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# Interlayer exciton dynamics in van der Waals heterostructures

## Abstract

Atomically thin transition metal dichalcogenides can be stacked to van der Waals heterostructures enabling the design of new materials with tailored properties. The strong Coulomb interaction gives rise to interlayer excitons, where electrons and holes are spatially separated in different layers. In this work, we reveal the time- and momentum-dependent elementary processes behind the formation, thermalization and photoemission of interlayer excitons for the exemplary MoSe2–WSe2 heterostructure. We identify tunneling of holes from MoSe2 to WSe2 on a ps timescale as the crucial process for interlayer exciton formation. We also predict a drastic reduction of the formation time as a function of the interlayer energy offset suggesting that interlayer excitons can be externally tuned. Finally, we explain the experimental observation of a dominant photoluminescence from interlayer excitons despite the vanishingly small oscillator strength as a consequence of huge interlayer exciton occupations at low temperatures.

## Introduction

A direct band gap in the optical range, efficient electron-light coupling and a remarkably strong Coulomb interaction make transition metal dichalcogenides (TMDs) highly interesting materials for both fundamental research and technological applications1,2,3,4,5,6. Tightly bound excitons, quasi-particles of Coulomb-bound electron-hole pairs, dominate the optical response of these materials7,8,9,10,11,12,13,14. They have binding energies that are one to two orders of magnitude larger than in conventional materials15,16. As a result, excitonic features are stable at room temperature and dominate the optical response and non-equilibrium dynamics in TMDs. Besides regular bright excitons, TMDs also exhibit a variety of optically forbidden dark excitons17,18,19,20,21,22,23,24, which cannot be addressed optically due to the required momentum transfer or spin flip. The fascinating exciton physics becomes even richer, when considering that atomically thin materials can be vertically stacked to form Van der Waals (vdW) heterostructures1. In these systems the strong Coulomb interaction gives rise to interlayer excitons, where the involved electrons and holes are located in different layers (Fig. 1)25,26,27,28,29. After optical excitation of a regular intralayer exciton (IaX), the hole can tunnel to the other layer forming an interlayer exciton (IeX). Due to an offset in the alignment of the monolayer band structures (type II heterostructures), these interlayer excitons lie energetically below the excitons confined within one layer29 (Fig. 1). Depending on spin and momentum of the states involved, interlayer excitons can be either bright or dark. VdW heterostructures present an emerging field of research, as evidenced by an increasing number of studies, in particular demonstrating the appearance of interlayer excitons in photoluminescence (PL) spectra25,26,28,30,31,32,33,34,35,36,37,38,39. In the low-temperature regime, a pronounced additional resonance is observed at an energy below the intralayer excitons of the single layers26,27,33,40,41. The PL intensity of this low-energy peak is very pronounced compared to the intralayer exciton in the weak excitation regime. This behavior can be traced back to formation of interlayer excitons that due to their spectral position are highly occupied. Furthermore, in time-resolved PL measurements a spectrally narrow resonance was observed exhibiting lifetimes of tens to hundreds of nanoseconds26,27,33,40,41. Theoretical studies of vdW heterostructures have so far been primarily restricted to static observables, such as screening42, excitonic binding energies29, and lattice mismatch effects35. Microscopic insights into the ultrafast dynamics of interlayer excitons have remained literally in the dark yet.

In this work, we present a comprehensive microscopic study of the ultrafast kinetics of excitons in vdW heterostructures, in particular revealing the elementary time-resolved and energy-resolved processes behind the formation, thermalization and decay of interlayer excitons. We predict the binding energy of 150 meV for the energetically lowest interlayer excitonic state in the exemplary MoSe2–WSe2 heterostructure, as well as a significant reduction of intralayer binding energies by about 30% due to the increased screening within a heterostructure. Moreover, we shed light on the exciton dynamics demonstrating that excited interlayer excitons are formed via hole tunneling on a picosecond timescale, followed by a phonon-driven relaxation to the ground state on a femtosecond timescale. The radiative interlayer recombination is by orders of magnitude slower reflecting the small oscillator strength of spatially separated excitons. Finally, we provide a microscopic explanation for the recently performed time-resolved photoluminescence experiments demonstrating the surprisingly dominant contribution of interlayer excitons at low temperatures.

## Results

### Theoretical approach

We investigate the exemplary MoSe2–WSe2 heterostructure on a typical SiO2 substrate. To bring the system into a non-equilibrium, we apply a laser pulse at the energy resonant to the 1s exciton of the MoSe2 layer. We restrict the study to 1s excitons consisting of Coulomb-bound electrons and holes located at the K point of the Brillouin zone.

To provide microscopic access to the dynamics of the coupled excitons, phonons and photons, we apply the density matrix formalism43,44,45,46 and derive the luminescence Bloch equations explicitly including excitonic effects9,44. The emitted luminescence intensity Iω(t) is obtained from the temporal change of the photon density and reads44,47.

$$I_\omega (t) \propto \omega \mathop {\sum}\limits_{l_{\mathrm{h}}l_{\mathrm{e}}} \left| {M_{\mathbf{0}}^{l_{\mathrm{h}}l_{\mathrm{e}}}} \right|^2\left( {\left| {P_{\mathbf{0}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)} \right|^2 + N_{\mathbf{0}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)} \right){\cal L}_\gamma ({\mathrm{\Delta }}E_\omega ^{l_{\mathrm{h}}l_{\mathrm{e}}})$$

Here, $$M_{\mathbf{0}}^{l_{\mathrm{h}}l_{\mathrm{e}}}$$ is the optical matrix element for excitons that are composed of electrons in layer le and holes in layer lh. The index 0 indicates that only excitons with vanishing center-of-mass momentum Q = 0 contribute to PL. While the optical matrix element can be obtained analytically using a tight-binding approach for TMD monolayers and adjusting the coupling strength to experimentally measured absorption9,22, first-principle calculations have been performed to determine the coupling for interlayer excitons. The Lorentzian $${\cal L}_\gamma$$ accounts for energy conservation, i.e., an exciton decays into a photon with the same energy. We calculate on a microscopic level the components of the dephasing rate γ stemming from exciton-phonon and exciton-photon interactions22,48. At lower temperatures disorder-induced dephasing becomes important and has been accounted for phenomenologically by using the full width at half maximum (FWHM) values measured in ref. 27.

The PL strength is determined by the excitonic polarization $$P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)$$ reflecting the optically driven coherence (often referred to as coherent excitons43,44) and by the exciton occupation $$N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}$$ describing the formation and thermalization of incoherent excitons induced by a non-radiative decay of the excitonic polarization. Equations of motion for $$P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)$$ and $$N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}$$ are derived in the low density regime taking into account exciton-photon coupling giving rise to a spontaneous radiative decay of excitons as well as exciton-phonon coupling allowing for a relaxation of the excited hot exciton distribution into a thermal equilibrium. Furthermore, we take into account the tunneling of carriers between layers. This process is included via the tunneling Hamilton operator $$H_{\mathrm{T}} = \mathop {\sum}\nolimits_{a,b} T^{ab}a_b^\dagger a_b$$, where a, b are compound indices containing layer, band and momentum of the electron. The coupling element T is given by the overlap integral 〈Ψa|VTb〉 of Bloch waves Ψ with the interlayer potential VT. The appearing spatial overlap of the wavefunctions from different layers is obtained from density functional theory calculations that have been carried out using the gpaw package49. More details on the calculation of the tunneling matrix element can be found in the methods section. Finally, applying the Heisenberg equation of motion, we obtain the luminescence Bloch equations for the excitonic polarization $$P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)$$ and the incoherent exciton occupation $$N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}$$, see Eqs. (2) and (3) in the methods section. Numerical solution of this coupled set of differential equations provides microscopic access to time-resolved and momentum-resolved formation, relaxation, and radiative decay dynamics of intralayer and interlayer excitons. It allows us to calculate the photoluminescence of heterostructures as a function of time and energy.

### Excitonic band structure

The excitonic eigenenergies and wave functions are obtained by solving the Wannier equation9,43,46 (Equation (1) in the methods section). To obtain the interaction potential between charge carriers in a heterostructure, we generalize the widely used Keldysh potential for the monolayer case9,50. We solve the Poisson equation for two aligned homogeneous slabs (Supplementary Fig. 1 and Supplementary Note 1 for more details). The obtained excitonic band structure and the corresponding excitonic wave functions are shown in Fig. 2 for the three energetically lowest excitonic states (1s, 2s, 3s) for intralayer and interlayer excitons. Here, the electronic band alignment has been extracted from PL measurements26,27. We find that the interlayer exciton is the energetically lowest state at 1.39 eV, while the intralayer excitons lie at 1.65 eV (MoSe2) and 1.75 eV (WSe2), Fig. 2. The resulting excitonic binding energies are displayed in Table 1. As expected, the binding energy for interlayer excitons is reduced (150 meV for 1s) compared to the value for intralayer excitons (205 meV for MoSe2 and 177 eV for WSe2), however it is still much larger than the thermal energy. Thus, interlayer excitons are expected to be stable at room temperature and significantly contribute to the PL. Our calculations also reveal that the binding energy of intralayer excitons is considerably reduced by roughly 30% when the monolayers are stacked into a heterostructure (Table 1). This is due to an increased screening of the Coulomb potential. These results are consistent with previous theoretical studies based on DFT calculations29,37.

