On the classification of integers by missing primes

On the classification of integers by missing primes

If p1,p2,p3,.. is the sequence of prime numbers then any positive integer can be written:

n=pi^a1*p2^a2*p3^a3...pq^aq

where q is a positive integer and ai are integers greater than or equal to zero.

Of course, if ai=0 then pi^ai=1 and we say that the prime pi is "missing" from n.

I wonder if it would be instructive to classify integers by their missing primes?
As a special case: if n is prime then it has the maximum possible number of missing primes.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Human Retina - Explained in 90 Seconds

The Human Retina - Explained in 90 Seconds

The "rod" cells in the retina can detect a single photon. These cells are responsible for peripheral vision and low light vision. They provide black/white/grey vision and cannot detect color.

Special proteins in the cells can absorb an individual photon and as a result change the cell's membrane potential enough to trigger the cell.

The retina contains about 100 million rod cells.

The "cone" cells in the retina provide color vision. The retina has about 6 million cone cells and almost all are concentrated in a tiny "dimple" on the retina called the macula. It's only a few mm wide. This minuscule structure gives us our central field vision and our color vision!!!!!

Cone cells can detect frequency of light and there are 3 types of cones, each sensitive to a different frequency range. The "B" cones are most sensitive to blue, the "G" cones are most sensitive to green and the "R" cones are most sensitive to red. So our color vision is called "tricolor vision". By comparing outputs from the 3 cell types the brain "mixes" the 3 color signals to detect any color!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Are Hidden Dimensions a Dead End?

Are Hidden Dimensions a Dead End?

Much talked about. Central to String Theory. Searched for at the LHC (with no success so far).

Now some think Gravitational Waves may be the key to finding them.

But suppose they did not exist for an incredibly simple reason: at short distances the actual concept of dimension itself simply breaks down and gets replaced by something VERY different.

So we're looking for something that does not exist - literally.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Optical Computers - When Photons Replace Electrons

Optical Computers - When Photons Replace Electrons

The electron totally dominates our current computing technology - with billions of electronic logic gates packed onto silicon chips. So if you want to build an optical computer the first thing you need to do is develop a photon logic gate.

Now a research group says they have done just that.. photon gates

Photons already dominate long haul data transmission (optical fiber). But a photon computer would be truly revolutionary!!!!!!

And let's not forget Quantum Mechanics. Photons are spin 1 bosons, while electrons are spin 1/2 fermions. Meaning photons play by a whole different set of rules than electrons.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Relationship Between Mathematics & Physics

The Relationship Between Mathematics & Physics

Understanding the relationship can eliminate much confusion.

In mathematics you are free to construct anything you like as long as it's consistent. So for example, in mathematics n-dimensional spaces really do exist. They are well defined mathematical objects.

Physics uses mathematical objects to model reality - with varying levels of success. The fact that physics may use a mathematical object to model reality does not mean that object now exists outside of mathematics. It doesn't. It simply means the object is useful in modeling reality.

So for example, do n-dimensional spaces "really" exist? The question is meaningless. They exist in mathematics, and they are useful in modeling reality. But to ask if they "really" exist is a meaningless question.

Heck, it's even conceivable that something other that mathematics will ultimately be better at modeling reality. Very unlikely, but conceivable.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Building 3D Space

Building 3D Space

Suppose I have a collection of identical balls. Each has 6 connectors on its surface - 2 red, 2 green and 2 blue.

Then I have cables that come in 3 varieties, red, green and blue. Red cables can only plug into red connectors, green cables can only plug into green connectors and blue cables can only plug into blue connectors.

Now I specify a simple constraint: ever ball must have all its connectors filled.

Bingo - the system is a 3D system. Or is it?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Inversion Symmetry (SIS)

Spin Inversion Symmetry (SIS)

SIS says every elementary particle in the Standard Model (except for the Higgs) of spin s has a partner of spin 1/s and vice versa. If you look at the Standard Model you'll realize SIS predicts the existence of 12 new spin 2 particles. But here's the story behind SIS..

I have no theoretical justification for SIS. But I do have a rather amusing story of how I came up with it.

I was imagining how you might model a spin 1/2 elementary particle with an everyday object. After a bit of thought I decided on a Mobius strip (strip of paper joined after applying 1 half turn). I figured 1 half turn = spin 1/2.

Then I made a strip with 4 half turns and discovered it will naturally "flip" into a double thickness Mobius strip (no cutting, folding, etc.)

In other words a strip with 4 half turns (spin 2) naturally flips into a strip with 1 half turn (spin 1/2) and vice versa.

So there you have it - a desktop version of SIS. All you need is paper, scissors and a touch of glue!!!!!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Metrics in Physics - An Unexplored Resource?

Metrics in Physics - An Unexplored Resource?

We could probably do a lot in Physics by simply "tweeking" the metric.

For example: Special Relativity uses the familiar metric x^2+y^2+z^2-t^2. But this is just one in an infinite family of metrics.. x^q+y^q+z^q-t^q where q=1,2,3,4,... And there are many other metrics.

Even General Relativity, THE metric theory, modifies the metric using coefficients but still retains q=2 and never even considers changing that!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Holographic Principle - An Interesting Example

The Holographic Principle - An Interesting Example

The Holographic Principle talks about regions of space where the surface area is just as important as the volume. So I decided to try and construct a specific example.

I did. And the result is both simple and elegant.

Consider objects that have their surface area equal to their volume.

For a circle of radius r in 2D we have surface area (circumference)= 2*pi*r and volume (area)=pi*r^2. Equating these two gives r=2. Which is the dimension of the space!

Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!

Is this always true? Yes!

Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.

So we can say in general..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

The radius of the sphere is equal to the dimension of the space!!!!

This result is related to the Holographic Principle because it equates surface area and volume.

Content written and posted by Ken Abbott abbottsystems@gmail.com

CERN - LHC to Get Major Upgrade

CERN - LHC to Get Major Upgrade

CERN has officially begun work on a major upgrade to the Large Hadron Collider (LHC) to boost luminosity.

When the upgrade is complete in 2026 the LHC be able to collect data at almost 10X its current rate.

CERN says..

"The secret to increasing the collision rate is to squeeze the particle beam at the interaction points so that the probability of proton-proton collisions increases. To achieve this, the HL-LHC requires about 130 new magnets, in particular 24 new superconducting focusing quadrupoles to focus the beam and four superconducting dipoles. Both the quadrupoles and dipoles reach a field of about 11.5 tesla, as compared to the 8.3 tesla dipoles currently in use in the LHC. Sixteen brand-new “crab cavities” will also be installed to maximise the overlap of the proton bunches at the collision points. Their function is to tilt the bunches so that they appear to move sideways – just like a crab."

Content written and posted by Ken Abbott abbottsystems@gmail.com

A Prime Number Conjecture in Binary

A Prime Number Conjecture in Binary

Take any prime number p and write it in binary format.

From this prime now generate a series of numbers using this simple algorithm:

You can insert a single digit (0 or 1) anywhere you wish. (But you can't insert a 0 at the beginning because that just gives p unchanged.)

If p is m digits long in binary you will generate 2*m+1 new binary numbers. (btw: some may are duplicates.)

CONJECTURE: At least one of these new numbers is prime.

Let me illustrate the process with a simple example.
Take the prime 5, which in binary is 101. It has 3 digits, so we know the algorithm will generate 2*3+1=7 numbers. Here they are (notice the duplicates)..

1010=10
1011=11
1011=11
1001=9
1101=13
1001=9
1101=13

So in this case we've generated two new prime numbers, 11 and 13.

A note on duplicates:
I think you will always get m duplicates. So if these are removed the algorithm generates just m+1 numbers and the conjecture says at least one of these is prime.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Could General Relativity Fail Completely?

Could General Relativity Fail Completely?

General Relativity sets gravity apart from all other fundamental interactions by claiming it's simply curvature of spacetime. It's not even a force!

This is why GR can't be reconciled with Quantum Mechanics.

So here's a strange thought. What if spacetime simply ceased to exist (as we currently know it) in certain circumstances?
Then General Relativity would fail completely.

What might these "certain circumstances" be?
I don't know. But a great place to start is the interior of a spinning black hole.

