Halting Problem & Quantum Entanglement 2020 Breakthrough result [MIP*=RE] @udiprod

udiprod This video explains the MIP*=RE result. We skip the proof details, just explain what the result means.

Please leave comments in the comment section if something is unclear.

The links mentioned in the video:

1) Proof that the halting problem can't be solved:
youtu.be/92WHN-pAFCs

2) Using the entanglement, see here: youtu.be/v7jctqKsUMA

Also, here's a general primer to quantum physics:
youtu.be/p7bzE1E5PMY

3) Proof that PSPACE contains P and is contained in EXP: TBA sometime

Chapters:
0:00 Part 1: Decision problems
3:13 Part 2: Complexity classes
7:32 Part 3: Verification
12:57 Part 4: More verification power
17:51 Part 5: Some implications

Some more explanations:

Part 3:

* We require the verifier to run in polynomial time. This has the usual definition with respect to the input's size. This constraints the size of the proof sent by the prover, since reading 1 character costs 1 time unit.

* In the video we make a distinction between honest and malicious provers. Sometimes it is defined differently: there's only one kind of prover, and it always wants to get the verifier to accept. For w that is in L, it gets the verifier to accept by cooperating and behaving nicely. For w outside of L, it tries to get the verifier to accept by cheating.

Part 4:

* MIP: In previous games, the verifier faces a single prover. Its goal was to be able to detect if the prover is lying or telling the truth. For languages inside PSPACE, the prover could do that. But languages outside PSPACE are more complicated. For these languages, the verifier can't tell if the prover is lying or telling the truth.
In MIP we help the verifier more by having two provers. The verifier can now judge if a response is a lie not just by examining the response itself, but by comparing what the two provers say. If they respond to the same question differently, one of the responses must be a lie, and the verifier can reject immediately.
This added ability to detect lies helps it verify more complicated languages - all languages in the large class called NEXP.

MIP*: Here there's something non-intuitive going on. We help the provers by giving them a quantum device, but it turns out it actually helps the verifier. The details here are complicated, but we may explore them further in future videos.

One frequent question is why the provers use the entanglement at all, even the honest ones. The reason is that the verifier forces them to use the entangelement.

To see why, consider the non-isomoprhism example: the verifier gives the prover a computational challenge. This challenge is designed such that the prover can win only if the graphs are non-isomorphic.

The quantum entanglement opens the door for new kinds of challenges that the provers can win. It wasn't obvious if any of them are useful for our purposes, but then one was discovered: there exists a challenge such that for a given program p:
1) The provers can only win if p halts.
2) And they can only win if they cooperate via the entanglement.
Condition (2) is logically important: without it the game would have been possible with MIP as well. The provers can choose not to use the entangelement, but then they'll lose. We assume the provers want to win.

updated 4 years ago

Halting Problem & Quantum Entanglement 2020 Breakthrough result [MIP*=RE]

udiprod 2021-11-14 | This video explains the MIP*=RE result. We skip the proof details, just explain what the result means.

Please leave comments in the comment section if something is unclear.

The links mentioned in the video:

1) Proof that the halting problem can't be solved:
youtu.be/92WHN-pAFCs

2) Using the entanglement, see here: youtu.be/v7jctqKsUMA

Also, here's a general primer to quantum physics:
youtu.be/p7bzE1E5PMY

3) Proof that PSPACE contains P and is contained in EXP: TBA sometime

Chapters:
0:00 Part 1: Decision problems
3:13 Part 2: Complexity classes
7:32 Part 3: Verification
12:57 Part 4: More verification power
17:51 Part 5: Some implications

Some more explanations:

Part 3:

* We require the verifier to run in polynomial time. This has the usual definition with respect to the input's size. This constraints the size of the proof sent by the prover, since reading 1 character costs 1 time unit.

* In the video we make a distinction between honest and malicious provers. Sometimes it is defined differently: there's only one kind of prover, and it always wants to get the verifier to accept. For w that is in L, it gets the verifier to accept by cooperating and behaving nicely. For w outside of L, it tries to get the verifier to accept by cheating.

