text:
How does the recent Chinese quantum supremacy claim compare with Google's?

Very recently, China claims achieving 'Quantum Supremacy' with the world's fastest Quantum Computer. Their computer was designed as a circuit of lasers, beam splitters and mirrors (see figure bellow) and to solve a very specific problem called Gaussian boson sampling.

The quantum computer from Google, for example, consists of microscopic circuits of superconducting metal that entangle 53 qubits in a complex superposition state. In the case of IBM, their quantum chips are formed by microwave resonators in combination with Josephson superconducting junctions. In case of the IBM, there is already, an open source SDK called Qiskit that expresses quantum computing concepts intuitively and concisely in Python. In both cases, it is possible to program the computers to solve different tasks.

My questions is: how honest is the Chinese claim of quantum supremacy?

"how honest is the Chinese claim of quantum supremacy?"

It's equally (or at least as) honest in comparison to Google's claim.

In a comment to this answer at Quantum Computing Stack Exchange, Craig Gidney (who works at Google and was a co-author on Google's Quantum Supremacy paper), confirmed that the classical computer would have been 2^(20*7/4) = 34359738368 times faster if Google's hardware used CZ gates instead of their gates. This would mean that the 10,000 years that they said it would take classical computers to do the calculation that Google did, would become 9 seconds.

Furthermore IBM pointed out that the estimations Google made about the time it would take the classical computer, assumed that the classical computer could only use RAM and no disk at all, and if you allow the classical computer to use disk (which in this case, since they used Oak Ridge National Lab's Summit supercomputer, has petabytes available rather than just terabytes) then even though I/O is slower on disk than RAM you will save so much time by not re-calculating things, that you will get several orders of magnitude further speed-up for the classical computer.

I'm not aware of anything like this for the Chinese experiment, but to be clear: boson sampling is not enough to do "universal" computation, however universal computation is not a requirement for quantum supremacy. Boson sampling allows us to solve #P problems for which no polynomially scaling classical algorithms are yet known.

Furthermore, Figure 1 of this paper shows that it wasn't hard for the authors to simulate 144 qubits on a classical computer. That paper was published 1 year earlier than Google's quantum "supremacy" claim! This was discussed in more detail here.

In any case, "quantum supremacy" is a buzzword and until people are actually using quantum devices to solve useful problems, classical computers are superior because people actually find them valuable enough to purchase.

• Re: "buzzword", I think that's why it's being dropped in favor of quantum advantage. Dec 14, 2020 at 19:10
• @stafusa The reason the phrase is being discouraged is much more because of the historical context of the word "supremacy" being associated with racism. My paragraph about it being buzzword would remain the same if it was called "quantum advantage" because truly there is zero computational advantage of using a quantum computer right now, for any real-world computation. I don't deny though, the benefits of pursuing quantum technologies, since if we can improve them, they may truly have advantages over classical technology. Dec 14, 2020 at 19:23
• In reference to the claim that "no one knows how to program anything with [the gate that was used]": Section VII-F of the supplement to the paper explains how to decompose arbitrary two qubit interactions into the gates that were used and this conversion is available in Cirq via the cirq.google.ConvertToSycamoreGates() optimizer. I certainly won't say it's a convenient gate set, but people have used it. Dec 16, 2020 at 18:41
• @CraigGidney That comment is very much appreciated, and I will incorporate it into the next edit of my answer, thanks so much! The decomposition of CZ into fSim gates seems even more complicated than I ever imagined. It seems if I really wanted to solve a small version of a real-world problem on Sycamore with gates I know (like CZ), there would be an overhead due to these extra gates that are needed, which will slow down Sycamore's speed, not by 2^35 but the classical computer also won't be simulating CZ either (it would be solving the problem directly). Real supremacy needs lots more qubits! Dec 16, 2020 at 19:01
• @NikeDattani There is actually a more direct decomposition of arbitrary 2 qubit interactions that doesn't go through the CZ gate, which uses at most 4 Syc gates (instead of the 6 you'd need using the CZ decomposition as an intermediary). I think this is what is done in cirq, though it is not explained in the paper. For comparison, decomposing into CZs you'd need at most 3 operations to do arbitrary interactions. Dec 16, 2020 at 19:13


text:
# Browse Classroom Capsules and Notes

You can filter the list below by selecting a subject category from the drop-down list below, for example by selecting 'One-Variable Calculus'. Then click the 'APPLY' button directly, or select a subcategory to further refine your results.

Displaying 1 - 10 of 1212

The author reviews the evidence of authorship of Cramers Rule and comes to the conclusion stated in the title of the paper.

The fact that, for an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the height of the triangle is shown visually.

The author considers two infinite decimals, where the $n$th digit is the last non-zero digit of $n!$, creating the number $F$, and using $n^n$, creating the number $P$. The author shows...

The geometry of Archimedes proof of a formula for a sum of squares is depicted visually.

In this note the authors present an elementary proof that the inequalities $(1+1/n)^n < e \leq (1+1/(m-1))^m$ hold for $n>0$ and $m>1$, using only the arithmetic-geometric mean...

This capsule uses the determinants of matrices to study Fibonacci numbers. Specifically, the sum property of the determinant is used to derive identities between Fibonacci numbers.

This capsule project starts with computing average class size from the institution and student points of view. The results are drastically different. The author then tries to explain why this...

This  project uses a sampling problem to compute certain probabilities. In the process, certain binomial identities are established.

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text:
+0

# volume

0
846
14

john wants to place a layer of mulch on top of his 12ft by 24ft vegetable garden. if a 1.5ft^3 bag of mulch costs $4, how much will it cost to place a 3-inch layer of mulch on the garden? Feb 20, 2015 ### Best Answer #13 +1036 +5 OK. The link now functions. Mar 1, 2015 ### 14+0 Answers #1 +109 +5 Alright, so I was curious to give this question the old college try, and here's what I came back with. Lol. Starting off, we know that John's vegetable garden is 12' by 24'. We also know the volume of the mulch bags he'll be using are 1.53ft, and that they cost$4/bag.

Garden Area = 12ft * 24ft = 288ft2

Volume of Mulch Bag = 1.53ft = 3.375ft3

To find out how much area one mulch bag covers, spread at a height of 3'', we'll permutate our cube's dimensions proportionately. We know that the volume does not change, the height is 3'', and that the width and length of our cube are equal to one another.

Width = x

Length = x

Height = 0.25' (Converted from inches to feet)

$$Volume = L * W * H = 3.375ft^3$$

Replacing what we know in the Volume equation gives us:

$$3.375 = 0.25 * X^2$$

Solving for X gives us:

$$X^2 = 13.5ft^2$$

$$X = sqrt(13.5) \approx 3.674ft$$

Now, we know that one bag of mulch covers 13.5ft2 at a height of three inches. We divide the area of John's garden, by the area of one bag of mulch at 3" tall.

$${\frac{{\mathtt{288}}\left[{{ft}}^{{\mathtt{2}}}\right]}{\left({\mathtt{13.5}}{\mathtt{\,\times\,}}\left({\frac{{{\mathtt{ft}}}^{{\mathtt{2}}}}{{\mathtt{bag}}}}\right)\right)}} = {\mathtt{21.33}}{\mathtt{\,\times\,}}{\mathtt{bags}}$$

Multiply this by John's cost per bag, giving us:

$$\left({\mathtt{21.33}}{\mathtt{\,\times\,}}{\mathtt{bags}}\right){\mathtt{\,\times\,}}{\mathtt{4}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{dollar}}}{{\mathtt{bag}}}}\right) = {\mathtt{85.33}}{dollar}$$

It'll cost $85.33. Feb 20, 2015 #2 +116078 +5 3 in= .25 ft So we have 12 x 24 x .25 = 72 cu ft Each bag covers 1.5 cu ft So 72 / 1.5 = 48 bags required x$4 per bag = \$192

Feb 21, 2015
#3
+1036
+3

Well, Sorasyn it’s a good thing you went to college where you leaned to permutate dimensions. I’ve never heard of it. The closest is permutate wood. This was a program control on a machine used by Purina to make termite food from shipping pallets.

Any way, "Mary, Mary, is quite contrary watching her garden grow. It seems her gardener didn’t buy enough mulch. It’s understandable though, her gardener has the blight of CDD. Permutating this we have DDC and DCD.

Well, until next time, Sorasyn Keep up the permutations.

Feb 21, 2015
#4
+89
0

Funny troll. Maybe Sorasyn formula permutate the cube to termite food instead of mulch. Maybe some DDT will k**l termites and cure CDD. Hahaha

Why he get 3 points for wrong answer?

Feb 21, 2015
#5
+109
+5

Well it was a shot. Lol.

Feb 21, 2015
#6
+1036
0

Root’s idea is a good one. How about it, Sorasyn, can you modify DDT to cure CDD? If so, you will be on the short list to win the Nobel Prize –or at least some permutation of it.

I would add a few more snarky comments, but Melody uses an algorithm that does a fair job of permutating snarky comments into Scooby snacks.

Why he get 3 points for wrong answer?

It balances out.  I sometimes get negative points for great answers. We trolls like them. It means we’re doing a good job.

Feb 21, 2015
#7
+116078
0

I gave Sorasyn 3 points for a good try.....and since I'm a mod....I have the power to giveth ....or taketh away.....

Feb 21, 2015
#8
+1036
+5

Well, Root, you have your answer: Mod = God ± 3 points. However, that’s not their greatest power. The mod version of Zeus’s lightning bolts are what you have to pay attention to. In an instant, a fine piece of troll art and labor will be vaporized and rendered invisible to all except the forum’s gods. Hera turns the best ones into tasty, baked Scooby snacks.

Yep, the gods have some fine art to admire while having a snack. Although, they do have to sweep up vast amounts of gutter clutter tossed about by the Morlock-classed (morlockus troglodytes) trolls. I suppose the private art collection and snacks are really just a minor gratuity. I’m happy to contribute to the gratuity jar and offering plate.

Feb 21, 2015
#9
+112513
+5

Yes well Nauseated, you are correct about one thing, troll droppings are the only private offerings that us moderator 'Gods' are ever given.