Content written and posted by Ken Abbott abbottsystems@gmail.com

A Special Sphere in n-Dimensional Space

A Special Sphere in n-Dimensional Space

It's really interesting to consider objects that have their surface area equal to their volume.

For a circle of radius r in 2D we have surface area (circumference)= 2*pi*r and volume (area)=pi*r^2.

Equating these two gives r=2. Which is the dimension of the space!

Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!

Is this always true? Yes.

Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.

So we can say in general..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

The radius of the sphere is equal to the dimension of the space! I think this is a neat result because it relates the dimension of the space to a certain class of spheres within it. It may also be related to the Holographic Principle because it equates surface area and volume.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Making Prime Numbers - A Conjecture

Making Prime Numbers - A Conjecture

Take any prime number. You are allowed to insert a single digit (i.e. 0-9) anywhere in the number, including the beginning and end (but we don't count adding a 0 at the beginning of the number because that just gives the original number.)

Conjecture: This process will always generate at least one new prime number.

Doing this in binary (base 2) means the only digits I can insert are 0 and 1 and it makes the process very simple. The conjecture of course remains the same.

If you write a program to test this conjecture I would love to hear about your results.

Examples in base 10
503 is prime. I decide to insert 2 before the 3 to get 5023, which is prime.
20129 is prime. I decide to insert 5 at the beginning to get 520129, which is prime.

Example in binary
Take the number 3 in binary (11), so the process generates 3 new numbers..

111=7
110=6
101=5

And in this case it has generated two new primes, 5 and 7.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Sphere in n-Dimensional Space

Sphere in n-Dimensional Space

Here's the elegant result clearly explained.

Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

Then the Theorem is simply this..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

The radius of the sphere is equal to the dimension of the space!!!! I think this is a neat result because it relates the dimension of the space to a certain class of spheres within it. It also has a "holographic" aspect because it equates surface area and volume.

If you're interested in the proof it goes like this..

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Day Einstein Got Lucky

The Day Einstein Got Lucky

When physicists describe a moving object they measure its position at different times, and obviously these positions have to be given relative to some reference points. These reference points are called a reference frame.

Special Relativity showed that the Laws of Physics are independent of the choice of reference frames - when the reference frames are moving with constant velocity.

It was a huge success. But how could Einstein generalize it?

Easy.. just do the same thing for reference frames that are accelerating. Makes total sense.

So Albert went about the usual business that had been so successful before. But then he suddenly realized something.

On a local basis acceleration is indistinguishable from gravity. Bingo. Albert could do his usual thing with reference frames and describe gravity at the same time.

A huge bonus!!!!

So he produced General Relativity. And it was a massive success. That is, until it met Quantum Mechanics. The two did not get along. And still don't.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Was Einstein Toying With Us?

Was Einstein Toying With Us?

First we invent the idea of a coordinate system. That's very helpful.

Then we frame all our physics in terms of it.

But we know that the laws of physics must be covariant - i.e. independent of our choice of coordinate system.

General Relativity showed that, and that alone would have been a very nice achievement.

But no, Einstein wove into it a description of gravity - placing gravity in a totally unique position.

No wonder we can't unify gravity with the rest of physics!!!!!!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Fluid Dynamics of Spacetime

The Fluid Dynamics of Spacetime

The unification of the two great theories of the 20th Century - Quantum Mechanic (QM) and General Relativity (GR) - still seems far away.

Why?

I think both theories will require major modification before they can be united.

The modification will be in how they treat spacetime.

In QM spacetime is a stage on which the action occurs. In GR spacetime is the action!

That's a massive difference.

Think of a fluid - at the macro level an excellent description is fluid dynamics - the Navier-Stokes Equations. But at the micro level molecules rule.

So, the EFE (Einstein Field Equations of General Relativity) are the "Navier-Stokes Equations" of spacetime - an excellent description at the macro level.

But at the micro level of spacetime we have what? Nobody knows. Our theories are not there yet.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Fermat's Last Theorem - The Search for a Generalization

Fermat's Last Theorem - The Search for a Generalization

One elegant approach is to write the theorem in the language of metric spaces..

Let (x1,x2,..,xn) be a point in a n-dimensional vector space.

Then r^q=|x1|^q+|x2|^q+...+|xn|^q defines a series of metrics for q=1,2,3,.. where r is the distance of the point from the origin and |xi| is the absolute value of xi.

This allows a generalized version of Fermat's Last Theorem to be written as follows..

"An integer point (x1,x2,..,xn) is never an integer distance (r) from the origin when q>n"

It's interesting to note that the generalization holds when the metric parameter (q) exceeds the dimensionality of the space (n).

The special case of n=2 was proved in 1994 by Andrew Wiles. It was an amazing achievement because mathematicians had been trying to prove it since Fermat first suggested it in 1637.

But is this generalized version true?

At first I though it was, but one of my readers pointed out a counter example. And one counter example is all you need to disprove a conjecture! However, I am now investigating ways the conjecture may be modified (or even generalized further) so it still holds.

Content written and posted by Ken Abbott abbottsystems@gmail.com

When Gravity Fails

When Gravity Fails

Einstein's General Theory of Relativity is now over 100 years old and still remains our best description of gravity.

But the elegant Tensor form of Einstein's Field Equations (EFE) masks a critical property - they are basically differential equations of derivatives with respect to spacetime i.e. with respect to the x,y,z and t coordinates.

But, as any mathematician will tell you, derivatives are only valid for "very smooth" functions. Meaning if spacetime becomes "granular" EFE will not apply - for the simple reason that the derivatives are not defined.

At some point this granularity of spacetime will become apparent - the center of a rotating black hole could be an example.

So, it's not that you can't solve the EFE in this situation - it's that you can't even write them down!!!!!!!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Size of the Universe

The Size of the Universe

It takes light about 100,000 years just to cross our Galaxy. And the Universe contains billions of Galaxies.

So it's huge, right?

Maybe, maybe not.

Suppose the speed of light was changed. So it took a second to cross our Galaxy. Then the Universe would be a much smaller place.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Taxicab Geometry

Taxicab Geometry

Imagine an integer lattice and the only way to move around is to jump from node to node by horizontal and vertical movements. Diagonal movements are not allowed.

Think of a Taxicab on the Manhattan street grid. It certainly can't drive diagonally through a block!!!

So the node point (m,n) is at a distance m+n from the origin (m and n are integers of course) and the metric on the space is therefore d(m,n)=m+n

A "circle" centered on the origin is a diamond and pi=4 exactly.

In this grid each node is connected to exactly 4 others and pi=4. Is this just a coincidence or a fundamental result?

Is there a grid where each node is connected to exactly 3 others? Yes, a hexagonal grid has this. And pi=3.

The General Case This geometry is not constrained to grids (rectangular, hexagonal or otherwise). So long as the connectivity is correct the whole thing could be a mesh piled in a giant heap on the floor!!!

There are just two rules: You can only move between nodes along a connection, and the (minimum) number of connections between 2 nodes is the distance between the nodes.

It's incredibly general. It's topological. It's all about connectivity. Examples: the "rectangular grid" is a mesh with connectivity=4 and the "hexagonal grid" is a mesh with connectivity=3.

A fun way to imagine this geometry: Think of space as a fishing net. The knots in the net are the points of space, the snippets of reality. And the rope between the knots are the connections. It does not matter how you handle the net - throw it on the ground in a heap if you wish - the topology is unchanged. So reality is unchanged.

Could we build physics on such a mesh and if so what would it look like?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Inversion Symmetry

Spin Inversion Symmetry

Spin Inversion Symmetry (SIS) is currently a conjecture.

It says says that every elementary particle of spin s has a dual particle of spin 1/s. Of course, SIS does not apply to the spin 0 Higgs. But it applies to all other elementary particles in the Standard Model and to the graviton.

For example, applying SIS to the 3 neutrinos (electron neutrino, muon neutrino, tau neutrino), and assuming charge is conserved, we get 3 neutral spin-2 particles.

Applying SIS to the 3 leptons (electron, muon, tau), and assuming charge is conserved, we get 3 charged spin-2 particles. The photon is interesting. It's spin-1 so the photon is its own dual. The same is true for the W and Z bosons that mediate the weak force, and also for the gluon that mediates the color force.

