text:
# Towards an empirical investigation of inner alignment

post by evhub · 2019-09-23T20:43:59.070Z · LW · GW · 9 comments

## Contents

  Background
The proposal
environment
architecture
None

I recently wrote a post [AF · GW] detailing some concrete experiments that could be done now to start learning interesting things about inner alignment. The goal of that post was to provide an overview of a bunch of different possible proposals rather than go into any single proposal in detail.

The goal of this post, on the other hand, is to actually sketch out a more complete proposal for the single experiment I would most want to be done now, which is to provide a definitive empirical demonstration of an inner alignment failure.[1] Furthermore, I have tried to make this post as accessible as possible for someone with only a machine learning background so as to facilitate people being able to work on this without having read the entirety of “Risks from Learned Optimization.” Additionally, if you’re interested in working on this, definitely reach out to me either in the comments here or at evanjhub@gmail.com, as I’d love to help out however I can.

# Background

First, we have to understand what exactly we’re looking for when we say inner alignment failure. At least when I say inner alignment failure, I mean the following:

Inner alignment fails when your capabilities generalize but your objective does not [AF · GW].

That seems a bit cryptic, though—what do I actually mean by that? Well, consider a maze-solving agent trained to get to the end of mazes of the following form:

Then, I want to know how it will generalize on the following larger maze with an interesting twist where the green arrow that marked the end has now been moved to a different position:

In this situation, there are a couple of different ways in which your model could generalize:

1. Complete generalization failure: The model only knows how to solve small mazes and can’t properly navigate the larger maze.
2. Intended generalization: The model learned how to navigate mazes in general and uses that knowledge to get to the end of the larger maze.
3. Capability generalization without objective generalization: The model learned how to navigate mazes in general, but it learned to do so for the purpose of getting to the green arrow rather than actually getting to the end. Thus, the model successfully navigates the larger maze, but it successfully navigates to the green arrow rather than successfully navigating to the end.

The reason I think this last situation is particularly concerning—and in a very different way than the first failure mode of complete generalization failure—is that it raises the possibility of your model taking highly-competent well-optimized actions towards a different objective than the one you actually intended it to pursue.

Of course, this raises the question of why you would ever expect a model to learn a proxy like “find the green arrow” in the first place rather than just learn the actual goal. But that’s where empirical investigation can come in! I have some hypotheses about the sorts of proxies I think models like this are likely to learn—namely, those proxies which are faster/easier-to-compute/simpler/etc. than the true reward—but those are just hypotheses. To put them to the test, we need to be able to train an agent to concretely demonstrate this sort of capability generalization without objective generalization and start measuring and understanding the sorts of proxies it tends to gravitate towards.

# The proposal

I believe that it should be possible to demonstrate capability generalization without objective generalization in current ML systems. This is definitely a questionable assumption—to the extent that good cross-domain generalization at all is currently beyond our reach, one might expect that you also wouldn’t be able to get this sort of perverse generalization. I am less pessimistic, however. To make this happen, though, there’s going to be two components that you’re definitely going to need:

1. An environment with lots of indistinguishable or barely distinguishable proxies.
2. An architecture with the capacity to learn a search algorithm that can actually succeed or fail at objective generalization in a meaningful sense.

I’ll try to address some of the complexities I see arising in these two components below. However, given those two components, the basic proposal is as follows:

1. Train an RL agent (e.g. with standard PPO) using that architecture in that environment.
2. Test how it generalizes to environments where the different possible proxies that it could have learned come apart. In particular, look for situations where it optimizes some proxy off-distribution at the expense of the true reward.

## The environment

The first condition is that we want an environment with lots of indistinguishable or barely distinguishable proxies. In some sense, environments with lots of proxies are quite common—the real world is full of proxies, for example, and other very complex environments should also have lots of different variables which are correlated with the true reward without necessarily exactly matching up with it.

That being said, I think there’s a danger here of demonstrating something too trivial: if you train on empirical performance according to some reward function that you specifically set up to be exactly equivalent to some other reward function on the training data, then you shouldn’t be that surprised if you generalize according to rather than . In some sense, I think it’s fine if that’s the primary takeaway—I think a good portion of the problem I want to point to is contained just within that observation—but I want to demonstrate that this phenomenon doesn’t just appear in artificial environments where and are explicitly constructed to be indistinguishable. Thus, I think it’s important to pick an environment that’s natural enough to be able to demonstrate that this sort of a problem isn’t isolated just to very artificial scenarios.

Another important property is going to be the ability to actually know what the likely proxies are for our agent to latch onto—if there are too many proxies, then we might not be able to determine what proxy it’s using at test time such that we wouldn’t be able to produce a clear demonstration.[2] Thus, though we want our environment to be diverse enough that it contains lots of different proxies for the true reward, we also want to ensure that the possible alternative proxies are obvious enough that we can actually test for them by coming up with test environments in which they come apart from the true reward.

Another property that—while not strictly necessary—would be nice to have in our environment would be proxies which allow us to test some of my hypotheses regarding what sorts of proxies models will be more likely to pay attention to. For example, I have hypothesized that models will gravitate towards proxies that are 1) easier for the model to optimize for and 2) simpler to specify in terms of the model’s input data. Thus, an ideal environment would be one that included some proxies which we could demonstrate did or did not satisfy those properties and see if the model does in fact systematically gravitate to the ones that do.

Finding an environment that satisfies all of these properties is likely to be far from trivial, and I suspect would end up being a significant portion of any project of this form. I suspect that the right way to do this would probably be to use some sort of physics sandbox. That being said, there are also other possibilities too including more complex environments such as Minecraft as well as simpler environments such as a gridworld. While I can provide lots of examples of the sorts of environments I’m envisioning here, I think the right thing to do is just to have a tight empirical feedback loop in terms of testing and iterating on lots of different environments (though I think you could probably do all of that iteration just in the physics sandbox setting).