Part 4:

* MIP: In previous games, the verifier faces a single prover. Its goal was to be able to detect if the prover is lying or telling the truth. For languages inside PSPACE, the prover could do that. But languages outside PSPACE are more complicated. For these languages, the verifier can't tell if the prover is lying or telling the truth.
In MIP we help the verifier more by having two provers. The verifier can now judge if a response is a lie not just by examining the response itself, but by comparing what the two provers say. If they respond to the same question differently, one of the responses must be a lie, and the verifier can reject immediately.
This added ability to detect lies helps it verify more complicated languages - all languages in the large class called NEXP.

MIP*: Here there's something non-intuitive going on. We help the provers by giving them a quantum device, but it turns out it actually helps the verifier. The details here are complicated, but we may explore them further in future videos.

One frequent question is why the provers use the entanglement at all, even the honest ones. The reason is that the verifier forces them to use the entangelement.

To see why, consider the non-isomoprhism example: the verifier gives the prover a computational challenge. This challenge is designed such that the prover can win only if the graphs are non-isomorphic.

The quantum entanglement opens the door for new kinds of challenges that the provers can win. It wasn't obvious if any of them are useful for our purposes, but then one was discovered: there exists a challenge such that for a given program p:
1) The provers can only win if p halts.
2) And they can only win if they cooperate via the entanglement.
Condition (2) is logically important: without it the game would have been possible with MIP as well. The provers can choose not to use the entangelement, but then they'll lose. We assume the provers want to win.

Visualization of tensors - part 3A

udiprod 2024-11-01 | This videos visualizes the direct sum space and starts explaining about tensor product spaces.
It shows a simple example from classical physics and a two qubit system as an example from quantum physics. We demonstrate how tensors are related to quantum entanglement.

Notes about confusing terminology:
--------------------------------------
* The term "tensor product" can refer to the combination of two vector spaces W=U*V, creating a tensor product space. But it also sometimes refers to the combination of two vectors u and v creating a simple tensor w=u*v. The latter is sometimes also called "tensor product mapping". In the video we used an outer product, which can perform the tensor product mapping given a coordinate system. Later in the video we refer to it as "tensor product welding".
* The term "tensor" is sometimes defined as an element in a tensor product space W=U*V. Sometimes though the definition is more restrictive, stating tensors are only elements of a tensor product space if the vectors spaces are the same, i.e., W=V*V (allowing also for dual spaces, something we haven't touched yet).
* The term "simple tensor" used in the video has some alternatives: "decomposable tensor", "pure tensor", or "elementary tensor".
* Quantum physics uses "Hilbert spaces" which is a vector space with some additional properties, among them the existence of an inner product operation. A postulate of quantum physics says every system has a Hilbert space as its state space.
* If two systems have state spaces U and V, then the combined system's space is their tensor product W=U*V, as explained in the video. For many purposes W is just another Hilbert space which acts as the state space of a system, hence its elements are still called "vectors". We need their tensor-ish properties only for some purposes.
* The video shows the distinction between separable states and entangled states. There is another distinction between "pure states" and "mixed states". A pure state corresponds to an actual physical state. The video only shows pure states. A mixed state reflects "classical uncertainty", i.e., the fact that we may not know the physical state in full. For example, say we put in two boxes an 'off' qubit and an 'on' qubit. Then we shuffle the boxes and pick one. We know this box contains a qubit in an 'off' state or 'on' state, with 50-50% distribution. This is a mixed state: the qubit is either off or on, we just don't know. This is a different from 50-50% superposition which is a pure state: the physical state itself contains uncertainty about the on-off property.
* Quantum physics has formalism for handling mixed states which includes an "outer product" operation, resulting with a "density matrix". This is not what's shown in the video. The video only refers to pure states.

Visualization of Radix sort

udiprod 2024-06-15 | A visualization of the Radix sort algorithm.
We start with a simpler algorithm: Pigeonhole sort (sometimes also called Bucket sort or Bin sort, see below). Then discuss stability of sorting algorithm, and finally Radix sort.