Doesn't really pay to be a 'God' does it.

Feb 22, 2015
#10
+116078
+5

Yeah....we "Gods" get droppings from "doGs".......this is only a slight permutation.....I don't think Sorasyn would mind.........

After all...."Nauseated" does almost permute to "Seated A**s".....well....give or take an "s" or two......Mmmmmm.....I'm beginning to see EXACTLY where those droppings are originating from, now.......

And before it's pointed out....yes "CPhill" almost permutes to "Phallic".....give or take an "a"...or two....which is only one step away from being  "Richard Cranium"......  I'll let the "slower ones" use Google Translate for further understanding.....

Feb 22, 2015
#11
+1036
+5

LMAO! You could be a standup comedy writer, CPhill --or at least a sit down comedy writer.

Who would ever think permutations could generate so much humor? But they do:

http://web2.0calc.com/questions/how-come-when-i-put-in-numbers-like-if-i-put-in-60-it-will-show-up-back-wards-like-06-what-did-i-freakin-do#r2

Feb 28, 2015
#12
+112513
0

Feb 28, 2015
#13
+1036
+5

Nauseated Mar 1, 2015
#14
+112513
0

You are right Nauseated.

Permutations can be highly amusing.

Thanks Chris and Nauseated.

Mar 1, 2015


text:
Proving the equivalence of two definitions of NFA acceptance

Recall that an NFA $$N = (Q,\Sigma,\delta,S,F)$$ accepts $$w=w_1w_2\ldots w_n$$, where $$w_i \in \Sigma$$, if one of the following holds:

(a) $$\hat\delta(S,w) \cap F \neq \emptyset$$, where $$\hat\delta\colon P(Q) \times \Sigma^* \to P(Q)$$ is given by $$\hat\delta(Q',w) = \begin{cases} Q' & w = \epsilon \\ \bigcup_{q \in \hat\delta(q',w_1\ldots w_{n-1})} \delta(q,w_n) & |w| = n > 0 \end{cases}$$

(b) $$\exists r_0,\ldots,r_n$$, where $$r_i \in Q$$, such that:

• $$r_0 = q_0$$

• $$r_n \in F$$

• $$r_{i+1} \in \delta(r_i,w_{i+1})$$

Show that (a) $$\leftrightarrow$$ (b)

I think I have proven from (b) to (a) but I'm not sure, and I seem to have no idea where to start the other way around. This is part of the first exercise in the computational models course I'm taking and I would really appreciate the help.

• First, generalize the problem: show that, for any $$q\in Q$$ and any word $$w$$, $$q \in\hat\delta(S,w)$$ if and only if $$\exists r_0,...,r_{|w|}\in Q$$ such that $$r_0\in S$$, $$r_{|w|}=q$$, and $$r_{i+1}\in\delta(r_i, w_{i+1})$$.
• Second, show this statement by induction on the length $$|w|$$ of $$w$$. This is trivial for the base case, and should be easy for the inductive case, just apply the induction hypothesis and the definition of $$\hat\delta$$.
• Finally, show that your equivalence follows from the general statement: apply it for any $$q\in F$$.


text:
Question

# If $$\triangle ABC$$ is right angled at $$C$$, then the value of $$\cos(A+B)$$ is

A
0
B
1
C
12
D
32

Solution

## The correct option is A $$0$$Since ABC is a right angled triangle right angled at CHence$$A+B=180^{0}-C=90^{0}$$Hence$$cos(A+B)$$$$=cos(90^{0})$$$$=0$$Mathematics

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0

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text:
# Neural Network

## The Fine Line between Linguistic Generalization and Failure in Seq2Seq-Attention Models (Weber et al., 2018)

We know that training a neural network involves optimising over a non-convex space, but using standard evaluation methods we see that our models…

## Attention Strategies for Multi-Source Sequence-to-Sequence Learning (Libovicky et al., 2017)

To apply attention across multiple input sources, it is best to apply attention independently and then have a second phase of attention over the summary vectors for each source.

## Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer (Shazeer et al., 2017)

Neural networks for language can be scaled up by using a form of selective computation, where a noisy single-layer model chooses among feed-forward networks (experts) that sit between LSTM layers.

## Dynamic Evaluation of Neural Sequence Models (Krause et al., 2017)

Language model perplexity can be reduced by maintaining a separate model that is updated during application of the model, allowing adaptation to short-term patterns in the text.

## Searching for Activation Functions (Ramachandran et al., 2017)

Switching from the ReLU non-linearity, $\text{max}(0, x)$, to Swish, $x \cdot \text{sigmoid}(x)$, consistently improves performance in neural networks across both vision and machine translation tasks.

## Attention Is All You Need (Vaswani et al., ArXiv 2017)

To get context-dependence without recurrence we can use a network that applies attention multiple times over both input and output (as it is generated).

## DeftNN: Addressing Bottlenecks for DNN Execution on GPUs via Synapse Vector Elimination and Near-compute Data Fission (Hill et al., MICRO 2017)

GPU processing can be sped up ~2x by removing low impact rows from weight matrices, and switching to a specialised floating point representation.


text:
# Difference between revisions of "Kerbin"

 Kerbin Kerbin as seen from orbit. Planet of Kerbol Orbital Characteristics Semi-major axis 13 599 840 256 m [Note 1] Apoapsis 13 599 840 256 m [Note 1] Periapsis 13 599 840 256 m [Note 1] Orbital eccentricity 0 Orbital inclination 0 ° Argument of periapsis 0 ° Longitude of the ascending node 0 ° Mean anomaly 3.14 rad (at 0s UT) Sidereal orbital period 9 203 545 s 426 d 0 h 32 m 24.6 s Synodic orbital period Not defined Orbital velocity 9 285 m/s Physical Characteristics Equatorial radius 600 000 m Equatorial circumference 3 769 911 m Surface area 4.5238934×1012 m2 Mass 5.2915158×1022 kg Standard gravitational parameter 3.5316000×1012 m3/s2 Density 58 484.090 kg/m3 Surface gravity 9.81 m/s2 (1 g) Escape velocity 3 431.03 m/s Sidereal rotation period 21 549.425 s 5 h 59 m 9.4 s Solar day 21 600.000 s 5 h 59 m 60 s Sidereal rotational velocity 174.94 m/s Synchronous orbit 2 863.33 km Sphere of influence 84 159 286 m [Note 1] Atmospheric Characteristics Atmosphere present ✓ Yes Atmospheric pressure 101.325 kPa 1 atm Atmospheric height 70 000 m 1.0×10-6 atm Temperaturemin -86.20 °C 186.95 K Temperaturemax 15 °C 288.15 K Oxygen present ✓ Yes Scientific multiplier Surface 0.3 Splashed 0.4 Lower atmosphere 0.7 Upper atmosphere 0.9 Near space 1 Outer space 1.5 Recovery 1 The distances are given from the bodies center, not from the surface (unlike ingame)

Kerbin is the home planet of the Kerbals, the location of the Space Center, and the main focus of Kerbal Space Program. It is also the Earth analog for the game and has two moons named Mun and Minmus.

Kerbin is the third planet in orbit around the star Kerbol. It is the third largest celestial body that orbits Kerbol, following Jool and Eve. Jool's moon Tylo has the same radius of Kerbin, though it may be classified as larger, as the highest point on Tylo is about 5 km higher than the highest point on Kerbin. However, Tylo has only 80 % of Kerbin's mass.

Reaching a stable orbit around Kerbin is one of the first milestones a player might achieve in the game. With the introduction of version 1.0.3, attaining low Kerbin orbit requires a Δv of approximately 3400 m/s (vacuum), though the exact amount depends on the efficiency of the ascent profile and the aerodynamics of the launch vehicle and payload. The only planet that requires a higher Δv to attain orbit is Eve. Many interplanetary missions expend over half of their Δv in reaching Kerbin orbit. The velocity required to escape a body from a given altitude is always exactly the square root of two times the velocity of a circular orbit around the body at that height, leading one observer to remark:

 “ If you can get your ship into orbit, you're halfway to anywhere.— Robert Heinlein , quoted on page 194 of A Step Farther Out by Jerry Pournelle ”

## In-game description

 “ A unique world, Kerbin has flat plains, soaring mountains and wide, blue oceans. Home to the Kerbals, it has just the right conditions to support a vast, seemingly undepletable population of the eager green creatures. Reaching a stable orbit around Kerbin is one of the first things budding space programs strive for. It is said that those who can get their ship into orbit are halfway to anywhere.'' ”

## Topography

Topographical representation of Kerbin's surface as of 0.18.2. Click for high resolution. by Zeroignite
Kerbal at Kerbin's highest peak

Kerbin has a roughly equal distribution of liquid surface water and solid land, with polar ice caps and scattered deserts. Some of its mountains exceed 6 km in height, with the tallest peak being 6764.1 m in altitude at the coordinates 46°20'17" E 61°35'53" N. The lowest point is almost 1.4 km deep and about 313° south-west of the Kerbal Space Center.

### Craters

Terrain model centered on Kerbin's most pronounced craters

Unlike other bodies in its system, Kerbin has few visible craters because its environment would erode craters from the few meteors that avoid the gravity or surface of its large moon and survive entry. Nevertheless, some geological formations indicate that bodies have violently collided with Kerbin: a planetary feature appears to be an impact crater, while a secondary rupture lies on the other side of the planet (made by the intense longitudinal, or P-wave earthquakes that ensue.) Both are in excess of 100 km diameter, and the main crater lies along the far-western coastline. The uplift is easily visible as a series of islands, and the feature has a central peak that pokes up through the water (also known as a rebound peak.) The other, and smaller of the two, is near the prime meridian in the northern hemisphere and is more easily missed, but its uplift rims are visible, and it also has a central rebound peak.