SIS is still a hunch.. or conjecture. But if true the implications are very deep. The universe is awash in spin-2 bosons and the particles we currently know in the Standard Model are only a fraction of the particles out there.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Does Hawking Radiation Exist?

Does Hawking Radiation Exist?

Is it possible that Hawking Radiation does not exist?

Hawing Radiation from a black hole is supposed to happen when the event horizon separates a pair of virtual particles. One falls into the black hole but the other escapes and appears as radiation.

Current theories assume space is a continuum, meaning the position of the event horizon can be specified with infinite accuracy. Suppose this was not the case. Suppose the event horizon was fuzzy. Simply not enough resolution to separate a virtual pair of particles. Then there would be no Hawking radiation.

And let's not forget, Hawking Radiation has never been experimentally verified.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Are there Laws of Information?

Are there Laws of Information?

Suppose I told you this..

"I weigh 175 lbs."

I just gave you some information. But what exactly is information, and can it be defined in a quantitive way to become a key concept in math and physics? Let's look more closely at what I just gave you..

It removed an uncertainty. Meaning, before you read the statement you were uncertain about my weight. Reading the statement removed your uncertainty. So perhaps information is simply the removal of uncertainty. Great, we've defined information!

Not so fast..

I gave you the information by publishing it in a blog. So you were not the only person to receive it, thousands of others did also. Perhaps some of those people were my family. But they already know this, so for them it did not remove an uncertainly. Which means for them it was not information!

So, our definition is true for some people but not for others. That's a bad definition. We need something better.

OK, let's forget this whole thing. I regret giving you the information, so I'll just delete it.

Not so easy. I can delete it from the blog, but thousands have read it and they remember it. I can't delete that. So perhaps we've discovered the first law of information..

"Once received, information can never be deleted."

Oh boy, information is getting complex. What kind of stuff is this?

It gets more interesting. The above statement implies there's some information that's never received. Can this be true? If it's never received, ever, then how do we know it exists? We don't. So perhaps we should restrict our definition of information to information that's received. Then our first law of information gets even simpler..

"Information can never be deleted."

Does this mean the information content of the universe is constantly increasing, for the simple reason that information cannot be deleted?

Let's look more closely at what happened when the above information was received. The person receiving the information scanned it and committed it to memory. So they can recall it anytime they want. We don't know exactly how memory works, but we do know that something must have changed in the brain's neural structure in order to store this information. And no change comes for free, a small amount of energy was required to process and store the information. So perhaps we have a second law of information..

"Receiving information expends energy."

This statement is interesting for two reasons. First, it shows there's a relationship between information and energy. Second, it indicates a possible quantitative definition of information.. perhaps we can relate the amount of information in a message to the amount of energy needed to receive the message.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Counting is Not Always Easy

Counting is Not Always Easy

Counting is not always easy, even for a small number of objects.

Here are the famous Borromean Rings. How many linkages are there?

Take any two rings and look at the linkage between them - there is none.

But you cannot pull these rings apart, so they must be linked. How many linkages are there?

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Nature of Spacetime

The Nature of Spacetime

We tend to think of space as nothing - simply a void with no properties. But General Relativity says otherwise. It says spacetime can get up to all sort of tricks!

Consider the case of light..

In a vacuum light travels at about 186,000 miles per second. That's incredibly fast, right?

No, it's incredibly slow.

For example, it takes light 100,000 years just to cross our Galaxy. And the Universe contains billions of Galaxies.

So why is light so slow?

Think of spacetime as something "tangible", something that provides "resistance to motion".

So light has difficulty plowing through spacetime.. which is why it's so slow.

And the situation with objects is even worse. Which is why you have to apply a force to move an object.

Moving through spacetime is like wading through molasses!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Man Who Counted Beyond Infinity

The Man Who Counted Beyond Infinity

Georg Cantor was a mathematician who proved something quite amazing - there are numbers bigger than infinity!

He called these numbers "transfinite numbers" and he even developed an arithmetic for working with them. He denoted them by the Hebrew letter "aleph".

So what did Cantor do?

He formalized counting. He started with the integers {1,2,3,...} and asked what other sets could be placed in 1-to-1 correspondence with the integers. Instead of just saying there are an infinite amount of integers he denoted the number of integers by aleph0 and developed an arithmetic that in many ways treated aleph0 as a regular number. But he went further..

He showed that the rational numbers (fractions) could be placed in 1-to-1 correspondence with the integers. So counterintuitively, there are only as many rational numbers as there are integers. Not more!

But when it comes to irrational numbers, there are many more. He called this number aleph1 and he showed that it was different and bigger than aleph0. He proved that the number of subsets of the set of integers {1,2,3,...} is also aleph1 and he produced this amazing result..

aleph1=2^aleph0

He even asked if there was an aleph number between aleph0 and aleph1.

In his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians!

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

Today Cantor's work is part of any university math curriculum and is regarded as one of the most beautiful pieces of mathematics ever created. It stands apart from most advanced math because you don't need to know much math to understand it. In fact, all you need to know is how to count!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Mechanical Spin - The most fundamental thing?

Quantum Mechanical Spin - The most fundamental thing?

What do you think it is?

Mass? Energy? Charge? Time? Dimension?



If you look at the list of elementary particles in the Standard Model you'll see that each has a property called spin. There are only a few values.. 0, 1/2, 1 (and the hypothetical graviton - not in the Standard Model - has spin 2).

Spin divides all elementary particles into two radically different groups. Spin 1/2 particles are Fermions. Spin 0,1,2 particles are Bosons.

And spin is conserved, there is no known process that can change the spin of an elementary particle.

So perhaps spin is the most fundamental thing.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Is String Theory a Dead End?

Is String Theory a Dead End?

Oh sure, it produces some very nice ideas.

But it requires "multiple dimensions". The last I heard was 26.

Hey, with 26 free parameters to adjust I could produce some nice results also!

Suppose multiple dimensions don't exist (even at the Planck level). Suppose all we had was 3. You know, like we currently have.

Is String Theory taking us down the wrong path?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Why is the Speed of Light so Slow?

Why is the Speed of Light so Slow?

In a vacuum light travels at about 186,000 miles per second. That's incredibly fast, right?

No, it's incredibly slow.

For example, it takes light 100,000 years just to cross our Galaxy. And the Universe contains billions of Galaxies.

So why is light so slow?

Think of space-time as something "tangible", something that provides "resistance to motion".

So light has difficulty plowing through space-time.. which is why it's so slow.

And the situation with objects is even worse. Which is why you have to apply a force to accelerate an object.

Plowing through space-time is like wading through molasses!

Content written and posted by Ken Abbott abbottsystems@gmail.com

My DNA Shock

My DNA Shock

I did a DNA test and the results totally shocked me!

So now I'm doing a second test with another company to see if the results agree.

This time I'm using AncestryDNA

I'm doing the basic ethnic origin test, not a medical analysis test. They don't do full DNA sequencing - they just sample your DNA at 700,000 points and then check you against their database.

This lets them find your ethnic origin i.e. the region of the world you were most likely from. Plus, they also give you "matches" - people who have very similar DNA and so could be related.

AncestryDNA Test - Get a $10 Discount.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Ramanujan - Indian Math Genius

Ramanujan - Indian Math Genius

On 16 January 1913, Srinivasa Ramanujan wrote to G. H. Hardy.

Ramanujan was a self taught mathematician from a small village in India. He had almost no formal training in mathematics. Hardy was professor of mathematics at Cambridge University and one of the leading mathematicians in the world.

The letter sent by Ramanujan contained a sampling of theorems he had discovered. Hardy later said, "the theorems defeated me completely; I had never seen anything in the least like them before" and added, "they must be true, because, if they were not true, no one would have the imagination to invent them."

What happened next would change both their lives. Hardy would later write about Ramanujan in his book "A Mathematician's Apology" and say that working with Ramanujan was the most significant event of his life.

Ramanujan produced some amazing infinite series, including several for π that converge extraordinarily fast and form the basis of today's computer algorithms used to calculate π.

His other results involved continued fractions. Ramanujan had a special love of continued fractions and used them to extraordinary effect.

Hardy worked hard to try and discover how Ramanujan produced his remarkable results. He never found out.

