1

New Theories / What makes NP problems hard to solve?

« on: 16/01/2025 11:52:47 »

In my earlier thread, I discussed the problem of Riemann's hypothesis, which is an unsolved math problem to the time it was started.

The Essentials of Problem Solving

An introduction to the psychology of problem solving. Featured problems: the towers of Hanoi, the Chinese ring puzzle, the Wason 4-card selection task, the candle problem, Roman matchstick problems, and toothpick shape problems.

00:00 A quick note
00:47 The problem state space and the towers of Hanoi
4:45 Problems of representation and the Chinese ring puzzle
6:42 Context and variations of the Wason 4-card selection task
9:42 Introduction to insight problems: the candle problem
11:05 Differences between insight and incremental problems
12:15 Barriers to insight: Roman matchstick problems
17:30 Insight problems: too big of a distinction?
19:08 Well-structured and ill-structured problems
21:11 Representation and argument
23:34 Becoming a better problem solver: toothpick problems
26:45 Domain-specific knowledge and strategy change
30:55 What transfers across problem-solving domains?

IMO, the biggest factor in solving the Riemann hypothesis is about barriers to insight.
The problem is well structured. There's some problems of representation, especially in the past, where computer aided visualization wasn't a thing. Although the state space is infinite, it must not be the main difficulty here, since other problems involving infinity have been solved before.

P vs NP problem is another unsolved math problem.  They are two classifications of problems.
P problems are easy to solve and verify.
NP problems are hard to solve but easy to verify.
What makes NP problems hard to solve?

IMO, the biggest factor in determining P vs NP problem is because the problem isn't well structured/defined. What's the definitive boundary between the two classes?

2

New Theories / What's the unit of Torque?

« on: 09/08/2024 03:27:10 »

This is a spin-off of my other thread. I thought the discussion would be short and would quickly come into conclusion, but it seems I was not the case. Let's discuss into more details here.

I asked Gemini for other Physical Measures with the Same Units

Excellent question! There are several other physical quantities that share the same units, even though they represent different concepts. Here are a few examples:
...
2. Torque and Energy
 * Both measured in newton-meters (Nm) or joules (J).
 * Torque is a rotational force, while energy is the capacity to do work.
...

While the other points are basically pairs of the same measures, the second point tells a different story. There is something missing here.

I told Gemini that energy is torque times the angle of rotation.

You're absolutely right!
Energy = Torque x Angle of Rotation
This equation is a fundamental relationship in rotational mechanics. Here's a breakdown:
 * Torque: A measure of the force that can cause an object to rotate around an axis. It's analogous to force in linear motion.
 * Angle of rotation: The amount of rotation an object undergoes around an axis.
When you multiply torque by the angle of rotation, you get the work done, which is a form of energy.
Key points:
 * This equation holds true for constant torque. If the torque varies, integration is required.
 * The units of torque (Newton-meter) multiplied by the unit of angle (radians) give the unit of energy (joules).

It means that the proper unit for torque is Joule/radian, or Newton meter/radian.
I found many people being confused about measurement units involving radian, which is often omitted for being a dimensionless quantity.
The unit radian for measuring angle of rotation is convenient since the traveling distance of a point due to 1 radian rotation equals the radius of rotation.

https://en.wikipedia.org/wiki/Radian

An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.

3

Just Chat! / Where do I get my confidence to challenge established theories?

« on: 14/02/2024 12:41:23 »

To be clear, everything I wrote here should not in any way change the validity of what I wrote in the other threads. They have their own independent justifications and falsifications.

I see the need for posting this thread to give a story behind what I've written in the other threads. Simply posting facts and figures do not seem enough to get people's attention and gain traction.

Usually, people are interested in something with story, especially when it's based on real life or related to their own lives.

Those who've read my threads in this forum may wonder, where do I get my confidence to challenge established theories, especially in physics, philosophy, and mathematics?

This thread is an anticipation to what may happen when my hypotheses and their justifications somehow get more attention and acceptance among science community and wider audience.

I'd like to prevent the spread of wild speculations and misinformation.\\

Like most people, I learned about science, technology, and philosophy from school/college, textbooks, and online sources, including wikipedia, quora, science web sites, and science forums.

To understand things, I tried to make a mental map of knowledge nodes with relationships among them. Once in a while I found something didn't add up. Something must be missing somewhere.

Sometimes the error turned out to be just a typo in textbooks or websites. Sometimes it was caused by miscommunication and misunderstanding in the part of authors of the article.

