text:
## Introduction

Thermally activated delayed fluorescence (TADF) organic molecules convert electricity to light more efficiently than traditional organic fluorophores, making them attractive materials for organic light-emitting diodes (OLEDs)1,2,3,4,5,6. Spin statistics for OLEDs dictate that 75% of electrically-generated excitons populate triplet (spin = 1) states (T1), which, in the case of traditional fluorophores, limits electroluminescence (EL) efficiency since radiative de-excitation of these states to the singlet (spin = 0) ground state (S0) is spin-forbidden7,8. Current commercial OLED devices employ organometallic chromophores based on iridium and platinum as red and green emitters. In such materials, enhanced spin–orbit coupling from the heavy element promotes direct emission from triplet excitons (phosphorescence), allowing up to 100% internal quantum efficiency (IQE) for EL9,10,11. Blue emitters based on the same approach have remained elusive, particularly because the long lifetimes of high-energy triplet excitons lead to OLED degradation12,13. Consequently, blue OLEDs typically employ organic fluorophores, with an enhancement of EL IQE achieved using triplet-triplet annihilation (TTA, T1 + T1 → S1 + S0), which can harvest up to half of the triplet excitons for light emission via singlets14,15,16. TTA-based OLEDs can in principle achieve up to 62.5% EL IQE. Within this context, TADF emitters present an alternative approach to achieving 100% EL IQE in devices, thereby offering a step-change in performance for blue OLED technologies17,18.

In TADF devices, indirect emission from dark triplet states via bright singlet excitons (T1 → S1 → S0) is activated by ambient thermal energy. There is general agreement that TADF emitters should be designed to have small singlet-triplet exchange energies that promote rapid forward and reverse intersystem crossing (ISC) between singlet and triplet excitons; such small exchange energies can be engineered by spatial separation of the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO)1,2,3,4,5,6. However, there is no agreement on the spin interactions and spin-flip mechanisms responsible for the singlet-triplet interconversion in fluorophores exhibiting TADF with intramolecular charge-transfer (CT) states19,20,21,22. Typical TADF emitters have low-lying singlet and triplet excitonic states with mixed local excitation (LE) and intramolecular CT character21,23. Emission from the predominantly CT singlet excitonic state (1CT) via prompt or delayed fluorescence is desirable. Therefore, the triplet excitons populating either CT or local excitation states (3CT and 3LE) must be efficiently converted to 1CT. ISC between the singlet and triplet manifolds of fluorophores can be driven by a spin-orbit coupling (SOC) mediated spin conversion, whereby an orbital transition between two states generates a torque capable of flipping an electron’s spin. Direct SOC (dependent only on the electronic character of the coupled states) will in general be suppressed between a singlet and triplet state if they possess similar orbital character (El Sayed’s Rule)24,25. Therefore, in order to achieve singlet-triplet ISC between two electronic states of the same orbital character (e.g. 1CT to 3CT) additional processes must occur, which are dependent on both the electronic character and nuclear coordinates of the coupled states. However, the presence of a significant change in orbital character upon spin-flip (e.g. 1CT to 3LE) is not in itself sufficient to guarantee rapid ISC and, furthermore, it has been shown that direct SOC alone cannot explain high rates of ISC observed26,27,28. Vibration-mediated SOC has been suggested to explain this discrepancy, and also to facilitate singlet-triplet conversion between states of similar orbital character18,29. Vibrational contributions from an additional, or ‘intermediate’, triplet state give rise to the ‘spin-vibronic mechanism’ for the efficient conversion of dark triplet excitons to emissive singlet excitons (reverse ISC) in TADF emitters30,31,32. Revealing the character and formation dynamics of intermediate triplet states is challenging, although critical to understanding the TADF mechanism33,34.

Here we use transient electron spin resonance (trESR) spectroscopy to study the role of spin-orbit coupling and vibrational perturbations in ISC for TADF. While optical spectroscopy has proved a powerful tool to examine the photophysics and spin-conversion rates in TADF30,31,35,36, trESR provides a complementary window into the underlying spin physics. TrESR is sensitive to paramagnetic states and can therefore probe the formation dynamics and coupling mechanisms of triplet excitons37, as highlighted by broad applications in studying triplet excitons in organic photovoltaics38,39,40, singlet fission41,42,43,44, photosensitizers37,45,46,47, and molecular electronics48,49. Because trESR is not sensitive to singlet excitons (due to their diamagnetism) it can only explicitly probe forward ISC. However, assuming microscopic reversibility, the triplet exciton formation dynamics revealed are applicable to TADF-relevant reverse ISC. We have previously shown using trESR that spin-orbit coupling interactions mediate ISC for two benchmark TADF molecules: 1,2,3,5-tetrakis(carbazol-9-yl)-4,6-dicyanobenzene (4CzIPN) and 1,2-bis(carbazol-9-yl)-4,5-dicyanobenzene (2CzPN)20. The trESR data suggested a dynamic picture of SOC-mediated ISC involving vibrational modes, although the excited-state ordering and coupling requirements for efficient ISC were not established. To reveal these requirements, here we have conducted trESR studies on a series of ten molecular organic fluorophores exhibiting delayed fluorescence with varying 3LE–3CT energy gaps; this allows us to probe the spin-vibronic mechanism directly and resolve the nature of intermediate triplet states. TrESR gives conclusive evidence that 3LE can be populated from the singlet manifold by the vibrational SOC ISC mechanism. However, when the 3LE–3CT gap is sufficiently small, trESR reveals that additional spin conversion pathways emerge, and 3CT is populated via the spin-vibronic ISC mechanism because of enhanced 3LE–3CT coupling. TrESR can crucially resolve these different ISC mechanisms, which are influenced by difficult to predict molecular properties beyond the excited-state character and energetics measured with optical spectroscopy. Our spin-sensitive measurements reveal direct evidence that distinct triplet excitonic states (T1 + T2) mediate the spin conversion responsible for delayed fluorescence. These findings uncover the fundamental mechanism of the spin-vibronic ISC pathway relevant in TADF emitters, providing detail on the nature of the process by probing distinct yet vibronically-accessible 3LE and 3CT states.

## Results

### Design and detection of multiple triplet excitonic states

Three donor-acceptor emitters sharing a common 4-(3,6-di-tert-butyl-9H-carbazol-9-yl)diphenyl sulfone (DTCz-DPS) core structure were synthesised. We employed inductively electron-withdrawing fluorine substituents on a phenyl group linking carbazolyl and sulfone moieties to lower the LUMO energy of the DTCz-DPS core, resulting in lowered CT state energies as in our previous study50. The three emitters, DTCz-DPS-1, -2, and -3 (Fig. 1b–d) share a fixed 3LE energy level, associated with the carbazole donor, but possess increasingly red-shifted 1CT and 3CT, from DTCz-DPS-1 to -3, consistent with time-dependent density-functional theory (TD-DFT) calculations (Supplementary Table 1), described in ‘Methods’. The calculated Gibbs free energy difference between the 3LE and 3CT minima (the 3LE–3CT gap), in their optimised geometries, is reduced from DTCz-DPS-1 to -3 (Table 1). Importantly, our calculations also found that the direct SOC matrix elements between 1CT and 3LE are relatively constant for the three emitters (Table 1). Therefore, our structural modifications of excited-state energies allow higher-order SOC effects to be studied due to changing of the 3LE–3CT energy gap and energetic landscape, while direct SOC matrix elements between 1CT and 3LE are fixed.

For the trESR measurements, pulsed laser excitation (355 nm) and cryogenic temperatures (30 K) were used to generate long-lived, molecular triplet excitons in dilute (500 μM), deoxygenated and flash-frozen toluene solutions. A variable applied magnetic field (200–500 mT) induced Zeeman energy separation of the triplet sublevels from the zero-field splitting (ZFS) eigenstates (Tx,y,z, Fig. 1a, e). A continuous wave microwave source (~9.7 GHz) induced triplet sublevel transitions (absorptive, A, and emissive, E) when resonant with the sublevel energetic spacing; the intensity of these transitions is dependent on the triplet sublevel population (Px,y,z), which is determined by the nature of the triplet formation (ISC) process. The magnetic field positions of the positive (absorptive) and negative (emissive) peaks in the trESR signal (also known as ‘polarisation pattern turning points’) are determined by the ZFS parameters, D and E (Fig. 1a). The spin–spin dipolar interaction contribution to ZFS can yield a D-dependence on spin-density delocalisation across the molecule that allows for more LE- and CT-type triplets to be distinguished; a smaller D-value typically correlates with greater delocalisation, indicative of a CT state in the molecules investigated here19,51. D, E, and Px,y,z are obtained from inspection, simulation and least-squares fitting (using EasySpin52) of the spin-polarised trESR signal. Following photoexcitation, the all-organic emitters have typically low triplet yield and hence data acquisition times of 4–12 h were required for each trESR spectrum.

Each emitter exhibited spin-polarised trESR signals with lifetimes of several microseconds, throughout which the spectral shapes were conserved. The trESR signal of DTCz-DPS-1 (Fig. 1f) has a 6-turning point EEEAAA polarisation pattern with a preferential population of Tx and Tz (Px,y,z = 0.38, 0.00, 0.62). The ZFS parameters (|D | , | E | = 108, 9 mT), the position of the half-field transition (161 mT, Supplementary Fig. 4), and EEEAAA polarisation pattern of DTCz-DPS-1 are all typical of triplet states in organic chromophores51. These parameters are consistent with the 3LE associated with the carbazole monomer53, leading us to attribute the trESR signal to a triplet exciton localised on the carbazole donor. The trESR signals of DTCz-DPS-2 and -3 (Fig. 1g, h) show spin signatures of two overlapping triplet signals. It is evident from the line shape of the DTCz-DPS-2 trESR spectrum that the wider triplet signal resembles that seen for DTCz-DPS-1. Furthermore, the prominent half-field transition peak in the DTCz-DPS-2 trESR spectrum (162 mT, Supplementary Fig. 4) matches well with that in DTCz-DPS-1. After subtracting the 3LE contribution from the trESR signal of DTCz-DPS-2, the resulting triplet is narrower (|D | , | E | = 23, 3 mT) and has a spin-polarisation pattern that indicates relative overpopulation of Tx and Ty (Px,y,z = 0.4, 0.6, 0.0). A reduced magnitude of D of this scale typically indicates an increased spatial separation of interacting spins; therefore, we attribute the narrow signal to the delocalised 3CT19,51.

