(SOLUTIONS FOR THE MILLENNIUM SEVEN PRIZE PROBLEMS)

A NOTICE BOARD(SOLUTIONS FOR THE MILLENNIUM SEVEN PRIZE PROBLEMS)

LABORATORY FOR MEASUREMENT OF ROTATIONAL ELECTROMAGNETIC WAVEIN MECHANICAL ENGINEERING DEPARTMENT OF AJOU UNIVERSITY SOUTH KOREA e-mail : [email protected]

1. We have apparatus for measuring life energy(rotational electromagnetic wave) for health relating products.

Patent ➀ South Korea 10-0631869➁ U.S.A. US 7,286,228 B2

2. We have the relating research papers.➀ Solution for the millennium seven(prize) problem http://blog.daum.net/hkoh/2 http://m.blog.daum.net/hkoh/2

➁ Many research papers about the rotating electromagnetic waves

➂ Gaussian Distribution, Life Energies http://blog.naver.com/hkoh10

Published Papers

(1) Exact Analytical Solution for Partial Differential Equilibrium Equations (Exact Solution of

Navier – Stokes’s Equation)

Hung – Kuk Oh, Yohan Oh, Jeunghyun Oh

Journal of Applied subtle Energy, 2013, Vol.11, No.1, PP 23~26

(2) Solutions for the Millennium Seven Problems

Hung – Kuk Oh, Yohan Oh, Jeunghyun Oh

Journal of Applied Subtle Energy, 2013. Vol.11, No.1, PP 12~15

Proceedings

(1) Hungkuk Oh, Yohan Oh, Jeunghyun Oh, “Exact Analytical Solution for Partial Differential

Equilibrium Equation (exact solution for Navier – Stokes’s equation)” , Proc. Appl. Subtle

Energy Vol.11, No.1, June 2013

(2) Hungkuk Oh, Yohan Oh, Jeunghyun Oh, “Solutions for the Millennium Seven Problem”,

Proc. Appl. Subtle Energy, Vol.11, No.1, June 2013


Equilibrium Equation (exact solution for Navier – Stokes’s equation)”, Proceedings of the

Korean Society for Industrial and Applied Mathematics, Vol.8, No.1, May 2013


Equilibrium Equation”, International Congress of Mathematicians Seoul 2014 Abstracts

Short Communications page 346

Published books

(1) “Solution for the Millennium Seven Problem”, Hung-Kuk Oh,

[ebook] http://www.upaper.net/hkoh/1025942

(2) “Solution for the Millennium Seven Problem”, Hung-Kuk Oh,

[ebook] http://book.daum.net/detail/book.do?bookid=DGT480130001639P

(3) “ORIENTAL PHYSICS IS THE SAME AS SOLUTIONS FOR THE MILLENNIUM SEVEN

PROBLEMS IN VIEW OF MODERN PHYSICS”, Hung-Kuk Oh,

[ebook] http://www.upaper.net/hkoh/1028379

http://www.upaper.net/hkoh/1025942

http://book.daum.net/detail/book.do?bookid=DGT480130001639P

http://www.upaper.net/hkoh/1028379

EXACT ANALYTICAL SOLUTION FOR PARTIAL DIFFERENTIAL

EQUILIBRIUM EQUATIONS

(Exact Solution of Navier – Stokes’s Equation)

Hungkuk Oh1*

, Yohan Oh2, Jeunghyun Oh

2

1 Dept of Mechanical engineering, Ajou University South Korea

2 Dept of life Sciences, Ajou University South Korea

Non-quantum state particle gave kinetic energy by the inertia interaction, while quantum state particle stores its potential energy produced by atomic (or molecular) bondings.

Schrödinger equations are those for the non-quantum state particle.

The equation for quantum state particles are derived for non-steady state and steady state. General relativity is completed by deriving the equations for quantum state particles.

The two dimensional stress tensors in the partial differential equilibrium equations can be converted to one dimensional tensors per unit volume, which generate Laplacian.

The Laplacian has exact analytical solution and needs boundary conditions.

It gives us exact solution of Navier – Stokes’s equation.

1. Free Particle

We assume that for a particle moving

freely in the ＋x direction is specified by

)/( vxtiwAe ···········(1)

where : wave function,

w : angular velocity, t : time,

v : velocity in x direction

*Corresponding author: Tel: +82(31)-219-2523.

