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DEX-SMI Workshop on
Quantum Statistical Inference

Dates: 2-4 March, 2009
Place: 12F, National Institute of Informatics (NII), Tokyo, Japan
Organizer: Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI), Grant-in-Aid for Scientific Research on Priority Areas, MEXT.

## Organizing Committee:

• Masahito Hayashi (Chair), Tohoku University
• Kae Nemoto, NII
• Hiroshi Imai, NII
• Keiji Matsumoto, NII

Contact: Masahito Hayashi

## Program:

2th, March, 2009

13:20-13:30 Opening: Masahito Hayashi

13:30-14:15 Masahito Hayashi (Tohoku Univ.)
Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing Presentation file
14:25-15:10 Masato Koashi (Osaka Univ.) Fumitaka Takenaga, Takashi Yamamoto, and Nobuyuki Imoto
Quantum nonlocality without entanglement in a pair of qubits

15:10-15:30 Tea break

15:30-16:15 Miguel Navascues (Imperial College London), Masaki Owari, Martin B. Plenio
Merging state estimation and entanglement theory
16:25-16:55 Yutaka Shikano (Tokyo Institute of Technology), Seth Lloyd (MIT), Akio Hosoya (Tokyo Institute of Technology)
Optimal Quantum Covariant Clock

16:55-16:15 Tea break

17:15-18:00 Giacomo Mauro D'Ariano (Pavia Univ.)
Probabilistic theories: what is special about Quantum Mechanics? Presentation file

3rd, March, 2009

10:00- 10:45 Masaki Owari (Imperial College London), M. B. Plenio, E. S. Polzik, A. Serafini, M. M. Wolf
Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states Presentation file

10:45-11:05 Tea break

11:05- 11:50 John Calsamiglia (Universitat Autonoma Barcelona)
Phase-Covariant Quantum Benchmarks for Quantum Storage and Teleportation Presentation file

11:50-13:30 Lunch Time

13:30- 14:15 Masahiro Takeoka (NICT) K. Tsujino, M. Sasaki, C. Wittmann, K. N. Cassemiro, G. Leuchs, and U. L. Andersen
Experimental implementation of near-optimal quantum measurements of optical coherent state Presentation file
14:25-15:10 Emilio Bagan (Universitat Autonoma Barcelona)
Phase eatimation with thermal Gaussian states Presentation file

15:10-15:30 Tea break

15:30-16:15 Dominic Berry (Macquarie University), B. L. Higgins, H. M. Wiseman, S. D. Bartlett, M. W. Mitchell, and G. J. Pryde
Phase Measurements at the Theoretical Limit Presentation file

16:15-16:35 Tea break

16:35-17:15 Hiroshi Imai (NII), Masahito Hayashi (Tohoku Univ.)
Fourier Analytic Approach to Phase Estimation Presentation file
17:25-17:50 Yu Matsumoto (Tohoku Univ.), Masahito Hayashi (Tohoku Univ.)
Identification of rotation axis in Bloch Sphere

4th, March, 2009

Local asymptotic normality in quantum statistics Presentation file

10:45-11:05 Tea break

11:05-11:50 Keiji Matsumoto (NII)
Monotone metrics on quantum channels

11:50-13:30 Lunch Time

13:30-14:00 Yu Watanabe (Tokyo Institute of Technology), Takahiro Sagawa (Univ. of Tokyo) and Masahito Ueda (Univ. of Tokyo)
Visualization of Quantum Operations Presentation file
14:10-14:55 Akihisa Hayashi (Fukui Univ.)
Unitary process discrimination with error margin Presentation file

14:55-15:15 Tea break

15:15-16:00 Paolo Perinotti (Pavia Univ.)
Estimation and discrimination of quantum networks Presentation file
16:00-16:45 Giulio Chiribella (Pavia Univ.)
Optimal quantum learning and multiround reference frame alignment Presentation file

16:45-16:50 Closing remarks

## Abstracts:

Masahito Hayashi (Tohoku Univ.)
Title: Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing Presentation file
Abstract: In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented. Related manuscript: http://arxiv.org/abs/0810.3380

Masato Koashi (Osaka Univ.) Fumitaka Takenaga, Takashi Yamamoto, and Nobuyuki Imoto
Title: Quantum nonlocality without entanglement in a pair of qubits
Abstract: We consider unambiguous discrimination of two separable bipartite states, one being pure and the other being a rank-2 mixed state. There is a gap between the optimal success probability under global measurements and the one achieved by generalized measurements with separable measurement operators. We show that even the latter success probability cannot be achieved via local operations and classical communication, leaving a nonzero gap in between.
Related manuscript: http://arxiv.org/abs/0709.3196

Miguel Navascues (Imperial College London) Masaki Owari, Martin B. Plenio
Title: Merging state estimation and entanglement theory
Abstract: In this talk, I will try to convince the audience that there exists a connection between state estimation and entanglement theory from which both fields can mutually benefit. On one hand, I will use recent results on probabilistic state estimation in order to analyze the speed of convergence of the Doherty-Parrilo-Spedalieri (DPS) criterion for entanglement detection. More concretely, I will show how to derive non trivial upper bounds for the distance between the states returned by the computer and the set of separable states, in terms of trace distance, operator norm distance and entanglement robustness. On the other hand, I will show how the DPS criterion can be applied to provide two converging numerical sequences of upper and lower bounds for the optimal fidelity in any pure state estimation problem.

Yutaka Shikano (Tokyo Institute of Technology), Seth Lloyd (MIT), Akio Hosoya (Tokyo Institute of Technology)
Title: Optimal Quantum Covariant Clock
Abstract: We cannot measure time directly but we can measure a clock and then infer the time from the position of the hand we read. This feature of measuring time is more acute in quantum mechanics because there is no self-adjoint time operator if the Hamiltonian is bounded below. The time in the Schroedinger equation is only a parameter but not a physical observable. Any physical object can be a clock but its quality may vary. It is reasonable to consider a clock is good if it has a high precision and its accuracy is maintained for a long time. We are going to explore the best quantum clock of the highest possible precision given a finite energy range, which keeps accuracy in an arbitrarily long time passage.

Giacomo Mauro D'Ariano (Pavia Univ.)
Title: Probabilistic theories: what is special about Quantum Mechanics? Presentation file
Abstract: Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. Here I will analyze the possibility of deriving QM as the mathematical representation of a "fair operational framework", i.e. a set of rules which allows the experimenter to make predictions on future "events" on the basis of suitable "tests", e.g. without interferences from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: "no-signaling from the future" (for the possibility of making predictions on the basis of past tests) FAITH: "existence of faithful states" (for the possibility of calibrating all tests and of preparing any state). I will show that all theories satisfying NSF admit a C*-algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are "no-signaling without interaction" (shortly "no-signaling")--another requirement for a fair operational framework. Postulate FAITH then implies the "local observability principle", along with the tensor-product structure for the linear spaces of states and effects. What is special about QM is that also "effects make a C*-algebra. However, whereas the sum of effects can be operationally defined, the notion of effect abhors any kind of composition. Based on another very natural postulate--AE: atomicity of evolution--along with a purely mathematical postulate--CJ: Choi-Jamiolkowski isomorphism--it is possible to identify effects with atomic events, through which we can then define composition.
Related manuscript: http://arxiv.org/abs/0807.4383([v3] Tue, 31 Mar 2009) appear in "Philosophy of Quantum Information and Entanglement" Eds A. Bokulich and G. Jaeger (Cambridge University Press, Cambridge UK)

Masaki Owari (Imperial College London), M. B. Plenio, E. S. Polzik, A. Serafini, M. M. Wolf
Title: Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states Presentation file
Abstract: We derive fidelity benchmarks for the quantum storage and teleportation of squeezed states of continuous variable systems, for input ensembles where the degree of squeezing $s$ is fixed, no information about its orientation in phase space is given, and the distribution of phase space displacements is a Gaussian. In the limit where the latter becomes flat, we prove analytically that the maximal classical achievable fidelity (which is 1/2 without squeezing, for $s=1$) is given by $\sqrt{s}/(1+s)$, vanishing when the degree of squeezing diverges. For mixed states, as well as for general distributions of displacements, we reduce the determination of the benchmarks to the solution of a finite-dimensional semidefinite program, which yields accurate, certifiable bounds thanks to a rigorous analysis of the truncation error. This approach may be easily adapted to more general ensembles of input states.
Related manuscript: New Journal of Physics, vol. 10, 113014, (2008) http://arxiv.org/abs/0808.2260

John Calsamiglia (Universitat Autonoma Barcelona)
Title: Phase-Covariant Quantum Benchmarks for Quantum Storage and Teleportation Presentation file
Abstract: We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of phase- covariant states and certifies that an experiment can not be emulated by a classical setup, i.e., by a measure-and-prepare scheme. We give an analytical solution for qubits, which shows important differences with standard state estimation approach, and compute the value of the benchmark for coherent and squeezed states, both pure and mixed.
Related manuscript: http://arxiv.org/abs/0807.5126

Masahiro Takeoka (NICT) K. Tsujino, M. Sasaki, C. Wittmann, K. N. Cassemiro, G. Leuchs, and U. L. Andersen
Title: Experimental implementation of near-optimal quantum measurements of optical coherent state Presentation file
Abstract: The optimal discrimination of optical coherent states is a fundamental task in quantum information theory as well as an important primitive in optical communication. We propose and experimentally demonstrate a new and simple near-optimal quantum measruement strategy capable of discriminating two coherent states with smaller average error probability than the homodyne detection. Our proof-of-principle experiment clearly shows the lower error probability than that of the homodyne limit after compensating quantum efficiency (QE) of the photon detector. We discuss the necessary QE to truly outperform the homodyne limit without compensating experimental imperfections and show that it is feasible with the current technology.
As a related topic, we discuss alternative figures of merit to evaluate the classical communication performance, that is reliability function and its cut-off rate. Interestingly, a variety of quantum measurements can attain the maximum (single-shot) cut-off rate while some of them are not optimal from the viewpoints of average error probability or mutual information.
Related manuscripts:
http://arxiv.org/abs/0809.4953 [Phys. Rev. Lett. 101, 210501 (2008)]
http://arxiv.org/abs/0706.1038 [Phys. Rev. A 78, 022320 (2008)]