The trESR signal of DTCz-DPS-3 (Fig. 1h) comprises a more intense central feature compared to DTCz-DPS-2. The total width of the DTCz-DPS-3 signal is unchanged and the same 161 mT half-field transition peak is recorded as in DTCz-DPS-1 (Supplementary Fig. 4); therefore, the same 3LE is assigned to the broad triplet signal. By subtracting the 3LE contribution from the total DTCz-DPS-3 trESR spectrum a narrower triplet signal ascribed to 3CT is obtained with Px,y,z = 0.6, 0.4, 0.0, |D | , | E | = 56, 19 mT and a half-field transition peak at 169 mT; the breadth of the DTCz-DPS-3 3CT signal compared to the DTCz-DPS-2 3CT signal is a point to which we will later return. These results show that with decreasing 3LE–3CT gap (from DTCz-DPS-1 to -3), the recorded trESR spectra progress from a relatively pure 3LE signal to one resulting from contributions of both 3LE and 3CT polarisation patterns. The triplet ZFS simulation parameters of all three emitters are shown in Table 2.

Similarly, we observe the emergence of unstructured 3CT emission, in addition to structured 3LE emission, with decreasing 3LE–3CT gap in the frozen (77 K) toluene phosphorescence spectra (solid lines in Fig. 2d–f)50,54. We note that the phosphorescence was recorded at 77 K, which is a higher temperature than used in the trESR experiment (30 K). Crucially, both these temperatures are well below the freezing point of toluene (178 K). Therefore, the energetic relaxation of the 1CT states (a common phenomenon undergone by intramolecular CT states in solution) observed when comparing the photoluminescence (PL) in frozen (77 K) and solution (295 K) toluene (Fig. 2a–f) is minimised at both 30 K and 77 K. The progression in trESR and phosphorescence shows that 3LE and 3CT are both populated when the 3LE–3CT gap is sufficiently small55. Time-resolved PL measurements confirmed that all three of the emitters exhibit intramolecular CT character in prompt and delayed fluorescence at room temperature (295 K) in deoxygenated toluene (Fig. 2, Table 1, and Supplementary Fig. 10). The contribution of delayed fluorescence to the total emission was determined from the integrated time-resolved PL and is listed in Table 1. The electroluminescence performance of DTCz-DPS-1 and DTCz-DPS-3 were also evaluated in preliminary solution-processed OLEDs and are presented in the Supplementary Information (Supplementary Fig. 11 and Supplementary Table 7).

### Density-functional theory calculations

Optimised geometries and molecular orbitals were established for the low-lying excitonic states of DTCz-DPS-1, -2, and -3 (Fig. 3a–c) using the range-separated LC-BLYP functional56. The calculations predicted both 3LE and 3CT in close proximity to the 1CT, supporting our experimental observation of both triplets being populated in trESR and PL measurements in frozen toluene. The relative energy and donor-acceptor dihedral angle was calculated in the 3LE, 3CT configurations, and also at the minimal-energy conical intersection where two electronic triplet states are equal in energy and electronically coupled (Fig. 3). The geometrical change between the 3LE and 3CT configuration is predominantly characterised by rotation around the donor-acceptor linker: see dihedral angle between shaded planes in Fig. 3d–f.

The ZFS tensors (D and E parameters) were calculated for each 3LE and 3CT (Table 2) using the DFT-based method reported by Neese et al.57,58 that considers contributions to ZFS from both spin–spin interactions and spin–orbit interactions (Supplementary Table 2). DFT predicted D < 0 in all cases, therefore, D was assumed to be negative in all trESR spectral simulations. The sign of E was chosen such that axes labelling conformed with molecular axes labelling convention, as opposed to ESR convention. The total D and E values calculated for the 3LE configurations are in good quantitative agreement with the values obtained from trESR measurements. The calculated contribution to total D from spin–spin interactions decreases from 3LE to 3CT, whereas the contribution from spin–orbit interactions does not decrease. The resulting 3CT total D and E values are overestimated in comparison to those obtained from trESR measurements, although the experimental observation that the ZFS D-value of DTCz-DPS-3 3CT is larger than DTCz-DPS-2 3CT is correctly predicted. The larger calculated D-value of DTCz-DPS-3 compared to DTCz-DPS-2 arises from a greater spin–spin interaction contribution to ZFS (Supplementary Table 2). Variation in spin–spin interaction can be rationalised by examining the 3CT molecular orbitals in Fig. 3: the electron-withdrawing effect of the additional fluorine atoms in DTCz-DPS-3 localises the LUMO more strongly on the fluorinated phenyl ring than in DTCz-DPS-2 where the LUMO is more delocalised over both phenyl rings. Closer average HOMO-LUMO proximity leads to reduced spin-spin distance and enhanced interaction, which typically manifests as larger D-value.

### Further transient electron spin resonance studies: molecular modifications and concentration series

We designed a number of additional experiments to test our interpretation that the trESR signals comprise polarisation patterns arising from 3LE and 3CT excitonic molecular states. First, we explored the effect of molecular substitutions on excited-state energies and subsequently the photophysics and ISC mechanism. We modified the donor groups of DTCz-DPS-2 and DTCz-DPS-3 by removing the tertiary butyl groups (to give Cz-DPS-1 and -2, respectively), or by replacing the tertiary butyl groups with additional carbazole units (to give 3Cz-DPS-1 and -2, respectively). The 3LE–3CT gap was decreased by fluorination on going from Cz-DPS-1 to −2 and from 3Cz-DPS-1 to -2; we observed with decreasing gap: (a) emergence of the 3CT trESR signals (Fig. 4) and 3CT phosphorescence (Supplementary Fig. 78) and (b) increase in contribution to total PL from delayed fluorescence. These observations are consistent with our findings for DTCz-DPS-1, -2, and -3. Photophysics and trESR simulation parameters can be found in the Supplementary Information (Supplementary Fig. 7, 8, and Supplementary Table 45). Trends in the calculated excited-state energies and direct SOC matrix elements between 1CT and 3LE for Cz-DPS-1 and -2 and 3Cz-DPS-1 and -2 (Supplementary Table 6) are also consistent with the trends identified for DTCz-DPS-1, -2, and -3. In non- or partially-fluorinated versions of the emitters (Cz-DPS-0, DTCz-DPS-0, 3Cz-DPS-0) the CT states were blue-shifted, widening both the 3LE–1CT and 3LE–3CT gaps such that no substantial delayed component of the time-resolved PL was detected at room temperature (Supplementary Fig. 9). The trESR spectra of the fluorescent emitters (Cz-DPS-0, DTCz-DPS-0, and 3Cz-DPS-0) exhibit only EEEAAA polarisation patterns, with signal widths and half-field peaks consistent with the 3LE associated with monomer carbazole, as before (Supplementary Fig. 9). The effects of changing the molecular environment by switching the solvent from toluene to 2-methyltetrahydrofuran were compared for DTCz-DPS-1 and 3Cz-DPS-2 and are discussed in the Supplementary Information (Supplementary Fig. 6). Finally, to rule out intermolecular interactions we reduced the concentration of DTCz-DPS-1, DTCz-DPS-3, and 3Cz-DPS-2 in toluene from 500 μM to 20 μM. No significant differences in trESR signal were observed, supporting our interpretation of the mechanisms responsible for triplet state formation as all being intramolecular processes, rather than concentration-dependent aggregate states or intermolecular energy transfer (Supplementary Fig. 5).

## Discussion

TADF facilitated by spin-orbit coupling, as described in the ‘Introduction’, can be interpreted within the following framework of ISC mechanisms26,59:

1. (I)

Direct spin–orbit coupling between 1CT and 3LE;

2. (II)

Vibrational spin–orbit coupling between 1CT and 3LE;

3. (III)

Spin-vibronic coupling between 1CT and 3CT via intermediate 3LE.

Vibrational modes can enhance ISC in (II) because spin–orbit coupling matrix elements for 1CT–3LE transitions depend on the nuclear degree of freedom, $${Q}_{\alpha }$$. The vibronic coupling in (III) drives rapid internal conversion between 3CT and 3LE, equilibrating their population; 3LE is then coupled to 1CT via either (I) or (II)26,30. 3LE is considered an intermediate in the sequence of state formation: 1CT–3LE–3CT (Fig. 1i). Any initial population of 1LE is assumed to rapidly convert to 1CT, as will be discussed later. Here, the 3CT–3LE vibronic coupling in (III) is assigned to modes associated with the donor–acceptor linker20,60,61. These three ISC processes are expressed together by the sum of their first non-zero terms:

$${\hat{H}}_{{SO}}=\left\langle {\varPsi }_{^1{CT}}|{\hat{H}}_{{SO}}|{\varPsi }_{^3{LE}}\right\rangle$$
(1)
$$+\mathop{\sum}\limits_{\alpha }\frac{\partial \left\langle {\varPsi }_{^1{CT}}|{\hat{H}}_{{SO}}|{\varPsi }_{^3{LE}}\right\rangle }{\partial {Q}_{\alpha }}{Q}_{\alpha }$$
(2)
$$+\frac{\left\langle {\varPsi }_{^1{CT}}|{\hat{H}}_{{SO}}|{\varPsi }_{^3{LE}}\right\rangle \left\langle {\varPsi }_{^3{LE}}|{\hat{H}}_{{vib}}|{\varPsi }_{^3{CT}}\right\rangle }{{E}_{^1{CT}}-{E}_{^3{LE}}}$$
(3)

where $$\varPsi$$ is the molecular wavefunction of states coupled either by the spin–orbit ($${\hat{H}}_{{SO}}$$) or vibronic ($${\hat{H}}_{{vib}}$$) Hamiltonian26. The vibronic coupling between two states generally increases as the energy gap between the states decreases and hence mechanism (III) only significantly contributes to the full spin–orbit interaction if the 3LE–3CT gap (ΔG) is sufficiently small (<0.5 eV). Furthermore, orbital and vibrational mode-dependent mechanisms are determined by the geometry of a molecule and differently affect ISC mechanisms (I), (II), and (III)22,25,26. TrESR polarisation patterns are fingerprints for the relative spin population of the triplet ZFS sublevels (x, y, z), which are also related to the molecular geometry62—Fig. 1 shows how the 3LE ZFS axes are related to the molecular axes (a, b, c) in DTCz-DPS-1, -2, and -3. These relationships mean that triplet formation by each ISC mechanism will yield a particular trESR polarisation pattern that can be distinguished by the ISC mechanism’s orbital and/or vibrational mode dependence on molecular geometry.