### Exciton dynamics

Solving the luminescence Bloch equations (the Methods section) we can resolve the dynamics of intralayer and interlayer excitons. Figure 3a shows the temporal evolution of exciton densities, i.e., momentum-integrated exciton occupations $$n = \mathop {\sum}\nolimits_{\mathbf{Q}} N_{\mathbf{Q}}$$, at the exemplary temperature of 77 K (Supplementary Fig. 3 and Supplementary Note 3 for 4 K and room temperature). The system is excited by a 100 fs long Gaussian pulse centered at 0.5 ps and a frequency resonant to the intralayer 1s exciton of the MoSe2 layer. We find that the optically excited coherent excitons (dashed red line in Fig. 3) decay on a timescale of hundreds of femtoseconds due to radiative emission and exciton-phonon scattering. The latter leads to the formation of incoherent intralayer excitons (IaX) through the so-called polarization-to-population transfer23,43,51,52. After about 10 ps, these intralayer excitons are completely transferred to interlayer excitons (IeX). This occurs through tunneling between energetically resonant states of the two layers and results in a transfer of holes to the WSe2 layer (Fig. 1). The subsequent phonon-induced relaxation of holes in the WSe2 layer to the valence band maximum effectively traps the holes within that layer, since tunneling back to MoSe2 is energetically forbidden. The resulting interlayer excitons have lifetimes orders of magnitudes longer than the intralayer excitons, since recombination mechanisms are suppressed due to the spatial separation of Coulomb-bound electrons and holes. We predict an interlayer exciton lifetime in the range of hundreds of microseconds at 77 K. Experimentally measured sub-nanosecond values27 suggest that the investigated radiative decay is not the dominant channel, but rather non-radiative decay e.g., induced by disorder might play the crucial role.

To provide the microscopic background of the exciton formation and thermalization process, we now show the time-resolved and energy-resolved dynamics of incoherent intralayer and interlayer excitons. Figure 4a illustrates how 1s intralayer excitons (IaX) are created in the MoSe2 layer after optical excitation at 0.5 ps. The process is followed by a phonon-assisted thermalization and tunneling-driven depletion of these excitons. The main features of these dynamics are illustrated in representative snapshots along the energy axis at fixed times (Fig. 4b). At 0.6 ps, a significant number of excitons is still located in the MoSe2 layer. The distribution is in a strong non-equilibrium due to the efficient polarization-to-population transfer, i.e., the excitonic polarization is converted into incoherent exciton occupations with non-vanishing center-of-mass momentum. This is the origin of the observed peaks (green line in Fig. 4b) corresponding to the position of intersections of exciton and phonon dispersion. After the coherence has decayed (1 ps), the occupation starts to thermalize into a Boltzmann distribution (blue line). The occupation within the light cone centered around the exciton dispersion minimum E0 = 1.65 eV is lowered as a result of radiative recombination. Furthermore, tunneling of holes to the WSe2 layer causes a considerable decay of the intralayer exciton occupation (black line). Finally, the temporal evolution of the exciton occupation at a fixed energy corresponding to the 1s resonance of intralayer excitons (1.65 eV) is shown in Fig. 4c. This occupation represents the optically active exciton with vanishing center-of-mass momentum. We see how intralayer excitons are created already during the optical excitation and how they decay on a ps timescale due to tunneling of holes into the neighboring WSe2 layer.

Figure 5a illustrates the corresponding dynamics of interlayer excitons (IeX). They first emerge at E = E0 corresponding to the energy of intralayer excitons in MoSe2. The exciton population generated at this discrete energy consequently relaxes to the band minimum of the interlayer dispersion mostly by emitting LO and TO phonons in MoSe2, which have approximately a constant energy of 36 meV. This yields a cascade-like redistribution of excitons through intermediate energy levels after a series of optical phonon emissions. The slow interaction with acoustic phonons, which provide a continuous range of energies, leads to a smoothing of the occupation distribution, but on a much larger time-scale. The energy conserving nature of the tunneling interaction eventually forces the entire system into a single Boltzmann-like distribution as can be seen in Fig. 5b, spanning over both intralayer and interlayer excitonic states.

Figure 5c is analogous to Fig. 4c, and depicts the temporal evolution of interlayer excitons at fixed energies. The green curve first shows how the interlayer exciton occupation at the energy 1.65 eV resonant to the intralayer exciton rises with its intralayer counterpart. Driven by scattering with phonons, excitons then accumulate at the minimum of the dispersion at 1.39 eV (orange curve).

It should be noted that the exciton-phonon scattering has an important effect on the effective tunneling rate. Since the tunneling interaction is symmetric, the effective interlayer tunneling rate is quenched by back-tunneling from energy-allowed states. On a short timescale, the impact is mostly determined by the rate of the phonon-induced down-scattering (Fig. 5). As the Boltzmann distribution is formed (Fig. 5b), the temperature starts becoming an important factor. The spectral width of the IeX distribution increases at enhanced temperatures, leading to more excitons available for back-tunneling. Nevertheless, for the investigated heterostructure the change in temperature has still a relatively small effect on the IeX formation time τ. We find a reduction of τ by approximately 200 fs from 4 K to 200 K, after which it remains nearly constant up to room temperature (Fig. 3b).

The interlayer energy offset ΔE in type II heterostructures (Fig. 1) turns out to be a crucial quantity for the IeX formation time. For the investigated MoSe2–WSe2 heterostructure, the offset reads 315 meV and results in an IeX formation time τ = 2.61 ps at room temperature. The latter is defined as the elapsed time between the pump pulse maximum and the IeX density reaching 1 − e−1 of its maximum value. Increasing (decreasing) the offset by 100 meV, the formation time becomes considerably enhanced to τ = 4.91 ps (reduced to τ = 0.69 ps). This behavior can be ascribed to the reduced (enhanced) tunneling rate reflecting the increased (decreased) momentum transfer, as the interlayer energy offset ΔE becomes larger (smaller). To put it in other words, the spectral overlap of the wavefunctions of the involved states in different layers is highly sensitive to ΔE. The larger the overlap (small ΔE), the more efficient is the tunneling and the faster is the formation time. The disorder-assisted momentum transfer function (Methods) also decreases with increasing ΔE. For this reason tunneling to the higher excitonic states might be non-negligible considering their lower binding energies, and will have to be addressed in a future study.

A modification of the interlayer energy offset could be e.g., achieved by introducing a doping of a single layer26 suggesting that the IeX formation time in heterostructures can be externally tuned. Furthermore, the interlayer wavefunction overlap and the tunneling rate could be tuned by modifying the substrate-induced screening. The smaller the dielectric constant, the stronger is the Coulomb interaction and the broader are the excitonic wavefunctions in the reciprocal space resulting in a more efficient tunneling. Finally, different layer stackings can also have a significant effect on the tunneling rate, as demonstrated by ref. 38 predicting formation times ranging from ps to sub-ps timescales.

### Photoluminescence

The exciton dynamics discussed above determines the light emission from the heterostructure. Figure 6a shows the time-resolved and energy-resolved PL spectrum at the exemplary temperature of 77 K in (Supplementary Fig. 4 and Supplementary Note 3 for 4 and 300 K). We find that in the first few ps, the emission from intralayer excitons (IaX) at 1.65 eV clearly dominates the PL. However, after approximately 41 ps, the contribution of interlayer excitons becomes pronounced. To better understand the underlying processes, we show again snapshots at fixed times and energies in Fig. 6b, c, respectively. The emission stemming from the intralayer exciton at 1.65 eV (red line in Fig. 6c) shows a maximum PL intensity in the first hundreds of femtoseconds originating from the efficient coherent emission (radiative decay of polarization on a fs timescale). The following slower decay on a time scale of tens of ps reflects the decrease of intralayer excitons due to hole tunneling to the WSe2 layer. In this time, interlayer excitons are formed (solid blue line in Fig. 6c). After approximately 41 ps, the contribution of the interlayer exciton surpasses the emission from the intralayer exciton. Snapshots of the energy-dependent PL along these characteristic times further demonstrate by far most pronounced coherent IaX emission (orange line, 0.6 ps), the reduced emission due to incoherent IaX (red, 1 ps), equal IaX and IeX emission (blue, 41 ps), and finally the dominant IeX emission (purple, 65 ps).

After 60 ps an equilibrium situation is reached that is characterized by a constant IaX–IeX intensity ratio of approximately 100 (dashed lines in Fig. 6c). This is determined by the ratio of the square of the corresponding optical matrix elements |MIeX|2/|MIaX|2 and exciton occupations $$N_0^{{\mathrm{IeX}}}/N_0^{{\mathrm{IaX}}}$$ within the light cone. Since the occupations are described by a Boltzmann distribution in equilibrium, we can explicitly calculate the temperature, at which the PL ratio is 1. We find that the interlayer exciton emission dominates until approximately 200 K. Above this temperature, the thermal occupation of energetically higher intralayer excitons increases and considering the significantly larger optical matrix element for intralayer emission, the intralayer peak exceeds the emission from the interlayer exciton.