## Biomes

The biomes on Kerbin as of 0.90

Before 0.90 Kerbin was one of the few bodies with multiple Biomes, Kerbin was second only to the Mun in number of biomes it has. Following the 0.90 update all celestial bodies have biomes. Science experiments can be performed at all biomes, though Kerbin's low multipliers result in less impressive results than more distant worlds. Kerbin's biomes show a loose correlation with Earth's biomes and geographic features. Uniquely, Kerbin has 33 location biomes at KSC, these are comprised of each building and their props, the crawlerway, the flag, and KSC itself; these give a jumpstart to gathering Science points in Career mode. With 1.2 Kerbin had destinct Northern and Southern Ice Shelf added

Kerbin biome map as of 1.2

### List of biomes

• Ice Caps
• Northern Ice Shelf
• Southern Ice Shelf
• Tundra
• Highlands
• Mountains
• Grasslands
• Deserts
• Shores
• Water

KSC mini biomes
• Baikerbanur
• Island Airfield
• Woomerang Launch Site
• KSC
• Astronaut Complex
• AstronautComplex Grounds
• Crawlerway
• FlagPole
• Mission Control
• MissionControl Grounds
• R&D
• R&D Central Building
• R&D Corner Lab
• R&D Main Building
• R&D Observatory
• R&D Side Lab
• R&D Small Lab
• R&D Tanks
• R&D Wind Tunnel
• Runway
• SPH
• SPH Main Building
• SPH Round Tank
• SPH Tanks
• SPH Water Tower
• Tracking Station
• Tracking Station Dish East
• Tracking Station Dish North
• Tracking Station Dish South
• Tracking Station Hub
• VAB
• VAB Main Building
• VAB Pod Memorial
• VAB Round Tank
• VAB South Complex(only exist when VAB is Level 2)
• VAB Tanks

## Atmosphere

Temperature and pressure of Kerbin's atmosphere as a function of altitude.
Drag force is proportionate to atmospheric density.

Kerbin has a thick, warm atmosphere with a mass of approximately 4.7×1016 kilograms, a sea level pressure of 101.325 kilopascals (1 atmosphere), and a depth of 70,000 meters. The atmosphere contains oxygen and can support combustion. Kerbin is the only planet or moon with a breathable atmosphere.

The average molecular weight of Kerbin air is 28.9644 g/mol, and its adiabatic index is 1.40. This suggests that Kerbin likely has an earthlike nitrogen-oxygen atmosphere. The air-fuel ratio of jet engines operating in Kerbin's atmosphere suggests that the percentage of oxygen is similar to that of Earth's atmosphere (about 21 %).

Like all other atmospheres in the game, Kerbin's atmosphere fades exponentially as altitude increases. The scale height varies with altitude, which is a change from pre-1.0 versions of the game. The pressure-altitude profile is globally constant and independent of temperature. The following table gives the atmospheric pressure and density at various altitudes above sea level. Temperature-altitude profile is not globally constant, therefore neither is the density-altitude profile, however variance is slight.

Altitude (m) Pressure (Pa) Pressure (atm) Density (kg/m^3)
0 101 325 1.000 1.225
2 500 69 015 0.681 0.898
5 000 45 625 0.450 0.642
7 500 29 126 0.287 0.446
10 000 17 934 0.177 0.288
15 000 6 726 0.066 0.108
20 000 2 549 0.025 0.040
25 000 993.6 0.010 0.015
30 000 404.1 0.004 0.006
40 000 79.77 0.001 0.001
50 000 15.56 0.000 0.000
60 000 2.387 0.000 0.000
70 000 0.000 0.000 0.000

Kerbin's atmosphere can be divided into three major layers, comparable to Earth's troposphere, stratosphere and mesosphere. In the lower and upper layers, temperature decreases as altitude increases, while the middle layer spans of a region of increasing temperature. The boundary between the lower and middle layers occurs at an altitude of about 16 km at low latitudes, and about 9 km at high latitudes. The boundary between the middle and upper layer occurs at an altitude of about 38 km.

Air temperatures vary with latitude and time of day. At the equator, sea level temperatures vary between a nighttime low of 32 °C and a daytime high of 41 °C. At the poles, the temperature varies between -35 °C and -30 °C. The globally averaged sea level temperature is approximately 13.5 °C. Since Kerbin has no axial tilt, there are no seasonal temperature variations.

The atmosphere of Kerbin is patterned after Earth's U.S. Standard Atmosphere (USSA), though with the vertical height scale reduced by 20 %. Kerbin's "base" temperature and atmospheric pressure can be very closely approximated using the equations of the USSA, where Kerbin's geometric altitude, z, is converted to Earth's geopotential altitude, h, using the equation:

${\displaystyle h={\frac {7963.75\cdot z}{6371+1.25\cdot z}}}$.

The base temperature is the temperature less latitudinal and diurnal adjustments; it is roughly equal to the global mean temperature.

### Atmospheric flight

The thickness of Kerbin's atmosphere makes it well suited for aerobraking from a high-speed interplanetary intercept. The periapsis altitude required for a successful aerocapture depends on the spacecraft's drag characteristics, its approach velocity, and the desired apoapsis of the resulting orbit. For most conditions, a periapsis altitude of about 30 km should result in an aerocapture.

Parachutes perform well in Kerbin's dense air, allowing landings on both land and water to be accomplished without the aid of propulsion.

Because of the presence of oxygen, jet engines can operate in Kerbin's atmosphere. And, together with it's thickness, Kerbin's atmosphere is ideally suited for aircraft flight.

A modified Orbiter 1A prepares to dock with a space station.

## Orbits

A Sputnik-derived satellite

A synchronous orbit is achieved with a semi-major axis of 3 463.33 km. Kerbisynchronous Equatorial Orbit (KEO) has a circularly uniform altitude of 2 863.33 km and a speed of 1 009.81 m/s. From a 70 km low equatorial orbit, the periapsis maneuver requires 676.5 m/s and the apoapsis maneuver requires 434.9 m/s. A synchronous Tundra orbit with eccentricity of 0.2864 and inclination of 63 degrees is achieved at 3799.7/1937.7 km. Inclination correlates with eccentricity: higher inclined orbits need to be more eccentric, while equatorial orbit may be circular, essentially KEO.

A semi-synchronous orbit with an orbital period of ½ of Kerbin's rotation period (2 h 59 m&nbsp34.7 s or 10774.7  seconds) is achieved at an altitude of 1 581.76 km with an orbital velocity of 1 272.28 m/s. A semi-synchronous Molniya orbit with eccentricity of 0.742[1] and inclination of 63 degrees can not be achieved, because the periapsis would be 36 km below the ground. The highest eccentricity of a semi-synchronous orbit with a periapsis of 70 km is 0.693 with an apoapsis of 3100.36 km.

The Hill sphere (the radius around the planet at which moons are gravitationally stable) of Kerbin is 136 185 km, or roughly 227 Kerbin radii.

## Delta-V Requirements

From the lowest possible stable orbit around Kerbin (70 km), the nominal amount of delta-V needed to reach other destinations is:

Body Δv
Mun ~860 m/s
Minmus ~930 m/s
Eve ~1033 m/s
Duna ~1060 m/s
Geostationary Orbit ~1120 m/s
Moho ~1676 m/s
Jool ~1915 m/s
Eeloo ~2100 m/s
Kerbol escape ~2740 m/s

## Reference frames

Time warp Minimum Altitude
Any
5× 70 000 m (above the atmosphere)
10× 70 000 m (above the atmosphere)
50× 70 000 m (above the atmosphere)
100× 120 000 m
1 000× 240 000 m
10 000× 480 000 m
100 000× 600 000 m

## Gallery

Spoiler: Easter Eggs Are Shown

## Changes

1.2
1.0.5
• Kerbin actually spun up to have a 6 h synodic day.
0.90
• Updated biomes.
0.22
0.21
• Terrain revised to produce more detailed and interesting landforms.
0.19.1
0.19
• New mesh for the launchpad and area (no launchtower anymore).
• New mesh for the runway, with lights and sloping edges for rovers.
0.18
• Terrain overhaul: Entire planet redo. Deserts, darker and greener grass, islands, darker ocean/water, snow capped mountains. Looks more realistic.
• Airport added to island off of KSC coastline. (Not a launching point)
0.17
• Improved atmosphere visuals.
0.15
0.14.2
• Much more varied and taller terrain added. Prior to this, some mountain ranges exceeded 600 m in height, but the tallest point was at an altitude of approximately 900 m.
0.12
0.11
• Terrain overhaul, oceans became wet.
0.10.1
• Atmosphere extended from ~34,500 m to ~69,000 m.
0.7.3
• Initial Release

## Trivia

Kerbin's continents are derived from libnoise[3], a coherent noise generating library, though they have been increasingly modified with time.

### Real-world comparison

Kerbin has approximately 1/113th the mass and a 1/10th the radius of Earth.

## Notes

1. Some Major Orbit Types” uses that, the Wikipedia article mentions 0.74105, and “Orbital Parameters of a Molniya Orbit” uses 0.72.
2. http://forum.kerbalspaceprogram.com/entry.php/247-A-Brave-New-World
3. http://libnoise.sourceforge.net/examples/complexplanet/


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7-9 November 2016 – Palo Alto, California, USA

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text:
# How do you simplify sqrt8/4?

##### 1 Answer
Sep 17, 2015

I found $\frac{\sqrt{2}}{2}$

#### Explanation:

You can write it as:
$\frac{\sqrt{8}}{4} = \frac{\sqrt{4 \cdot 2}}{4} = \sqrt{4} \frac{\sqrt{2}}{4} = 2 \frac{\sqrt{2}}{4} = \frac{\sqrt{2}}{2}$


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# Measurements of the $$\chi$$$$_c$$ and $$\chi$$$$_b$$ quarkonium states in $$\rho$$$$\rho$$ collisions with the ATLAS experiment

Chisholm , Andrew Stephen (2014). Measurements of the $$\chi$$$$_c$$ and $$\chi$$$$_b$$ quarkonium states in $$\rho$$$$\rho$$ collisions with the ATLAS experiment. University of Birmingham. Ph.D.