Ramanujan died on 26 April 1920. He was 32 years old. Hardy died many years later, on 1 December 1947 at the age of 70.

The Ramanujan story is now part of mathematics legend and his notebooks are still being studied today.

The movie "The Man Who Knew Infinity" tells the story of Ramanujan with Dev Patel in the lead role.

Content written and posted by Ken Abbott abbottsystems@gmail.com

2,500 Years of English History - Explained in 90 Seconds

2,500 Years of English History - Explained in 90 Seconds

This is not exactly math or physics, but I thought my readers would like it, and it does relate to my earlier posts about DNA. In fact, it was my DNA analysis that led me to research this.

500 BC
The Celts show up from Central Europe and bring the Iron Age. They occupy all of England.

500 YEARS LATER
The Romans arrive and the Celts scatter to Wales, Scotland, Ireland, and the far tip of Cornwall.

500 YEARS LATER
After making absolutely sure the Romans have gone the Anglo-Saxons invade.

250 YEARS LATER
The Viking take a shot, but their invasion and occupation is limited.

250 YEARS LATER
The Normans mount their famous 1066 invasion and bring the Cross Bow and Garlic. This is the last time England is invaded.

1,000 YEARS LATER
Ken Abbott does DNA test and discovers he's from the original Celtic invasion.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics - A New Elementary Particle Symmetry?

Physics - A New Elementary Particle Symmetry?

Spin Inversion Symmetry (SIS) is currently a conjecture.

It says says that every elementary particle of spin s has a dual particle of spin 1/s. Of course, SIS does not apply to the spin 0 Higgs. But it applies to all other elementary particles in the Standard Model and to the graviton.

For example, applying SIS to the 3 neutrinos (electron neutrino, muon neutrino, tau neutrino), and assuming charge is conserved, we get 3 neutral spin-2 particles.

Applying SIS to the 3 leptons (electron, muon, tau), and assuming charge is conserved, we get 3 charged spin-2 particles. The photon is interesting. It's spin-1 so the photon is its own dual. The same is true for the W and Z bosons that mediate the weak force, and also for the gluon that mediates the color force.

SIS is still a hunch.. or conjecture. But if true the implications are amazing.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Could there be 3 Gravitons?

Could there be 3 Gravitons?

Spin Inversion Symmetry (SIS) is currently a conjecture.

It says says that every elementary particle of spin s has a dual particle of spin 1/s. Of course, SIS does not apply to the spin-0 Higgs Boson. But it applies to all other elementary particles in the Standard Model and to the graviton.

So, applying SIS to the 3 neutrinos (electron neutrino, muon neutrino, tau neutrino) we get 3 neutral spin-2 bosons. A trio of gravitons?

It's still a hunch. But if true the implications are amazing.

Is there a physical principle behind SIS?

I'm not sure, but here's an interesting model that provides some guidance. Hint: Think String Theory and associate one half twist with spin-1/2.

A model for SIS

Content written and posted by Ken Abbott abbottsystems@gmail.com

Elementary Particles - Spin Inversion Symmetry

Elementary Particles - Spin Inversion Symmetry

At the moment I must call this a conjecture.

Spin Inversion Symmetry (SIS) says that every elementary particle of spin s has a dual particle of spin 1/s. Of course, SIS does not apply to the spin 0 Higgs. But it applies to all other elementary particles in the Standard Model and the hypothetical graviton.

So, assuming charge is conserved, the spin 1/2 neutrino has a neutral dual of spin 2. The spin 1 photon is its own dual. And the spin 1/2 electron has a charged spin 2 dual.

So SIS predicts a slew of new particles. Notice that the neutral spin 2 particles could also explain dark matter. However, as of now I don't have any underlying physics to justify SIS.

It's still a hunch.. or conjecture. But if true the implications are amazing.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Is General Relativity a Dead End?

Is General Relativity a Dead End?

Don't get me wrong, General Relativity is brilliant. It's probably the greatest insight ever produced by a single individual. And it's built from an amazing observation..

That acceleration is locally equivalent to a gravitational field. That's clever to say the least.

But we can regard an acceleration as a stress on space-time. So what happens in the Quantum limit when the acceleration is truly huge? Does it fracture space-time into discrete units or some underlying fundamental constituents?

If so that's where GR breaks down. And the rest of GR was a nice macroscopic average.

And don't forget, GR assumes space-time is a continuum. That's a classical concept which simply cannot be true at very small distances.

Bottom line: GR has to break down at some point. It's great at making predictions at the classical level. But it cannot move to the future. It's a dead end. An incredibly clever dead end. But a dead end none the less.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Cambridge University - Remembering the Old Cavendish Laboratory

Cambridge University - Remembering the Old Cavendish Laboratory

When I was doing my physics PhD at Cambridge my office was in the original Cavendish Laboratory on Free School Lane.

My window overlooked the courtyard which contained Kapitza's magnetics lab. It was built in the courtyard due to lack of space.

And of course I had a clear view of the famous Crocodile carved into the wall of Kapitza's Lab. He used it to tease Rutherford, "The Crocodile" being Kapitza's pet name for Rutherford.

The Cavendish moved to its new spacious location in West Cambridge in 1974.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Gravity and Dark Matter

Quantum Gravity and Dark Matter

At the moment Physics recognizes 4 fundamental forces:

-The Color Force (mediated by a spin 1 Boson - the Gluon)
-The Electromagnetic Force (mediated by a spin 1 Boson - the Photon)
-The Weak Force (mediated by 2 spin 1 Bosons - the Z Boson and W Boson)
-The Gravitational Force (mediated by the hypothetical spin 2 Boson - the Graviton)

But if we apply the SIO (Spin Inversion Operator) to the spin 1/2 neutrinos (electron, muon and tau) we get spin 2 neutral particles. Could these be "partners" of the Graviton in a new force that's closely related to gravity. And if these new particles have high mass the new force would have very short range - making it an ideal candidate for "quantum gravity".

So just as the Electromagnetic force gets extended to the Electroweak force by the addition of new spin 1 mediating particles, so Gravity gets extended to "quantum gravity" by the addition of new spin 2 mediating particles.

Plus, could these new particles also account for Dark Matter?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Remembering Physicist Otto Frisch

Remembering Physicist Otto Frisch

Otto Frisch was one of the great physicists of the 20th Century. I met Otto at the Cavendish Laboratory. We used to eat "digestive biscuits" together at teatime.

He was a messy eater and showered crumbs all over the place and I called him out on it, much to my regret. His autobiography is "What Little I Remember". A very humble man. It's worth reading.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Predicting New Gravitons

Predicting New Gravitons

At the moment Physics recognizes 4 fundamental forces:

-The Color Force (mediated by a spin 1 Boson - the Gluon)
-The Electromagnetic Force (mediated by a spin 1 Boson - the Photon)
-The Weak Force (mediated by 2 spin 1 Bosons - the Z Boson and W Boson)
-The Gravitational Force (mediated by the hypothetical spin 2 Boson - the Graviton)

But if we apply the SIO (Spin Inversion Operator) to the spin 1/2 neutrinos (electron, muon and tau) we get spin 2 neutral particles. Could these be "partners" of the Graviton in a new force that's closely related to gravity. And if these new particles have high mass the new force would have very short range - making it an ideal candidate for "quantum gravity".

So just as the Electromagnetic force gets extended to the Electroweak force by the addition of new spin 1 mediating particles, so Gravity gets extended to "quantum gravity" by the addition of new spin 2 mediating particles.

Plus, could these new particles also account for Dark Matter?

Content written and posted by Ken Abbott abbottsystems@gmail.com

LED Surprise

LED Surprise

An LED (Light Emitting Diode) is just a solid state junction that emits light in response to an applied voltage.

These junctions can be switched amazingly fast - making them ideal for data transmission.

So forget Wi-Fi, the talk is now about Li-Fi networks. Li-Fi is short for light fidelity. What an interesting idea.. tiny LED lights transmitting data with massive bandwidth.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Mechanics - Efimov States Explained

Quantum Mechanics - Efimov States Explained

The Efimov effect is an effect in quantum mechanics that was predicted by the Russian physicist Vitaly Efimov in 1970. It has only recently been observed.