Fax: +82(31)-219-1610, E-mail: [email protected]

Replacing w in the above formula by v2

gives

)/(2 xvtiAe ·············(2)

where :frequency, : wave length, A :

constant

This is convenient since we already know what

v and are in terms of the total energy E and

moment p of the particle being described by

(ref.1).

Journal of Applied Subtle Energy,

2013. Vol. 11, No. 1, pp. 15-22

http://www.soase.org/

Because

2 hE

and pp

h

2

we have free particle

))(/( pxEtiAe ·······(3)

2. Non-quantum State Particle

2-1 Time dependent Schrödinger equation

We begin by differentiating equation(3) twice

with respect to x, which gives

2

222

xp

· ········(3-1)

Differentiating equation(3) with respect to t

gives

tiE

· ··········· (3-2)

At speeds small compared with that of light,

),(2

2

txUm

pE ·······(4)

Multiplying both sides of equation(4) by the

wave function gives

Um

pE

2

2

which gives time-dependent Schrödinger

equation

U

zyxmti )(

2 2

2

2

2

2

22 ····(5)

2-2 Steady-state Schrödinger Equation

Substituting equation (3-2) into equation(5)

0)(2

22

2

2

2

2

2

UE

m

zyx ····(6)

where : position dependent function.

3. Quantum State Particle and General

Relativity(Nagative mass and Positive

Mass)

General relativity equation between E and p is

Kcmmc 2

0

2 ··········(7)

)1( 00

r

rh

r

rhh ········(8)

Kpcr

rh )1( 0

22202 )1( cpr

rE ········(9)

where r

r0 : general relativity factor

Multiplying both sides of equation(9) by the

wave function gives

22202 )1( cpr

rE ·······(10)

Substituting equation(3-1) and equation(3-2)

into equation(10),

).()1(2

22220

xc

tir

rE

which gives

)11()(

)1(

2

2

2

2

2

222

20

zyxc

ti

r

rE

Substituting tiEe )/( into equation (11)

)12()(

)()1(

)/(

2

2

2

2

2

222

)/(20

tiE

tiE

ezyx

c

eiE

ir

rE

which gives

)13()(

)1(

2

2

2

2

2

222

202

zyxc

r

rE

Equation (11) is the general relativity quantum

mechanical time-dependent equation while

equation (13) is the general relativity quantum

mechanical steady-state equation.

4. Exact Analytical solution for Partial

Differential Equilibrium Equations and

Discussions (Exact Solution of Navier –

Stokes’s Equation)

The partial differential equilibrium equations

are as follows

𝜕σ𝑥𝑥

∂x+

𝜕τ𝑥𝑦

∂y+

𝜕τ𝑥𝑧

∂z= 0

𝜕τ𝑦𝑥

∂x+

𝜕σ𝑦𝑦

∂y+

𝜕τ𝑦𝑧

∂z= 0

𝜕τ𝑦𝑥

∂x+

𝜕τ𝑧𝑦

∂𝑦+

𝜕𝜎𝑧𝑧

∂z= 0 ············ (14)

Where σ and τ are axial and shear stresses.

The stresses are two dimensional tensors.

They can be converted into one dimensional

tensors as follows.

𝜕2Ψ

∂x2+

𝜕2Ψ

∂y2+

𝜕2Ψ

∂x2=0·····················(15)

The equation (13) becomes equation (15) when

(1 −𝑟0

𝑟) = 0.

The one dimensional tensor ψ can be as

follows.

ψ = 𝐽1 + 𝑖𝑖𝐽2····································(16)

where 𝐽1 = 𝜎1 + 𝜎2 + 𝜎3

𝐽2 =1

4{(𝜎1 − 𝜎2)

2 + (𝜎2 − 𝜎3)2 + (𝜎3 − 𝜎1)

2}

Where 1,2,3 are principal axes and 𝐽1 , 𝐽2

have spherical symmetries and are potential

energies per unit volume (ref.3).

The Laplacian has exact analytical solutions

and needs boundary conditions.

The strains are calculated from the flow rule

and Levy-Mises relations.

The ductile fracture criterion comes from 𝐽1

and 𝐽2 at any deformation zones (ref.4).