Emilio Bagan (Universitat Autonoma Barcelona)
Title: Phase eatimation with thermal Gaussian states Presentation file
Abstract: We consider two classes of Gaussian states: displaced thermal states and squeezed thermal states; to perform phase estimation. We find that while for displaced thermal states an increase in temperature gives rise to a reduction in the precision, for squeezed thermal states it goes the other way around. In the multiple-copy scenario we analyze the most paradigmatic measurement strategies (including homodyne and heterodyne measurements) and discuss their performance as compared to the optimal schemes. We apply our results to investigate the influence of losses in an optical metrology experiment.
Related manuscript: http://arxiv.org/abs/0811.3408

Dominic Berry (Macquarie University), B. L. Higgins, H. M. Wiseman, S. D. Bartlett, M. W. Mitchell, and G. J. Pryde
Title: Phase Measurements at the Theoretical Limit Presentation file
Abstract: The interferometric measurement of optical phase is a vital tool for precision measurement. The performance of phase measurements is usually quantified by how the uncertainty scales in terms of the number of resources, N. Standard measurement techniques result in an uncertainty scaling as O(1/sqrt(N)) _ the standard quantum limit (SQL). In contrast, the fundamental limit on phase measurements is the Heisenberg limit of O(1/N). The SQL results from the independent use of resources, so methods of beating the SQL usually involve special nonclassical resource states, such as squeezed states or NOON_ states. Similar results can also be obtained using multiple passes of single photons.
Problems with using nonclassical states or multiple passes are that they may only work over a small range of phases, require initial knowledge of the phase or give ambiguous phase information. We have developed a range of techniques to eliminate these problems, providing unambiguous phase measurements with accuracy close to the theoretical limit. We have experimentally demonstrated a scheme that uses an adaptive technique and multiple passes of single photons to implement the quantum phase estimation algorithm (QPEA), as well as a generalised form of this algorithm.
The QPEA only achieves the SQL due to fat tails in the distribution. Simply combining the QPEA with standard measurements can improve the scaling, although not achieve the theoretical limit. In contrast, our generalised scheme uses multiple measurements to eliminate the fat tails, and achieve the theoretical limit with an overhead factor of only about 1.6. We have now further improved this scheme to achieve the theoretical limit without requiring adaptation, albeit with a slightly increased overhead factor of about 2. Reducing or removing adaptive measurements should make practical implementation of precision phase estimation far easier.

Hiroshi Imai (NII), Masahito Hayashi (Tohoku Univ.)
Title: Fourier Analytic Approach to Phase Estimation Presentation file
Abstract: For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.
http://arxiv.org/abs/0810.5602

Yu Matsumoto (Tohoku Univ.), Masahito Hayashi (Tohoku Univ.)
Title: Identification of rotation axis in Bloch Sphere
Abstract:

Title: Local asymptotic normality in quantum statistics Presentation file
Abstract: Quantum statistics deals with the processing of statistical information carried by quantum systems. One of the central problems in quantum statistics is that of estimating an unknown quantum state by performing general measurements on a (large) number of identically prepared quantum systems.
We will show how to optimally estimate a completely unknown quantum state by developing the quantum analogue of the classical statistical concept of "local asymptotic normality".
The latter means that the statistical model described by the collective state of n identically prepared quantum systems converges in a statistical sense to a model consisting of a classical Gaussian model and a quantum Gaussian state of d(d-1)/2 harmonic oscillators. Both Gaussian models have fixed variance and unknown mean which can be easily estimated by means of standard (heterodyne/homodyne) measurements. This optimal measurement can be pulled back to an optimal strategy for estimating the quantum state.
http://arxiv.org/abs/0804.3876

Keiji Matsumoto (NII)
Title: Monotone metrics on quantum channels
Abstract:

Yu Watanabe (Tokyo Institute of Technology), Takahiro Sagawa (Univ. of Tokyo) and Masahito Ueda (Univ. of Tokyo)
Title: Visualization of Quantum Operations Presentation file
Abstract: Recently many technologies of controlling quantum states have developed. In particular, methods of transferring arbitrary quantum states from one system to another have been proposed and some of them have been experimentally implemented. These technologies are important for making quantum computers and quantum networks. However there is no general measure of accuracy of transferring quantum states which can be applied for any experimental results.
By considering some examples, we relate the accuracy of transferring quantum states to that of estimating input states from output states. The accuracy of estimation is measured by Fisher information. We think that we can characterize any quantum operations (which is a mathematical formulation of quantum state transformations) by using Fisher information. In this talk, we introduce Fisher information for estimating input states from output states of any quantum operations and propose a method of visualizing quantum operation by quantifying the information.

Akihisa Hayashi (Fukui Univ.)
Title: Unitary process discrimination with error margin Presentation file
Abstract: Unitary process discrimination with general error margin m is considered. This interpolates two standard discrimination problems, minium-error discrimination for m=1 and unambiguous discrimination for m=0. We will report two solvable cases: discrimination between two unitary processes and discrimination among unitary processes generated by a group projective representation.

Paolo Perinotti (Pavia Univ.)
Title: Estimation and discrimination of quantum networks Presentation file
Abstract: We review the theory of quantum combs for description of quantum networks, and the theory of testers for measurements of network parameters. After an overview of the problems that can be tackled by means of the new theoretical framework, we show three important results recently obtained through its application.
The first problem is optimal discrimination of two transformations with a finite number of available uses. The measuring party has a fixed number of identical black boxes that perform one unknown transformation and must determine what transformation the single black box performs, out of two. In this case the theory simplifies the problem of taking into account all possible dispositions of black boxes, (e. g. in parallel or in a sequence), and provides the ultimate solution in the case of unitary transformations.
The second problem is optimal covariant estimation of unitary channels having a finite number of available uses. This problem is similar to the first one, but here the possible transformations are the whole unitary representation of a group. In this case the theory allows to prove under very general hypotheses that the parallel disposition is always optimal.
The third problem is optimal tomography of channels and quantum operations. One wants an experimental scheme that allows to reconstruct the unknown and completely free input/output transformation performed by a black box, using it and measuring it an arbitrary number of times, independently. Also in this case the optimal setup can be derived exploiting the general theory, providing a solution that exhibits some non trivial aspects that will be discussed.
Related manuscripts:
[1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 060401 (2008). http://arxiv.org/abs/0712.1325
[2] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180501 (2008).
[3] A. Bisio, G. Chiribella, G. M. D\$B!G(BAriano, S. Facchini, and P. Perinotti, Phys. Rev. Lett. 102, 010404 (2009).

Giulio Chiribella (Pavia Univ.)
Title: Optimal quantum learning and multiround reference frame alignment Presentation file
Abstract: Quantum combs and testers entail a new paradigm of quantum information processing where the input of transformations and measurements are quantum channels rather then states. In this talk, I will present two remarkable applications of this paradigm. The first application is the optimal automated learning of an unknown qubit unitary from a finite training set of N examples. The examples are first exploited in an optimal storing network, whose output state is then sent to a retrieving machine that optimally reproduces the unknown unitary M times. The second application is the optimal alignment of reference frames with multiple rounds of quantum and classical communication. In this case, quantum combs and testers provide a simple proof that the precision of reference frame alignment only depends on the total number of exchanged qubits, and that a single round of forward quantum communication is sufficient.
Related manuscripts:
[1] G. Chiribella, G. M. D'Ariano, P. Perinotti, Quantum Circuits Architecture, Phys. Rev. Lett. 101, 060401 (2008), http://arxiv.org/abs/0712.1325
[2] G. Chiribella, G. M. D'Ariano, P. Perinotti, Optimal covariant quantum networks, http://arxiv.org/abs/0812.3922
[3] A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, and P. Perinotti, Optimal quantum learning of an unknown qubit unitary, in preparation


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# # Crossover from hydrogen to chemical bonding

The bonds in aqueous FHF^− are neither simple hydrogen nor covalent bonds

During chemical conversion, covalent bonds often have to convert to weaker H-bonds [1], and proton transport in water relies on the continuous interconversion of covalent and H-bonds.

How to define the strength of a H-bond?

The abstract of ref. [1] says the strength of "classical H-bond" is around 3-5 kcal/mol. What's it stands for?

A proton sandwiched between two water molecules, is an example of a hybrid between covalent and hydrogen bonding but is extremely short-lived in liquid water. 'Short-lived' does not mean 'weak'. The F-H-F$^-$ anion provides a long-lived proxy to study this structure. An extensive conputational study revealed that the bonds are neither simply covalent nor a simple H-bond. The existence of a hybrid covalent-hydrogen bonded state not only challenges our current understanding of what a chemical bond exactly is but also offers the opportunity to better understand chemical reactions, where “intermediate reaction states” are often invoked but rarely studied directly. This findings also have direct implications for proton transfer reactions in proton transfer in water. [2] Simulating an excess proton in liquid bulk water has proved to be immensely challenging.[2]

TIP

The ablility to obtain a electron of F is much stronger than O.

