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# Post-Deposition Wetting and Instabilities in Organic Thin Films by Supersonic Molecular Beam Deposition

## Abstract

We discuss the formation and post-deposition instability of nanodrop-like structures in thin films of PDIF-CN2 (a perylene derivative) deposited via supersonic molecular beam deposition technique on highly hydrophobic substrates at room temperature. The role of the deposition rate on the characteristic lengths of the organic nanodrops has been investigated by a systematic analysis of atomic force microscope images of the thin films and through the use of the height-height correlation function. The nanodrops appear to be a metastable configuration for the freshly-deposited films. For this reason, post-deposition wetting effect has been examined with unprecedented accuracy throughout a year of experimental observations. The observed time scales, from few hours to months, are related to the growth rate, and characterize the thin films morphological reordering from three-dimensional nanodrops to a well-connected terraced film. While the interplay between adhesion and cohesion energies favors the formation of 3D-mounted structures during the growth, wetting phenomenon following the switching off of the molecular flux is found to be driven by an instability. A slow rate downhill process survives at the molecular flux shutdown and it is accompanied and maybe favored by the formation of a precursor layer composed of more lying molecules. These results are supported by simulations based on a non-linear stochastic model. The instability has been simulated, for both the growth and the post-growth evolution. To better reproduce the experimental data it is needed to introduce a surface equalizer term characterized by a relaxation time taking into account the presence of a local mechanism of molecular correlation.

## Introduction

Thin or ultra-thin film coating of a solid surface is a technological issue for a wide number of applications from flexible optoelectronics1,2 to adhesives3 and other emerging applications4. The study of aggregation, condensation and growth processes in the solid state continues to be of fundamental interest for applications through the control of the coating formation and its chemico-physical properties. For example, the electrical performances of an organic thin-film transistor (OTFT) is strictly related to the final morphology, the molecular assembly and the microscopic structure of the organic film which, in turn, is directly linked to various constructive parameters such as deposition technique, deposition rate or molar concentration, temperature of the substrate and the thermodynamic properties of the surface5,6.

In the past decades, a lot of studies have focused on the stability and self-organization of coated surfaces. Meso-structure modification induced by dewetting7,8,9 and spreading effects10 result ubiquitous for thin-film technology which can either be deleterious, destabilizing the thin-film structure, or advantageous, leading for example to the controlled formation of an array of nanometric islands11 or micro-nano-patterning12,13.

In the framework of vapor-deposited thin-films, for metals and inorganic semiconductors, the knowledge of the kinetic growth mechanism by vapor condensation is well assessed since the mass transport between layers is primarily ascribed to the Ehrlich-Schwöbel (ES) energy barrier. This approach gives an accurate control of the film properties and the instabilities are mainly due to the relaxation of strain elastic energy in hetero-epitaxial growth14. In contrast, the growth mechanism of organic thin films remains poorly understood. Several studies have tried to cover the gap, following the scaling theories15 and assuming that the principles ruling the inorganic growth can be universally transposed to weakly interacting systems such as organic semiconductors. It is however important to remark that the intrinsic anisotropy and the internal degree of freedom of the molecules respect to single atoms induce a series of activation barriers for crossing the steps that depend on configuration and orientation of the diffusing species16. This feature leads to a strong tendency to roughening (or “rapid roughening”) and to a transition from the layer-by-layer growth mode of the early stage to a 3D-mounded morphology17,18,19,20. Furthermore, the strongly anisotropic molecule-molecule interaction, mediated by the interaction with the solid surface, induces a multiple energetic minima in the possible molecular packing enabling polymorphs and surface induced phases21. We point out that the energetic non-equilibrium thermodynamic conditions of the growth do not always stabilize when the flow of molecules ceases. Post-deposition relaxation phenomena can intervene to change the film micro-structure and cracks or dewetting processes can be observed typically favored by the presence of temperature gradients or external fields. Spinodal dewetting due to capillary waves or instability22,23 and elastic or strain energy relaxation are the main intrinsic perturbation in solid-state dewetting of thin films24. Conversely, the reverse phenomena is more difficult to observe: spontaneous wetting process of a solid surface results very intriguing and unconventional in molecular thin films, even if it is common in liquids or viscous polymers10.