A recent DFT study37 on MoSe2/WSe2 heterostructures suggests that the interlayer coupling might shift the Λ valley below the K valley in both layers, giving rise to a momentum mismatch between the lowest conduction and the highest valence band in the heterostructure. The resulting enhanced occupation of momentum-forbidden dark interlayer excitons and their impact on luminescence spectra (e.g., via phonon-assisted recombination) is of particular interest for future studies. In particular, for tungsten-based heterostructures the inclusion of other high-symmetry points of the Brillouin zone is expected to be important to capture the entire interlayer exciton dynamics41,53,54

## Discussion

In conclusion, we have presented a microscopic view on the dynamics of inter- and intralayer excitons in van der Waals heterostructures. Solving the Wannier equation, we report a binding energy of 150 meV for the interlayer exciton as well as a clear 30% reduction of the binding energy of the intralayer excitons due to the enhanced screening within a heterostructure. Solving the luminescence Bloch equations, we reveal the time-resolved and energy-resolved processes behind the formation, thermalization, and decay of interlayer excitons. We predict that tunneling of holes from the optically excited into the neighboring layer is the dominant formation channel occurring on a picosecond timescale. We demonstrate the crucial role of phonons in both their formation and subsequent relaxation dynamics. Furthermore, we suggest a possibility to externally tune the formation time of interlayer excitons by changing the interlayer energy offset in the type II heterostructure. Finally, we show that although the light emission from interlayer excitons is strongly quenched due to the spatial separation of charge carriers, they actually dominate the photoluminescence spectrum of heterostructures at temperatures below 200 K. The gained insights will trigger new experimental studies on van der Waals heterostructures. In particular, the predicted formation dynamics of interlayer excitons can be experimentally addressed by pumping and probing the intralayer exciton transition in different layers, where a clear bleaching is expected due to the efficient interlayer tunneling.

## Methods

### Binding energies

To model the interlayer exciton dynamics in heterostructures, we derive the luminescence Bloch equations for the excitonic polarization $$P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t) = \mathop {\sum}\nolimits_{\mathbf{q}} \varphi _{\mathbf{q}}^{ \ast l_{\mathrm{h}}l_{\mathrm{e}}}\left\langle {v_{{\mathbf{q}} + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{h}}}c_{{\mathbf{q}} - \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{e}}}} \right\rangle$$ and the incoherent exciton occupation $$N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t) = \mathop {\sum}\nolimits_{{\mathbf{q,q}}\prime } \varphi _{{\mathbf{q}}\prime }^{ \ast l_{\mathrm{h}}l_{\mathrm{e}}}\phi _{\mathbf{q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\delta \left\langle {c_{{\mathbf{q}} - \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{e}}}v_{{\mathbf{q}} + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{h}}}v_{{\mathbf{q}}\prime + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{h}}}c_{{\mathbf{q}}\prime - \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{e}}}} \right\rangle (t)$$ corresponding to electron-hole pair correlations44. Here, we have introduced φq as excitonic wavefunctions in momentum space, $$v^{(\dagger )},c^{(\dagger )}$$ as operators for annihilation (creation) of valence- and conduction band electrons as well as relative q and center-of-mass momenta Q with the coefficients α = me/(mh + me) and β = mh/(mh + me) describing the relative electron and hole masses.

To obtain the excitonic binding energies and wave functions, we solve the Wannier equation9,43,46

$$\frac{{\hbar ^2{\mathbf{q}}^2}}{{2\mu ^{l_{\mathrm{h}}l_{\mathrm{e}}}}}\varphi _{\mathbf{q}}^{l_{\mathrm{h}}l_{\mathrm{e}}} - \mathop {\sum}\limits_{\mathbf{k}} V_{{\mathbf{k}} - {\mathbf{q}}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\varphi _{\mathbf{k}}^{l_{\mathrm{h}}l_{\mathrm{e}}} = E_b^{l_{\mathrm{h}}l_{\mathrm{e}}}\varphi _{\mathbf{q}}^{l_{\mathrm{h}}l_{\mathrm{e}}},$$
(1)

where μ is the reduced mass, Eb the excitonic binding energy, and $$V_{\mathbf{k}}^{l_{\mathrm{h}}l_{\mathrm{e}}}$$ the Coulomb matrix element for the electron-hole interaction. Solving the Poisson equation for two aligned homogeneous slabs, we obtain an effective 2D Coulomb potential $$V_{\mathbf{k}}^{l_{\mathrm{h}}l_{\mathrm{e}}} = \frac{{e_0^2}}{{k\varepsilon _0\varepsilon ^{l_{\mathrm{h}}l_{\mathrm{e}}}({\mathbf{k}})}}$$ with a dielectric function $$\varepsilon ^{l_{\mathrm{h}}l_{\mathrm{e}}}({\mathbf{k}})$$, depending on the momentum transfer k and the overall composition of the heterostructure, Supplementary Note 1 for more details.

### Dynamics

Equations of motion for the exciton polarization and the exciton occupations are derived taking into account all relevant interaction mechanisms in the low density regime. The carrier-phonon matrix elements are taken from DFT calculations55 and are treated in analogy to refs. 22,23,52. The interaction mechanism that distinguishes the exciton dynamics in a vdW heterostructure from the dynamics in a bare TMD monolayer is the tunneling of carriers between layers. This process is included via the tunneling Hamilton operator $$H_T = \mathop {\sum}\nolimits_{a,b} T^{ab}a_b^\dagger a_b$$, where a, b are compound indices containing layer, band and momentum of the electron. The coupling element T is given by the overlap integral 〈Ψa|VTb〉 of Bloch waves Ψ with the interlayer potential VT. The latter can be separated into an out-of-plane component Vz and an in-plane disorder potential Vρ56. The first is given by a step function, which is only non-zero within the region between the two layers and its value was fixed to 5 eV corresponding to the ionization energy of TMD monolayers57,58. The tunneling matrix element can then be expressed as Tab = Vρ(|kb − ka|)Vzua|ubuc, where u are the lattice-periodic parts of the Bloch waves, which are integrated over one unit cell (uc). This integral was obtained from density functional theory calculations, yielding an overlap of approximately 1 × 10−2. The calculations were carried out using the gpaw package49. The wave function was expanded on a grid and exchange-correlation effects were represented using the PBE exchange-correlation functional59. The details behind this process are written in Supplementary Note 4. Finally, the Fourier transform of the in-plane component of the disorder potential reads56 $$V_\rho (|{\mathbf{k}}_b - {\mathbf{k}}_a|) = \sqrt \pi L_{\mathrm{C}}/\left( {1 + \frac{{|{\mathbf{k}}_b - {\mathbf{k}}_a|^2L_{\mathrm{C}}^2}}{2}} \right)^{\frac{3}{4}}$$ with LC as the correlation length. It has been set to 1 nm in accordance with the excitonic Bohr radius60, an approximation that applies for short-range disorder61,62.

Applying the Heisenberg equation of motion, we obtain the luminescence Bloch equations for vdW heterostructures

$$\dot P_\alpha = \frac{1}{{i\hbar }}E_\alpha P_\alpha + i{\mathrm{\Omega }}_\alpha - \left( {\gamma _{\mathrm{r}}^\alpha + \frac{1}{2}\mathop {\sum}\limits_\beta {\mathrm{\Gamma }}_{\mathrm{P}}^{\alpha \beta }} \right)P_\alpha ,$$
(2)
$$\dot N_\alpha = \mathop {\sum}\limits_\beta \left( {{\mathrm{\Gamma }}_{\mathrm{P}}^{\beta \alpha }\left| {P_\beta } \right|^2 + {\mathrm{\Gamma }}_{\mathrm{T}}^{\alpha \beta }{\mathrm{\Delta }}N_{\alpha \beta }} \right) - 2\gamma _{\mathrm{r}}^\alpha N_{\alpha ,{\mathbf{0}}} + N_\alpha ^{{\mathrm{scatt}}},$$
(3)

where the excitonic compound indices α, β contain center of mass momentum and electron/hole layers. The dynamics of the exciton polarization Pα(t) is determined by the Rabi frequency Ωα containing the driving optical pump pulse and the decay processes stemming from radiative damping $$\left( {\gamma _{\mathrm{r}}^\alpha } \right)$$ and exciton-phonon interaction $$\left( {\Gamma _{\mathrm{P}}^{\alpha \beta }} \right)$$. The latter has been obtained by truncating the electron-phonon interaction by a second order Born-Markov approximation44,46, giving rise to energy-dependent scattering rates. Those incorporate all exciton-phonon scattering channels, i.e., all energy allowed exciton transitions, weighted by the excitonic scattering cross-sections, electron-phonon couplings and the occupations of involved phonon modes. The dynamics of the incoherent exciton occupation Nα(t) is determined by formation processes driven by phonon-assisted decay of the excitonic polarization, the radiative decay, and exciton-phonon scattering $$\left( {N_\alpha ^{{\mathrm{scatt}}}} \right)$$ driving the excited system towards an equilibrium Boltzmann distribution23,52. Note that that radiative decay scales with Nα,0 = NαδQ,0, where only states within the light cone with a nearly zero center-of-mass momentum Q = 0 can contribute. Finally, $$\Gamma _{\mathrm{T}}^{\alpha \beta }$$ describes resonant tunneling between different layers. It depends on the occupation difference (ΔNαβ = Nβ − Nα) of the involved excitonic states and causes the formation of interlayer excitons. It was obtained by applying the same methodology as for $$\Gamma _{\mathrm{P}}^{\alpha \beta }$$. The explicit form of the scattering rates is given in Supplementary Note 5.