Future work will undoubtedly focus on the dynamics of the interconversion in different systems.Many exciting questions remain open, including what role the environment (for example, the fluctuating solvent) plays in such interconversion, what molecular motions trigger these interconversions, and how these systems can be steered in a desired direction. Forthcoming answers to these questions should be relevant for proton transport in biology and technologies such as fuel-cell membranes.

Questions

What causes the proton transfer in water? To answer this quesiton, I can find lots of proton transfer processes. Then analysis the dynamic properties of this processes. Maybe I can conclude the answer. How these systems can be steered in a desired direction? This question is about how to control those processes, and also need to better understand the properties of this process.

Considering short strong H-bonds (SHBs) [3], they can have interaction strengths similar to those of covalent bonds, protons appearing shared between D and A, and could involve subsitantial electronic redistribution across the bonded atoms. Where do hydrogen bonds end and covalent bonds begin?

Where do hydrogen bonds end and covalent bonds begin?

Maybe this is a question about when, but they use where.

[1] R. Srinivasan, J. S. Feenstra, S. T. Park, S. Xu, A. H. Zewail, J. Am. Chem. Soc. 126, 2266 (2004)
[2] D. Marx, M. E. Tuckerman, J. Hutter, M. Parrinello, Nature 397, 601 (1999)
[3] Dereka et al., Science 371, 160–164 (2021)

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## February 25, 2008

### A Question or Two

#### Posted by David Corfield

Points of a set, $X$, correspond to certain maps from the Boolean algebra of subsets, $P(X)$, to $2$, namely those corresponding to prime ideals of the algebra.

Points of a space, $Y$, correspond to certain functors from the topos of locally constant sheaves to Set, via evaluation at a point again. Is there a way to construe this by analogy to the prime ideal story? Is there a ‘spectrum’ around?

How does one characterise the fibre functors to Set which correspond to points? Is there something ‘ideal’ going on?

Cartier’s Mad Day’s Work paper seems to suggest there is such a story going on here. In the same paragraph (p. 404) as the description of the fundamental group as the automorphism group of a fibre functor, he speaks of the Galois group of a field extension in terms of the fields’ spectra.

Posted at February 25, 2008 11:43 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1610

### Re: A Question or Two

Here are some quick thoughts on this.

First, a nitpick: if you start with a set $X$, then points $x \in X$ do of course give prime ideals of $P X$ (kernels of Boolean ring maps $eval_x: P X \to \mathbf{2}$). But if $X$ is infinite, then by the axiom of choice, there are other prime ideals too: namely, maximal ideals containing the prime ideal consisting of finite subsets of $X$. These are in bijection with nonprincipal ultrafilters on $X$, which can be considered “ideal points at infinity” that are adjoined to $X$ to form its Stone-Cech compactification.

But we can get $X$ precisely if we change the language, to include arbitrary joins instead of just finite ones. Then an “ideal” $I$ of $P X$ would be defined to be closed under arbitrary joins (and would be a submodule of $P X$ under the meet operation); the prime ideals in that case do correspond to points of $X$. In other words, work in the category of (Boolean) locales, not just (Boolean) rings.

Actually, this is not just a nitpick; I think the change in point of view is important. The idea is that the concept of Grothendieck topos is a categorification of the concept of locale. The analogue of locale map (one which preserves arbitrary joins and finite meets) would be a functor which preserves arbitrary colimits and finite limits, i.e., the left adjoint part of a geometric morphism.

So, for example, suppose we start with a topological space $X$. Under a fairly mild assumption (that $X$ is sober), those functors

$Sheaves(X) \to Set$

which preserve colimits and finite limits correspond exactly to points $x$ of $X$; the corresponding “evaluation” functor $eval_x$ sends a sheaf to its stalk at $x$ (or its fiber, restricting to locally constant sheaves as you mention).

So a first answer to how to characterize these fiber functors is to say they’re the cocontinuous finitely continuous ones to $Set$.

The analogue to ‘ideal’ is less clear-cut. In ring theory, the notion of ideal is a simplification of an appropriate notion of congruence relation, a simplification which is possible because of the presence of additive inverses (i.e., we say $a$ is congruent to $b$ modulo $I$ if $a - b \in I$). Since such congruence relations are kernel pairs of ring maps $f: A \to B$, one could categorify that by taking the kernel pair = weak pullback of a left exact left adjoint $f: E \to F$ against itself:

$E \times_F E \stackrel{\to}{\to} E.$

This kind of thing is studied by the topos theorists, especially in conjunction with descent theory, starting with the Joyal-Tierney monograph An Extension of the Galois Theory of Grothendieck. (I would have to reread that monograph to remind myself of the precise way in which all this is connected with Grothendieck’s Galois theory, and anyway the explanation wouldn’t fit in the margin here.)

In a rather different direction, Steve Awodey in his thesis pursued the following analogy: just as a commutative ring can be reconstituted as the ring of global sections of a sheaf of local rings on its space of prime ideals, so in the 2-category of elementary toposes and logical morphisms, a small topos $E$ could be reconstituted by taking global sections of a sheaf [or actually stack] of [“hyper”]local toposes over a certain site. The first such site is relatively tautological: it is the category $E$ itself equipped with the finite covering topology. The second is more sophisticated: the site is a certain topological space! Awodey goes on to give some interesting applications to higher-order logic, but as I say this direction would seem to be a little different to the one you want to go in.

Posted by: Todd Trimble on February 25, 2008 4:03 PM | Permalink | Reply to this

### Re: A Question or Two

Thanks!

…this direction would seem to be a little different to the one you want to go in.

The thing is you may well have a better idea of this direction than I do.

Perhaps the Spec idea isn’t worth pursuing to the extent that I seem to be looking in a ‘rig’-like direction, though Bertrand Toën et Michel Vaquié seem to be able to do things with Spec $\mathbb{N}$, see p. 4 of Au-dessous de Spec $\mathbb{Z}$.

[If $A$ is an associative, unitary, commutative monoid object in a certain kind of symmetric monoidal category $C$, so an object in $Comm(C)$, then $Spec(A)$ is the corresponding object in $Comm(C)^{op}$. $\mathbb{N}$ is a commutative monoid object in $Comm(\mathbb{N}-Mod)$, the category of commutative monoids.]

Posted by: David Corfield on February 25, 2008 5:42 PM | Permalink | Reply to this

### Re: A Question or Two

The general question is a bit too abstract for my level of competence, but there are much more restrictive situations in which the issues are very deep. The one I’m familiar with is when X is a compact curve of higher genus defined over Q with a rational point b. On the category of finite covering spaces of X, b determines a fiber functor F_b that’s defined over Q’ in a suitable sense. There is a conjecture that proposes that *all* fiber functors that are defined over Q should come from rational points. This is a version of Grothendieck’s section conjecture.

Some details are contained in my colloquium talk at Leeds

MK

Posted by: Minhyong Kim on February 26, 2008 3:11 PM | Permalink | Reply to this

### Re: A Question or Two

I get the odd glimpse of what’s going on from your exposition, but would need it to be Baezified a little more.

The remarkable upshot of this formulation is that the study of solutions to equations is subsumed into the study of maps whose very nature compels us to consider as the most basic in all of in mathematics. (p. 3)

Which word(s) are missing here?

Does anything $\pi_2$-like go on in Diophantine geometry?

Posted by: David Corfield on February 28, 2008 12:22 PM | Permalink | Reply to this

### Re: A Question or Two

I know my limitations well-enough! Baezification is infinitely harder than categorification.

Higher homotopy is interesting and has been used in various aspects of arithmetic geometry. Higher *rational* homotopy has even a motivic nature. But curiously enough, there are no applications so far to Diophantine problems. That is, at the moment \pi_1 seems to be much more powerful.

In the paragraph you cite, no words are missing. The end of the sentence should be …in all of mathematics.’

The meaning of it has to do with the intuition that algebras that are finitely generated over \Z or \Q are much more basic than manifolds over the real or complex numbers, while maps between the corresponding schemes is the usual preoccupation of Diophantine geometry. In their more fanciful moods, arithmetic geometers like to speculate that the real or complex numbers need to be understood much better in a `bottom up’ fashion, and that a proper such understanding might eventually express spaces modeled on complete fields also as rather constructive limits of Diophantine spaces. In that sense, Diophantine geometry should be a foundation to the study of *all* spaces.

Posted by: Minhyong Kim on February 28, 2008 11:51 PM | Permalink | Reply to this

### Re: A Question or Two

Upon re-reading the paragraph, I realize that limitations in my English are probably showing up as well. Oh well. I hope the meaning was clear enough.

MK

Posted by: Minhyong Kim on February 29, 2008 12:42 AM | Permalink | Reply to this

### Re: A Question or Two

The inclusion of ‘them’ would do the trick:

…maps whose very nature compels us to consider them as the most basic in all of mathematics.

That’s interesting about Diophantine geometry as a ‘foundation’, in presumably the conceptual sense.

It is striking how philosophers seem wedded to the idea that numbers are some of the simplest mathematical entities around, when we know how number theory has generated such a wealth of key ideas, as Barry Mazur pointed out in Number Theory as Gadfly.

I remember also a couple of essays by André van Es on the generative power of number theory. This is the only one I can find online:

…mathematically speaking, arithmetic and geometry may be heavily intertwined. Nonetheless, the spirit of this essay is that arithmetic has very different ‘parameters of evolution’. These are not the ones described above as the isolation of aspects by abstraction and generalization. One could describe them alternatively as the isolation of aspects by analogy.

And later,

There exists a natural fusion of aspects in the objects of arithmetic. Analogy serves to isolate these aspects in separate analogous contexts.

Posted by: David Corfield on February 29, 2008 9:04 AM | Permalink | Reply to this

### Re: A Question or Two

Ugh. I do have occasional blind spots in English, not having been born into the language. But is the construction really *incorrect*? I considered the following sequence of transformations:

…sentences that David compels us to consider as incorrect.

