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Mahan excitons in room-temperature methylammonium lead bromide perovskites

Abstract

In a seminal paper, Mahan predicted that excitonic bound states can still exist in a semiconductor at electron-hole densities above the insulator-to-metal Mott transition. However, no clear evidence for this exotic quasiparticle, dubbed Mahan exciton, exists to date at room temperature. In this work, we combine ultrafast broadband optical spectroscopy and advanced many-body calculations to reveal that organic-inorganic lead-bromide perovskites host Mahan excitons at room temperature. Persistence of the Wannier exciton peak and the enhancement of the above-bandgap absorption are observed at all achievable photoexcitation densities, well above the Mott density. This is supported by the solution of the semiconductor Bloch equations, which confirms that no sharp transition between the insulating and conductive phase occurs. Our results demonstrate the robustness of the bound states in a regime where exciton dissociation is otherwise expected, and offer promising perspectives in fundamental physics and in room-temperature applications involving high densities of charge carriers.

Introduction

In the quest for highly efficient light-energy conversion schemes in semiconductors, the manipulation of excitons is playing an increasingly central role. In photovoltaics, the formation of bound electron–hole (e–h) pairs upon photon absorption has dramatic effects on the charge transport, hindering efficient e–h dissociation at heterojunctions. In light-emitting devices, strongly bound excitons are instead desirable to reach high e–h capture rates for radiative recombination1,2. With these perspectives in mind, it is key to reveal the strength of e–h correlations in different classes of semiconductors and clarify how excitons react to the stimuli present in optoelectronic devices.

One such stimulus is the free carrier density injected via chemical doping or photoexcitation. Tuning this parameter results in a plethora of single-particle and many-body phenomena that profoundly influence the excitonic states3. For example, exchange and correlation effects modify the underlying electronic structure through bandgap renormalization (BGR). The free carriers also perturb the energy- and momentum-dependence of the material’s dielectric background, screening the exciton binding energy (Eb). Finally, the single-particle states become partially filled, which leads to the weakening of the exciton oscillator strength and to the Burstein–Moss shift of the absorption onset. As the carrier concentration increases further, exciton dissociation processes turn the exciton gas into an ionized e–h plasma (scenario (i) in Fig. 1a). This is the essence of the paradigmatic insulator-to-metal (Mott) transition of band semiconductors4, which manifests itself with the broadening and complete suppression of the excitonic peak (Fig. 1b, blue curve) at the so-called Mott density (nM)5.

While this picture of the Mott transition is correct to a first approximation, in 1967 Mahan proposed that bound excitonic states can still survive at doping levels above nM6. In his seminal paper, Mahan considered a degenerate Fermi sea involving one type of carriers, akin to a chemically-doped semiconductor (scenario (ii) in Fig. 1a). In this high-density phase, an excited e–h pair can still interact with the electron gas, giving rise to exciton-like bound states—the so-called Mahan excitons. These states cause a singularity of the absorption at the Fermi edge, enhancing the above-gap oscillator strength (Fig. 1b, red curve). The phenomenon is known as excitonic enhancement or Fermi edge singularity and it can also emerge in metals7,8. While the original prediction discussed the quasi-equilibrium of chemically-induced carriers5, joint theoretical-experimental studies extended this framework to photoexcited systems by defining quasi-Fermi levels for the e–h plasma (scenario (iii) in Fig. 1a)9,10,11,12. As a result, Mahan excitons can still be defined when intraband cooling is complete and the carriers have established a quasi-equilibrium within the bands. This scenario holds in a window of excitation densities between nM and the density required for population inversion, over which the Mahan excitons are still influenced by the simultaneous action of single-particle and many-body effects. Eventually, the subtle interplay between these effects can also lead the correlated e–h system to exhibit a rich variety of collective phenomena, typically at low temperature. Notable examples include the emergence of correlated metallicity13, the instability towards nonequilibrium excitonic insulator phases (involving the Bose condensation of photoexcited e–h Cooper pairs)3,14, and the nucleation of e–h droplets15,16.

Discovering unambiguous signatures of Mahan exciton physics would open intriguing avenues in many-body theory and applied research. However, despite extensive efforts, this problem is far from being settled. So far, the majority of studies have focused on chemically- or photodoped semiconductors using steady-state spectroscopies9,10,11,12,17,18,19,20,21,22. Under static conditions, conventional lineshape analysis fails to disentangle the concurrent nonlinearities in the optical spectrum and advanced calculations are required to isolate the possible fingerprints of Mahan excitons21,23. In chemically-doped materials, this analysis is further complicated by the presence of donors/acceptors, which introduce random impurity-like potentials and significantly broaden any feature stemming from Mahan exciton physics10.

To overcome these limitations, a promising strategy is exploring a nonequilibrium time-dependent setting through pump–probe spectroscopy. This allows for discriminating different phenomena on the basis of their timescale and unveiling the possible persistence of excitonic correlations above nM. Preliminary results on indirect bandgap materials (Si and Ge) at cryogenic temperatures reported the robustness of the Wannier excitons above nM by monitoring the 1s–2p intraexciton transition through ultrafast terahertz spectroscopy24,25. However, the low probe photon energy used could not assess the presence of any Mahan exciton-related physics, i.e. the enhancement of the absorption continuum at high energy. The case of direct-gap materials in time-resolved experiments has been even more elusive to date, with no clear evidence for the existence of Mahan exciton states above nM26,27,28. Due to the reduced scattering channels for their excitonic states compared to indirect gap materials29, direct bandgap semiconductors hosting strongly bound excitons are the ideal platform in the search for Mahan excitons.

Among all such direct-gap semiconductors, the organic-inorganic lead-halide perovskites have recently attracted huge interest due to their outstanding photovoltaic performances30 and high photoluminescence quantum efficiencies1,31,32,33,34,35,36,37. Numerous studies focused on understanding the extent to which the exciton concept describes the photophysics of these systems38,39,40,41. In CH3NH3PbI3, Eb of the order of the thermal energy was reported at room temperature (RT), suggesting the coexistence of weakly bound excitons and free carriers40,42,43. In contrast, the smaller dielectric constant of CH3NH3PbBr3 single crystals results in a well-resolved excitonic peaks at RT, with Eb ranging from 60 to 72 meV44,45,46.

Here, we use methylammonium lead tribromide (CH3NH3PbBr3) single crystals at RT to reveal direct evidence for Mahan exciton physics through a combination of time-resolved spectroscopy and advanced many-body calculations. We accurately evaluate nM through theory, and track the optical response of the material at photoexcitation densities well above nM. A broadband optical probe allows us to uncover the persistence of the Wannier exciton peak, as well as the birth of the excitonic enhancement. We rationalize these observations through a model based on the semiconductor Bloch equations (SBEs), which confirms the absence of a sharp insulator-to-metal transition in the system and witnesses a crossover between the two phases. These results highlight the crucial role of e–h correlations in a regime where complete exciton dissociation is otherwise expected, paving the way towards a deeper understanding of the many-body phenomena emerging in semiconductors upon intense illumination47.

Results

Estimation of the nominal Mott density

A common approach to evaluate nM that is used in semiconductor physics directly compares the exciton Bohr radius and the screening length induced by the free carriers. This approximate method yields for CH3NH3PbI3 values of nM on the order of 1016–1018 cm−35,48,49. A similar estimate can be given for our sample of CH3NH3PbBr3 (Fig. 1c), setting for nM an upper limit of approximately 2 × 1018 cm−3 (see Supplementary Notes 1 and 2 for the estimation of Eb and nM, respectively). However, these values of nM do not account for many-body effects and are rather inaccurate. Therefore, only a sophisticated theory can provide a reliable description of the interaction between excitons and free carriers. To this aim, here we use a many-body treatment of the exciton-plasma conglomerate at a given temperature and excitation density using ionization equilibrium theory50 (see Supplementary Note 3). It is based on the assumption that optically-excited e–h pairs in a semiconductor can form either a plasma of unbound (yet correlated) carriers, or a gas of bound excitons, with all possible states of coexistence between the two phases depending on the experimental parameters. This approach yields the fraction of carriers forming the plasma (nfree) or the excitons (nX). The fraction of ionized e–h pairs αeh = nfreen, where n = nfree + nX is the total density of e–h pairs, is shown as red dots in Fig. 1d. At low carrier densities, αeh closely follows the ionization ratio predicted by the Saha equation (solid line in Fig. 1d), which assumes the classical thermal equilibrium between free and bound pairs, and predicts the formation of excitonic species with increasing n (see Supplementary Note 3). However, at about 5 × 1017 cm−3, the two curves abruptly deviate, and αeh increases to 1. Many-particle renormalization and Coulomb screening (CS) lead to full exciton ionization at nM approximately 8 × 1017 cm−3 for CH3NH3PbBr3 single crystals. αeh is calculated using the experimental values for Eb, dielectric constant, and e–h effective masses, i.e. the uncertainty in nM is related to that of these quantities, which remains below 10%.