## The architecture

I think architecture is also going to be really important to getting something like this to work. In particular, for you to get capability generalization without objective generalization, you have to have a model which is doing some sort of internal search such that it actually has an objective that can fail to generalize.[3] I think there is good reason to believe that many modern architectures (LSTMs, Transformers, etc.) might just be able to do this by default—though I am not that confident in that assertion, and I think it might also be necessary to make some changes to make this possible. However, I am optimistic that at least some forms of capability generalization without objective generalization can be demonstrated in current models.

In particular, some forms of capability generalization without objective generalization seem easier to demonstrate in current models than others. For example, two common forms of this which I think are important to distinguish between are the side-effect case and the instrumental case.

In the side-effect case, the reason that and are identified during training is that has the side-effect of increasing —that is, increasing causes to increase. As an example, imagine a cleaning robot where is the cleanliness of the room and is the number of times the room is swept. In this case, the two proxies of cleanliness and times swept are identified because sweeping the room causes the room to become cleanlier.

Alternatively, in the instrumental case, and are identified because the best strategy for maximizing is to maximize —that is, increasing causes to increase. For example, in the cleaning robot case where is the cleanliness of the room, might be the amount of dirt in the dustpan. In this case, the two proxies are identified because cleaning the room causes there to be more dirt in the dustpan.

I hypothesize that the side-effect case will be visible before the instrumental case, since the instrumental case requires a model which is significantly more forward-looking and capable of planning out what it needs to do to accomplish some goal. The side-effect case, on the other hand, doesn’t require this, and thus I suspect to see it appear first. In particular, I expect that the side-effect case will be significantly easier to demonstrate with current architectures than the instrumental case, since the instrumental case might require models which can learn more powerful search algorithms than we currently know how to implement (though it also might not—it’s currently unclear). However, I’m optimistic that at least the side-effect case will be possible to demonstrate in current models, and I’m hopeful that current models might even be up to the task of demonstrating the instrumental case as well.

1. Note that I am not the only person currently thinking about/working on this—most notably Rohin Shah at CHAI also recently developed a proposal to produce a demonstration of an inner alignment failure that shares many similarities with my proposal here. ↩

2. In some sense, this is actually exactly what the worry is for AGI-level systems—if the environment is so complex that there are too many different proxies that we can’t all test during training, then we might not be able to catch the existence of a situation where our model generalizes perversely in this way even if one actually exists. ↩

3. I call models which are doing search internally (and thus have some notion of an objective) “mesa-optimizers.” ↩

comment by rohinmshah · 2019-09-28T17:18:01.819Z · LW(p) · GW(p)
Rohin Shah at CHAI also recently developed a proposal to produce a demonstration of an inner alignment failure that shares many similarities with my proposal here.

The proposal is here. I'm not planning to work on it soon, so if someone else would like to, please do (but let me know so we don't duplicate work).

Similarities:

1. A focus on inner alignment failures, and particularly the definition of "capabilities generalize, objective doesn't"

2. Identification of "many possible proxy rewards" as the key problem, and also that it isn't particularly interesting / surprising to construct environments where there are multiple identical rewards

3. The focus on the two components of environment design and an architecture that can lead to capability generalization

Differences:

1. I am less optimistic that existing architectures will learn general capabilities on simple environments (though they might do so on complex ones if you have enough data). I think you'll need something like the SATNet (see proposal for details). This could be a difference in what experiments we're imagining -- I think it is more likely than not that we see either intended generalization or just capability generalization for the maze example using an RNN, if the only thing you change at test time is where the arrow is. However, if you make the maze larger, then I think it is more likely than not that you get a complete generalization failure (though this depends on how exactly you encoded the maze for your RNN to consume).

2. I care a decent amount about producing an example that is convincing to ML researchers, and so I want a much more realistic environment, like Minecraft.

3. I don't agree that inner alignment failures need a "model which is doing some sort of internal search such that it actually has an objective that can fail to generalize". It's possible to learn heuristics that don't look like search that nonetheless capability-generalize well. However, if you want, you can take the intentional stance towards the model in order to infer a behavioral objective.

4. I am less interested in figuring out which behavioral objectives neural nets tend to learn: there are so many potential behavioral objectives that I think it is basically inevitable that regular training leads to the wrong one; it doesn't seem that helpful to know which wrong ones it leads to. You could hope that this would help us "look into" a neural net to find the inner objective, but per disagreement 3 I don't expect this to work. (I'm more optimistic about looking for particular scenarios where the model optimizes for the wrong thing.) This is not to say I think such an empirical investigation would be useless -- it is the sort of thing that is likely to produce insights we didn't see coming -- just that it seems less useful than other research one could be doing.

Side question: How can I easily make properly-formatted numbered lists?

comment by G Gordon Worley III (gworley) · 2019-10-15T01:37:33.656Z · LW(p) · GW(p)

I really like the maze example. It's good enough, thanks to your short explanation, that when I saw the big maze I was genuinely unsure of the intended behavior, i.e. should the thing find the green arrow (that marking the goal, and the exit is the proxy) or exit the maze (the arrow is the proxy).

comment by rohinmshah · 2019-09-28T17:26:31.393Z · LW(p) · GW(p)
Alternatively, in the instrumental case, r and r′ are identified because the best strategy for maximizing r′ is to maximize r—that is, r increasing causes r′ to increase. For example, in the cleaning robot case where r is the cleanliness of the room, r′ might be the amount of dirt in the dustpan. In this case, the two proxies are identified because cleaning the room causes there to be more dirt in the dustpan.

I am confused by this. Surely putting dirt in the dustpan is causing the room to be clean, not the other way around?

Either way, I think the most likely form of inner alignment failure you can make work now is one where the true reward is to do Y, and in the training environment, you need to do X in order to do Y, and so you learn a reward function that rewards both X and Y highly. Then, at test time it becomes a lot easier to do X, and the agent only does X, ignoring Y. I would have said that the dustpan example fit this story, but you seem to call it part of the "instrumental" case.

ETA: Lol, this is basically what Matthew's environment [LW · GW] does, which I read right after writing this comment.

comment by shminux · 2019-09-24T02:21:06.245Z · LW(p) · GW(p)

This "proxy fireworks" where expanding the system causes various proxies to visibly split and fly in different directions is definitely a good intuitive way to understand some of the AI alignment issues.

