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# Characterization of the 1S–2S transition in antihydrogen

## Abstract

In 1928, Dirac published an equation1 that combined quantum mechanics and special relativity. Negative-energy solutions to this equation, rather than being unphysical as initially thought, represented a class of hitherto unobserved and unimagined particles—antimatter. The existence of particles of antimatter was confirmed with the discovery of the positron2 (or anti-electron) by Anderson in 1932, but it is still unknown why matter, rather than antimatter, survived after the Big Bang. As a result, experimental studies of antimatter3,4,5,6,7, including tests of fundamental symmetries such as charge–parity and charge–parity–time, and searches for evidence of primordial antimatter, such as antihelium nuclei, have high priority in contemporary physics research. The fundamental role of the hydrogen atom in the evolution of the Universe and in the historical development of our understanding of quantum physics makes its antimatter counterpart—the antihydrogen atom—of particular interest. Current standard-model physics requires that hydrogen and antihydrogen have the same energy levels and spectral lines. The laser-driven 1S–2S transition was recently observed8 in antihydrogen. Here we characterize one of the hyperfine components of this transition using magnetically trapped atoms of antihydrogen and compare it to model calculations for hydrogen in our apparatus. We find that the shape of the spectral line agrees very well with that expected for hydrogen and that the resonance frequency agrees with that in hydrogen to about 5 kilohertz out of 2.5 × 1015 hertz. This is consistent with charge–parity–time invariance at a relative precision of 2 × 10−12—two orders of magnitude more precise than the previous determination8—corresponding to an absolute energy sensitivity of 2 × 10−20 GeV.

## Main

The transition of interest here, between the ground state and the first excited state of antihydrogen, has an energy of about 10.2 eV. The frequency of this transition in hydrogen has been measured8 to a few parts in 1015. We previously demonstrated7 the existence of the transition in antihydrogen, localizing the frequency to a few parts in 1010. Here we characterize the spectral line shape of the transition to the limits of precision of our current apparatus.

Matter and antimatter annihilate each other, so antihydrogen must be synthesized and then held in ultrahigh vacuum, in isolation from matter, to be studied. The ALPHA-2 apparatus at CERN (Fig. 1) combines antiprotons from the antiproton decelerator9 with positrons from a positron accumulator10, 11 to produce and trap12 atoms of antihydrogen. Antihydrogen can be trapped in ALPHA-2’s magnetic multipole trap if it is produced with a kinetic energy of less than 0.54 K in temperature units. The techniques that we use to produce antihydrogen that is cold enough to trap are described elsewhere12,13,14. In round numbers, a typical trapping trial in ALPHA-2 involves mixing 90,000 antiprotons with 3,000,000 positrons to produce 50,000 antihydrogen atoms, about 20 of which will be trapped. The anti-atoms are confined by the interaction of their magnetic moments with the inhomogeneous magnetic field. The cylindrical trapping volume for antihydrogen has a diameter of 44.35 mm and a length of 280 mm.

The key to anti-atomic spectroscopy, as developed so far7, 15, 16, is to illuminate a sample of trapped antihydrogen atoms with electromagnetic radiation (microwaves or laser photons) that causes atoms to be lost from the trap if the radiation is on resonance with the transition of interest. ALPHA-2’s silicon vertex detector17 (Fig. 1) affords us single-atom detection capability for the annihilation events associated with lost antihydrogen atoms or antiprotons that encounter the walls of the apparatus. The silicon vertex detector tracks the charged pions from the antiproton annihilation, and various reconstruction algorithms are used to determine the location (vertex) of each annihilation and to distinguish antiprotons from cosmic-ray background using multivariate analysis18 (Methods).

To excite the 1S–2S transition, we use a cryogenic, in vacuo enhancement cavity (Fig. 1) for continuous-wave light from a 243-nm laser system (Methods) to boost the intensity in the trapping volume. Long interaction times are possible, because the anti-atoms have a storage lifetime of at least 60 h in the trap. Two counter-propagating photons can resonantly excite the ground-state atoms to the 2S state. Absorption of a third photon ionizes the atom, leading to loss of the antiproton from the trap. Atoms that decay from the 2S to the 1S state via coupling to the 2P state may also be lost, owing to a positron spin-flip19.

Referring to the energy-level diagram of hydrogen in Fig. 2, there are two trappable, hyperfine substates of the 1S ground state (labelled ‘c’ and ‘d’). In practice, we find that these states are, on average, equally populated in our trap: Nc = Nd = Ni/2, where Ni is the number of ground-state atoms that are initially trapped in an experimental trial. The 2S state has corresponding hyperfine levels, and we refer to the transitions between the two manifolds as d–d (Fig. 2) and c–c (not pictured).

For each experimental trial, we first accumulate antihydrogen atoms from three mixing cycles or ‘stacks’13 and then remove any leftover charged particles using pulsed electric fields. After a wait of about 10 s to allow any excited atoms to decay to the ground state, the trapped population is exposed to laser radiation at a fixed frequency for 300 s. The frequencies used here were chosen to probe only the d–d transition (Fig. 2). Following the laser exposure, we use microwave radiation to remove the 1Sc state atoms by driving a resonant spin-flip15, 16. The microwave frequency is scanned over 9 MHz in 32 s; these parameters and the injected power level (160 mW at the vacuum feed-through) are chosen to eject anti-atoms quickly while minimizing the perturbation of the vacuum and cryogenic environment. The silicon vertex detector is used to detect annihilations of antihydrogen atoms that are lost during the laser and microwave exposures. Finally, the atom-trap magnets are ramped down in 1.5 s, so that any surviving anti-atoms would be released and their annihilations detected. If the microwave removal of 1Sc-state atoms is 100% effective, then the surviving particles would be only 1Sd-state atoms that were not removed by laser action.

We collected data for nine different laser frequencies in four sets. Each set involved four distinct frequencies and 21 (or 23, see below) trials at each of these frequencies. In each set, two of the frequencies were always the calculated hydrogen on-resonance frequency at zero laser power (zero detuning) and a far-off-resonance frequency (−200 kHz detuning at 243 nm), as used previously7. The other two frequencies in each set were chosen to address various detunings in the neighbourhood of the d–d resonance. The data are summarized in Table 1. The repetition of the points at −200 kHz and zero detuning was intended to address variations in laser power and trapping number between sets. The repetition at + 25 kHz was a check of reproducibility. During the accumulation of data for each set, the four frequencies were interleaved in a varying order and the operators were blinded as to the identity of each frequency setting. The power of the enhancement cavity (about 1 W) was monitored by measuring the transmitted power outside of the vacuum chamber (Fig. 1). Each set was preceded by a thermal cycle of the apparatus to regenerate the cryo-pumping surface.

The background-corrected numbers in Table 1 are calculated from raw detector events using the measured, overall efficiencies of the silicon vertex detector. These efficiencies depend on the particular multivariate analysis algorithm that was used to distinguish antiproton annihilations from cosmic rays (Methods) in the relevant time window. The efficiencies and background rates are listed in Table 2.

The number of initially trapped atoms Ni for a trial is unknown a priori, but was typically about 60 at the beginning of a measurement set. In Table 1, the total number of atoms for each group of trials is assumed to be the sum L + M + S of the numbers of atoms lost during laser (L) or microwave (M) exposure and the number of surviving atoms (S) (see Table 1). The trapping rate declined slowly but reproducibly during each set (Extended Data Fig. 1). The third set has 23 trials at each frequency because of a hardware failure in an early block of four trials; extra trials were added to compensate for the excluded data.

To examine the general features of the measurement results, we plot (Fig. 3a) the four datasets on one graph by using a simple scaling. The points at zero (on-resonance) and −200-kHz detuning (at which no signal is expected7), repeated for each set, are used for the scaling. For the laser exposure (‘appearance’) data, we define a scaled response at detuning D within each set: rl(D) = L(D)/L(0). Similarly, for the surviving population (‘disappearance’ data), we use rs(D) = [S(−200 kHz) − S(D)]/[S(−200 kHz) − S(0)]. The uncertainties shown are due to Poissonian counting errors only. For comparison, we also plot the results of a simulation19 based on the expected behaviour of hydrogen in our trap for a cavity power of 1 W, scaled to the zero-detuning data point. We see that the peak position and the width of the scaled spectral line are consistent with the calculation for hydrogen and that the experiment generally reproduces the predicted asymmetric line shape. There is also good agreement between the appearance and disappearance data (Fig. 3a).

The simulation involves propagating the trapped atoms in an accurate model of the magnetic trap. When an atom crosses the laser beam, which has a waist of 200 μm at the cavity centre, we calculate the two-photon excitation probability, taking into account transit-time broadening, the a.c. Stark shift and the residual Zeeman effect. The simulation determines whether excited atoms are lost owing to ionization or to a spin-flip event. The variable input parameters for the simulation are the cavity power and the laser frequency. The modelled response is asymmetric in frequency owing to the residual Zeeman effect19. The width of the line, for our experimental parameters, is dominated by transit-time broadening, which contributes about 50 kHz full-width at half-maximum (FWHM) at 243 nm. For 1 W of cavity power, the a.c. Stark shift is about 2.5 kHz to higher frequency and the ionization contributes about 2 kHz to the natural line width.

