1

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

« on: Yesterday at 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 to be helpful.

Visualizing the Riemann zeta function and analytic continuation

2

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?

3

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.

4

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.

5

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?

6

Physiology & Medicine / What's the most practical method to detect sleep deprivation?

« on: 16/01/2022 15:02:55 »

This article reminds me of some incidents on the road and industrial work place caused by sleep deprivation of the driver or machine operator, which could have been prevented.
https://neurosciencenews.com/sleep-deprivation-serotonin-2a-19897/
Is there a practical method to detect cognitive decline caused by sleep deprivation or intoxication? Perhaps something like blood pressure, heart rate, or brain wave detector? Non-invasive methods would be preferred. And the result should be available within few minutes.

7

Physics, Astronomy & Cosmology / How can we see ultraviolet light in Balmer series?

« on: 28/12/2021 11:24:30 »

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

The "visible" hydrogen emission spectrum lines in the Balmer series. H-alpha is the red line at the right. Four lines (counting from the right) are formally in the visible range. Lines five and six can be seen with the naked eye, but are considered to be ultraviolet as they have wavelengths less than 400 nm.

How can we see ultraviolet light in Balmer series?

8

Just Chat! / Can we be sure of our own existence

« on: 13/11/2021 06:39:04 »

If you add enough "necessary axioms", you can end up believing whatever made them necessary. Hence religion, a flat earth supported on turtles, and other kinds of foolishness.

Better to start with observations.

The only thing we can be sure of is our own existence.  Any other things can be deceiving, including our observations.
An assumption is necessary if it can somehow be related to our existence.

9

General Science / What's 0^0 ?

« on: 02/11/2021 03:16:16 »

What's 0 to the power of 0?

10

New Theories / How Many Numbers Exist?

« on: 29/09/2021 04:53:12 »

How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.

There are an infinite number of infinities. Which one corresponds to the real numbers?

An Infinity of Infinities
Yes, infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.

Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number ₀ (“aleph-zero”).

But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.


Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.

Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ₁, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.

His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ₁ real numbers. In other words, the cardinality of the continuum immediately follow ₀, the cardinality of the natural numbers, with no sizes of infinity in between.

But to Cantor’s immense distress, he couldn’t prove it.

In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.

To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.

The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove.

As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.

These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.

In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.

https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/

What do you think about this continuum hypothesis?

11

New Theories / Is there a better way to explain light?

« on: 12/07/2021 02:12:39 »

This thread is a follow up of my previous thread discussing and criticizing existing theories about light.
https://www.thenakedscientists.com/forum/index.php?topic=68595.0

Here I'll try to figure out if there is a way to improve it. If there is, what will it look like?

I just become aware that a similar topic has been created by CrazyScientist.
https://www.thenakedscientists.com/forum/index.php?topic=82373.0
What Is The Nature Of Photons & EM Radiation?

He has his own reasoning to come to his conclusion, which has some differences and similarities than my current understanding of this matter. If I have something to say about his reasoning, I'll post it there. But to avoid complication, I'll post my own reasoning here.

To avoid getting unexpected results, I'll try to avoid making false assumptions, especially the hidden ones, which are likely hard to identify. Any assumptions put into the model should be stated explicitly, along with the reasons why they can't be dismissed. This can significantly slow down the process, but I guess it worths the efforts to resolve spookiness in science.

12

Just Chat! / Who wants to take part in #VeritasiumContest?

« on: 03/07/2021 22:31:21 »

Welcome to the Veritasium Science Communication contest! I recently won a $10,000 bet with a UCLA physics professor over a wind-powered car (here’s the video). Now, the team and I have decided to pass the $10,000 on by holding a contest to highlight science communicators.
What you need to do to enter is create a science communication video that is one minute or less in length, and post it on YouTube or TikTok with the hashtag #VeritasiumContest. You will also need to include your email address in the video description or clearly on your profile, so we can contact you if you’ve won. You must be subscribed to Veritasium on YouTube or following Veritasium on TikTok. We are looking for videos that clearly and creatively explain complex or counterintuitive concepts in the fields of Science, Technology, Engineering, and Mathematics.