Next, we map the ultrafast response of the material up to e–h densities above nM. Specifically, we use intense above-bandgap excitation with an ultrashort laser pulse and monitor the pump-induced changes in the exciton optical response with a delayed continuum probe (details are given in the Methods section). Since the correct estimate of the photoexcited carrier density is central in this study, we precisely characterize all the uncertainties that may affect the experimental parameters (see Supplementary Note 4) and find that the error on the reported excitation densities is approximately 5%. A different choice of the excitation volume yields a variation in the density of less than a factor 2.

Figure 2a shows the color map of the transient reflectivity (ΔRR) as a function of probe photon energy and time delay between pump and probe for the excitation density of 5 × 1018 cm−3. We observe a derivative-like spectral response, which is positive above 2.40 eV and negative below. For time delays more than 1 ps, this signal agrees with previous reports at lower densities (5 × 1017 cm−3), where it was assigned to the bleaching of the excitonic feature by band filling (BF)51. A closer inspection of our data reveals a complex evolution occurring in the first picoseconds. Specifically, the signal amplitude rises in the first 500 fs with an isosbestic point around 2.40 eV (Fig. 2b), while at later times it undergoes an asymmetric evolution around the inversion point, accompanied by a sizeable redshift of both the zero-crossing energy and the positive shoulder maximum (Fig. 2c). This trend is reproduced at different excitation densities above nM (5, 7.5 and 10 × 1018 cm−3, see Supplementary Fig. 4).

To retrieve the absorption spectra at the different excitation densities and time delays, we combine our steady-state and time-resolved optical data (see Supplementary Note 5). We model the reflectivity spectrum with a set of Lorentz oscillators, and describe its spectral changes through the variation of the model parameters. By iterating the fit for each time delay, we extract the time evolution of the oscillator strength (ΔOS), peak energy (ΔEX), and linewidth (ΔΓ) (Fig. 3a–c), as well as the time-dependent absorption coefficient α(ωt) (Fig. 3d, e). For the lowest density (n = 5 × 1018 cm−3), we observe only a 10% decrease of the oscillator strength (Fig. 3a), accompanied by a slight redshift of EX (Fig. 3b) and a net increase of Γ (Fig. 3c). All these effects occur in less than 500 fs. The same procedure is applied to the data measured at higher excitation densities (Supplementary Fig. 7), resulting in the absorption spectra in Supplementary Fig. 8. We observe that renormalization effects become more pronounced with increasing n; nevertheless, the Wannier exciton resonance remarkably persists, retaining a finite oscillator strength at all time delays. In addition, spectral weight is transferred from the exciton peak to the above-bandgap region, providing an enhancement of the absorption that has never been reported before. These results are summarized in Fig. 4a, which shows the RT absorption spectrum at three different excitation densities and at a representative time delay. The persistence of the excitonic feature in the absorption spectra up to 1019 cm−3 is further confirmed by a separate lineshape analysis, presented in Supplementary Note 6. These observations support Mahan’s prediction6 and deviate from the conventional scenario of the Mott transition4, in which an abrupt transformation from an insulating to a metallic phase occurs above nM, along with a disappearance of the excitonic correlations in the absorption spectrum.

Semiconductor Bloch equations

In order to rationalize the fate of the excitons above nM, we use the SBE52,53,54 and calculate the absorption spectrum of CH3NH3PbBr3 in the presence of photoexcited carriers. The SBEs describe the two-particle optical response based on the single-particle band structure, e–h occupancies, dipole matrix elements, and Coulomb interaction-induced bound states. Many-body effects that shape the optical response in the high-excitation regime, consistent with the theory of ionization equilibrium, can be systematically included. Since excitons are neutral particles, they do not cause significant BGR or CS. Thus, we consider only the influence of free carriers on the absorption spectrum. In addition, while spin–orbit splitting and Rashba effects are not explicitly included in the SBE, the effective masses utilized in our model originate from previous GW calculations that effectively account for the spin–orbit splitting55. More details are given in Supplementary Note 7.

The solution of the SBE under these assumptions yields the theoretical absorption spectra plotted in Fig. 4b for different photoexcited carrier densities (0.1 to 3.2 × 1018 cm−3). Here, the traces at the lowest densities are taken as a reference for the absorption coefficient at n = 0 cm−3, since no sizeable changes are observed in the optical spectra up to n approximately 3.2 × 1017 cm−3 (see Supplementary Note 7). The computed spectra also show the persistence of the excitonic feature above nM (8 × 1017 cm−3), along with the enhancement of the above-bandgap absorption with increasing carrier density (black arrow), in excellent qualitative agreement with the experimental data. The theory predicts a blueshift of the absorption edge with increasing density that is also apparent in the experimental spectra. The size of the blueshift in the measured data (see Supplementary Fig. 8) originates from the different temporal evolution of the optical nonlinearities (BF, BGR, and CS) acting on the Wannier exciton peak. The mismatch between the results in Fig. 4a, b is governed by the assumptions underlying our theoretical model. In particular, the GW self-energy tends to overestimate renormalization effects, and our prediction of the critical density might be slightly smaller. In the SBE, the description of CS and BGR can be improved by introducing a vertex correction to the self-energy. However, such a correction is computationally challenging and results in a small variation of nM, which should in any case not be larger than the experimental e–h densities.

Therefore, the qualitative agreement between theory and experiment is remarkable and it confirms the persistence of the bound states at the Mott transition contrary to their disappearance, which would result in optical properties resembling those of non-interacting e–h pairs (Fig. 1b, blue curve). The origin of the above-bandgap absorption increase can be explained by considering the distribution of e–h populations upon photoexcitation (scenario (iii) in Fig. 1a), where the excitonic enhancement arises from the strong e–h correlations in the Fermi sea above nM. Therefore, our calculations clarify that the Mahan excitons here detected do not stem from single electrons (holes) interacting with a degenerate Fermi sea of holes (electrons). Moreover, the robustness of the low-energy Wannier exciton provides additional evidence that the excitonic correlations are not completely screened in the highly-photoexcited state. These findings strongly indicate the existence of a crossover between the insulating and conductive phases in highly-excited CH3NH3PbBr3 rather than a phase transition. In this scenario, e–h pairs still lead to exciton-like states, but in the form of Mahan excitons5,12,21,25. Finally, to investigate the role of the material parameters in this process, we solve the SBE for different values of dielectric constant (3.5, 7, and 10) and e–h effective masses (0.5, 0.75, and 1.5 times the effective masses indicated in Supplementary Note 2), which in turn yields different values of Eb and nM. The theoretical absorption spectra of CH3NH3PbBr3 in the presence of increasing n are shown in Supplementary Fig. 12. These results suggest that the presence of a large Eb is key to the emergence of Mahan exciton physics, irrespective of whether this originates from a small dielectric constant or a large exciton effective mass (bottom graphs in Supplementary Fig. 12b, c). In contrast, for small Eb, the exciton peak promptly disappears as the e–h density overcomes nM, indicating that under these conditions a Mott scenario still holds and the excitons become fully ionized.

We point out that there is no conflict between the exciton ionization ratio estimated in thermal equilibrium and the e–h correlations surviving above nM in the SBE solution. In fact, the theory of exciton ionization equilibrium is the most elaborate way to estimate the Mott density in a semiconductor. It assumes that the excited carrier density influences only the quasiparticle gap, but not the 1s-exciton energy. Under these conditions, nM is reached when the BGR compensates the exciton Eb. The SBE act as a complementary strategy that relies on the equilibrium state of the photoexcited carriers as an input. The SBEs can display the fate of possible excitonic correlations on top of the continuum by considering a density-dependent exciton energy in a particular parameter space of Eb and n.

Discussion

The present results show the peculiar excitonic character of the fundamental photoexcitations in CH3NH3PbBr3 even beyond the densities at which, in the Mott scenario, the e–h plasma is expected to form. This observation is rather surprising and raises the question why similar fingerprints of Mahan excitons have not been reported before in other classes of direct-gap semiconductors. In the following, we ascribe our observations to the careful choice of experimental parameters in this study. First, the use of single crystals (instead of polycrystalline thin films) is beneficial to the visibility of the excitons, which are known to be extremely sensitive to grain boundaries and inhomogeneities41. Second, transient reflectivity represents the most accurate experimental methodology to map the existence and behavior of an exciton immediately after its creation. Alternative techniques like ultrafast photoluminescence would probe the exciton properties immediately after its annihilation, which is suitable neither to accurately track exciton ionization in semiconductors nor to identify the spectral signatures of Mahan excitons56,57 (in particular the above-gap absorption enhancement). Third, the combination of large excitation densities above nM and RT: While lowering the temperature leads to a sharper quasi-Fermi surface, it also induces structural phase transitions58, fine-structure splittings59, and polaronic sidebands60 that render the isolation of the spectroscopic fingerprint due to Mahan excitons very challenging. The final reason lies in the choice of the present material. Indeed, in the hybrid perovskite of the CH3NH3PbI3 type (where Eb is very small), a prompt disappearance of the bound exciton peak was reported using ultrafast THz spectroscopy and assigned to the dynamic screening of the exciton population49, similarly to what observed in GaAs61. In CH3NH3PbBr3, the large Eb is key to hindering the full ionization of the exciton gas and therefore allows Mahan exciton-related spectral features to emerge at high carrier densities.