### Code availability

The code used to calculate the findings of this study is available from the corresponding author upon request.

## Data availability

The data that supports the findings of this study is available from the corresponding author upon request.

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## Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 696656. Furthermore, we acknowledge financial support from the Swedish Research Council (VR), Stiftelsen Olle Engkvist, and the Deutsche Forschungsgemeinschaft (DFG) through SFB 951 and the School of Nanophotonics (SFB 787).

## Author information

Authors

### Contributions

E.M. conceived and supervised the project. S.O. derived the theory and developed the code, based on previous work by S.B. and M.S. C.L., M.K., and P.E. performed the DFT calculations. S.O. wrote the main bulk of the manuscript, with the help of multiple revisions by mainly S.B. and E.M. T.K. provided fruitful discussions and experimental data to compare to.

### Corresponding author

Correspondence to Simon Ovesen.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

## Rights and permissions

Reprints and Permissions

Ovesen, S., Brem, S., Linderälv, C. et al. Interlayer exciton dynamics in van der Waals heterostructures. Commun Phys 2, 23 (2019). https://doi.org/10.1038/s42005-019-0122-z

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s42005-019-0122-z

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• ### Large area chemical vapour deposition grown transition metal dichalcogenide monolayers automatically characterized through photoluminescence imaging

• Toby Severs Millard
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npj 2D Materials and Applications (2020)

• ### Twist-angle dependence of moiré excitons in WS2/MoSe2 heterobilayers

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• ### Interplay between spin proximity effect and charge-dependent exciton dynamics in MoSe2/CrBr3 van der Waals heterostructures

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• D. Gillard
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• ### Interlayer excitons in van der Waals heterostructures: Binding energy, Stark shift, and field-induced dissociation

• Høgni C. Kamban
• Thomas G. Pedersen

Scientific Reports (2020)


text:
# mcmcse v1.3-2

0

0th

Percentile

## Monte Carlo Standard Errors for MCMC

Provides tools for computing Monte Carlo standard errors (MCSE) in Markov chain Monte Carlo (MCMC) settings. MCSE computation for expectation and quantile estimators is supported as well as multivariate estimations. The package also provides functions for computing effective sample size and for plotting Monte Carlo estimates versus sample size.

## Functions in mcmcse

 Name Description mcse.mat Apply mcse to each column of a matrix or data frame of MCMC samples. mcse.multi Multivariate Monte Carlo standard errors for expectations. confRegion Confidence regions (ellipses) for Monte Carlo estimates ess Estimate effective sample size (ESS) as described in Gong and Felgal (2015). mcse.q Compute Monte Carlo standard errors for quantiles. mcse.q.mat Apply mcse.q to each column of a matrix or data frame of MCMC samples. estvssamp Create a plot that shows how Monte Carlo estimates change with increasing sample size. mcmcse-package Monte Carlo Standard Errors for MCMC mcse Compute Monte Carlo standard errors for expectations. mcse.initseq Multivariate Monte Carlo standard errors for expectations with the initial sequence method of Dai and Jones (2017). minESS Minimum effective sample size required for stable estimation as described in Vats et al. (2015). multiESS Effective Sample Size of a multivariate Markov chain as described in Vats et al. (2015). qqTest QQplot for Markov chains No Results!

format: plain


text:
Applied Mathematics Seminar, Spring 2022

## Mondays 4:00 PM - 5:00 PM MT (unless otherwise noted), Hybrid format: In-Person in LCB 219 (unless otherwise noted), and Online (zoom information will be provided before the seminars)

• Please direct questions or comments about the seminar to Farhan Abedin (abedinf (at) math.utah.edu) or to Will Feldman (feldman (at) math.utah.edu) or to Akil Narayan (akil (at) sci.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizers for information about how to subscribe to this list.

January 4 (Tuesday). In-Person LCB 215
Speaker: Michael Lindsey, Courant Institute of Mathematical Sciences, New York University
Title: Optimization and variational methods for quantum many-body systems
Abstract: A fundamental quantity of interest in the study of a quantum many-body system is the ground-state energy, which is the lowest eigenvalue of an exponentially high-dimensional Hermitian operator. Some approximation frameworks for determining this energy are variational in that they furnish a guaranteed upper or lower bound on the energy. In this talk we discuss recent work on approaches that are variational in this sense. First we discuss new semidefinite relaxations for the ground-state problem, which yield variational lower bounds and can be interpreted via convex duality as quantum embedding methods. We also introduce a scalable solver for these relaxations based on the embedding interpretation. Second we discuss a new optimization approach in variational Monte Carlo (VMC), which yields variational upper bounds by direct optimization over a parametric class of wavefunctions. Of particular interest are recently introduced neural-network-based parametrizations. Time permitting, we also discuss a new optimization approach for computing excited states in VMC.

January 7 (Friday). In-Person, LCB 219
Speaker: Jingni Xiao, Department of Mathematics, Rutgers University
Title: The Anisotropic Calderón Problem for Nonlocal Operators
Abstract: The Calderón problem asks whether we can determine the conductivity of a medium by making voltage and current measurements at the boundary. The information is encoded in the Dirichlet-to-Neumann map that sends the Dirichlet data to the Neumann data of solutions of the conductivity equation. We consider in this talk a nonlocal analog of this problem. More precisely we consider a zeroth order perturbation of a fractional operator associated to a general second order elliptic equation and show that we can recover the perturbation from the corresponding Dirichlet-to-Neumann map. An example is the fractional Laplacian associated to an anisotropic conductivity.

January 11 (Tuesday). In-Person, LCB 219
Speaker: Junshan Lin, Department of Mathematics and Statistics, Auburn University
Title: A Mathematical Perspective on Resonant Diffraction and Super-resolution Imaging by Subwavelength Holes
Abstract: Since the discovery of the extraordinary optical transmission through the nanohole array by Thomas Ebbesen, a wealth of research has been sparked in the experimental and theoretical investigation of anomalous diffraction by subwavelength structures. In this talk, I will present rigorous mathematical theories for several anomalous diffractions in metallic gratings with subwavelength slit holes. The diffraction problem in three configurations will be investigated. Based upon the layer potential technique, Gohberg-Sigal type theory, asymptotic analysis and the homogenization theory, quantitative analysis of the diffraction anomaly in each configuration will be presented. I will also present a new imaging modality with illumination patterns generated from an array of resonant slit holes. At resonant frequencies, the patterned illuminations through the subwavelength holes encompass both low frequency and highly oscillatory waves, which allow for probing both the low and high spatial frequencies components of the imaging sample to achieve super-resolution. The imaging setup, the underlying mathematical framework and the computational results will be discussed.

January 14 (Friday), 2:30pm. In-Person, LCB 215
Speaker: Alice Nadeau, Department of Mathematics, Cornell University
Title: Rate-induced Tipping Points: Bifurcations and Heteroclinic Connections in Time
Abstract: Qualitatively, a tipping point in a dynamical system is when a small change in system inputs causes the system to move to a drastically different state. The discussion of tipping points in climate and related fields has become increasingly urgent as scientists are concerned that different aspects of Earth's climate could tip to a qualitatively different state without sufficient warning (e.g., loss of summer sea ice in the Arctic). Often a system that has tipped is difficult---or even impossible---to return back to its original state, making the study of predicting and preventing tipping phenomena extremely important. In this talk, I will briefly overview three main mathematical causes for tipping in dynamical systems as well as examples of each in different climate models. I will discuss the recent endeavors to put one of these causes--rate-induced tipping--on firm mathematical footing. I will show how viewing this type of tipping from a nonautonomous framework allows us to extend the types of systems we can analyze for tipping phenomena.

January 26 (Wednesday), 11am. Zoom
Speaker: Jody Reimer, Leeds University
Title: Stochasticity and uncertainty in ecological models
Abstract: It is far easier to come up with examples of apparent randomness in biology than of reliably deterministic processes. Seeds are dispersed by the wind, taking root only if they happen to land in favorable soil. Unpredictable combinations of temperature and precipitation create fluctuations in animal survival. Diffusion and collision of ice floes form the highly heterogeneous environment at the center of polar marine ecology. I will discuss several examples of the various ways that I have represented this uncertainty in mathematical models, including applications of the Central Limit Theorem for scaling from individuals to populations, Galton-Watson branching processes, stochastic differential equations, and uncertainty quantification approaches for parametric uncertainty. Embracing stochasticity and uncertainty is necessary if we are to fully exploit the power of mathematics for understanding the natural world by connecting models to data.

February 28. In-Person, LCB 219
Speaker: Kamala Liu, Department of Mathematics, University of Utah
Title: LOCAL WELL-POSEDNESS OF A NONLINEAR FOKKER-PLANCK MODEL
Abstract: In this talk I present a nonlinear Fokker-Plank model which emerges from our study of grain growth dynamics of polycrystalline materials. In particular, the model corresponds to inhomogeneous absolute temperature and obey a special energy law. The main part of this presentation is about our results of local existence and uniqueness of the solution. I'll also present some preliminary numerical simulations of this model, in particular its long-time asymptotic behaviors. This is joint work with Yekaterina Epshteyn, Chun Liu and Masashi Mizuno.