Often times, comparing different sources for the same topic can help identify where the errors come from, and how to fix it.

But there are times when I need to get a first-hand information by doing my own research and experiments.

In my workplace, I'm often involved in process safety review, feasibility studies for CAPEX and OPEX, also incident investigations, either as lead investigator or a contributor or adviser.

Those roles give me experience and train my neurons to understand things deeper than what's shown in the surface. The justification for the correct investigation is rather simple.

The error stops occurring, and no repeated case for a long duration of time.

4

New Theories / Is there any relationship between Planck's law and inverse gamma distribution?

« on: 14/01/2024 08:17:55 »

I've been interested in Planck's law for years, and tried to understand the equation using various interpretations by trying to relate it to various statistical distribution curves. I was intrigued by the resemblance of Planck's equation to inverse gamma distribution.
https://en.m.wikipedia.org/wiki/Planck's_law
https://en.m.wikipedia.org/wiki/Inverse-gamma_distribution

5

Science Experiments / How does electricity propagate through conductors?

« on: 13/12/2023 14:36:04 »

The story started with Veritasium's video, which generated many responses among science Youtubers.

The Big Misconception About Electricity

The misconception is that electrons carry potential energy around a complete conducting loop, transferring their energy to the load.

6

New Theories / What is non-returning twin paradox?

« on: 17/09/2023 08:30:13 »

This is a slightly modified twin paradox to distinguish the effects of relative speeds and acceleration.
Twin A started a journey to Alpha Centauri 4 light years away in a space ship moving at 0.4c. He is expected to arrive 10 years later, according to earth observer.
Twin B stayed home to improve the space ship, so he can go to Alpha Centauri 5 years later at 0.8 c.
Classical physics calculation predicts that they'll arrive at Alpha Centauri simultaneously. Does special theory of relativity predict the same?
How old are they when they meet up at Alpha Centauri?

7

Science Experiments / How many types of capacitance are there?

« on: 15/09/2023 05:26:25 »

Wikipedia distinguishes capacitance into two different types, i.e. self capacitance and mutual capacitance.
https://en.wikipedia.org/wiki/Capacitance

Capacitance is the capability of a material object or device to store electric charge. It is measured by the charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: self capacitance and mutual capacitance.[1]: 237?238  An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component designed to add capacitance to an electric circuit.

The capacitance between two conductors is a function only of the geometry; the opposing surface area of the conductors and the distance between them, and the permittivity of any dielectric material between them. For many dielectric materials, the permittivity, and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.

The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates.[2] The reciprocal of capacitance is called elastance.

Are they mutually exclusive to each other, without overlap?
Are there any other types of electric capacitance beside them?

8

Just Chat! / What's your advice on writing a series of story?

« on: 05/08/2023 15:45:20 »

For years, I've been plotting a long story following a main character growing up from childhood into adulthood. It has something similar to Harry Potter series. For Asian kids, perhaps Naruto is more familiar to them. Older kids may be more familiar with Dragon Ball or Kungfu Boy. They are extremely popular among children, and some adults too. There's something we can learn from them.

Unlike those examples, the genre of my story is science fiction, and the goal is to introduce and socialize /popularize critical thinking, science literacy, and basic ethics to children so they could be equipped with mental tools to face the world in front of them.

My plot also covered the main characters in adulthood, where their decisions have higher stake and affect many people's lives.

9

Science Experiments / How do electromagnetic couplings work?

« on: 21/05/2023 22:18:39 »

Let's Experiment with an Electromagnetic Chain!

Have you ever seen an electromagnetic chain? Its a series of electric and magnetic toroid  cores linked in the form of a chain transferring power from electricity to magnetism and back again as it works it way along the device. Its a great hands-on illustration of Maxwell's equations as well as an illustration of transformer action taken to extremes.

10

Science Experiments / What phenomena can be demonstrated by a cheap high voltage generator?

« on: 27/03/2023 16:39:27 »

I've got one of it, and recorded some videos with it. It can generate spark 1 to 2 cm long in open air. Some dielectric barrier can prevent the formation of spark. I think there are many interesting things can be shown with it.

11

New Theories / How does Hamdani explain the twins paradox?

« on: 23/01/2023 14:30:59 »

Time dilation is not a function of acceleration and there's no mathematical formula expressing it in terms of acceleration.

If acceleration has no effect on time dilation, then the observation of the accelerated twin should be equally valid as non-accelerated one.
Alternatively, there must be something else which makes their situations different, such as their relative motion against an external object, like CMB.