TADF emitters generally comprise all-organic electron donor and acceptor units. Individually the donor and acceptor units, on which LE states are localised, typically possess at least a C2 symmetry axis; symmetry does not persist in the delocalised CT states. Under these symmetry conditions the triplet sublevel populations describing the trESR polarisation pattern for each ISC mechanism in a TADF emitter is as follows:

1. (I)

Overpopulation along the C2 symmetry axis (x here);

2. (II)

Overpopulation of the in-plane axes (x and z here);

3. (III)

Redistribution of the population from the in-plane (x and z here) to the out-of-plane (y here) axes.

The difference between ISC mechanisms (II) and (III) is the CT-like transition from 3LE to 3CT upon internal conversion. In a donor–acceptor emitter, the electron density in the 3LE configuration is localised on either the acceptor or, in this case, the donor unit and hence the triplet ZFS (x, y, z) and molecular (a, b, c) axes will be well-aligned. In contrast, the electron density in a 3CT configuration will be delocalised across both the donor and acceptor units (Fig. 3). The donor-acceptor twist, integral to TADF design, results in the rotation of the 3CT ZFS axes relative to the 3LE ZFS axes63. Therefore, the spin-density redistribution upon charge-transfer internal conversion from 3LE to 3CT rotates the triplet ZFS axes such that overpopulation of the out-of-plane axes (relative to the 3LE ZFS axes) can arise. This additional out-of-plane population causes the difference between the population of Tx,y,z in ISC mechanism (II) and (III)64,65. The donor–acceptor twist is a fundamental TADF molecular design rule because the spatial separation of HOMO and LUMO is required to reduce singlet-triplet offset and enable reverse ISC. Therefore, we can expect the relationship between molecular structure, ISC mechanism, and trESR pattern to be applicable to many classes of TADF emitters, as well as to donor–acceptor CT-type molecules in general.

Applying the framework outlined above to the triplet sublevel populations from experimental spectra (Table 2), we can infer the ISC mechanisms responsible for the population of 3LE and 3CT in each emitter. The 3LE states of DTCz-DPS-1, -2, and -3 (and Cz-DPS-1 and -2, 3Cz-DPS-1 and -2) are populated via vibrational SOC with the photoexcited 1CT – mechanism (II) – apparent from the overpopulation of Tx and Tz45,51,62,66,67. On the other hand, the triplet sublevel corresponding to the out-of-plane molecular axis (Ty) is relatively overpopulated in the 3CT polarisation patterns of DTCz-DPS-2 and -3 (and Cz-DPS-2 and 3Cz-DPS-2). The distinct triplet trESR spectra, therefore, reveal that spin-vibronic coupling between 1CT and 3CT via intermediate 3LE – mechanism (III) – is responsible for 3CT population20,59,60. The ability to resolve these ISC mechanisms from one another is a crucial advantage of using trESR; it is not possible to predict how excited states interact based purely on optical measurements of their separation energies or excitation character because additional factors arising from molecular structure affect ISC.

Our computational results qualitatively support the experimental triplet sublevel population assignments of the trESR spectra, and the involvement of vibrational and vibronic enhancement to SOC. First, the calculated $$\left\langle {\varPsi }_{^1{CT}}|{\hat{H}}_{{SO}}|{\varPsi }_{^3{LE}}\right\rangle$$ at the optimised 3LE geometries (described in the ‘Methods’) confirms that if direct SOC between 3LE and 1CT – mechanism (I) – were the dominant mechanism we would indeed observe overpopulation predominantly in the 3LE x plane (Supplementary Table 36). The DFT calculations also predict the relationship between the 3LE ZFS and in-plane molecular axes (overpopulation in Tx and Tz). A rotation of the in-plane xz frame upon charge-transfer internal conversion would redistribute population from Tz to Ty, consistent with the experimental observations (Table 2). The most significant mode identified in the calculations that are associated with the 3LE–3CT conversion is the rotation around the single bond between the DTCz donor and DPS acceptor in DTCz-DPS-1, -2, and -3 (Fig. 3d–f); this torsional mode likely drives vibronic coupling20.

The D and E values predicted by DFT for 3LE in each emitter agree well quantitively with those obtained from the measured trESR (Table 2). However, only the qualitative trend in relative magnitude is well-predicted in DFT calculations of 3CT; the calculated D and E values are overestimated compared to those obtained from the measured trESR (Table 2). As discussed, we would expect spin–orbit interactions related to 3CT configurations to be weaker compared to their 3LE counterparts; no such reduction in the contribution to D from spin–orbit interactions is predicted in the DFT calculations (Supplementary Table 2). We suggest that the mismatch between predicted and measured 3CT D and E values is a result of this overestimation of spin–orbit interactions between 3CT and other nearby electronic states. This discrepancy highlights the current limitations of common computational calculations and the necessity for experimental trESR to reveal the complex nature of multiple triplet states, which underpin the spin physics of TADF-type organic emitters.

To reiterate: our findings show that it is indirect, not direct, coupling of 3CT to the singlet manifold that facilitates ISC. This assertion is supported by the following pieces of evidence, which rule out direct ISC between 3CT and either 1CT or 1LE. Direct ISC from 1CT to 3CT would need to be mediated by hyperfine coupling because the lack of orbital change suppresses SOC. Hyperfine-mediated ISC facilitates the transfer from the singlet state to only the energetically closest high-field triplet sublevel and consequently has a trESR polarisation pattern that is distinct from the SOC-mediated ISC patterns described above, and it is not observed in our measurements19,39. If 1LE to 3CT ISC were prevalent, we would not observe a dependence of the 3CT trESR signal on the 3LE–3CT gap, and would instead observe the 3CT spectra in all measurements, which we do not. Further, we would not expect a substantial population of 1LE for two reasons. First, 1LE to 1CT internal conversion is rapid (ps) compared with ISC (ns)22,45,68,69,70,71. Second, in the experiment, the 1CT absorption band was selectively excited (and is spectrally distinct from the 1LE absorption) in order to mimic operational OLED conditions where the lowest energy excitonic states are electrically generated (Fig. 2a–c solid lines and Supplementary Table 1). To summarise, the progression from a purely 3LE polarisation pattern to a combination of both 3LE and 3CT polarisation patterns with decreasing 3LE–3CT gap is evidence that a small 3LE–3CT gap regime exists in which strong vibronic coupling persists, even at low temperatures (kT < 3 meV), such that 3LE and 3CT are simultaneously populated. It is worth noting that, as TADF development is predominantly focused on the critical task of achieving efficient blue emission, the above outlined spin-vibronic mechanism is described in the context of the energetic regime attainable in blue emitters and shown in Fig. 1i: 1CT > 3CT > 3LE6,22. However, the same principles can be applied to a donor–acceptor emitter exhibiting a different excited-state ordering. We note that the unusual observation of multiple, distinct triplet signals in trESR spectra has also been reported for twisted donor–acceptor triplet photosensitizers, although the mechanism through which both triplets were populated was not rationalised;45,46,47 we postulate that the spin-vibronic mechanism may be active in such triplet photosensitizers.

Finally, we return to discuss the photophysical results in light of the spin-vibronic mechanism determined from trESR. The increase of 3CT spectral contribution with decreasing 3LE–3CT gap observed here (and in other systems55,69,72,73) reinforces the direct evidence for the spin-vibronic mechanism from trESR. Furthermore, the emergence of the additional 3CT trESR spectra, reflecting the strength of vibronic coupling, correlates with increased contribution to total emission from delayed fluorescence. However, as the 3LE–3CT gap decreases so does the 3LE–1CT gap and therefore their combined effect on delayed fluorescence cannot be decoupled. We note that the dynamics of excited state couplings are also affected by time- and temperature-dependent molecular reorganisation, as is evident when comparing the 295 K fluorescence with the 77 K fluorescence (dashed lines in Fig. 2a–c versus dashed lines in Fig. 2d–f). The spectral redshift undergone by charge-transfer states from 77 K to 295 K (due to post-photoexcitation molecular reorganisation) increases from DTCz-DPS-1 to -3. The differing reorganisation complicates the temporal evolution of energetic spacing between 1CT, 3LE, and 3CT, and hence coupling dynamics at room temperature are not directly comparable to frozen solution measurements.

TrESR has been used previously to probe triplet excitons in twisted donor-acceptor emitters that, due to their large T1–T2 energy gap (>0.6 eV), are non-interacting and exhibit delayed fluorescence through either the hot-exciton74 or hyperfine pathways75. Our experimental results for the present series of donor–acceptor molecules, supported by computational calculations, show that coupling between the singlet and triplet manifold is facilitated by the spin-vibronic mechanism when 3LE is in energetic proximity (<100 meV) to 3CT. This coupling drives ISC, and therefore TADF-relevant reverse ISC (assuming microscopic reversibility). We have demonstrated that trESR can reveal mechanistic details of ISC, providing a critical, complementary approach to optical spectroscopy methods33,76,77 and have established a trESR framework to study the spin-vibronic mechanism in TADF emitters. We have demonstrated that the mechanism for ISC within an emitter can be tuned through chemical modifications; engineering and measuring the spin properties are critical for the informed development of efficient TADF emitters. We have characterised the nature of triplet excitations in the chemically modified set of delayed fluorescence emitters, resolving two distinct triplet excitonic states (T1 and T2) involved in spin-vibronic coupling. This distinction supports the basis for achieving spin-vibronic TADF by engineering distinct yet vibronically-accessible 3LE and 3CT states. As discussed, the mechanism of 3CT population by vibronic coupling with 3LE may find further application in the development of heavy-atom free triplet photosensitizer design45,46,47. Tuning spin–orbit coupling in molecular systems and their subsequent trESR signatures, as demonstrated here, is not only of paramount importance to blue OLED development but may also find resonance more broadly in the development of organic materials for light-controlled quantum information and spintronics78,79,80,81,82.

## Methods

### Synthesis of emitters

The emitters were generally synthesised by nuclophilic substitution reactions between the appropriate carbazole derivatives and partially fluorinated derivatives of diphenylsulfone in dimethylsulfoxide in the presence of potassium carbonate, as described in the detail in the Supplementary Information. The fluorinated diphenylsulfones were obtained by reaction of lithio(fluoroarenes), obtained through bromine-lithium exchange or deprotonation, with 1,2-diphenyldisulfane, followed by oxidation with 3-chlorobenzoperoxoic acid.