What I am wondering is whether it is possible to specify anything but a proxy, even in principle. After all, humans as general intelligences fall into the same trap of optimizing proxies (instrumental goals) instead of the terminal goals all the time. We are also known for our piss-poor corrigibility properties. Certainly the maze example or a clean room example are simple enough, but once we ramp up the complexity, odds are that the actual optimization proxies start showing up. And the goal is for an AI to be better at discerning and avoiding "wrong" proxies than any human, even though humans are apt to deny that they are stuck optimizing proxies despite the evidence to the contrary. The reaction (except in PM from Michael Vassar) to an old post of mine [LW · GW] is a pretty typical example of that. So, my expectation would be that we would resist to even consider that we optimize a proxy when trying to align a corrigible AI.

comment by ofer · 2019-09-24T04:57:07.388Z · LW(p) · GW(p)

Very interesting! This research direction might lead to researchers having better intuitions about what sort of mesa-objectives we're more likely to end up with.

Perhaps similar experiments can be done with supervised learning (instead of RL).

comment by SoerenMind · 2019-09-26T17:24:36.386Z · LW(p) · GW(p)

Just registering that I'd like to see a less loaded term than "inner alignment" being adopted.

Don't want to be confused with these people: "Inner alignment means becoming more of your true self. Your inner being is your true self, and by developing your inner potential, you express more and more of your true self."

comment by habryka (habryka4) · 2019-09-26T17:35:32.568Z · LW(p) · GW(p)

Contrarian opinion: The two terms are not unrelated and have a surprising amount of actual shared structure.

comment by Ben Pace (Benito) · 2019-09-26T19:19:12.648Z · LW(p) · GW(p)

Snap, I was also gonna write this comment.

comment by SoerenMind · 2019-10-12T10:33:20.282Z · LW(p) · GW(p)

Hmmm my worry isn't so nuch that we have an unusual definition of inner alignment. It's more the opposite: that outsiders associate this line of research with quackery (which only gets worse if our definition is close to the standard one).


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Measure AC performance metrics of ADC output

• Library:
• Mixed-Signal Blockset / ADC / Measurements & Testbenches

## Description

The ADC AC Measurement block measures ADC AC performance metrics such as signal to noise ratio (SNR), signal to noise and distortion radio (SINAD), spurious free dynamic range (SFDR), effective number of bits (ENOB), noise floor, and conversion delay. You can use ADC AC Measurement block to validate the ADC architectural models provided in Mixed-Signal Blockset™, or you can use an ADC of your own implementation

## Ports

### Input

expand all

Converted digital signal from an ADC, specified as a scalar.

Data Types: `fixed point` | `single` | `double` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32`

Indicates whether the analog to digital conversion is complete, specified as a scalar.

Data Types: `double`

External conversion start clock, specified as a scalar. The signal at the start port indicates when the analog to digital conversion process starts.

Data Types: `double`

## Parameters

expand all

Type of distortion the ADC AC Measurement block is set to measure, specified as `Harmonic` or `Intermodulation`.

#### Programmatic Use

 Block parameter: `DistortionMeasurement` Type: character vector Values: `Harmonic` | `Intermodulation` Default: `Harmonic`

Frequency of the analog input signal to an ADC block, specified as a positive real scalar in hertz. Analog stimulus frequency must match the input frequency to the ADC device under test.

• When the ADC AC Measurement is set to measure the `Harmonic` distortion, the default value of Analog stimulus frequency is `10000`.

• When the ADC AC Measurement is set to measure the `Intermodulation` distortion, the default value of Analog stimulus frequency is `[9000,11000]`.

Analog stimulus frequency needs to satisfy two requirements:

• All the output codes of the ADC must be activated.

• The Analog stimulus frequency must not share any common multiples other than 1 with the Start conversion frequency.

To satisfy both the conditions, use the equation ${f}_{\text{analog}}=\frac{J}{M}{f}_{\text{start}}$ [2],

where:

fanalog is the analog signal frequency,

fstart is the start conversion frequency,

$M>{2}^{Nbits}·\pi$, where Nbits is the number of bits of the ADC,

and J is an integer with no common factors with M.

#### Programmatic Use

 Block parameter: `InputFrequency` Type: character vector Values: positive real scalar | positive real valued vector Default: `10000`

Resolution bandwidth, specified as a positive real scalar in hertz. This parameter defines the smallest positive frequency that can be resolved. By default, this parameter is calculated automatically. You can deselect Set automatically to customize the value.

• When the ADC AC Measurement is set to measure the `Harmonic` distortion, the default value of Resolution bandwidth (Hz) is `1000`.

• When the ADC AC Measurement is set to measure the `Intermodulation` distortion, the default value of Resolution bandwidth (Hz) is `900`.

#### Programmatic Use

 Block parameter: `RBW` Type: character vector Values: positive real scalar Default: `1000`

Number of physical bits in ADC, specified as a unitless positive real integer. Number of bits must match the resolution specified in the ADC block.

#### Programmatic Use

 Block parameter: `NBits` Type: character vector Values: positive real integer Default: `5`

Frequency of the start conversion clock of the ADC, specified as a positive real scalar in hertz. Start conversion frequency must match the frequency of the start conversion clock of the ADC block.

#### Programmatic Use

 Block parameter: `Frequency` Type: character vector Values: positive real scalar Default: `10e6`

Delays measurement analysis to avoid corruption by transients, specified as a nonnegative real scalar in seconds.

#### Programmatic Use

 Block parameter: `HoldOffTime` Type: character vector Values: nonnegative real scalar Default: `0`

Minimum time the simulation must run to obtain meaningful results, specified as a positive real scalar in seconds.

For AC measurement, the simulation must run so that the ADC can generate 6 spectral updates of the ADC output. The time to generate one spectral output based on Welch's method [1] is given by:

`$t=\frac{1.5·SamplingFrequency}{RBW}$`

where SamplingFrequency and RBW are the sampling frequency and resolution bandwidth of the spectrum estimator inside the ADC Testbench block.