Among the broad variety of molecular compounds studied in molecular-beam-deposited thin-films, perylene derivatives alike DIP (diindenoperilene)17,25,26 or PTCDA (perylene-3,4,9,10-tetracarboxylic-dianhydride)27 have been intensively studied as prototype molecules to better understand the specific growth mechanism of extended planar molecules. PDIF-CN2 (N,N′-1H,1H-perfluorobutil dicyanoperylene-3,4,9,10-bis dicarboximides see Fig. 1a for a sketch) has established as n-type organic semiconductor in high performance OTFTs28,29,30,31,32. We have already discussed in ref.33 the fabrication of PDIF-CN2 n-type organic thin-film transistors (OTFTs) by supersonic molecular beam deposition (SuMBD), where high mobility devices were obtained keeping the substrate at room temperature during the organic film growth. Here SiO2 substrate acts both as dielectric barrier and growth surface and is made hydrophobic by HMDS (Hexamethyldisilazane) monolayer coating34 needful to optimize the device performances31. SuMBD technique has demonstrated to be a powerful approach to grow molecular films and to explore unconventional growth regimes, thanks to its specificity in decoupling the deposition rate (R) of the molecular flux from the impinging kinetic energy (Ek)35,36. Besides, deposition at high Ek (in the order of electronvolts) induces a reduction of the nucleation center density (n)37. This is similar to what happens in the case of increased molecular diffusivity38, but without increasing the substrate temperature.

More recently, we have demonstrated that PDIF-CN2 molecules deposited by SuMBD show a characteristic growth regime, where the interplay between the highly Ek value and the highly hydrophobic substrate at room temperatures drives to a three-dimensional (Volmer-Weber-like) growth with the formation of nanodrops-like islands. We observed that the layer composed of nanometric drops were unconnected, however after a long time (i.e., from days to months) a several orders of magnitude increase in the field-effect charge carrier mobility was achieved. This was related to the observation of an unconventional spreading or ripening effect of the condensate over the time39. It is important to remark that, as illustrated in the control experiment discussed in ref.39, no dewetted growth and post-deposition spreading are observed in the case of bare (less hydrophobic) SiO2 substrate. The specific combination of surface hydrophobicity, molecular structure, and hyperthermal deposition conditions brings the system in a mesophase like a liquid crystal that relaxes with a spontaneous wetting process occurring over very long time scales typical of a viscous phase.

Moreover, this terraced wetting dynamics was found to depend on the initial growth rate and therefore to the as-grown density and geometrical distribution of the nanodrops. In particular, the process is negligible at deposition rate lower than 0.05 nm/min and a stable nanodrops array was formed in that case. The spreading process is clearly observed at R around 0.08 nm/min with a faster evolution than that observed for even higher R39.

In this paper, in order to understand the mechanism inducing this critical wetting effect, we report an accurate study of the anomalous growth and relaxation effect of PDIF-CN2 thin films deposited by SuMBD. We present a systematic analysis of the growth and relaxation processes by measuring statistical correlation functions of the surface morphology, just after the deposition as well as over time, of two series of films at different thicknesses (10 and 20 nm). Finally, a non-linear stochastic model based on differential equations has been used to analyze both the growth and the post-deposition relaxation dynamics focusing on the ES barrier effect.

## Results and Discussion

Using atomic force microscopy (AFM) in non-contact mode, we have investigated the morphology of our films during the time. Figure 1 shows the AFM topographies of a 10 nm-thick film 1 hour, 1 month and 1 year after the end of the deposition process. A clear morphology change is observed passing from a full nanodrop-like morphology (in Fig. 1c) to a mixed one (in Fig. 1d) and ending with a more flat and compact surface (in Fig. 1e).

The theoretical framework to describe and analyze the growth processes of molecular films is well assessed and the height-height correlation function H(r) is the most effective tool to extract the scaling exponents and establish a morphology-growth mechanism correlation15. Unfortunately, no general approach is actually consolidated to describe the post-deposition instability observed here. We have decided to use H(r) as the key function to extract statistical quantities and classify the film morphologies.