### Transient electron spin resonance spectroscopy

Dilute solutions were prepared at 500 μM in toluene in a nitrogen environment. The solutions were contained within 3.8 mm O.D. clear-fused quartz tubes and degassed by 4 freeze-pump-thaw cycles before being flame-sealed. Measurement of transient continuous wave ESR (trESR) was done in the Centre for Advanced ESR (CAESR) in the Department of Chemistry of the University of Oxford, using a Bruker BioSpin EleXSys I E680 at X-band (9.7 GHz, 0.2 mW) with an ER 4118X-MD5W resonator. The temperature was controlled with an Oxford Instruments CF935O cryostat under liquid helium flow and an ITC-503S controller. An Opotek Opolete HE355 LASER was used for optical excitation of the samples and it was synchronised to the spectrometer by a Stanford Research DG645 delay generator. Use of a Stanford Research SR560 low-noise preamplifier of the Schottky diode-detected CW Mode signal at 3–300 kHz was in substitute of the video amplifier, following verification against the quadrature mix-down DC-AFC Transient Mode signal of 20 MHz bandwidth setting. Spectra were recorded for between 4 and 12 h. Data was processed with MatLab (The Mathworks, Natick, N.J.) and ESR simulations made use of the EasySpin routines53.

### Density-functional theory calculations

The linear-response TD-DFT calculations were performed using the LC-BLYP/6-31 G(d) within the Tamm–Dancoff approximation83 implemented in the Gaussian 16 programme. The geometry optimisation of the minimum-energy conical intersection between the 3LE and the 3CT was performed with the GRRM17 programme84, which refers to the energy and gradient calculated by the Gaussian 16 programme. $$\left\langle {\varPsi }_{^1{CT}}|{\hat{H}}_{{SO}}|{\varPsi }_{^3{LE}}\right\rangle$$ was calculated using the TD-DFT LC-BLYP/6-31 G(d) with the scalar relativistic zeroth-order regular approximation (ZORA) Hamiltonian85,86 implemented in the ORCA 4.2.1 programme. The ZFS parameters of the 3LE and the 3CT were simulated using the DFT PBE0/6–31 G(d) with the spin-spin term calculated from the spin-densities derived from the unrestricted natural orbitals57 and the spin-orbit term calculated by the coupled-perturbed SOC approach58 implemented in the ORCA 4.2.1 programme.

### Steady-state absorption and emission spectra

Dilute solutions were prepared at 100 μM in toluene. UV−vis absorption spectra were measured using a Cary 5000UV−vis−NIR spectrophotometer. PL spectra were recorded on a Horiba FL3-2i Fluorometer equipped with a liquid nitrogen attachment in solution at ambient temperature or either as steady-state or gated spectra at 77 K.

### Time-resolved photoluminescence

Dilute solutions were prepared at 100 μM in deoxygenated toluene in a nitrogen environment and subsequently measured in a nitrogen environment. Time-resolved PL spectra were recorded using an electrically-gated intensified charge-coupled device (ICCD) camera (Andor iStar DH740 CCI-010) connected to a calibrated grating spectrograph (Andor SR303i). Pulsed 325 nm photoexcitation was provided at a repetition rate of 1 kHz. Temporal evolution of the PL was obtained by stepping the ICCD gate delay with respect to the excitation pulse. The minimum gate width of the ICCD was 5 ns. Recorded data was subsequently corrected to account for camera sensitivity.


text:
# CO oxidation on Pt(111)¶

We present here the CO oxidation on Pt(111) which is published along with the oxidation on Pt nanoparticles in: M. Jørgensen and H. Grönbeck, ACS Catal., 7, 5054-5061 (2017) .

For the CO oxidation the following five chemical reactions have to be considered:

$\begin{split}O_2 + 2^* & \longleftrightarrow 2O^* \\ CO + * & \longleftrightarrow CO* \\ CO^* + O^* & \longrightarrow CO_2 \\ CO^* + * & \longleftrightarrow * + CO^* \\ O^* + * & \longleftrightarrow * + O^*\end{split}$

which are dissociative adsorption of oxygen, adsorption of CO, reaction of adsorbed O and CO to CO2 and CO and O diffusion. In the model it is assumed that the reaction to CO2 is not reversible and the reaction product is directly desorbing.

In this example, species 0 denotes empty sites, 1 is CO and 2 refers to O. All energies were obtained using density functional theory and are given in the paper mentioned above.

## Define constants and parameters¶

Any global constants can be defined in user_constants.py. In this example this file stores physical constants but also the calculated vibrations of the various adsorbed species and of the CO2 formation transition state and their physical properties.

Next, the simulation parameters must be set and stored in a dictionary:

T = 800.  # Temperature (K)
pCO = 2E3  # CO pressure (Pa)
pO2 = 1E3  # O2 pressure (Pa)
tend = 1E-3  # End time of simulation (s)

Here the Temperature is set to 800 K, and the pressure ratio of CO to O2 to 2:1. The pressure is given in Pa. Also the end time is defined, but because we are interested mainly in steady-state properties, the simulation is run until steady-state is reached and usually stopped before the given end time reached.

## Define sites and system¶

One site is defined for each surface atom using an Ase.Atoms object. We are using the ase.build class to construct our (111) surface. It consists of only one layer, because its only purpose is to simplify the site-connectivity set-up. Therefore, the used lattice-parameter is also not related to any results of density functional theory calculations or experiments. Thus starting with an empty list of sites we construct a $$10x10$$ surface consisting of 100 atoms:

from ase.build import fcc111
from user_sites import Site

sites = []
a = 4.00  # Lattice Parameter (not related to DFT!)
Ncutoff = a / np.sqrt(2.) + 0.05  # Nearest neighbor cutoff
atoms = fcc111("Pt", a = a, size = (10,10,1))
atoms.write('surface.traj')

We also have written out the prepared surface as .traj file to see if it looks as intended. As next step we have to add a site for each surface atom. Although we know that the oxygen preferably adsorbs on fcc sites and the CO on top sites, we assume here all sites to be equal (coarse-graining) but use the corresponding energies of the preferred sites.

# Create a site for each surface-atom:
for i in range(len(atoms)):
sites.append(Site(stype=0,
covered=0, ind=i))

Here, the block ind=i stores the index of the atom in the ASE.Atoms object on the NeighborKMC.user_sites.Site object, which can be later on used to write out .traj-files during the simulation.

Now the NeighborKMC.user_system.System object can be defined from the collection of sites:

from user_system import System
p = System(atoms=atoms, # store ASE.Atoms as well
sites=sites)

In addition to the overall system, we also need to define neighbor lists. It is simplest to define this according to the nearest neighbor distances. Thus we call in the main definition file of our kMC simulation:

p.set_neighbors(Ncutoff,pbc = True)

which sets the global neighbor list based on distances for our System. Because we are using a surface, periodic boundary conditions are desirable. The actual function is then defined in NeighborKMC.user_sites.Site and looks as follows:

def set_neighbors(self, Ncutoff, pbc=False):

if self.atoms is None:
raise Warning("Tried to set neighbor-distances in user_system.set_neighbors() with self.atom = None")

for i, s in enumerate(self.sites):  #For all sites
for j, sother in enumerate(self.sites): #Check all the other sites
dcur = self.atoms.get_distance(s.ind, sother.ind, mic=pbc) # use ase function get_distance
if dcur < Ncutoff and j != i:
s.neighbors.append(j)        #add site j to neighbor list of site i
if len(self.neighbors) == 0:             #check if neighbors exists -- otherwise the site will not interact with each other
self.neighbors = [s.neighbors for s in self.sites]
self.verify_nlist()

## Define reaction energies and entropies¶

In this step, the reaction energies, or methods to calculate these, are defined in user_energy.py and the entropies in user_entropy.py. That makes it simple to do all the book-keeping accordingly.

We used the from density functional theory obtained energies for the adsorption of oxygen on the fcc site and of CO on the top site and diffusion constants for oxygen to diffuse from fcc to fcc site and for CO for from top to top site. In this example, the adsorption energies are defined to be positiv in contradiction to the more generally used negative notation in theoretical papers.

EadsCO = 1.36

EdiffCO = 0.08 # CO diffusion barrier
EdiffO = 0.58 # O diffusion barrier

From the adsorption energies the activation energy for the CO2 formation is calculated from the BEP relations according to:

def get_Ea(ECO, EO):
ETS = 0.824 * (-EO -ECO) + 0.168 + 0.47238  # How much larger is the energy of CO and O wrt Pt(111)
Ea = ETS + ECO + EO  # Translate the barriers relative to Pt(111)
return Ea

The reason why not one single activation energy is used, are the repulsive adsorbate-adsorbate interactions which depend locally on the coverage of the neighbors of the reaction site:

def get_repulsion(cov_self, cov_NN, stype):
repulsion = 0.
ECOCO = 0.19  #  How CO affects CO
EOO = 0.32  # How O affects O
ECOO = 0.3  # How CO affects O
EOCO = 0.3  # How O affects CO
HInttwo = [[0., 0., 0.], [0., ECOCO, EOCO],
[0., ECOO, EOO]]  # Two body interaction Hamiltonian 3x3 because 0 = empty.
for j in cov_NN:  # For each covered Neighbor, give a repulsion:
repulsion += HInttwo[cov_self][j]

return repulsion

The in user_entropy.py stored entropies are included in the calculation of the rates as will be shown in the next section. Generally we define here the entropy for gas-phase CO and oxygen, as well as a method to calculate harmonic adsorbate entropy. The definitions of this functions can be looked up in this file directly.

## Define events¶

Here event-types are defined, which are stored in user_events.py. For each possible type of event, a class is derived from NeighborKMC.base.events.EventBase. For example take the $$\mathrm{CO_2}$$ formation. This event is defined in user_events.py as follows.

First we import the necessary functions, classes, and constants:

from base.events import EventBase
from user_entropy import get_Zvib, get_Z_CO, get_Z_O2
modes_TS_COOx, modes_COgas, modes_O2gas, kB, eV2J, s0CO, s0O, h,

Now we derive a class to contain the event:

class COOxEvent(EventBase):
def __init__(self, params):
Zts = get_Zvib(params["T"], modes_TS_COOx)
EventBase.__init__(self, params)

The constructor __init__(self,params) attaches relevant parameters to the object, and self.Zratio is the ratio between the partition functions in the initial state and transition state, used to calculate the rate constant in transition state theory. We need a function possible(self,system, site, other_site) that returns True if the event is possible on the current site-pair:

def possible(self, system, site, other_site):
if (system.sites[site].covered == 1 and system.sites[other_site].covered == 2) or \
(system.sites[site].covered == 2 and system.sites[other_site].covered == 1):
return True
else:
return False

Thus, for the event to be possible, the site needs to be covered by 1 (CO) and the neighbor site by 2 (O) or the other way round. That is originated by the use of single neighbor site pairs. Thus a pair with the indexes (10,11) would be the same as (11,10) in the code to avoid double counting in the time list.