…sentences David compels us to consider as incorrect.

…maps Grothendieck compels us to consider as basic.

…maps whose nature compels us to consider as basic.

and so on. Sorry for cluttering up the discussion with silly questions on grammar.

Now back to mathematics.

In fact, my own tendency is to think of Diophantine geometry as foundational in a very *physical* sense, rather than a conceptual one. In particular, generative power at a conceptual level and analogies, while undoubtedly important, seem to receive too much emphasis.

Some feeling for this I explained in a talk for graduate students long ago. What I wrote there appears painfully naive, but it’s probably self-deception to think I could express matters any better now. I should point out however that the second half of that talk dealing with Matiyasevich is actually much less relevant to the present discussion than the first bit.

Posted by: Minhyong Kim on February 29, 2008 12:05 PM | Permalink | Reply to this

### Re: A Question or Two

Your way is possible. It’s just that there’s something about the singular ‘nature’ requiring the ‘s’ in ‘compels’ and the plural ‘maps’ which interferes with my parsing.

I’ll have to look at it afresh later.

Posted by: David Corfield on February 29, 2008 2:01 PM | Permalink | Reply to this

### Re: A Question or Two

The last of those transformations (to “maps whose nature…”) is illegitimate. Putting the “that” back makes this obvious (“maps that whose nature…”). I’m not certain of the precise nature of the problem, but it’s to do with restrictions on what relative pronouns are allowed to do. You can’t have the relative possessive “whose” referring either to the relative pronoun “that” that heads its clause (particularly if the latter isn’t actually there…), or to the antecedent of the relative clause in which it is embedded (that is, “maps”). So I suspect this is due to a general restriction on relatives in embedded clauses in English whose precise details I can’t remember just now. “Maps that their nature compels us to consider as…” would be OK.

Posted by: Tim Silverman on March 1, 2008 12:06 AM | Permalink | Reply to this

### Re: A Question or Two

Thanks! Your explanation definitely demystifies the strange transformation whose very nature compelled me to overlook the error of.

MK

Posted by: Minhyong Kim on March 1, 2008 1:11 AM | Permalink | Reply to this

### Re: A Question or Two

“Diophantine geometry should be a foundation”

I wonder if this would not imply that one finds arithmetic or function field analogies to all constructions in geometry, e.g. the mathematics behind the geometrisation theorem or Connes’ noncommutative geometry?

Posted by: Thomas Riepe on February 29, 2008 2:25 PM | Permalink | Reply to this

### Re: A Question or Two

In fact, this is the usual direction of research. The Weil conjectures form a well-known example and Connes believes in arithmetic non-commutative geometry with characteristic passion. In his anabelian letter to Faltings, Grothendieck alludes to the rigidity of hyperbolic geometry as inspiration for anabelian geometry.

But own fanciful view is that such macroscopic phenomena should, in some sense, *originate* in Diophantine structures.

Posted by: Minhyong Kim on February 29, 2008 4:19 PM | Permalink | Reply to this

### Re: A Question or Two

Thanks for pointing to that conference! I looked the past days a bit into NC-Geometry, because of the apparent links with modular forms (else I doubt that Dixmier traces etc. would ever have interested me). In case you know articles connecting NC-Geometry with F1, would you please post the bibliographic infos?

Posted by: Thomas Riepe on February 29, 2008 7:48 PM | Permalink | Reply to this

### Re: A Question or Two

Minhyong, if you’d be willing to put your PR hat on and explain in your opinion the best case for an interesting connection between Connes-style non-commutative geometry and arithmetic algebraic geometry, I’d be very interested.

(To put all my cards on the table, I’m tad skeptical. But I’m much more ignorant than skeptical, so I’m keeping an open mind. In any event, I’m truly interested and not setting anyone up for an ambush.)

Posted by: James on February 29, 2008 11:17 PM | Permalink | Reply to this

### Re: A Question or Two

That PR hat is much too big for me, since both my ignorance and skepticism may well run deeper than your own. I think I mentioned once that neither reference nor quotation signify endorsement.

However, I have acquired over the years a great deal of respect for Connes the person on account of the absolute devotion and seriousness with which he pursues his scientific convictions. They certainly make me want to sympathize with him, even when my understanding is rather dim. There is a similar situation with another F_1-theorist, Shinichi Mochizuki.

For this reason, I may make an attempt to fulfill your request in some version of the future, regardless of my own beliefs (or lack of them).

Posted by: Minhyong Kim on March 1, 2008 1:32 AM | Permalink | Reply to this


text:
A surprising result

Calculus Level 4

$\large \displaystyle \int_{0}^{1} \dfrac{2x - 3x^2 - x^4}{x^6 + 2x^3 + 1} \, dx$

If the above integral can be expressed in the form $$\dfrac{a}{2016}$$, find $$a$$.

×


text:
Version:

# Guidelines for Writing Error Messages in Open 3D Engine

In this section, you will learn how to write great error messages for O3DE. Why are error messages so important? Shouldn’t the user figure out why their code failed or their operation didn’t work? Not at all. The users expect a seamless experience without any errors. Nevertheless, errors occur and error messages can ease and help users overcome them.

Great error messages can salve the sting of an unexpected error. Poor ones can lead to negative feedback or cause the customer to abandon your product completely. An effective and helpful error message acknowledges the inconvenience on the user and reassures them with guidance to work around it.

Handling errors well and writing good error messages requires more effort, but produces the following worthwhile advantages:

• Reduces user frustration and improves productivity throughout the product’s experience.
• Leads users towards better behaviors in code and UI development and use.
• More informative and actionable bug reports.
• Better code and UI implementation overall.

Example

Consider the following pair of examples for the same issue.

ERROR: Cannot complete operation. Maximum range value exceeded.

Good example:

This operation requires positive integer values between 0 and 127. Please choose a value in this range before rebuilding the scene.

The first example is discouraged because it forces the user to figure out the problem through trial and error without providing any information that leads them towards the correct solution. The second example makes it immediately clear to the user what they did wrong and provides guidance on how to prevent the error next time. This error message helps the user and may save them from future frustration. Additionally, bespoke and well-designed error messages implicitly communicates consideration for the user.

Poor error messages erode trust and increase frustration. Great error messages enable the user to continue work and provide implicit guidance around improved practices.

## Writing principles

The most common mistake when presenting errors to a user is displaying low-level errors and exceptions thrown by code directly to the user without editing them to make it user friendly. The most effective error messages are ones that any user and developer can understand. O3DE is a tool for builders, so it’s important that error messages can reach a wide audience.

The writing principles on this page will help you think strategically about developing good error messages and practice good development habits for code and UI.

To start, we’ll look at two categories of error message: error messages for code operations and error messages for application users (UI).

### Error messages for code operations

Error messages first appear as code-level error descriptions. A poor error message simply passes this error message to the end-user in a dialog widget or console interface. Instead, you can improve the user’s overall experience with your code by authoring your error messages that’s suitable for all audiences, not just technical experts.

The following points help you author good error messages for code operations:

• Be clear and concise. Short messages are better than long ones, but don’t leave out any essential details. Make every word count! Use basic, proper English grammar. Avoid complex phrasing or idioms.

• Consistently use a passive voice and past-tense. For an example of passive voice, use “An error occurred…”, instead of “We found an error…”. This mitigates judgement in the tone of voice. Additionally, use past-tense, such as “occurred”, because the error has already happened and is not in the process of occurring.
• Don’t pass the default description of a low-level error to the end user. An effective error message may contain technical information transcribed for end-users of varying levels of knowledge. This is especially important if the default error message is inscrutable or useless by itself. Every little detail helps.

• Format your error messages for readability and clarity. For example, white space or a pair of quotation marks can clarify text and reduce user irritation.

• Don’t dump code into an error message. Code takes up space in the error message and makes the message harder to scan. Rather, the user can see the code in their integrated development environment (IDE).

• Write error messages with the least technical context and terminology possible. For example, errors that raise up to the user interface should concern the user’s interaction and not problems at the code-level. Choose words that best fit that context and avoid deeper code-level jargon whenever possible.

For example, suppose a developer encounters an error with the user interface, such as a secondary menu failure. An unhelpful message may be STACKOVERFLOWEXCEPTION: SYSTEM STATE UNKNOWN". This message is highly alarming to the user and does not help them understand the problem. A better message may be “ERROR: A stack overflow error occurred when updating the menu element tree. Please report this error and restart the application.” Users may need additional context, so consider adding a comment that this message should be rewritten depending on the context if it can appear in the UI.

• Avoid “compiler-speak”. Use the most viable and least specific terms that describe the error. Focus on the way the user perceives the error, not the compiler or the run-time. For example, a message with compiler jargon may be “ERROR: The operation cannot complete due to unsafe type usage when unmarshalling thread context”. This is confusing and concerning to the user. A better message may be “ERROR: The operation cannot complete as the context cannot be determined. Check to see if any types used to recreate this work are unsafe.” Also, this makes it easier for UI writers to craft an appropriate UI-level message.
• Signal next steps that the system takes or for the user to take wherever viable.

• Provide hints to code definitions if the context of the message is appropriate. For example, “Expected a type HANDLE_T * for the first parameter in ProcessBlobSync(). HANDLE_T is defined as a shared macro in common.h.” This message is appropriate to developers, but may be inscrutable on the UI-level.

• Avoid judgement in the tone of the message, especially if the error originates from a third-party library or binary. Instead of blaming the code, state the problem clearly and objectively. For example, “The operation could not complete due to a misconfiguration between O3DE and the CoolGraphicsV3 Gem. Please report this issue on GitHub”.