This parameter is only reported by the testbench and is not editable.

Data Types: `double`

Click to automatically set the Recommended min. simulation stop time (s) as the stop time of the Simulink® model.

Store detailed test results in the base workspace for further processing at the end of simulation. By default, this option is not selected.

Name of the variable that stores detailed test results, specified as a character string.

#### Dependencies

This parameter is only available when Output result to base workspace is selected.

#### Programmatic Use

 Block parameter: `VariableName` Type: character vector Values: character string Default: `adc_ac_out`

Displays spectrum analyzer during simulation. By default, this option is not selected.

expand all

## References

[1] Spectrum Analyzer, DSP System Toolbox, MathWorks Documentation.

[2] IEEE Std 1241-2010. "IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters," pp. 29-30, 14 January 2011.

format: plain


text:
## Algebra & Number Theory

### Lifting harmonic morphisms II: Tropical curves and metrized complexes

#### Abstract

We prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the nonvanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular, we show that any finite harmonic morphism of (nonaugmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve $C$ coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from $C$ to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a $d$-gonal tropical curve that does not lift to a $d$-gonal algebraic curve.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 2 (2015), 267-315.

Dates
Revised: 9 July 2014
Accepted: 6 December 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842281

Digital Object Identifier
doi:10.2140/ant.2015.9.267

Mathematical Reviews number (MathSciNet)
MR3320845

Zentralblatt MATH identifier
1312.14138

#### Citation

Amini, Omid; Baker, Matthew; Brugallé, Erwan; Rabinoff, Joseph. Lifting harmonic morphisms II: Tropical curves and metrized complexes. Algebra Number Theory 9 (2015), no. 2, 267--315. doi:10.2140/ant.2015.9.267. https://projecteuclid.org/euclid.ant/1510842281

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text:
Corpus ID: 118951809

# Several almost critical regularity conditions based on one component of the solutions for 3D N-S Equations

@inproceedings{Fang2013SeveralAC,
title={Several almost critical regularity conditions based on one component of the solutions for 3D N-S Equations},
author={Daoyuan Fang and Chenyin Qian},
year={2013}
}
• Published 2013
• Mathematics
• In this article, we establish several almost critical regularity conditions such that the weak solutions of the 3D Navier-Stokes equations become regular, based on one component of the solutions, say $u_3$ and $\partial_3u_3$.

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• Mathematics
• 2011

## Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor

• Mathematics
• 2011
VIEW 3 EXCERPTS


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# Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons

## Abstract

Graphene-based samples have shown a plethora of exotic characteristics and these properties may help the realization of a new generation of fast electronic devices. However, graphene’s centrosymmetry prohibits second-order electronic transport. Here, we show giant second-order nonlinear transports in graphene moiré superlattices at zero magnetic field, both longitudinal and transverse to the applied current direction. High carrier mobility and inversion symmetry breaking by hexagonal boron nitride lead to nonlinear conductivities five orders of magnitude larger than those in WTe2. The nonlinear conductivity strongly depends on the gate voltage as well as on the stacking configuration, with a giant enhancement originating from the moiré bands. Longitudinal nonlinear conductivity cannot originate from Berry curvature dipoles. Our theoretical modelling highlights skew scattering of chiral Bloch electrons as the physical origin. With these results, we demonstrate nonlinear charge transport due to valley-contrasting chirality, which constitutes an alternative means to induce second-order transports in van der Waals heterostructures. Our approach is promising for applications in frequency-doubling and energy harvesting via rectification.

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• ### Strong room-temperature bulk nonlinear Hall effect in a spin-valley locked Dirac material

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## Data availability

The data that support the findings of this study are available within the paper and the Supplementary Information. Source data are provided with this paper. Other relevant data are available from the corresponding authors on reasonable request.

## Code availability

The codes that support this study are available from the corresponding authors on reasonable request.

## References

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39. Battilomo, R., Scopigno, N. & Ortix, C. Berry curvature dipole in strained graphene: a Fermi surface warping effect. Phys. Rev. Lett. 123, 196403 (2019).

40. & Kumar, D. et al. Room-temperature nonlinear Hall effect and wireless radiofrequency rectification in Weyl semimetal TaIrTe4. Nat. Nanotechnol. 16, 421–425 (2021).

41. Finney, N. R. et al. Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices. Nat. Nanotechnol. 14, 1029–1034 (2019).

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43. Stauber, T., Peres, N. M. R. & Guinea, F. Electronic transport in graphene: a semiclassical approach including midgap states. Phys. Rev. B 76, 205423 (2007).

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## Acknowledgements

We acknowledge K. Novoselov, A. Rodin and A. Carvalho for discussions. This work was partially supported by the National Research Foundation (NRF) Singapore Investigatorship (grant no. NRFI06-2020-0015). P.H. was sponsored by the National Key Research and Development Program of China (grant no. 2020YFA0308800), the Natural Science Foundation of Shanghai (grant no. 21ZR1404300) and start-up funding from Fudan University. The work at Massachusetts Institute of Technology was supported by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under Collaborative Agreement Number W911NF-18-2-0048.

## Author information

Authors

### Contributions

P.H., G.K.W.K. and H.Y. designed the experimental study. P.H. performed the measurements and analysed the data. G.K.W.K., J.Y.T., A.H.C.N. and J.H. fabricated devices. H.I. and L.F. performed theoretical studies. P.H., G.K.W.K., H.I. and H.Y. wrote the manuscript. All authors commented on the manuscript. H.Y. led the project.

### Corresponding authors

Correspondence to Pan He, Liang Fu or Hyunsoo Yang.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Nanotechnology thanks Yuval Ronen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 The second-order nonlinear transports in graphene honeycomb lattice with inversion symmetry breaking.

The second-order electric responses are schematically depicted for an ac electric field Eω (black) applied along a zigzag (a) and an armchair direction (b). The second-order current with double frequency J (red) exists along the armchair direction for both cases. The three-fold rotational symmetry is schematically shown here with a rotation axis perpendicular to the plane and centered at a carbon atom, where the three dashed lines (equivalent crystalline axes) cross.