The present study supports a scenario in which a larger Eb in a direct-bandgap semiconductor leads to a stronger persistence of the e–h correlations above nM, irrespective of whether the enhanced Eb stems from a small dielectric constant or from large effective masses. Our joint experimental-theoretical effort points toward rational-design strategies to search for signatures of Mahan exciton physics in other photoexcited direct-gap semiconductors with comparable Eb. Notable examples include wide-gap oxides such as ZnO (Eb = 60 meV) and nitrides such as GaN (Eb = 30 meV). These materials have been studied under steady-state conditions using chemical doping exceeding nM. While features that could be attributed to Mahan excitons were highlighted21,22, the spectra remained ambiguous because of inhomogeneities and the static nature of the measurement technique. In this respect, the application of pump–probe spectroscopies appears as a promising avenue to solve long-standing questions about the insulator-to-metal transition in band semiconductors. Preliminary results on highly-photoexcited ZnO seem to confirm this direction62.

On the fundamental side, these findings deepen our knowledge of the exciton Mott transition and open intriguing possibilities towards the use of CH3NH3PbBr3 as a suitable platform for Bose-Einstein condensation of Wannier exciton-polaritons in ad hoc designed microcavities63. On the applications side, the exceptional stability of the excitons highlights the potential of this hybrid perovskite in RT devices that demand high injected carrier densities, such as light-emitting devices and lasers1,2.

Methods

Synthesis of CH3NH3PbBr3 single crystals

We synthesized high-purity crystals of CH3NH3PbBr3 via solution growth as described by ref. 64. Freshly cleaved surfaces of a few mm2 were obtained by applying mechanical stress on the crystals with the tip of a blade. This results in the clean and shiny surface indicated by the black arrow in Supplementary Fig. 1a. The high crystallinity of the sample is demonstrated by the X-ray diffractogram in Supplementary Fig. 1b, which shows sharp lines for the oriented lattice planes and is free from the characteristic lines of degradation products.

Spectroscopic ellipsometry measurements

Steady-state spectroscopic ellipsometry measurements were performed to characterize the optical properties of the crystal at RT. The ellipsometric quantities Ψ and Δ were measured for incident angles of 60, 75, and 78 in the energy range 1.00–5.50 eV, and used to obtain the complex dielectric function ε = ε1 + iε2 (Supplementary Fig. 2a). To calculate the absorption spectrum (Fig. 1c) and the reflectivity (Supplementary Fig. 2b) we used the relations:

$$\alpha =\frac{2\kappa \omega }{c},\qquad R=\frac{{(1-n)}^{2}+{\kappa }^{2}}{{(1+n)}^{2}+{\kappa }^{2}},$$
(1)

where n and κ are, respectively, the real and imaginary parts of the complex refractive index $$\tilde{n}=n+{\rm{i}}\kappa$$:

$$n={\left(\frac{{\varepsilon }_{1}+\sqrt{{\varepsilon }_{1}^{2}+{\varepsilon }_{2}^{2}}}{2}\right)}^{\frac{1}{2}},\qquad \kappa ={\left(\frac{-{\varepsilon }_{1}+\sqrt{{\varepsilon }_{1}^{2}+{\varepsilon }_{2}^{2}}}{2}\right)}^{\frac{1}{2}}.$$
(2)

Transient reflectivity measurements

Femtosecond broadband transient reflectivity (ΔRR) experiments were performed using the setup described in ref. 65. A Ti:sapphire oscillator, pumped by a continuous-wave Nd:YVO4 laser, delivers sub-50 fs pulses at 1.55 eV with a repetition rate of 80 MHz. The output of the oscillator seeds a cryo-cooled Ti:sapphire amplifier, which is pumped by a Q-switched Nd:YAG laser. The amplified laser system provides 45 fs pulses at 1.55 eV and a repetition rate of 6 kHz. One third of the output, representing the probe beam, is sent to a motorized delay line to set a controlled delay between pump and probe. The 1.55 eV beam is focused onto a 3 mm-thick CaF2 cell using a combination of a lens with short focal distance and an iris to limit the numerical aperture of the incoming beam. The generated continuum covers the 2.00–2.80 eV spectral range. The probe is subsequently collimated and focused onto the sample through a pair of parabolic mirrors under an angle of 15. The remaining two thirds of the amplifier output, representing the pump beam, is frequency-doubled to 3.10 eV with a β-barium borate (BBO) crystal and directed towards the sample under normal incidence. Along the pump path, a chopper with a 60 slot plate is inserted, operating at 1.5 kHz and phase-locked to the laser system. Both pump and probe beams are focused onto the sample on spots of dimensions 155 μm × 125 μm and 60 μm × 30 μm, respectively. The sample is mounted inside a closed-cycle cryostat. The reflected probe is dispersed by a fiber-coupled 0.3 m spectrograph and detected on a shot-to-shot basis with a complementary metal-oxide-semiconductor linear array. In a typical experiment, the acquisition of each data set is repeated multiple times to improve the statistics of the measurement. Hence, the experiment strongly relies on the repeatability of the scans, which requires stability of the sample under laser light illumination for several hours. Before the data analysis, the ΔRR matrix has to be corrected for the group velocity dispersion of the probe. Since the probe beam is not dispersion-compensated after generation of the white light continuum, the probe pulses arrive at the sample stretched to a duration of few ps. This is beneficial for the experiment, because it significantly reduces the instantaneous probe light intensity on the sample. It is important to note that the probe beam dispersion is not a limiting factor for the time resolution of the setup, which is given on the detection side by the much smaller effective pulse duration per detector pixel. As such, the time resolution of the setup for all probe photon energies is 50 fs. Finally, we devoted particular attention in maintaining the sample under controlled environmental conditions to avoid its degradation or contamination. To this aim, all measurements were performed at RT, while keeping the crystal in the cryostat under a pressure less than 10−8 mbar66,67. The actual excitation densities impinging on the sample are calculated as n = (1 − R)F∕(hνλp) where F is the pump fluence, hν is the pump photon energy, λp = 1∕α the penetration depth, and R is the reflectivity of the sample. The final excitation density also takes into account the double reflection of the pump beam due to the cryostat window. All the parameters are evaluated at the pump photon energy of 3.10 eV.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

The authors thank Prof. Giulia Grancini for useful discussions. T.P., E.B., A.A., and M.C. acknowledge financial support from the Swiss NSF via the NCCR:MUST, the contract 154056 (PNR 70, “Energy turnaround”), and the project PP00P2_170544. A.S. and F.J. acknowledge financial support from the Deutsche Forschungsgemeinschaft (RTG 2247 “Quantum Mechanical Materials Modelling”) as well as resources for computational time at the HLRN (Hannover/Berlin). M.K., E.H., and L.F. were supported by the ERC Advanced Grant “Picoprop” (670918).

Author information

Authors

Contributions

T.P. and E.B. performed the transient reflectivity experiments, A.S. and F.J. performed the theoretical calculations. T.P. and A.A. performed the spectroscopic ellipsometry measurements, M.K., E.H., and L.F. prepared and characterized the sample. T.P. and E.B. analyzed the experimental data, T.P. and E.B. contributed to the data interpretation, T.P., E.B., and M.C. wrote the final manuscript. All authors participated in the final version of the article. T.P., E.B., and M.C. conceived the study.

Corresponding authors

Correspondence to Edoardo Baldini or Majed Chergui.

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Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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

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Reprints and Permissions

Palmieri, T., Baldini, E., Steinhoff, A. et al. Mahan excitons in room-temperature methylammonium lead bromide perovskites. Nat Commun 11, 850 (2020). https://doi.org/10.1038/s41467-020-14683-5

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text:
@Jonathan, thank you for pointing me towards your post! I had a few questions as I was reading it and was wondering if you could clarify.

First you wrote
> in other words, we have \\((x, y) \le (x', y')\\) iff \\(x \le x' \land y \le y'\\)

I assume you used \\(\land\\) to mean "and" and not "join". Is that correct?

Second I am still unclear about what ordering you are getting on \\(\mathbb R ^2 \\) from product. You wrote:
> this lifts to the product straightforwardly

Any chance you could make that explicit?

format: plain


text:
An Efficient Bound for the Condition Number of the Matrix Exponential

Al-Mohy, Awad H. (2015) An Efficient Bound for the Condition Number of the Matrix Exponential. [MIMS Preprint]

A new bound for the condition number of the matrix exponential is presented. Using the bound, we propose an efficient approximation to the condition number, denoted by $\kappa_g(s,X)$, that \emph{avoids} the computation of the Fr\'echet derivative of the matrix exponential that underlies condition number estimation in the existing algorithms. We exploit the identity $e^X=(e^{X/2^s})^{2^s}$ for a nonnegative integer $s$ with the properties of the Fr\'echet derivative operator to obtain the bound. Our cost analysis reveals that considerable computational savings are possible since estimating the condition number by the existing algorithms requires several invocation of the Fr\'echet derivative of the matrix exponential whose single invocation costs as twice as the cost of the matrix exponential itself. The bound and hence $\kappa_g(s,X)$ only involve Fr\'echet derivative of a monomial of degree $2^s$, which can be computed exactly in $2s$ matrix multiplications. We propose two versions of the scaling and squaring algorithm that implement $\kappa_g(s,X)$. %that represents the squaring phase of %the scaling and squaring method. Our numerical experiments show that $\kappa_g(s,X)$ captures the behavior of the condition number and moreover outperforms the condition number in the estimation of relative forward errors for a wide range of problems.


text:
# Find the point on the line y = 5x + 3 that is closest to the origin.?