• Use an empathetic tone as though the error message appears in the user interface or the IDE, and not the command-line. To help you establish this tone, imagine that you are encountering this error for the first time as a new developer. Ask yourself whether or not you would find this error message clear and helpful, or opaque. Remember, it may not be an experienced developer who sees your error message.

### Error messages in UI

Writing informative and actionable error messages for end users who are working in a UI is an art form. Too often, developers send error messages from the code directly to the user interface without writing a simplified and helpful message that all users can comprehend. This is important because not all users are experienced developers or developers at all.

The following rules will help you write better messages in the UI:

• Avoid jargon and technical terms unless the context is immediately apparent or if the terminology directly applies to the user’s code or actions. Don’t introduce new technical ideas that aren’t covered in the immediate context. If it is a low-level error and some code or algorithm jargon is unavoidable, provide a clause or a sentence that clarifies it in more accessible terms.
• Describe the error in terms of the action that the user took. Avoid a tone that blames the user. You can achieve this by using a passive voice, rather than an active voice. For example:
• BAD (active voice): “You entered an incorrect value for Gradient Range.”
• GOOD (passive voice): “The Gradient Range input field requires a value between 0 and 63.”
• To respect the user’s cognitive focus, don’t overwrite the error message or provide too much detail. Aim to shorten error messages to less than 200 characters, if possible. Shorter is often better, but some errors may require detailed explanation — try to limit the number of long error messages.
• If the message reflects a composite or aggregated error, format it clearly and clarify the issues that occurred and its causality.
• If the error is common, provide a clickable link to a troubleshooting document, if one exists. Providing a link reassures the user that the problem is common and they can find help from a documentation.
• If the error requires a lot of detail to understand and resolve, consider requesting a troubleshooting documentation for it. Don’t try to fit all the information in the UI message.
• Separate messaging from any essential technical details (such as a file name, path, or other programmatic data).

## Error message guidelines

Error messages commonly occur in low-level code, in APIs, and in the UI. Writing error messages at each levels can vary as they appear at different levels of technical context and for different user roles.

Overall, observe these guidelines when writing error messages at any level:

• Include the following elements in the error message:

• Description of error
• Technical details (optional)
• Next steps or links to further assistance (optional)
• Use passive voice.

• Be specific.

• Don’t use all uppercased text.

• Use proper U.S. English syntax.

• Try to keep error message at a maximum of 200 characters. Short messages are better, but use your judgement when longer error messages are needed. Optimize your character space by simplifying sentences where possible.

• Use simple grammar. Target a U.S. 8th grade reading level.

• Avoid technical jargon in end user UI messages. Be considerate of the developer’s understanding when using jargon in code-level error messages. Consider including a clause of sentence to explain jargon if helpful.

• Focus on the end user’s immediate needs, not your own.

• Reread your work to check for spelling or grammatical errors.

### Example

The following examples demonstrate how to apply these guidelines in an error message.

Example 1

Consider the context in which a user fails to provide correct credentials to a service.

ERROR: CREDENTIALS NOT FOUND. WE CANNOT AUTHORIZE YOU.

REWRITTEN:

Oops! The password you provided didn't match our records. Can you try again?

Example 2

Consider an error that flows from low-level code to developers or other end users at the API level and UI level

An error occurred at line 258, col 8 in arglebargle.cpp. Pointer returned null.

REWRITTEN (API level):

An error occurred when the operation received a null pointer. A null pointer is not allowed for the target data when calling RenderToTarget(handle, target, options).

Details:
- File: arglebargle.cpp
- Line: 258
- Column: 8

REWRITTEN (UI level):

Oh no! The target for rendering your scene was not defined. This is a serious error in our code. Please report it at <link to GitHub issues> and provide any relevant details.
`

The examples above demonstrate how you can provide clear, informative, and actionable details, and adapt it for a specific audience. Always think about who might see your error message and when it might occur.

## Best practices

To summarize the key points when writing error messages:

• Above all else, aim for clarity. Ask yourself: “If I got this message while doing my job and I was new to this product or code, what would help me move forward or avoid this error in the future?” Never make the user guess. If you can get further details from the run-time or the existing code, do so, even if it seems tedious. Your extra effort will save your users countless hours in the future.

• If you are working on developer-focused code (as opposed to UI code), provide detailed code comments around your error handling and messages. This will help UI error message writers better understand the context and create better end-user messages.

• Use a natural, friendly voice. Write error messages with empathy, as though you were advising a fellow developer or a new user. It’s okay to lead with “Sorry” or “Oops!” if you feel that the error might be in a really disruptive context. Consider localization before you employ sympathetic idioms. Don’t assume the user’s technical knowledge. Even terminology that seems standard for developers can feel unnatural and frustrating, especially to developers or users who are learning the product. Be aware of including “programmer” text, which makes the error feel arbitrary and cryptic, as though it’s not intended for the user to see. Remember the purpose of an error message and write them with respect to that: an error is an issue we knew might occur and we’re alerting you, the user, to help you out.

• Avoid hard-coding error messages. Instead, store error message in data objects, such as a JSON file, a CSV, or a flat-text model of your own implementation. This acts as a “dictionary” file that you can refer to and access using a unique token or handle. This system makes it easier for developers to do code reviews and maintenance without risking build errors or regressions. It also allows for more straightforward and cheaper localization. Document your schema or format to help others understand and maintain it.

• Ask a technical writer or UX expert to review your error messages. When receiving a review in a GitHub pull request (PR), call out your updated error strings and add comments to provide any relevant information. This improves collaboration between you and your reviewer and eases the process to get your updates approved.

## More resources

For more reading, check out Jakob Nielsen’s guidance on writing good error messages .


text:
Marie wants to make a cone-shaped witch's hat for a masquerade ball. How much material will it need if it counts on an annular rim with diameters of 28cm and 44cm? Hat side length is 30cm. Add 5% of the material to the bust. Round to cm2.

Result

S =  2335 cm2

#### Solution:

$D_{1}=28 \ \text{cm} \ \\ D_{2}=44 \ \text{cm} \ \\ \ \\ s=30 \ \text{cm} \ \\ q=1 + \dfrac{ 5 }{ 100 }=\dfrac{ 21 }{ 20 }=1.05 \ \\ \ \\ r_{1}=D_{1}/2=28/2=14 \ \text{cm} \ \\ r_{2}=D_{2}/2=44/2=22 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r_{2}^2 - \pi \cdot \ r_{1}^2=3.1416 \cdot \ 22^2 - 3.1416 \cdot \ 14^2 \doteq 904.7787 \ \text{cm}^2 \ \\ \ \\ S_{2}=\pi \cdot \ r_{1} \cdot \ s=3.1416 \cdot \ 14 \cdot \ 30 \doteq 1319.4689 \ \text{cm}^2 \ \\ \ \\ S=q \cdot \ (S_{1}+S_{2})=1.05 \cdot \ (904.7787+1319.4689) \doteq 2335.46 \doteq 2335 \ \text{cm}^2$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Do you want to convert length units?

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text:
# If the atmosphere is (very nearly) an ideal gas, how can it ever be barotropic?

A barotropic fluid is a fluid whose density is a function of pressure only, i.e. $\rho=\rho(p)$. My understanding is that, in the atmosphere, in general, the tropics are barotropic whereas the extratropics are baroclinic.

If the atmosphere is an ideal gas, then $\rho =p/R T$. Therefore, density is a function of pressure and temperature. How, then, can the atmosphere ever be barotropic?

• Possibly better suited for Chemistry SE?
– L.B.
Mar 10, 2017 at 19:06
• @L.B. I disagree. If this was primarily a question about thermodynamics, I would agree. However, the barotropic assumption is usually made in atmospheric dynamics. Mar 10, 2017 at 21:21

You are correct, the atmosphere cannot ever be barotropic. It is important to note that the ideal gas law is relevant only as a state equation; it does not describe the gas as a whole. The barotropic assumption effectively states $P=\rho RT_0$. Effectively, it says "hold temperature constant" You may note this is particularly important when discussing the hydrostatic and hypsometric/barometric equations.


text:
# 2.4.17. 1D - Schrödinger-Poisson - A comparison to the tutorial file of Greg Snider’s code¶

In this tutorial we calculate the self-consistent solution of Schrödinger-Poisson equations using nextnano++ and another code provided by Greg Snider (University of Notre Dame). We compare the two results and see the agreement of them.

We also discuss about the basic concept of the Schrödinger-Poisson solution.

The related input files are followings:

• Greg_Snider_MANUAL_1D_nn*.in

• Greg_Snider_MANUAL_1D_analytic_nn*.in

• Greg_Snider_MANUAL_2D_nn*.in

• Greg_Snider_MANUAL_2D_analytic_nn*.in

These are available in the sample file folder. The files which have analytic in their names use analytic doping function.

We appreciate that Greg Snider provided his code, the manual and the input files free of charge, so that we could use it here as a test case. His 1D Poisson/Schrödinger code can be obtained from this link. This tutorial is based on his manual (1D Poisson Manual.pdf, MANUAL.EX).

## Structure¶

We simulate a structure consisting of the following matrerials and doping profile. The additional doping profile based on LSS Theory is explained in the next section.

 surface Schottky barrier of 0.6V z = 0 ~ 15 nm GaAs n-type doped (1018 cm-3) z = 15 ~ 35 nm Al0.3 Ga0.7 As n-type doped (1018 cm-3) z = 35 ~ 39.5 nm Al0.3 Ga0.7 As z = 39.5 ~ 54.5 nm GaAs quantum well z = 54.5 ~ 105 nm Al0.3 Ga0.7 As z = 105 ~ 355 nm Al0.3 Ga0.7 As p-type doped (1017 cm-3) substrate
• The grid resolution is 1 nm with the exception of the 250 nm layer which has a resolution of 5 nm and the material interfaces of the quantum well which has a resolution of 0.5 nm.