March 14. In-Person
Speaker: Wesley Hamilton, Department of Mathematics, University of Utah
Title: Some Spectral Partitioning Problems
Abstract: Spectral partitioning generally refers to clustering data using spectral tools, such as a graph Laplacian built on the data. In this talk I'll present a number of other data partitioning problems I work on that all have distinct spectral aspects, including some discrete-to-continuum shape optimization problems. I'll also present on recent work on computational redistricting that incorporate these tools.

March 21. Online
Speaker: Grey Ballard, Department of Computer Science, Wake Forest University
Title: Efficient Algorithms for Computing Tensor Decompositions
Abstract: Multidimensional data, coming from applications such as brain imaging, combustion simulation, and parameter-dependent PDEs, can often overwhelm the memory or computational resources of a single workstation. In this talk, I'll describe three tensor decompositions used to represent and approximate multidimensional data: CP, Tucker, and Tensor Train. Algorithms for computing these decompositions are based upon solving fundamental matrix problems such as linear least squares and the matrix SVD. I'll talk about how to solve these problems using parallel algorithms for the bottleneck computations. The algorithms are scalable, able to process terabyte-sized tensors and maintain high computational efficiency for 100s to 1000s of processing nodes. The open-source software libraries we have developed are designed for clusters of computers and have been employed on various supercomputers.

April 4. In-Person
Speaker: Emily Evans, Department of Mathematics, Brigham Young University
Title: Force-based models of cell-extracellular interaction
Abstract: To predict, alter and control wound healing and pathological conditions biologists need a better understanding of cell-cell and cell-extracellular interactions. We model these interactions as a system of stochastic differential equations representing the location of cell centers and extra-cellular matrix attachment sites. Numerical simulations and analysis show that when the duration of attachment to the extracellular matrix is a memoryless and force independent random process, the speed of the cell is independent of the force these attachment sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment to the extra-cellular matrix is key to predicting cell speed. To better understand the relationship between attachment dynamics and cell speed, we consider the problem in the context of two related (but simpler) models of cell motion. In this talk, we will also show the full results showing that expected average cell speed is independent of force and dependent on attachment dynamics.

April 11. In-Person
Speaker: Mahdi Soltanolkotabi, Departments of Electrical and Computer Engineering and Computer Science, University of Southern California
Title: Towards Stronger Foundations for AI and its Applications to the Sciences
Abstract: Despite wide empirical success, many of the most commonly used learning approaches lack a clear mathematical foundation and often rely on poorly understood heuristics. Even when theoretical guarantees do exist they are often too crude and/or pessimistic to explain their success in practical regimes of operation or serve as a guiding principle for practitioners. Furthermore, in many scenarios such as those arising in scientific applications they require significant resources (compute, data, etc.) to work reliably.

The first part of the talk takes a step towards building a stronger theoretical foundation for such nonconvex learning. In particular, I will focus on demystifying the generalization and feature learning capability of modern overparameterized learning where the parameters of the learning model (e.g. neural network) exceed the size of the training data. Our result is based on an intriguing spectral bias phenomena for gradient descent, that puts the iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Notably this analysis overcomes a major theoretical bottleneck in the existing literature and goes beyond the "lazy" training regime which requires unrealistic hyperparameter choices (e.g. very small step sizes, large initialization or wide models). In the second part of the talk I will discuss the challenges and opportunities of using AI for scientific applications and medical image reconstruction in particular. I will discuss our work on designing new architectures that lead to state of the art performance and report on techniques to significantly reduce the required data for training.

April 18. In-Person
Speaker: Mark Allen, Department of Mathematics, BYU
Title: Sharp quantitative Faber-Krahn inequalities
Abstract: We present new sharp quantitative Faber-Krahn inequalities. We use a notion of distance that accounts for sets with zero measure but with positive capacity. By employing the selection principle, this new notion of distance leads to a critical perturbation of the classical Alt-Caffarelli functional rather than a lower-order perturbation. By utilizing our Faber-Krahn inequality on the sphere, we establish a quantitative form of the Alt-Caffarelli-Monotonicity formula. This is joint work with Dennis Kriventsov and Robin Neumayer.

April 25. In-Person
Speaker: N. Benjamin Erichson, Department of Mechanical Engineering and Materials Science, University of Pittsburgh
Title: Continuous Networks for Sequential Predictions
Abstract: Deep learning is playing a growing role in many areas of science and engineering for modeling time series. However, deep neural networks are known to be sensitive to various adversarial environments, and thus out of the box models are often not suitable for mission critical applications. Hence, robustness and trustworthiness are increasingly important aspects in the process of engineering new neural network architectures and models. In this talk, I am going to view neural networks for time series prediction through the lens of dynamical systems. First, I will discuss novel continuous-time recurrent neural networks that are more robust and accurate than other traditional recurrent units. I will show that leveraging classical numerical methods, such as the higher-order explicit midpoint time integrator, improves the predictive accuracy of continuous-time recurrent units as compared to using the simpler one-step forward Euler scheme. Then, I will discuss a connection between recurrent neural networks and stochastic differential equations, and extensions such as multiscale ordinary differential equations for learning long-term sequential dependencies.

Seminar organizers: Farhan Abedin (abedinf (at) math.utah.edu), Will Feldman (feldman (at) math.utah.edu), and Akil Narayan (akil (at) sci.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148


text:
+0

0
274
1

2 times the square root of 5

over------------------------------------------------------------------

2 times the square root of 7 + 3 times the square root 3

Guest Apr 2, 2017
#1
+7324
+2

We want to get the radicals out of the denominator.

Like this:

$$\frac{2\sqrt5}{2\sqrt7+3\sqrt3}*\frac{2\sqrt7-3\sqrt3}{2\sqrt7-3\sqrt3}=\frac{2*2\sqrt{5*7}-2*3\sqrt{5*3}}{2*2\sqrt{7*7}-3*3\sqrt{3*3}} \\~\\ = \frac{4\sqrt{35}-6\sqrt{15}}{4*7-9*3}= \frac{4\sqrt{35}-6\sqrt{15}}{1}=4\sqrt{35}-6\sqrt{15}$$

If you want an approximate answer, it is $$\approx 0.426$$

hectictar  Apr 2, 2017


text:
# Supposedly simple double integral

1. Nov 3, 2007

### raynoodles

double integral of xy dA
in the triangular region of (0,0), (3,0), (0,1).
my problem that I am having is finding the limits I am suposed to find dx or dy in. I figure I should use 0 to 3 for dx, but then i do dy from 0 to what? Help appreciated.

2. Nov 3, 2007

### Avodyne

Try drawing a picture of the region. Then, for a given value of x, what values of y lie within the region? This gives you the limits of integration for y, given x. (Of course, you must then do the y integral before you do the x integral.)

3. Nov 3, 2007

### raynoodles

so then the parameters for y would be: 0 to x/3?

4. Nov 4, 2007

### HallsofIvy

Staff Emeritus
Yes, because the upper boundary is the line y= x/3.

It is a very good exercise to "swap" the limits of integration. Suppose you wanted to integrate with respect to x first and then y? Clearly to cover the entire triangle, you must take y going from 0 to 1. For each y, then, x must go from the left boundary, x= 0, to the "right" boundary which is still that line y= x/3. That is, x must go from x= 0 to x= what? Do the integral of xy both ways and see if you get the same thing.

5. Nov 4, 2007

### raynoodles

still not getting the right answer.

I used the parameters dy= 0 to 1 and dx= 0 to -3y+3 and got 2.375.


text:
# How To Calculate Active Power

Filter Type:

### circuit analysis - Calculate the active power - Electrical

Details: The formula of P is = U * I * sin (alpha) Not in this universe. That formula is for "so-called" reactive power but, you don't have any reactive components so it's null and void. The RMS of a triangle or saw-tooth waveform is this: -. So, without the diode half wave rectifier, the power is: -. [ V P 3] 2 R. active power formula

› Verified 1 days ago

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### DQ0-transformation: calculate active and reactive power

Details: The final step is to calculate the active, reactive, and apparent power using the DQ0-components as described in the computational implementation above. As expected, in the first period it is purely active power, while in the second period it is a combination of active and reactive power. how to calculate reactive current

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### Watts, VAR , VA (Active, Reactive, Apparent) Power Calculator

Details: Apparent Power in an AC circuit is the resultant of the active and reactive powers. In complex number, S = P + j Q. Where the S is the apparent power, P is active power and the Q is reactive power. Active Power or True power, P = V x I CosΦ W. Reactive Power, Q = V x I SinΦ VAR. Apparent Power, S = V x I VA var calculator power

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### AC - Active, Reactive and Apparent Power

Details: The required power supply to an electric circuit depends on the. active power - real electrical resistance power consumption in circuit; reactive power - imaginary inductive and capacitive power consumption in circuit; The required power supply is called the apparent power and is a complex value that can be expressed in a Pythagorean triangle relationship as indicated in the … active reactive and apparent power

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### Active & Reactive Power - MathWorks

Details: Description. The Active & Reactive Power block measures the active power P and reactive power Q associated with a periodic voltage-current pair that can contain harmonics. P and Q are calculated by averaging the V I product with a running average window over one cycle of the fundamental frequency, so that the powers are evaluated at fundamental frequency. active and reactive power

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### 3-Phase Power Calculator + Formula (kW To Amps, Amps To kW)

Details: Quite similarly, a higher power factor is proportional to higher power output. You can use this example to see how the 3-phase power calculator works: A 100 amps motor on a 240V 3-phase circuit with a 0.9 power factor produces 37.41 kW of electrical power. Insert these 3 quantities in the calculator, and you should get the same result. real reactive and apparent power

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### How to calculate active subscriptions with an - Power BI

Details: Measure = Calculate ( DATEDIFF (Inicial Date,Final Date , DAY), Filter ( Table Name , Type = "A" ) Try this one it will give u the days between intial date to final date with type is Active . Chnage Table and column name in the measure . Let me know it doesn't help u . i … reactive and apparent power

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### Active, Reactive and Apparent Power - Electrical Concepts

Details: Active Power: Active power is the real power consumed in an electrical circuit. It is the useful power which can be converted into other form of energy like heat energy in heater, light energy in bulb etc. It is also known as true or real power and measured in terms Watt, kW (Kilo Watt) or MW (1 Mega Watt = 106 Watt).

