Let's make an experiment where the positively charged test particle is at rest, while the electrons in a wire move to the left at v m/s, and the metal atoms move to the right at v m/s. Will the test particle accelerate? In what direction?
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Is it a problem? It's often required that the positive charges don't and can't spread out despite a Lorentz contraction. The positive charges are the metal atoms and they are locked into a lattice. Even when Lorentz contraction puts them closer together and suggests there should be increased repulsion between them, the metal atoms cannot move apart.Yes, it is a problem. If the difference is due to the formation of crystal lattice, then we would be able to distinguish the different response in liquid metals like mercury, or ionic solutions like some acids, bases, or salts.
However the electrons are not like that, they are free to move around and can spread out.
In the past, if some axioms lead to ridiculous conclusion, we would reject them, or modify some of them. But identifying those false assumptions could be tricky, especially when they are hidden, and we are not aware of making them in the first place, like the featureless and infinitely rigid blocks of single or double slit aperture.Some modern scientists seem to be too confident in their own scientific knowledge. So when their predictions differ from observations, they just declare that objective reality may not exist, instead of scrutinizing their assumptions more thoroughly.
That reality might be in the eye of the observer is a very peculiar aspect of the physical reality in the quantum domain, and the mystery itself shows no signs of abating, both researchers agree.
In order to model the future evolution of society completely, you will need a complete model of every individual plus a predictive model of the climate and all natural disasters.With finite data and computational resources to run a model of the world, we must work smart, using Pareto principle, and address most significant things first.
It is for example difficult to imagine how science would have evolved if Newton was not in quarantine. Or if someone had studied the antibiotic effect as thoroughly as Fleming (who just noticed the accidental contamination of a culture) in time to cure Henry VIII's syphilis: No Anglican church → vastly different history of Britain and America....
This has been the mainstream view of electric and magnetic fields for quite a while: Changing frames of reference can make an Electric field look like a Magnetic Field and vice versa.The problem identified here is the asymmetric response between the movement of positive and negative charges in the wire. If only electrons that move, there's no force. If only the positively charged metal lattice moves, there's a force.
This zeta function satisfies the functional equation
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers.
The fact that
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = Ĺ
It is also known that no zeros lie on a line with real part 1.
Hardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.
It is known that the zeros are symmetrically placed about the line I(s)=0. This follows from the fact that, for all complex numbers s,
1. s and the complex conjugate s* are symmetrically placed about this line.
2. From the definition (1), the Riemann zeta function satisfies zeta(s*)=zeta(s)*, so that if s is a zero, so is s*, since then zeta(s*)=zeta(s)*=0*=0.
It is also known that the nontrivial zeros are symmetrically placed about the critical line R(s)=1/2, a result which follows from the functional equation and the symmetry about the line I(s)=0. For if s is a nontrivial zero, then 1-s is also a zero (by the functional equation), and then 1-s* is another zero. But s and 1-s* are symmetrically placed about the line R(s)=1/2, since 1-(x+iy)*=(1-x)+iy, and if x=1/2+x', then 1-x=1/2-x'.
No, it just shows that society is dynamic and evolutionary (except in the USA).It shows that the models don't take dynamics of the society into account, which makes them inaccurate and can only be good for a short period of time.
Complex Integration and Finding Zeros of the Zeta Function
In this video we examine the other half of complex calculus: integration. We explain how the idea of a complex line integral arises naturally from real definite integrals via Riemann sums, and we examine some of the properties of this new sort of integral. In particular, we consider some complications that arise when trying to apply the fundamental theorem of calculus to complex functions.
We then bring these ideas to the central focus of this series: the zeta function and the Riemann hypothesis. By the end of the video, we will be able to use complex integrals to approximate the location of the zeroes of the zeta function (or those of any other complex function for that matter)!
Analytic Continuation and the Zeta Function
Where do complex functions come from? In this video we explore the idea of analytic continuation, a powerful technique which allows us to extend functions such as sin(x) from the real numbers into the complex plane. Using analytic continuation we can finally define the zeta function for complex inputs and make sense of what it is the Riemann Hypothesis is claiming.