Links:
Quick sort vs Bubble sort: youtu.be/aXXWXz5rF64
Merge sort vs Quick sort: youtu.be/es2T6KY45cA
Heap sort: youtu.be/H5kAcmGOn4Q
Insertion sort vs Bubble sort: youtu.be/TZRWRjq2CAg
Stooge sort and Bogo sort: youtu.be/bfzYj-qGw7U
Shell sort vs Insertion sort: youtu.be/g06hNBhoS1k

About Radix sort:
It dates back to Hollerith's sorting machines from around 1890.
The machines only did Pigeonhole sort, and the human operators were instructed how to use this as a step in Radix Sort.
Initial instructions were for MSD Radix Sort, but apparently an anonymous human operator discovered LSD Radix Sort is easier.

About Pigeonhole sort:
Sometimes it is called Bucket sort or Bin sort. But usually these two refer to an algorithm where each 'bucket' or stack contains a range of possible values, and not just one. Each bucket is then sorted using some algorithm. If each bucket is sorted using Bucket sort recursively, then we get MSD Radix sort.
Also Counting sort is pretty similar to Pigeonhole sort, except it just counts the number of values in each bucket, instead of actually moving them to the bucket.

See more details: udiprod.com/radix-sort

Riddle: jumping spiders in a square

udiprod 2024-04-29 | A riddle about 4 jumping spiders arranged in a square.

Visualization of tensors - part 2B

udiprod 2024-01-02 | Part 2 is devoted to the electromagnetic tensor and deals mostly with this example. You can safely skip to part 3 for a more general tensor discussion.

Part 2B continues explaining how a sphere with arrows visualizes the electromagnetic tensor. This time we move to a 4-dimensional space, and show how now it can represent both the electric and the magnetic field. We show how different coordinate systems can cause it to change from electric to magnetic and vice versa.

Visualization of tensors - part 2A

udiprod 2023-10-17 | Part 2 is devoted to the electromagnetic tensor and deals mostly with this example. You can safely skip to part 3 for a more general tensor discussion.

Part 2A explains how our sphere-with-arrows visualization can represent magnetic fields. Part 2B extends it to electric fields as well.

A relativistic electromagnetism example

udiprod 2023-06-03 | This video shows an example of a particle moving in a uniform electric field.

1. First we use the classical law of motion: F=ma. According to this law uniform force means uniform acceleration, and the particle can reach the speed of light, and continue to accelerate. According to relativity, this can't be true.

2. The classical law can also be formulated: F=d/dt mv. Meaning: the change of momentum is proportional to the force. This works also in relativity, only this time the mass m is changing. It grows as the velocity grows. This makes it harder for it to increase its velocity further. This prevents it from ever reaching the speed of light.

3. Now we switch to a different point of view: an observer moving at 60% the speed of light to the left. This observer views the same scene, but from this point of view the particle and the hoop have an initial velocity of 60% the speed of light to the right. If we used the F=ma law, then the particle would have accelerated downwards, while v_x would remain at a steady 0.6. But with the relativistic law something different happens: the x momentum, m v_x, remains constant, because there's no force along the x direction. But m grows, so it means v_x diminishes. The hoop however floats steadily at 0.6, so the particle falls short. This can't be true of course, the second observer is watching the same scene: the particle should pass through the hoop.

4. The 4th scene fixes that. According to relativity, when we switch between points of view we should transform the electromagnetic field. In our example, this means the electric field grows by 25%, and a magnetic field shows up. The 'x' marks shown in the video are a common convention for a vector field pointing into the screen, along a Z axis that is not shown. The magnetic force is perpendicular to the magnetic field and to the particle's velocity. It provides just the right amount of pull along the x-axis to keep v_x steady.

5. The 5th scene is a sneak peek into visualization of tensors part 2, due in August.

Visualization of tensors - part 1

udiprod 2022-12-22 | This video series visualizes tensors using a unique and original visualization of a sphere with arrows.

Part 1 introduces the concept using the Cauchy stress tensor.