So, what is it?

It's a bound state of 3 identical bosons that happen even if the force between any two of the bosons is not strong enough to cause binding.

In other words, these states only appear when you have 3 bosons, take any one away and the whole system disappears.

Efimov States have an infinite series of excited three-body energy levels.

They are often depicted symbolically by the Borromean rings - no two rings are linked, but all 3 rings are linked!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Permutahedron

The Permutahedron

This is the 4-permutahedron. It's a geometric representation of the ways to rearrange the numbers 1,2,3,4.

Two permutations are connected by an edge if one can be transformed into the other by swapping two consecutive numbers.

And this is just for 4 numbers, I wonder what the 100-permutahedron would look like?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Driverless Cars - The Problem

Driverless Cars - The Problem

Oh sure, you'll see driverless cars on the road. A few here and a few there. But they will never happen in a major commercial way for a very simple reason: people love to drive.

I do. It's a skill that took me a while to learn. I'm proud of my skill and I enjoy using it. A driverless car takes that pleasure away. Can you imagine being in a car that obeys all traffic laws including speed limits?

That would be incredibly infuriating and amazingly boring.

But there's a much bigger problem. On todays aggressive roads driverless cars are outright dangerous!!!!!!

Go ahead, plod along at the 30 mph speed limit while 18 wheeler trucks swerve past you at 55 mph. No thanks.

Or your driverless car hits and kills an innocent person. While you watch. Because the algorithms were not smart enough to evaluate the situation.

But there's one good thing about driverless cars - as more come on the road they will put my driving skills to the test in order to avoid them!

Driverless cars are putting me to sleep. Wake me up when we have driverless national train services.

You gotta love the driverless car industry - a solution to a problem that doesn't exist.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Difference Calculus - Explained in 90 Seconds

Difference Calculus - Explained in 90 Seconds

Most people have heard of Differential Calculus, but few have heard of Difference Calculus. In the rare times it does get mentioned it's described as an "approximation" to Differential Calculus. That's a pity, because Difference Calculus deserves better!

So what is it?

It's easily explained if you remember the definition of a derivative in Differential Calculus. For a function f(x) the derivative df(x)/dx is defined as..

df(x)/dx=Limit as c approaches 0 of (f(x+c)-f(x))/c

It turns out that for this limit to exist f(x) has to be a very "well behaved" function. Not all functions are, so df(x)/dx does not exist for many functions. Which means that Differential Calculus cannot be used with these functions.

Difference Calculus gets around this problem in an incredibly simple way, it just removes the limit from the definition of df(x)/dx to define the "difference" Df(x) as..

Df(x)=(f(x+c)-f(x))/c

But this definition is not as simple as it looks. Notice that in the definition of df(x)/dx the c never appears in the result, it's simply used to define the limit. This is not true for Df(x), where c actually appears in the result, and so different choices of c will give different values for Df(x). This is not a bad thing, but it's something that should be noted. To be strictly accurate we should denote the difference as Dcf(x) because it depends on c, but it's usually just denoted by Df(x) and some value of c is assumed.

Of course, for the case of c=1 we get the simple expression..

Df(x)=f(x+1)-f(x)

Let's try an example..
The function f(x)=e^x is famous in Differential Calculus because it's equal to its own derivative, df(x)/dx=f(x)

This is not true in Difference Calculus, because the difference is..

Df(x)=f(x)*(e^c-1)

As we noted above, the result depends on c. If we want to get the same result as differential calculus, Df(x)=f(x), we have to use a specific value of c such that..

(e^c-1)=1 which means c=ln(2)=0.69314718056..

This is a hint that Difference Calculus has its own properties and is more than just an "approximation" to Differential Calculus.

For functions of an integer variable, f(n) where n=1,2,3,4,.. the value c=1 is natural and the difference is Df(n)=f(n+1)-f(n)

Equations involving Df(x) are called difference equations and are the equivalent of differential equations in Differential Calculus. Here's a simple example..

Df(x)=-2*f(x) for c=1, which is the simple difference equation f(x+1)+f(x)=0

One solution to this difference equation is f(x)=a*cos(pi*x), where a is an arbitrary constant. So even a very simple difference equation has a "wave-like" solution! This is another hint that Difference Calculus has its own properties distinct from Differential Calculus.

In Differential Calculus we can define higher order derivatives. Can we do this in Difference Calculus? Yes! For example, the second order difference is just the difference of the first order difference..

D(Df(x))=(f(x+2*c)-2*f(x+c)+f(x))/(c^2) and for the case of c=1 this is the simple expression f(x+2)-2*f(x+1)+f(x)

Higher order differences simply use the value of the function at more points. The first order difference uses x and x+1. The second order differences uses x, x+1 and x+2. This continues for higher order differences.

Difference Calculus has very broad applicability. In fact, I like to think of Differential Calculus as just a special case of Difference Calculus!

More to explore: "The Theory of Finite Differences" by C. Jordan, first published in Budapest in 1938. This book give a complete mathematical treatment of Difference Calculus.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Plot a Function - Google Plot is Fast and Free

Plot a Function - Google Plot is Fast and Free

Need to plot a function?

Just punch your request into Google to get an instant plot. The Google plot feature is powerful, fast and free! It's a great way to teach yourself functions, or to make math homework easy!

Example: to plot the function f(x)=x^2 just enter plot x^2 into Google. Of course, it can also handle more complex functions, for example try 5*(x-4)^2+5 or x^3+(1/x) or try one of my favorites, just enter plot 1/(1+x^2)

Notice the symbols used by the Google Plot function: x^n means raise x to the power n and of course * is used to denote multiplication.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Problem with General Relativity

The Problem with General Relativity

General Relativity had its 100th birthday in 2015 and is still the #1 way to describe gravity. But change may be on the horizon.

The theory is probably the most brilliant physics insight ever produced by a single individual. But it can never be correct at the quantum level and must be replaced or seriously modified.

Totally new thinking is needed, especially concerning space time coordinates and dimension.

Sorry Albert, but heck, 100 years is a pretty good run!

Content written and posted by Ken Abbott abbottsystems@gmail.com

What is Physics?

What is Physics?

A reality check, food for thought, and perhaps a good topic of discussion..

Physics is a bunch of procedures that let us predict a small subset of nature with various levels of accuracy.

That's it.

There is no truth.

The so called "Laws" of Physics are not laws, but simply a great name devised by physicists to make their bunch of procedures sound impressive.

And who said physicists are no good at marketing!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics - The Number 3 Strikes Again

Physics - The Number 3 Strikes Again

It seems that Physics likes the number 3 at all levels, from macroscopic down to the quantum level. Consider this..

- There are 3 spacial dimensions.

- The Standard Model has 3 families of fermions.

- The Color Force has 3 charges, RGB.

- And we now have the recently discovered "Efimov States" which are bound groups of 3 bosons.

Coincidence? Maybe. Maybe not.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Information and The Dimension of Space

Information and The Dimension of Space

A simple way to think of the gravitational field of an object is to imagine a fixed number of "lines of force" that radiate from the object evenly into space. The density of the lines at any given point in space represents the strength of the gravitational field at that point.

With this simple model in mind, we write down Newton's famous formula for the gravitational force f between two masses m and q at a distance r apart..

f=G*m*q/(r^2)

G is just a constant which goes by the impressive name "universal gravitational constant", but it's not really interesting. Its value depends on which unit system we're using to measure f, m and r and we can pick a unit system so that G=1, then..

f=m*q/(r^2)

We can make q a unit mass, q=1, and imagine we're using q to measure the gravitational field generated by m, so..

f=m/(r^2)

This is the essence on Newton's theory of gravitation. But what is this equation really saying? If we think about our lines of force model in a 1-dimensional space and then in a 2-dimensional space we realize that this equation is actually..

f=m/r^(n-1)

where n=number of dimensions of the space containing the mass m.

We can solve this equation for n..

n=1+(log(m/f)/log(r))

The expression on the right is an experimentally measurable quantity. But it's the dimension of space!

Can we push our luck further? Hey, why not..