Next we need to define a function get_rate(self, system, i_site, other_site) that returns the rate constant:

def get_rate(self, system, i_site, other_site):
if system.sites[site].covered == 1:
Ncovs_CO = [system.sites[n].covered for n in system.neighbors[site] ]
Ncovs_O = [system.sites[n].covered for n in system.neighbors[other_site]]
else:
Ncovs_CO = [system.sites[n].covered for n in system.neighbors[other_site] ]
Ncovs_O = [system.sites[n].covered for n in system.neighbors[site]]
ECO -= get_repulsion(1, Ncovs_CO, 0)
EO -= get_repulsion(2, Ncovs_O, 0)
Ea = max(0., get_Ea(ECO, EO)) # No negative energy barriers
return self.alpha * self.Zratio * np.exp(-Ea/(kB * self.params['T'])) * kB * self.params['T'] / h

The adsorption energies are fixed, but weaken in the case of any repulsions. A call is made to get_Ea(ECO, EO) to obtain the reaction energy barrier. The rate constant is multiplied with self.alpha, because self.alpha is the slowing-down factor that is adjusted dynamically for each event during simulation.

Also the event requires a method do_event(self,system, site, other_site) to perform modifications to the site-occupations when fired:

def do_event(self, system, site, other_site):
system.sites[site].covered = 0
system.sites[other_site].covered = 0

In this case, the two sites containing CO and O are simply emptied. At the end we define the method get_involve_other(self) to define if the neighboring sites are actually involved in the event:

def get_involve_other(self):
return True

After giving this example for one event, the other events can be defined similarly. How to define single site and dissociative adsorption was shown before. Only the rates have to be adjusted according to transition state theory. Having all events for each type of reaction defined in this order:

1. NeighborKMC.user_events.COAdsEvent for CO adsorption.
1. NeighborKMC.user_events.CODesEvent for CO desorption.
1. NeighborKMC.user_events.OAdsEvent for O2 dissociative adsorption.
1. NeighborKMC.user_events.ODesEvent for O2 desorption.
1. NeighborKMC.user_events.CODiffEvent for CO diffusion.
1. NeighborKMC.user_events.ODiffEvent for O diffusion.
1. NeighborKMC.user_events.COOxEvent for CO+O -> CO2.

we can now store the event-class references in a list for the NeighborKMC simulations and define accordingly a list of reverse events. In this example the CO (O) adsorption and desorption are reverse to each other. In addition are the diffusion processes reverse to themselves. The CO oxidation itself is not reversible. Thus it isn’t defined in this list.

events = [COAdsEvent, CODesEvent, OAdsEvent,
ODesEvent, CODiffEvent,
ODiffEvent, COOxEvent]
reverse_events = {0: 1, 2: 3, 4: 4, 5: 5}

The numbering of events is determined by the order in the list events defined here.

## Define and run simulation¶

Now the simulation object NeighborKMC.user_kmc.NeighborKMC can be defined and the simulation performed:

# Instantiate simulator object.
sim = NeighborKMC(system=p, tend=tend,
parameters=parameters,
events=events,
rev_events=reverse_events)
sim.run_kmc()
print("Simulation end time reached ! ! !")

## Analyze results¶

The results are analyzed by reading in the code output. For example, if we need to calculate the CO and O coverage as a function of time for the entire system:

import numpy as np

evs_exec = np.loadtxt("evs_exec.txt")

format: plain


text:
# Tag Info

For example, if I write a 2000 character answer that has one reference to the molecular formula H2O, is it necessary for this to be changed to $\ce{H2O}$? In my opinion, no. As long as the chemical ...