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## Abstract

The chi_b bottomonium states are observed through the reconstruction of the radiative decays chi_b(nP)→Upsilon(1S) gamma using 4.4/fb of pp collision data collected at sqrt(s) = 7 TeV with the ATLAS experiment. The production of the chi_b(1P) and chi_b(2P) bottomonium states is observed in addition to a candidate for a new bottomonium state, consistent with theoretical expectations for the chi_b(3P) states. The production-averaged mass barycentre for the chi_b(3P) candidate is measured to be 10541$$\pm$$11 (stat.) $$\pm$$ 30 (syst.) MeV. The consequences of this discovery for our understanding of bottomonium production phenomenology in hadronic collisions is reviewed. The production of the chi_c1 and chi_c2 charmonium states has been measured in sqrt(s) = 7 TeV pp collisions with the ATLAS experiment using a data sample representing an integrated luminosity of 4.5/fb. The prompt and non-prompt production cross sections for the chi_c1 and chi_c2 states are measured within the region | y(J/psi)| < 0.75. These measurements suggest that 24-28% of prompt J/psi are produced in feed-down from radiative chi_c1 and chi_c2 decays. The production of the chi_c2 state, relative to the chi_c1 state, is measured for both prompt and non-prompt production processes. This collection of measurements is compared to a number of theoretical predictions for chi_cJ production at the LHC. The branching fraction B(B$$\pm$$→chi_c1 K$$\pm$$) = (4.9$$_pm$$$$\pm$$ (stat.) $$\pm$$0.6 (syst.))x 10^-4 is also measured with the same dataset and chi_c event selection.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Hawkes, ChrisUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Physics and Astronomy
Funders: None/not applicable
Subjects: Q Science > QC Physics
URI: http://etheses.bham.ac.uk/id/eprint/5056

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In the artificial intelligence field, learning often corresponds to changing the parameters of a parameterized function. A learning rule is an algorithm or mathematical expression that specifies precisely how the parameters should be changed. When creating an artificial intelligence system, we must make two decisions: what representation should be used (i.e., what parameterized function should be used) and what learning rule should be used to search through the resulting set of representable functions. Using most learning rules, these two decisions are coupled in a subtle (and often unintentional) way. That is, using the same learning rule with two different representations that can represent the same sets of functions can result in two different outcomes. After arguing that this coupling is undesirable, particularly when using artificial neural networks, we present a method for partially decoupling these two decisions for a broad class of learning rules that span unsupervised learning, reinforcement learning, and supervised learning.
In this paper we study the graph parameter of burning number, introduced by Bonato, Janssen, and Roshanbin (2014). We are particular interested in determining the burning number of Circulant graphs. In this paper, we find upper and lower bounds on the burning number of classes of circulant graphs of degree at most four. The burning number is found exactly for a family of circulant graphs of degree three, and two specific families of circulant graphs of degree four. Finally, we given upper and lower bounds on the burning number of families of circulant graphs of higher degree.
Neural machine translation has meant a revolution of the field. Nevertheless, post-editing the outputs of the system is mandatory for tasks requiring high translation quality. Post-editing offers a unique opportunity for improving neural machine translation systems, using online learning techniques and treating the post-edited translations as new, fresh training data. We review classical learning methods and propose a new optimization algorithm. We thoroughly compare online learning algorithms in a post-editing scenario. Results show significant improvements in translation quality and effort reduction.
This paper aims at one-shot learning of deep neural nets, where a highly parallel setting is considered to address the algorithm calibration problem – selecting the best neural architecture and learning hyper-parameter values depending on the dataset at hand. The notoriously expensive calibration problem is optimally reduced by detecting and early stopping non-optimal runs. The theoretical contribution regards the optimality guarantees within the multiple hypothesis testing framework. Experimentations on the Cifar10, PTB and Wiki benchmarks demonstrate the relevance of the approach with a principled and consistent improvement on the state of the art with no extra hyper-parameter.
We consider the problem of training generative models with a Generative Adversarial Network (GAN). Although GANs can accurately model complex distributions, they are known to be difficult to train due to instabilities caused by a difficult minimax optimization problem. In this paper, we view the problem of training GANs as finding a mixed strategy in a zero-sum game. Building on ideas from online learning we propose a novel training method named Chekhov GAN 1 . On the theory side, we show that our method provably converges to an equilibrium for semi-shallow GAN architectures, i.e. architectures where the discriminator is a one layer network and the generator is arbitrary. On the practical side, we develop an efficient heuristic guided by our theoretical results, which we apply to commonly used deep GAN architectures. On several real world tasks our approach exhibits improved stability and performance compared to standard GAN training.
While the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) are powerful tools for model selection in linear regression, they are built on different prior assumptions and thereby apply to different data generation scenarios. We show that their respective assumptions can be unified within an augmented model-plus-noise space and construct a prior in this space which inherits the beneficial properties of both AIC and BIC. The performance of our ‘Noncentral Information Criterion’ (NIC) matches or exceeds that of the AIC and BIC both for weak and strong signal cases.
Anomalies in time-series data give essential and often actionable information in many applications. In this paper we consider a model-free anomaly detection method for univariate time-series which adapts to non-stationarity in the data stream and provides probabilistic abnormality scores based on the conformal prediction paradigm. Despite its simplicity the method performs on par with complex prediction-based models on the Numenta Anomaly Detection benchmark and the Yahoo! S5 dataset.
We consider the problem of quickest change-point detection in data streams. Classical change-point detection procedures, such as CUSUM, Shiryaev-Roberts and Posterior Probability statistics, are optimal only if the change-point model is known, which is an unrealistic assumption in typical applied problems. Instead we propose a new method for change-point detection based on Inductive Conformal Martingales, which requires only the independence and identical distribution of observations. We compare the proposed approach to standard methods, as well as to change-point detection oracles, which model a typical practical situation when we have only imprecise (albeit parametric) information about pre- and post-change data distributions. Results of comparison provide evidence that change-point detection based on Inductive Conformal Martingales is an efficient tool, capable to work under quite general conditions unlike traditional approaches.
With the goal of making high-resolution forecasts of regional rainfall, precipitation nowcasting has become an important and fundamental technology underlying various public services ranging from rainstorm warnings to flight safety. Recently, the convolutional LSTM (ConvLSTM) model has been shown to outperform traditional optical flow based methods for precipitation nowcasting, suggesting that deep learning models have a huge potential for solving the problem. However, the convolutional recurrence structure in ConvLSTM-based models is location-invariant while natural motion and transformation (e.g., rotation) are location-variant in general. Furthermore, since deep-learning-based precipitation nowcasting is a newly emerging area, clear evaluation protocols have not yet been established. To address these problems, we propose both a new model and a benchmark for precipitation nowcasting. Specifically, we go beyond ConvLSTM and propose the Trajectory GRU (TrajGRU) model that can actively learn the location-variant structure for recurrent connections. Besides, we provide a benchmark that includes a real-world large-scale dataset from the Hong Kong Observatory, a new training loss, and a comprehensive evaluation protocol to facilitate future research and gauge the state of the art.
Designing an auction that maximizes expected revenue is an intricate task. Indeed, as of today–despite major efforts and impressive progress over the past few years–only the single-item case is fully understood. In this work, we initiate the exploration of the use of tools from deep learning on this topic. The design objective is revenue optimal, dominant-strategy incentive compatible auctions. We show that multi-layer neural networks can learn almost-optimal auctions for settings for which there are analytical solutions, such as Myerson’s auction for a single item, Manelli and Vincent’s mechanism for a single bidder with additive preferences over two items, or Yao’s auction for two additive bidders with binary support distributions and multiple items, even if no prior knowledge about the form of optimal auctions is encoded in the network and the only feedback during training is revenue and regret. We further show how characterization results, even rather implicit ones such as Rochet’s characterization through induced utilities and their gradients, can be leveraged to obtain more precise fits to the optimal design. We conclude by demonstrating the potential of deep learning for deriving optimal auctions with high revenue for poorly understood problems.
Variable selection plays a fundamental role in high-dimensional data analysis. Various methods have been developed for variable selection in recent years. Some well-known examples include forward stepwise regression (FSR), least angle regression (LARS), and many more. These methods typically have a sequential nature in the sense that variables are added into the model one-by-one. For sequential selection procedures, it is crucial to find a stopping criterion, which controls the model complexity. One of the most commonly used techniques for controlling the model complexity in practice is cross-validation (CV). Despite its popularity, CV has two major drawbacks: expensive computational cost and lack of statistical interpretation. To overcome these drawbacks, we introduce a flexible and efficient testing-based variable selection approach that could be incorporated with any sequential selection procedure. The test is on the overall signal in the remaining inactive variables using the maximal absolute partial correlation among the inactive variables with the response given active variables. We develop the asymptotic null distribution of the proposed test statistic as the dimension tends towards infinity uniformly in the sample size. We also show the consistency of the test. With this test, at each step of the selection, we include a new variable if and only if the $p$-value is below some pre-defined level. Numerical studies show that the proposed method delivers very competitive performance in terms of both variable selection accuracy and computational complexity compared to CV.
Ensemble methods are arguably the most trustworthy techniques for boosting the performance of machine learning models. Popular independent ensembles (IE) relying on naive averaging/voting scheme have been of typical choice for most applications involving deep neural networks, but they do not consider advanced collaboration among ensemble models. In this paper, we propose new ensemble methods specialized for deep neural networks, called confident multiple choice learning (CMCL): it is a variant of multiple choice learning (MCL) via addressing its overconfidence issue.In particular, the proposed major components of CMCL beyond the original MCL scheme are (i) new loss, i.e., confident oracle loss, (ii) new architecture, i.e., feature sharing and (iii) new training method, i.e., stochastic labeling. We demonstrate the effect of CMCL via experiments on the image classification on CIFAR and SVHN, and the foreground-background segmentation on the iCoseg. In particular, CMCL using 5 residual networks provides 14.05% and 6.60% relative reductions in the top-1 error rates from the corresponding IE scheme for the classification task on CIFAR and SVHN, respectively.
Recent work has explored the syntactic abilities of RNNs using the subject-verb agreement task, which diagnoses sensitivity to sentence structure. RNNs performed this task well in common cases, but faltered in complex sentences (Linzen et al., 2016). We test whether these errors are due to inherent limitations of the architecture or to the relatively indirect supervision provided by most agreement dependencies in a corpus. We trained a single RNN to perform both the agreement task and an additional task, either CCG supertagging or language modeling. Multi-task training led to significantly lower error rates, in particular on complex sentences, suggesting that RNNs have the ability to evolve more sophisticated syntactic representations than shown before. We also show that easily available agreement training data can improve performance on other syntactic tasks, in particular when only a limited amount of training data is available for those tasks. The multi-task paradigm can also be leveraged to inject grammatical knowledge into language models.
With a rapid increase in volume and complexity of data sets there is a need for methods that can extract useful information in these data sets. Dimension reduction approaches such as Partial least squares (PLS) are increasingly being utilized for finding relationships between two data sets. However these methods often lack a probabilistic formulation, hampering development of more flexible models. Moreover dimension reduction methods in general suffer from identifiability problems, causing difficulties in combining and comparing results from multiple studies. We propose Probabilistic PLS (PPLS) as an extension of PLS to model the overlap between two data sets. The likelihood formulation provides opportunities to address issues typically present in data, such as missing entries and heterogeneity between subjects. We show that the PPLS parameters are identifiable up to sign. We derive Maximum Likelihood estimators that respect the identifiability conditions by using an EM algorithm with a constrained optimization in the M step. A simulation study is conducted and we observe a good performance of the PPLS estimates in various scenarios, when compared to PLS estimates. Most notably the estimates seem to be robust against departures from normality. To illustrate the PPLS model, we apply it to real IgG glycan data from two cohorts. We infer the contributions of each variable to the correlated part and observe very similar behavior across cohorts.
Object detection is a core problem in computer vision. With the development of deep ConvNets, the performance of object detectors has been dramatically improved. The deep ConvNets based object detectors mainly focus on regressing the coordinates of bounding box, \eg, Faster-R-CNN, YOLO and SSD. Different from these methods that considering bounding box as a whole, we propose a novel object bounding box representation using points and links and implemented using deep ConvNets, termed as Point Linking Network (PLN). Specifically, we regress the corner/center points of bounding-box and their links using a fully convolutional network; then we map the corner points and their links back to multiple bounding boxes; finally an object detection result is obtained by fusing the multiple bounding boxes. PLN is naturally robust to object occlusion and flexible to object scale variation and aspect ratio variation. In the experiments, PLN with the Inception-v2 model achieves state-of-the-art single-model and single-scale results on the PASCAL VOC 2007, the PASCAL VOC 2012 and the COCO detection benchmarks without bells and whistles. The source code will be released.
Along with the recent advances in scalable Markov Chain Monte Carlo methods, sampling techniques that are based on Langevin diffusions have started receiving increasing attention. These so called Langevin Monte Carlo (LMC) methods are based on diffusions driven by a Brownian motion, which gives rise to Gaussian proposal distributions in the resulting algorithms. Even though these approaches have proven successful in many applications, their performance can be limited by the light-tailed nature of the Gaussian proposals. In this study, we extend classical LMC and develop a novel Fractional LMC (FLMC) framework that is based on a family of heavy-tailed distributions, called $\alpha$-stable L\'{e}vy distributions. As opposed to classical approaches, the proposed approach can possess large jumps while targeting the correct distribution, which would be beneficial for efficient exploration of the state space. We develop novel computational methods that can scale up to large-scale problems and we provide formal convergence analysis of the proposed scheme. Our experiments support our theory: FLMC can provide superior performance in multi-modal settings, improved convergence rates, and robustness to algorithm parameters.
We present an efficient block-diagonal ap- proximation to the Gauss-Newton matrix for feedforward neural networks. Our result- ing algorithm is competitive against state- of-the-art first order optimisation methods, with sometimes significant improvement in optimisation performance. Unlike first-order methods, for which hyperparameter tuning of the optimisation parameters is often a labo- rious process, our approach can provide good performance even when used with default set- tings. A side result of our work is that for piecewise linear transfer functions, the net- work objective function can have no differ- entiable local maxima, which may partially explain why such transfer functions facilitate effective optimisation.
Verification determines whether two samples belong to the same class or not, and has important applications such as face and fingerprint verification, where thousands or millions of categories are present but each category has scarce labeled examples, presenting two major challenges for existing deep learning models. We propose a deep semi-supervised model named SEmi-supervised VErification Network (SEVEN) to address these challenges. The model consists of two complementary components. The generative component addresses the lack of supervision within each category by learning general salient structures from a large amount of data across categories. The discriminative component exploits the learned general features to mitigate the lack of supervision within categories, and also directs the generative component to find more informative structures of the whole data manifold. The two components are tied together in SEVEN to allow an end-to-end training of the two components. Extensive experiments on four verification tasks demonstrate that SEVEN significantly outperforms other state-of-the-art deep semi-supervised techniques when labeled data are in short supply. Furthermore, SEVEN is competitive with fully supervised baselines trained with a larger amount of labeled data. It indicates the importance of the generative component in SEVEN.
Variational Autoencoder (VAE) is an efficient framework in modeling natural images with probabilistic latent spaces. However, when the input spaces become complex, VAE becomes less effective, potentially due to the oversimplification of its latent space construction. In this paper, we propose to integrate recurrent connections across channels to both inference and generation steps of VAE. Sequentially building up the complexity of high-level features in this way allows us to capture global-to-local and coarse-to-fine structures of the input data spaces. We show that our channel-recurrent VAE improves existing approaches in multiple aspects: (1) it attains lower negative log-likelihood than standard VAE on MNIST; when trained adversarially, (2) it generates face and bird images with substantially higher visual quality than the state-of-the-art VAE-GAN and (3) channel-recurrency allows learning more interpretable representations; finally (4) it achieves competitive classification results on STL-10 in a semi-supervised setup.
Unsupervised learning of low-dimensional, semantic representations of words and entities has recently gained attention. In this paper we describe the Semantic Entity Retrieval Toolkit (SERT) that provides implementations of our previously published entity representation models. The toolkit provides a unified interface to different representation learning algorithms, fine-grained parsing configuration and can be used transparently with GPUs. In addition, users can easily modify existing models or implement their own models in the framework. After model training, SERT can be used to rank entities according to a textual query and extract the learned entity/word representation for use in downstream algorithms, such as clustering or recommendation.
The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. Experiments on two machine translation tasks show these models to be superior in quality while being more parallelizable and requiring significantly less time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-to-German translation task, improving over the existing best results, including ensembles by over 2 BLEU. On the WMT 2014 English-to-French translation task, our model establishes a new single-model state-of-the-art BLEU score of 41.0 after training for 3.5 days on eight GPUs, a small fraction of the training costs of the best models from the literature. We show that the Transformer generalizes well to other tasks by applying it successfully to English constituency parsing both with large and limited training data.


