Kuan Peng

Hilbert's Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true.

The apparent non-gravitational acceleration that the extra-solar-system 'Oumuamua 6 exhibits is puzzling. We find that when the position and velocity of the Sun are correctly set in 7 computing the predicted orbit, 'Oumuamua's trajectory can be explained with gravity and we 8 have reproduced the unexpected gap by computation. We also propose to sea...

The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive.

Cantor's diagonal argument shows that ℝ is uncountable. But our analysis shows that ℝ is in fact the set of points on the number line which can be put into a list. We will explain what the property that supports uncountability really is and propose a new scheme to count points.

An analysis of Cantor's theory of uncountable sets: The logic of his proofs has some weaknesses. Cantor assumes for both his proofs that all real numbers (set R) are in a list (list L). Considering L as a set this assumption assumes R belongs to L. This makes the claim "a real number is constructed but is not in the list L" questionable. We propose...

Event horizon could have a thickness but be not a boundary.

The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics. The gravitational force on m approaches infinity near M. But the speed of m never reaches the speed of light c, which is true for any force however strong it is and for time of acceleration however long it is. How the...

Today's methods for computing orientation are quaternion and rotation matrix. However, their efficiencies are tarnished by the complexity of the rotation matrix and the counterintuitivity of quaternion. A better method is presented here. It uses complex multiplication for rotating vectors in 3D space and can compute orientation without angle and tr...

Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional compl...

The orientation of body in space is defined 3 by angles. The step by step rotation process and chain of three-dots multiplication give an easy way to compute pile of rotations in 3D and high dimensional space and give a general orientation system. A visualization of quaternion is proposed.

Special relativity does not deal with acceleration, general relativity does not deal with non gravitational acceleration, which leave the theory of relativity imperfect. We will demonstrate some relativistic dynamical laws that specify relativistic acceleration, force and kinetic energy. Also, based on equivalence principle does gravitational mass...

The rotation curves of disc galaxies are flat and dark matter is speculated as explanation. Alternatively, the gravity of material disk could explain the flat curves. Using the gravitational force that a disk exerts on a body in the disk, we have computed the the rotation curves of disc galaxies and the curve of their mass densities. The numerical...

Orbital precession of the planet Mercury was computed using Space-Time curvature which is extremely difficult to understand. This article exposes an analytical orbit equation for relativistic gravity and explains how it is derived without Space-Time geodesics. This orbit equation makes relativity in gravitational field much easier to understand. Th...

To clearly explain the contradiction between the constant flow of ticks delivered by clocks and the relativistic time dilation.

I find that in theory the weird Speed Boost of the interstellar object 'Oumuamua should be 0.217 mm/s above the prediction and that 'Oumuamua should slow down less than prediction, in proportion of which the difference is 4.28×10-8 near the Sun. For Pioneer anomaly I have computed the gap between real and predicted acceleration and found the value...

Like in Newtonian kinematics, the relativistic change of reference frame must be a vector system of transformation laws for position, velocity and acceleration. In special relativity, when changing the reference frame the coordinates of a moving point is transformed using Lorentz transformation. But the velocity-addition formula that transforms vel...

The theory of Time relativity explains well the weird behavior of the interstellar object 'Oumuamua. I find that the real solar mass is slightly higher than today's value, which caused the mysterious Speed Boost of which the value should be 0.217 mm/s above the prediction at perihelion. Time relativity confirms that 'Oumuamua should slow down less...

Analytical derivation of relativistic velocity, mass, momentum and kinetic energy of an accelerated object.

The relativistic transformation of velocity creates a discrepancy. A discrepancy-free transformation of velocity is derived using the Time relativity transformation of coordinates.

Without length contraction, time relativity transformation solves paradoxes and explains incongruent relativistic experiments.

There are 2 types of length contraction: Object contraction and Distance contraction. Each has a different physical meaning.

Einstein's derivation of the Lorentz Transformation is purely theoretical. This study shows how it is related to the physical phenomenon of time dilation and length contraction.

Relativity of simultaneity destroys synchronization of GPS satellites.

Relativity of simultaneity can be measured with clocks of GPS satellites.

Astrophysical jets are flows of matter that moves at relativistic speed. They are opportunity to see length contraction in action. An astrophysical jet is analyzed to explain the length contraction effect on it.

Relativistic length contraction is theoretically predicted but not directly tested, which lead to incorrect interpretation of the theory illustrated by Bell's spaceship paradox and Ehrenfest paradox. But these paradoxes can help us designing experiments to test length contraction.

We analyze the mathematical mechanism that slows the time of the traveler in the twin paradox and explain what distinguishes the traveler's frame from the Earth's frame.

When counting a set, we can plot a graphic that represents the members of the set on the plane (x, y) to observe visually the counting. Also, graphic of counting of infinite set helps us to understand infinite natural number.

A counting set is the set of natural numbers with which a countable set is put in bijection. Is the counting set for the power set of ℕ the set that builds it?