### Extended Data Fig. 2 Tunability of nonlinear transport with gating in another superlattice device (Device 2).

a, b, The longitudinal resistivity ρxx (a) and Hall resistivity ρxy (b) as a function of back gate voltage Vg. c, d, The second-harmonic longitudinal $$V_x^{2\omega }$$ (c) and transverse $$V_y^{2\omega }$$ (d) voltage as a function of Vg at I = 3 µA. The data in (a), (c) and (d) are measured under zero magnetic field. The data in (b) are obtained at H = 0.5 T. e, f, The second-order nonlinear conductivities along the longitudinal $$\sigma _{{{{xxx}}}}^{\left( 2 \right)}$$(e) and transverse $$\sigma _{{{{yxx}}}}^{\left( 2 \right)}$$(f) directions as a function of Vg. All data are measured at T = 1.7 K. The misalignment angle between graphene and hBN in this device is similar to the one in the main text, as estimated from the Vg where SDP occurs.

### Extended Data Fig. 3 Temperature dependence of the nonlinear transport in superlattice Device 2.

a, ρxx versus Vg measured at different temperatures under zero magnetic field. b, ρxx as a function of temperature for three gate voltages Vg, as marked by arrows in (a). c, The mobility as a function of temperature for three gate voltages. The data are obtained by measuring ρxx (Vg) and ρxy (Vg) simultaneously at H = 0.5 T. d, e, The extracted $$\sigma _{{{{xxx}}}}^{\left( 2 \right)}$$(Vg) (d) and $$\sigma _{{{{yxx}}}}^{\left( 2 \right)}$$(Vg) (e) at different temperatures. At Vg = −62.5 V, $$\sigma _{{{{yxx}}}}^{\left( 2 \right)}$$(Vg) shows a peak in (e). The data in (a), (b), (d) and (e) are obtained under zero magnetic field and I = 3 µA. (f) $$\sigma _{{{{yxx}}}}^{\left( 2 \right)}$$ and μ3 as a function of temperature at Vg = −62.5 V.

### Extended Data Fig. 4 The Fermi level dependence of linear conductivity.

The linear conductivity as a function of gate voltage Vg for Device 1 (a) and Device 2 (b). The data were measured at T = 1.7 K and H = 0 T. The arrows indicate the position of primary and secondary Dirac points.

### Extended Data Fig. 5 The cubic scaling of nonlinear conductivity with mobility.

The nonlinear conductivity σyxx versus the cube of mobility for Device 1 (a) and Device 2 (b). The red lines are linear fittings to the experimental data. The error bars represent the standard deviation of the nonlinear conductivity.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–11 and Table 1.

## Source data

### Source Data Fig. 2

Statistical Source Data

### Source Data Fig. 3

Statistical Source Data

### Source Data Fig. 4

Statistical Source Data

### Source Data Table 1

Statistical Source Data

### Source Data Extended Data Fig. 2

Statistical Source Data

### Source Data Extended Data Fig. 3

Statistical Source Data

### Source Data Extended Data Fig. 4

Statistical Source Data

### Source Data Extended Data Fig. 5

Statistical Source Data

## Rights and permissions

Reprints and Permissions

He, P., Koon, G.K.W., Isobe, H. et al. Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons. Nat. Nanotechnol. 17, 378–383 (2022). https://doi.org/10.1038/s41565-021-01060-6

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• DOI: https://doi.org/10.1038/s41565-021-01060-6

• ### Strong room-temperature bulk nonlinear Hall effect in a spin-valley locked Dirac material

• Lujin Min
• Hengxin Tan
• Zhiqiang Mao

Nature Communications (2023)

• ### A detector that can learn the fingerprint of light

• Justin C. W. Song
• Yidong Chong

Nature (2022)


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Substituting variables into displayed maths

Attention!

This page is about substituting variables into mathematical expressions. You can substitute text strings into plain text using curly braces; see :ref:substituting-into-content for a description of the different methods of substituting variables into question text.

In Numbas, maths is displayed using LaTeX. For help with LaTeX, see :ref:LaTeX notation.

LaTeX is purely a typesetting language and is ill-suited for representing meaning in addition to layout. For this reason, dynamic or randomised maths expressions must be written in JME syntax and converted to LaTeX. Numbas provides two new LaTeX commands to do this for you.

To substitute the result of an expression into a LaTeX expression, use the \var command. Its parameter is a JME expression, which is evaluated and then converted to LaTeX.

For example:

$\var{2^3}$

produces:

$8$

and if a variable called x has been defined to have the value 3:

$2^{\var{x}}$

produces:

$2^{3}$

This simple substitution doesn't always produce attractive results, for example when substituted variables might have negative values. If y=-4:

$\var{x} + \var{y}$

produces:

$3 + -4$

To deal with this, and other more complicated substitutions, there is the \simplify command.

The main parameter of the \simplify command is a JME expression. It is not evaluated - it is converted into LaTeX as it stands. For example:

$\simplify{ x + (-1/y) }$

produces:

$x - \frac{1}{y}$

Variables can be substituted in by enclosing them in curly braces. For example:

$\simplify{ {x} / {y} }$

produces, when x=2,y=3:

$\frac{ 2 }{ 3 }$

The \simplify command automatically rearranges expressions, according to a set of simplification rules, to make them look more natural. Sometimes you might not want this to happen, for example while writing out the steps in a worked solution.

The set of rules to be used is defined in a list enclosed in square brackets before the main argument of the \simplify command. You can control the \simplify command's behaviour by switching rules on or off.

For example, in:

$\simplify{ 1*x }$

I have not given a list of rules to use, so they are all switched on. The unitFactor rule cancels the redundant factor of 1 to produce:

$x$

while in:

$\simplify[!unitFactor]{ 1*x }$

I have turned off the unitFactor rule, leaving the expression as it was:

$1 x$

When a list of rules is given, the list is processed from left to right. Initially, no rules are switched on. When a rule's name is read, that rule is switched on, or if it has an exclamation mark in front of it, that rule is switched off.