There are cash prizes for this competition—with first place receiving $5,000, second place receiving $3,000, and third place receiving $2,000.

https://www.veritasium.com/contest

Is anyone interested to participate?

13

Science Experiments / Can we go downwind faster than the wind?

« on: 14/06/2021 04:24:34 »

Veritasium made a video showing that we can build a wind powered car that goes faster than the wind itself.
Alexander Kusenko, a UCLA Physics Prof. disagree.

https://twitter.com/veritasium/status/1403130178197278720

Big News! UCLA Physics Prof, @alexkusenko bet me $10,000 that I'm wrong about going downwind faster than the wind.

Our wager was witnessed by
@neiltyson
@BillNye
@seanmcarroll
If I win, to what charitable cause should I donate the funds?

Here is the response from Alexander Kusenko trying to debunk the claim in Veritasium's video.

Here is a set of 10 slides that (1) explain what is seen in the video (2) point out the errors theoretical arguments, and (3) provide a complete solution to the problem.

https://docs.google.com/presentation/d/1xuN-9C1Gs6MAVTJkhh1vMJw0atmSsTsKDkhEfN898K4/edit#slide=id.gdc0eb9892c_0_223

Replying to
@thephysicsgirl
 @alexkusenko
 and 3 others
Come on! If someone emails me saying I’m wrong, first I try to convince them that I’m right. Then if we continue to disagree I suggest we make it interesting. The key is to get to the truth. The best, scientifically accurate explanation should win.

14

Physics, Astronomy & Cosmology / what would happen if gravitational mass were different than inertial mass?

« on: 06/05/2021 23:09:51 »

Currently accepted gravitational theory demands that they are the same.
What's expected to happen if gravitational mass were twice of inertial mass? What if it's only a half?

15

New Theories / Is there a better explanation for interference pattern produced in single slit

« on: 11/04/2021 05:36:10 »

Single slit experiment is usually explained using Huygen's principle,  like in these videos.

Is it really the best explanation we can provide?

16

New Theories / Where does quantization of energy of electromagnetic radiation come from?

« on: 08/12/2020 13:15:09 »

According to Planck's Law, energy of radiation is quantized.
E = n.h.f
n is integer.
h is Planck's constant
f is frequency

Dimensional analysis tells us that energy has mass and length in it. In the equation above, they reside in Planck's constant. So the quantification of energy must come from some things that determine the value of Planck's constant.

We have learned that electromagnetic radiation comes from moving electrically charged particles. They have kinetic energy according to the formula
Ek=½.m.v²
m is the mass and v is the speed
The particles also have potential energy which depends on their electric charge.

Electrostatic potential energy of q due to Q1 and Q2 charge system:

https://en.wikipedia.org/wiki/Electric_potential_energy#One_point_charge_q_in_the_presence_of_n_point_charges_Qi
Those particles have discrete values of mass and electric charge. So it comes naturally that their electromagnetic radiation would come in quantified amounts.

17

New Theories / Can Fields Induce Other Fields in Vacuum?

« on: 27/11/2020 13:50:53 »

I did random Google search to collect materials for my planned experiments using radio wave. Then I bumped into this interesting article.
Trouble with Maxwell’s Electromagnetic Theory: Can Fields Induce Other Fields in Vacuum?
by Ionel DINU, M.Sc., a Physics Teacher
https://vixra.org/pdf/1206.0083v8.pdf

Abstract
The purpose of this article is to point out that Maxwell’s electromagnetic theory,
believed by the majority of scientists a fundamental theory of physics, is in fact built
on an unsupported assumption and on a faulty method of theoretical investigation.
The result is that the whole theory cannot be considered reliable, nor its conclusions
accurate descriptions of reality. In this work it is called into question whether radio
waves (and light) travelling in vacuum, are indeed composed of mutually inducing
electric and magnetic fields.

Introduction
This study is addressed to that small percent of students and researchers who suspect
that there is something wrong with the way in which we understand nowadays how radio
waves are generated and how they propagate in space.
I know that there is always a feeling of distrust amongst students when university
professors obtain the equation of a wave from the four Maxwell’s equations. I felt that
myself as a student and I have seen it again in the open courses made available on the
Internet by prestigious universities of the world. Students ask pertinent questions but the
professor fails to address the issue.
[See

min. 0:35:00].