To make a more quantitative comparison of the experimental results with the expectations for hydrogen, it is necessary to scrutinize differences between the four datasets. The overall response should be linear in the number of atoms addressed, so it is possible to normalize for this. However, the line width depends on the stored power in the cavity, as does the frequency of the peak (Fig. 3b). The cavity power is difficult to measure in our geometry because the amount of transmitted light depends sensitively on the small transmission from the output coupler (about 0.05%) and on absorption in the optical elements through which the transmitted light exits (Fig. 1). We observe that the transmitted power can degrade, owing to accumulated ultraviolet damage to the window and mirror substrate, whereas the finesse of the cavity does not change.

A modelling approach that self-consistently accounts for fluctuations in experimental parameters is a simultaneous fit in which we allow the four sets to have distinct powers (P1–4), but the same frequency shift with respect to the hydrogen calculation (Methods). We require that the average powers for the appearance and disappearance data within a set are the same. We find the parameters that best reproduce the data to be: P1 = 1135(50) mW, P2 = 904(30) mW, P3 = 1123(43) mW, P4 = 957(31) mW and δf = −0.44 ± 1.9 kHz, where δf is the difference (at 243 nm) between the resonant frequency inferred from the fit and the resonant frequency of hydrogen expected for our system, both at zero power. The uncertainties represent the 68% confidence interval of a least-squares fit and do not take into account systematic uncertainties. The fit uses the five variables identified above, and the individual data points at each frequency are weighted by their Poissonian counting errors. We include an uncertainty of 3.8 kHz (Table 3) in the final resonance frequency to represent statistical and curve-fitting uncertainties.

Considering systematic effects, the microwave removal procedure for the 1Sc-state atoms provides a reproducibility check on the strength of the magnetic field at the centre of the trap. At the beginning of each data-taking shift, the magnetic field of the external solenoid magnet was reset to a standard value using an electron cyclotron resonance technique16. For the complete dataset, we find that the variations in the magnetic field at the minimum field of about 1 T are about 3.2 × 10−5 T (1 s.d.). This corresponds to a resonance frequency shift19 of only about 15 Hz at 243 nm for the d–d transition. (At 1 T, the c–c transition is about 20 times more sensitive to magnetic field shifts, which is why the d–d transition is more attractive here.) The laser frequency was tuned with respect to the minimum of the magnetic well, such that the resonance condition should be met in the centre of the trap for zero detuning in the limit of zero laser power. The accuracy of the magnetic-field determination corresponds to an uncertainty of 300 Hz in the 243-nm laser frequency.

Including all of the statistical and systematic uncertainties that we have identified (Table 3, for 121 nm), our fit of the experimental data to the hydrogen model yields

$${f}_{{\rm{d}}-{\rm{d}}}=\mathrm{2,466,061,103,079.4}(5.4)\,{\rm{k}}{\rm{H}}{\rm{z}}$$

The value (Methods) for hydrogen calculated at the minimum field in our system (1.03285(63) T) is

$${f}_{{\rm{d}}-{\rm{d}}}=\mathrm{2,466,061,103,080.3}(0.6)\,{\rm{k}}{\rm{H}}{\rm{z}}$$

where the uncertainty is determined by the experimental error in measuring the field.

Owing to the motion of the antihydrogen atoms in the inhomogeneous trapping field, this comparison is necessarily model-dependent. We therefore conclude that the measured resonance frequency for this transition in antihydrogen is consistent with the expected hydrogen frequency to a precision of about 2 × 10−12. Although the precision of our measurement is still a few orders of magnitude short of the state of the art with a cold hydrogen beam8, the modern frequency reference permits the accuracy of our experiment to exceed that achieved with trapped hydrogen20 as recently as the mid-1990s. We used a total of about 15,000 antihydrogen atoms to obtain this result, compared to 1012 trapped atoms in the analogous matter experiment. Our dataset was accumulated over a period of ten weeks, illustrating that the antihydrogen trapping procedure is robust and that systematic effects are manageable. ALPHA’s emergent antihydrogen production, storage and detection techniques, together with advances in ultraviolet laser technology and frequency metrology, pioneered by Hänsch and colleagues, enable precision anti-atom spectroscopy.

Precision experiments at the antiproton decelerator have recently constrained the properties of the antiproton through studies in Penning traps21, 22 or with antiprotonic helium23. For example, the antiproton charge-to-mass ratio is known to agree with that of the proton to 69 parts per trillion21, equivalent to an energy sensitivity of 9 × 10−27 GeV. The ratio of the antiproton mass to the electron mass has been shown to agree with its proton counterpart23 to 8 × 10−10, and antihydrogen has been shown to be neutral24 to 0.7 parts per billion. Our measurement of antihydrogen probes different and complementary physics at a precision of a few parts per trillion, or an energy level of 2 × 10−20 GeV. This already exceeds the precision (4 × 10−19 GeV) in the mass difference of neutral kaons and antikaons25, which has long been the standard for particle-physics tests of charge–parity–time invariance.

Near-term improvements in the ALPHA-2 apparatus will include a larger waist size for the radiation in the optical cavity to reduce transit-time broadening, operation at lower magnetic fields and operational improvements to accelerate data acquisition and to reduce statistical uncertainties. Future measurements will require an upgrade to our frequency reference to exceed a fractional precision of 8 × 10−13 (Methods). The rapid progress detailed here confirms that, in principle, there is nothing to prevent the achievement of hydrogen-like precision in antihydrogen and the associated very sensitive test of charge–parity–time symmetry in this system.

## Methods

### Time evolution of the dataset

The time evolution of the atoms detected in one of the datasets is depicted in Extended Data Fig. 1.

### Laser system for 243-nm light

A Toptica TA-FHG pro laser system uses a pair of frequency-doubling cavities to generate 150 mW of 243-nm light from a 972-nm extended cavity diode laser (ECDL). The 243-nm beam is mode-matched to the 1S–2S enhancement cavity and sent along a 7-m-long path with active beam-pointing stabilization between the laser laboratory and the ALPHA-2 apparatus. The enhancement cavity is locked to the laser frequency using a single piezoelectric actuator located behind the output coupler mirror26 to feedback on an error signal generated via the Pound–Drever–Hall technique27. The light transmitted through the cavity is monitored using a photodiode that is located outside the vacuum system. The cavity has a measured finesse of 250 and achieves a circulating power of approximately 1 W.

The 972-nm ECDL is frequency-stabilized (also using the Pound–Drever–Hall technique) to a Menlo Systems ultralow-expansion cavity via an acousto-optic modulator, which shifts the light from the 1S–2S transition frequency of the laser to the closest resonance of the ultralow-expansion cavity. The resonance frequency of the cavity is monitored continuously using a Menlo Systems femtosecond frequency comb, which is referenced to atomic time using a K + K Messtechnik GPS-disciplined quartz oscillator.

The measured difference between the ultralow-expansion resonance frequency and a comb line with a known frequency is fed forward to the control of the acousto-optic modulator with an averaging time of 20 s to remove long-term drifts. The uncertainty of the frequency difference over the 20-s averaging period corresponds to an Allan deviation28 of 75 Hz at 972 nm (300 Hz at 243 nm). One of the frequency-comb counters is used to measure the signal from a Symmetricom CS4000 caesium clock to confirm correct operation of the quartz oscillator and the radio-frequency chain of the frequency comb. The count reaches a fractional Allan deviation of 8 × 10−13 after 1,000 s of averaging, which corresponds to fluctuations of 250 Hz at 972 nm (1 kHz at 243 nm).

An independent, identical, 972-nm ECDL frequency stabilized to an independent, identical, ultralow-expansion cavity is used to evaluate the short-term line width of the spectroscopy laser. The beat note generated between the two 972-nm lasers has a spectrum composed of individual lines, each with a line width of less than 1 Hz, within a 300-Hz (1.2 kHz at 243 nm) FWHM Gaussian envelope. The source of the broadening is thought to be acoustic noise within the laser laboratory; work is ongoing to reduce the broadening effect.

### Suppression of cosmic-ray background

To determine the signal events in the (a) 1.6-s, (b) 32-s and (c) 300-s observation windows, we require three different suppression techniques. (The 1.6-s window extends to 0.1 s after the magnet rampdown is complete.) We tune the multivariate analysis (MVA) for each of the three windows to optimize the statistical significance of the estimated signal. Annihilation events are distinguished from background events (primarily cosmic rays) by their distinctive topologies. Fourteen selection variables that are sensitive to the difference between annihilation and background events were used as inputs to an MVA package18. The variables included are: (i) the total number of channels registering ‘hits’ by charged particles; (ii) the radial coordinates of the reconstructed annihilation vertex; (iii) the sum of the squared residual distances of hits from a fitted straight line; six topological variables (iv–ix); and five additional variables (x–xiv). The topological variables are: (iv) a sphericity variable; (v) the cosine of the angle between the event axis and the detector axis; (vi) the angle between the event axis and the vertical direction in the xy plane; (vii) the number of reconstructed tracks; (viii) the number of three-hit combinations used as track candidates; (ix) the distance of closest approach of the tracks. The additional variables are: (x) the minimum and (xi) mean values of the track radius in canonical form; (xii) the minimum and (xiii) mean values of the pitch of the helical track in canonical form; and (xiv) an integer sum of the sense of curvature (left = −1 or right =  + 1) for all of the tracks in the event.