00:00 zetamath does puzzles
02:40 Bombelli and the cubic formula
08:45 Evaluating real functions at complex numbers
12:33 Maclaurin series
21:22 Taylor series
27:19 Analytic continuation
35:57 What goes wrong
48:19 Next time
The Basel Problem Part 1: Euler-Maclaurin Approximation
This is the first video in a two part series explaining how Euler discovered that the sum of the reciprocals of the square numbers is π^2/6, leading him to define the zeta function, and how Riemann discovered the surprising connection between the zeroes of the zeta function and the distribution of the primes, leading ultimately to his statement of the Riemann Hypothesis. This video focuses on how Euler developed a method to approximate this sum to 17 decimal places, as well as how the Bernoulli numbers naturally appear as part of this problem.
The Basel Problem Part 2: Euler's Proof and the Riemann Hypothesis
In this video, I present Euler's proof that the solution to the Basel problem is pi^2/6. I discuss a surprising connection Euler discovered between a generalization of the Basel problem and the Bernoulli numbers, as well as his invention of the zeta function. I explain Euler's discovery of the connection between the zeta function and the prime numbers, and I discuss how Riemann's continuation of Euler's work led him to state the Riemann hypothesis, one of the most important conjectures in the entire history of mathematics.
Sections of this video:
01:24 Euler's Basel proof
23:20 The zeta function and the Bernoulli numbers
32:01 Zeta and the primes
48:15 The Riemann hypothesis
A Youtube channel seems to be dedicated to explain this problem, and I find it as one of the best explanation online.If you are serious to understand this problem, this video is a good place to start.
Factorials, prime numbers, and the Riemann Hypothesis
Today we introduce some of the ideas of analytic number theory, and employ them to help us understand the size of n!. We use that understanding to discover a surprisingly accurate picture of the distribution of the prime numbers, and explore how this fits into the broader context of one of the most important unsolved problems in mathematics, the Riemann Hypothesis.
Hamdani,Have you built the motor yourself, or do you know someone who has built it? Do you know a working prototype of it?
I am so pleased that you are investigating magnetic force fields. We also need to explain the magnoflux spin effect of the magnetic field please.
So the question now is, what's all the fuss? Who would deny that?It seems that some people are convinced that consciousness can have effects on matter without physical interface which can be analyzed, modeled, and manipulated. This would make consciousness looks magical. But that's how magic tricks are usually done. Show audiences the beginning and the end of a process while hiding something in between. Some misdirections can amplify the effects of mythical confusions.
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
. Many consider it to be the most important unsolved problem in pure mathematics. 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
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.
Visualizing the Riemann zeta function and analytic continuation
Even if a highly ultimate technology, would comunicante trough physical means.We usually call them interfaces. They can be biological neurons, implanted electrodes, or non-invasive brain wave sensors.
There a wire or even light it's not different from my hands.
Science is less concerned about why or what some ultimate truth might be and more concerned with having an explanation and a model that is useful for making predictions.What does science say that can be used to answer the question in the op?
Lenin, Mao and Hitler had very good models of reality that helped them persuade millions of individuals to do all sorts of things that you might consider immoral.The collapse of the systems that they built show that their models of reality are not as good as you'd like to think.
Here is the visualization of the second experiment, which start from the first as described before. If the charged particle is stationary to the wire, no magnetic force is received.The difficulty in working with electrically charged particles/objects is that they are attracted to even neutral objects due to electric displacement. An electrically charged metal ball is attracted to the plastic hose even when it's empty and electrically neutral.
Next, the wire is zoomed to show the electrons and metal atoms inside.
From the picture above, the electrons inside the wire move to the left with speed v, but particle q doesnít receive magnetic force.
Now if the wire is moved to the right with speed v, the speed of electrons becomes 0, while the speed of the metal atoms = v. It is shown that magnetic force F is produced downward.
The picture above is equivalent to the picture from previous post.
Here we can conclude that electronís movement is not responded by the particle, while atomís movement produces magnetic force to the particle. It seems that for a long time we had missed the difference between atoms and free electrons which cause electric current and produce magnetic force.
For the second experiment, we will study the effect of the movement of charged particles inside a conductor (or convector) toward the test particle. We will study the hypothesis that magnetic force is not only affected by the magnitude of electric charge that moves inside a conductor (or convector), but also affected by the mass of the particle.
Electric current in a copper wire is produced by the flow of electrons inside. The charge and mass of electrons are always the same, so we need some other particles as electric current producers to get reference. For that we will replace the conductor by a hose filled by electrolyte solution that contains ions, since ions are also electrically charged and have various masses. Some of electrolytic solutions that will be used are NaCl, H2SO4, HCl, CuSO4, FeCl3.