12

Science Experiments / How to build DIY lasers?

« on: 24/12/2022 10:31:11 »

I just found a great Youtube channel named Les' Lab, explaining how to build DIY laser. I'd like to share the videos here. I hope you find them useful.

Nitrogen Laser tear down, power up, and Home Made Dye Laser Demonstration

13

Technology / How to learn process control system in practice?

« on: 16/11/2022 13:39:57 »

We were just finished a project commissioning and starting up a new process plant. One of the most frequently asked questions during this phase, which came from production operators, shift leaders, interns, as well as engineers from other fields, is how to tune parameters of PID controllers. But it is just the tip of an iceberg. To do it properly, we need to understand the fundamental basics first.

I summarized them in my videos. Here's the first one, splitting a system into a process model and control model. The splitting concept was very useful when I need to build a simulation of a process control system to let users get first-hand experience in tuning a PID control without causing upset in real process. It lets users see for themselves how a system response to disturbances as well as changes in PID parameters.

14

Physics, Astronomy & Cosmology / How much precession is observed in Parker Solar Probe's orbit?

« on: 15/09/2022 15:17:51 »

Parker Solar Probe's orbital period is similar to Mercury's. But the eccentricity is significantly different. How does it affect its precession?

https://en.wikipedia.org/wiki/Parker_Solar_Probe
Semi-major axis   0.388 AU (58.0 million km; 36.1 million mi)
Perihelion altitude   0.046 AU (6.9 million km; 4.3 million mi; 9.86 R☉)[note 1]
Aphelion altitude   0.73 AU (109 million km; 68 million mi)[5]
Inclination   3.4°
Period   88 days

https://en.wikipedia.org/wiki/Mercury_(planet)
Aphelion   0.466697 AU;  69,816,900 km
Perihelion   0.307499 AU;  46,001,200 km
Semi-major axis   0.387098 AU;  57,909,050 km
Eccentricity   0.205630[3]
Orbital period (sidereal)   87.9691 d;  0.240846 yr

The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to Earth, or 574.10±0.65 arcseconds per century[132] relative to the inertial ICRF.

15

Science Experiments / How to demonstrate polarization of light?

« on: 26/08/2022 11:07:09 »

Polarization of electromagnetic wave can be easily demonstrated using microwave and radio wave due to the macroscopic size of their wavelengths and the antennae. In this thread I'll show how to demonstrate the phenomenon using light.

Investigation on Polarization of Light 1 : Brewster's Angle

Demonstration of  polarization of light using reflection by dielectric material.

16

New Theories / What makes Riemann's Hypothesis Hard to Prove?

« on: 27/06/2022 15:40:12 »

What makes Riemann's Hypothesis Hard to Prove?
First, an introduction.

https://en.wikipedia.org/wiki/Riemann_hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part
1/2
. Many consider it to be the most important unsolved problem in pure mathematics.[1] It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:

The real part of every nontrivial zero of the Riemann zeta function is
1/2
.

Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers
1/2 + i t, where t is a real number and i is the imaginary unit.

You may find the visualization in this video by 3blue1brown to be helpful.

Visualizing the Riemann zeta function and analytic continuation

17

New Theories / Can black hole be electrically charged?

« on: 28/05/2022 06:08:22 »

If electrons are continuously shot into a black hole (from beta radiation or electron gun), will it be electrically charged?
Can the charge be sensed from outside, i.e. the Coulomb force?
How does the black hole mass affect the strength of Coulomb force, by modifying the distance from an outside object?

18

Science Experiments / How does induction heater work?

« on: 30/03/2022 11:40:06 »

How does induction heater work? This video can provide a starting point for us to understand the device better.

In this video I will show you how a common induction heater works, what kind of materials it can heat up and how you can easily create your own.

I've got a cheap ZVS (zero voltage switch) induction heater to explore its behavior. It has 120W maximum power, and working at 5-12VDC. I've tested its functionality, and I'm planning to investigate further on its characteristics, and how it reacts to different things.

19

Science Experiments / How does microwave superheat water?

« on: 16/03/2022 13:52:07 »

I just found old raw videos I took while discussing in this thread.
https://www.thenakedscientists.com/forum/index.php?topic=78517

I guess it's time to edit and upload them, so the information that they contain becomes available to more people.

20

Science Experiments / What's electro-magneto hydro dynamics?

« on: 08/02/2022 09:07:28 »

This experiment is about Magnetohydrodynamics

While this experiment is about Electrohydrodynamics

I think to myself, why not combine those effects in the same experiment?