exponent swhen we used f(x) = 1.2. When
our exponent was 1
s, and our function was f(x) = 1, we got 1,1.57, and 2.
We now used the exponent log21
12s, the function f(x) = 1, and got
1.3,1.475, and 1.6.
All the conditions for us to have a proper extension seem to have been sat-
isfied, so that this definition seems to be as good as the one we had before
it,which is what we wanted to verify with this paper.
The replacement has been proposed because ideally we would have the same
height and shape for the limiting curve in both the negative and the non-
negative case, but such fact was not being verified with the previous defini-
tion, as seen in [8].
More studies are necessary to reach a final conclusion. Perhaps two more
papers.
We need to worry more about the details involved in all.
7
VI. References
[1] M. R. Pinheiro. Minima Domain Intervals and the Sconvexity, as well
as the Convexity, Phenomenon. Advances in Pure Mathematics, vol. 3,
2013.
[2] M. R. Pinheiro. First Note on the Definition of S2convexity. Advances
in Pure Mathematics, vol. 1, pp. 1-2, 2011.
[3] L. S. Hush. 1995. Continuous Functions. Visual Calculus.
http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html.
Accessed on the 8th of December of 2011.
[4] Planetmath authors. From
http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html. Ac-
cessed on the 9th of December of 2011.
[5] E. W. Weisstein. Indeterminate. From MathWorld - A Wolfram Web Re-
source. http://mathworld.wolfram.com/Indeterminate.html. Accessed on
the 12th of December of 2010.
[6] E. W. Weisstein. Arc Length. From MathWorld - A Wolfram Web Re-
source. http://mathworld.worlfram.com/ArcLength.html. Accessed on the
5th of December of 2010.
[7] G. P. Michon. Perimeter of an ellipse. From Numericana.com.
http://www.numericana.com/answer/ellipse.htm. Accessed on the 12th of
December of 2011.
[8] Pinheiro, M. R. Summary and importance of the results involving the
definition of S-Convexity, International Journal of Pure and Applied Math-
ematics, vol. 106, no. 3, pp. 699 − −713, 2016.
[9] Pinheiro, M. R. Third Note on the Shape of S-Convexity, International
Journal of Pure and Applied Mathematics, vol. 93, no. 5, pp. 729 − −739,
2014.
[10] Pinheiro, M. R. Second Note on the Shape of S-convexity, International
Journal of Pure and Applied Mathematics, vol. 92, no. 2, pp. 297–303, 2014.
8
[11] Pinheiro, M. R. Second Note on the Definition of S1-Convexity. Ad-
vances in Pure Mathematics, vol. 5, pp. 127ˆa130, 2015.
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Perimeter of an ellipse. From Numericana.com
  • G P Michon
G. P. Michon. Perimeter of an ellipse. From Numericana.com. http://www.numericana.com/answer/ellipse.htm. Accessed on the 12th of December of 2011.
Continuous Functions. Visual Calculus
  • L S Hush
L. S. Hush. 1995. Continuous Functions. Visual Calculus. http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html. Accessed on the 8th of December of 2011.
From http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html. Accessed on the 9th of
  • Planetmath Authors
Planetmath authors. From http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html. Accessed on the 9th of December of 2011.