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A Counterexample to the First Zassenhaus Conjecture

# A Counterexample to the First Zassenhaus Conjecture

Florian Eisele  and  Leo Margolis Florian Eisele
Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, United Kingdom
Leo Margolis
Departamento de Matemàticas, Universidad de Murcia, 30100 Murcia, Spain
###### Abstract.

Hans J. Zassenhaus conjectured that for any unit of finite order in the integral group ring of a finite group there exists a unit in the rational group algebra of such that for some . We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order whose integral group ring contains a unit of order which, in the rational group algebra, is not conjugate to any element of the form .

2010 Mathematics Subject Classification. Primary 16S34. Secondary 16U60, 20C11, 20C05.Key words and phrases. Unit Group, Group Ring, Zassenhaus Conjecture, Integral Representations.The first author was supported by the EPSRC, grant EP/M02525X/1. The second author was supported by a Marie Curie Individual Fellowship from EU project 705112-ZC and the FWO (Research Foundation Flanders).

## 1. Introduction

Let be a finite group and denote by the group ring of over a commutative ring . Denote by the unit group of . In the 1970’s Zassenhaus made three strong conjectures about finite subgroups of (cf. [Seh93, Section 37]). These conjectures, often called the first, second and third Zassenhaus conjecture and sometimes abbreviated as (ZC1), (ZC2) and (ZC3), had a lasting impact on research in the field. All three of these conjectures turned out to be true for nilpotent groups [Wei91], but metabelian counterexamples for the second and the third one were found by K. W. Roggenkamp and L. L. Scott [Sco92, Kli91]. Later M. Hertweck showed that there are counterexamples of order as small as [Her04, Section 11]. Unlike its siblings, the first Zassenhaus conjecture seemed to stand the test of time. Since it was the only one of the three to remain open, people in recent years started referring to it as the Zassenhaus conjecture, and we will do the same in this article.

###### Zassenhaus Conjecture.

If is a unit of finite order, then there is an such that for some .

This conjecture first appeared in written form in [Zas74] and inspired a lot of research in the decades to follow. The first results on the conjecture were mostly concerned with special classes of metabelian groups, [HP72, PM73, AH80, LB83, RS83, PMS84, Mit86, PMRS86, SW86, MRSW87, LT90, LP92, LS98, JPM00, BHK04, RS06]. Almost all of these results were later generalised by Hertweck [Her06, Her08a]. Hertweck proved that the Zassenhaus conjecture holds for groups which have a normal Sylow -subgroup with abelian complement or a cyclic normal subgroup such that for some abelian subgroup of . The latter result was further generalised in [CMR13], proving that the Zassenhaus conjecture holds for cyclic-by-abelian groups. In a different vein, A. Weiss’ proof of the conjecture, or even a stronger version of it, for nilpotent groups [Wei88, Wei91], was certainly a highlight of the study. The conjecture is also known to hold for a few other classes of solvable groups [Fer87, DJ96, BKM16, MR17b, MR17c, MR17a], as well as for some small groups. In particular, the conjecture holds for groups of order smaller than [HK06, HS15, BHK17].

Progress on non-solvable groups was initially lagging. For many years the conjecture was only known to hold for the alternating and symmetric group of degree [LP89, LT91] and the special linear group [DJPM97]. This state of affairs changed when Hertweck introduced a method to tackle the conjecture involving Brauer characters [Her07]. Nevertheless, results are still relatively far and between [Her07, Her08b, BH08, Gil13, BM16, KK17, BM17], and, for instance, the only non-abelian simple groups for which the conjecture has been verified are the groups where , [BM16] or where is a Fermat or Mersenne prime [MRS16].