The states of stresses and strains can be found

from the exact analytical solutions of the

Laplacian.

The invariant stresses may be the dependent

variables of the Laplacian.

It gives us exact solution of Navier – Sokes’s

equation.

References

1. Arthur Beiser, concepts of modern physics,

fifth edidtion, Mcgraw-hill, Inc., page 165-170

2. Hungkuk Oh, Yohan Oh, Jeunghyun Oh, “General

Relativity and Quantum Mechanics, Journal of

Applied Micromagnetic Energies, 2007, vol.5, no.2,

pp14-15

3. Hungkuk Oh, Yohan Oh, Jeunghyun Oh, “Lecture

Development of Engineering Plasticity for 3-1

Undergraduate students (Invariant and Spherical

Symmetry), Journal of Applied Subtle Energy,

2012, vol.10, no.2, pp17-18

4. Hungkuk Oh, “A note on a proposed general

criterion for brittle and ductile fracture in

materials and structure, Journals of materials

processing technology 99(2000) 275-276

SOLUTIONS FOR THE MILLENNIUM SEVEN PROBLEMS

Hungkuk Oh1*

, Yohan Oh2, Jeunghyun Oh2

1 Dept of Mechanical engineering, Ajou University South Korea

2 Dept of life sciences, Ajou University South Korea

Navier - Stokes’s equation is resolved if the stresses are looked for from the energy balance equation.

Rieman hypothesis is resolved by the fact that the zero point energy includes many kinds of frequencies.

Birch – Swinnerton – Dyer Conjecture is resolved by the fact that the specific heat of hydrogen gas at low temperature has the zero point energy.

Poincare conjecture is resolved by the fact that energy have a duality between vector and complex numbers.

Hodge conjecture is resolved by the fact that space is composed of four rotational electro – magnetic waves.

P vs NP problem is resolved by the fact that space is composed of the four rotational electro – magnetic waves.

Yang – Mills theory and mass Gap Hypothesis is resolved by the fact that the two dot products at the same time between 𝑙𝑝 and 𝑟𝑝 and between 𝑙𝑛 and 𝑟𝑛 make the energy confinement and mass gap

in the dual plane(𝑙𝑝:leftwise positive , 𝑙𝑛:leftwise negative 𝑟𝑝:rightwise positive , 𝑟𝑛:rightwise negative).

1.Special Relativity and Energy Vectors

The kinetic energy of an object is equal to

the increase in its mass due to its relative

motion multiplied by the square of the speed

of light(c).

It may be written

…………………(1)

Where m is the total mass, m0 is the rest

mass, and k is the kinetic energy of the object

(ref.1).

*Corresponding author: Tel: +82(31)-219-2523

Fax: +82(31)-219-1610, E-mail: [email protected]

The equation is integrated along the

momentum direction.

We can derive from the relativistic formulas for

the total energy( mc2 ) and the linear

momentum(p)

……………(2)

If we let

jieiieE

21

jipiipP

21 ··········(3)

jikiikK

21

KPE

········(4)

Journal of Applied Subtle Energy,

2013. Vol. 11, No. 1, pp. 23-26

http://www.soase.org/

0KP

·······(5)

The equation (1) is

jicmmciimcE

2

0

22 2

jicmiicmP

2

0

2

0 ····(6)

jkiikiK

The equation (2) is

jpciiicmE

2

0

jiiicmP

02

0 ······(7)

jpciiiK

0

We can seek for life energies as follows

(ref.2) rotational electromagnetic waves.

From equation (6),

leftwise positive + rightwise positive,

leftwise negative + rightwise negative,

leftwise positive + leftwise negative,

rightwise positive + rightwise negative.

From equation (7),

leftwise positive, rightwise positive,

leftwise negative, rightwise negative.

2. Zero Point Energy

Zero-point energy is the lowest possible

energy that a quantum mechanical physical

system may have and it is energy of its

ground state.

All quantum mechanical systems undergo

fluctuations even in their ground state and

have an associated zero-point energy, a

consequence of the Heisenberg cencertainty

principle.