The signal data and background data used for MVA training and testing comprise a set of 580,846 annihilation events and 3,740,613 background events. The signal events were produced during antiproton and positron mixing in the apparatus and contain less than 1% background. Background events were collected during times when there were no antiprotons in the apparatus.

The 1.6-s observation window. A classifier cut was chosen to optimize the significance for an expected 200 counts of signal and 350 counts of background. The analysis gives a background rate of 0.191 ± 0.001 s−1 and an efficiency of 0.852 ± 0.002 (statistical error only) annihilations per detector trigger.

The 32-s observation window. The analysis was chosen to optimize the significance for an expected 400 counts of signal and 3,500 counts of background. The analysis gives a background rate of 0.033 ± 0.0006 s−1 and an efficiency of 0.801 ± 0.002 (statistical error only) annihilations per detector trigger.

The 300-s observation window. A classifier cut was chosen to optimize the significance for an expected 250 counts of signal and 330,000 counts of background. The analysis gives a background rate of 0.0010 ± 0.0001 s−1 and an efficiency of 0.472 ± 0.001 (statistical error only) annihilations per detector trigger.

### Fitting the data using the hydrogen simulation

The build-up of laser power in the enhancement cavity is one of the primary experimental parameters that influence the data in Table 1. The main effect of a change in laser power is on the amplitude of the measured line, but there is also an effect on the peak position through the a.c. Stark shift and on the line width owing to depletion effects. In our set-up, there is considerable uncertainty in measuring the absolute intra-cavity laser power; relative measurements show that although the constancy of laser power within any single measurement set is good, there are variations between the sets.

To reflect this experimental reality in our analysis of the data, the χ2 statistic for the full dataset is minimized with respect to a function that, aside from an overall frequency shift, allows a unique laser power in each set and incorporates the effects of those laser powers on the amplitude, line width and line centre based on the simulation of hydrogen in our experiment.

The construction of the fit function therefore starts by running a detailed simulation of hydrogen in the ALPHA-2 magnetic trap for an array of input laser powers and frequencies that spans the experimentally relevant values, in this case from −200 kHz to + 300 kHz in laser detuning and from 0.7 W to 1.25 W in laser power. We simulate a total of 365,000 atoms in this array, after which we interpolate to obtain continuous values in both laser detuning and power. The interpolation in power is a linear regression at each detuning in the array, based on the observed linear behaviour. For interpolation in detuning, a fit to a piecewise-analytic function that provides a good approximation to the simulation data is used. An error associated with this fit is included in Table 3. The discrete simulated points and the smooth interpolation are plotted in Extended Data Fig. 2.

### Calculation of the resonant frequency for hydrogen

The frequency fd–d is calculated from corrections to the centroid-to-centroid frequency f1S2S:

$$\begin{array}{c}{f}_{{\rm{d}}-{\rm{d}}}(B)={f}_{1{\rm{S}}2{\rm{S}}}-\frac{1}{4}[{f}_{{\rm{H}}{\rm{F}}}(1)-{f}_{{\rm{H}}{\rm{F}}}(2)]+[{\mu }_{{\rm{e}}}(2)-{\mu }_{{\rm{e}}}(1)]\frac{B}{h}\\ \phantom{\rule{27pt}{0ex}}-[{\mu }_{{\rm{p}}}(2)-{\mu }_{{\rm{p}}}(1)]\frac{B}{h}+{(\frac{m}{\mu })}^{3}\frac{13{e}^{2}{a}_{0}^{2}}{4mh}{B}^{2}\end{array}$$

where h is Planck’s constant, fHF(n) is the hyperfine splitting of the state with principle quantum number n, μe and μp are the magnitudes of the magnetic moments of the electron and proton, respectively, μ is the reduced mass of the electron, m is the electron mass, e is the fundamental charge, a0 is the Bohr radius for an infinite-mass nucleus and B is the magnetic field.

The first correction describes the difference in the hyperfine splittings of the 1S and 2S states. The second (third) correction describes the difference in the magnetic moment of the electron (proton) in these states. The fourth correction describes the difference in the diamagnetic shift.

The magnetic moment of the bound electron is (equation (84))29

$${\mu }_{{\rm{e}}}(n)={\mu }_{{\rm{e}}}^{{\rm{free}}}\left[1-\frac{{\alpha }^{2}}{3{n}^{2}}+\frac{{\alpha }^{4}}{2{n}^{3}}\left(\frac{1}{2n}-\frac{2}{3}\right)+\frac{{\alpha }^{3}}{4{\rm{\pi }}{n}^{2}}+\frac{{\alpha }^{2}}{2{n}^{2}}\frac{m}{M}\right]$$

where α is the fine-structure constant, $${\mu }_{{\rm{e}}}^{{\rm{free}}}$$ is the free-electron dipole moment and M is the proton mass; the dependence on n is described elsewhere30, 31. The magnetic moment of the bound proton is (equation (87))29

$${\mu }_{{\rm{p}}}(n)={\mu }_{{\rm{p}}}^{{\rm{free}}}\left[1-\frac{{\alpha }^{2}}{3{n}^{2}}+\frac{{\alpha }^{2}}{6{n}^{2}}\frac{m}{M}\left(\frac{3+4\times 1.793}{1+1.793}\right)\right]$$

where $${\mu }_{{\rm{p}}}^{{\rm{free}}}$$ is the free-proton dipole moment.

Using current CODATA values32 for the fundamental constants, the frequency is

$$\begin{array}{c}{f}_{{\rm{d}}-{\rm{d}}}={f}_{1{\rm{S}}2{\rm{S}}}-\mathrm{310,712.229}\,{\rm{k}}{\rm{H}}{\rm{z}}+186.071B\,{\rm{k}}{\rm{H}}{\rm{z}}{{\rm{T}}}^{-1}\\ \,-0.283B\,{\rm{k}}{\rm{H}}{\rm{z}}{{\rm{T}}}^{-1}+387.678{B}^{2}\,{\rm{k}}{\rm{H}}{\rm{z}}{{\rm{T}}}^{-2}\end{array}$$

### Sample size

No statistical methods were used to predetermine sample size.

### Data availability

The datasets generated and analysed during this study are available from the corresponding author on reasonable request.

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Mohr, P. J., Newell, D. B. & Taylor, B. N. CODATA recommended values of the fundamental physical constants: 2014. Rev. Mod. Phys. 88, 035009 (2016).

## Acknowledgements

All authors are members of the ALPHA Collaboration. This work was supported by: the European Research Council through its Advanced Grant programme (J.S.H.); CNPq, FAPERJ, RENAFAE (Brazil); NSERC, NRC/TRIUMF, EHPDS/EHDRS, FQRNT (Canada); FNU (Nice Centre), Carlsberg Foundation (Denmark); ISF (Israel); STFC, EPSRC, the Royal Society and the Leverhulme Trust (UK); DOE, NSF (USA); and VR (Sweden). We are grateful for the efforts of the CERN Antiproton Decelerator team, without which these experiments could not have taken place. We thank J. Tonoli (CERN) and his staff for extensive, time-critical help with machining work. We thank the staff of the Superconducting Magnet Division at Brookhaven National Laboratory for collaboration and fabrication of the trapping magnets. We thank C. Marshall (TRIUMF) for his work on the ALPHA-2 cryostat. We thank D. Tommasini and A. Milanese (CERN) for the fabrication of conventional magnets for ALPHA-2. We acknowledge the influence of T. Hänsch on the methodology and the hardware used here. J.S.H. and C.L.C. acknowledge early encouragement from D. Kleppner (MIT) in these endeavours. We thank F. Besenbacher (Aarhus) for support in procuring the ALPHA-2 external solenoid. We thank J. Serrano (CERN) for making the atomic clock available to us.

### Reviewer information

Nature thanks D. Horvath, K. Jungmann and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

Authors

### Contributions

This experiment was based on data collected using the ALPHA-2 antihydrogen trapping apparatus, designed and constructed by the ALPHA Collaboration using methods developed by the entire collaboration. The entire collaboration participated in the operation of the apparatus and the data-taking activities. The laser and internal cavity system was conceived, implemented, commissioned and operated by W.B., N.M., J.S.H., S.E., C.Ø.R., S.A.J., C.L.C., B.X.R.A. and G.S. F.R., C.Ø.R., J.F. and N.M. developed the simulation program for laser interaction with magnetically trapped atoms. Analysis of the spectral line shapes was done by C.Ø.R., N.M. and J.S.H. Detailed analysis of the antiproton annihilation detector data was done by J.T.K.M. and A.O. Implementation of the microwave system and analysis of the microwave data was done by T.F. and M.E.H. The positron accumulator is the responsibility of C.J.B., M.C., C.A.I. and D.P.v.d.W. The manuscript was written by J.S.H., N.M., C.Ø.R., S.A.J. and J.T.K.M., with help from A.O., C.L.C. and S.E. The manuscript was then edited and improved by the entire collaboration.