In the present article we show that the Zassenhaus conjecture is false by providing a series of metabelian groups such that contains a unit of finite order not conjugate in to any element of the form for .

Let us describe these groups. To this end, let and be odd primes, an odd divisor of and , the additive group , and let and be primitive elements in the multiplicative groups and , respectively. Consider the abelian group

 A=⟨a,b,c | ap2−1d=bq2−1d=1, cd=a⋅b⟩

There is an action of on given by

 (x,y)a=(αd⋅x,y), (x,y)b=(x,βd⋅y),(x,y)c=(α⋅x,β⋅y)

and we may form the semidirect product , which we denote by . The following are our main results:

###### Theorem A.

Let , where is a root of the polynomial over and is a root of over . There exists a unit of order such that is not conjugate in to any element of the form for . In particular, the Zassenhaus conjecture does not hold for .

###### Theorem B.

Let be an odd positive integer, and let be arbitrary. There exist infinitely many pairs of primes and such that, for any admissible choice of and , for there are , each of order , such that neither one of the is conjugate in to an element of the form for , or to any other for . In particular, the Zassenhaus conjecture does not hold for such a group .

A more precise version of Theorem B, specifying lower bounds for and as well as the rational conjugacy classes of the , can be found in Corollary 7.3. The idea that groups like might be good candidates for a counterexample to the Zassenhaus conjecture was noted in [MR17a]. Looking at the various positive results mentioned above, it seems that metabelian groups would have been the next logical step, and people working in the field certainly attempted to prove the Zassenhaus conjecture for metabelian groups, to no avail. What is more, the class of metabelian groups provided E. Dade’s counterexample to R. Brauer’s question, which asked whether for all fields implies that and are isomorphic [Dad71]. The second Zassenhaus conjecture mentioned above, which asked if different (normalised) group bases of are conjugate in , fails for metabelian groups as well [Kli91]. On the other hand, metabelian groups were one of the first classes of groups for which the isomorphism problem on integral group rings was known to have a positive answer [Whi68].

Here is an outline of our strategy to prove Theorems A and B:

1. If is a cyclic group of order , then a unit of order corresponds to a certain -module , called a “double action module”. This is the well-known double action formalism explained in Section 2, and the defining property of double action modules is that their restriction to is a free -module of rank one. This principle works for any commutative ring .

2. Once we fix a conjugacy class of units of order in , or equivalently a -double action module corresponding to it, we need to find a -lattice in whose restriction to is free.

3. Let denote the localisation of at the prime ideal . We provide a fairly general construction of double action modules over for groups of the form , where is abelian. This is done in Section 5, and, of course, subject to a whole list of conditions. The double action modules we construct are direct sums of direct summands of permutation modules (see Definition 5.4), and as a consequence the local version of the counterexample is fairly explicit (see Proposition 7.11 at the end).

4. The problem of turning a family of “compatible” -lattices in with free restriction to into a -lattice in with the same property can be solved using a rather general local-global principle, provided the centraliser of the unit is big enough (think of as already being fixed up to conjugacy in ). This is done in Section 6.

5. In the last section we study groups of the form as defined above. All of the more general results of the preceding sections become explicit and elementarily verifiable in this situation. We use the general result of that section, Theorem 7.2, to prove Theorems A and B.

In regard to future research, it seems worth pointing out that many variations and weaker versions of the Zassenhaus conjecture remain open. An overview of the weaker forms of the conjecture can be found in [MR17c]. In particular, the question if the orders of torsion units of augmentation one in coincide with the orders of elements in remains open. It also might still be true that if is a torsion unit then is conjugate in to for some , where is some larger group containing .

Going in a different direction, the -version of even the strongest of the three Zassenhaus conjectures remains open. This variation asks if it is true that any -subgroup of consisting of elements of augmentation one is conjugate in to a subgroup of . This is sometimes called “(p-ZC3)” or the “Strong Sylow Theorem” for . An overview of results relating to this problem can be found in [BKM16]. For the counterexample to the Zassenhaus conjecture presented in the present article it is of fundamental importance that the order of the unit is divisible by at least two different primes.

Throughout the paper we are going to use the following notation, most of which is quite standard.

###### Notation and basic definitions.
1. Let be a finite group and let be a subgroup of . For a character of we write for the induced character, and for a character of we write for the restriction to . The trivial character of is denoted by .

2. For a prime number we denote by a Sylow -subgroup of and by the -part of an element . The conjugacy class of is denoted by . We also use to denote a -Hall subgroup of (this is used only for nilpotent ) and for the -part of .

3. Let be a finite group and let and be subgroups of such that . If and are characters of and , respectively, then denotes the corresponding character of . Similarly, if and are -modules for some commutative ring , denotes with the natural -module structure.

4. We write “” to denote a sum ranging over a set of representatives of the conjugacy classes of . If acts on a set we write “” for the sum ranging over representatives of the -orbits in .

5. If and are groups, and acts on by automorphisms, we denote by the set of -orbits of irreducible rational characters of . We write “” for a sum ranging over representatives of these orbits.

6. For a cyclic group and an element we define

 (1) [g]=⟨(g,c)⟩≤G×U

We will often use the fact that and for all primes .

7. If is a ring and is an -module we write for the direct sum of copies of .

8. If is a ring, is an -module and is an arbitrary subset, we write for the -module generated by .

9. Let be a commutative ring and let be an arbitrary element of . Then

 εgG(u)=∑h∈gGrh

is called the partial augmentation of at .

## 2. Double action formalism

The “double action formalism” (see, for instance, [Seh93, Section 38.6]) is a commonly used way of studying the Zassenhaus conjecture and other questions relating to units in group algebras via certain bimodules, the so-called “double action modules”. In this section we give a short (but complete, at least for our purposes) overview of this formalism. For the rest of this section let be a finite group and let be a cyclic group of order . By we denote an arbitrary commutative ring.

###### Definition 2.1.
1. Given a unit satisfying we define an -module as follows: as an -module, is equal to , and the (right) action of is given by

 (2) u(RG)G×(G×U)⟶u(RG)G: (x,(g,ci))↦(u∘)i⋅x⋅g

where the product on the right hand side of the assignment is taken within the ring , and denotes the standard involution on . We call this -module the double action module associated with the unit .

2. An -module is called -regular if is free of rank one as an -module (that is, it is isomorphic to considered as a right module over itself).

A double action module is clearly -regular, but it turns out that the converse is true as well:

###### Proposition 2.2.
1. If is a -regular -module, then for some unit with .

2. If are two units satisfying , then

 u(RG)G≅v(RG)G

if and only if and are conjugate inside .

###### Proof.

Assume that is -regular. Then we may choose an isomorphism of -modules , where we view as a right module over itself. As before, let denote the standard involution of the group algebra, and define . Then we have, for all ,

 φ(m⋅c)=φ(φ−1(1)⋅φ(m)⋅c)=φ(φ−1(1)⋅c)⋅φ(m)=u∘⋅φ(m)

where we made use of the fact that . It now follows immediately from the above that , and the map is easily seen to be an isomorphism of -modules.

Let us now prove the second part of the proposition. To this end, fix an isomorphism . Then . As an equation purely in the ring this yields . So it only remains to show that is an invertible element of , which follows from the fact that generates as an -module. ∎

As we have seen so far, double action modules are in one-to-one correspondence with conjugacy classes of elements of whose order divides . Evidently this means that each property of torsion units should have a counterpart in the language of double action modules. An important tool in the study of the Zassenhaus conjecture is the criterion given in [MRSW87, Theorem 2.5]. It states that a unit of finite order and augmentation one is conjugate in to an element of if and only if for all and all . In particular, finding a counterexample to the Zassenhaus conjecture is equivalent to finding a unit of finite order and augmentation one which has a negative partial augmentation. Hence it is important for us to have a way of recovering the partial augmentations of a unit from the corresponding double action module.

###### Proposition 2.3.

Let be a unit satisfying . Let

 (3) θu: G×U⟶R

denote the character of the -module . Then

 (4) θu((g,ci))=|CG(g)|⋅εgG(ui)
###### Proof.

Let us first calculate the trace of the linear map for arbitrary . This trace is equal to the number of such that , or, equivalently, . If then this number is equal to , otherwise it is zero. Now, if

 ui=∑g∈Gαg⋅g

then the linear endomorphism of induced by is equal to . The character value is the trace of this map, which is equal to

 ∑h∈Gαh⋅Tr(μ(h,g))=∑h∈gGαh⋅|CG(g)|=|CG(g)|⋅εgG(ui)

as claimed. ∎

We now turn our attention to the case of rational coefficients, i.e. . In that situation -regularity of -modules can readily be checked on the level of characters, and Proposition 2.3 can be used to ascertain whether the corresponding torsion unit in is indeed not conjugate to an element of .

###### Proposition 2.4.

Let be pair-wise non-conjugate elements of whose order divides , and let such that . Assume that

 (5) θ=k∑i=1ai⋅1↑G×U[gi]

is in fact a character of , rather than just a virtual character. Then is the character of for some satisfying . Moreover for all and whenever is not conjugate to any of the .

###### Proof.

Let us first prove that can be realised as the character of a -module, rather than just a -module. By definition can be written as the difference of the characters of two -modules, say and . Without loss of generality we may assume that and share no isomorphic simple direct summands. But then , which implies . That is, and share no isomorphic simple direct summands, which means that can only be a proper character if , which means that is a -module affording .