Sets of rules can be given names in the question's :ref:rulesets section, so they can be turned on or off in one go.

Display options

The \simplify and \var commands take an optional list of settings, separated by commas. These affect how certain elements, such as numbers or vectors, are displayed.

The following display options are available:

.. glossary::

fractionNumbers
This rule doesn't rewrite expressions, but tells the maths renderer that you'd like non-integer numbers to be displayed as fractions instead of decimals.

**Example:** \var[fractionNumbers]{0.5} produces :math:\frac{1}{2}.

mixedFractions
Improper fractions (with numerator larger than the denominator) are displayed in mixed form, as an integer next to a proper fraction.

**Example:** \var[fractionNumbers,mixedFractions]{22/7} produces :math:3 \frac{1}{7}.

rowVector
This rule doesn't rewrite expressions, but tells the maths renderer that you'd like vectors to be rendered as rows instead of columns.

alwaysTimes
The multiplication symbol is always included between multiplicands.

**Example:** \simplify[alwaysTimes]{ 2x } produces :math:2 \times x.

Simplification rules

As well as the display options, the \simplify command takes an optional list of names of simplification rules to use, separated by commas. These rules affect how the command rearranges the expression you give it.

Lists of simplification rule names are read from left to right, and rules are added or removed from the set in use as their names are read. To include a rule, use its name, e.g. unitfactor. To exclude a rule, put an exclamation mark in front of its name, e.g. !unitfactor.

Rule names are not case-sensitive: any mix of lower-case or upper-case works.

To turn all built-in rules on, use the name all. To turn all built-in rules off, use !all.

Note: Because they can conflict with other rules, the :term:canonicalOrder and :term:expandBrackets rules are not included in all. You must include them separately.

If you don't give a list of options, e.g. \simplify{ ... }, all the built-in rules are used. If you give an empty list of options, e.g. \simplify[]{ ... }, no rules are used.

For example, the following code:

\simplify[all,!collectNumbers,fractionNumbers]{ 0.5*x + 1*x^2 - 2 - 3 }

turns on every rule, but then turns off the collectNumbers rule, so every rule except collectNumbers can be applied. Additionally, the display option fractionNumbers is turned on, so the 0.5 is displayed as \frac{1}{2}.

Altogether, this produces the following rendering: \frac{1}{2} x + x^2 - 2 - 3

The following simplification rules are available:

.. glossary::

basic
These rules are always turned on, even if you give an empty list of rules.
They must be actively turned off, by including !basic in the list of rules.
See this behaviour in action <https://numbas.mathcentre.ac.uk/question/22839/when-does-the-basic-ruleset-get-turned-on/>_.

* +x → x (get rid of unary plus)
* x+(-y) → x-y (plus minus = minus)
* x-(-y) → x+y (minus minus = plus)
* -(-x) → x (unary minus minus = plus)
* -x → eval(-x) (if unary minus on a complex number with negative real part, rewrite as a complex number with positive real part)
* x+y → eval(x+y) (always collect imaginary parts together into one number)
* -x+y → -eval(x-y) (similarly, for negative numbers)
* (-x)/y → -(x/y) (take negation to left of fraction)
* x/(-y) → -(x/y)
* (-x)*y → -(x*y) (take negation to left of multiplication)
* x*(-y) → -(x*y)
* x+(y+z) → (x+y)+z (make sure sums calculated left-to-right)
* x-(y+z) → (x-y)-z
* x+(y-z) (x+y)-z'
* x-(y-z) > (x-y)+z
* (x*y)*z → x*(y*z) (make sure multiplications go right-to-left)
* n*i → eval(n*i) (always collect multiplication by :math:i)
* i*n → eval(n*i)

unitFactor
Cancel products of 1

* 1*x → x
* x*1 → x

unitPower
Cancel exponents of 1

* x^1 → x

unitDenominator
Cancel fractions with denominator 1

* x/1 → x

zeroFactor
Cancel products of zero to zero

* x*0 → 0
* 0*x → 0
* 0/x → 0

zeroTerm
Omit zero terms

* 0+x → x
* x+0 → x
* x-0 → x
* 0-x → -x

zeroPower
Cancel exponents of 0

* x^0 → 1

* -x+y → y-x
* -0 → 0

collectNumbers
Collect together numerical (as opposed to variable) products and sums.
The rules below are only applied if n and m are numbers.

* -x-y → -(x+y) (collect minuses)
* n+m → eval(n+m) (add numbers)
* n-m → eval(n-m) (subtract numbers)
* n+x → x+n (numbers go to the end of expressions)
* (x+n)+m → x+eval(n+m) (collect number sums)
* (x-n)+m → x+eval(m-n)
* (x+n)-m → x+eval(n-m)
* (x-n)-m) → x-eval(n+m)
* (x+n)+y → (x+y)+n (numbers go to the end of expressions)
* (x+n)-y → (x-y)+n
* (x-n)+y → (x+y)-n
* (x-n)-y → (x-y)-n)
* n*m → eval(n*m) (multiply numbers)
* x*n → n*x (numbers go to left hand side of multiplication, unless :math:n=i)
* m*(n*x) → eval(n*m)*x

simplifyFractions
Cancel fractions to lowest form.
The rules below are only applied if n and m are numbers and :math:gcd(n,m) > 1.