• let the point (m , n) on the line y = 5x + 3 is closest to origin (0,0)

substituting x = m and y = n in eqn of line

n = 5m + 3

so The point is (m . 5m + 3)

The distance, s between [ m , (5m + 3) ] and (0, 0) is

s = √[m^2 + (5m+3)^2 ]

= √(m^2 + 25m^2 + 30m + 9)

= √(26m^2 + 30m + 9)

differentiate and equate it to zero

(1/2)[ 52m + 30] / √(26m^2 + 30m + 9) = 0

=> 52m + 30 = 0

m = -30 / 52

= -15/26

n = 5m + 3

= -(75/26) + 3

= 3/26

The closest point to origin = (-15/26, 3/26)


text:
# Fourier transform of a Fourier transform

I generate a Gaussian noise and then I filter it with a passband FIR Kaiser window filter. When I perform the Fourier transform of the output of the filter and plot its magnitude spectrum, it is filtered in the correct frequency band. Now, I perform a second Fourier transform on the output of the first Fourier transform (I'm performing it on a complex vector this time and in the frequency domain), and when I plot the magnitude spectrum I see something very similar to the original filtered noise. I know this isn't time domain, I know this spectrum is not even because the input signal is complex. But as the time plot of the filtered noise and the absolute value plot of the Fourier transform (of the Fourier transform of the noise) are visually similar a doubt arised on me.

1. Why is the amplitude spectrum of the complex vector obtained after performing a Fourier tranform on the filtered noise once again occupying the whole vector (the whole frequency band) as if they were time samples spread in the time vector?
2. Is the negative symbol in the inverse Fourier transform of the Fourier transform of the noise having no effect so that it is visually similar performing a Fourier transform of Fourier transform of noise and performing an inverse Fourier transform of Fourier transform of noise?

This doubt arised to me when I was trying to filter on a matched filter implemented in Matlab using IFFT(FFT(signal_1).*FFT(signal_1_inverted)) and I accidentally filtered the output of an already Fourier transformed signal and coudn't notice the difference (not on the output of the filter but on the output of the first FFT stage). And this leads me to another question.

1. Suppose I would like to detect the likelihood of 2 signals by comparing their spectrums and I want to measure the likelihood by using a matched filter. Knowing that the convolution in time is the multiplication in frequency, is the convolution in frequency the multiplication in time and I get the same result if I use a matched filter (sort of convolution operation) on the spectrums than if I perform a sample by sample multiplication in time domain of the signals whose spectrums likelihood I want to determine?
• Related: this answer. – Matt L. Apr 3 '16 at 9:40
• If F(F(signal(n))) = signal(-n) is what wonders you, I suggest you to look at Fractional Fourier. – Valentin Tihomirov Apr 3 '16 at 16:17

I don't know what you've done wrong but applying the Discrete Fourier Transform (via FFT) twice to a time domain signal, should produce the original time domain signal in a circularly time reversed and linearly scaled by N form, as the Inverse Discrete Fourier transform is very similar to Forward Discrete Fourier transform except in a simple weight by N and the sign of the complex exponential, which reflects itself in the time reversal of the original signal...which is known as the Duality Property of DFT

Looking at the Inverse Discrete Fourier Transform of $X[k]$ we see: $$x[n] = \frac {1}{N} \sum_{k=0}^{N-1} {X[k]e^{j \frac{2\pi}{N} k n}}$$

and by just changing the sign of the exponent $e^{j \frac{2\pi}{N} k n}$ into negative by a change in the sign of the $n$ of $x[n]$ what you get is:

$$N x[(-n)_N] = \sum_{k=0}^{N-1} {X[k]e^{-j \frac{2\pi}{N} k n}}$$

which is N times the circularly reversed (obtained via modulus operator denoted by $(.)_N$) version of the original signal $x[n]$ that is obtained by an apparently forward but effectively inverse Discrete Fourier Transform applied to $X[k]$ treated as a time sequence, which is what you do at your second stage of applying forward FFT to $X[k]$ of $x[n]$

• What about question number 3? – VMMF Apr 5 '16 at 12:45
• good question! If your intention is to perform a matched filtering on the "frequency domain signal" which I havent heard of much, then you would perform a convolution in frequeny domain and this is indeed equivalent to a multiplication in time domain...Select signals length properly to avoid circular aliasing. – Fat32 Apr 5 '16 at 14:15
• Fat, if you understand the DFT and the DFS to be the same exact thing (which they are), then the DFT maps a discrete and periodic function of length $N$ to another discrete and periodic function of length $N$ and the iDFT maps it back. so $x[n+N] = x[n] \quad \forall n \in \mathbb{Z}$ and the same for $X[k]$. the "circularly reversed" is just a manifestation of that circularity. we can just say that the DFT of the DFT is $N \cdot x[-n]$ this is why some folks like the $\frac{1}{\sqrt{N}}$ scaling on both the DFT and iDFT. then applying the DFT twice simply reverses your input around $n$=0 – robert bristow-johnson May 23 '16 at 20:10
• @robert bristow-johnson In one book the author describes DFT, X[k], as an aperiodic sequence defined for k=0 to k=N-1 for which a negative index of -k or -n is mapped into a valid ranged index by the use of modulus operator $(-n)_N$ so that the user can mathematically stay in the DFT domain without referring back to the original DFS as you stated. So if one always invokes the inherent relation between DFS and DFT (as he should), then $x[-n]$ is the choice (as I did in my original answer) but soon I realised that the reader may misinterpret it for an aperodic signal and then added the modulus – Fat32 May 23 '16 at 20:29
• i think we're in agreement. i just sorta wanna repeat that when using the DFT, one is essentially periodically extending the finite-length (and as such, aperiodc) input to the DFT. in this question i get a little more fascistic about my insistence about formally modeling this periodic extension. – robert bristow-johnson May 23 '16 at 20:47

The difference between an IFFT and an FFT is (merely) in the direction of the result vector (indexed forwards or backwards/reversed in time), and usually a scaling factor of N or 1/N. So you should see nearly the same thing if the input is so close to symmetrical looking that it doesn't look too different when reversed, and your plot auto-scales.

• it's really just the difference between "$+j$" and "$-j$". although they are negatives of each other, so they add to zero, qualitatively they are exactly the same. both have equal claim to squaring to be $-1$. – robert bristow-johnson Apr 3 '16 at 5:14
• What about question number 3? – VMMF Apr 5 '16 at 12:45
• (3) is unclear as stated exactly what you are asking. If clarified, it likely should be separate question, not mixed in one about IFFTs. – hotpaw2 Apr 5 '16 at 13:25


text:
# Trim leading zeroes from second component of a string

I have a collection of strings as follows:

44.01
44.02
44.03
44.04
44.12
46.05
46

... and so on.

I want to remove leading zeros from the component of the string after the full stop.

As you can see from above, not all strings have a sub-component and not all sub-components will have leading zeros.

Is there a more elegant way of doing that from my attempt?

    List<string> codesWithSubCodes = new List<string>
{
"44.01", "44.02", "44.03", "44.04", "44.05", "44.06", "44.07", "44.08"
};

var codes = new List<string>();
foreach (var code in codesWithSubCodes)
{
if (code.Contains('.'))
{
var prefix = code.Split('.')[0];
var subCode = code.Split('.')[1].TrimStart('0');
codes.Add(prefix + '.' + subCode);
}
else
{
codes.Add(code);
}
}

foreach (var code in codes)
{
Debug.WriteLine(code);
}

• Are you trying to turn them into numbers? – Der Kommissar Jul 8 '15 at 17:14
• No, they will always remain as strings. – Ciaran Gallagher Jul 8 '15 at 17:15
• Is there ever more than 1 0? (46.002) – Der Kommissar Jul 8 '15 at 17:17
• No, the second component of each string (if there is a 2nd component) will always have 2 characters. – Ciaran Gallagher Jul 8 '15 at 17:20
• In that case, I just recommended two methods that could help you out. – Der Kommissar Jul 8 '15 at 17:31

## 2 Answers

The biggest thing I can see, which may or may not be an issue, is the manner in which you process the strings.

    if (code.Contains('.'))
{
var prefix = code.Split('.')[0];
var subCode = code.Split('.')[1].TrimStart('0');
codes.Add(prefix + '.' + subCode);
}
else
{
codes.Add(code);
}

This could be rewritten as:

var codeSections = code.Split('.');
var code = codeSections[0];

if (codeSections.Length == 2)
code += "." + codeSections[1].TrimStart('0');

codes.Add(code);

This saves you LoC, and probably makes it faster. (You could also possible make this a StringBuilder as well.) And in my opinion, this is more readable/concise/descriptive.