Note that this series talks about the term 'tensor' as used in physics and math. In the field of AI the term 'tensor' was borrowed with a simplified meaning. In AI it simply means a multi-dimensional array. So for example the array "double a[4][6]" (4 by 6 matrix of doubles) is called a second-order tensor, but there's no special relationship to vector spaces, as shown in the video.

Visualization of Einsteins special relativity [HD]

udiprod 2022-08-10 | This is a remake of my video from 2008, rendered in HD, with narration and minor changes.

This video demonstrates the effects of Einstein's special relativity on objects that move at high velocities due to the Lorentz transformation.

The Lorentz transformation was already known a few years before relativity, but mostly in attempts to justify the classical conception of Ether and absolute space.

The theory of relativity was created based on two principle:
* The principle of relativity: there's no absolute space, and physics works equally correct for different observers.
* Light speed is a fundamental property of physics.
From these two the Lorentz transformation follows as the only possible transformation.

The video shows a 3-dimensional view containing 2 dimensions of space and one dimension of time.

Shell sort vs Insertion sort

udiprod 2022-05-22 | Introduction of Shell sort, and a match with Insertion Sort.

For an introduction of Insertion sort, see:
youtu.be/TZRWRjq2CAg

Choosing the sequence 9-6-1:
For a list of size 10, the gaps can be any number 1,2,....,9, and the sequence must end with 1.
So each of the gaps 2,3,...,9 can be included in the sequence, or not included. So there's a total of 2^8=256 possible gap sequences. For each we checked the average number of comparisons for all possible 10! permutations.
Here are the 3 best sequences:
9-6-1: 25.512 comparisons
4-1: 25.516 comparisons
6-1: 25.539 comparisons

We could have also checked which has the highest probability of performing less comparisons than insertion sort. Here are the top 3 in this respect:
4-1: prob=0.72
9-4-1: prob=0.704
6-1: prob=0.701

Analyzing Shell sort variants more generally: See it explained in my home page:
udiprod.com/shell-sort/#analysis

Why did Shell sort lag behind in the second match in comparisons per second? You are welcome to post answers. Or read answer here:
udiprod.com/shell-sort/#timing

Gamification of Bells Theorem

udiprod 2022-03-07 | This video shows a gamified version of Bell's Theorem called the "CHSH Game".

The theorem proves the non-local nature of quantum physics, known sometimes as "Spooky action at a distance". The gamified version further shows that quantum entanglement can be useful.

See more about the usefulness of quantum entanglement in this video about the relation of such games and interactive proofs, and a recent result related to the halting problem:
youtu.be/2H8629BCbkM

Some history:
Bell's Theorem is due to John Stuart Bell in 1964.
The CHSH inequality, by John Clauser, Michael Horne, Abner Shimony, and Richard Holt, improved on it in 1969.
The CHSH game by R. Cleve, P. Hoyer, B. Toner and J. Watrous in 2004.

About "Bell Locality":
The formal statement of it is this requirement of the probability distribution:
P(x,y|a,b,particle1,particle2)=P(x|a,particle1)P(y|b,particle2)
This means that the outcomes are independent given the input bits and the respective particle statements.
Another way to say that is as stated in the video: that given the particle the top player has, we can specify the table P(x|a,particle1), and it makes no difference what happens with the bottom player (or the other way around of course).
Bell's theorem proves this is not what's happening.

As stated in the closing slide, this conclusion relies on a few more background assumptions:
* Each measurement has one outcome. According to the "many-worlds" theory a measurement can have multiple outcomes, and Bell locallity may still hold with this interpretation.
* The referee can choose random bits. According to "superdeterminisim" world view, there's no such thing as randomness, and it might be that the referee chooses bits somehow pre-determined to match the particles held by the players.

Some more links:
A primer on quantum physics: youtu.be/p7bzE1E5PMY
A more rigorous explanation of quantum spin (Physics Videos by Eugene Khutoryansky): youtu.be/3k5IWlVdMbo

Chapters:
0:00 Part 1: Decision problems
1:56 Part 2: Classical bound
6:20 Part 3: Quantum spin
13:44 Part 4: A Quantum Strategy
16:10 Part 5: Local realism
18:48 Part 6: No signaling
22:45 Part 7: Bell locality

[Confetti] Firing Squad synchronization problem

udiprod 2021-04-10 | This video presents a problem known as "Firing squad synchronization".