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› Get more:  How To Use

### How to calculate active power from waveform. Forum for

Details: I am simulating a reactive load in LTspice. I got a voltage and current waveform with some phase delay due to reactive nature of the circuit. I want to calculate the total power given to the load. I have attached a voltage and current waveform measured across load.

› Verified 4 days ago

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### What is Active, Reactive, Apparent and Complex Power?

Details: What is Active Power: (P) Active Power is the actual power which is really transferred to the load such as transformer, induction motors, generators etc and dissipated in the circuit.. Alternative words used for Real Power (Actual Power, True Power, Watt-full Power, Useful Power, Real Power, and Active Power) and denoted by (P) and measured in units of Watts …

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### Power Factor How To Calculate - ElectricalSells

Details: Power factor is an indicator of the amount of power that is used to create actual by the load. power factor closer to 1(one) mean power supplied by the source is mostly consumed by the load. And losses as reactive power is minimum. for factor how to calculate, let first understand the basic term which is going to be used in it.

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### Compute active and reactive powers of voltage-current pair

Details: The Power block computes the active power (P), in watts, and the reactive power (Q), in vars, of a voltage-current pair at fundamental frequency. To perform this computation, the block first determines the fundamental values (magnitude and phase) of the two input signals V & I. The following P and Q are then calculated:

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### 3 Steps to Calculate Required Reactive Power for PF

Details: Let’s say you have a 100kW induction motor whose current power factor is 0.7, and you want it to be 0.95. So, we’ll do our calculations to improve the power factor of this motor. The formula which we are going to use to calculate required reactive power, is given below. Required reactive power = P x [ tan (cos -I Ø1) – tan (cos -I Ø2)]

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### MEASURING REACTIVE POWER IN ENERGY METERS

Details: Method 1: Power triangle The Power triangle method is based on the assumption that the three energies, apparent, active and reactive, form a right-angle triangle as shown in Figure 1. The reactive power can then be processed by estimating the active and apparent energies and applying: Although this method gives excellent results

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### Three-phase power calculator jCalc.net

Details: Power factor (cosΦ). Enter the power factor of the load. Its between 0 and 1. How to calculate 3-phase power? The 3-phase power formula for active power is: $$P_{kW}=\dfrac{\sqrt{3} \cdot V_{LL} \cdot I \cdot \cos \phi } { 1000 }$$ Where, $$V_{LL}$$ is the 3-phase line voltage, $$I$$ is the current, and $$\cos \phi$$ is the power factor

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### Real Reactive and Apparent Power Calculator - CreatifWerks

Details: The Power Triangle. Refer the Power Triangle below: Real Reactive and Apparent Power Calculator The Formulae and How to Calculate : Apparent Power = True Power / PF. Reactance Power = sin θ x Apparent Power. Power Factor = True Power / Apparent Power. True Power = Apparent Power x Power Factor . Check out convert 4mA to 20mA Calculator

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### How do I measure active and reactive power consumption in

Details: Hello, There is not a block that will calculate P and Q for you automatically, you will need to apply this logic yourself. But we have several demo models that show how this can be done, for example the "Microgrid in Island Operation", "STATCOM Cascaded H-Bridge Converter", and "Neutral-Point Clamped Converter" models.

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### KVAR(Reactive Power): KVAR Calculation, formula & KVAR

Details: The power factor formula is the following: active power x 100 / apparent power. If you are looking for a way to save on energy consumption costs, then think about having capacitors installed. Use of the right capacitors will decrease the reactive power component by the Utility, which will lead to a drop in the apparent power component.

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### How to measure AC Real Power and Apparent Power with

Details: Apparent Power can be calculated by multiply root-mean-square (RMS) voltage value and (RMS) current value. The second term is the Active / Real Power. It is the actual power consumed by the load. Active Power can be calculated by averaging the product of instantaneous Voltage and Current waveform values.

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### What is apparent power definition and formula

Details: Different types of power dissipate in AC power systems i.e. Active, Reactive, Apparent, and complex powers. The measure of alternating current (AC) power is obtained from the multiplication of the root-mean-square voltage and the root-mean-square (RMS) current. In DC circuits there is only resistance, the Voltage and currents are in phase.While in AC circuit …

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### Solved: Active Employees per Period - Microsoft Power BI

Details: What I am trying to calculate is. the running total of people having started in a certain period, and. the running total of still active people (i.e. 'Contract End Date' empty or in the future) This is my Measure for the active ones: Active Employees = CALCULATE ( COUNTA (crm_employees [Employee]); crm_employees [Status Code] = "Active" ) And

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### how to calculate apparent power How to find angle

Details: we are going to see about apparent power calculation and finding angle between v and I without using scientific calculator. Formula for apparent power = V*IF

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### Free Complex Number Power Calculator - DCBA Online

Details: Complex number power calculator is programmed to calculate up to 10th power of a complex number. User needs to input the real part of complex number (a), and the imaginary part of the complex number (b). The power can either which is maximum up to 10 can be selected from the dropdown or manually input by the user.

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### Sinusoidal Steady-State Power Calculations - Technical

Details: Instantaneous Power. We begin our exploration of sinusoidal power calculations with the genaric circuit in Fig. 1.1. In here, v and i are steady-state sinusoidal signals. By using the passive sign convention (PSC), the power at any instant of time is given by: p = vi p = v i (1.1) Figure 1.1 Representation of a circuit used for calculating power.

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### 3 Steps to Calculate Required Reactive Power for PF

Details: Use this simple 3 steps to calculate reactive power required by you. 3 simple steps to calculate reactive power for power factor correction. Easiest and easy

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### What is Active, Reactive and Apparent Power - definition

Details: Active Power. Definition: The power which is actually consumed or utilised in an AC Circuit is called True power or Active power or Real power.It is measured in kilowatt (kW) or MW. It is the actual outcomes of the electrical system which runs the electric circuits or load.

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### Reactive Power and Compensation Solution Basics

Details: Why we don’t like reactive power. The total power, the so-called apparent power, of a transmission network is composed of active and reactive power (Figure 1).While the power consumers connected into supply transform the active power into active energy, the reactive energy pertaining to the reactive power is not consumed.

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### Determine consumed power from active-reactive components

Details: Dear Mahmmood, welcome, by definition the active power Pa= IV cos phi for single phase source, and the corresponding reactive power= Pr= …

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### Power factor - PF (COS φ)

Details: Power factor can be defined as the ratio of real power (Active power) to apparent power. It can also be defined as the absolute value of the cosine of the phase shift between the voltage and current in an AC circuit. It is denoted by the Greek alphabet λ (Lambda). Power factor (λ) = Active power/Apparent power. = VI.COS φ/ VI.

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### Unbalanced Three Phase Power Calculations All About Circuits

Details: How to calculate total three phase Active Power, Reactive Power and Apparent Power if active, reactive and apparent power of each phase are given: Active power phase1: 2432.96 W Active power phase2: 2012.01 W Active power phase3: 1843.34 W Reactive Power phase1: 1339.46 VAR Reactive Power

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### Calculating Reactive Power in AC Circuits

Details: Another way to calculate reactive power is to calculate the inductive power and capacitive power and subtract the smaller from the larger. Either one of these formulas will work. The formula you use depends upon the values you are given in a circuit. For example, find the reactive power of the circuit shown in figure 4-10.

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### Electrical Power in AC Circuits and Reactive Power

Details: Power in AC Circuits. In a DC circuit, the power consumed is simply the product of the DC voltage times the DC current, given in watts. However, for AC circuits with reactive components we have to calculate the consumed power differently. Electrical power is the “rate” at which energy is being consumed in a circuit and as such all

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### How to calculate required reactive power supplied by a

Details: Answer (1 of 3): As I recall, an electric motor would not improve the power factor from 0.95 lagging, it would just make it worse because motors are generally inductive loads. You need something that draws a capacitive reactance load. There are suppliers of power-factor correction capacitors who

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### How To Calculate Project Status In Power BI

Details: Create Calculated Column in Power BI. From the above ribbon, > at Home tab, > Add a new Calculated Column. A column formula would be shown to write the Project Status formula based on the Project start and end date. Type the column formula based on your data source name and column name as below. The Project Status equals “Not Started“, in case.

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### How to Calculate 3 Phase Power Sciencing

Details: Performing a 3-phase power calculation means using a simple formula relating the power to the voltage, current and a power factor describing the efficiency of the circuit. The key difference between working out power for 3-phase and single-phase systems is a factor of the square root of three.