• The dopants are assumed to be fully ionized.

• The temperature is 300 K.

• The Schrödinger equation will be solved between 5 nm and 195 nm.

## Doping¶

We consider two further impurity profile resulting from ion implantation using LSS Theory. The donor and acceptor profiles are written out of the file density_acceptor/acceptor.dat and look as follows:

The relevant parameters are:

 implant dose [cm-2] projected range $$R_p$$ [nm] projected straggle Delta $$\sigma_p$$ [nm] donor 2×10 12 86 44 acceptor 1×10 11 75 20

For further details on the LSS theory (ion implantation) and on the doping profiles, please check the relevant keyword doping{} for nextnano++ (or \$doping-function for nextnano³).

## Conduction and valence band edges¶

The following figure shows the conduction and valence band edges as well as the Fermi level (which is constant and has the value of 0 eV) for the structure specified above. These bands are the solutions of the self-consistent Schrödinger-Poisson equation.

Both codes, nextnano++ and Greg Snider’s “1D Poisson” lead to the same results.

## Electron eigenstates and eigenfunctions¶

Inside the GaAs quantum well there are three confined electron states. The ground state is below the Fermi level and thus occupied. The following figure shows a zoom of the GaAs Quantum well.

The wave functions as calculated with nextnano++ are nearly identical to Greg Snider’s “1D Poisson” code, as well as the energies. However, there are tiny differences which is not too suprising as the conduction band profile is not completely identical.

 Electron states nextnano++ nextnano³ Greg Snider’s “1D Poisson” code E1 [meV] -3.1 -3.0 -1.3 E2 [meV] 43.4 43.5 44.0 E3 [meV] 117.4 117.5 117.8

## Electron and hole densities¶

The electron and hole densities are depicted in this figure, there is also nice agreement between the two codes.

• The integrated electron density in the GaAs quantum well region is 0.667 * 1012 cm-2. (Greg Snider’s result: 0.636 * 1012 cm-2)

• The integrated hole density in the right most Al0.3Ga0.7As region is 1.033 * 1012 cm-2. (Greg Snider’s result: 1.085 * 1012 cm-2)

The relevant output files are:

• integrated_density_electron.dat

• integrated_density_hole.dat

This tutorial shows very nicely that both codes, nextnano++ and Greg Snider’s “1D Poisson” lead to the same results. Greg Snider’s 1D Poisson/Schrödinger code can be obtained from here: http://www.nd.edu/~gsnider/

## 2D simulations¶

• Greg_Snider_MANUAL_2D_nn*.in

We can also calculate the 2D schrödinger-Poisson equation for the same structure where the y direction has been assumed to be of length 100 nm with periodic boundary conditions.

## Self-consisent Schrödinger-Poisson solution¶

Here we briefly discuss about the basic concept of the method used to get the above results.

In this section, we refer to

Self-consistent calculation of Schrödinger-Poisson equations is one way to treat the manybody effects associated with Coulomb repulsion.

For example, suppose we calculate Schrödinger equation to obtain the energy eigenvalues and eigenstates for a quantum well only one time. If we add a further test electron into the system, the potential that the test electron feels is the band-edge potential plus Coulomb potential which is caused by the original electrons in the system. In most cases, the carrier density in a single quantum well is so high that it is important to take this additional potential into consideration. (6.67*1012 cm-2 for the GaAs quantum well in this tutorial.)

In order to obtain the solution which involves this effect, the potential used in Schrödinger equation for the electrons and the charge distribution which is based on the energy eigenstates from that Schrödinger equation must satisfy Poisson equation. This solution is described as self-consistent, rather like Hartree’s approach to solving many electron atoms.

The process for obtaining self-consistent solution of Schrödinger-Poisson equations is as follows:

1. Solve Schrödinger equation using band-edge potential $$V_{be}(\mathbf{r})$$ and obtain the eigenstates of an electron $$\Psi_{\alpha, E}^{el}(\bf{r})$$ and hole $$\Psi_{\beta, E}^{hole}(\bf{r})$$ . Here $$\alpha$$ is the conduction band number, $$\beta$$ is the valence band number and $$E$$ represents the eigenvalue.

2. Calculate the density distribution of the particles $$n(\mathbf{r})$$ using local density of state $$\rho^{el}(\mathbf{r}, E):=\sum_{\alpha}|\Psi_{\alpha, E}^{el}(\mathbf{r})|^2$$, $$\rho^{hole}(\mathbf{r}, E):=\sum_{\beta}|\Psi_{\beta, E}^{hole}(\mathbf{r})|^2$$ and fermi distribution $$f(E):= \frac{1}{e^{(E-E_f)/k_BT}+1}$$ .

\begin{align}\begin{aligned}n^{el}(\mathbf{r}):=\int dE \rho^{el}(\mathbf{r}, E) f(E)\\n^{hole}(\mathbf{r}):=\int dE \rho^{hole}(\mathbf{r}, E) f(E)\end{aligned}\end{align}
3. Solve Poisson equation and obtain the potential distribution $$\phi(\mathbf{r})$$ caused by the distributed electrons, holes, and ions.

$\nabla\cdot\big(\epsilon_s(\mathbf{r})\nabla\big)\phi(\mathbf{r}) = \frac{-e[n^{hole}(\mathbf{r}) - n^{el}(\mathbf{r}) + N_D(\mathbf{r}) - N_A(\mathbf{r})]}{\epsilon}$

where $$\epsilon_s$$ is the dielectric constant, $$N_D(\mathbf{r})$$ is the donor concentration and $$N_A(\mathbf{r})$$ represents the acceptor concentration.

4. Using the new potential

$V_{new}(\mathbf{r}) := -q\phi(\mathbf{r}) + V_{be}(\mathbf{r})$

which consists of the result of 3. and band-ege potential, solve Schrödinger equation.

5. Check whether the energy eigenvalues converged or not. Then

• Yes $$\longrightarrow$$ End

• No $$\longrightarrow$$ Go to 2.

The process is iterated until the energy eigenvalues converge. At last, the potential used in Hamiltonian and one calculated from charge distribution which is from Schrödinger equation will be identical.


text:
2019
Том 71
№ 8

# Vorob'ev N. N.

Articles: 4
Article (Russian)

### On the modularity of a lattice of $τ$-closed $n$-ultiply $ω$-composite formations

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 453–463

Let $n ≥ 0$, let $ω$ be a nonempty set of prime numbers and let $τ$ be a subgroup functor (in Skiba’s sense) such that all subgroups of any finite group $G$ contained in $τ (G)$ are subnormal in $G$. It is shown that the lattice of all $τ$-closed $n$-multiply $ω$-composite formations is algebraic and modular.

Article (Ukrainian)

### Constructive characteristic of continuous functions defined on smooth arcs

Ukr. Mat. Zh. - 1968. - 20, № 6. - pp. 750–758

Article (Ukrainian)

### Simultaneous approximation of functions and their derivatives in a complex domain

Ukr. Mat. Zh. - 1968. - 20, № 1. - pp. 113–119

Article (Ukrainian)

### Approximation of continuous functions of certain classes given on Jordan curves

Ukr. Mat. Zh. - 1967. - 19, № 3. - pp. 90–95


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# Concerned about financial well-being of its elderly

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Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 00:52
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Concerned about financial well-being of its elderly citizens, the government of Runagia decided two years ago to increase by 20 percent the government-provided pension paid to all Runagians over 65. Inflation in the intervening period has been negligible, and the increase has been duly received by all eligible Runagians. Nevertheless, many of them are no better off financially than they were before the increase, in large part because ________.

Which of the following most logically completes the passage?

A. They rely entirely on the government pension for their income
B. Runagian banks are so inefficient that it can take up to three weeks to cash a pension check
C. They buy goods whose prices tend to rise especially fast in times of inflation
D. The pension was increased when the number of elderly Runagians below the poverty level reached an all-time high
E. In Runagia children typically supplement the income of elderly parents, but only by enough to provide them with a comfortable living
[Reveal] Spoiler: OA

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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 02:28
Ans is C.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 05:02
I choose A.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 05:10
bpattnaik wrote:
Ans is C.

The passage states that inflation was negligible, so this appears to be irrelevant.

What if the original pension was a meager $100 a month. If the COL in Runagia is equal to that of the USA, then obviously$120 a month really isnt going to make a big difference.

(A)

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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 11:14
A. possible, but kind of strong in its word usage..."rely entirely" will keep for now and see if a better one appears...

B. out of scope, never does the passage mention banks...

C. buying stuff that rises with inflation, but in the passage it states the inflation issue. "It has been negligible"

D. possible, because if the new program came into effect when the poverty was an all time low, the program may not have helped much in improving the financial state of the elderly.

E. definetly out of scope...don't care.

Down to A and D...

because the final statement states:
Nevertheless, many of them are no better off financially than they were before the increase, in large part because ________.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 15:37
I second to (D). great explaination. acfuture.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 16:06
2
KUDOS

children supplement the income of elderly parents by just enough to provide the parents a comfortable living. So, if the pension amount has increased, the amount of money these elderly people receive from their kids would decrease. So, overall they would still be getting the same money.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 17:36
I choose E. I almost had to guess on this one but I just had a moment of clarity. As shoonya correctly point out: children only supply enough to provide their parents with a comfortable living...this implies that they will reduce their giving appropriately thereby leaving their parents at the same utility level.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 19:18
1
KUDOS
A. They rely entirely on the government pension for their income

======> If they rely entirely on pension then 20% increase will be helpful, even if its a little help.