Another alternative:

code = code.Replace(".0", ".");

Both should have the same effect on the cases you provided.

However, if, for whatever reason, one of the cases ends up as 46.00, then the second method will only make it 46.0. Choose that method with great caution.

As the OP pointed out in an edit that should have been a comment, this code can all be done as LINQ:

var codes = codesWithSubCodes.ConvertAll(code => code.Replace(".0", "."));

If you want to remove any number of leading zeros from the second component of your string then you can do:

var query = codesWithSubCodes.Select(r=> new
{
SplitResult = r.Split('.'),
OriginalValue = r
})
.Select(r=>
r.SplitResult.Length == 2 ?
r.SplitResult.First() + "." + r.SplitResult.Last().TrimStart('0')
: r.OriginalValue
);

This is how this query works:

• First Split the string on delimiter and store it temporarily so that you don't have to do Split multiple times
• Later it checks if the result of Split is a 2 element array then return concatenation of first element , delimiter and last element with removing all the 0s from the beginning.
• If the Split results in length not being 2, then return the original element.

With respect to your code, you can improve following thing:

• Do not check for delimiter Contains, instead Split the string and check length. That way you only Split the element once.

So the iteration part could look like:

foreach (var code in codesWithSubCodes)
{
string[] SplitResult = code.Split('.');
if (SplitResult.Length == 2)
{
var prefix = SplitResult[0];
var subCode = SplitResult[1].TrimStart('0');
codes.Add(prefix + '.' + subCode);
}
else
{
codes.Add(code);
}
}

If you are always going to have a single zero after the . then you can use this simple LINQ query:

var codes = codesWithSubCodes.Select(r => r.Replace(".0", ".")).ToList();

• This is a very intriguing answer, I never thought to use LINQ like that for this one. – Der Kommissar Jul 9 '15 at 14:03


text:
# Solve the given differential equation by undetermined coefficients y''+7y'+6y=30

Solve the given differential equation by undetermined coefficients
$y{}^{″}+7{y}^{\prime }+6y=30$
You can still ask an expert for help

## Want to know more about Laplace transform?

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Stella Calderon
Consider the differential equation
$y{}^{″}+7{y}^{\prime }+6y=30$...(1)
Rewrite the equation,$\left({D}^{2}+7D+6\right)y=30$ where $D=\frac{d}{dx}$
Auxiliary equation is,
$f\left(m\right)=0$
${m}^{3}+7m+6=0$
${m}^{2}+6m+m+6=0$
$m\left(m+6\right)+1\left(m+6\right)=0$
$\left(m+6\right)\left(m+1\right)=0$
$m=-1,-6$
Hence the complementary solution is,
${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}$
###### Not exactly what you’re looking for?
Medicim6
Can I get full answer, please. I think tha is not full.
###### Not exactly what you’re looking for?
RizerMix

Oh, sure, sorry
Apply the undetemined coefficient method to find the particular solution as follows:
${y}_{p}=Ax+B$
Find the first and second derivatives,
${y}_{p}^{\prime }=A,$
${y}_{p}^{″}=0$
Substitute these values in equation (1)
$0+7A+6\left(Ax+B\right)=30$
$\left(7A+6B\right)+6Ax=30$
Compare the coeficients of constant and x terms,
$6A=0$
$7A+7B=0$
Solve the above two equations,

Hence the particular solution is,
${y}_{p}=0\left(x\right)+5$
$=5$
The general solution is,
$y={y}_{c}+{y}_{p}$
${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}+5$
Therefore, the required general solution is ${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}$


text:
# Blender output file size versus Zoom recording file size

Sometimes I need to tweak a video recording made in Zoom. According to Zoom, it is an MP4 with M4A audio after conversion (the raw data is, I think, some propriety raw data). However, a 13 minute Zoom recording resulted in 91MB after converting (within Zoom).

When I made my edit (silenced audio for 1 second where I cursed oops), the rendered output from Blender was 471MB!!

I have tried reducing the audio sampling (32 instead of the 192 default) and the result was still several hundred MB. I tried removing the audio entirely and the output was about 150MB, still significantly larger. The recording is a full-screen (2880x1800) so as is the Blender output.

I feel like I'm missing something. I realize Blender doesn't output M4A, but I can't bring myself to believe an AAC or MP3 track should result in a file 4-5X as large.

FYI, I'm using Blender 2.79, but I get similar results in 2.83.1.

Ideas?

-C64Whiz

Edit 2020/09/22: I tried the 'ffmpeg' solution offered up by @Husch and the resulting file was 55MB smaller!!!! WTH? I cut out 1 second of audio, that was it!

Also, I tried again in Blender 2.79 after "Factory Reset" and no audio rendering...and still much larger than the original zoom MP4.

I've downloaded Blender 2.90 (Linux), cleared out all Blender settings I could find in my home dir (including within .config, .share, .etc) and tried rendering with no special settings: after 2% done I already had 10MB of data. Extrapolate and the final file would be over 500MB...i.e. no change.

I will say this, I keep having to set the render resolution to 2880x1800 as the default even after I add the video to an empty sequencer. I don't "import" a video. I "add" one; could this be an issue??

Edit 2020/09/23: Attempted a fresh install of Blender 2.90 on Debian 10 (though a slow 2-core system), and didn't modify the added video...just rendered with settings allegedly identical to the source. Resulting render was 511MB, ~5X the original 95MB Zoom file. I repeated the render changing only the % of the resolution setting to 50% (X,Y was still 2880x1800). Resulting file size was 178MB.

To @susu's comment about compression, I have it set for MPEG-4 container, Video codec H.264, Medium Quality, Good Speed, and audio AAC codec, mono, and 32000 sample rate, 192 bitrate. The codecs are the same as the source as indicated by 'exiftool', as is the 32000 sample rate. Not sure about the 192 bitrate and obviously no clue about the video quality level and encoding speed...

...then I did some investigation and found the source, according to 'exiftool', uses 16 "Audio Bits per Sample". So I tried changing Blender's 192 default bitrate for audio to 16 but it won't let me go any lower than 32...so I kept it at 32. Resulting rendered file size with Resolution % back at 100% - 494MB.

Sound doesn't seem to be much of a factor but I can't understand why I have to set Video % to 50 when the original shows no indication of it. (Honestly, I don't even know what % means if the resolution is X,Y.)

I'm stuck. :(

• The size of the file will depend on the compression settings (codec, data rate, number of channels of audio, etc) – susu Sep 22 '20 at 21:42
• In ffmpeg use the copy stream flag 'c:v copy' so that the video portion is not re-encoded, but just copied. – susu Sep 22 '20 at 21:53
• video.stackexchange.com/questions/29258/… – susu Sep 23 '20 at 14:33

ffmpeg -i zoom_0.mp4 -af "volume=enable='between(t,0,10)':volume=0, volume=enable='between(t,20,30)':volume=0" zoom_silenced.mp4


text:
Learn more about the benzene reactions at vedantu.com. It does not readily react with organic substances. Halogenation is an example of electrophillic aromatic substitution. However, halogens are not electrophillic enough to break the aromaticity of benzenes, which require a catalyst to activate. With aluminum bromide as a Lewis acid, we can mix Br2 with AlBr3 to give us Br+. The elements present in organic compounds are carbon and hydrogen. The presence of Br+ is a much better electrophile than Br2 alone. In electrophilic aromatic substitutions, a benzene is attacked by an electrophile which results in substition of hydrogens. . Considering the exothermic rates of aromatic halogenation decreasing down the periodic table in the Halogen family, Flourination is the most exothermic and Iodination would be the least. Bromination is acheived with the help of AlBr3 (Lewis acid catalysts) as it polarizes the Br-Br bond. The catalyst is either aluminum chloride (or aluminum bromide if you are reacting benzene with bromine) or iron. Benzene reacts with bromine or chlorine in an electrophilic substitution reaction only in the presence of a catalyst which is either chloride or iron. Step 2: The π electrons of aromatic C=C behave as a nucleophile which attacks the electrophilic Br and displaces iron tetrabromide. Facts and mechanism for the halogenation (chlorination and bromination) of benzene - an electrophilic substitution reaction between benzene and chlorine or bromine THE HALOGENATION OF BENZENE This page gives you the facts and a simple, uncluttered mechanism for the electrophilic substitution reaction between benzene and chlorine or bromine in the presence of a catalyst such as … For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Watch the recordings here on Youtube! Benzene reacts with chlorine or bromine in an electrophilic substitution reaction, but only in the presence of a catalyst. Benzene reacts with bromine or chlorine in an electrophilic substitution reaction only in the presence of a catalyst which is either chloride or iron. With aluminum bromide as a Lewis acid, we can mix Br2 with AlBr3 to give us Br+. To menu of electrophilic substitution reactions. The halogenation of benzene Substitution reactions Benzene reacts with chlorine or bromine in the presence of a catalyst, replacing one of the hydrogen atoms on the ring by a chlorine or bromine atom. 2Fe +3Cl2→ 2FeCl3 2Fe + 3Br2→2FeBr3 Step 3: The proton is removed from sp3 C and bears the Bromo group that reforms C=C and aromatic system generates HBr and regulates active catalyst. $C_6H_6 + Br_2 \rightarrow C_6H_5Br + HBr$. However, in the case of elemental fluorine (F2) we will need some proper apparatus and fulfil certain conditions. Hence it being an electrophilic aromatic SUBSTITUTION. (where X= Br or Cl, we will discuss further in detail later why other members of the halogen family Flourine and Iodine are not used in halogenation of benzenes). 1. Example, the addition of bromine to ethane. Step 1: The bromine reacts with Lewis acid to create a complex that makes bromine more electrophilic. This allows the other bromine atom to leave with the AlBr3 as a good leaving group, AlBr4-. 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# two samples comparison with cyclic variables

How to check if two samples are from the same distribution, when the variable is cyclic?