Cantor's theorem states that the power set of ℕ is uncountable. This article carefully analyzes this proof to clarify its logical reasoning.

Theoretical explanation of longitudinal magnetic force and its practical application in high field magnet. Although longitudinal force is not explained in classical theory, its action has been demonstrated by several experiments long time ago. For example Nasilowski effect. But why is it not recognized in theory? The reason is that it shows no sign...

Theoretical explanation of tangential magnetic force and the experiment of rotating coil. Tangential magnetic force is tangent to the current on which it acts. For the classical theory this force does not exist. However, my experiment « Continuous rotation of a circular coil experiment » showed that a force tangent to the current must be there. If...

Plasma is confined in fusion reactors using magnetic force. Coulomb magnetic force law for plasma is derived.

The relativistic length contraction effect and changing distance effect produce 2 different magnetic forces. Together they form complete magnetic force.

The motion of electrons in current modifies Coulomb force between charges by changing the distance between them. This effect creates magnetic force.

Experiment that shows continuous rotation of a coil in its plane revealing the action of a magnetic force that is parallel to current

Tangential magnetic force is shown in this experiment.

Derivation of the expression for the magnetic force between parallel current elements from relativistic length contraction formula.

Formula for magnetic force between 2 arbitrary current elements derived from Coulomb's law and relativistic length contraction formula.

Creation of binary lists, discussion about the power set of ℕ, the diagonal argument, Cantor's first proof and uncountability.

Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.

This article discusses the uncountability of the power set of ℕ proven by using the out-indexes subset contradiction.

This article clarifies the kind of infinity used to quantify the number of digits of irrational numbers and try to check the cardinality of decimal numbers. The value of a decimal number depends on the number of its digits. For irrational numbers that have infinity of digits, their values seem to be definitive. However, the meaning of infinity is a...

This article uncovers a hidden assumption that the diagonal argument needs, then, explains its implications in matter of infinity. The use of the diagonal digits imposes a condition unnoticed until now. If this assumption were found false, the conclusion of the diagonal argument should be rewritten.

This article explains why the cardinality of a set must be either Aleph0 or |ℝ|. 1. Rational numbers are discrete 2. Real numbers are continuous 3. Collectively exhaustive and mutually exclusive events 4. Continuum hypothesis 5. Cardinality of discontinuous subsets of real numbers

1. Counting the fractional binary numbers 2. Fractional binary numbers on the real line 3. Countability of BF 4. Set of all binary numbers, B 5. On Cantor's diagonal argument 6. On Cantor's theorem 7. On infinite digital expansion of irrational number 8. On the continuum hypothesis

This experiment shows the magnetic force on a coil exerted by the magnetic field of a long solenoid that should be zero. Aharonov–Bohm effect shows that electron beam can be deflected in the middle plane of a long solenoid where the magnetic field is zero for classical electromagnetic theory. If the deflection is due to a magnetic force, it would p...

This theorem states that for any n except 2, the equation + = is not true for all positive integer triplets X, Y and Z. Fermat's " I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. " has fascinated mathematicians from 1637 but no one has found what his proof was. Let us try to understand this theorem bet...

The magnetic field of the earth can rotate a flat coil in its plane. This is explained in Earth's magnetic field and parallel action. Will a disc magnet rotate a flat coil the same way? The magnetic field of a disc magnet is central symmetric and the field lines are contained in median planes (See Figure 1). If a flat coil is coplanar with a median...

In Q: Parallel action with a solenoid being zero, will it make a coil rotate in its plan? This experiment a long solenoid. I have done this experiment and here is my result.

The magnetic field of the earth is uniform on its surface. The resultant Lorentz force a uniform magnetic field exerts on a coil of any shape is zero. The torque perpendicular to a flat coil is also zero. So, a current carrying coil in the magnetic field of the earth should stay immobile. However, my experiment shows that the test coil rotates in i...

Nearly two years ago I proposed a CRT experiment to test the magnetic field of a solenoid, see Non-Lorentzian Magnetic force and Aharonov-Bohm effect in CRT, Blogspot, Word published June-27-2013. Since its consequence is great (see Consequences of macroscopic Aharonov-Bohm effect, Blogspot, word), I thought glory-searching physicists would rush on...

The Induced conductor net problem could be solved if we can determine the induced voltage in part of the circuit. But is it possible that a voltage exist in no-loop wire? Let us see what the induced voltage is in a circular loop formed with 4 segments separated by resistors (Figure 1).

My article Coil and resistor induction paradox shows that the potential energy of electrons in an induced coil is not correctly described by Faraday's law. This is illustrated by Figure 1 in which a circular circuit with a resistor A is induced in a magnetic field; the resistance of the wire P is negligible. The inconsistency is that the electric f...

In my article Faraday's Law Paradox, I have explained that in a magnetically induced coil the line integral of electric field is paradoxically zero in contraction with Faraday's law. This paradox has aroused a long discussion which also reveals that the air gap between the coil's terminals confused the understanding of the paradox. The closed coil...