Albert Einstein and Otto Stern(1913)

published a paper of great significance in

which they suggested for the first time

the existence of a residual energy that all

oscillators have at absolute zero. They called

this residual energy as zero-point energy. They

carried out an analysis of the specific heat of

hydrogen gas at low temperature, and

concluded that the data are best represented

if the vibrational energy is (ref.2)

21

hv

e

hvE

Tkhvb

······(8)

According to this expression, an atom system

at absolute zero retains an energy of hv2

1.

An energy vector is organized for equation(8)

as follows

ijhviiE

2

1

ij

e

hvhvii

e

hvhvP

Tkhv

Tkhv

bb

12

1

2

1

12

1

2

1

ij

e

hvii

e

hvK

Tkhv

Tkhv

bb 12

1

12

1

………(9)

where E

:total energy vector

ii, ij: unit rectangular vectors

h : Plank constant

v: frequencies

r0: 02 r (wave length)

r: radial distance

kb: Boltzman constant

T: absolute temperature

hv2

1 is derived from

r

rhv 0

as follows

hvr

rhv

r

rd

r

rhv

2

1

2

1 1

0

2

01

0

00

··(10)

where r change from to r0.

We knows that r

rhv 0

is a potential energy,

which is converted from a kinetic energy.

We can conclude from equation(9) that space

is composed of 4 kinds of rotational

electromagnetic waves (leftwise positive, leftwise

negative, rightwise positive and right negative).

3. Electromagnetic waves

Electromagnetic wave is a motion of a

harmonic free particle.

It retains conservations of momentum and

energy. It shows that the electromagnetic

wave can convert the kinetic energy into a

potential energy by energy vector each other.

The potential energy propagates from

rightwise to leftwise or from leftwise to

rightwise as in Fig.1 (ref.6).

The energy vector for an electromagnetic

wave is as follows

ijhviiE 0

ijr

rhvhvii

r

rhvhvP

00

2

1

2

1

ijhvhviiK2

1

2

1

……..(9)

Fig. 1 Representation of the electric field ( line ) and

the magnetic field ( dots and crosses ) in a plane

containing and oscillating electric dipole.

During one period the inner loop moves out and

expands to become the outer loop.

The total energy of a electromagnetic energy

is hv only and also its static mass is zero,

which is called a harmonic free particle motion.

4. The Millernium Seven Problems

4-1 Navier – Stokes’s Equation

The dependent variables(mass velocity and

stress) have no relationships between them.

The stresses are looked for from the quantum

mechanical equations (ref.9). If the stresses are

resolved from outside independently, Navier –

Stokes’s equations can be solved very easily.

The invariant stresses have spherical symmetry

and therefore they can be searched for from

the quantum mechanical equation very easily

and boundary conditions. The engineering

solutions needed an approximation of the

stress distributions. The axial and shear

stresses are calculated from the invaniants and

principal values. The velocities can also be

given from the Newton’s laws of friction.

Navier – Stokes’s equation is one of force

distribution while quantum mechanics is one

of energy distribution. Stresses are energies

per unit volume. The energy state is at

P=0 ,x=x(or 𝑖𝑥 = 𝑖𝑥)in later eq.(11).

4-2 Rieman Hypothesis

Space is composed of four rotational electro

– magnetic waves and life energies is also

composed of four rotational electrongentic

waves. Wave function (probability function in

quantum mechanics) is given as follows.

Ψ = 𝑒(−𝑖ℏ)(𝐸𝑡−𝑝𝑥)

······……………………(11)

Ψ : wave function(probability function)

ℏ : h/2π ( P lanck constant)

E : total energy

p : momentum

χ : position

The four rotational electro – magnetic waves

have zero momentum.(P=0 or x=0)

Ψ = 𝑒−(𝑖

ℏ)(𝐸𝑡)

…(12)

The total energy(E) has complex quantity.

E =1

2ℎ𝑣 + 𝑖

ℎ𝑣

𝑒ℎ𝑣/𝑘𝑏𝑇 − 1

······……………………(13)

Where 𝑘𝑏 and T are Boltman constant and

absolute temperature.

The 1

2ℎ𝑣 in equ(13) is come from equation(10).