text:
SMFT (Soft Majority Fault Tolerance) Consensus
EK has a BFT-based consensus protocol. It has 4 phases (approve, vote, pre-commit and commit) in order to ensure the consensus is reached. Basic assumptions of BFT consensus is that at least ⅔ of all network validators are honest. It means that a collusion of more than ⅓ of validators can result in a network halt. If more than ⅔ of validators are colluded then network corruption can occur.
When a validator detects a non-valid block, it broadcasts a Reject message.
One of the problems of any BFT consensus-based network, is that in order to reach a consensus, it has to pass all the protocol phases, which results in network delays because of the number of messages the nodes need to exchange. In the case of Catchain it is n*log(n) messages where n is the number of nodes in the network.
Thus, any BFT based consensus, while assuming that a communication could be malicious at any given time, is also assuming some large percentage of its participants is honest.
First of all, the assumption that the majority of participants in a social construct are honest, does not necessarily mean that they are right. A lot of times it is a minority that is both honest and correct. The more complicated the subject of a decision to be made is, the more chances there will be groups with different opinions. Therefore, the network Validators consensus should remain in a very simple and limited capacity, and mostly related to technical aspects of network operations. Validators' own stakes should be subject to slashing by non-validators' stakes ( for more on that see DePools).
Second, why is this assumption made at all? What exactly stimulates the majority of network nodes to be honest in the first place? There must be some motivation of network participants to behave one way or another. This motivation lies outside of any particular protocol and relates to the economics of that protocol.
In the Proof-of-Stake network the validator's motivation to approve correct blocks is secured by their fear of losing their stake if they do not. There is no sound ground to believe that a particular percentage of validators should be honest, and why, apart from game theory arguments.
“Note that there have been some recent attempts to develop consensus algorithms based on traditional Byzantine fault tolerance theory; however, all such approaches are based on an M-of-N security model, and the concept of “Byzantine fault tolerance” by itself still leaves open the question of which set the N should be sampled from. In most cases, the set used is stakeholders, so we will treat such neo-BFT paradigms as simply being clever subcategories of “proof of stake”.”
Therefore, the whole point of viewing BFT consensus having, say, ⅔ of honest nodes as theoretically secure is misleading at best. And game theory arguments are weak, because they can not prove anything outside of a closed system and to view a blockchain as a closed game system is naive.
Let us assume that there are two coins and a market for them with derivatives and margin trading (of course there are many such markets in existence). It is quite easy to prove that holding a large stake in one network and being able to short another may provide a better incentive which will destroy the security motivation of the latter network validators. Large stake holder may short the other network token using his long position as collateral with margin and pay say half of the premium to any validators who will prove to destroy the second network value. It will be enough to have ⅓ of the stakes of the second network to stop it completely and destroy its token value. Having a large number of network participants helps only in the sense of communication and coordination hurdle such attacks could carry, but as recent examples of coordinate stock market pumps shows — this is by far not a security guarantee.
The problem multiplies when the network is partitioned. If there is a subset of validators which could corrupt a thread, it will, by proxy, reflect on the security of the whole network.
As a general rule if a simple majority of token holders wants the network to continue — nobody should be able to prevent that. It is not currently so in any proof-of-stake networks.
In a BFT consensus algorithm, network participants agree upon a block. In the case of a single chain of blocks (no sharding, no threading) this block agreement is final, unless a fisherman detects a problem in it and blames validators for dishonesty. Remember that fishermen are not a part of the consensus. In fact we can say that fisherman is a representative of all other network participants who are not part of the validator consensus, but who want to secure the network.
A fisherman is someone who is motivated to secure the network by a potential reward. Fishermen do not need to be a token holder. Of course fishermen can not guarantee anything because there is no guarantee that fishermen will do anything. There are many more problems with fishermen we discuss in "Practical Byzantine Dynamic Slashing (PBDS)". For the purpose of this discussion what is interesting is that the security of the network relies on a party which does not participate in its "consensus".
The notion of Time is critical. It is safe to assume that a block approved and included in the chain even by a single node 10 years ago would most probably be correct, or not, but if nobody cares why is it important? Therefore, the block validity is a function not only of a number of consensus participants but of time. There is simply a threshold function of time/consensus participants that other network participants are ready to accept as an agreement on block finality.
This is similar to Bitcoin transaction subjective finality, where a participant will wait for some number of blocks to be mined on top of the block in question for it to ensure its validity (since each mined block increases the hash power spent, it may be prohibitively expensive to reverse the transactions after a certain number of blocks (usually 6)).
What is the problem with such an approach? The finality time. We need to wait 6 Bitcoin blocks (about an hour) or 32 Solana slots (about 13 sec.) for a transaction to be probabilistically secured.
The BFT finality is faster since it takes about ~5 sec. to agree on a block by ⅔ of 100 nodes. But is 100 nodes enough to be considered a decentralized network and is even 5 sec. acceptable?
Theoretically we would like to have a consensus protocol that can produce blocks as fast as 500 ms. and have a sub-second finality, with the security guarantees provided by a large set of nodes, say north of 10,000 assuming just ½+1 are honest. Is that even possible?
Remember that practically speaking most of the time most of the validators are honest. Indeed it is hard to imagine a network which would constantly pay fishermen success fees. Such a network would halt after several blocks. Occasionally there are bad actors which believe they would defy the protocol and try to corrupt it. But all current protocols all of the time are optimised for the event that is extremely rare.
Let's make a protocol that would come to a consensus using the Soft Majority Voting mechanism with some modifications. Remember that in soft majority voting the consensus is reached after some small number of positive Votes, say 20%, while no single negative vote is received. If at least one negative vote is received, the consensus threshold is increased until the point where the majority threshold is set (51% for simple majority).
Of course this will lead to the following problems:
1. 1.
If a collator collides with just 20% of the nodes, it will not transmit blocks to other validators of the set, quickly approve the block and send it to masterchain without even a single node being able to send a Reject.
2. 2.
The same malicious set can create a message from thin air and write it into that rogue block. The message will be viewed as valid by another shard that will execute it within another smart contract of that shard.
3. 3.
Because Everscale is very fast, all those operations will happen in a few seconds time frame (masterblock validation ~4 sec, shard block creation ~3 sec) and that time won’t be enough to stop the attack.
4. 4.
In a paradigm of distributed programming, when the smart contract is an object which interacts with other objects asynchronously the chain of interactions between those objects could be very long. Since it is impossible to predict what messages smart contracts will emit before executing them it will be very hard to correct any corruption in a block in the middle of such a transaction chain.
5. 5.
Once executed, such a transaction could allow attackers to exit with large sums of money using decentralized exchanges and bridges with other networks which could render all the staking security guarantees useless.
6. 6.
In which event the fishermen will be too late for the party. The fork of a blockchain (whether in a form of vertical block or just fork) will recover a very insignificant portion of the attacker's reward. The blockchain will suffer catastrophic perception damage. \
Let’s emphasize that the idea that we could correct such an action retrospectively is false. Forks, vertical blocks or any other such mechanism won’t be able to mitigate such a catastrophic event. Therefore the whole security assumption relying on fishermen is wrong.
Let also emphasize that guaranteeing a security of a native token in a shard does not provide enough security guarantees as it should also guarantee the smart contract execution correctness on that shard, facing catastrophic consequences if not.\
Let us present a Soft Majority Fault Tolerance protocol (or SMFT for short):
Will leave the Catchain BFT protocol intact as of the leadership selection and consensus phases. The Thread validators will validate the block as they are now, just with the BFT threshold lowered to say 51% signatures, which will be used for preventing natural forking and slashing in case of an attack.
In addition, let's have a protocol that samples some random nodes out of the WorkChain Validator set and requires them to validate every block (let’s call them “Verifiers”). The randomness must be calculated after the block candidate has been propagated. The protocol must be non-interactive, succinct and low latency.
Let us concentrate on the block collation phase, before the consensus, because in order for malicious validators to validate the malicious block it should be collated in the first place. If we prove that the malicious collation will be impractical, it won’t matter how many potentially malicious validators the thread has.
Instead of broadcasting the block that has been already validated, we will ask collator to broadcast the block candidate to all nodes in the WorkChain. If a collator won’t do this, the block header won’t be included into Masterchain and the Collator will be slashed for data withholding.
Before we describe how blocks are verified let’s prove that the block has indeed been broadcast. Without such proof the malicious set of actors could siphon the block only to themselves, “mining” the necessary outcome.
Let some validators be defined as Broadcast Protectors (BPs for short).
When receiving a new block broadcast from the collator, the validators calculate a set of BPs from a hash of the block and list of validators. Say, taking the Workchain Validator Set divided by some number.
Validators will send a Block Receipt — a block hash signed by a BLS signature to all BPs. When a BP collects 51% of Receipts it will broadcast a multiplied Receipts and a numbered list of validators it got receipts from. The Validators that are not sending Block Receipts over some period of time or within a particular delay must be slashed.
The MC will verify the broadcast proof message by decrypting the block hash using multiplication of pubkeys of validators which provided receipts and comparing it with the collator’s block hash. Passing the check for 51% of validators will finalize the block if no NACK messages have been received so far. Assuming there are 200 neighbors and that there are less than 50% of malicious validators, getting 50%+1 receipt guarantees that there is
$2^{-200}$
probability for broadcast not to happen.
Now that we proved 51% of validators got the block, let's choose a set of verifiers.
In order to do this let’s choose a random number and a secret that only the verifier knows. Let’s then calculate if the validator is a verifier based on those two numbers which would be random and could not be calculated by an outsider.
When receiving the block candidate broadcast, the WorkChain validators will calculate Ω from a thread ID, block candidate sequence number and hash of a masterchain block signed by a verifier private key. Calculate Hv = hash (Ω). Let N be a number of validators in WorkChain, R is a predefined parameter in a workchain’s config. Then T is a multithreading factor equal to [N/R]. If a remainder of Hv/T division = 0 then the Validator is a Verifier for a thread block.
If the Validator is a Verifier it will verify the block immediately and send the result (ACK or NACK) to all Masterchain Validators and to a certain number of validators in all other Workchains together with the hash of the block candidate and the proof of correct Verifier selection.
If a Verifier will not submit either Nack or Ack it should be slashed. In order to accommodate that we could imagine a protocol that would reveal the true verifiers after the block has been finalized. Several such schemes could be imagined where the Verifier will have to calculate its participation from some key that must be revealed in another context. Since all ACKS and NACKS are included in the masterchain block it is available publicly. The Broadcast protection receipt can include a validator signature that would reveal the key from which the Verifier has been calculated.
Now Masterchain validators will need to wait for a thread block signed by 51% of that thread consensus and broadcast proof from at least one of Broadcast Protectors. If no NACK has been received for a certain period of time (which in reality is the broadcast proof creation time), the block header will be included into the Masterchain. If at least 1 NACK message has been received, the Masterchain will issue a special verification procedure.
Since in every round the probability of any particular validator to become a verifier is 1/T, the probabilities for each verifier are discrete.
Malicious validators (M) attack could only be successful if the majority of thread validators are dishonest, none of the honest validators are chosen as Verifier and their message delivery fails. Disregarding the network failure rate the probability of this event was compared with failure of Bitcoin “6 block finality”.
Let’s compare this with a probability of similar attack on Bitcoin for C =4, R = 15, N = 900
We have proved that by having a mechanism of block verification by a random validator set from a global set we can decrease the threshold of the consensus validation in the thread (effectively to zero), reducing the amount of time necessary for block finality while simultaneously greatly improving the thread security. Applying the correct economics to that model we have also proved that it will be completely impractical to corrupt any block data.
We can also prove that by reducing the BFT assumptions in the thread we mitigate most attacks on that consensus. In BFT consensus if the thread validators could not come to the consensus regarding the block, i.e. more than ⅓ of its validators are malicious, they could be easily slashed and excluded from the thread validator set by masterchain validators based on the verifiers' attestations of the collate candidate. Moreover we could extend the use of that mechanism to the Masterchain blocks as well and therefore ensure that even the masterchain could not be attacked by a ⅓ of malicious validators. In fact, the only thing BFT is good at is fast coordination of an execution, providing necessary redundancy.
By proving that only one node could be enough to Reject a malicious block to stop the attack we demonstrate great improvement in security assumptions of proof-of-stake networks making it a more secure network than a proof-of-work.
Question remains, could we make it even more radical and provide a zero probability network security guarantee? What if we add a full node capability of producing Nack messages as in Dynamic Fisherman with one critical difference. In fisherman the proof of a block corruption is supplied post factum, rendering the whole idea practically useless, as we discussed above. Adding the ability of a full node to deliver a Nack message within the block producing delay changes everything. Now we only need to prevent DDOS attacks on both the full node itself or by a full node. In order to do this we can imagine a simple commit reveal scheme with a deposit bond. The full node which wishes to check blocks for validity (lets call it a Watchdog) will create a contract in the Masterchain using some private key and post a bond there. Once a malicious block candidate is detected, such Watchdog will send a Nack message and a bond contract address, signed with the private key of a bond contract holder. Now if proved wrong Masterchain validators would know who to slash and nobody would not know who are the holders of the bond contracts before the Nack is actually received.
Such a mechanism would not be incentified and the fisherman-type incentives are useless because there would be no corrupted blocks detected ever, since the attack will have practically 0 chance of succeeding, as we demonstrated in this chapter. Yet since there are many parties which hold a lot of tokens, altruistic behaviour is expected in this case as it will guarantee the security of their tokens to be virtually 100%.


text:
# How to define acos in relation to cos?

Hey,

Does anyone know how to calculate the formula for acos in relation to cos?

We know that cos (∏/3) = 0.5
and acos(0.5) = ∏/3

However, I assumed I could calculate acos(x) in relation to cos(x) as: 1/cos(x).
Well, I assumed wrong :(

Is there any way to do this?

Thanks


text:
# Homework Help: A question about analysis

1. Oct 20, 2012

### Artusartos

http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw7-solns.pdf

In number 3, the text says "By the denseness of Q in R, there exists a sequence $(r_n)$ of rationals converging to x."

1) Why/how does denseness imply that there exists a sequence? Becuase Q is dense in R, we know that any two real numbers have a rational number between them. But how does that tell us that there is a "sequence" of rational numbers?

2) Do we have to assume that x is an irrational number?

3) Does this statement mean that for every irrational number, there is a sequence of rational number converging to it?

2. Oct 20, 2012

### jbunniii

Well, that's not all that denseness tells you. Between any two even integers is an odd integer, but that doesn't mean that the odd integers are dense in the even integers. What is the definition of denseness?

The first paragraph proves that f(x) = x for any rational x. The second paragraph uses that fact, but you don't have to assume that x is irrational in the second paragraph. The argument also works for rational x, because given a rational number, you can certainly find a sequence of rationals converging to that number.

I'm not sure what "this statement" refers to, but it's true that for every irrational number, there is a sequence of rational numbers converging to it.

If $x$ is irrational, start by picking any rational, say $r_1$. Since $x \neq r_1$, the distance between these numbers is positive: $|x - r_1| > 0$. Choose a rational $r_2$ such that $|x - r_2| < (1/2)|x - r_1|$, i.e. $r_2$ is less than half the distance from $x$ as $r_1$.

Then just keep repeating this process. At the n'th step, you will choose a rational $r_n$ such that $|x - r_n| < (1/2)^n |x - r_1|$. Clearly the sequence $r_n$ converges to $x$.

Another, more concrete (but less rigorous) way to see this is to consider the decimal expansion of $x$. Any real number has a decimal expansion. You can form a sequence of rationals converging to the real number by simply taking more and more digits, e.g.

$x_1 = 3$
$x_2 = 3.1$
$x_3 = 3.14$
$x_4 = 3.141$
$x_5 = 3.1415$
$x_6 = 3.14159$
and so on

3. Oct 20, 2012

### Artusartos

This is the definition that my textbook gives us "If a,b in R and a < b, then there is a rational r in Q such that a < r < b".

But I think I understand why sequences exist, from the answer you gave to my 3rd question. Thanks a lot. :)


text:
# Gas station without pumps

## 2021 May 4

### Resonance with nonlinear impedances

Filed under: Circuits course — gasstationwithoutpumps @ 08:24
Tags: , ,

My book uses nonlinear impedance $Z=(j\omega\;1\,s)^\alpha M$ for modelling loudspeakers and electrodes.  The loudspeaker models include an inductor-like component with $\alpha>0$ and the electrode models include a capacitor-like component with $\alpha<0$. Standard linear components are special cases of the nonlinear impedance: inductors have $\alpha=1$, resistors have $\alpha=0$, and capacitors have $\alpha=-1$.

This past week I was thinking about the resonance that I see in some loudspeakers around 4MHz, which can be modeled with a capacitor in parallel with the main nonlinear impedance.  How can I estimate the capacitance to get initial values for model fitting?  In general, when do I get resonance with two nonlinear impedances?