In physics we know that the log function shows up in discussions of information entropy. But we've expressed dimension in terms of the log function, so perhaps dimension is derived from the information content of space. Of course, we need to define "information content", but if that could be done we would have a concept more fundamental than dimension. Now that would be a breakthrough!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Mathematics of Gun Deaths

The Mathematics of Gun Deaths

It happens with depressing frequency in the United States - there's a mass killing and the gun debate jumps to the front. There's much hand waving on both sides for a few days. Then it fades until the next time.

And there's always a next time.

So I decided to do some quantitative analysis. Turns out it's hard to understand guns. I'm no slouch at data analysis, but this turned into a mystery..

I looked at the AR-15, which seems to be the weapon of choice for mass killers. It's available for civilian ownership in many countries. Some make you jump through more hoops than others, but basically if you want one you can get one.

Then I looked at deaths by guns. The statistics include all types of gun deaths not just mass killings. The numbers are quoted as deaths per 100,000 of population per year. I used the United Kingdom as my baseline.

United Kingdom 0.23

United States 10.54

That makes gun deaths in the United States 4,582% higher than in the United Kingdom.

Of course, the United States is not the worst. The highest number I found was Honduras at 67.18 and the lowest was South Korea at 0.08

But still, I think most people identify the United States closely with the United Kingdom. I know I do.

So the mystery is why gun deaths in the United States are 4,582% higher than in the United Kingdom.

4,582% is a massive gap between two close allies.

Data source: Firearm Death Rates by Country

Content written and posted by Ken Abbott abbottsystems@gmail.com

Rethinking Black Holes

Rethinking Black Holes

Here's a solution to the "Black Hole Information Paradox" that nobody seems to have considered..

Hawking Radiation does not exist.

Reason: Current theories assume the continuum, meaning the position of the event horizon can be specified with infinite accuracy. Suppose this was not the case. Suppose the event horizon was fuzzy. Simply not enough resolution to separate a virtual pair of particles. Then there would be no Hawking radiation.

And let's not forget, Hawking Radiation has never been experimentally verified.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Borromean Rings

Borromean Rings

Here's a nice piece of topology.

Notice that no two rings are linked, yet all 3 rings are linked.

You could use this a a model for the proton - the rings representing the 3 quarks that make up a proton and "ring linkage" representing the color force.

Then we see that the color force is a very unusual force. It's a 3-force, meaning one that only appears when you have 3 objects, the force between any 2 objects being zero.

This model even shows the famous "asymptotic freedom" and "confinement" of bound quarks. When the rings are very close together you can move them around as if they were free (asymptotic freedom), but when you try to pull them apart a strong force appears to stop you (confinement).

Content written and posted by Ken Abbott abbottsystems@gmail.com

A New Goldbach Conjecture

A New Goldbach Conjecture

The Goldbach Conjecture says.. "every even number greater than two can be written as the sums of 2 primes."

The Twin Prime Conjecture says.. "there are an infinite number of twin primes."

Twin primes are two primes separated by just one number. For example: (11,13) are twin primes, so are (1049,1051)

Neither conjecture has ever been proven or disproven and they remain the holy grail of mathematical research.

But what if they were connected?

I did a search of numeric computer tests and found this gem..

Let a t-prime be a prime that's a member of a twin prime pair. Then..

"Every even number greater than 4208 is the sum of two t-primes."

Content written and posted by Ken Abbott abbottsystems@gmail.com

Goldbach Conjecture Revisited

Goldbach Conjecture Revisited

If you could show that the Goldbach Conjecture is "decoupled" from the Axioms of Number Theory then it could never be proved or disproved within Number Theory.

I wonder if there are ways to check if a mathematical statement is "decoupled" from the Axioms of a system?

The Goldbach Conjecture is a very simple statement, so surely it must be easy to check if it's decoupled. Methinks not!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Twin Prime Pair (1049,1051)

The Twin Prime Pair (1049,1051)

Let's write twin primes in binary.

For example, the twin prime pair (1049,1051)=(10000011001,10000011011).

Now concatenate the binary bit streams to get 1000001100110000011011=2149403 which is a prime!

Primes generating primes. Quite amazing.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Google Math Tip

Google Math Tip

Need to plot a function?

Just punch your request into Google to get an instant plot.

Example: to plot the function f(x)=x^2 just enter plot x^2 into Google. Of course, it can also handle much more complex functions, for example try plot x^3+1/x or try one of my favorites, just enter plot 1/(1+x^2)

Content written and posted by Ken Abbott abbottsystems@gmail.com

Your chance of winning the Lottery

Your chance of winning the Lottery

We all know the probability is low. But how low?

Let's take the Powerball, an incredibly popular lottery in the USA. Last week someone won $571 million in the Powerball.

So what are your chances of winning?

If your 5 numbers plus the Powerball match the winning six numbers drawn, then you win or share the Grand Prize. If the jackpot is not won in any drawing, the First Prize Pool Money is carried forward and is added to the next Powerball Jackpot.

Your chance of getting 5 + Powerball - Grand Prize is 1 in 292,201,338

Powerball offers 9 ways to win, and here are the probabilities..

So does this mean you should not buy a lottery ticket? If you don't buy your chance of winning is absolutely zero!

Good luck.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Twin Primes - A Surprising Result

Twin Primes - A Surprising Result

Let's write twin primes in binary. For example the prime pair (281,283)=(100011001,100011011).

Now concatenate the two binary string to get 100011001100011011=144155 which is not a prime. But just reverse the order of concatenation to get 100011011100011001=145177 and this is a prime!

Here's an even more impressive example.

The prime pair (1049,1051)=(10000011001,10000011011) then concatenate the binary bit streams to get 1000001100110000011011=2149403 which is a prime!

Primes generating primes! Quite amazing. Please contact me with your results.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Mechanics - Explained in 90 Seconds

Quantum Mechanics - Explained in 90 Seconds

If I was forced to summarize Quantum Mechanics in one sentence I would say this.. "small objects behave very differently than big objects".

It sounds like an innocent statement. But it's not. It's a profound discovery of how nature works.

An example will help..

Spin is something we're all familiar with. We can make any object spin. And not just a top, spin is used to great effect in many sports such as tennis, baseball and cricket. And of course you could never throw a frisbee without spin.

We can spin an object at any speed we please, and as soon as it starts spinning it defines an axis about which the spin occurs.

But that's the spin of a big object, meaning an object we can handle. What about the spin of a really small object such as an electron?

It turns out the electron spins just like a tiny top - but with two big surprises..

- What about the speed of spin?
The electron spins at a fixed rate that can never be changed. There is no known process that can change the spin rate of the electron. This makes electron spin a fundamental quantity.

- What about the axis of spin?
You can measure the spin along any axis you want and you'll always get the same result, +1/2 or -1/2 (these two values just correspond to the electron spinning clockwise or counter clockwise). So the electron behaves as if it's spinning about every axis at the same time!

So electron spin is totally different than the spin of a big object such as a top.

Physicists call the electron a "spin 1/2 particle". And it's not just the electron, all elementary particles have spin except for the recently discovered Higgs boson.

If you plan to study Quantum Mechanics pay attention to spin. It's a wonderful example of how "small objects behave very differently than big objects".

Content written and posted by Ken Abbott abbottsystems@gmail.com

Transcendental Numbers - A Simple Definition

Transcendental Numbers - A Simple Definition

Take any number and ask if it can be converted to a rational number by raising it to an positive integer power. If it can we say it's coupled to the integers. If not we say it's decoupled from the integers.

All rational numbers are coupled by definition. But so are many irrational numbers, for example sqrt(2) is irrational, but (sqrt(2))^2=2. Even many complex numbers are coupled, for example i^2=-1

So it seems most numbers are coupled to the integers, making integers fundamental. Not true. In fact Cantor showed the opposite is true, most numbers are decoupled from the integers. These are the (still mysterious) transcendental numbers. And the name is good, transcendental numbers "transcend" the integers.

Content written and posted by Ken Abbott abbottsystems@gmail.com

What is a Prime Number?

What is a Prime Number?

Here's a nice simple definition of a prime number..

Suppose you have a set of objects and I ask you to divide them into subsets of equal size. The key thing here is "equal size".

This can often be done. But there are some sets where it cannot be done.

When it cannot be done we call the number of objects in the set a prime number.