Ψ = 𝑒−(

2𝜋𝑖ℎ

)(12ℎ𝑣+𝑖

ℎ𝑣

𝑒ℎ𝑣/𝑘𝑏𝑇−1)𝑡

······……………………(14)

Ψ = (1

1𝑠+

1

2𝑠+

1

3𝑠+

1

4𝑠+

1

5𝑠+ ⋯)

······……………………(15)

Where S =1

2+ 𝑖

1

𝑒ℎ𝑣/𝑘𝑏𝑇−1

The complex number(S) has always the real

number (1

2) at Ψ=0.

The time (t) in equation (14) is the inward time

of the composed rotational electro – magnetic

waves.

This is the origin of the third one in Newton’s

laws of motion , which is originated from the

structure of space.

Numbers are orginated from the space

structure.

It is a meeting between physics and

mathematics. They have the same origin.

4-3 Birch – Swinnerton – Dyer conjecture

The elliptical rational numbers are from The

eqation(13) in the space can be written again

as follows.

E = hv(1

2+ 𝑖

1

𝑒ℎ𝑣𝑘𝑏𝑇 − 1

)

······……………………(16)

While the parameter T(absolute temperature)

in eq.(16) varies between zero and infinite in

the space, the elliptical rational numbers can

vary infinitely as in(eq.15) at the elliptical

curve(eq.(11) at P=0 and x≠0 or 𝑖𝑥 ≠ 0 ).

4-4 Poincare Conjecture

The E in eq.(11) can be rewritten as follows.

E⃗⃗ = ℎ𝑣 (𝑟𝑜𝑟) 𝑖𝑖 + ℎ𝑣 (1 −

𝑟𝑜𝑟) 𝑗𝑗

E = h𝑣 (ror) + 𝑖ℎ𝑣(−

𝑟𝑜𝑟)

······……………………(17)

If r has infinity , it has straight line motion. If

r has any value between 𝑟𝑜 and infinity, it has

curved motion. If 𝑟𝑜 has any value between

𝑟𝑜 and zero, it makes an approach to the point.

If 𝑟𝑜 returns from zero to 𝑟𝑜 , it returns from

the point to any sphere.

Poincare conjecture is resolved by the fact that

energy has a duality between vector and

complex number (ref.5,ref.12). The three cases

are equal,(P=0,X=0),(P≠0 or 𝑖𝑃 ≠ 0 ,X=0) and

(P=0,X≠0 or 𝑖𝑥 ≠ 0) by eq.(11) and eq.(15).

4-5 Hodge Conjecture

Space is composed of four rotational electro –

magnetic waves and then any configuration(ge

ometry) in the space can be represented by a

combination of the four rotational electro -

magnetic waves(eq.(10), eq.(11), eq.(14), eq.(15)

and eq.(16)). Any geometry can be configured

by eq.(15).

4-6 P vs NP Problem

Because space is composed of four rotational

electro – magnetic waves, any recognized easy

problems in the space may naturally be easy

ones originally. Any events, occurrences and

happenings are composed of the four

rotational electro – magnetic waves, and also

any geometric configurations in the space are

composed of the four rotational electro –

magnetic waves. Any thinking , idea and spirit

are composed of the four rotational electro –

magnetic waves. The four rotational electro –

magnetic waves can be expressed by

mathematical methods(eq.(10), eq.(11), eq.(14),

eq.(15) and eq.(16)). Any NP can be expressed

into P by(eq.15). NP only repeats P.

4-7 Yang – Mills Theory and Mass Gap

Hypothesis

Vector dot product between leftwise positive

rew(rotational electro – magnetic wave) and

rightwise positive one induces space

contraction, while the one between leftwise

negative one and rightwise negative one

makes space expansion. The two dot products

(Fig.2)at the same time makes mass gap and

energy confinement(ref.5,ref.6,ref.8,ref.9,ref.11).

The confined rotational energies have

spherical symmetry (the quamtum mechanical

structure)(ref.5) and greatly contracted

energy(ref.5,ref.9,ref.11).

Fig.2 dual plane between vector and complex

number in the rotational electro – magnetic

waves

The mass gap is between X=0, P=P(or 𝑖𝑝 = 𝑖𝑝)

and X=0, P=0. It repeats Poincare conjecture.

Solution for Yang – Mills theory is in ref.9. The

minimum energy level exists(ref.5, ref.10). Any

particle has integer electric charge by eq.(15).