Summary
 In conclusion, in this article it was shown that Maxwell’s theory of electromagnetic
waves contains an unfounded assumption, a faulty method of theoretical investigation and
makes a prediction that is contrary to observations.
These are:
(i) the unfounded assumption that a changing magnetic field B creates (induces) an
electric field E (a.k.a. Faraday’s law of electromagnetic induction). In fact, a changing
magnetic field B is observed to produce an electric current J, not an electric field E and
there is a great difference between an electric current J and an electric field E.
(ii) the assumption that a changing electric field E creates (induces) a magnetic field B
(a.k.a. Maxwell’s correction to Ampere’s Law). This was derived by Maxwell through a
faulty method of theoretical investigation, no such effect was known in Maxwell’s time
and no experiment has been made since then that proves this assumption.
(iii) the prediction that radio waves and light are composed of entangled electric and
magnetic waves that create (induce) one another in vacuum. No experiment revealed that
radio waves and light have a structure containing electric and magnetic fields.
Although it seemed an easy and straightforward matter to accomplish, Faraday failed
in his attempt to change the plane of polarization of light travelling in vacuum by the
application of strong electric and magnetic fields. Only when the polarized beam of light
passed through glass of great density could this be accomplished, and even then by the
application of a magnetic field only.
Furthermore, Faraday initially applied the magnetic field perpendicular to the ray,
believing that this would change the direction of the plane of polarization. Not obtaining
any positive result, he then placed the magnetic field parallel to the direction of the ray,
and he finally obtained the change he was looking for. But then how can this result be
reconciled with the theory in which light is considered to be composed of two transverse
magnetic and electric fields? It does not seem that the magnetic field applied by Faraday
and the magnetic field of the light-ray vibrating perpendicular to it give a resultant in a
different plane.

What do you think is wrong with his argumentation? Do you agree with him instead?
What kind of experiments can determine which one is the more accurate model to physical reality?

18

Science Experiments / How does radio wave behave?

« on: 23/11/2020 07:37:57 »

Continuing my effort to understand the behavior of microwave in another thread, here I'd like to share experiments showing how radio wave behaves. I've prepared some experimental equipment, but didn't have enough time to execute the experiments yet, so for now I'll put this well made video from Harvard Natural Sciences Lecture Demonstrations first.

19

New Theories / How much information can we get from a single photon?

« on: 22/10/2020 04:14:31 »

According to QM, a photon carries a unit of energy according to its frequency through formula E=hf. Since h is a constant, no new information is obtained in calculating the energy because it's a dependent variable.
But we know that photon has other characteristics independent of its frequency, such as phases and polarization states.
Increasing the amplitude of EM wave is usually interpreted as increasing the number of photon with the same frequency, phase, and polarization state.
 
Are there other known characteristics of a photon?

20

Physics, Astronomy & Cosmology / What is the best explanation for Three Polarizer “Paradox”?

« on: 09/10/2020 05:40:19 »

Google search for "three polarizer paradox" gives these results:

Three Polarizer “Paradox”
If the polarizers are opposed at a 90° angle, the polarized light from the first polarizer is stopped by the second. If a third polarizer is sandwiched between the two opposed polarizers at a 45° angle some light gets through the last polarizer.

http://www.users.csbsju.edu/~frioux/polarize/POLAR-sup.pdf
https://faculty.csbsju.edu/frioux/polarize/POLAR-sup.pdf

A beam of unpolarized light illuminates a vertical polarizer and 50% of the light emerges vertically polarized. This light beam encounters a diagonal polarizer oriented at a 45 degree angle to the original vertical polarizer and 50% of it emerges as diagonally polarized light. Finally 50% of the diagonally polarized light passes a horizontally oriented polarizer. In other words 12.5% of the light illuminating the first vertical polarizer passes the final horizontal polarizer. However, if the diagonal polarizer sandwiched between the vertical and horizontal polarizers is removed, no light emerges form the final horizontal polarizer.