Example: A set of 12 objects can be divided into subsets of equal size (actually, in several different ways), so 12 is not a prime number. A set of 13 objects can never be divided into subsets of equal size so 13 is a prime number.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Mathematics of Drinking

The Mathematics of Drinking

First we define a fundamental unit - the "standard drink". It's a drink that needs one hour for its alcohol content to be metabolized. In other words it takes one hour to get out of your system.

Then using this fundamental unit we can measure drinks..

Liquor (80 Proof - 40% Alcohol by volume)

“Bottle” 750ml = 17 Standard Drinks
“Martini” 200ml = 4.5 Standard Drinks
“Miniature” 50ml = 1 Standard Drink

Wine (13% Alcohol by volume)

“Bottle” 750ml = 5.5 Standard Drinks
“Big Glass” (10 fluid ounces) = 2 Standard Drinks
“Standard Glass” (5 fluid ounces) = 1 Standard Drink

Beer (5% Alcohol by volume)

“Pint” (16 fluid ounces) = 1.2 Standard Drinks
“Bottle” (12 fluid ounces) = 1 Standard Drink

I think the biggest shock here is the Martini. Wow!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Information and Dimension - Are They Related?

Information and Dimension - Are They Related?

This is an extreme example, but it shows there may be a fundamental relationship between the availability of information and the concept of dimension.

Sitting on your desk is a sphere. As in mathematics, let's assume an axiom - your sphere is an information void. What's that? It's a closed surface which allows absolutely no knowledge of its interior. The interior of the object is off limits to any form of investigation. Think of its surface as an information barrier that stops any information about the interior from getting out.

So what properties does your sphere have?

You can't break it open to look at the interior, that would violate the axiom, so the object must be infinitely strong.

You can measure its diameter - just use a ruler. You can measure its surface area - just take a small unit square and see how many times you can paste it on the surface.

What about volume? Be careful, knowing volume means you have information about the interior and that's impossible. Of course you can use the formula volume=(4/3)*pi*r^3 where r is the radius. But this is not a measurement of the interior, so the best you can do is call this volume the "external volume". You have no knowledge of the geometry or topology of the interior, so the "internal volume" could be very different!

Now, if you can never have any knowledge of the interior you come to a simple conclusion, the object is totally defined by whatever is on its surface. The object certainly looks 3D but can be fully described as if it was 2D. Interesting object!

Your sphere has no reality inside - just like bubbles in a liquid have no liquid inside. If you own one please handle it carefully!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Prime Cores - The Core of a Prime Number

Prime Cores - The Core of a Prime Number

What's a Prime Core? Take a prime number in binary, then strip off the first and last digits (which, for all primes except 2 are always 1's) then interpret the binary string you have left as an integer, and that's the prime core.

Example, the prime 79 in binary is 1001111 so its core is 00111 which is 7. So using C to represent the prime core operation, we have C(79)=7.

Then here's an interesting question: "when is the core of a prime also a prime?"

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Problem with Blockchain

The Problem with Blockchain

Don't get me wrong. It's clever. The mathematics behind it is very clever. It would make a great PhD thesis. But it's a solution looking for a problem.

Blockchain is many things to many people. But let's take Bitcoin, the most famous blockchain app.

A payment method? Really? VISA can process up to 47,000 transactions/second. Bitcoin maxes out at 7 transactions/second. Blockchain is the bottleneck.

A peer-to-peer system? Really? Nobody cares about peer-to-peer. Most people like a central authority.

So what is blockchain? It's an example of technology in love with itself. And with no concept of consumer marketing or psychology.

Watch Bitcoin Trading in Realtime

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Computing Gets Closer

Quantum Computing Gets Closer

Studying computer programming in hopes of getting a great job?

An experienced programmer looking to boost your career?

Perhaps the really hot jobs in the future will be in quantum computing.

It's worth keeping up to speed in this fast developing field, so take some time to learn quantum programming!

Here's a nice quantum computing tutorial. It teaches you step-by-step how to write a program for these amazing machines.

How to Program a Quantum Computer

Also, IBM is very active in quantum computing. Their main quantum computer research Lab is the Watson Research Center in Yorktown Heights. They have a quantum computer up and running and provide developer tools to let you write and run programs. It's called the "IBM Q experience".

IBM Q experience

Content written and posted by Ken Abbott abbottsystems@gmail.com

Hydrogen Cars - Explained in 90 Seconds

Hydrogen Cars - Explained in 90 Seconds

Hydrogen cars are very different from all other cars (gasoline, battery, hybrid). Here's how they work..

A hydrogen car is a 100% electric car, but instead of a battery it uses hydrogen fuel cells. These are simple, feed them hydrogen gas and they generate electricity. They are like a battery that never need recharging. There is no combustion, nothing burns, hydrogen gas is simply fed to the fuel cells to produce electricity.

So where do we get the hydrogen? The cars carry high pressure tanks of hydrogen gas to supply their fuel cells. The tanks get refilled at a hydrogen gas station, and the filling process is very similar to regular gasoline filling. The hydrogen gas station produces its hydrogen on the spot. How? By electrolysis of water.

Yes, the raw material to generate hydrogen gas is water!

So, the hydrogen gas station makes hydrogen from water. Cars refill their hydrogen tanks. The hydrogen goes through the car's fuel cells and generates electricity to drive the car. And the exhaust? The car's exhaust is water vapor. There is zero pollution.

The whole thing is a water to water cycle!

The range of a hydrogen car is about the same as a gasoline car, and the refueling time is about the same.

Also, hydrogen is the most abundant element in the universe. So I doubt we'll run out!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Simple Definition of a Prime Number

Simple Definition of a Prime Number

Suppose you have a set of objects and I ask you to divide them into subsets of equal size. The key thing here is "equal size".

This can often be done. But there are some sets where it cannot be done.

When it cannot be done we call the number of objects in the set a prime number.

Example: A set of 12 objects can be divided into subsets of equal size (actually, in several different ways), so 12 is not a prime number. A set of 13 objects can never be divided into subsets of equal size so 13 is a prime number.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Lineland - Life in 1 Dimension

Lineland - Life in 1 Dimension

You're a point mass and you live on the x axis. That's your entire world. It's "Lineland". What's your life like?

First, as regards moving, you only have two directions, forward and backwards. And if you meet another point mass you cannot pass. So you can only know two other masses. You have just two friends maximum!

You have no reason to count objects beyond two, so you might be slow in developing the concept of integers. Or perhaps you never develop the concept at all. You simply have no need for it.

What about Physics in Lineland?

Consider Newtonian Gravity. You're a point mass, so you have mass, let's say m. Another point mass could have a different mass, say M. So at least Newtonian gravity exists, right? It does, but it has a strange form. Newton's formula for the gravitational force F between two masses m and M is..

F=G*M*m/(r^2)

where G is a constant and r is the distance between the two masses.

The r^2 term is good in a 3D space, but in general it's r^(n-1) where n is the dimension of the space. Putting n=1 for Lineland we get..

r^(1-1)=r^0=1 so F=G*M*m

Which means F is independent of distance! Gravity has the same strength no matter how far apart the objects are. So physics in Lineland is very different.

This is Lineland on the x axis. What if Lineland is the circumference of a circle? That's even more interesting. Would you be aware that Lineland had a "curvature"? What does gravity do now that Lineland is a closed loop?

What happens if Lineland is a closed loop that intersects itself at several points? What happens at these intersection points and how do they contribute to gravity?

How do things change as the number of point masses in Lineland changes?

It turns out that even 1 dimension can be very complex!

Just think, there's probably a 4D world somewhere with physics teachers looking for a nasty problem to set on an exam. Finally they come up with one, "explain how physics would have developed if our world was constrained to only 3 dimensions".

Content written and posted by Ken Abbott abbottsystems@gmail.com

Bitcoin Trading Explained Fast

Bitcoin Trading Explained Fast

Bitcoin trades 24/7 around the world on many exchanges. So there's no one Bitcoin price and arbitrage opportunities exist.

If you plan to trade Bitcoins the choice of exchange is very important and should be researched carefully before you open an account.

Watch Bitcoin Trading in Realtime

Here's one of many Bitcoin Exchanges. This one trades Bitcoin in US Dollars, which is called BTCUSD.