All the correlation functions can be calculated through the height profile h(r, t):

1. 1.

the interface width w(t) defined from the equation: $$w(t)\equiv {[\langle |h({\bf{x}},t)-\langle h({\bf{x}},t)\rangle {|}^{2}\rangle ]}^{\frac{1}{2}}$$;

2. 2.

the autocorrelation function $$R({\bf{r}},t)\equiv {w}^{-2}(t) < h({\bf{r}}+{\bf{x}},t)h({\bf{x}},t) >$$;

3. 3.

the height-height correlation function $$H({\bf{r}},t)\equiv < {[h({\bf{r}}+{\bf{x}},t)-h({\bf{x}},t)]}^{2} > =2{w}^{2}(t)[1-R({\bf{r}},t)]$$.

At a given time t, it is possible to give a functional form of H(r) for a mounded surface that takes into account the presence of a periodic recurrence with wavelength λ, adding an oscillatory term to the exponential model commonly used for self-affine surfaces15 as:

$$H(r,t)=2{w}^{2}(t)\,[1-\exp [-{(\frac{r}{\xi (t)})}^{2\alpha }]\,{J}_{0}\,(\frac{2\pi r}{\lambda (t)})],$$
(1)

where r is the radial distance from a generic point of the surface, J0 is the zeroth-order Bessel function. The coefficient α affects the behavior of H for small r and can be related to the local fractal dimensionality of the surface. Furthermore, w (interface width), λ (recurrence wavelength) and ξ (correlation length) can be interpreted as the surface roughness, the average peak-to-peak distance between islands and the average diameter of the mounded islands (in our case the nanodrops as illustrated in Fig. 1b). It is common to observe a power-law behavior for these characteristic lengths as a function of the deposition time: $$w(t)\sim {t}^{\beta }$$, $$\xi (t)\sim {t}^{\mathrm{1/}z}$$ and $$\lambda (t)\sim {t}^{p}$$, where β, z and p are the scaling exponents characterizing the dynamic scaling theory15.

In the framework of the dynamic scaling theory, we extract height-height correlation data (HHCD) by applying a numerical algorithm (see Gwyddion open source manual) to the AFM images (see Fig. 2) for films deposited at different rate (from 0.056 nm/min to 0.218 nm/min). The morphologies are acquired just after the flux switching off (t = tc) with a delay due to experimental timing (about 2.5 hours, i.e. 0.1 days). The HHCD are fitted with Eq. (1). From this fit we have extracted the functional dependency of the characteristic lengths on R. These results are shown in Fig. 3a for both the series of samples with effective thickness of 10 and 20 nm. It is important to observe the exponential behavior, where both λ and ξ rapidly decrease with R until to a more flat trend at rate values greater than 0.1 nm/min. A similar behavior is plotted in Fig. 3b for the difference of (λ − ξ). Assuming this quantity proportional to the average distance between the nanodrops, we get evidence that it decreases with the increase of the deposition rate without great variations changing the nominal thickness.

The obtained results can be easily discussed via geometrical evaluations considering that fixing the thickness (d) of the films we are fixing the deposited volume (V) too. Being the morphology consisting of spherical caps separated by an average distance, we can write the following equation:

$$d=\frac{V}{S}=\frac{AN{\xi }^{3}}{S}=An{\xi }^{3},$$
(2)

where S is the total surface, A is a geometrical factor, N the total number of nanodrops and n their density. Considering that $$n\sim {(R)}^{q}$$ (where q is a coefficient related to the dimension of the critical nucleus)38 and substituting it in eq. 2, a relation of proportionality between R and ξ is obtained assuming the molecular surface diffusivity fixed by the choice of the kinetic energy of the impinging molecules and the temperature of the substrate: $$\xi \sim {R}^{-q\mathrm{/3}}$$. Similarly, the recurrence wavelength λ of the nanodrop structure has a dependence from the molecular flux as $$\lambda \sim {n}^{-\mathrm{1/2}}\sim {R}^{-q\mathrm{/2}}$$. These considerations allow to claim that, lowering the flux at fixed film thickness, the correlation length ξ will increase more slowly than the λ, while reducing the thickness at the same deposition rate the expected ξ and λ values get reduced too. Now, being R = d/tc where tc is the deposition time, we can assume a dependence with tc: $$\xi \sim {t}_{c}^{q\mathrm{/3}}$$ as for Ostwald ripening40 and $$\lambda \sim {t}_{c}^{q\mathrm{/2}}$$.