### Corresponding author

Correspondence to J. S. Hangst.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

## Extended data figures and tables

### Extended Data Fig. 1 Time evolution of the dataset.

The integrated number of atoms is plotted against the trial number for the four detunings D (−200 kHz, −100 kHz, 0 kHz and 100 kHz) in set 1. The error bars are 1-s.d. counting uncertainties.

### Extended Data Fig. 2 Simulation fitting functions.

The points (crosses) from the numerical simulation are plotted for various cavity powers. The solid lines represent fits to the simulation by a piecewise-analytic function. The coloured surface represents the interpolation used to fit the experimental data.

## Rights and permissions

Reprints and Permissions

Ahmadi, M., Alves, B.X.R., Baker, C.J. et al. Characterization of the 1S–2S transition in antihydrogen. Nature 557, 71–75 (2018). https://doi.org/10.1038/s41586-018-0017-2

• Accepted:

• Published:

• Issue Date:

• ### Tests of discrete symmetries

• M S Sozzi

Journal of Physics G: Nuclear and Particle Physics (2020)

• ### Doppler-free two-photon cavity ring-down spectroscopy of a nitrous oxide ( N2O ) vibrational overtone transition

• Gang Zhao
• , D. Michelle Bailey
• , Adam J. Fleisher
• , Joseph T. Hodges
•  & Kevin K. Lehmann

Physical Review A (2020)

• ### Exotic helium atom lit up

Nature (2020)

• ### The QUPLAS experimental apparatus for antimatter interferometry

• A. Ariga
• , S. Cialdi
• , G. Costantini
• , A. Ereditato
• , R. Ferragut
• , M. Giammarchi
• , M. Leone
• , G. Maero
• , L. Miramonti
• , C. Pistillo
• , M. Romé
• , S. Sala
• , P. Scampoli
•  & V. Toso

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment (2020)

• ### Stimulated decay and formation of antihydrogen atoms

• T. Wolz
• , C. Malbrunot
• , M. Vieille-Grosjean
•  & D. Comparat

Physical Review A (2020)


text:
# Implement Applicative Maybe

Typeclassopedia presents the following exercise:

Maybe can easily be made an instance of Applicative; writing such an instance is left as an exercise for the reader.

Here's my implementation:

data Option a = Some a
| None deriving Show

instance Functor Option where
fmap _ None     = None
fmap f (Some x) = Some (f x)

instance Applicative Option where
pure x                = Some x
(Some f) <*> (Some x) = Some (f x)
_ <*> _               = None

How does this look? In particular, I'm curious if the second line of the Applicative Option looks good.

Should I be using fmap on the right-hand side instead of Some (f x)?

Looks good, follows the laws of Applicative AFAICT, and I'd leave it like this, with fmap you'd need to write more.
• But, by using fmap, I'd be re-using already written code - for Functor Option. Isn't that a meaningful code re-use? – Kevin Meredith Dec 1 '14 at 22:29
• Compare fmap f (Some x) with Some (f x), I'd argue that the latter one is easier to understand, which is more important than reuse IMO. If you had that twice or more times in a row I'd agree. – ferada Dec 2 '14 at 9:47

format: plain


text:
# Multiply square root of x+3* square root of x-3

Combine using the product rule for radicals.
Multiply square root of x+3* square root of x-3


text:
# Why does $\lim_{x \to 3} (x^2-5x+4)^{x-3}$ not exist? [closed]

Why does the following limit not exist? $$\lim_{x \to 3} \left(x^2-5x+4\right)^{x-3}$$

The base tends to $$-2$$ while the exponent tends to $$0$$. According to me the limit should be $$1$$, but the solution given for the problem says that the limit does not exist. Which is the correct answer?

• The limit does not exist because the function is not defined on a neighborhood of 3. – Did Dec 21 '18 at 16:39
• Because the base is negative in a neighborhood of 3 hence raising it to a noninteger power cannot be done. – Did Dec 21 '18 at 16:42
• @Dr.SonnhardGraubner: For any choice of the branch of log to be used, the limit would be $1$, but we don't know if we can use complex analysis or not. Certainly no mention of complex analysis was made. – robjohn Dec 21 '18 at 16:49
• @Did Technically, $3$ is a limit point of and belongs to the domain. The $\varepsilon-\delta$ definition of limit doesn't require $x\mapsto (x^2-5x+4)^{x-3}$ to be defined in the entire deleted $\delta$ neighbourhood of $3$ – Shubham Johri Dec 21 '18 at 18:11
• If you interpret the function as having domain $(-\infty, 1) \cup [(1, 4) \cap \mathbb{Z}_{(2)}] \cup [4, \infty)$ then the limit as $x\to 3$ does not exist (nor does either one-sided limit): 3 is a cluster point of the numbers $p/q$ with $p$ even and $q$ odd for which the limit approaches 1, and also a cluster point of the numbers $p/q$ with $p$ odd and $q$ odd for which the limit approaches $-1$. – Daniel Schepler Dec 21 '18 at 19:26

The quadratic $$x^2-5x+4$$ factorises as $$(x-4)(x-1)$$. If $$x$$ is in a small neighbourhood of $$3$$ then $$x-4<0$$ and $$x-1>0$$, so $$x^2-5x+4 < 0$$. This means that we run into issues with even defining the quantity $$(x^2-5x+4)^{x-3}$$ when, say, $$x = 3 \pm \dfrac{1}{2n}$$ for some positive integer $$n$$. This is bad news for the limit as $$x \to 3$$.
• @Pranav: The issue is that the base is negative, since in the limit you are required to take fractional powers in a neighbourhood of the limit. You run into the same issue if you replace the $x-3$ by $x-2$ (or $x-a$ for any $a$), so it is not the case that $\lim_{x \to 3} (x^3-5x+4)^{x-2} = -2$. You can evaluate $(x^3-5x+4)^{x-2}$ when $x=3$, and you get $-2$, but that is not the limit as $x \to 3$, since the quantity isn't even defined everywhere in an open interval about $2$ (or indeed any real number). – Clive Newstead Dec 21 '18 at 16:50
• ...this is ignoring complex numbers, of course. Taking the limit over $\mathbb{C}$ you need to consider different branch cuts of the complex plane to figure out how to define the quantity with non-integral exponents. – Clive Newstead Dec 21 '18 at 16:53
• @CliveNewstead am I right in thinking magnitude limits to $1$ although the sign is ambiguous? So the absolute value of the expression has a limit. – samerivertwice Dec 23 '18 at 4:25


text:
## Remember, when it comes to getting offended, we all decide what offends us and how to get offended by what someone says. TRUE empowerment means you have the control within yourself. We don’t have to allow things to offend us simply because someone says something, and this also doesn’t mean everyone is going to be mean to us all the time, this is an unsubstantiated fear.