If you want to try to solve it, note it's not easy. Also, make sure you understand the rules. See the first paragraph here: udiprod.com/firing-squad-synchronization

The video had an earlier laser version. This version is now unlisted:
youtu.be/xV1aKUdlljU

[Laser] Firing squad synchronization problem

udiprod 2021-03-26 | The video shows a theoretical problem in computer science and a solution for it. It is presented as a riddle, but note it's not easy to solve.

The shown solution is based on the first solution proposed for this problem. It requires ~3n steps, and 17 states.

See list of states and some additional information here: udiprod.com/firing-squad-synchronization

SVM with polynomial kernel visualization (HD)

udiprod 2021-02-07 | NOTE: This is a new version in HD of my video from 2007. A brand new video is expected in next month.

A visual demonstration of the kernel trick in SVM.

This short video demonstrates how vectors of two classes that cannot be linearly separated in 2-D space,
can become linearly separated by a transformation function into a higher
dimensional space.

The transformation used is:
f([x y]) = [x y (x^2+y^2)]

Visit my homepage udiprod.com

Slow sorting: Stooge sort and Bogo sort

udiprod 2020-07-25 | Watch sorting algorithms compete for the title of the slowest sorting algorithm.

If you don't want to watch the whole thing:
0:16 Stooge sort
5:13 Bogo sort
38:36 Discussion about shuffling algorithms

See more comments here:
udiprod.com/slow-sorting

Previous matches with bubble sort:
Vs. Insertion sort
youtu.be/TZRWRjq2CAg
Vs. Quick sort
http://www.youtube.com/watch?v=aXXWXz5rF64

Visualization of Tidal Forces

udiprod 2019-12-26 | An explanation of how tidal forces are generated using a flat model of the Earth with beads attached to it.
This is the source of tides on Earth, though the actual mechanism for generating tides is more complicated with many subtleties.

See more details about the simulations shown in the video here:
udiprod.com/tidal-forces

A video visualizing why the Coriolis force deflects moving objects:
http://www.youtube.com/watch?v=49JwbrXcPjc

Insertion Sort vs Bubble Sort + Some analysis

udiprod 2017-11-11 | A visual demonstration of insertion sort, competition with bubble sort, and performance analysis including these two and quick sort.

See more notes and a more accurate analysis of the algorithms performance:
udiprod.com/insertion-sort

Previous matches:

Heaps sort vs merge sort
youtube.com/watch?v=H5kAcmGOn4Q

Merge sort vs quick sort
http://www.youtube.com/watch?v=es2T6KY45cA

Quick sort vs bubble sort
http://www.youtube.com/watch?v=aXXWXz5rF64

Visualization of Quantum Physics (Quantum Mechanics)

udiprod 2017-01-31 | This video visually demonstrates some basic quantum physics concepts using the simple case of a free particle.

All the simulations here are based on real equations and laws. See more information here: udiprod.com/quantum-physics

The mathematics involved was taken from this excellent book:
amazon.com/exec/obidos/ASIN/0387989293/visualquantummec

Visualization of conservation of energy

udiprod 2015-12-15 | See explanations for this video in udiprod.com/energy

Heaps and Heap Sort

udiprod 2015-01-18 | A demonstration of heaps, heap sort, and a competition with merge-sort.
See here udiprod.com/heap-sort a more detailed discussion of the properties of heap sort.

Note that the procedures mentioned in the video, "sift-down", "heapify", and "sift-up", may be named differently in different descriptions of heapsort. The implementation is the same though.

Links:
---------

More details about this video: udiprod.com/heap-sort

Previous matches:
Merge Sort vs Quick Sort: http://www.youtube.com/watch?v=es2T6KY45cA
Quick Sort vs Bubble Sort: http://www.youtube.com/watch?v=aXXWXz5rF64

Next match: youtube.com/watch?v=TZRWRjq2CAg

Visit my homepage: udiprod.com

Merge Sort vs Quick Sort

udiprod 2014-01-17 | A demonstration of merge sort and a two round competition between merge sort and quick sort.