A simple answer for non-cyclic variable is the Kolmogorov-Smirnov test. What is the equivalent for cyclic variables? Is there other powerful similar tests specific for cyclic variables?

In my specific case I have two samples of an angle in $[0, 2\pi]$. The two distributions are nearly uniform, but not perfectly.

• N..H. Kuiper and G.S. Watson proposed tests based on the empirical cdf. – Nick Cox May 24 '16 at 20:56
• Ruggero: To my recollection Watson's 1961 paper deals with the one-sample case, but Watson's 1962 Biometrika paper (Goodness of fit tests on a circle, II) deals with the two-sample version of his test, which is what you're asking for here. Kuiper's paper - just having checked now - deals with both cases. – Glen_b May 24 '16 at 23:09


text:
# Throwing Block

Jump to: navigation, search
Throwing Block
Throwable Item

A block you can throw!

Throwing Block is a construction block that can be thrown by the player. Found most often in Apex Miniknog Bases in bundles of 5, the block will stick to other blocks upon contact when thrown as demonstrated in the image below. There is also a small chance of finding 20 Throwing Blocks in any container.

Throwing Blocks have a base power of 1 and a 0.4-second delay between consecutive throws with the same hand, producing a base one-handed DPS of 2.5.

## Data

### File Details

Spawn Command /spawnitem throwingblock throwingblock.thrownitem assets\items\throwables


text:
Solved

# Add line number to text file

Posted on 2013-06-17
Medium Priority
579 Views
Last Modified: 2013-06-17
I have a text file without line numbers and i need to insert a line number for each file.

Text File

aaaaaa
bbbbb
cccccc
.....

With line numbers

1 aaaaaa
2 bbbbb
3 cccccc
0
Question by:rflorencio
[X]
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2 Comments

LVL 42

Assisted Solution

sedgwick earned 600 total points
ID: 39253353
        Dim lines = File.ReadAllLines("c:\temp\1.txt")
Dim list As List(Of String) = New List(Of String)
For Each l In lines
list.Add((list.Count + 1).ToString & " " & l)
Next

File.WriteAllLines("c:\temp\1.txt", list.ToArray)

0

LVL 75

Accepted Solution

käµfm³d   👽 earned 1400 total points
ID: 39253398
Depending on how large your files are, it may be more prudent to use StreamReaders and StreamWriters.

e.g.

Imports System.IO

...

Using reader As New StreamReader("C:\path\to\your\file.txt")
Using writer As New StreamWriter("C:\path\to\temp\file.txt")
Dim i As Integer = 1

While Not reader.AtEndOfStream
writer.WriteLine("{0} {1}", i.ToString(), reader.ReadLine())
i += 1
End While
End Using
End Using

You can move (i.e. rename) the temp file to the original file to effectively overwrite the old file.
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# A good book of functional analysis

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics). What I am looking for is good books that I could understand to go deeper in this areas, what do you recommend? (I can read in Spanish, English, French and German)

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Can you be a little more specific in what you are interested in? If you are interesting in spectral theory, the classic 4 volume treatise by Reed and Simon, modern methods of mathematical physics is definitely recommended. ;-) – Helge Aug 9 '11 at 1:50
You seem to be getting plenty of answers anyway, but I for one don't know what a "credit" means in your country. Maybe you could list the most advanced topics in functional analysis and topology that you've covered in class; then we'd all know what it would mean to "go deeper". – Tom Leinster Aug 9 '11 at 12:15
In class we covered Banach spaces (until Hahn-Banach theorem), Hilbert spaces (until Riesz Representation theorem) and Lp spaces. Then we did some work (three days) on Banach algebras (how to proof a subspace is dense in them). A credit in the European Union is between 20 and 25 hours of work in class and home. – dan232 Aug 9 '11 at 13:05
Not a textbook, but I recommend that you read Dieudonné's History of Functional Analysis if you can find it. Seeing the historical motivation for the basic concepts was quite enlightening for me. – Qiaochu Yuan Aug 9 '11 at 14:40
Thanks, Dan. In case you don't know, you can edit your original question using the "edit" button below it. And by the way, the credit system is not uniform through the EU: e.g. here in Scotland, a credit corresponds to about 75 minutes of lecture time. – Tom Leinster Aug 9 '11 at 15:17

I am an algebraist and not an analyst, however my favourite book on this area is "Walter Rudin: Functional Analysis".

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Wow, I guess Rudin is THE right answer! All the more interesting since I think the book is dull as dishwater. – Igor Rivin Aug 9 '11 at 2:45
Maybe "an algebraist and not an analyst" wants the shortest presentation possible, so they can return to less painful topics. – Gerald Edgar Aug 9 '11 at 14:16
Indeed, while Rudin is useful, he treats many topics (e.g., vector-valued integrals) as merely obligatory, rather than useful (v.valued-integrals disappear after their brief sighting). Sobolev spaces are horribly shorted, and treated clumsily when they briefly appear. Quasi-completeness is ignored entirely. These are not odd, isolated topics. – paul garrett Aug 9 '11 at 15:53
Perhaps it would be constructive to have an MO CW question asking us to construct a list of "Good mathematics books that are actually bad" together with reasons for the opinions collected. Certain books, when taken biblically by an advisor for example, can serve to dampen mathematical intuition to a criminal degree. I'm NOT saying Rudin's book has this property (I've found this book a very clear and useful reference recently). I'm not going to ask such a question because it really wouldn't be right to do so...but I'd like to have the list of answers, nevertheless! – Jon Bannon Aug 9 '11 at 18:02
This is not a bad book. It is just "brief" like all of Rudin's books. At every point there is the shortest, most elegant, proof. Not necessarily the proof that the students would be likely to think of themselves. @Jon: most good books are probably like that: some people like them, others hate them. Unlike run-of-the-mill books which most people never notice. – Gerald Edgar Aug 12 '11 at 13:31

I am not an analyst of any sort, so you do not need to listen to me, but I really like Lax's "Functional Analysis".

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Yes! A really great book... – Igor Rivin Aug 9 '11 at 1:32
A magnificent book that often gives new insights on classical results. – Alekk Aug 9 '11 at 9:37
AWESOME book by a master and no list of textbooks in this area would be complete without it's mention. And it's clearly designed for students with some familiarity with the concepts,so it may be just what you're looking for, Dan. – The Mathemagician Aug 12 '11 at 18:30

Some of the good books are:

• Elements of the Theory of Functions and Functional Analysis by Kolmogorov, Fomin.

• Functional Analysis, by F.Riesz and Nagy.

• Functional Analysis : Spectral Theory by V.Sunder. Freely available here.

• Analysis now. By Gert Kjeargård Pedersen. (As suggested by Theo Buehler at $\textbf{Math.SE}$

• Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press. ISBN 0691113874.

• Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) by Haim Brezis.

• Elementary Functional Analysis by Georgi E. Shilov.

• Introductory Functional Analysis with Applications by Erwin Kreyszig.

• Notes on Functional Analysis by Rajendra Bhatia. (Hindustan Book Agency.)

• Functional Analysis by S.Kesavan. ( Hindustan Book Agency.)

• Elementary functional analysis By Barbara D. MacCluer

• Functional analysis: an introduction By Yuli Eidelman, Vitali D. Milman, Antonis Tsolomitis (AMS)

• Principles of functional analysis By Martin Schechter. (AMS)

• You may also want to see this thread: Problem books in Functional Analysis

$\textbf{Note.}$ The books which are written in Italics are the ones which I have read partially. The ones which are not in Italics are the ones which I have come to know (by friends, teachers) are good books in Functional Analysis. Also, I really don't know which publisher actually publishes the book in foreign edition written by Kesavan and Bhatia.