You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
I might be scrapping my motor and going back to the drawing board. Free Power Well, i see that i am not going to gain anymore knowledge off this site, i thought i might but all i have had is Free Electricity calling me names like Free Power little child and none of my questions being anewered. Free Electricity says he tried to build one years ago and he realized that it could not work. Ok tell me why. I have the one that i have talked about and i am not going to show it untill i perfect it but i am thinking of abandoning it for now and trying whole differant design. Can the expert Free Electricity answer shis? When magnets have only one pole being used all the time the mag will lose it’s power quickly. What will happen if you use both poles in the repel state? Free Electricity that ballance the mag out or drain it twice as fast? How long will Free Power mag last running in the repel state all the time? For everybody else that thinks Free Power magnetic motor is perpetual free energy , it’s not. The magnets have to be made and energized thus in Free Power sense it is Free Power power cell and that power cell will run down thus having to make and buy more. Not free energy. This is still fun to play with though.
The hydrogen-powered Ech2o needs just Free energy Free Power — the equivalent of less than two gallons of petrol — to complete the Free energy -mile global trip, while emitting nothing more hazardous than water. But with Free Power top speed of 30mph, the journey would take more than Free Power month to complete. Ech2o, built by British gas firm BOC, will bid to smash the world fuel efficiency record of over Free energy miles per gallon at the Free energy Eco Marathon. The record is currently…. Free Power, 385 km/per liter [over Free Electricity mpg!]. Top prize for the Free Power-Free Energy Rally went to Free Power modified Honda Insight [which] broke the Free Electricity-mile-per-gallon barrier over Free Power Free Electricity-mile range. The car actually got Free Electricity miles-per gallon. St. Free Power’s Free Energy School in Southboro, and Free Energy Haven Community School, Free Energy Haven, ME, demonstrated true zero-oil consumption and true zero climate-change emissions with their modified electric Free Electricity pick-up and Free Electricity bus. Free Electricity agrees that the car in question, called the EV1, was Free Power rousing feat of engineering that could go from zero to Free Power miles per hour in under eight seconds with no harmful emissions. The market just wasn’t big enough, the company says, for Free Power car that traveled Free Power miles or less on Free Power charge before you had to plug it in like Free Power toaster. Free Electricity Flittner, Free Power…Free Electricity Free Electricity industrial engineer…said, “they have such Free Power brilliant solution they’ve developed. They’ve put it on the market and proved it works. Free Energy still want it and they’re taking it away and destroying it. ”Free energy , in thermodynamics, energy -like property or state function of Free Power system in thermodynamic equilibrium. Free energy has the dimensions of energy , and its value is determined by the state of the system and not by its history. Free energy is used to determine how systems change and how much work they can produce. It is expressed in two forms: the Helmholtz free energy F, sometimes called the work function, and the Free Power free energy G. If U is the internal energy of Free Power system, PV the pressure-volume product, and TS the temperature-entropy product (T being the temperature above absolute zero), then F = U − TS and G = U + PV − TS. The latter equation can also be written in the form G = H – TS, where H = U + PV is the enthalpy. Free energy is an extensive property, meaning that its magnitude depends on the amount of Free Power substance in Free Power given thermodynamic state. The changes in free energy , ΔF or ΔG, are useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained from thermodynamic processes involving chemical or other types of reactions. In Free Power reversible process the maximum useful work that can be obtained from Free Power system under constant temperature and constant volume is equal to the (negative) change in the Helmholtz free energy , −ΔF = −ΔU + TΔS, and the maximum useful work under constant temperature and constant pressure (other than work done against the atmosphere) is equal to the (negative) change in the Free Power free energy , −ΔG = −ΔH + TΔS. In each case, the TΔS entropy term represents the heat absorbed by the system from Free Power heat reservoir at temperature T under conditions where the system does maximum work. By conservation of energy , the total work done also includes the decrease in internal energy U or enthalpy H as the case may be. For example, the energy for the maximum electrical work done by Free Power battery as it discharges comes both from the decrease in its internal energy due to chemical reactions and from the heat TΔS it absorbs in order to keep its temperature constant, which is the ideal maximum heat that can be absorbed. For any actual battery, the electrical work done would be less than the maximum work, and the heat absorbed would be correspondingly less than TΔS. Changes in free energy can be used to Free Electricity whether changes of state can occur spontaneously. Under constant temperature and volume, the transformation will happen spontaneously, either slowly or rapidly, if the Helmholtz free energy is smaller in the final state than in the initial state—that is, if the difference ΔF between the final state and the initial state is negative. Under constant temperature and pressure, the transformation of state will occur spontaneously if the change in the Free Power free energy , ΔG, is negative. Phase transitions provide instructive examples, as when ice melts to form water at 0. 01 °C (T = Free energy. Free energy K), with the solid and liquid phases in equilibrium. Then ΔH = Free Power. Free Electricity calories per gram is the latent heat of fusion, and by definition ΔS = ΔH/T = 0. Free Power calories per gram∙K is the entropy change. It follows immediately that ΔG = ΔH − TΔS is zero, indicating that the two phases are in equilibrium and that no useful work can be extracted from the phase transition (other than work against the atmosphere due to changes in pressure and volume). Free Power, ΔG is negative for T > Free energy. Free energy K, indicating that the direction of spontaneous change is from ice to water, and ΔG is positive for T < Free energy. Free energy K, where the reverse reaction of freezing takes place.
Building these things is easy when you find the parts to work with. That’s the hard part! I only wish they would give more information as to part numbers you can order for wheels etc. instead of scrounging around on the internet. Wire is no issue because you can find it all over the internet. I really have no idea if the “magic motor” as you call it is possible or not. Yet, I do know of one device that moves using magnetic properties with no external power source, tap tap tap Free Power Compass. Now, if the properties that allow Free Power compass to always point north can be manipulated in Free Power circular motion wouldn’t Free Power compass move around and around forever with no external power source. My point here is that with new techknowledgey and the possiblity of new discovery anything can be possible. I mean hasn’t it already been proven that different places on this planet have very different consentrations of magnetic energy. Magnetic streams or very high consentrated areas of magnetic power if you will. Where is there external power source? Tap Tap Tap Mie2centsHarvey1Thanks for caring enough to respond! Let me address each of your points: Free Power. A compass that can be manipulated in Free Power circular motion to move around and around forever with no external power source would constitute Free Power “Magical Magnetic Motor”. Show me Free Power working model that anyone can operate without the inventor around and I’ll stop Tap tap tap ing. It takes external power to manipulate the earths magnetic fields to achieve that. Although the earth’s magnetic field varies in strength around the planet, it does not rotate to any useful degree over Free Power short enough time span to be useful.
The torque readings will give the same results. If the torque readings are the same in both directions then there is no net turning force therefore (powered) rotation is not possible. Of course it is fun to build the models and observe and test all of this. Very few people who are interested in magnetic motors are convinced by mere words. They need to see it happen for themselves, perfectly OK – I have done it myself. Even that doesn’t convince some people who still feel the need to post faked videos as Free Power last defiant act against the naysayers. Sorry Free Power, i should have asked this in my last post. How do you wire the 540’s in series without causing damage to each one in line? And no i have not seen the big pma kits. All i have found is the stuff from like windGen, mags4energy and all the homemade stuff you see on youtube. I have built three pma’s on the order of those but they don’t work very good. Where can i find the big ones? Free Power you know what the 540 max watts is? Hey Free Power, learn new things all the time. Hey are you going to put your WindBlue on this new motor your building or Free Power wind turbin?
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.

And solar panels are extremely inefficient. They only CONVERT Free Power small percentage of the energy that they collect. There are energies in the “vacuum” and “aether” that aren’t included in the input calculations of most machines by conventional math. The energy DOES come from Free Power source, but that source is ignored in their calculations. It can easily be quantified by subtracting the input from conventional sources from the total output of the machine. The difference is the ZPE taken in. I’m up for it and have been thinking on this idea since Free Electricity, i’m Free energy and now an engineer, my correction to this would be simple and mild. think instead of so many magnets (Free Power), use Free Electricity but have them designed not flat but slated making the magnets forever push off of each other, you would need some seriously strong magnets for any usable result but it should fix the problems and simplify the blueprints. Free Power. S. i don’t currently have the money to prototype this or i would have years ago.
NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!

Although I think we agree on the Magical Magnetic Motor, please try to stick to my stated focus: — A Magnetic Motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets and that can operate outside the control of the Harvey1 kimseymd1 Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! @Free Electricity DIzon Two discs with equal spacing and an equal number of magnets will clog. Free Electricity place two magnets on your discs and try it. Obviously you haven’t. That’s simple understanding. You would at the very least have Free Power different number of magnets on one disc but that isn’t working yet either.
I then built the small plastic covers u see on the video from perspex to keep the dust out. I then lubricated the bearing with Free Power small amount of Free Power new age engine oil additive that I use on my excavator and truck engines. Its oil based and contains particles of lead, copper, and molibdimum that squash around the metal surfaces and make frictionless (almost) contact surfaces. Geoff, your patience is exceptional. I’m glad you stick it out. Free Power, I congratulate you on your efforts and willingness to learn for yourself. All of this reminds me of my schooling. Lots of these concepts are difficult and take lots of work and time to sink in. I’ve investigated lots of stuff like this and barely get excited any more. I took Free Power look at your setup. You’ve done well. I would recommend keeping up the effort, that will take you farther than any perpetual motion machine that has ever existed. Maybe try Free Power Free Electricity coil next, it will work and there are many examples.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the Free Electricity physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the Free Energy physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy ’ for the expression E − TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions, specifically constant temperature. [Free Electricity]:Free Power.


text:
## Precalculus (6th Edition) Blitzer

vertical asymptote $x=3$ no horizontal asymptote. slant asymptote $y=x+5$ See graph.
Step 1. Rewrite the function as $y=\frac{x^2+2x-3}{x-3}=\frac{x^2-3x+5x-15+12}{x-3}=x+5+\frac{12}{x-3}$ Step 2. The vertical asymptote can be found as $x=3$ Step 3. There is no horizontal asymptote. Step 4. A slant asymptote can be found as $y=x+5$ Step 5. See graph.


text:
# How to access an object value using variable key in JavaScript?

JavascriptObject Oriented ProgrammingFront End Technology

We know that an object value can be accessed by using a Dot notation or Bracket notation. But we can also access the value using a variable key. Let's look over them in a nutshell.

## Using Dot and Bracket Notations

### Example

In the following example, object value is accessed using Dot and Bracket notation. Using a bracket notation is nothing but using a string key.

Live Demo

<html>
<body>
<script>
let me = {
name: 'javascript'
};
document.write(me.name);
document.write("</br>");
document.write(me['name']);
</script>
</body>
</html>

### Output

javascript
javascript

## Using Variable Key

### Example

In the following example, instead of Dot and Bracket notations, a variable key is used to access the value of an object.