See more details here: udiprod.com/ms-vs-qs

Previous match: http://www.youtube.com/watch?v=aXXWXz5rF64
Next match: youtube.com/watch?v=H5kAcmGOn4Q

Proof That Computers Cant Do Everything (The Halting Problem)

udiprod 2013-09-27 | If you disagree or get confused by this video, read this FAQ: udiprod.com/halting-problem/#faq
Visit my home page: udiprod.com

This video gives an informal presentation of Alan Turing's Halting Theorem, a serious, highly influential result in computer science.

A few more comments on this video:

1) This video skips a lot of technicalities for sake of simplicity. There are many rigorous descriptions of this proof easily found on the web.

2) There really is an unbeatable checkers machine. See here: http://en.wikipedia.org/wiki/Draughts#Computer_draughts

Zuto: The Adventures of a Computer Virus - Book trailer

udiprod 2013-01-30 | An adventure story that takes place inside a computer.

Visit home page for more information:
http://www.zutopedia.com

Purchase from Amazon - Paperback:
http://www.amazon.com/Zuto-The-Adventures-Computer-Virus/dp/1477683305

Purchase from Amazon - Kindle edition:
http://www.amazon.com/Zuto-Adventures-Computer-Virus-ebook/dp/B008JSGFOU

Written by Udi Aharoni, illustrated by Gil Troitsa.
Trailer by the author (still images are by Gil).

The "Miscellaneous reviews" refer to the reviews of the Hebrew edition, which can be read (in Hebrew) here: http://www.zuto.co.il

A Visual Riddle (The Epitaph of Stevinus)

udiprod 2012-12-26 | A physics riddle following the "Epitaph of Stevinus".
A string of beads is placed on top of a triangular prism. One side of the string is longer, but the other is steeper, which raises the question: to which side the string will slide to, if at all?

This question was asked and answered by Simon Stevin over 400 years ago, in a thought experiment that proves
the mechanical advantage of inclined planes. I.e., it demonstrates that inclined planes can be employed amplify the force exerted by small weights, allowing them to lift heavier objects.

Read about the history of this riddle here: http://en.wikipedia.org/wiki/Inclined_plane#History

Visit my homepage, udiprod.com, or read about my latest book http://www.zutopedia.com

Visualization of Quick sort (HD)

udiprod 2012-09-22 | An animated demonstration of sorting algorithms.

Read about my new book, "Zuto: The Adventures of a Computer Virus", http://www.zutopedia.com, or visit my homepage, udiprod.com

This video shows two comparison based sorting algorithms: Bubble sort and Quick sort.

The algorithms are demonstrated by robots sorting balls by hue. Comparison based sorting algorithms must make decisions based solely on pairwise comparison results. This is visualized by the robot's short-sightedness, which forces them to perform pairwise comparisons in a very pronounced manner.

The video culminates in a contest. Each algorithm is presented with an identical series of balls and they compete to see which finishes the sorting task faster. In addition to raw speed, the number of comparisons they perform is also measured.
Advanced viewers will notice bubble sort performs comparisons quicker than quick sort. This is because quick sort has to move around a lot between comparisons. Quick sort aims at reducing the number of comparisons, but in domains where movement is costly, it becomes inferior to other algorithms, which aim to reduce movement as well.

Cars: engine power and transmission - 3D animation

udiprod 2011-12-31 | The physical principles behind car gears.

Visit my homepage, udiprod.com, or read about my latest book http://www.zutopedia.com

This video offers an unusual perspective to the relation between engine power, RPM and acceleration.

Some comments:

1) Torque - a gear of radius R that exerts force F is said to have a torque of T=R*F. Engines do not produce force per se, but torque. So if the engine produces a torque of T, and that is transmitted unchanged to a gear of radius R, you a get a force of T/R.
The video offers an alternative view, where the relation between RPM, power and accelartion is explained without torques.
See more about torques in this video: http://www.youtube.com/watch?v=Sm4pV3xyJRE

2) Efficiency - this video assumes and ideal world with 100% efficiency. No energy is wasted on friction or any other mechanical losses.