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@Theo: Thanks for correcting the spelling mistakes. – S.C. Aug 9 '11 at 11:53
The Stein, Shakarchi series of books have been great thus far. One thing that I enjoy, at least with their Real Analysis book is that it is not as "clean" as some of the other texts that have been around for so long and therefore forces some extra thinking on the part of the reader. Some might not like this, but as a student it helped me. – Andrew Aug 9 '11 at 13:49
@Andrew: I heard that their Real Analysis text is much better than the complex analysis one. – S.C. Aug 9 '11 at 14:01
I am also not an analyst, hence recommend generally user friendly books rather than "experts only" books. I agree especially with Riesz-Nagy, and would mention also the very readable book Spectral Theory by Lee Lorch. This was the text for a course at Harvard I took with Barry Simon, who was I think then an undergraduate. – roy smith Mar 5 '14 at 5:28

You might be interested in "Analysis Now" by Pedersen. A very nice book on graduate level analysis in my opinion. It covers some areas of functional analysis as well.

-

I would suggest the book by Haim Brezis: Analyse fonctionnelle, theorie et applications. It was recently translated into English and you can find the information for the English translation here.

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I haven't seen this book before, but it looks good. :-) – Anand Aug 9 '11 at 7:23
It's a good book (I do have it at home), but it stays quite elementary : the same author covers things more deeply in "Functional Analysis, Sobolev Spaces and Partial Differential Equations" (which is mentioned in the answer I upped). Bonus points for being short and elementary, though ;-) – Julien Puydt Aug 9 '11 at 17:25
@ Snark: I think that's the same book. It's the English translation of the French book ;-) – Mahdi Majidi-Zolbanin Aug 9 '11 at 18:37
@MahdiMajidi-Zolbanin The English version contains the translation of the French book but also exercises for almost every chapter with partial correction and problems (with solutions also). I think that the English translation is actually the translation of the French book + the translation of French book of exercises and problem that has never been published. Also the French edition : 240 pages or so. English edition : 600 pages or so... – user37238 Feb 27 '14 at 10:41
THIS IS THE BEST OF ALL BOOKS.IT TEACHES ANALYSIS IN ACTION – Koushik Apr 9 '14 at 9:36

Since you read German, my favorite is Funktionalanalysis by Dirk Werner. It's not necessarily comprehensive, but it covers a lot, has extensive historical remarks, and is extremely well-written -- I find it more readable than most math books in English (my first language).

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+1 for mentioning this book! – Suvrit Aug 9 '11 at 16:37
I have been learning a lot of stuff from this book the last weeks and it is much better than I first thought it would be. Definitely something to look at! – Konrad Voelkel Sep 1 '11 at 20:01

John B. Conway's "A course in functional analysis" is also pretty decent.

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There's no reason to listen to me either, but for delving a bit deeper, you might want to check out T. W. Körner's Fourier Analysis. The book consists of very short (often just a couple of pages) chapters which contain gems like computing the age of the Earth.

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I love this book! – Olga Jan 25 at 10:10

When I was studying, I was influenced much by K. Yosida's "Functional Analysis".

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More operator theory oriented. :-) I like to use this book as a reference. To study from front cover to back cover might be difficult. – Anand Aug 9 '11 at 7:25
I used it as a source book for a course I taught. I found it very well-organised and thorough. – Amritanshu Prasad Aug 9 '11 at 9:07
This book is dense. It covers a lot in one book, but it's not something you can read quickly. – John D. Cook Aug 9 '11 at 11:04

I'd recommend the Dunford and Schwartz. It's a classic. It's huge -- three volumes. But you don't have to read the whole series cover-to-cover. If you read half of the first volume, you'll learn about as much as reading many other books on functional analysis. Volume 1 alone is big, but it's easy to read for a book on its subject.

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Apart from the classics already mentioned (Yosida, Brezis, Rudin), a good book of functional analysis that I think is suitable not only as a reference but also for self-study, is Fabian, Habala et al. Functional Analysis and Infinite-Dimensional Geometry. It has a lot of nice exercises, it's less abstract than the usual book and provides a lot of "concrete" theorems.

And I'm not sure about it, but I heard there is a spanish translation (the original is of course in english).

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Zimmer's Essential Results of Functional Analysis is a very interesting read, specially if you already know some basic stuff in functional analysis.

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As many here I was schooled on the Rudin's book. However, for a person like me who doesn't work directly in functional analysis but by the nature of his business (dynamical systems) needs often quickly to refresh his memory I warmly recommend Zimmer's book. Probably an appropriate title for the book would be a "Functional Analysis for a working mathematician". It should be on everybody shelf. – Predrag Punosevac Sep 1 '11 at 18:03

The book on Functional Analysis by Meise and Vogt

is quite comphrensive and contains beside standard functional analysis more advanced sections on the theory of locally convex spaces. There is also a German version if you want to improve your German by reading both together.

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I personally like a recent book of Helemskii Lectures and Exercises on Functional Analysis. One of the differences with other books on the subject is that it uses the categorical point of view. The author starts with a very brief introduction to the category theory and uses this language throughout the book. It's a sort of modern core of FA book, with a sidelines to some physics applications and of historic nature, a terse advertisement of the quantum functional analysis and so on (but there is no measure theory, Radon Nikodym theorem etc. which are elaborated in many excellent old textbooks.) Also it gives somewhat broader picture of FA sketching some directions and stating from time to time theorems without proofs 'that every student should know'.

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I like this one very much as well. The Russians not only know mathematics well, they certainly take pride in teaching it. – The Mathemagician Aug 12 '11 at 18:27

I have learnt Functional Analysis from Peter Lax himself. His book is his notes. Exactly the same notes as the ones he handed to us. Every chapter consists of one two-hours long lecture in a two semester graduate course on Functional Analysis. There are a few mistakes here and there, but this book is really ALIVE! It is as if you are in a class of Peter Lax! (I should note here that, Lax's book is published a long time after I left Courant, and at that time the recommended textbook was Yosida's book together with Dunford & Schwartz.)

Having said all these, I should add that as an undergraduate student, I had taken two semesters of Functional Analysis which covered a part of Rudin's book. I still use this book sometimes, as some topics are presented in a beautiful way, but I believe that it is far from introductory, as it starts with Topological Vector Spaces, and it takes a while before normed spaces are mentioned.

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In a course I'm taking now, we're using Gerald Teschl's "Topics in Real and Functional Analysis". It seems like you may already know the first few chapters, however.

It's quite well written, and is free: http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html

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Let me add to this list of suggestions the nice and recent book by J. Cerda "Linear Functional Analysis", Graduate Studies in Mathematics, Vol. 116, AMS-RSME, 2010.

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My preferred text(s) for functional analysis are

(1) B.V. Limaye, Functional Analysis

(2) P.D. Lax, Functional Analysis

(3) R. Larsen, Functional Analysis

Book by Larsen introduces seminormed linear spaces, which in my opinion has enough generali

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NEVER EVER TOUCH LIMAYE. – Koushik Apr 9 '14 at 9:37
@Koushik why would you recommend not touching Limaye's Functional Analysis text? I am curious because I was thinking of getting it. – Vishal Belsare Apr 21 '15 at 11:28
IS there any book limaye ever wrote without mistakes? – Koushik Apr 21 '15 at 11:55


text:
Prove that 1 + 2 + 2
Question:

Prove that $1+2+2^{2}+\ldots+2^{n}=2^{n+1}-1$ for all $n \in \mathbf{N}$.

Solution:

Let $\mathrm{p}(n): 1+2+2^{2}+\ldots+2^{n}=2^{n+1}-1 \forall n \in \mathbf{N}$

Step I: For $n=1$,

$\mathrm{LHS}=1+2^{1}=3$

$\mathrm{RHS}=2^{1+1}-1=2^{2}-1=4-1=3$

As, $\mathrm{LHS}=\mathrm{RHS}$

So, it is true for $n=1$.

Step II : For $n=k$,

Let $\mathrm{p}(k): 1+2+2^{2}+\ldots+2^{k}=2^{k+1}-1$ be true $\forall k \in \mathbf{N}$

$\mathrm{LHS}=1+2+2^{2}+\ldots+2^{k}+2^{k+1}$

$=2^{k+1}-1+2^{k+1} \quad($ Using step II $)$

$=2 \times 2^{k+1}-1$

$=2^{k+1+1}-1$

$=2^{k+2}-1$

$\mathrm{RHS}=2^{(k+1)+1}-1=2^{k+2}-1$

As, LHS = RHS

So, it is also true for $n=k+1$.

Hence, $1+2+2^{2}+\ldots+2^{n}=2^{n+1}-1$ for all $n \in \mathbf{N}$.


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Streams in C++

This topic is 5898 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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How do I get the whole line when I input something from a file to a variable?

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use cin::getline(string,size):

where string is the string you read into, and size is the size of the string

  #include main(){ char somestring[256]; cout << "type a string" << endl; cin.getline(somestring,sizeof(somestring)); cout << "you typed: " << somestring << endl;}

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No, use getline(basic_istream&, basic_string&), declared in string.