Live Demo

<html>
<body>
<script>
let me = {
name: 'javascript'
};
let key = 'name'
document.write(me[key]);
</script>
</body>
</html>

### Output

javascript
Published on 30-Jul-2019 08:51:57


text:
# Surface charge distribution of two metal spheres

1. Nov 19, 2008

### jdstokes

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

A total charge Q is shared by two metal spheres of small radii R1 and R2, that are connected by a long thin wire of length L. Fin (a) the charge on each sphre and (b) the tension in the wire.

Source: Haliday 4th edn chapter 26 q. 91, p. 738

3. The attempt at a solution

I'm a bit confused by this. I assume that since the potential must be uniform for a conductor, that we can make the equation

$\frac{rQ}{4\pi\epsilon_0 R_1} = \frac{(1-r)Q}{4\pi\epsilon_0 R_2}$

from which we obtain

$Q_1 = \frac{Q}{1 + R_2/R_1}$.

But I suppose this is only valid if the length of L is sufficiently large that we can ignore the interaction energy of the two spheres right?

2. Nov 19, 2008

### jdstokes

Perhaps I am correct, and merely

$\mathrm{Tension} = \frac{Q_1 Q_2}{4\pi \epsilon_0 (L + R_1 +R_2)^2}$?

3. Nov 19, 2008

### Redbelly98

Staff Emeritus
You're correct for both parts.

However, they probably want tension in terms of the given parameter Q, not in terms of Q1 and Q2.


text:
# How often do supernovae occur?

Type Ia supernova are used as standard candles. But they also are transient events. This means that to determine the distance of a galaxy using supernovae, you have to wait for one to occur. How often do type Ia supernovae typically occur in a galaxy ?

• Some things really..... en.wikipedia.org/wiki/History_of_supernova_observation – Carl Witthoft Jan 21 at 18:49
• Don't think the Wikipedia article answers the OP's question, it seems to mostly discuss very local supernovae. I read it as asking for the supernova rate in galaxies generally (which is a function of redshift, metallicity and galaxy type) rather than in the Milky Way specifically – astrosnapper Jan 21 at 19:48

There is a lot of scope to provide a very detailed answer here. The rate depends on what sort of a galaxy you are considering and when, what its star formation rate is (or was) and what its total stellar mass is.

A good reference is the Annual Review of Astrophysics article by Maoz et al. (2019). This says that for a Sbc galaxy like the Milky Way, the specific rate ( the rate per unit of stellar mass in the galaxy) of type Ia supernovae is $$10^{-13}$$ per year per solar mass of stars. For a Milky Way stellar mass of $$\sim 6\times 10^{10} M_\odot$$, this means a rate of 0.006 per year, or one type Ia SNe every $$\sim 200$$ years.

In the local universe this rate scales as about $$M^{-0.5}$$, where $$M$$ is the galaxy mass (valid over about 5 orders of magnitude for $$M>10^7 M_\odot$$; Brown et al. 2018). So the specific rate of type Ia supernovae is larger in smaller galaxies.

The rate also decays after a galaxy is formed, or after a burst of star formation as something like $$t^{-1}$$.

• This answer hurt my brain with all the math and astronomy going on here, especially when contemplating that we have actually calculated this sort of thing and can use it to calculate subsequent events. I like it. – Doktor J Jan 22 at 4:38
• Interestingly, nasa.gov says "Astronomers believe that about two or three supernovas occur each century in galaxies like our own Milky Way." – Victor Stone Jan 22 at 23:53
• @VictorStone the rate of type Ia supernovae must of course be smaller than the rate of (all) supernovae And is probably smaller than the rate of core collapse (type II) supernovae. – Rob Jeffries Jan 23 at 0:03

The rate is evolving with time (and consequently with redshift). According to a paper in 2005 by Evan Scannapieco & Lars Bildsten (https://arxiv.org/abs/astro-ph/0507456), the rate of SNe Ia is 1 to 3.5×104 per yr per Mpc3 at z=2, which is the peak rate.

You can very roughly consider it as 1 to 3.5 events in a galaxy in 10,000 years. Considering there are so many galaxies in our observable universe, you can still observe some in each year. With more powerful telescopes and strategies, there will more and more SNe Ia be observed. They are dimmer, but more. Probably, more than once per day.

I have a much different answer. I'm taking a course in Astrophysics and the instructors took the approach of using the amount of iron in our galaxy to calculate the number of SN since galaxy was formed. They found how many SN it would take to produce that much iron. According to their calculations you need about 1 SN every 50 years from the Milky Way.

• I don't think that's "much different" from the other answers. Can you show the calculation? – pela Jan 23 at 20:11
• This is indeed a valid and more direct cross-check of results for the Milky Way (since we have not been able to observe enough Type Ia supernovae in the MW to count directly). However, could you give a reference for the calculation and be clear whether this results applies to Type Ia supernovae (which is what the question is asking about) or all supernovae. Another point is that this presumably then is an average rate over the lifetime of the Galaxy? If so, the current rate is likely to be significantly lower. – Rob Jeffries Jan 23 at 21:06
• The course is a series of video lectures. they are located here: courses.edx.org/courses/course-v1:ANUx+ANU-ASTRO1x+2T2016/…. This links takes you to the video before the one of interest. So scroll to the top of page and select the next lecture. – jmh Jan 24 at 18:46
• The calculation is for type II sn which they state accounts for 1/3 of all the iron, and yes they use the lifetime of the universe. – jmh Jan 24 at 18:54
• So @jmh, it isn't an answer to this question. – Rob Jeffries Jan 25 at 17:57


text:
# Entropy of surroundings and heat

Apparently the entropy change of a system can be found by

$dS=\frac{\delta q_{rev}}{T}$

I am slightly confused about this because I have seen this being used when the temperature of the system and the surroundings are not the same. However for the heat exchange to be reversible between the system and surroundings, need not the temperature of the system and surroundings be the same?

I have a feeling this might have to do with reversibility with respect to the system and the surroundings. I do not know about this and had not heard of it before I read it in a response to this post:

"So if the actual process is irreversible, the irreversibility (entropy generation) is all considered to take place in the system, rather than in the surroundings (in the case of an ideal infinite reservoir)"

Could someone please elaborate on this?

• The temperature of the system and surroundings always match at their interface, irrespective of whether the process is reversible. I would be interested in seeing an example of when that equation is used in a case when the temperature of the system away from the interface is different from the temperature of the surroundings. – Chet Miller Feb 23 '17 at 13:30

Yes, that is a perfectly acceptable formula for entropy change even outside of thermal equilibrium.

And, as you say, it's related to the irreversibility of heat transfer. The point about ideal thermal reservoirs always exchanging heat reversibly is not necessary to understand this, so let's park it outside for a moment.

Let's imagine a body in an environment. We'll call the temperature of the body $T_b$ and the temperature of the environment $T_e$. If they exchange heat, their respective entropies, $S_b$ and $S_e$, change according to the formula you mentioned:

$dS_b=\cfrac{\delta q}{T_b}$

$dS_e=\cfrac{- \delta q}{T_e}$

Note the sign (the heat flowing into the body is leaving the environment, or vice versa).

Also note, and this is critical, that the magnitude of $\delta q$ is equal in both equations (one receives exactly the same amount of heat as the other loses), but, if $T$ is different, the entropy change won't be the same. The change in entropy (positive or negative) will depend on the temperature of the body or the environment.

Let's have a look at how the total entropy of the system ($S=S_b+S_e$) changes.

If the body and the environment are at thermal equilibrium, in other words, their temperature is the same ($T_b=T_e=T$),

$dS=dS_b+dS_e=\cfrac{\delta q}{T}-\cfrac{\delta q}{T}=0$

In other words, transferring heat between bodies at thermal equilibrium does not change the total entropy of the system.

However, if they are at different temperatures,

$dS=\cfrac{\delta q}{T_1}-\cfrac{\delta q}{T_1}=\delta q \left( \cfrac{1}{T_1}-\cfrac{1}{T_2}\right)$

Now, the sign of the entropy change depends on the relative values of $T_1$ and $T_2$:

If $T_1>T_2 \rightarrow 1/T_1 < 1/T_2 \rightarrow 1/T_1 - 1/T_2 < 0 \rightarrow dS < 0$

If $T_1<T_2 \rightarrow 1/T_1 > 1/T_2 \rightarrow 1/T_1 - 1/T_2 > 0 \rightarrow dS > 0$

Note that I chose the signs of $\delta q$ assuming that the flux of heat went from the environment to the body. So, this means that if the body is hotter than the environment ($T_1 > T_2$), this would imply a reduction in the entropy of the whole system ($dS < 0$) - which the second principle explicitly forbids. And, to the contrary, if the body is colder than the environment, that flux will be spontaneous ($dS > 0$).