Although I initially looked at the capacitor case, I realized that the general case of two nonlinear impedances in series gives me simple math that I can easily generalize to special cases and to parallel connections, so let’s look at $Z=(j\omega\;1\,s)^\alpha M + (j\omega\;1s)^\beta N$.  The first thing to do is to replace $j$ by the polar form $e^{j\pi/2}$, getting $Z=e^{j \alpha\pi/2}(\omega\;1\,s)^\alpha M + e^{j \beta\pi/2}(\omega\;1s)^\beta N$.  The we can apply Euler’s formula to get the real and imaginary parts:

$\Re(Z) = \cos(\alpha\pi/2)(\omega\;1\,s)^\alpha M + \cos(\beta\pi/2)(\omega\;1s)^\beta N$.

$\Im(Z) = \sin(\alpha\pi/2)(\omega\;1\,s)^\alpha M +\sin(\beta\pi/2)(\omega\;1s)^\beta N$.

We will get resonance whenever the imaginary part goes to zero:

$0= (\omega\;1\,s)^{\alpha-\beta} M\sin(\alpha\pi/2) +N \sin(\beta\pi/2)$, or

$\omega = \left(\frac{ -N \sin(\beta\pi/2)}{M\sin(\alpha\pi/2)} \right)^{1/(\alpha-\beta)} s^{-1}$.

The special case of an inductor and a capacitor sets $\alpha=1$, $M = L / 1\,s$, $\beta = -1$, and $N= 1\,s/C$,  yielding $\omega= \left(\frac{N}{M}\right)^{1/2} s^{-1} = \sqrt{\frac{1}{LC}}$, which is the standard result.

We get a resonance whenever $\alpha$ and $\beta$ have opposite signs.

We can deal with parallel rather than series impedances by looking at the sum of admittances instead of the sum of impedances.  To get the admittances, the exponents $\alpha$ and $\beta$ get negated and the coefficients $M$ and $N$ inverted, giving us

$\omega = \left(\frac{ -M \sin(\beta\pi/2)}{N\sin(\alpha\pi/2)}\right)^{1/(\beta-\alpha)} s^{-1}$.

Note: this post is a much simpler analysis than last year’s in Resonance for non-linear impedance, because here I am just looking for where the phase goes to zero, rather than where the magnitude of impedance is minimized.

Update 2021 May 4: The two definitions of resonance I’ve used (minimum $|Z|$ and $\Im(Z)=0$) are not the same—I tried doing a parametric plot of the magnitude vs. the phase for one asymmetric example ($\alpha=0.6$ and $\beta=-0.2$) and saw that the minimum magnitude did not occur at 0°.  So I’ll need to think some more about what I want “resonance” to mean for nonlinear impedances.

## 2021 January 9

### One week into new quarter

We’re one week into the new quarter (10% of the way through!) and the course is going ok. Most of the students have finished the first-week lab, which consists of installing a lot of software and soldering headers onto a Teensy LC board.

The software they had to install was

Of course, each piece of software has its own installation idiosyncracies, different on Windows, macos, and Linux.  Some people even bumped into some problems because of running old versions of macos or Python (which were luckily cleared by upgrading to slightly newer versions).

The soldering was a bigger problem, because many students plugged in their cheap irons and left them on for a long time without tinning the tips.  The result was a sufficient build-up of corrosion that that they could not then tin the tips—even using a copper ChoreBoy scrubber to clean the tips didn’t help in some cases. In the in-person labs, I often spent most of the first week labs cleaning soldering iron tips that students had managed to mess up, but I can’t do that online.  This was not such a problem last quarter, as most of the students knew how to care for soldering irons from the first half of the course, but it may be a bigger problem this quarter, as most of the students have never touched a soldering iron before.  Some of the ones who are living here in town may be contacting the lab staff to see if they can get access to tip tinner or get some help cleaning their irons.  Those further away may be buying tip tinner on their own—I had not included it in kits, because I nad not expected so many to need it and it costs $8 apiece. Grading is going fairly well. My grading team and I have had two Zoom meetings so far (for Homeworks 1 and 2) and I graded Quiz 1 by myself, so we are keeping up with the grading. He have Homework 3 and Prelab 2a (there is no Prelab 1) both due Monday morning, and we’ll try getting them graded Monday night. We’re having to do most of our grading in the evening, because one of the graders is living in China, 15 time zones away, and none of us in California is an early morning person. In other news, I’ve finally finished clearing the blackberries and ivy from behind the garage (a project I started about 2 years ago). I’ll probably find some more when I cut back the kiwi vine (an annual winter project, in addition to frequent minor pruning during the summer). I think I either need to get some female kiwi vines and an arbor for them or uproot the male kiwi. There is really not much point to having just a male kiwi intent on taking over a big chunk of the yard. There are still a lot of blackberry roots out there that will sprout new vines. I’ll try uprooting them where I have access (not where they are coming through the cracks in the concrete), but I’ll probably have to do a monthly sweep of the yard to remove blackberries for the rest of my life in this house. ## 2020 October 28 ### Analog Discovery 2 power-supply noise Filed under: Circuits course — gasstationwithoutpumps @ 11:38 Tags: , , , Last night and this morning I spent some time investigating the noise on the power supplies of the Analog Discovery 2, because some students were having trouble with power-supply noise on their audio amplifiers (an inherent problem with biasing the microphone with just a bias resistor to the power supply). I looked at the positive supply set to +3.3V using oscilloscope Channel 1, and saw a fluctuation in voltage that was not too surprising for a switching power supply (though the switching frequency seemed ridiculously low). The power supply is specified to stay within 10mV of the desired voltage, and the voltage seemed to be doing that. I know that some switching power supplies shut themselves off under low-load conditions, to retain efficiency at the cost of adding low-frequency ripple to the output, so I tried running the power supply with different load resistors. I did the sampling at 400kHz and took FFTs of the signal (exponential averaging of RMS with weight 100, Blackman-Harris window). Here are the signals: The signals show quite a bit of oscillation without a load, but decreasing with increasing load. Here are the spectra from the Fourier transform (removing the DC spike): The spike around 57.2kHz is present with all loads and remains at the same frequency even if I change the sampling rate, so is probably the underlying frequency of the switching power supply. The rather large fluctuations in the audio range are probably the result of the power supply shutting itself off when there is low current draw. Drawing 10 mA is not quite enough to prevent this shutdown, but 27.5mA seems to be enough. So there seem to be at least three solutions for students having problems with power-supply noise: • Taking enough current from the power supply that the power supply doesn’t shut itself down. This is a rather fragile technique, as other sources of power-supply noise (such as noise injected by the power-amplifier stage in a later lab) will not be eliminated. • Using a transimpedance amplifier instead of a bias resistor to bias the mic. The bias-voltage input to the transimpedance amplifier can have a low-pass filter to keep it clean. • Putting a low-pass filter (with a small resistor and large capacitor) between the power supply and the bias resistor. The resistance of the filter adds to the resistance for the DC bias calculation, but not to the resistance for the i-to-v conversion. ## 2020 October 5 ### First Zoom lab Filed under: Circuits course,Uncategorized — gasstationwithoutpumps @ 20:25 Tags: , I had my first remote lab session today, using Zoom to supervise pairs of students working from home. It went more smoothly than I expected, but not perfectly. I had pre-assigned lab partners to groups using a CSV file, following the instructions at https://support.zoom.us/hc/en-us/articles/360032752671-Pre-assigning-participants-to-breakout-rooms. A couple minutes into the lab time, when most of the students had shown up, I opened the breakout rooms, and everyone managed to get into their rooms. Except those students who showed up late—they had to be manually assigned to their rooms, and of course they did not remember what group number they were in, so I had look for them in my list of email addresses (which was not easy, because the name they were showing on the screen might have no more than 1 letter in common with their email address). I wish that Zoom could remember the pre-assignment even for those who are late! Once I finally got everyone into their breakout rooms, I started going from room to room, looking over the shoulders of the students and asking if they had any questions. On the second or third room, the students couldn’t get screen sharing to work (though others had in other rooms). I tried setting all the screen sharing options, but nothing seemed to work. I left that group to answer a question in another room, which also turned out to be about screen sharing, but reactivating it for them worked! So I went back to the room that first had trouble, and reactivated screen sharing for them, and this time it worked. After that I mostly answered questions for a group until some other group asked for help, then I moved over and answered questions there. It was very similar to the experience I had with the live labs, except that it was hard to see their breadboards. Most of the questions were about setting up Waveforms on the Analog Discovery 2 to collect the data or about gnuplot scripts to plot and model the data. A couple of times students had to quit Zoom and re-enter, and I had to reassign them to their breakout rooms. It turns out that this can be done while in a breakout room, so I did not have to go back to the main room. Again, I wish Zoom could remember their assignments! There were a few times when I was free to float between breakout rooms, and I think I managed to touch base with each group at least once, but I’m not 100% sure of that. I was pretty burned out after 2 hours of being constantly “on”, but that is not so different from a usual lab session. I did not, however, feel like recording another video tonight. ## 2020 September 11 ### Edition 1.1 released today! Filed under: Circuits course — gasstationwithoutpumps @ 11:49 Tags: , , , , I finally released the new version of the textbook today! (https://leanpub.com/applied_analog_electronics). The book is only slightly longer than the previous edition: 659 pages 337 figures 14 tables 515 index entries 162 references The chapter on design report guidelines is available free as a separate publication: https://leanpub.com/design_report_guidelines At the same time as I released the new edition, I eliminated my COVID-19 sale, so the minimum price is now$7.99. I will still provide coupons for free copies to instructors who are considering using the textbook for a course.

I may have to do another version before January, as I have not checked the labs for BME 51A yet to see what modifications are needed for doing the labs at home. For example, I haven’t decided whether it is worth buying more blood-pressure cuffs and extra tubing, to have enough to ship one to everyone. I’ll probably have to give up on the drill-press instruction. I’d rather not skip the micrometer instruction, but that would mean buying a lot more micrometers, as we generally share 5 for the whole class.

One nice thing about selling through Leanpub is that purchasers get all future editions published through Leanpub as part of the price—the company is trying to encourage authors to publish book drafts through them, rather than waiting until the book is completely polished. That means that students who got earlier versions of the book will get this release for free, and anyone who buys now will get the benefit of future releases.

Next Page »


text:
# How TeX places text horizontally and vertically?

Am I right that TeX processes text in a manner like this? Since the very first character it is forming an hbox of textwidth width. When the first hbox is filled in its width, TeX makes a hyphenation, adds this hbox to a vbox (for a paragraph) and starts to form the second hbox and so on. So the vbox is like a holder for several hbox'es (lines)? Is it like a vertical linked list of horizontal linked lists?