  #include #include int main(void){ using namespace std; string str; cout << "Type something and then press return: "; getline(cin, str); cout << "You typed " << str << endl;}

Edited by - DrPizza on February 24, 2002 6:37:06 AM

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(Someone has got to tell me how to do those cool code block things above)

I think he asked about getting input from a file? In that case, you can try the following:

------------------------
if( (fFile = fopen( "Data.txt", "r" )) == NULL )
exit (EXIT_FAILURE);

fread( pData, sizeof( char ), nCharCount, fMapFile );

fclose( fMapFile );
------------------------

But, for the above you need to know nCharCount (the amount of characters(bytes) in the file). You could also use:

------------------------
if( (fFile = _open( "Data.txt", _O_RDONLY )) == -1 )
exit(EXIT_FAILURE);
bytesread = _read( fh, buffer, nbytes ) );
------------------------

The above code will run and read through the file until it hits a CTRL_Z character, which signifies EOF (end of file.)

We know nothing

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I''m sure he did -- but one istream is much like any other istream, so it doesn''t matter a hoot.

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YOu type "source" and "/source" in square brackets to get the source thingy.

Anyway, I was wondering if anyone knew how youcoul code your own stream functions... Rather than just for input/export?

Is it possible and what kind of things would see them applied?

Oli

All the best problems start with C

.:: K o s m o s E n g i n e ::.

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What i want to know is how i input a whole line from a file to a variable, not to let the user input something.

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quote:
Original post by peter86
What i want to know is how i input a whole line from a file to a variable, not to let the user input something.

Just do the same thing with a file stream instead of cin.

  #include void main(){ ifstream f; string str; f.open( "myFile"); getline( f, str); f.close(); }

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It dosen´t work =(!

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What does "it doesn''t work" mean? What did you try? Did it fail to compile, fail to run, do something unexpected, crash? As a point of interest, I''ve not seen any mention in this thread of which language you are using. Since this forum is about "General Programming", it could be any language. Most people seem to have assumed you are using C++, but is that the case? Ask questions the smart way, and state exactly what you are trying to do and, if something goes wrong, state what the symptoms are.

--

The placement of a donkey''s eyes in its head enables it to see all four feet at all times.


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I’ve just uploaded to the arXiv my paper “A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential“, submitted to Dynamics of PDE. This paper continues some earlier work of myself in an attempt to understand the soliton resolution conjecture for various nonlinear dispersive equations, and in particular, nonlinear Schrödinger equations (NLS). This conjecture (which I also discussed in my third Simons lecture) asserts, roughly speaking, that any reasonable (e.g. bounded energy) solution to such equations eventually resolves into a superposition of a radiation component (which behaves like a solution to the linear Schrödinger equation) plus a finite number of “nonlinear bound states” or “solitons”. This conjecture is known in many perturbative cases (when the solution is close to a special solution, such as the vacuum state or a ground state) as well as in defocusing cases (in which no non-trivial bound states or solitons exist), but is still almost completely open in non-perturbative situations (in which the solution is large and not close to a special solution) which contain at least one bound state. In my earlier papers, I was able to show that for certain NLS models in sufficiently high dimension, one could at least say that such solutions resolved into a radiation term plus a finite number of “weakly bound” states whose evolution was essentially almost periodic (or almost periodic modulo translation symmetries). These bound states also enjoyed various additional decay and regularity properties. As a consequence of this, in five and higher dimensions (and for reasonable nonlinearities), and assuming spherical symmetry, I showed that there was a (local) compact attractor $K_E$ for the flow: any solution with energy bounded by some given level E would eventually decouple into a radiation term, plus a state which converged to this compact attractor $K_E$. In that result, I did not rule out the possibility that this attractor depended on the energy E. Indeed, it is conceivable for many models that there exist nonlinear bound states of arbitrarily high energy, which would mean that $K_E$ must increase in size as E increases to accommodate these states. (I discuss these results in a recent talk of mine.)

In my new paper, following a suggestion of Michael Weinstein, I consider the NLS equation

$i u_t + \Delta u = |u|^{p-1} u + Vu$

where $u: {\Bbb R} \times {\Bbb R}^d \to {\Bbb C}$ is the solution, and $V \in C^\infty_0({\Bbb R}^d)$ is a smooth compactly supported real potential. We make the standard assumption $1 + \frac{4}{d} < p < 1 + \frac{4}{d-2}$ (which is asserting that the nonlinearity is mass-supercritical and energy-subcritical). In the absence of this potential (i.e. when V=0), this is the defocusing nonlinear Schrödinger equation, which is known to have no bound states, and in fact it is known in this case that all finite energy solutions eventually scatter into a radiation state (which asymptotically resembles a solution to the linear Schrödinger equation). However, once one adds a potential (particularly one which is large and negative), both linear bound states (solutions to the linear eigenstate equation $(-\Delta + V) Q = -E Q$) and nonlinear bound states (solutions to the nonlinear eigenstate equation $(-\Delta+V)Q = -EQ - |Q|^{p-1} Q$) can appear. Thus in this case the soliton resolution conjecture predicts that solutions should resolve into a scattering state (that behaves as if the potential was not present), plus a finite number of (nonlinear) bound states. There is a fair amount of work towards this conjecture for this model in perturbative cases (when the energy is small), but the case of large energy solutions is still open.

In my new paper, I consider the large energy case, assuming spherical symmetry. For technical reasons, I also need to assume very high dimension $d \geq 11$. The main result is the existence of a global compact attractor K: every finite energy solution, no matter how large, eventually resolves into a scattering state and a state which converges to K. In particular, since K is bounded, all but a bounded amount of energy will be radiated off to infinity. Another corollary of this result is that the space of all nonlinear bound states for this model is compact. Intuitively, the point is that when the solution gets very large, the defocusing nonlinearity dominates any attractive aspects of the potential V, and so the solution will disperse in this case; thus one expects the only bound states to be bounded. The spherical symmetry assumption also restricts the bound states to lie near the origin, thus yielding the compactness. (It is also conceivable that the localised nature of V also restricts bound states to lie near the origin, even without the help of spherical symmetry, but I was not able to establish this rigorously.)

This weekend I was (once again) in San Diego, this time for the Southern California Analysis and PDE (SCAPDE) meeting. I gave a talk on “The asymptotic behaviour of large data solutions to NLS”, which is based on two of my previous papers on what solutions to focusing nonlinear Schrödinger equations behave like as time goes to infinity. (Note that this is a specialist conference, and this talk will be a bit more technical than some of the general-audience talks that I have blogged about previously.)


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# zbMATH — the first resource for mathematics

A finite element method for equal width equation. (English) Zbl 1088.65084
Summary: A numerical solution of the equal width (EW) equation is obtained using the space-splitting technique and the quadratic B-spline Galerkin finite element method. Solitary wave motion, interaction of two solitary waves, wave undulation and wave generation are studied using the proposed method. Comparisons are made with analytical solutions and with some spline finite element method calculations at selected times. Accuracy and efficiency are discussed by computing the numerical conserved laws and $$L_{2}, L_{\infty}$$ error norms.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35L75 Higher-order nonlinear hyperbolic equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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##### References:
 [1] Prenter, P.M., Splines and variational methods, (1975), John Wiley New York · Zbl 0344.65044 [2] Olver, P.J., Euler operators and conservation laws of the BBM equation, Math. proc. camb. philos. soc., 85, 143-159, (1979) · Zbl 0387.35050 [3] Morrison, P.J.; Meiss, J.D.; Carey, J.R., Scattering of RLW solitary waves, Physica, 11D, 324-336, (1984) · Zbl 0599.76028 [4] Gardner, L.R.T.; Gardner, G.A., Solitary waves of the equal width wave equation, J. comput. phys., 101, 218-223, (1992) · Zbl 0759.65086 [5] Garcia-Archilla, B., A spectral method for the equal width equation, J. comput. phys., 125, 395-402, (1996) · Zbl 0848.65069 [6] Gardner, L.R.T.; Gardner, G.A.; Ayoub, F.A.; Amein, N.K., Simulations of the EW undular bore, Commun. numer. meth. eng., 13, 583-592, (1997) · Zbl 0883.76048 [7] Zaki, S.I., A least-squares finite element scheme for the EW equation, Comput. methods appl. mech. eng., 189, 587-594, (2000) · Zbl 0963.76057 [8] Saka, B.; Irk, D.; Dağ, İ., A numerical study of the equal width equation, Hadronic J., Suppl. 18, 99-116, (2003) · Zbl 1087.65598 [9] Irk, D.; Saka, B.; Dağ, İ., Cubic spline collocation method for the equal width equation, Hadronic J., Suppl. 18, 201-214, (2003) · Zbl 1087.65596 [10] Dağ, İ.; Saka, B., A cubic B-spline collocation method for the EW equation, Math. comput. appl., 90, 381-392, (2004) · Zbl 1084.65098 [11] Esen, A., A numerical solution of the equal width wave equation by a lumped Galerkin method, Appl. math. comput., 168, 270-282, (2004) · Zbl 1082.65574 [12] Doğan, A., Application of galerkin’s method to equal width wave equation, Appl. math. comput., 160, 65-76, (2005) · Zbl 1063.65104
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