So, outside of thermal equilibrium, you still can use that expression to calculate entropy changes - as long as the heat flux goes in the right direction, from hot to cold. It will also apply to heat transfers in equilibrium (and therefore reversible) - although in that case you'll be having some other energy transfer process of equivalent magnitude siphoning that energy off - otherwise the heat would increase the temperature of part of the system, breaking thermal equilibrium (that won't happen).

That formula is used when you have the $Q_{reversible}$ for a system. However, most of the time you don't have $Q_{reversible}$ for the system, but always remember that the $Q$ of the system is actually $-Q_{reversible}$ for the surroundings. So therefore you can use that formula to always find the entropy of the surroundings!


text:
# Question

Jessica Porter works in both the jewelry department and the hosiery department of a retail store. Porter assists customers in both departments and arranges and stocks merchandise in both departments. The store allocates Porter’s \$ 30,000 annual wages between the two departments based on a sample of the time worked in the two departments. The sample is obtained from a diary of hours worked that Porter kept in a randomly chosen two- week period. The diary showed the following hours and activities spent in the two departments. Allocate Porter’s annual wages between the two departments.
Selling in jewelry department . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 hours
Arranging and stocking merchandise in jewelry department . . . . . . . . . . . . . . . . . . 6 hours
Selling in hosiery department . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 hours
Arranging and stocking merchandise in hosiery department . . . . . . . . . . . . . . . . . . . 7 hours
Idle time spent waiting for a customer to enter one of the selling departments . . . . 4 hours

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text:
## Wednesday, March 18, 2015

As I am researching the π, I find interesting ways ancient indian researchers described their findings. One such methodology used is Katapayadi System of verses. It is basically an system of code so that things can be defined in a way so that people can remember. The code is as follows.

 1 2 3 4 5 6 7 8 9 क ख ग घ ङ च छ ज झ ट ठ ड ढ ण त थ द ध प फ ब भ म य र ल व श ष स ह क्ष
With the above key in place, Sri Bharathi Krishna Tirtha in his Vedic Mathematics gives following verse.
गोपी भाग्य मधुव्रात  श्रुङ्गिशो दधिसन्धिग  |
खलजीवित खाताव गलहालारसंधार |
If we replace the code from the above table in the above verse, here is what we get.
31 41 5926 535 89793
23846 264 33832792
That gives us $\frac{π}{10} = 0.31415926535897932384626433832792$

Apprantly this methodology of remembering digits of π has a name. It is called Piphilology.
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. The word is a play on the word "pi" itself and of the linguistic field of philology.


text:
# Writing an Equation to Represent a Proportional Relationship Online Quiz

Following quiz provides Multiple Choice Questions (MCQs) related to Writing an Equation to Represent a Proportional Relationship. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using Show Answer button. You can use Next Quiz button to check new set of questions in the quiz.

Q 1 - Write an equation to represent the proportional relationship given in the table.

 x 5 6 8 15 20 y 20 24 32 60 80

### Explanation

Step 1:

From given table of values

$\frac{y}{x} = \frac{20}{5} = \frac{24}{6} = \frac{32}{8}… = \frac{4}{1}$

Step 2:

So, the equation representing this proportional relationship is $y = \frac{4}{1} \times \frac{x}{1} = \frac{4x}{1} = 4x$

or $y = 4x$

Q 2 - Write an equation to represent the proportional relationship given in the table.

 m 6 12 18 22 24 n 9 18 27 33 36

### Explanation

Step 1:

From given table of values

$\frac{n}{m} = \frac{9}{6} = \frac{18}{12} = \frac{27}{18}…= \frac{3}{2}$

Step 2:

So, the equation representing this proportional relationship is $n = \frac{3}{2} \times \frac{m}{1} = \frac{3m}{2}$

or $n = \frac{3m}{2}$

Q 3 - Write an equation to represent the proportional relationship given in the table.

 c 3 9 15 21 33 d 4 12 20 28 44

### Explanation

Step 1:

From given table of values

$\frac{d}{c} = \frac{4}{3} = \frac{12}{9} = \frac{20}{15}… = \frac{4}{3}$

Step 2:

So, the equation representing this proportional relationship is $d = \frac{4}{3} \times \frac{c}{1} = \frac{4c}{3}$

or $d = \frac{4c}{3}$

Q 4 - Write an equation to represent the proportional relationship given in the table.

 k 3 12 15 27 36 l 7 28 35 63 84

### Explanation

Step 1:

From given table of values

$\frac{l}{k} = \frac{7}{3} = \frac{28}{12} = \frac{35}{15}… = \frac{7}{3}$

Step 2:

So, the equation representing this proportional relationship is $l = \frac{7}{3} \times \frac{k}{1} = \frac{7k}{3}$

or $l = \frac{7k}{3}$

Q 5 - Write an equation to represent the proportional relationship given in the table.

 y 6 18 24 39 48 z 14 42 56 91 112

### Explanation

Step 1:

From given table of values

$\frac{y}{z} = \frac{14}{6} = \frac{42}{18} = \frac{56}{24}… = \frac{7}{3}$

Step 2:

So, the equation representing this proportional relationship is $y = \frac{7}{3} \times \frac{z}{1} = \frac{7z}{3}$

or $y = \frac{7z}{3}$

Q 6 - Write an equation to represent the proportional relationship given in the table.

 a 5 7 8 9 11 b 15 21 24 27 33

### Explanation

Step 1:

From given table of values

$\frac{b}{a} = \frac{15}{5} = \frac{21}{7} = \frac{24}{8}… = \frac{3}{1}$

Step 2:

So, the equation representing this proportional relationship is $b = \frac{3}{1} \times \frac{a}{1} = \frac{3a}{1} = 3a$

or b = 3a

Q 7 - Write an equation to represent the proportional relationship given in the table.

 p 6 10 13 14 18 q 18 30 39 42 54

### Explanation

Step 1:

From given table of values

$\frac{q}{p} = \frac{18}{6} = \frac{30}{10} = \frac{39}{13}… = \frac{3}{1}$

Step 2:

So, the equation representing this proportional relationship is $q = \frac{3}{1} \times \frac{p}{1} = \frac{3p}{1} = 3p$

or q = 3p

Q 8 - Write an equation to represent the proportional relationship given in the table.

 r 10 20 30 40 50 s 6 12 18 24 30

### Explanation

Step 1:

From given table of values

$\frac{s}{r} = \frac{6}{10} = \frac{12}{20} = \frac{18}{30}… = \frac{3}{5}$

Step 2:

So, the equation representing this proportional relationship is $s = \frac{3}{5} \times \frac{r}{1} = \frac{3r}{5}$

or $s = \frac{3r}{5}$

Q 9 - Write an equation to represent the proportional relationship given in the table.

 i 10 20 30 40 50 j 8 16 24 32 40

### Explanation

Step 1:

From given table of values

$\frac{j}{i} = \frac{8}{10} = \frac{16}{20} = \frac{24}{30}… = \frac{4}{5}$

Step 2:

So, the equation representing this proportional relationship is $j = \frac{4}{5} \times \frac{i}{1} = \frac{4i}{5}$

or $j = \frac{4i}{5}$

Q 10 - Write an equation to represent the proportional relationship given in the table.

 u 2 16 24 32 40 v 7 56 84 112 140

### Explanation

Step 1:

From given table of values

$\frac{v}{u} = \frac{7}{2} = \frac{56}{16} = \frac{84}{24}… = \frac{7}{2}$

Step 2:

So, the equation representing this proportional relationship is $v = \frac{7}{2} \times \frac{u}{1} = \frac{7u}{2}$

or $v = \frac{7u}{2}$

writing_equation_represent_proportional_relationship.htm


text:
# Think Positive

Algebra Level 3

$\large{\begin{cases} a + 2b -12 > 0 \\ 3a - b -17 > 0 \\ 80 - 9a -4b > 0 \end{cases}}$

Given that $$a$$ and $$b$$ are integers satisfying system of the inequalities above, compute $$a^2 + b^2$$.

×


text:
User frederic palesi - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:01:53Z http://mathoverflow.net/feeds/user/21084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87354/flips-of-triangulations-on-non-orientable-surfaces Flips of triangulations on non-orientable surfaces Frederic Palesi 2012-02-02T18:26:05Z 2012-02-24T22:14:51Z
Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures.

I'm looking at ideal triangulations of the surface, namely maximal collection of pairwise disjoint ideal arcs (joining punctures) on the surface.

Given a triangulation and an arc $\alpha$ in that triangulation, we can perform a flip along $\alpha$ : the arc belongs to two triangles forming a quadrilateral, the flip along $\alpha$ gives the new triangulation where the arc has been replaced by the other diagonal of the quadrilateral, and all the other arcs remains unchanged. (When the arc belongs to only one triangle, we cannot do the flip.)

A classical result in the orientable case states that given any two triangulations of an orientable surface, there is a sequence of flips that transform one into the other (Hatcher, On triangulations of surfaces). So the so-called triangulation graph, whose vertices are triangulations and edges are flips, is connected.