3) Another simplifying assumption is that the engine produces a constant torque, and thus a constant force. As mentioned in one of the comments in the end, this is not the case with real engines. In most engines the torque rises as RPM rises, peaks at some RPM, and then starts to fall. Because of that the power does not steadily increase with RPM. It increases in the beginning, and continues to increase a little bit after the torque peaks. Then as the torque starts to dimish significantly, the power reduces as well.

4) Measurments - here are the measurments used in the simulation:
16 teeth gear radius: 18cm = 0.6 ft
Force exerted by this gear: 196N = 44.1 lb
Car wheel radius: 21cm=0.7 ft
Car weight: 352kg = 775 lb

5) The car wheels are slightly bigger than the 16-teeth gear, and this is adds up to the gear ratio effect. In the first car, for example, while the transmission box has a gear ratio of 1:1, the larger wheels cause an overall gear ratio of 1.18:1. This can be seen at time 2:25, where the wheel's power visualization is slightly elongated (a bit more speed and a bit less force).

Visualization of Torques (Moments)

udiprod 2010-12-27 | This video demonstrates the concept of Torque, a.k.a. Moment or Moment of force.

Visit my homepage, udiprod.com, or read about my latest book http://www.zutopedia.com

Here are some notes:

1) Torques are often mentioned in context of engine power and transmission gears. Engine manufacturers usually specify the maximal Torque (T) their engine is capable of producing. If you know the radius (R) of the gear attached to the engine, you can compute the force it exerts: T*R. Transmission gears along the way to the car wheels can increase or decrease this force.
The following video explains that using the concept of engine power: http://www.youtube.com/watch?v=3-ilzxawUAs

2) Many scales don't have a straight lever, but instead a downward bending lever. This produces the same effect of placing the points where the weight's forces are applied below the pivot, as explained at 1:37-1:55.

3) The video gives a visual definition of "cross-product". It is slightly different, though equivalent of course, than the traditional one. In particular, the vectors are placed one following another instead of placing both at the origin, and the right-hand rule is shown differently.

4) The last scene in space: this is a more general example since the object is not attached to a pivot, so it can arbitrarily move and rotate. The situation where there sum of forces is zero is a special one: In this case the object will not move but it may rotate, depending on the torques. Also special in this case that one may calculate the torques about any point (i.e., any point may be chosen as the pivot) and the torque vector result will be the same. From the symmetry of the object shown we now the true pivot will be in the middle.

Visit our sponsors: http://www.therockerspaniels.com

מיון מהיר

udiprod 2009-02-28 | סרטון המדגים את אלגוריתם המיון "מיון מהיר".

הסרטון נעשה עבור האתר
http://www.zuto.co.il
שמלווה את הספר "זוטו - הרפתקאותיו של וירוס מחשבים"

הסרטון מדגים שני אלגוריתמי מיון:
מיון מהיר ומיון בועות.
שניהם אלגורתמים מבוססי השוואות, כלומר הם פועלים על סמך השוואות בלבד. לכן הם יכולים למיין כל סוג של עצמים שאפשר להשוות בינהם: מספרים, מילים, כדורים צבעוניים...

Visualization of Quick sort

udiprod 2009-02-21 | See a new version of this video: http://www.youtube.com/watch?v=aXXWXz5rF64 (HD + narration)

A demonstration of the Quick Sort sorting algorithm.

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It demonstrates two comparison sorting algorithms: Bubble sort and Quick sort.
Comparison sorting algorithms are only allowed to 'see' the data through a sequence of pair-wise comparisons, therefore they are applicable to any type of comparable objects: numbers, strings, colored balls, etc

Bubble sort is very simple but has poor performance. A comparison sorting algorithm's performance is usually measured by the number of comparisons it makes. Bubble sort performs on the order of n^2 comparisons to sort n elements.