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### AC power - Wikipedia

Details: Active, reactive, apparent, and complex power in sinusoidal steady-state. In a simple alternating current (AC) circuit consisting of a source and a linear time-invariant load, both the current and voltage are sinusoidal at the same frequency. If the load is purely resistive, the two quantities reverse their polarity at the same time.At every instant the product of voltage and current is

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### Star Connection (Y): 3 Phase Power, Voltage & Current Values

Details: In a three phase AC circuit, the total True or Active power is the sum of the three phase power. How to Calculate the Battery Charging Time & Battery Charging Current – Example. How To Calculate Your Electricity Bill. Electric Bill Calculator with Examples.

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### Count Amount of Active Employees by - Power BI User Group

Details: I'm new to Power BI, I'm assuming this is because I have to create a Measure ahead of time for each filter I might use and don't allow the user to change the filters? Here's the current formula: Count of Active Employees Measure =

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### Subscription Pattern in Power BI - DAX Measure for Active

Details: One of the common patterns of a data model is the subscription pattern. In this pattern, we have subscriptions (or tickets, or issues, or whatever else you want to call it) open date and close date. The way that this data is stored in a table makes it a bit challenging to get informative insight Read more about Subscription Pattern in Power BI – DAX Measure for …

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### dax - How to calculate headcount in PowerBI - Stack Overflow

Details: I have the following visuals: Visual for HR Dashboard. The headcount code is as follow: Headcount = CALCULATE ( DISTINCTCOUNT ( FactTable [EmpID]), FILTER ( ALL (FactTable), FactTable [EmploymentStatus] = "Active" ) ) The problem is, that the visual doesn't slice on the seniority. Need help. powerbi dax. Share.

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### Motor Efficiency and Power Factor

Details: Power Factor Correction Electrical loss 2 between power plant and useful work output of motor was the transmission and transformer I2R and hysteresis losses resulting from the imaginary component of the power. This loss applies whenever an imaginary component is present- not just for motors. Power factor (PF) correction can reduce loss by

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### Harmonic Filter Calculator – Voltage Disturbance

Details: See: Power Factor Calculator. The harmonic filter calculator below can be used to calculate the filter parameters needed to tune to a particular frequency. Harmonic Filter Calculation. The following are the basic steps involved in design of single tuned or detuned power system harmonic filter calculation. These passive power filters are used in

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### Power Factor - TheElectricalGuy

Details: Now, it is important to understand where the capacitors are to be installed? So, if you want to get the details you need to watch the video. Improve power factor using these 3 ways. Know about the 3 most commonly used ways of improving power factor i.e. capacitor banks, synchronous condenser & Phase advancer.

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text:
×

# Base 10 $$Dictionary$$

There was once a Base-10 dictionary, giving out 10 letters from the alphabet, namely $$P,L,K,E,M,N,A,G,J,B$$, in this order : P as the 1st word, L as the 2nd word, K as the 3rd word...... ,J as the 9th word and B as the 10th word. There are some "basetenword"s in this dictionary, where you pick out 1 to 5 different letters to form a "basetenword", such as $$PPPPP, PLLLL, PLKKK, PLKMM, PLEJB$$. In which sequence is $$NBMBG$$, and find $$\sqrt[(NBMBG^{NBMBG})]{NBMBG}$$, correct to 2 decimal places.

**Note: There are $$10^{5}-10^{4}$$ "basetenword" in this Base 10 $$Dictionary$$

e.g. MPJBN is in the $$52006^{th}$$ "basetenword"

Note by Bryan Lee Shi Yang
2 years, 1 month ago


text:
# How to pick a basis for the result of a non-linear function given a basis for its argument

I am trying to represent the result of a non-linear function in a small basis, given another small basis that does a good job a representing the argument of the function.

More specifically, there is a non-linear map, $$f: \left< r \middle| \rho \middle| r \right> \longrightarrow \left< r \middle| V \middle| r \right>$$ or alternatively $f[\rho(r)] = V(r)$, where $\rho$ and $V$ are Hermitian and $V$ is diagonal in real-space. I already have a great (small) basis $u_i$ that represents $\rho$ perfectly: $$\sum_i \left< r \middle| \rho \middle| r \right> \left<r\middle|u_i\right> \left<u_i\middle|r\right> = \left< r \middle| \rho \middle| r \right>$$

With access to $f$, is there anything I can do to the basis $u_i$ to make it represent $V$ more completely?

$f$ generally has terms that look like $\rho(r)^{1/3}$, plus some linear terms, but can be more general. I don't need to evaluate $f$ in this basis; I just want to be able to transform $V$ into it after it is computed.

Scientific back-story: $\rho$ is an electron density and $V$ is the local potential. Real-space is really big, so I'm trying to get out of it as quickly as possible.

-

This sounds pretty close to what people doing reduced order modeling for optimization of nonlinear differential equations are working on. They have a reduced basis for expressing (and computing) an approximation $\tilde y$ of the solution $y$, but evaluation of the nonlinearities $f(\tilde y)$ in the equation still have to be performed in the full space. To circumvent this, they construct a reduced model $\widetilde{f(\tilde y)}$ from a basis $\{f(\tilde y_1),f(\tilde y_2),\dots\}$.


text:
Learning to play guitar is loads of fun, though playing chords may seem a little intimidating at first. Fear not, it is not much different than playing single notes: you're just playing them all at once! This article will walk you through the process of working out the fingering, and show you how to play some common chords. Pull out your axe, and rock on!
Classical guitars, also known as "Spanish" guitars [11] , are typically strung with nylon strings, plucked with the fingers, played in a seated position and are used to play a diversity of musical styles including classical music. The classical guitar's wide, flat neck allows the musician to play scales, arpeggios, and certain chord forms more easily and with less adjacent string interference than on other styles of guitar. Flamenco guitars are very similar in construction, but they are associated with a more percussive tone. In Portugal, the same instrument is often used with steel strings particularly in its role within fado music. The guitar is called viola, or violão in Brazil, where it is often used with an extra seventh string by choro musicians to provide extra bass support.
In music, a guitar chord is a set of notes played on a guitar. A chord's notes are often played simultaneously, but they can be played sequentially in an arpeggio. The implementation of guitar chords depends on the guitar tuning. Most guitars used in popular music have six strings with the "standard" tuning of the Spanish classical-guitar, namely E-A-D-G-B-E' (from the lowest pitched string to the highest); in standard tuning, the intervals present among adjacent strings are perfect fourths except for the major third (G,B). Standard tuning requires four chord-shapes for the major triads.
Português: Tocar Acordes de Guitarra, Español: tocar acordes de guitarra, Deutsch: Akkorde auf der Gitarre spielen, Italiano: Suonare gli Accordi con la Chitarra, Français: jouer des accords à la guitare, Русский: играть аккорды на гитаре, 中文: 弹吉他和弦, Nederlands: Gitaarakkoorden spelen, हिन्दी: गिटार बजाएँ, Tiếng Việt: Chơi Hợp âm Ghita, ไทย: จับคอร์ดกีตาร์, 日本語: ギターでコードを弾く, العربية: عزف كوردات الجيتار
The ratio of the spacing of two consecutive frets is {\displaystyle {\sqrt[{12}]{2}}} (twelfth root of two). In practice, luthiers determine fret positions using the constant 17.817—an approximation to 1/(1-1/ {\displaystyle {\sqrt[{12}]{2}}} ). If the nth fret is a distance x from the bridge, then the distance from the (n+1)th fret to the bridge is x-(x/17.817).[15] Frets are available in several different gauges and can be fitted according to player preference. Among these are "jumbo" frets, which have much thicker gauge, allowing for use of a slight vibrato technique from pushing the string down harder and softer. "Scalloped" fretboards, where the wood of the fretboard itself is "scooped out" between the frets, allow a dramatic vibrato effect. Fine frets, much flatter, allow a very low string-action, but require that other conditions, such as curvature of the neck, be well-maintained to prevent buzz.
Once you've got your categories narrowed down, then you can start getting into the nitty-gritty differences between strings. For instance, electric guitars will give you the choice between nickel (for authentic vintage sound) and stainless steel (for maximum durability). Some string manufacturers have exotic material options with their own unique characteristics, like Ernie Ball's Slinky Cobalt strings. With so many subtle differences separating guitar strings, you owe it to yourself to browse carefully and look at all the choices on the table before making a decision.


text:
# Interesting Problem

• July 11th 2006, 03:15 AM
nath_quam
Interesting Problem
The unit square ABCD where A(1,0), B(1,1), C(2,1), D(2,0)

i) A line, l , passing through the origin with gradient m, cuts the sides AB and CD at P and Q respectively
is the answer to this part
$
m < \frac{1}{2}\
$

i am unsure on how to approach these two parts any help would be most appreciated.
ii) For what value(s) of m does the line, l, divide the area of the square in the ratio 2:1??

iii) Another like, k, passes through the origin with gradient, n, and cuts the square through the sides AB and Bc at S and T respectively.