Below is an illustration of what I want to clarify. In the first line it makes an indent and hyphens as ma-chines. In the second one I tried to explicitly represent this. Does TeX do the same (ideologically)?

\documentclass{article}

\begin{document}
Computer science is the study of algorithmic processes, computational machines and computation itself.

\vbox{%
\hbox{\indent Computer science is the study of algorithmic processes, computational ma-}%
\hbox{chines and computation itself.}}
\end{document}

No TeX does not typeset line by line. Once a paragraph starts, horizontal mode is entered and the entire paragrah is set in a horizontal list (similar to an \hbox) then the entire paragraph is passed through a line breaking algorithm that does a least cost optimisation trying to minimise badness (actually, demerits) of the paragraph. The resulting sequence of hboxes and vertical glue and penaltiues is then added to a similarly infinite vertical list and later a similar but simpler algorithm splits a page off this main vertical list.


text:
# Leetcode 002: Add Two Numbers

## Idea:

Since the length of two linked list may be different, we firstly add up each node in the two list until one reaches the end.

Then because we do not know which linked list reaches the end, we iterate both list (the finished list will not enter the loop statement), and add up the node with the carry c.

In the end, we need to check whether the carry c is 0 or 1. If it is 1, then we need to add one more node to the result.

This problem is not difficult and requires a bit more carefulness.


text:
# Comparing effects y1 ON x vs. y2 ON x in different regressions

Let's assume we have predictors $$x_1-x_5$$ and dependent variables $$y_1-y_9$$ in one dataset. We have a certain hypothesis about $$x_1$$: it should have differently strong effects on our 9 dependent variables $$y_1-y_9$$.

We perform 9 regressions, and find that $$x_1$$ significantly predicts all y with $$p < .001$$. Now we want to find out whether these (highly significant) effects are different from each other (just because they are significant effects does not mean they are equally strong). Two questions:

1. What information that a regression provides us with would give us insight into this question? Unstandardized beta? Standardized beta? t?

2. Is there a statistical test we can perform to find out whether the strengths of predictions are different between (y1 ON x1) and (y2 ON x1) and (y3 ON x1)?

Software I can use is R and SPSS; field is psychology/medicine.

• Is that $y_1 \lt -x_1$ or $y_1 \leftarrow x_1$ or possibly the R code y1 <- x1? Nov 15 '12 at 7:15
• With y <- x I mean the regression effect of x on y. y ON x, as MPLUS puts it. Nov 15 '12 at 15:09
• Torvon, are you asking whether there is a way to compare which of the three outcomes, $y_1,y_2,y_3$ that $x_1$ has the greatest effect on? Also: do $y_1,y_2,$ and $y_3$ all have the same units? Nov 20 '21 at 17:44


text:
# Relation between a group's cardinality and number of subgroups

Why are these following situations not possible?

A. An infinite group has finite number of subgroups

B. An uncountable group has countable number of subgroups.

Any infinite group that I can think of now has infinite number of subgroups.But what is the logic behind it ?And why the number has to be uncountable if the group is uncountable?

• For the uncountable case, note that any element of the group generates a (cyclic) subgroup which is finite or countably infinite. – André Nicolas Aug 2 '15 at 20:40
• In fact if $G$ is infinite, there are precisely $|G|$ cyclic subgroups. – whacka Aug 2 '15 at 20:45
• Question A. is old - see here. Question B is similar to A. – Dietrich Burde Aug 2 '15 at 20:47
• math.stackexchange.com/questions/1909385 – Watson Oct 24 '16 at 17:43

If an infinite group has an element of infinite order (that is, a subgroup isomorphic to $\Bbb Z$) then it has an infinite number of subgroups (because $\Bbb Z$ does).
Otherwise, every element is of finite order, in which case, if we only have a finite number of subgroups $G = \bigcup\limits_k \langle g_{i_k}\rangle$, which is finite, contradiction.
Say $G$ is infinite. Let $S(x)$ denote the subgroup generated by $x\in G$. If there exists $x$ such that $S(x)$ is infinite then $G$ has infinitely many subgroups, since an infinite cyclic group has infinitely many subgroups. On the other hand if every $S(x)$ is finite then there must be infinitely many distinct subgroups of the form $S(x)$, since $G=\bigcup_{x\in G}S(x)$.
If $G$ is uncountable a simpler argument works; every $S(x)$ is at most countable, so there must be uncountably many distinct $S(x)$.


text:
We think you are located in United States. Is this correct?

# Report data

## 20.4 Report data

As you have learnt, the data handling process is about gathering, recording, representing and analysing information, and then presenting it in a way that is easily understood and used by other people. People carry out this process when they want a question answered or to find something out. They might want to do research on a particular topic, make better decisions, improve a business’s success, or improve people’s lives.

The results and findings from the data need to be communicated clearly, so that the people who asked the question or wanted something researched can easily understand what the results of the data show.

When you write a report, remember to bear in mind the person who will be reading the report. This will help you to use the appropriate language and words so that the reader will be able to understand the report. Use clear, simple sentences and organise your findings into logical paragraphs. At the beginning of the report, tell the reader what you are trying to find out or what question you are hoping to answer with insights from the data. At the end of the report, write a short summary of the key findings from the data, and draw conclusions about what the data tells you.

# Test yourself now

High marks in science are the key to your success and future plans. Test yourself and learn more on Siyavula Practice.

Exercise 20.1

The table shows the estimated population for each province in South Africa in 2022.

Province Estimated Population
Free State $$2\ 921\ 611$$
Northern Cape $$1\ 308\ 734$$
Gauteng $$16\ 098\ 571$$
KwaZulu-Natal $$11\ 538\ 325$$
Western Cape $$7\ 212\ 142$$
Eastern Cape $$6\ 676\ 691$$
Limpopo $$5\ 941\ 439$$
Mpumalanga $$4\ 720\ 497$$
North West $$4\ 186\ 984$$

Data source: https://www.statssa.gov.za/publications/P0302/P03022022.pdf

1. How many provinces does South Africa have?
2. Which province has the largest estimated population?
3. Which province has the smallest estimated population?
4. What is the total estimated population of South Africa?
5. Calculate the average estimated population for the provinces.

1. $$9$$ provinces
2. Gauteng $$16\ 098\ 571$$
3. Northern Cape $$1\ 308\ 734$$
4. $60\ 604\ 994$
5. The average population for the provinces is $$6\ 733\ 888$$.

The pie chart and bar graph below show the data from Exercise 20.1.

1. Which graph do you think makes the data easier to read and interpret?
2. Which graph do you think makes it easier to compare between the estimated populations of different provinces?
3. What do you think are some of the disadvantages of a pie chart?
4. How would you show the average population on the bar graph?
5. Use the graphs and your calculations in Exercise 20.1 to summarise the data in a few short sentences.
1. Design a data collection tool for recording the favourite types of music of each of your classmates or friends.
2. Record, organise, summarise and represent data on the favourite types of music of each person you ask.
3. Analyse the data to determine which types of music are the most popular and which are the least popular.
4. Write a short report summarising your findings from the data. Present your findings to the rest of the class.


text:
# Approximating a log-power function

I can't figure out how the following approximation has been done, I would appreciate any guidance: $$y=-60+10\log_{10}\left[\frac{\left(\frac{99}{100}\right)^m}{\frac{1}{11}\left(\frac{1}{3}\right)^m+\frac{10}{11}\left(\frac{1}{6}\right)^m}\right]$$ is approximated to: $y=-49.6+4.73m$ when $m>5$ and $y=-60+7.73m$ when $m<3$. Thanks in advance.

-

Let's put $y$ in logarithm and exponential form (factorizing $\left(\frac 16\right)^m=\left(\frac 13\right)^m\left(\frac 12\right)^m$) : \begin{align} y&=-60+\frac {10}{\ln(10)}\;\ln\left[\frac{e^{m\ln\left(\frac{99}{100}\right)}}{\frac{1}{11}e^{m\ln\left(\frac{1}{3}\right)}\left(1+\frac {10}{2^m}\right)}\right]\\ &=-60+\frac {10}{\ln(10)}\;\left[m\,\ln\left(\frac{99}{100}\right)-m\,\ln\left(\frac{1}{3}\right)-\ln\left(1+\frac {10}{2^m}\right)+\ln(11)\right]\\ \end{align}
At this point observe that : \begin{align} a&:=\frac {10}{\ln(10)}\left(\ln\left(\frac{99}{100}\right)+\ln(3)\right)\approx 4.73\\ b&:=-60+\frac {10}{\ln(10)}\;\ln\left(11\right)\approx -49.6\\ \end{align} If $\;m\gg 1\;$ then $\ \ln\left(1+\frac {10}{2^m}\right)\sim \frac {10}{2^m}\$ may be neglected and you get $\ b+a\,m\$ as wished.
For small $m$ rewrite $y$ as : \begin{align} y&=-60+\frac {10}{\ln(10)}\;\ln\left[\frac{e^{m\ln\left(\frac{99}{100}\right)}}{\frac{10}{11}e^{m\ln\left(\frac{1}{6}\right)}\left(1+\frac {2^m}{10}\right)}\right]\\ \end{align} so that using : \begin{align} c&:=\frac {10}{\ln(10)}\left(\ln\left(\frac{99}{100}\right)+\ln(6)\right)\approx 7.74\\ d&:=-60-\frac {10}{\ln(10)}\;\ln\left(\frac{10}{11}\right)\approx -59.6\\ \end{align}
you will get the approximation $\ d+c\,m\$ (this one is not very good for 'moderate' values since valid only for $m\ll 1$.


text:
## Thinking Mathematically (6th Edition)

{x | x is a country in which 7.6% $\leq$ percentage having used marijuana $\lt$ 9%} From the graph, we can see that the contries for which this is true are: {Spain, Ireland, Switzerland}


text:
# What does axially symmetric mean mathematically?

1. Dec 6, 2009

### AxiomOfChoice

What does "axially symmetric" mean mathematically?

If we, for example, say that a magnetic field $$\vec B$$ is axially symmetric, does that mean that (in cylindrical coordinates) we have $$\frac{\partial \vec B}{\partial \phi} = 0$$, where $$\phi$$ is the azimuthal angle?

2. Dec 6, 2009

### Mapes

Re: What does "axially symmetric" mean mathematically?

Yes, exactly. (Sometimes called "axisymmetric.")