To verify that is the character of for some satisfying , it suffices to show that is equal to the regular character of , which is equal to . Note that by Mackey’s theorem we have

 1↑G×U[gi]|G=∑x1↑G[gi]x∩G

where ranges over a transversal for the double cosets . Since , there is just one such double coset, and therefore

 1↑G×U[gi]|G=1↑G[gi]∩G=1↑G{1}

independent of . Combining this fact with (5) we get

 θ|G=k∑i=1ai⋅1↑G{1}=1↑G{1}

All that is left to prove now is our claim on the partial augmentations. We know by now that for some , with as defined in Proposition 2.3. Thus, Proposition 2.3 yields

 (6) εgG(u)=θ((g,c))|CG(g)|=1|CG(g)|k∑i=1ai⋅1↑G×U[gi]((g,c))

The character evaluated on is equal to the number of such that . Since the order of divides , which is the order of , it follows that the projection from to is an isomorphism, and hence is the only element of whose projection to is . This implies that if and only if . It follows that is equal to zero if , and equal to if . Plugging this back into (6) yields the desired result for the partial augmentations of . ∎

## 3. Local and semi-local rings of coefficients

Let be the ring of integers in an algebraic number field . For a maximal ideal of we let denote the localisation of at the prime . If is a finite collection of maximal ideals of , we define

 Rπ=⋂p∈πR(p),

which is a semi-local ring whose maximal ideals are precisely for . Testing whether a particular module is a double action module of a unit is particularly easy over , as the following proposition shows:

###### Proposition 3.1.

Let be an -module such that is projective and is -regular. Then is -regular.

###### Proof.

By assumption is projective and its character is equal to the character of the regular -module. We need to show that this implies that is isomorphic to the regular -module. This follows from the fact that two projective -modules are isomorphic if and only if their characters are the same, a consequence of the fact that the decomposition matrix of a finite group has full row rank (see [CR81, Corollary 18.16] for the precise statement we are using). ∎

Constructing a -regular -module is actually equivalent to constructing a -regular -module for each in such a way that all of these modules have the same character.

###### Proposition 3.2.

Let be an -order in a finite-dimensional semisimple -algebra and let be a finite-dimensional -module.

1. Assume that we are given full -lattices for each . Then

 L=⋂p∈πL(p)

is an -lattice in with the property that for each .

2. Given two -lattices and in , we have if and only if for each .

###### Proof.

is clearly a -module, and in order to show that it is a lattice it suffices to show that it is contained in some -lattice. If is an arbitrary full -lattice in , then for each there is a number such that . Let be a number such that for each . Then for each , and therefore , which is an -lattice.

Now let us prove that for each . Clearly . On the other hand, if , then there is an integer such that , for all with , and (we can take to be contained in a product of sufficiently large powers of the maximal ideals in different from ). By definition, we now have , and since is invertible in we also have . This implies , which completes the proof of the first point. For the second point see [Rei75, Exercise 18.3]. ∎

## 4. Locally free lattices and class groups

As in the previous section let be the ring of integers in an algebraic number field . Let be a finite-dimensional semisimple -algebra, and let be an -order in . Throughout this section we adopt the following notational convention: if is a maximal ideal of , and is an -module, then denotes the -adic completion of . In particular, is a complete field with valuation ring , is a finite-dimensional -algebra and is an -order in .

Let us first check that no information is lost in passing from the localisations considered in the previous section to the completions we are going to consider now. If we keep the notation for the localisation of at and , then and can also be viewed as the -adic completions of and , respectively.

###### Proposition 4.1 ([Cr87, Proposition 30.17]).

Let and be finitely generated -modules. Then

 M≅Nif and only ifMp≅Np

In particular, if and are finitely generated -modules, then as -modules if and only if as -modules.

Now let us define the protagonist of this section: the locally free class group of .

###### Definition 4.2 (cf. [Cr87, §49a]).
1. A right -lattice is called locally free of rank if

 Lp≅Λ⊕np

as right -modules for all maximal ideals of .

2. If and are right -lattices, we say that and are stably isomorphic if

 L⊕Λ⊕n≅L′⊕Λ⊕n

for some .

3. The locally free class group of , denoted by , is an additive group whose elements are the stable isomorphism classes of locally free right -ideals in . The group operation on is defined as follows: if and are locally free right -ideals in , then there is a locally free right -ideal such that

 X⊕Y≅Z⊕Λ

as right -modules. We define the sum to be equal to .

Note that the unit element of is . For the purposes of this article, class groups serve as a means to prove that certain -lattices are free. The reason this works is that most group algebras satisfy the Eichler condition relative to , which guarantees that we can infer from :

###### Definition 4.3 (cf. [Cr87, Remark 45.5 (i)]).

We say that satisfies the Eichler condition relative to if no simple component of is isomorphic to a totally definite quaternion algebra.

###### Theorem 4.4 (Jacobinski Cancellation Theorem [Cr87, Theorem 51.24]).

If satisfies the Eichler condition relative to , then any two locally free -lattices which are stably isomorphic are isomorphic.

###### Theorem 4.5 ([Cr87, Theorem 51.3]).

If is a finite group which does not have an epimorphic image isomorphic to either one of the following:

1. A generalised quaternion group of order where .

2. The binary tetrahedral group of order .

3. The binary octahedral group of order .

4. The binary icosahedral group of order .

then satisfies the Eichler condition relative to .

We now turn our attention to the problem of deciding whether a given locally free -ideal is trivial in .

###### Definition 4.6 ([Cr87, (49.4)]).
1. We define the idèle group of as

 J(A)={(αp)p∈∏pU(Ap) ∣∣∣ αp∈U(Λp) for all except finitely many p}

where ranges over all maximal ideals of . If and are two elements of , then their product in is defined as .

2. We identify with the subgroup of consisting of constant idèles.

3. Define

 U(Λ)={α∈J(A) | αp∈U(Λp) % for all p}

This is also a subgroup of .

Even though it is not immediately obvious from the definition, does not depend on the order (in fact, if is another -order in , then for all except finitely many ).

###### Theorem 4.7 (Special case of [Cr87, Theorem 31.18]).

There is a bijection between the double cosets

 U(A)∖J(A)/U(Λ)

and isomorphism classes of locally free right -ideals in given by

 U(A)⋅α⋅U(Λ)↦A∩⋂pαpΛp

where . We denote the right hand side of this assignment by .

As shown in [CR87, Theorem 31.19] we have, for arbitrary , an isomorphism . This shows that there is an epimorphism of groups

 θ: J(A)↠Cl(Λ): α↦[αΛ]

Since is commutative by definition, we certainly have . In [Frö75], A. Fröhlich gave an explicit characterisation of the kernel of , which will be very useful to us later.

###### Definition 4.8 (Reduced norms).

Let be a field, and let be a finite-dimensional semisimple -algebra. Then there is a decomposition

 B=B1⊕…⊕Bn

where each is a simple -algebra. We may view as a central simple algebra over its centre . In each component we have a reduced norm map

 nrBi/Z(Bi): Bi⟶Z(Bi)

obtained by embedding into for some field extension which splits , followed by mapping isomorphically onto a full matrix ring over of the appropriate dimension and then taking the determinant. We can then define a reduced norm map on component-wise. This map will take values in . That is, we get a multiplicative map

 nrB/Z(B): B⟶Z(B)
###### Definition 4.9.

Define

 J0(A)={(αp)p∈J(A) | nrAp/Z(Ap)(αp)=1 for all p}

This is a normal subgroup of .

###### Theorem 4.10 ([Frö75, Theorem 1 and subsequent remarks]).

The map

 J(A)J0(A)⋅U(A)⋅U(Λ)\lx@stackrel∼⟶Cl(Λ): α⋅J0(A)⋅U(A)⋅U(Λ)↦[αΛ]

is an isomorphism of groups.

For our purposes it will suffice to know that for any the corresponding element is trivial.

## 5. Semi-local counterexamples

After these general sections we will now start to work with a more concrete class of groups which will ultimately provide our counterexample. Let be a finite group of the form

 (7) G=N⋊A

where is an abelian group. Moreover, let be a cyclic group such that the exponent of and the exponent of coincide. Let

 ε: G⟶Z: g↦εgG

be a class function which vanishes outside of (the notation for is deliberately chosen to resemble our notation for partial augmentations). When we say that the partial augmentations of a unit are given by , for some commutative ring , we mean that for all . Define

 (8) χ=∑gGεgG⋅1↑G×U[g]

Note that, a priori, is only a virtual character of . Assume that all of the following hold:

1. .

2. If for some , then for all and all primes dividing the order of .

3. For each prime dividing the order of we have a decomposition

 (9) χ|N×U=∑n∈Npξn⊗1↑Np×Up[n]p

where is a proper character of for each .

The aim of this section is to prove the following theorem:

###### Theorem 5.1.

Let be a finite collection of primes. Then, under the above assumptions, there exists a -regular -lattice with character . Moreover, the partial augmentations of the associated unit are given by .

By [CW00, Theorem 3.3] the condition 3 actually implies that there exists a -lattice which is locally free over . We will not use this fact, but it provided the original motivation for this construction. The condition is also studied in [MR17a].

###### Remark 5.2.

In (9), the are uniquely determined. Namely,

 ξn=∑m∈Np′ε(m⋅n)G⋅1↑Np′×Up′[m]p′
###### Proof.

By Mackey decomposition we have

 1↑G×U[h]|N×U=[CG(h):N]∑m∈hG1↑N×U[m]

for every . Thus we obtain

 χ|N×U=∑h∈N[CG(h):N]⋅εhG⋅1↑N×U[h]

Note that our assumptions imply that is equal to whenever . Hence

 χ|N×U=∑h∈NεhG⋅1↑N×U[h]

So, setting

 ξn=∑m∈Np′ε(m⋅n)G⋅1↑Np′×Up′[m]p′

for every certainly ensures that (9) holds.

To prove that the ’s are uniquely determined as virtual characters, it suffices to show that they can be recovered from . Let . Then

 1↑Np×Up[m]p((n,cp))={0if n≠m|Np|otherwise

Hence, if and , then

 χ((h⋅n,c))=|Np|⋅ξn((h,cp′))

which shows that is determined uniquely by . ∎

###### Definition 5.3.

For a group we define

 (10) e(X)=1|X|∑x∈Xx
###### Definition 5.4.

Assume that is a prime dividing the order of and let be any prime not dividing the order of (in particular, is a possible choice for ). Let be a subgroup such that is cyclic. Let denote the primitive idempotent in the rational group algebra corresponding to the unique faithful irreducible representation of over , and denote its preimage in by (we may choose in such a way that ).

We define a -lattice

 (11) M0(X,p,q)=Z(q)↑Np×UpXp⊗(Z(q)↑Np′×Up′Xp′⋅ˆe)

as well as a -lattice

 (12) M(X,p,q)=M0(X,p,q)↑G×UN×U
###### Proposition 5.5.