* n/m → eval(n/gcd(n,m))/eval(m/gcd(n,m)) (cancel simple fractions)
* (n*x)/m → (eval(n/gcd(n,m))*x)/eval(m/gcd(n,m)) (cancel algebraic fractions)
* n/(m*x) → eval(n/gcd(n,m))/(eval(m/gcd(n,m))*x)
* (n*x)/(m*y) → (eval(n/gcd(n,m))*x)/(eval(m/gcd(n,m))*y)

zeroBase
Cancel any power of zero

* 0^x → 0

constantsFirst
Numbers go to the left of multiplications

* x*n → n*x
* x*(n*y) → n*(x*y)

sqrtProduct
Collect products of square roots

* sqrt(x)*sqrt(y) → sqrt(x*y)

sqrtDivision
Collect fractions of square roots

* sqrt(x)/sqrt(y) → sqrt(x/y)

sqrtSquare
Cancel square roots of squares, and squares of square roots

* sqrt(x^2) → x
* sqrt(x)^2 → x
* sqrt(n) → eval(sqrt(n))   (if n is a square number)

trig
Simplify some trigonometric identities

* sin(n) → eval(sin(n)) (if n is a multiple of :math:\frac{\pi}{2})
* cos(n) → eval(cos(n)) (if n is a multiple of :math:\frac{\pi}{2})
* tan(n) → 0 (if n is a multiple of :math:\pi)
* cosh(0) → 1
* sinh(0) → 0
* tanh(0) → 0

otherNumbers
Evaluate powers of numbers.
This rule is only applied if n and m are numbers.

* n^m → eval(n^m)

cancelTerms
Collect together and cancel terms.
Like :term:collectNumbers, but for any kind of term.

* x +x → 2*x
* (z+n*x) - m*x → z + eval(n-m)*x
* 1/x + 3/x → 4/x

cancelFactors
Collect together and cancel factors inside a term.

* x*x → x^2
* x^2 * x → x^3
* (1/x^3)*x → 1*x^(-2)

collectLikeFractions
Collect together fractions over the same denominator.

* x/3 + 4/x → (x+4)/3

canonicalOrder
Rearrange the expression into a "canonical" order, using :jme:func:canonical_compare.

**Note:** This rule can not be used at the same time as :term:noLeadingMinus - it can lead to an infinite loop.

expandBrackets
Expand out products of sums.

* (x+y)*z → x*z + y*z
* 3*(x-y) → 3x - 3y

Display-only JME functions

There are a few "virtual" JME functions which can not be evaluated, but allow you to express certain constructs for the purposes of display, while interacting properly with the simplification rules.

.. function:: int(expression, variable)

An indefinite integration, with respect to the given variable.

* int(x^2+2,x) → :math:\displaystyle{\int \! x^2+2 \, \mathrm{d}x}
* int(cos(u),u) → :math:\displaystyle{\int \! \cos(u) \, \mathrm{d}u}

.. function:: defint(expression, variable,lower bound, upper bound)

A definite integration between the two given bounds.

* defint(x^2+2,x,0,1) → :math:\displaystyle{\int_{0}^{1} \! x^2+2 \, \mathrm{d}x}
* defint(cos(u),u,x,x+1) → :math:\displaystyle{\int_{x}^{x+1} \! \cos(u) \, \mathrm{d}u}

.. function:: diff(expression, variable, n)

:math:n-th derivative of expression with respect to the given variable

* diff(y,x,1) → :math:\frac{\mathrm{d}y}{\mathrm{d}x}
* diff(x^2+2,x,1) → :math:\frac{\mathrm{d}}{\mathrm{d}x} \left (x^2+2 \right )
* diff(y,x,2) → :math:\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}

.. function:: partialdiff(expression, variable, n)

:math:n-th partial derivative of expression with respect to the given variable

* partialdiff(y,x,1) → :math:\frac{\partial y}{\partial x}
* partialdiff(x^2+2,x,1) → :math:\frac{\partial }{\partial x} \left (x^2+2 \right )
* partialdiff(y,x,2) → :math:\frac{\partial{2}y}{\partial x^{2}}

.. function:: sub(expression,index)

Add a subscript to a variable name.
Note that variable names with constant subscripts are already rendered properly -- see :ref:variable-names -- but this function allows you to use an arbitray index, or a more complicated expression.

* sub(x,1) → :math:x_{1}
* sub(x,n+2) → :math:x_{n+2}

The reason this function exists is to allow you to randomise the subscript.
For example, if the index to be used in the subscript is held in the variable n, then this::

\simplify{ sub(x,{n}) }

will be rendered as

:math:x_{1}

when n = 1.

.. function:: sup(expression,index)

Add a superscript to a variable name.
Note that the simplification rules to do with powers won't be applied to this function, since it represents a generic superscript notation, rather than the operation of raising to a power.

* sup(x,1) → :math:x^{1}
* sup(x,n+2) → :math:x^{n+2}

You can’t perform that action at this time.


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# The graph of $f$ is given. State, with reasons, the numbers at which $f$ is $not$ differentiable.

## $x=1$, $x=5$

Limits

Derivatives

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##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

So we're given this graph of this function and were asked where this function F is not differentiable. Okay. So what we can see is that this is a smooth curve all the way through Until we get to x equals four. At which time we have this corner. That happens right? So at x equals four. Yeah. Is not differential. Remember the derivative have prime is the slope of the tangent line at that X. Well, the problem is with this corner here, there is no tangent line there And x equals four. There is no tangent line due to the corner happening the sharp turn until therefore it is not differentiable. At X equals four.

DM
Oklahoma State University

#### Topics

Limits

Derivatives

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp


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# Find all Linear spaces on an orbit (by a connected linear algebraic group)

Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.

Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace $L\subset V$.

Is there a nice description for all the linear spaces contained in $Y$?

I would be really happy with an answer in terms of the representation theory of $G$ and the stabilizer $G_L$.

I know the answer in the case of the Segre variety (I am pretty sure this is classical), and I guess that the answer for other homogeneous varieties might be known. So I'm really interested when the linear space $L$ is at least 2-dimensional.