I am looking for the similar result for non-orientable surfaces. Does anyone have a reference ? My guess is that the result should also hold but I did not find anything on the litterature.
 http://mathoverflow.net/questions/87354/flips-of-triangulations-on-non-orientable-surfaces/89449#89449 Comment by Frederic Palesi Frederic Palesi 2012-02-27T11:29:31Z 2012-02-27T11:29:31Z Thank you very much for this detailed answer.


text:
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 Trudy MIAN: Year: Volume: Issue: Page: Find

 Tr. Mat. Inst. Steklova, 2019, Volume 304, Pages 252–256 (Mi tm3970)

Singular Sets of Extremal Controls in Optimal Control Problems

M. S. Nikol'skii

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: Some optimal control problems are considered with an integral functional (to be minimized), fixed motion time of the controlled object, and free right endpoint. In these problems, the set of degeneracy points of the Pontryagin maximum principle (singular set) for the extremal control is studied. For a wide class of linear control systems, sufficient conditions are obtained under which the singular set is either empty or consists of a finite set of points. In addition, an example of a control system is constructed in which the singular set has a very general form.

DOI: https://doi.org/10.4213/tm3970

Full text: PDF file (144 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 304, 236–240

Bibliographic databases:

UDC: 517.977
Revised: October 24, 2018
Accepted: December 24, 2018

Citation: M. S. Nikol'skii, “Singular Sets of Extremal Controls in Optimal Control Problems”, Optimal control and differential equations, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin, Tr. Mat. Inst. Steklova, 304, Steklov Math. Inst. RAS, Moscow, 2019, 252–256; Proc. Steklov Inst. Math., 304 (2019), 236–240

Citation in format AMSBIB
\Bibitem{Nik19} \by M.~S.~Nikol'skii \paper Singular Sets of Extremal Controls in Optimal Control Problems \inbook Optimal control and differential equations \bookinfo Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin \serial Tr. Mat. Inst. Steklova \yr 2019 \vol 304 \pages 252--256 \publ Steklov Math. Inst. RAS \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3970} \crossref{https://doi.org/10.4213/tm3970} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3951623} \elib{http://elibrary.ru/item.asp?id=37461011} \transl \jour Proc. Steklov Inst. Math. \yr 2019 \vol 304 \pages 236--240 \crossref{https://doi.org/10.1134/S0081543819010164} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000470695400016} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066784752}


text:
# A uniform characterization of the octonions and the quaternions using commutators

Erwin Kleinfeld, Yoav Segev

Research output: Working paper/PreprintPreprint

## Abstract

Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We prove that $R$ has no zero divisors. Furthermore, if $\text{char}(R)\ne 2,$ then the localization of $R$ at its center is an octonion division algebra, if $R$ is alternative and a quaternion division algebra, if $R$ is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.
Original language English Published - 8 Dec 2021

## Keywords

• math.RA
• math.GR
• Primary: 12E15, 17D05, Secondary: 11R52, 17A35

## Fingerprint

Dive into the research topics of 'A uniform characterization of the octonions and the quaternions using commutators'. Together they form a unique fingerprint.


text:
Advent of Code 2021 in Kotlin - Day 11

Introduction

The Day 11 problem requires form us writing some simulation of flashing in described manner. We can implement this in some readable way by using custom data structures and redefining some operators in Kotlin - let’s begin and see how to express some complicated code in a few lines of text.

Solution

We defined the operators for modifying internal values of the map to get extra functionality by calculating posOf i.e. the map from values to the positions where we can find every of them on the map. Thanks to this function we can always ask about the places with energy equal to 10 (in constant time) and propagate the energy from them in single flash execution.

Let’s notice that we defined the flash function as tailrec fun so it is optimized by the compiler to standard loop. We did that because it’s usually more readable to see that was our intention when writing this code - we want to mark current positions as flashed, then for every of them propagate the energy to its neighbours and then try to flash again from the positions on which the maxVal appeared in current step of flashing.

It’s worth mentioning also how we defined the LazyDefaultMap<K, V> (and modified DefaultMap<K, V>) when solving this task. Now we ended up with the definition, that allows to compute the default value lazily, but also it sets the computed value to the backing map. It seemed to be more reasonable approach as we think about the value extracted from map as it would be in it - so when it’s a mutable value, then after its modifications, we should observe this modification in map.

class LazyDefaultMap<K, V>(
private val default: () -> V,
private val map: MutableMap<K, V> = HashMap()
) : MutableMap<K, V> by map {
override fun get(key: K): V = map.getOrDefault(key, default()).also { map[key] = it }
}

Don’t miss the definitions that uses Sequence<T> in today code - they both were defined to improve code readability and keep the performance in my opinion - let me know what you think about such “functional” approaching to the problems 😉.

Day11.kt

object Day11 : AdventDay() {
override fun solve() {
val map = reads<String>()?.toEnergyMap() ?: return

with(map.copy()) {
(1..100).fold(0) { sum, _ -> sum + simulateStep() }.printIt()
}
with(map.copy()) {
generateSequence(1, Int::inc).first { simulateStep() == 100 }.printIt()
}
}
}

private fun List<String>.toEnergyMap() =
EnergyMap(maxVal = 10, map { line -> line.map { it.digitToInt() }.toMutableList() })

private data class Pos(val x: Int, val y: Int)

private data class EnergyMap(val maxVal: Int, private val values: List<MutableList<Int>>) {

val indices = values.flatMapIndexed { y, row -> row.indices.map { Pos(it, y) } }
private val posOf = LazyDefaultMap(::mutableSetOf,
indices.groupBy { this[it] }.mapValues { it.value.toMutableSet() }.toMutableMap()
)

fun copy() = EnergyMap(maxVal, values.map { it.toMutableList() })
operator fun get(p: Pos): Int = with(p) { values[y][x] }
operator fun set(p: Pos, v: Int) {
val newVal = v.coerceAtMost(maxVal)
posOf[this[p]].remove(p)
values[p.y][p.x] = newVal
}

fun neighbours(of: Pos) = sequence {
for (x in -1..1) for (y in -1..1) yield(Pair(x, y))
}
.filterNot { (x, y) -> x == 0 && y == 0 }
.map { (x, y) -> Pos(of.x + x, of.y + y) }
.filter { it.isValid() }

fun simulateStep(): Int {
val flashed = mutableSetOf<Pos>()
tailrec fun flash(flash: Set<Pos>) {
flashed += flash
flash.toList().forEach { pos -> neighbours(pos).forEach { this[it] = this[it] + 1 } }
if (flash.isNotEmpty()) flash(posOf[maxVal] - flashed)
}
indices.forEach { this[it] = this[it] + 1 }
flash(posOf[maxVal])
indices.forEach { this[it] = this[it] % maxVal }
return flashed.size
}

private fun Pos.isValid() = y in values.indices && x in values[y].indices
}

Extra notes

Let’s see that we solved each of the parts of the task on the copy of the input map. That’s because the map internal state is mutable (it changes during the simulation) but we need to have a fresh map when trying to solve the second part of the problem.

We used with Kotlin construct here, that simplified our code and introduced the context of current copy of EnergyMap. That’s one of the really cool Kotlin constructs that is in fact an inline function, so it brings no extra overhead on runtime while really simplifying our code.

Also in this day we introduced tests running with expected outputs of days. We used for it some tricky and simple function

fun catchSystemOut(action: () -> Unit) = ByteArrayOutputStream().also {
val originalOut = System.out
System.setOut(PrintStream(it))
action()
System.setOut(originalOut)
}.toString()

which is capable of catching the System.out value and returning it as simple String (that is later cmopared with the expected output in tests and Advent days are defined to print their solutions).

Student of Computer Science

My interests include robotics (mainly with Arduino), mobile development for Android (love Kotlin) and Java SE/EE applications development.


text:
Lemma 28.24.1. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be an open subscheme. The following are equivalent:

1. $U$ is retrocompact in $X$,

2. $U$ is quasi-compact,

3. $U$ is a finite union of affine opens, and

4. there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ such that $X \setminus U = V(\mathcal{I})$ (set theoretically).

Proof. The equivalence of (1), (2), and (3) follows from Lemma 28.2.3. Assume (1), (2), (3). Let $T = X \setminus U$. By Schemes, Lemma 26.12.4 there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subscheme structure on $T$. Note that $\mathcal{J}|_ U = \mathcal{O}_ U$ which is an $\mathcal{O}_ U$-modules of finite type. By Lemma 28.22.2 there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_ U = \mathcal{J}|_ U$. Then $X \setminus U = V(\mathcal{I})$ and we obtain (4). Conversely, if $\mathcal{I}$ is as in (4) and $W = \mathop{\mathrm{Spec}}(R) \subset X$ is an affine open, then $\mathcal{I}|_ W = \widetilde{I}$ for some finitely generated ideal $I \subset R$, see Lemma 28.16.1. It follows that $U \cap W = \mathop{\mathrm{Spec}}(R) \setminus V(I)$ is quasi-compact, see Algebra, Lemma 10.29.1. Hence $U \subset X$ is retrocompact by Lemma 28.2.6. $\square$

There are also:

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