The character of the -lattice is equal to

 ψM0(X,p,q)=1↑Np×UpXp⊗φ

where is the unique irreducible rational character of with kernel .

###### Proof.

This follows immediately from the fact that is afforded by the -lattice . ∎

###### Remark 5.6.

The following description of the character of for certain is useful for explicit computations, even though we do not use it in this article:

1. If , then

 (13) ψM(X,p,q)=1↑G×UX
2. If is cyclic of order for some prime , then

 (14) ψM(X,p,q)=1↑G×UX−1↑G×UX⋅(Np′×Up′)
###### Proposition 5.7.

If , then is projective.

###### Proof.

This follows from the Mackey formula, as is a direct summand of :

 (15) Z(q)↑G×UX|G≅⨁(g,u)Z(q)↑GX(g,u)∩G=⨁(g,u)Z(q)↑G(X∩G)(g,u)

where the summation index runs over a transversal of the double cosets . Each summand on the right hand side is induced from a -subgroup of , and therefore is projective. ∎

###### Lemma 5.8.

Let be a prime dividing the order of and let be some -element of . Let be the following character of :

 (16) χn=∑CG(n)⋅g∈CG(n)∖G1↑Np×Up[ng]p⊗ξgn

where is the character of defined in the beginning of this section (in particular is stabilised by ).

Then, for any prime not dividing the order of , is the restriction to of the character of the -lattice

 (17) L=⨁φ∈IrrQ(Np′×Up′)/CG(n)M([n]p×Ker(φ),p,q)⊕μ(φ,n)

with

 (18) μ(φ,n)=(φ,ξn)(φ,φ)⋅1[CG(n)∩NG(Ker(φ)):N]∈Z


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HomeUncategorizedcondition for orthogonal eigenvectors

\ker(A) = \ker(A^TA) = \ker(AA^T) = \ker(A^T) = \im(A)^\perp is a properly normalized eigenstate of ËA, corresponding to the eigenvalue a, which is orthogonal to Ïa. We The partial answer is that the two eigenvectors span a 2-dimensional subspace, and there exists an orthogonal basis for that subspace. A sucient condition â¦ \$4pt] \dfrac{2}{L} \int_0^L \sin \left( \dfrac{2}{L}x \right) \sin \left( \dfrac{3}{L}x \right) &= ? It happens when A times A transpose equals A transpose. Thus, if two eigenvectors correspond to different eigenvalues, then they are orthogonal. The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. Then any corresponding eigenvector lies in \ker(A - \lambda I). Because x is nonzero, it follows that if x is an eigenvector of A, then the matrix A I is In Matlab, eigenvalues and eigenvectors are given by [V,D]=eig(A), where columns of V are eigenvectors, D is a diagonal matrix with entries being eigenvalues. This equality means that $$\hat {A}$$ is Hermitian. The reason why this is interesting is that you will often need to use that given a hermitian operator A, there's an orthonormal basis for the Hilbert space that consists of eigenvectors of A. hv;Awi= hv; wi= hv;wi. ABÎ. If a matrix A satifies A T A = A A T, then its eigenvectors are orthogonal. (Thereâs also a very fast slick proof.) \endgroup â Arturo Magidin Nov 15 '11 at 21:19 To prove that a quantum mechanical operator $$\hat {A}$$ is Hermitian, consider the eigenvalue equation and its complex conjugate. Multiply Equation $$\ref{4-38}$$ and $$\ref{4-39}$$ from the left by $$ψ^*$$ and $$ψ$$, respectively, and integrate over the full range of all the coordinates. @Shiv Setting that aside (indeed, one can prove the existence of SVD without the use of the spectral theorem), we have AA^T = A^TA \implies V^T\Sigma^2 V = U^T \Sigma^2 U, but it is not immediately clear from this that U = V. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. I used the definition that U contains eigenvectors of AA^T and V contains eigenvectors of A^TA. \[\hat {A}^* \psi ^* = a^* \psi ^* = a \psi ^* \label {4-39}$, Note that $$a^* = a$$ because the eigenvalue is real. initial conditions y 1(0) and y 2(0). Note that $$ψ$$ is normalized. @Shiv As I said in my comment above: this result is typically used to prove the existence of SVD. Therefore $$\psi(n=2)$$ and $$\psi(n=3)$$ wavefunctions are orthogonal. Legal. Matrix Ais diagonalizable (A= VDV1, Ddiagonal) if it has nlinearly independent eigenvectors. And those matrices have eigenvalues of size 1, possibly complex. Find $$N$$ that normalizes $$\psi$$ if $$\psi = N(φ_1 − Sφ_2)$$ where $$φ_1$$ and $$φ_2$$ are normalized wavefunctions and $$S$$ is their overlap integral. Since the two eigenfunctions have the same eigenvalues, the linear combination also will be an eigenfunction with the same eigenvalue. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. ~v i.~v j = 0, for all i 6= j. In linear algebra, eigenvectors are non-zero vectors that change when the linear transformation is applied to it by a scalar value. If A is symmetric and a set of orthogonal eigenvectors of A is given, the eigenvectors are called principal axes of A. $\textbf {\mathrm {AB\Gamma}}$. times A. 6.3 Orthogonal and orthonormal vectors Definition. Note that $\DeclareMathOperator{\im}{im}$ To prove this, we start with the premises that $$ψ$$ and $$φ$$ are functions, $$\int d\tau$$ represents integration over all coordinates, and the operator $$\hat {A}$$ is Hermitian by definition if, $\int \psi ^* \hat {A} \psi \,d\tau = \int (\hat {A} ^* \psi ^* ) \psi \,d\tau \label {4-37}$. We prove that eigenvalues of orthogonal matrices have length 1. The eigenvalues and orthogonal eigensolutions of Eq. Eigenvalue and Eigenvector Calculator. The above proof of the orthogonality of different eigenstates fails for degenerate eigenstates. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. I have not had a proof for the above statement yet. By the way, by the Singular Value Decomposition, A = U Î£ V T, and because A T A = A A T, then U = V (following the constructions of U and V). If the eigenvalues of two eigenfunctions are the same, then the functions are said to be degenerate, and linear combinations of the degenerate functions can be formed that will be orthogonal to each other. i.e. Definition: A symmetric matrix is a matrix $A$ such that $A=A^{T}$.. In fact, the skew-symmetric or diagonal matrices also satisfy the condition $AA^T=A^TA$. \label{4.5.1}\]. Suppose that $\lambda$ is an eigenvalue. Usually the fact that you are trying to prove is used to prove the existence of a matrix's SVD, so your approach would be using the theorem to prove itself. This proposition is the result of a Lemma which is an easy exercise in summation notation. Applying T to the eigenvector only scales the eigenvector by the scalar value Î», called an eigenvalue. the dot product of the two vectors is zero. It can be seen that if y is a left eigenvector of Awith eigenvalue , then y is also a right eigenvector of AH, with eigenvalue . of the new orthogonal images. If we computed the sum of squares of the numerical values constituting each orthogonal image, this would be the amount of energy in each of the But how do you check that for an operator? Such eigenstates are termed degenerate.  Proposition 3 Let v 1 and v 2 be eigenfunctions of a regular Sturm-Liouville operator (1) with boundary conditions (2) corresponding â¦ It is straightforward to generalize the above argument to three or more degenerate eigenstates. But in the case of an inï¬nite square well there is no problem that the scalar products and normalizations will be ï¬nite; therefore the condition (3.3) seems to be more adequate than boundary conditions. For instance, if $$\psi_a$$ and $$\psi'_a$$ are properly normalized, and, $\int_{-\infty}^\infty \psi_a^\ast \psi_a' dx = S,\label{ 4.5.10}$, $\psi_a'' = \frac{\vert S\vert}{\sqrt{1-\vert S\vert^2}}\left(\psi_a - S^{-1} \psi_a'\right) \label{4.5.11}$. Eigen Vectors and Eigen Values. they satisfy the following condition (13.38)dTi Adj = 0 where i â  j Note that since A is positive definite, we have (13.39)dTi Adi > 0 4. Given a set of vectors d0, d1, â¦, dn â 1, we require them to be A-orthogonal or conjugate, i.e. Watch the recordings here on Youtube! Have questions or comments? We saw that the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. But again, the eigenvectors will be orthogonal. The eigenvalues of operators associated with experimental measurements are all real. By the way, by the Singular Value Decomposition, $A=U\Sigma V^T$, and because $A^TA=AA^T$, then $U=V$ (following the constructions of $U$ and $V$). Denition of Orthogonality We say functions f(x) and g(x) are orthogonal on a


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A formal model of reaction systems was a few years ago introduced in [4]. The original purpose was to model interactions between biochemical reactions. The reference [4] contains some of the original motivation and initial setup. Each reaction is characterized by its set of reactants, each of which has to be present for the reaction to take place, by its set of inhibitors, none of which is allowed to be present, and by its set of products, each of which will be present after a successful reaction. Thus, a single reaction is based on facilitation and inhibition. Everything is defined within a fixed finite background set S. The sets of reactants, inhibitors and products, R, I and P, are nonempty subsets of S, the sets R and I being disjoint. A reaction system RS consists of finitely many such triples (R, I, P), called reactions. The application of RS to a subset Y of S (to be explained in detail below) produces another subset Y ′ of S and, thus, we have a function from subsets of S to subsets of S. Iterating the function we get a sequence of subsets of S, referred to as states of the sequence. Altogether this gives a new kind of mechanism for generating functions and sequences over a finite set.The model of reaction systems turned out to be suitable in different setups. Then also many variants of the systems and additions to them were introduced. The reference [1] constitutes a recent survey. However, in this paper we are concerned with the basic variant only. We now outline the contents of this paper. The exposition is largely selfcontained. The basic definitions are given in Section 2, together with some illustrative examples. The exposition is focused on the needs of the next sections. Section 3 discusses propositional formulas with monotonic truth-functions. Section 4 generalizes the discussion and proves the material needed in Section 5, where the main NP-completeness results are established. The paper is concluded with considerations concerning the connection between many-valued propositional logic and reaction systems

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