Quick sort is only slightly more complicated but usually performs much better (as demonstrated in the video). It performs on average an order of n log(n) comparisons to sort n elements. This is much lower than n^2 for large values of n. However, if the algorithm makes some 'unlucky' choices it might require n^2 comparisons after all.

Other algorithms exist that guarantee the number of comparisons will not exceed n log(n), however, in practice Quick sort usually out-performs all other comparison sorting algorithms due to its simplicity.

If other operations other than pair-wise comparisons are allowed, then a broader range of algorithms can be used. Some of them can perform much faster than Quick sort, but they are limited to a particular type of elements, e.g., numbers is a certain range.

Visualization of Einsteins special relativity

udiprod 2008-01-28 | A visual demonstration of the effects of Einstein's relativity.
See a new version of this video (HD + narration): youtu.be/CS7aLVHpeTs

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This video demonstrates the effects of Einstein's special relativity on objects that move at high velocities. More particularly, it visualizes the Lorentz transformation.

The video shows a 3-dimensional view containing 2 dimensions of space and one dimension of time. This view is used to demonstrate the difference between classical physics and Einstein's relativity, and why the latter was necessary to understand experimental results.

Visualization of the Coriolis and centrifugal forces

udiprod 2007-12-29 | A visual demonstration of the effects of the Coriolis and Centrifugal forces.

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This clip demonstrates the effects of the Coriolis and Centrifugal forces, by viewing various scenes from both rotating and stationary cameras. (The Coriolis force is also known as the Coriolis effect).

The first example shows a cannon fixed to a rotating disc. The cannonballs
fly in straight lines since once shot no force acts on them. When
this scene is viewed from the disc's frame of reference (i.e., as would
be seen by a viewer that stands on the disc) the cannonballs seem to
fly in a curved path. This demonstrates that in a rotating frame of reference
one must take into account the Coriolis and Centrifugal forces.
(Read more about them in Wikipedia:
http://en.wikipedia.org/wiki/Centrifugal_force,
http://en.wikipedia.org/wiki/Coriolis_effect)

The second example shows a pendulum swinging over a rotating disc.
A pendulum swinging through a small angle approximates what is called "harmonic motion"
in which the ball is pulled to the center by a force proportional to
its distance to the center. In the pendulum, the string exerts a force
whose vertical component balances gravity and the horizontal component
(shown in the clip) causes the harmonic motion (approximately).
The disc and the pendulum has the same period,
meaning both complete a cycle at the same time.

When viewed from the disc's frame of reference the centrifugal and Coriolis
forces appear yet again. This time the centrifugal force balances
the string's horizontal component. This is because the centrifugal force is
also proportional to the distance from the center but pushing outward instead
of inward. The equal periods make the factor of proportion the same for both.
This leaves the Coriolis force alone to act on the ball. Since it is always
perpendicular to the object's path, this creates a perfectly circular path.

Finally, we decrease the pendulum's period to be 2/3 of the disc's period.
Now the centrifugal force no longer balances that of the string and the
motion becomes more complicated.

Note that there's a small scaling error in the visualization: the coriolis force shown should actually be double in length.

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The Rocker Spaniels cross the street

udiprod 2007-09-15 | Visit my homepage http://www.zutopedia.com/udia.html, or read about my latest book "Zuto: The Adventures of a Computer Virus", http://www.zutopedia.com

In this short home made video you can see The Rocker Spaniels, the greatest dog band in the history of rock, crossing the street toward their recording studio carrying they musical instruments (as always). See more in http://www.therockerspaniels.com

SVM with polynomial kernel visualization

udiprod 2007-02-05 | See a new version of this video in HD: youtu.be/OdlNM96sHio

A visual demonstration of the kernel trick in SVM.

This short video demonstrates how vectors of two classes that cannot be linearly separated in 2-D space,
can become linearly separated by a transformation function into a higher
dimensional space.

The transformation used is:
f([x y]) = [x y (x^2+y^2)]

Visit my homepage udiprod.com, or read about my latest book "Zuto: The Adventures of a Computer Virus", http://www.zutopedia.com