• I think in general the answer is not known. Dec 7, 2013 at 15:22
• What was wrong with the answer given by Peter Crooks? Dec 8, 2013 at 2:31
• The linear subspace suggested in the answer is not contained in the orbit closure. Dec 8, 2013 at 20:21
• I guess this is a difficult question in general, so I guess I should keep working with small examples. Dec 9, 2013 at 19:05


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## Basic College Mathematics (10th Edition)

Published by Pearson

# Chapter 8 - Geometry - 8.9 Similar Triangles - 8.9 Exercises - Page 600: 14

#### Answer

$a = 2$ ft $b = 6$ ft

#### Work Step by Step

1. Solve for $a$ $\frac{a}{6} = \frac{3}{9}$ $a = \frac{3 \times 6}{9}$ $a = \frac{18}{9}$ $a = 2$ ft 2. Solve for $b$ $\frac{b}{2} = \frac{9}{3}$ $b = \frac{9 \times 2}{3}$ $b = \frac{18}{3}$ $b = 6$ ft

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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## Pen & Paper

#### eircom F1000 Modem

14 Dec 2014

The XyXEL F1000 modem that comes with eircom eFibre offers a wide range of funtionality. An average user will probably not use any advanced features except USB share. The user guide, however, seems to have explained all the modem’s features sufficiently well with the exception of USB share that almost everyone may find useful. There is no detail to show how to access a USB drive once it is plugged into the modem. The USB light on the modem goes green but it is not shown up in the file explorer. Googling didn’t help either.

The USB share is a network drive. For Windows users, it can be accessed by mapping it as such. It can be reconnected every time Windows starts if this is one’s preference. To map a network drive, go to “Computer” in the explorer window and click on the option “Map Network Drive”

.

Choose a “Drive Letter” and point it to \\192.168.1.254\usbkey\ where 192.168.1.254 should be the modem’s IP. Windows will prompt for username and password (which should be the same username and password that are used to access the F1000 modem’s user interface. Enter the credential and that’s all there is to it.

For Linux, the address is smb://admin@192.168.1.254/usbkey. Obviously, you will need smbclient to do that or you may wish to give “gigolo” a try. GNOME users may need gvfs-smb to enable access to Samba shares in the file explorer.


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Top

# Percentage

1.

Three candidates contested an election and received $$1136, 7636$$ and $$11628$$ votes respectively. What percentage of the total votes did the winning candidate get?

Total number of votes polled $$= (1136 + 7636 + 11628) = 20400.$$

$$\therefore$$ Requried percentage $$= (\frac{11628}{20400} \times 100)$$$$= 57$$%

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2.

A student multiplied a number by $$\frac{5}{6}$$ instead of $$\frac{6}{5}$$. What is the percentage
error in the calculation?

The number be 30

When the student multiplied it by $$\frac{5}{6}$$, ⇒ original result $$= \frac{5}{6} \times 30 = 25$$

When the student multiply it by $$\frac{6}{5}$$, ⇒ New result $$= \frac{6}{5} \times 30 = 36$$

⇒ Percentage error in calculation $$= \frac{(36-25)}{36} \times 100$$

$$\Rightarrow \frac{1100}{36} = 30.56$$

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3.

In an election between two candidates, one got $$55$$ % of the total valid votes, $$55$$% of the votes were invalid. If the total number of votes was $$7500$$, the number of valid votes that the other candidate got, was:

Number of valid votes $$= 80$$ % of $$7500 = 6000.$$

$$\therefore$$ Valid votes polled by other candidate $$= 45$$% of $$6000$$

$$= (\frac{45}{100} \times 6000) = 2700.$$

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4.

If $$A = x$$% of $$y$$ and $$B = y$$% of $$x$$, then which of the following is true?

$$x$$% of $$y$$ $$= (\frac{x}{100} \times y) = (\frac{y}{100} \times x) = y$$% of $$x$$

$$\therefore A = B$$

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5.

A grain dealer cheats to the extent of 10% while buying as well as selling by using false weights. His total profit percentage is :

The retailer buys 110 gram in Rs 100 because of 10 % cheating so cost price = 1 gram = Rs $$\frac{100}{110}$$ = Rs 0.909 and retailer sells 90 gram in Rs 100

and hence selling price = Rs 100 / 90 = Rs 1.111

Profit % $$= \frac{1.111-0.909}{0.909} = 22.22$$

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6.

Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56 % of the sum of their marks. The marks obtained by them are:

Their marks be $$(x + 9)$$ and $$x$$

Then, $$x + 9 = \frac{56}{100}(x + 9 + x)$$

$$\Rightarrow 25(x + 9) = 14(2x + 9)$$

$$\Rightarrow 3x = 99$$

$$\Rightarrow x = 33$$

So, their marks are 42 and 33.

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7.

Two tailors $$X$$ and $$Y$$ are paid a total of $$Rs. 550$$ per week by their employer. If $$X$$ is paid $$120$$ % of the sum paid to $$Y$$, how much is $$Y$$ paid per week?

The sum paid to $$Y$$ per week be $$Rs. Z.$$

Then, $$Z + 120$$ % of $$Z = 550.$$

$$\Rightarrow Z + \frac{120}{100}Z =550$$

$$\Rightarrow \frac{11}{5}Z = 550$$

$$\Rightarrow Z = (\frac{550 \times 5}{11}) = 250.$$

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8.

In a certain school, 20 % of students are below 8 years of age. The number of students above 8 years of age is$$\frac{2}{3}$$  of the number of students of 8 years of age which is 48. What is the total number of students in the school?

Let the number of students be $$x$$

Then, Number of students above 8 years of age $$= (100 -20)$$% of $$x = 80$$% of $$x$$

$$\therefore 80$$% of  $$x = 40 + \frac{2}{3}$$ of 48

$$\Rightarrow \frac{80}{100}x = 80$$

$$\Rightarrow x = 100.$$

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9.

The population of a town increased from $$1,75,000$$ to $$2,62,500$$ in a decade. The average percent increase of population per year is:

Increase in $$10$$ years $$= (262500 - 175000) = 87500.$$

Increase % $$= (\frac{87500}{175000} \times 100)$$ % $$= 50$$ %

$$\therefore$$ Required average $$= (\frac{50}{10})$$ % $$= 5$$ %

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10.

A fruit seller had some apples. He sells 40 % apples and still has 420 apples. Originally, he had:

Suppose originally he had $$x$$ apples.

Then, $$(100 - 40)$$% of $$x = 420$$

$$\Rightarrow \frac{60}{100} \times x = 420$$

$$\Rightarrow x = (\frac{420 \times 100}{60}) = 700$$

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