Bitcoin Trading Platform

Content written and posted by Ken Abbott abbottsystems@gmail.com

Can Reality Vanish?

Can Reality Vanish?

Quantum Mechanics deals with incredibly small objects and this makes it difficult to visualize what's happening. But let's bring a quantum object up to desktop size and see how it might behave.

Sitting on your desk is a sphere. As in mathematics, let's assume an axiom - your sphere is an information void. What's that? It's a closed surface which allows absolutely no knowledge of its interior. The interior of the object is off limits to any form of investigation. Think of its surface as an information barrier that stops any information about the interior from getting out.

So what properties does your sphere have?

You can't break it open to look at the interior, that would violate the axiom, so the object must be very strong.

You can measure its diameter - just use a ruler. You can measure its surface area - just take a small unit square and see how many times you can paste it on the surface.

What about volume? Be careful, knowing volume means you have information about the interior and that's impossible. Of course you can use the formula volume=(4/3)*pi*r^3 where r is the radius. But this is not a measurement of the interior, so the best you can do is call this volume the "external volume". You have no knowledge of the geometry or topology of the interior, so the "internal volume" could be very different!

Now, if you can never have any knowledge of the interior you come to a simple conclusion, the object is totally defined by whatever is on its surface. The object certainly looks 3D but can be fully described as if it was 2D. Interesting object!

Your sphere has no reality inside - just like bubbles in a liquid have no liquid inside. If you own one please handle it carefully!

Do such objects really exist? I'm not sure, but a Black Hole may be a good approximation!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Shannon Entropy - Explained in 90 Seconds

Shannon Entropy - Explained in 90 Seconds

Shannon Entropy (also called Information Entropy) is a concept used in physics and information theory. Here's the scoop..

Suppose you have a system with n states i.e. whenever you make an observation of the system you find it's always in one of the n possible states.

Now make a large number of observations of the system, then use them to get the probability pi that if you make an observation the system is in state i. So for every state of the system you have a probability pi.

Now construct this crazy sum = p1*log(p1) + p2*log(p2) +... + pn*log(pn) where the sum is over all the states of the system.

If the log is base 2 then (-1)*sum is called the "information entropy" of the system.

Note that "information entropy" applies to a complete system, not individual states of a system.

Here's a simple example..

My system is a penny and a table.
I define the system to have 2 states.. penny lying stationary on the table with heads up or with tails up.

My experiment is to throw the penny and then observe which state results.

I throw the penny many times and make notes. It lands heads up 1% of the time and tails up 99% of the time (it's biased).

The crazy sum is 0.01*log(0.01) + 0.99*log(0.99) = 0.01*(-6.643856) + 0.99*(-0.0145) = -0.08079356

So the information entropy of the system is (-1)*(-0.08079356) = 0.08079356

Content written and posted by Ken Abbott abbottsystems@gmail.com

Fractions Explained in 5 Minutes

Fractions Explained in 5 Minutes

The hardest thing about learning fractions is adding them. Multiplication is easy, but addition is usually taught as a complex multi-step process. It's a real pain.

But it doesn't have to be that way. Take the two fractions to be added..

a c
-+-
b d

Take the diagonal numbers (top left and bottom right) and multiply them, a*d

Do the other diagonal, c*b

Add the two results (a*d)+(c*b)

That's the "top" (numerator) of the answer.

The "bottom" (denominator) of the answer is much easier, it's just b*d

You're done!

Did you notice the pattern? It's two diagonals and a base, like this..

X
_

A lot of mathematics involves patterns.

To multiply fractions just multiply the tops and multiply the bottoms. So..

a c
-*-
b d

is (a*c)/(b*d)

That's it. Fractions done in a few minutes. With the time saved why not study more interesting math topics!

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Content written and posted by Ken Abbott abbottsystems@gmail.com
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Elementary Particle Physics - The Spin Inversion Operator

Elementary Particle Physics - The Spin Inversion Operator

I just had an idea for a "Spin Inversion Operator". It takes an elementary particle and inverts its spin. Example: the Graviton with spin 2 would map into a spin 1/2 particle. Likewise a spin 1/2 particle would map into a spin 2 particle.

The Photon is interesting, it has spin 1, so the Spin Inversion Operator does not change the spin. It's invariant.

Coming back to the Graviton again, assuming charge is conserved, the Spin Inversion Operator maps it into a neutral spin 1/2 particle. The only one known is the Neutrino. So is there some deep connection between the Graviton and the Neutrino?

The Spin Inversion Operator works for the Graviton and all elementary particles in the Standard Model - except for the Higgs Boson.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Simple Harmonic Oscillator - A Different Approach

Simple Harmonic Oscillator - A Different Approach

The Simple Harmonic Oscillator is a famous system in physics. Its equation of motion is written as a second order differential equation which is then solved to give the characteristic "wave" solution. But there's an alternate method which does not need differential equations!

Consider a function of an integer variable defined by this..

f(n)=k*f(n-1)-f(n-2)

where n=2,3,4,5,.. and k is a constant.

If k, f(0) and f(1) are given then f(n) can be calculated for any n.

This equation is an example of a difference equation. An area of mathematics called the Theory of Finite Differences or Difference Calculus tells how to solve these equations. The solution is a nice surprise..

f(n)=sin(n*a)

It's the famous sine function, where a is a constant and k=2*cos(a). So the solution is a wave. If you plot f(n) for n=2,3,4,5,... you get the beautiful sine wave, and the three numbers k, f(0) and f(1) determine the amplitude, wavelength and phase of the wave.

What about n? It plays the role of time, because at time=n the function f(n) is the displacement from the origin.

So the difference equation f(n)=k*f(n-1)-f(n-2) replaces the second order differential equation used to describe the Simple Harmonic Oscillator. It's a nice example of Difference Calculus in action.

Of course, you may have noticed that things are not exactly the same.. time is no longer a continuous variable!

Content written and posted by Ken Abbott abbottsystems@gmail.com



Differential Calculus Explained in 5 Minutes

Differential Calculus Explained in 5 Minutes

Differential calculus is one of the two branches of calculus, the other is integral calculus. Most mathematicians refer to both branches together as simply calculus.

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Let's take a simple function, say f(x)=x^2

What's the value of this function at a specific point, say x=a? That's easy, it's f(a)=a^2. But now we ask an interesting question, can we possibly know anything else about the function at point a? At first glance this seems impossible, the value of the function at a is f(a), so surely that's all we can know, right? Wrong. It turns out there's a process called "differentiation" that can tell us more. Here's how it works..

Take a very small number, say q, and ask what the function is doing at a+q, in other words at a point very close to a..

f(a+q)=(a+q)^2=a^2+2*a*q+q^2

But we can make q as small as we please, which means q^2 is much smaller, so to a good approximation we can ignore it, and we get..

f(a+q)=(a+q)^2=a^2+2*a*q

Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..

f(a+q)=f(a)+2*a*q

Which means..

f(a+q)-f(a)=2*a*q

And so..

(f(a+q)-f(a))/q=2*a

What is the meaning of the expression on the left? If you draw a diagram you'll see that the term on the left is simply the slope of the curve f(x) close to x=a. So this gives us some valuable information about what's going on near a. Now all we need to do is keep making q smaller so we get closer and closer to a. In fact, we can use the concept of a limit to say..

Limit(f(a+q)-f(a))/q as q goes to zero is 2*a

Of course we could do this for any point a, so in general..

Limit(f(x+q)-f(x))/q as q goes to zero is 2*x

This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..

The derivative of the function f(x)=x^2 is 2*x and is written df/dx=2*x and it's the slope of the f(x) curve at x. Of course, a slope is simply a rate of change, so we can also say that df/dx=2*x is the rate of change of the function f(x).

Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x

To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..

df/dx=n*x^(n-1)

So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9

So, the essence of differential calculus is this.. in addition to knowing the value of a function f(x) at x=a we also know the rate of change (slope) of the function at a. Differential calculus gives us an extra piece of information!

Much of differential calculus is simply finding ways to differentiate different functions. This can get boring, so why bother? Because the derivative of a function is a really useful thing for solving all sorts of problems. It's especially useful in physics and many laws of physics are written as differential equations.

Content written and posted by Ken Abbott abbottsystems@gmail.com