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.

PDF Graphic of set counting and infinite number https://pengkuanonmaths.blogspot.com/2018/11/graphic-of-set-counting-and-infinite.html

or Word https://www.academia.edu/37766761/Graphic_of_set_counting_and_infinite_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?

Please read the article at

PDF Building set and counting set https://pengkuanonmaths.blogspot.com/2018/10/building-set-and-counting-set.html

or Word https://www.academia.edu/37590687/Building_set_and_counting_set

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

Please read the article at

PDF Analysis of the proof of Cantor’s theorem http://pengkuanonmaths.blogspot.com/2018/09/analysis-of-proof-of-cantors-theorem.html
or Word https://www.academia.edu/37356452/Analysis_of_the_proof_of_Cantors_theorem

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 significant effect on practical devices, so no physicist is interested in exploring these experiments. But I have found a huge effect of longitudinal force in high field resistive magnets which could improve their performance.

Please read the article at
Longitudinal magnetic force and high field magnet
PDF http://pengkuanem.blogspot.com/2018/06/longitudinal-magnetic-force-and-high.html
or
Word https://www.academia.edu/36787024/Longitudinal_magnetic_force_and_high_field_magnet

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 tangential magnetic force exists, why was it not detected in almost 200 years?

Please read the article at
Showing tangential magnetic force by experiment
PDF http://pengkuanem.blogspot.com/2018/05/showing-tangential-magnetic-force-by.html
or
Word https://www.academia.edu/36652163/Showing_tangential_magnetic_force_by_experiment

Nuclear fusion reactors use strong magnetic field to confine plasma in reaction chambers. The magnetic field is so designed that plasma should follow field lines which do not encounter the chambers’ wall. But it seems that a mysterious force pushes plasma off its track. For explaining this force, Coulomb magnetic force law for plasma is derived.

Please read the article at
Plasma under Coulomb magnetic force
PDF http://pengkuanem.blogspot.com/2018/04/plasma-under-coulomb-magnetic-force.html
or
Word https://www.academia.edu/36379490/Plasma_under_Coulomb_magnetic_force

The relativistic length contraction effect and changing distance effect produce 2 different magnetic forces. Together they form complete magnetic force.
I have derived 2 magnetic forces with Coulomb’s law and charges’ velocity. The first force dFlc is derived in «Length-contraction magnetic-force between arbitrary currents». The second force dFcd is derived in «Changing distance effect». dFlc and dFcd are added together to give the expression for complete magnetic force dFcm.

Please read the article at
Coulomb magnetic force
PDF http://pengkuanem.blogspot.com/2018/03/coulomb-magnetic-force.html
or
Word https://www.academia.edu/36278169/Coulomb_magnetic_force

PDF Changing distance effect http://pengkuanem.blogspot.com/2018/03/changing-distance-effect.html or
Word https://www.academia.edu/36272940/Changing_distance_effect

There is a long standing debate about whether tangential magnetic force exists. In «Tangential magnetic force experiment with circular coil» I discussed this force and presented an experiment that showed the action of this force. But, as the rotation of the coil in that experiment was limited to a small angle, it does not show that tangential force exists all over the coil. So, I have carried out the present experiment that shows continuous rotation of the coil to make clear that tangential force has the same value around the coil

Please read the article at

PDF Continuous rotation of a circular coil experiment http://pengkuanem.blogspot.com/2017/06/continuous-rotation-of-circular-coil.html
or Word with video https://www.academia.edu/33604205/Continuous_rotation_of_a_circular_coil_experiment

If magnetic force is to respect Newton’s third law, there should be a recoil force on the vertical current which is Ft. This force is tangent to the current I1 and called tangential magnetic force. Some physicists claim that tangential magnetic force exists, this claim is supported by some experiments such as the rail gun recoil force shown by Peter Graneau and Ampère’s hairpin experiment, see Lars Johansson’s paper. But these experiments did not convince the main stream physicists and tangential magnetic force is rejected. I have carried out an experiment to show tangential magnetic force acting on a circular coil.
Please read the article at

PDF Tangential magnetic force experiment with circular coil http://pengkuanem.blogspot.com/2017/06/tangential-magnetic-force-experiment.html
or Word with video https://www.academia.edu/33353400/Tangential_magnetic_force_experiment_with_circular_coil_with_video_

In ≪Relativistic length contraction and magnetic force≫ I have explained the mechanism of creation of magnetic force from Coulomb force and relativistic length contraction. For facilitating the understanding of this mechanism I used parallel current elements because the lengths are contracted in the direction of the currents. But real currents are rarely parallel, for example, dIa and dIb of the two circuits in Figure 1. For correctly applying length contraction on currents in any direction, we will consider conductor wires in their volume and apply length contraction on volume elements of the wires.
Please read the article at

PDF Length-contraction-magnetic-force between arbitrary currents http://pengkuanem.blogspot.com/2017/05/length-contraction-magnetic-force.html
or Word https://www.academia.edu/32815401/Length-contraction-magnetic-force_between_arbitrary_currents