Figure 4 shows the ξ, λ and w parameters extracted from the acquired images as function of the time. We have chosen three samples deposited at different deposition rate values (R = 0.056, 0.086, 0.218 nm/min) for which we observe a strong difference in the behavior of the wetting dynamics. The experimental data in Fig. 4 are characterized by the presence of a cusp, in correspondence of the external molecular flow ending. Indeed, when the molecular flux is active, all the characteristic parameters of the superficial morphology tend to grow with a power law, as expected in the scaling theory typically used for the description of the thin-film growth mechanism. Switching the flux off, on the other hand, these parameters relax and evolve with a different behavior, because of the system wetting relaxation effect towards a more flat surface. We find that the ξ values, after an initial reduction, converge to the same value over long observation times for all the different flux rates. The effect of the film micro-structure modification from nanodrops to a flat continuous film, instead, is reflected in the large increase of λ with a timing depending by R as already discussed. A peculiar behavior is observed for the roughness w. After an initial reduction of the value over a period of few days, the roughness increases slowly and the dynamics seems to asymptotically tend to a final value different for each starting time tc, i.e. for each R value. The observed discontinuities in Fig. 4 are, ultimately, the direct representation of the two regimes ruling the film formation with and without the presence of the external molecular flow.

From Fig. 4 (see the red lines) the scaling coefficients are obtained fitting the experimental data: β = 0.27 ± 0.08, 1/z = 0.41 ± 0.06, and p = 0.42 ± 0.06.

To complete our experimental observation, we show in Fig. 5a the AFM image of an ultra-thin film of about 3 nm thick acquired after a long time from the end of the deposition process. Here the formation of an uncompleted layer with terraced islands gives us the opportunity to analyze the dynamics effect on the molecular system in proximity of the substrate. In Fig. 5b, the height distribution of the AFM topography is shown and the average height of the terraces of (1.9 ± 0.2) nm was measured. This is compatible with the typical PDIF-CN2 morphology with terrace heights comparable with the c-axis length of the crystal structure of 1.96 nm41, obtained in thin films when the molecules are in the stand-up configuration respect to the surface31,42,43.

Less conventional is the anomalous distance measured between the substrate peak and the first layer peak of (1.1 ± 0.5) nm. This could be ascribed to the presence of a surface induced phase (SIP) like a polymorph or a different molecular orientation21. In our case, the molecules seem to be in a lying down configuration for which the molecular core is closer to the substrate, as proposed in Fig. 5c. This observation suggests that the relaxation process is accompanied by the formation of a wetting foot around the islands acting as adhesion layer on which molecules and clusters can diffuse more effectively thanks to the weakening of the effective substrate corrugation inducing with time the observed effects of coarsening and eventually terracing. The formation of the terraces and maybe a partial dewetting can justify the increase of roughness w and the weak increase of the correlation length ξ in the final part of the morphology evolution.

Furthermore, the behavior of the observed slow wetting dynamics is strongly related to the as-grown-deposited film structure and hence to the deposition rate. As showed in Fig. 3b, the average distance between the nanodrops, proportional to (λ − ξ), results to decrease with the increase of the deposition rate. Respect to the spontaneous spreading effect of the mounds observed for the samples characterized by deposition rate values exceeding 0.05 nm/min, the experimental results suggest that for $$R < 0.05$$ nm/min the interaction between mounds is heavily depressed by the pronounced interstitial space. We interpret this observation as due to the fact that the precursor feet expand on the surface for a distance lower than the average distance between the nanodrops. With the reduction of (λ − ξ), the wetting foot becomes able to cover all the gap between the nanodrops making faster and more efficient the wetting process (in the range 0.07–0.10 nm/min). On the other hand, when (λ − ξ) reduces more and more at increasing R, the reduced space between the nanodrops could hinder and quench the formation of the wetting foot making less efficient the reformation process and producing an increase in the timing as observed.