5. Discussions and conclusion

Space is composed of four rotational electro

– magnetic waves. Matter and spirit are

therefore composed of them. Seven

millennium problems may be resolved if

mathematics and numbers are expressed by

the energies and space structure. Life

energies(four rotational electro – magnetic

waves.) are derived from the special nelativity.

They have dual property (vector and complex

number).

The zero point energy includes many kinds of

frequencies.

Navier – Stokes’s equation is resolved if the

stresses are looked for from the energy

balance equation.

Rieman hypothesis is resolved by the fact that

the zero point energy include many kinds of

frequencies.

Birch – Swinnerton – Dyer conjecture is

resolved by the fact that the specific heat of

hydrogen gas at low temperature has the zero

point energy.

Poincare conjecture is resolved by the fact that

(+)𝑙𝑝

(-i)𝑙𝑛

(i)𝑟𝑝

(-)𝑟𝑝

(negative imaginary axis) (positive imaginary axis)

(negative real axis)

(positive real axis)

𝑙𝑝:leftwise positive

𝑟𝑝 :rightwise positive

𝑙𝑛:leftwise negative

𝑟𝑛:rjghtwise negative

energy have a duality between vector and

complex numbers.

Hodge conjecture is resolved by the fact that

space is composed of four rotational electro –

magnetic waves.

P vs NP Problem is resolved by the fact that

space is composed of the four rotational

electro – magnetic waves.

Yang – Mills theory and Mass Gap Hypothesis

are resolved by the fact that the two dot

products at the same time between 𝑙𝑝 and 𝑟𝑝

and between 𝑙𝑛 and 𝑟𝑛 make the energy

confinement and mass gap in the dual

plane(𝑙𝑝:leftwise positive , 𝑙𝑛:leftwise negative ,

𝑟𝑝 :rightwise positive , 𝑟𝑛:rightwise negative).

Reference

1. Hung – Kuk Oh, Yohan Oh, Jeunghyun Oh,

2007, ”Nature of light and Energy

Vector(Connection Between Modern Physics

and life Energies”, Journal of Applied

Micromagnetic Energy , Vol.5, No.2, PP.1~6

2. Hung – Kuk Oh, Yohan Oh , Jeunghyun Oh,

2007, “measuring and characterization of Eight

kinds of Rotational Electro – Magnetic

Waves(Life Energies), “Jounal of Applied Micro

– Magnetic Energy, Vol.5, No.2, PP.24~29

3. Hung – Kuk Oh, “Method and Apparatus for

measuring circularly polarized rotating electro

– magnetic wave using magnetic field”, Patent

No.10 – 0631869, South Korea, US 7,286,228

B2 USA.

4. Hung – Kuk Oh, Yohan Oh, Jeunghyun Oh,

“New Method for making Starch”, Journal of

Applied Subtle Energy, 2010, Vol.8, No.2,

PP.25~27

5. Hung – Kuk Oh, 1999, “some observation

on the cavity of creation for cold fusion and

the generation of heat”, Journal of material

processing technology”, Vol.94, PP.60~65

6. Hungkuk Oh, Yohan Oh, Jeunghyun Oh,

“Space is composed of four kinds of rotational

electro – magnetic waves”, Journal of Applied

subtle Energy, 2011, Vol.9, No.2, PP.6~9

7. HungKuk Oh, Yohan Oh ,Jeunghyun

Oh, ”relationship Between Modern Physics and

Hado, Naksue and Eight Elements of the Book

of changes, Taekuk and rotational electro –

magnetic waves”, Journal of Applied Subtle

Energy, 2012, Vol.10,No.1, PP.31~34


“Duality Between Vector and complex

Numbers in Rotational Electro – magnetic

waves”, Journal of Applied subtle Energy”,

2012, Vol.10 , No.1, PP.19~22


“Exact Solution for Partial Differential Equation

(exact solution for Navier – Stokes’s

equation)” , Journal of Applied Subtle Energy,

2013, Vol.11, No.1


“Quark does not appear because it has not

integer electric charge of electron”, Journal of

Applied Subtle Energy, 2012, Vol.10, No.2,

PP.14~16


“General Relativity and Quantum Mechanics”,

Journal of Applied Micromagnetic Energy,

2007, Vol.5, No.2, PP.14~15

(SOLUTIONS FOR THE MILLENNIUM SEVEN PRIZE PROBLEMS)

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