B. Runagian banks are so inefficient that it can take up to three weeks to cash a pension check

======> Then the cycle should be same.

C. They buy goods whose prices tend to rise especially fast in times of inflation

======> As stated in passage, inflation is negligible.

D. The pension was increased when the number of elderly Runagians below the poverty level reached an all-time high

======> Even if number of people are more everybody is getting the increased amount. As passage says "...paid to all Runagians over 65"

E. In Runagia children typically supplement the income of elderly parents, but only by enough to provide them with a comfortable living

======> CORRECT. This means children have reduced their support and the increase in pension was offset by decrease in support by children. Why children are decreasing the support? See the bold part in E.
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 23:02
ps_dahiya wrote:
A. They rely entirely on the government pension for their income

======> If they rely entirely on pension then 20% increase will be helpful, even if its a little help.

B. Runagian banks are so inefficient that it can take up to three weeks to cash a pension check

======> Then the cycle should be same.

C. They buy goods whose prices tend to rise especially fast in times of inflation

======> As stated in passage, inflation is negligible.

D. The pension was increased when the number of elderly Runagians below the poverty level reached an all-time high

======> Even if number of people are more everybody is getting the increased amount. As passage says "...paid to all Runagians over 65"

E. In Runagia children typically supplement the income of elderly parents, but only by enough to provide them with a comfortable living

======> CORRECT. This means children have reduced their support and the increase in pension was offset by decrease in support by children. Why children are decreasing the support? See the bold part in E.

"comfortable" in E seems very vague. One mans comfort is another man`s luxury...
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Re: Concerned about financial well-being of its elderly [#permalink]  15 Jul 2006, 23:15
Agree with shoonya and ps_dahiya's explanations.
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Re: Concerned about financial well-being of its elderly [#permalink]  16 Jul 2006, 14:29
Gauss wrote:
Concerned about financial well-being of its elderly citizens, the government of Runagia

decided two years ago to increase by 20 percent the government-provided pension

paid to all Runagians over 65. Inflation in the intervening period has been negligible,

and the increase has been duly received by all eligible Runagians. Nevertheless,

many of them are no better off financially than they were before the increase, in large

part because ________.

Which of the following most logically completes the passage?

A. They rely entirely on the government pension for their income

B. Runagian banks are so inefficient that it can take up to three weeks to cash a pension check

C. They buy goods whose prices tend to rise especially fast in times of inflation

D. The pension was increased when the number of elderly Runagians below the poverty level reached an all-time high

E. In Runagia children typically supplement the income of elderly parents, but only by enough to provide them with a comfortable living

My choice is E.
A: Even if they rely entirely on the govenrnment, if the pension has increased by 20%, they should be getting 20% more, so they should be better of.
B: Irrelevant. It clearly says that the pension was received.
C: Irrelevant. It clearly says inflation was negligible
D: So what? 20% increase is a 20% increase.
E: In the beginning they received 10 dollars from Government. Children gave them 5 dollars more to live comfortably. Then Government gave them 2 dollars more. Children now gave them 3 dollars to live comfortably. At the end of the day they still had the same, and therefore were no better off than before.
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Re: Concerned about financial well-being of its elderly [#permalink]  17 Jul 2006, 07:45
OA E
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Re: Concerned about financial well-being of its elderly [#permalink]  17 Jul 2006, 11:03
ps_dahiya wrote:
A. They rely entirely on the government pension for their income

======> If they rely entirely on pension then 20% increase will be helpful, even if its a little help.

B. Runagian banks are so inefficient that it can take up to three weeks to cash a pension check

======> Then the cycle should be same.

C. They buy goods whose prices tend to rise especially fast in times of inflation

======> As stated in passage, inflation is negligible.

D. The pension was increased when the number of elderly Runagians below the poverty level reached an all-time high

======> Even if number of people are more everybody is getting the increased amount. As passage says "...paid to all Runagians over 65"

E. In Runagia children typically supplement the income of elderly parents, but only by enough to provide them with a comfortable living

======> CORRECT. This means children have reduced their support and the increase in pension was offset by decrease in support by children. Why children are decreasing the support? See the bold part in E.

Wow... good one I selected D
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Re: Concerned about financial well-being of its elderly [#permalink]  17 Jul 2006, 21:27
E for me
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Re: Concerned about financial well-being of its elderly [#permalink]  19 Jul 2006, 04:34
Good explanation shoonya
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Re: Concerned about financial well-being of its elderly [#permalink]  18 Oct 2013, 11:22
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Re: Concerned about financial well-being of its elderly [#permalink]  12 Jul 2015, 23:24
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text:
# Segal's 1999 Stanford lecture notes on TQFT, where to find them?

I am trying to track down a copy of Graeme Segal's 1999 lecture notes on topological field theory. These are sometimes referred to as the "Stanford lectures" or something similar.

For many years these unpublished notes were available at Duke's mathematics website via the link: http://www.cgtp.duke.edu/ITP99/segal/

However this link is now dead. Every reference I have found by googling seems to lead back to this dead link. It would be a tragedy if these lecture notes were lost to the mathematics community. I am hoping that somebody out there has retained a copy.

• @joro has found the lectures note that were at the dead link, but lectures 4 and 6 are missing. I am still wondering if there were ever any notes produced from those missing lectures? – Chris Schommer-Pries Sep 2 '15 at 8:12
• Chris, archive.org has many snapshots over the years. Check another year on archive.org (possibly the last alive snapshot). – joro Sep 2 '15 at 8:18

• Very nice and helpful command @joro. But, what is the number $2000....$? How can we find it? – Shahrooz Janbaz Sep 2 '15 at 8:07


text:
# Ancilla-assisted discrimination of quantum gates

Publisher:
Rinton Press
Publication Type:
Journal Article
Citation:
Quantum Information and Computation, 2010, 10 (1), pp. 160 - 177
Issue Date:
2010-01
The intrinsic idea of superdense coding is to find as many gates as possible such that they can be perfectly discriminated. In this paper, we consider a new scheme of discrimination of quantum gates, called ancilla-assisted discrimination, in which a set of quantum gates on a $d-$dimensional system are perfectly discriminated with assistance from an $r-$dimensional ancilla system. The main contribution of the present paper is two-fold: (1) The number of quantum gates that can be discriminated in this scheme is evaluated. We prove that any $rd+1$ quantum gates cannot be perfectly discriminated with assistance from the ancilla, and there exist $rd$ quantum gates which can be perfectly discriminated with assistance from the ancilla. (2) The dimensionality of the minimal ancilla system is estimated. We prove that there exists a constant positive number $c$ such that for any $k\leq cr$ quantum gates, if they are $d$-assisted discriminable, then they are also $r$-assisted discriminable, and there are $c^{\prime}r\textrm{}(c^{\prime}>c)$ different quantum gates which can be discriminated with a $d-$dimensional ancilla, but they cannot be discriminated if the ancilla is reduced to an $r-$dimensional system. Thus, the order $O(r)$ of the number of quantum gates that can be discriminated with assistance from an $r-$dimensional ancilla is optimal. The results reported in this paper represent a preliminary step toward understanding the role ancilla system plays in discrimination of quantum gates as well as the power and limit of superdense coding.


text:
The HoTT reals coincide with the Escardó-Simpson reals

Posted on 2017-06-20

Escardó and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy this universal property. We show that this is indeed the case in the category of sets of any universe. We also show that the type of HoTT reals is the least Cauchy complete subset of the Dedekind reals containing the rationals.


text:
# Submanifolds as Inverse Images

1. Feb 25, 2008

### Kreizhn

1. The problem statement, all variables and given/known data

Show that there exists no smooth function $f : \mathbb{RP}^2 \rightarrow \mathbb{R} \text{ such that } f^{-1}(q) = \mathbb{RP}^1$ for some regular value q of f.

2. The attempt at a solution

So we can quite simply show that $\mathbb{RP}^1$ is a submanifold of $\mathbb{RP}^2$, which is what the first part of the question actually asked but I have omitted here since I don't think it's important. I've tried looking at a few things, but I have one method that really seems to scream out at me but I'm not sure how to use it.

Theorem: Let $f : M\rightarrow N$ be a smooth manp and $P \subseteq N$ be a submanifold of N. If $f \pitchfork P$, then $f^{-1}(P)$ is a submanifold of M, provided that $f^{-1}(P)\neq \emptyset$.
Here $f\pitchfork P$ means f is transverse to P. I want to use contradiction, but this theorem seems to have the implication in the wrong direction in order for me to fully utilize it. Any ideas?


text:
## Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0.(English)Zbl 1218.37079

The authors consider the following second order Hamiltonian system $u''(t)+A(t)u(t)+ \nabla H(t,u(t))=0,\quad t\in \mathbb{R},$ where $$A(\cdot )$$ is a continuous $$T$$-periodic symmetric matrix, $$H:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}$$ is $$T$$-periodic in its first variable, $$H(t,x)$$ is continuous in $$t$$ for each $$x\in \mathbb{R}^{N}$$, and continuously differentiable in $$x$$ for each $$t\in [0,T]$$. The main result of the paper shows that if $$0$$ lies in a gap of $$\sigma(B)$$, where $$B=-\frac{d^2}{dt^2}-A(t)$$, and the technical work assumptions $$H_0 -H_6$$ hold, then the considered Hamiltonian system has at least one ground state $$T$$-periodic solution (Theorem 1.1).
The authors illustrate by examples that the assumptions $$H_0 -H_6$$ are reasonable and there are situations in which the well-known Ambrosetti-Rabinowicz superquadratic condition is not satisfied.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
Full Text:

### References:

 [1] Antonacci, F., Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign, Nonlinear anal., 29, 12, 1353-1364, (1997) · Zbl 0894.34036 [2] Bartsch, T.; Willem, M., Periodic solutions of nonautonomous Hamiltonian systems with symmetries, J. reine angew. math., 451, 149-159, (1994) · Zbl 0794.58037 [3] Benci, V., Some critical point theorems and applications, Comm. pure appl. math., 33, 2, 147-172, (1980) · Zbl 0472.58009 [4] Barletta, G.; Papageorgiou, N.S., Nonautonomous second order periodic systems: existence and multiplicity of solutions, J. nonlinear convex anal., 8, 3, 373-390, (2007) · Zbl 1158.34030 [5] Ding, Y.; Lee, C., Periodic solutions of Hamiltonian systems, SIAM J. math. anal., 32, 3, 555-571, (2000) · Zbl 0973.34035 [6] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electron. J. differential equations, 2002, 08, 1-12, (2002) · Zbl 0999.37039 [7] Girardi, M.; Matzeu, M., Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear anal., 7, 475-482, (1983) · Zbl 0512.70021 [8] Girardi, M.; Matzeu, M., Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear anal., 10, 371-382, (1986) · Zbl 0607.70018 [9] Guo, Z.; Yu, J., Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China ser. A, 46, 4, 506-515, (2003) · Zbl 1215.39001 [10] Gasiński, L.; Papageorgiou, N.S., Nonsmooth critical point theory and nonlinear boundary value problems, (2005), Chapman & Hall/CRC Press Boca Raton, FL · Zbl 1058.58005 [11] He, X.; Wu, X., Periodic solutions for a class of nonautonomous second order Hamiltonian systems, J. math. anal. appl., 341, 2, 1354-1364, (2008) · Zbl 1133.37025 [12] Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a landesman-lazer type problem set on $$R^N$$, Proc. roy. soc. Edinburgh sect. A, 129, 787-809, (1999) · Zbl 0935.35044 [13] Kandilakis, D.; Kourogenis, N.C.; Papageorgiou, N.S., Two nontrivial critical points for nonsmooth functionals via local linking and applications, J. global optim., 34, 2, 219-244, (2006) · Zbl 1105.49019 [14] Long, Y.M., Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. amer. math. soc., 311, 2, 749-780, (1989) · Zbl 0676.34026 [15] Li, S.J.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 1, 6-32, (1995) · Zbl 0820.58012 [16] Luan, S.; Mao, A., Periodic solutions for a class of nonautonomous Hamiltonian systems, Nonlinear anal., 61, 8, 1413-1426, (2005) · Zbl 1071.34039 [17] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York · Zbl 0676.58017 [18] Motreanu, D.; Motreanu, V.V.; Papageorgiou, N.S., Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential, Topol. methods nonlinear anal., 24, 269-296, (2004) · Zbl 1070.34068 [19] Motreanu, D.; Motreanu, V.V.; Papageorgiou, N.S., Two nontrivial solutions for periodic systems with indefinite linear part, Discrete contin. dyn. syst., 19, 1, 197-210, (2007) · Zbl 1146.34011 [20] Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. pure appl. math., 31, 2, 157-184, (1978) · Zbl 0358.70014 [21] Rabinowitz, P.H., Multiple critical points of perturbed symmetric functionals, Trans. amer. math. soc., 272, 2, 753-769, (1982) · Zbl 0589.35004 [22] Struwe, M., Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, (2000), Springer-Verlag Berlin · Zbl 0939.49001 [23] Szulkin, A.; Zou, W., Homoclinic orbits for asymptotically linear Hamiltonian systems, J. funct. anal., 187, 1, 25-41, (2001) · Zbl 0984.37072 [24] Schechter, M.; Zou, W., Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM control optim. calc. var., 9, 601-619, (2003) · Zbl 1173.35482 [25] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. funct. anal., 257, 12, 3802-3822, (2009) · Zbl 1178.35352 [26] Tao, Z.-L.; Yan, S.; Wu, S.-L., Periodic solutions for a class of superquadratic Hamiltonian systems, J. math. anal. appl., 331, 1, 152-158, (2007) · Zbl 1123.34311 [27] Willem, M., Minimax theorems, (1996), Birkhäuser · Zbl 0856.49001 [28] Willem, M.; Zou, W., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana univ. math. J., 52, 109-132, (2003) · Zbl 1030.35068 [29] Yang, M.; Chen, W.; Ding, Y., Solutions for discrete periodic Schrödinger equations with spectrum 0, Acta appl. math., 110, 3, 1475-1488, (2010) · Zbl 1191.35260 [30] Yang, M.; Chen, W.; Ding, Y., Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, J. math. anal. appl., 364, 2, 404-413, (2010) · Zbl 1189.35108 [31] Yang, M., Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear anal., 72, 5, 2620-2627, (2010) · Zbl 1184.35139
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Home » Add percentages to your axes in R’s ggplot2 (and set the limits)

# Add percentages to your axes in R’s ggplot2 (and set the limits)

Tags:

Something that took me a while to do properly in ggplot2 is adding the percentage sign as a suffix to your tick labels, controlling decimals and at the same time still being able to set the limits of your axis.

I’ll show an example using the iris data set. Let’s say I want to show the mean sepal length per species, as a percentage of the maximum sepal lenth in the dataset.

In the first chunk of code I load in the data set and I make the required transformation.

library(datasets)
library(scales)
library(data.table)
library(ggplot2)

dt <- as.data.table(iris)

dt[,Sepal.Length := Sepal.Length / max(Sepal.Length)]
dt <- dt[,.(Species, Sepal.Length)]

We can make the visualization as follows. By setting the labels in ggplot2’s scale_y_continuous() function, I can process all the values through a function that turns every value into a percentage.

ggplot(dt, aes(x = Species, y = Sepal.Length, fill = Species)) +
geom_bar(stat = 'summary', fun.y = 'mean') +
scale_y_continuous(labels = function(x) paste0(x * 100, '%'))

But there is an easier way, using the scales library, by setting the accuracy parameter, you can control how many decimals you would like to show.

ggplot(dt, aes(x = Species, y = Sepal.Length, fill = Species)) +
geom_bar(stat = 'summary', fun.y = 'mean') +
scale_y_continuous(labels = scales::percent_format(accuracy = 1))

Finally, another thing I struggled with is setting the limits of my y axis. Let’s say, you only want to show the range from 50% to 100%. Using the limits parameter in scale_y_continuous or if you use the lims() or ylim() function, you will break the scale and you will have an empty visualization. However, if you use coord_cartesian() function, you will be able to do it flawlessly.

ggplot(dt, aes(x = Species, y = Sepal.Length, fill = Species)) +
geom_bar(stat = 'summary', fun.y = 'mean') +
scale_y_continuous(labels = scales::percent_format(accuracy = 1)) +
coord_cartesian(ylim = c(0.5,1))

Here’s the final viz.

By the way, if you’re having trouble understanding some of the code and concepts, I can highly recommend “An Introduction to Statistical Learning: with Applications in R”, which is the must-have data science bible. If you simply need an introduction into R, and less into the Data Science part, I can absolutely recommend this book by Richard Cotton. Hope it helps!

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Technologies get updated, syntax changes and honestly… I make mistakes too. If something is incorrect, incomplete or doesn’t work, let me know in the comments below and help thousands of visitors.

## 3 thoughts on “Add percentages to your axes in R’s ggplot2 (and set the limits)”

1. thaaanks for this post! Useful and clearly explained.

1. You’re welcome!

2. Thanks for the feedback! That’s an interesting point. I suppose there are many reasons, internal and external, why we do not accomplish all that we might. Regardless, I am thankful for the Lord’s grace in that. Thanks again!


text:
# Sh:1135

• Raghavan, D., & Shelah, S. (2019). Two results on cardinal invariants at uncountable cardinals. In Proceedings of the 14th and 15th Asian Logic Conferences, World Sci. Publ., Hackensack, NJ, pp. 129–138.
• Abstract:
We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncoutable regular cardinal \kappa, \mathfrak{b}(\kappa) = \kappa^+ implies \mathfrak{a}(\kappa) = \kappa^+. This improves an earlier result of Blass, Hyttinen and Zhang [3]. It is also shown that if \kappa \ge \beta_\omega is an uncountable regular cardinal, then \mathfrak{d} (\kappa) \le \mathfrak{r}(\kappa). This result partially dualizes an earlier theorem of the authors [6]
• published version (6p)
Bib entry
@inproceedings{Sh:1135,
author = {Raghavan, Dilip and Shelah, Saharon},
title = {{Two results on cardinal invariants at uncountable cardinals}},
booktitle = {{Proceedings of the 14th and 15th Asian Logic Conferences}},
year = {2019},
pages = {129--138},
publisher = {World Sci. Publ., Hackensack, NJ},
mrnumber = {3890461},
mrclass = {03E17},
doi = {10.1142/9789813237551_0006},
note = {\href{https://arxiv.org/abs/1801.09369}{arXiv: 1801.09369}},
arxiv_number = {1801.09369}
}


text:
# Angular acceleration problem

## Homework Statement

angular acceleration of line a= -2w^2 rad/s

what is angular velocity when theta= pi/3?

chain rule

## The Attempt at a Solution

w= e^-2theta is my answer, but I know this is wrong

I know that this question is really easy, but I just don't know how to get it

rock.freak667
Homework Helper
Well you should know that α= -2ω2 means that dω/dt = -2ω2

which is an easily solvable DE once you 'separate the variables' i.e. put all the things with ω on one side and t on the other then integrate both sides.

But then I end up with t=1/2w, how do I find the rest of the answer?