In order to quantitatively simulate by an unified model the dynamics of growth and the following wetting process, a full-time-scale continuum-equation approach has been adopted starting from the assumption that both the processes are driven by instabilities in the growth front and molecular diffusion currents. The overall dynamics is modeled by the following equation15,44:

$$\frac{\partial h}{\partial t}=-\,\nu \nabla \cdot (\frac{\nabla h}{1+|\nabla h{|}^{2}})-\,k{\nabla }^{4}h+\eta ({\bf{r}},t)+F(t)+{(\frac{\partial h}{\partial t})}_{r},$$
(3)

where the first term describes the uphill growth due to the ES effect ( is the gradient operator), the second one is the Mullins diffusion term modeling the overall effect of surface diffusion. We remark that in the first term there is the sign −, therefore the ES effect introduces an instability in the growth process favoring the formation of mounds. The stochastic term η(r, t) is present just during the growth and modeled as Gaussian noise. When the molecular flux F(t) is switched off at time t = tc the noise is no longer active (for a more exhaustive description see the Supplementary Information (SI)).

The last term of Eq. (3) provides a characteristic relaxation time τ:

$${(\frac{\partial h}{\partial t})}_{r}=-\,\frac{[h({\bf{x}},t)-d]}{\tau },$$
(4)

where d is the film thickness. This relaxation term becomes important after the flux switching off and works like a surface equalizer.

The starting point of the self-consistent approach is the observation that:

$$\langle |\nabla h{|}^{2}\rangle ={m}^{2}(t),$$
(5)

where m(t) is the surface slope. In order to solve the stochastic equation, we retain all the temporal fluctuations of the non-linear contribution in the first term of Eq. (3), approximating its spatial fluctuations (see SI):

$$\nabla \cdot (\frac{\nabla h}{1+|\nabla h{|}^{2}})\simeq \frac{{\nabla }^{2}h}{[1+{m}^{2}(t)]}\mathrm{.}$$
(6)

In the simulation we focus on the thickness of d = 20 nm. With the parameters ν and k present in Eq. (3)(we obtain the Mullins coefficient and the diffusion coefficient in equation 3 respectively: k = √2l04/2t0 = 1.7 × 10−26 cm4/s and v = √2l02/t0 = 7 × 10−16 cm2/s. The value of v is compatible with typical values of diffusivity in large cluster diffusion processes at room temperature for thin film of metals as reported in45,46), one can obtain the length $${l}_{0}=\mathrm{1/}{q}_{0}=\sqrt{2k/\nu }$$ and the time $${t}_{0}={l}_{0}^{3}/\sqrt{\nu k}=2\sqrt{2}k/{\nu }^{2}$$. The quantities l0 and t0 provide a term $${D}_{0}={l}_{0}^{4}/{t}_{0}=\sqrt{2}k$$, which has the same dimensions as the term D, which measures the strength of the deposition noise η(r, t). For the dynamics analyzed in this paper, we set the following values of l0 and t0 as unit length and time, respectively: l0 = 77 nm and t0 = 1.38 days. Then, the free parameters of the theory are R (or tc), D, given in terms of D0 and τ, expressed in terms of t0. For the dynamics analyzed in this paper, D is of the order of 0.001 − 0.01 D0, while the relaxation time τ is assumed to be quite large being of the order of 6 − 7 t0.

Applying the model of Eq. (3) with the above-mentioned parameters, we extract the simulated characteristic lengths. In Fig. 6a, we plot the theoretical values of the correlation length ξ and the recurrence wavelength λ as a function of the flux rate, calculated at the molecular beam switch-off, in order to make a comparison with experimental data in Fig. 3a. As thoroughly discussed in the SI, the theoretical estimates for the exponents are the following: β = 0.291, 1/z = 0.405, and p = 0.433. The values of the exponent β indicate a slight deviation from the situation of dynamic scaling and a reduced instability, as found in the literature neglecting the correction m2(t) in the ES term of the dynamical equation (Eq. 6)47, (since up to times of the order of a few hours, the values of m2(t) are always smaller than 1, they provides weak perturbations because the denominator in Eq. (6) is close to 1 shrinking the term to the more simple, but typically used, Edwards-Wilkinson one). All these results are in good agreement with the experimental ones, suggesting that the model well simulates the unconventional hyperthermal deposition conditions.

To shed light on the interesting wetting dynamics observed after the flux switching off, we simulate the relaxation dynamics applying the model in Eq. 3 from the flux switching off up to times larger than 20 days. The theoretical approach shows that the asymptotic behavior of ξ at very large times is independent of the rate reaching, we can thus assume the average size of a mound as a stable quantity in the time. The λ, instead, increases as a function of time, reinforcing the agreement with the experiments (see images in SI).