Using the figure below vector algebra will be used to analyze this so-called "three-polarizer paradox." The paradox being that it is surprising that the insertion of the diagonal polarizer between crossed polarizers allows photons to pass the final horizontal polarizer.

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Tutorials_(Rioux)/Quantum_Optics/268%3A_The_Three-Polarizer_Paradox

If you take two crossed polarizers (for example, a horizontal and vertical one), no light will get through them. Yet when you insert a third polarizer between the two, oriented diagonally, then some photons make it through. How does adding that polarizer (which will block some photons) cause photons to get through?


Say that the first polarizer is horizontal. Any photons that make it through that one are then horizontally polarized. If the vertical polarizer comes next, it will block all of these photons. When the diagonal polarizer is in place, however, it will let half of them through and these transmitted photons will then be diagonally polarized. When these diagonally polarized photons arrive at the vertical polarizer, now half of them will get through—they have no "memory" of ever having been horizontally polarized.

https://www.scientificamerican.com/article/quantum-eraser-answer-to-three-polarizer-puzzle/

Dirac Three Polarizers Experiment
In his 1930 textbook The Principles of Quantum Mechanics, Paul Dirac introduced the uniquely quantum concepts of superposition and indeterminacy using polarized photons.
Dirac's examples suggest a very simple and inexpensive experiment to demonstrate the notions of quantum states, the projection or representation of a given state vector in another basis set of vectors, the preparation of quantum systems in states with known properties, and the measurement of various properties.

Albert Einstein said of Dirac and polarization,
"Dirac, to whom, in my opinion, we owe the most perfect exposition, logically, of this [quantum] theory, rightly points out that it would probably be difficult, for example, to give a theoretical description of a photon such as would give enough information to enable one to decide whether it will pass a polarizer placed (obliquely) in its way or not." Maxwell's Influence on the Evolution of the Idea of Physical Reality...1931, Ideas and Opinions, p.270

Any measuring apparatus is also a state preparation system. We know that after a measurement of a photon which has shown it to be in a state of vertical polarization, for example, a second measurement with the same (vertical polarization detecting) capability will show the photon to be in the same state with probability unity. Quantum mechanics is not always uncertain. There is also no uncertainty if we measure the vertically polarized photon with a horizontal polarization detector. There is zero probability of finding the vertically polarized photon in a horizontally polarized state.
Since any measurement increases the amount of information, there must be a compensating increase in entropy absorbed by or radiated away from the measuring apparatus. This is the Ludwig-Landauer Principle.

The natural basis set of vectors is usually one whose eigenvalues are the observables of our measurement system. In Dirac's bra and ket notation, the orthogonal basis vectors in our example are | v >, the photon in a vertically polarized state, and | h >, the photon in a horizontally polarized state. These two states are eigenstates of our measuring apparatus.

The interesting case to consider is a third measuring apparatus that prepares a photon in a diagonally polarized state 45° between | v > and | h >.

Dirac tells us this diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes."

Thus,

| d > = ( 1/√2) | v > + ( 1/√2) | h >          (1)

Note that vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities, as first proposed by Max Born in 1927.

When these complex number coefficients are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers represent the probabilities of finding the photon in one or the other state, should a measurement be made. Dirac's bra vector is the complex conjugate of the corresponding ket vector.

It is the probability amplitudes that interfere in the two-slit experiment. To get the probabilities of finding a photon, we must square the probability amplitudes. Actually we must calculate the expectation value of some operator that represents an observable. The probability P of finding the photon in state |ψ> at location (in configuration space) r is

P(r) = < ψ | r | ψ >
No single experiment can convey all the wonder and non-intuitive character of quantum mechanics. But we believe Dirac's simple examples of polarized photons can teach us a lot. He thought that his simple examples provided a good introduction to the subject and we agree.
...

We animated Dirac's idea of introducing an oblique polarizer between the two crossed polarizers A and B that are blocking all light. Adding this filter actually allows more photons to pass through, which is counter-intuitive.

https://www.informationphilosopher.com/solutions/experiments/dirac_3-polarizers/