Show that it is not possible for k to divide the area of the square in the ratio 2:1

Thanks Nath
• July 11th 2006, 04:17 AM
malaygoel
Quote:

Originally Posted by nath_quam
The unit square ABCD where A(1,0), B(1,1), C(2,1), D(2,0)

i) A line, l , passing through the origin with gradient m, cuts the sides AB and CD at P and Q respectively
is the answer to this part
$
m < \frac{1}{2}\
$

Here is first.
Let the coordinates of P be (1,a) and Q be (2,b).Let O be the origin.
m is the slope.
$m=\frac{a}{1}=\frac{b-a}{1}$
it gives $m = a = \frac{b}{2}$
Since B lies on CD $b < 1$
Hence $m<\frac{1}{2}$
• July 11th 2006, 04:25 AM
malaygoel
Quote:

Originally Posted by nath_quam
The unit square ABCD where A(1,0), B(1,1), C(2,1), D(2,0)

i) A line, l , passing through the origin with gradient m, cuts the sides AB and CD at P and Q respectively
is the answer to this part
$
m < \frac{1}{2}\
$

i am unsure on how to approach these two parts any help would be most appreciated.
ii) For what value(s) of m does the line, l, divide the area of the square in the ratio 2:1??

I will continue with the notation of the previous post.
Let the area of trapezium ADQP be x
Then there are two cases:
$\frac{x}{1-x}=2,\frac{1}{2}$
$x=\frac{2}{3},\frac{1}{3}$
From the diagram, $x=\frac{1}{2}(a+b)=\frac{3a}{2}$
Hence values of slope are $m=\frac{2}{9},\frac{4}{9}$

Keep Smiling
Malay
• July 11th 2006, 04:37 AM
nath_quam
Thanks Malay i am understanding any idea on the last?
• July 11th 2006, 04:39 AM
malaygoel
Quote:

Originally Posted by nath_quam
The unit square ABCD where A(1,0), B(1,1), C(2,1), D(2,0)

iii) Another like, k, passes through the origin with gradient, n, and cuts the square through the sides AB and Bc at S and T respectively.

Show that it is not possible for k to divide the area of the square in the ratio 2:1

Join O and C.
The square is divided into two parts.
Area of the smaller part(containing the vertex B) is 1/4.
Now if the line passes through AB and BC, the triangle formed by it has area less than 1/4. But for the ratio to be 2:1, it has to be either 1/3 or 2/3(see previous post), which is not possible since 1/3>1/4 and 2/3 >1/4.

Keep Smiling
Malay
• July 11th 2006, 04:46 AM
nath_quam
Quote:

Originally Posted by malaygoel
$\frac{x}{1-x}=2,\frac{1}{2}$
$x=\frac{2}{3},\frac{1}{3}$
From the diagram, $x=\frac{1}{2}(a+b)=\frac{3a}{2}$
Hence values of slope are $m=\frac{2}{9},\frac{4}{9}$

Keep Smiling
Malay

i get sorta lost here u jump steps too quickly would u be able to explain it
• July 11th 2006, 05:07 AM
malaygoel
Quote:

Originally Posted by nath_quam
i get sorta lost here u jump steps too quickly would u be able to explain it

I used two equations of the previous post.
b=2a
and m=a

Keep Smiling
Malay
• July 16th 2006, 05:03 AM
nath_quam
Quote:

Originally Posted by malaygoel
Here is first.
Let the coordinates of P be (1,a) and Q be (2,b).Let O be the origin.
m is the slope.
$m=\frac{a}{1}=\frac{b-a}{1}$
it gives $m = a = \frac{b}{2}$
Since B lies on CD $b < 1$
Hence $m<\frac{1}{2}$

and also greater than zero??
• July 16th 2006, 05:16 AM
malaygoel
Quote:

Originally Posted by nath_quam
and also greater than zero??

Yes.
You could see it this way also: Line rises towards right, hence it has a positive slope. It cannot be zero because it will imply that line doesn't rise towards right.

Keep Smiling
Malay
• July 16th 2006, 07:29 AM
Rich B.
Show that it is not possible for k to divide the area of the square in the ratio 2:1

Hi:

If the line passes through side(BC), then its slope must be greater than 1/2. But, as Malay has demonstrated, the slope must have value 2/9 or 4/9, neither of which is greater 1/2.

Regards,

Rich B.


text:
# RdUtils

0th

Percentile

##### Utilities for Processing Rd Files

Utilities for converting files in R documentation (Rd) format to other formats or create indices from them, and for converting documentation in other formats to Rd format.

Keywords
utilities
##### Usage
R CMD Rdconv [options] file
R CMD Rd2pdf [options] files
##### Arguments
file

the path to a file to be processed.

files

a list of file names specifying the R documentation sources to use, by either giving the paths to the files, or the path to a directory with the sources of a package.

options

further options to control the processing, or for obtaining information about usage and version of the utility.

##### Details

R CMD Rdconv converts Rd format to plain text, HTML or LaTeX formats: it can also extract the examples.

R CMD Rd2pdf is the user-level program for producing PDF output from Rd sources. It will make use of the environment variables R_PAPERSIZE (set by R CMD, with a default set when R was installed: values for R_PAPERSIZE are a4, letter, legal and executive) R_PAPERSIZE (default a4, also legal and executive) and R_PDFVIEWER (the PDF previewer). Also, RD2PDF_INPUTENC can be set to inputenx to make use of the LaTeX package of that name rather than inputenc: this might be needed for better support of the UTF-8 encoding.

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maths > fractions

Equivalent Fractions and Simplest form of a fraction

what you'll learn...

overview

In this page, equivalent fractions and simplest form of a fraction are explained

•  Two or more fractions that specify the same amount with different place values are equivalent fractions.

•  Among the set of equivalent fractions, the fraction with lowest denominator (representing lowest number of cuts of the whole) is the simplest form of a fraction.

split into more or less

The whole is split into $4$$4$ pieces and $2$$2$ pieces are considered. The fraction represented by the colored part is $\frac{2}{4}$$\frac{2}{4}$

Whole is split into $2$$2$ pieces and $1$$1$ piece is considered. The fraction represented by the colored portion is $\frac{1}{2}$$\frac{1}{2}$.

The picture shows three parts in three different colors.

•  The gray part shows the whole

•  The orange part shows the fraction $\frac{2}{4}$$\frac{2}{4}$

•  The purple part shows the fraction $\frac{1}{2}$$\frac{1}{2}$

The fraction $\frac{2}{4}$$\frac{2}{4}$ and $\frac{1}{2}$$\frac{1}{2}$ represent the same amount and the same number.

The numerator and denominator of one fraction is a multiple of numerator and denominator of another fraction respectively. Such fractions are called equivalent fractions.

The word "equivalent" means equal in value or amount. The equivalent fractions are fraction of equal amount.

simplify

The picture shows four fractions , $\frac{1}{2}$$\frac{1}{2}$, $\frac{2}{4}$$\frac{2}{4}$, $\frac{3}{6}$$\frac{3}{6}$, and $\frac{4}{8}$$\frac{4}{8}$. These are equivalent fractions.

The fractions $\frac{2}{4}$$\frac{2}{4}$, $\frac{3}{6}$$\frac{3}{6}$, and $\frac{4}{8}$$\frac{4}{8}$ have numerator and denominator with common factors $2$$2$, $3$$3$ and $4$$4$ respectively.

It is noted that
•  in $\frac{1}{2}$$\frac{1}{2}$, the whole is cut into two pieces,
•  in $\frac{2}{4}$$\frac{2}{4}$ the whole is cut into four pieces,
•  And so on for $\frac{3}{6}$$\frac{3}{6}$ and $\frac{4}{8}$$\frac{4}{8}$, with six and eight pieces respectively.
All these fractions represent the same amount and among these, $\frac{1}{2}$$\frac{1}{2}$ has the smallest number of cuts. It is the simplest form of all fractions.

It is noted that, only the fraction $\frac{1}{2}$$\frac{1}{2}$ has no common factor between numerator and denominator.
A fraction without a common factor between the numerator and the denominator is the simplest form of the equivalent fractions.

The word "simplest" means: without complexity, plain, simple and elementary. A fraction without a common factor between the numerator and the denominator is the simplest form of the equivalent fractions.

Fractions that specify the same amount with different place values are equivalent fractions.

The fraction that specifies the same amount with lowest denominator possible is the simplest form of equivalent fractions.

Equivalent Fraction: Two fractions $\frac{p}{q}$$\frac{p}{q}$ and $\frac{l}{m}$$\frac{l}{m}$ are equivalent if the numerators $l$$l$, $p$$p$ and the denominators $m$$m$, $q$$q$ have a common factor $a$$a$.

That is
$\quad l = a \times p$ and
$\quad m = a \times q$.
$a$$a$ is an integer.
Note that, $\frac{l}{m}$$\frac{l}{m}$ and $\frac{p}{q}$$\frac{p}{q}$ specify the same amount with different place values.

Simplest Form of a fraction: A fraction $\frac{p}{q}$$\frac{p}{q}$ without a common factor between the numerator and the denominator is the simplest form of the equivalent fractions. That is, $p\ne a×q$$p \ne a \times q$ for any integer $a$$a$.

Given fractions $\frac{1}{2}$$\frac{1}{2}$and $\frac{2}{4}$$\frac{2}{4}$. Which one is the simple form of the given equivalent fractions?
'$\frac{1}{2}$$\frac{1}{2}$' is the simple form of the two fractions. .

summary

»  Equivalent Fractions
→  fractions that specify the same amount with different place values
»  Simplest Form
→  fraction that specify the amount in lowest denominator possible
→  eg: Simplest form of $\frac{2}{4}$$\frac{2}{4}$ is $\frac{1}{2}$$\frac{1}{2}$

Outline