In Fig. 6b and c, we show the results of the model concerning the less obvious behavior of the interface width w with time assuming $$\mathrm{1/}\tau \simeq 0$$ and a finite τ value, respectively.

In the case of a very large τ (i.e. $$\mathrm{1/}\tau \simeq 0$$, Fig. 6b), the relaxation dynamics is characterized by a decrease of w just after the flux switching off followed by an increase after a transient time; qualitatively confirming the experimental observation. These results can be discussed considering that the mound instability on the scale of l0, with a so big value of t0, is always active in the system even in the absence of the flux; the nonlinear ES term of the dynamical Eq. (3) strongly affects the relaxation. However, we point out that the asymptotic behavior of w is found to be independent on the rate for very large times, consequently the system seems to lose memory of the different starting state incoherently with the observed phenomenology (for a more exhaustive description see the Supplementary Information (SI)).

In the case of a finite τ (Fig. 6c), we optimize the calculation of w in order to simulate the experimental behavior using different value of τ for each deposition rate. In fact, while the values of w at the flux switch-off in Fig. 6b and c can be corrected just changing the value of the noise strength D, the agreement of the model with experimental behavior of w after tc is improvable only if we operate on τ. While the behavior of λ and ξ does not change so much, the effect of τ is able to extend the relaxation tail at larger times and to speed up the simulated evolution of w lowering its value. In particular, the flux rate R = 0.086 nm/min corresponds to the situation where the dynamics was observed to be fastest and for our simulation to impose the smallest τ value indicating how the relaxation process could be easily represented by quite rigid addition of the term 4 in the Eq. 3.

We emphasize that the simulation well reproduces the experimental observations without strong “ad hoc” hypotheses, but we need to introduce the global parameter τ, in order to reproduce the saturation effect at long times of w depending on the deposition rate R. This indicates that the system does not lose spatial coherence with the relaxation accomplishment taking memory of the original state; an effect maybe connected to the presence of an interaction between the islands that we ascribe to the formation of the adhesion layer and related to the distances (λ − ξ) of the nanodrops at the end of the growth process.

In the observed phenomenology, we suppose that the side fluoro-alkylic chains of the molecule can play a key role in determining the molecular mobility. As recently demonstrated for a parent compound by friction force microscopy measurements, the side alkylic chains pack into a compact low-surface-energy overlayer with a periodic heterogeneity of the chains bending properties and subsurface anchoring48. The formation of a mesophase can break this periodicity also in our molecular system by strongly modifying the adhesion/cohesion energy ratio and the equilibrium between nematic elasticity and the long-range van der Waals forces and so triggering the observed wetting phenomena49. However other mechanisms alike some relaxation of HMDS layer, molecular cross-Linking, glass-like relaxation effects or isotropic-nematic phase transitions cannot be excluded.

## Conclusion

In this paper, we analyze the dynamic effect of the critical wetting of PDIF-CN2 nano-structured thin-films deposited at room temperature by SuMBD on hydrophobic silicon dioxide. Atomic force microscopy is used to acquire surface morphology monitoring post-deposition transition from partial wetting to a complete wetting during one year. The images were analyzed extracting the characteristic morphological lengths (w, ξ and λ). In parallel, a continuum-equation approach is used to simulate both the growth and post-deposition relaxation. In order to classify the growth, scaling exponents were derived from both the experimental data and the simulations and we observe a slight deviation from typical values of mounded structures. The simulations are in good agreement with the experimental results and the proposed model can be useful, in the framework of the scaling theory, not only to simulate the growth, but also to the description of the solid-on-solid relaxation phenomena and dewetting effects. The need to consider in the non-linear stochastic model a surface equalizer term indicates the presence of a local mechanism of molecular correlation related to the balance between nematic elasticity and long-range van der Waals forces. The change in the molecule-surface interaction likely induces the formation of the adhesion layer on the uncovered part of the surface driving the wetting phenomena.

## Methods

### Material and Thin films Deposition

Every sample has been fabricated via SuMBD deposition from PDIF-CN2 powder (Polyera ActivInk N1100) used without any further purification. The organic precursor has been deposited on the top of a square shaped multilayer wafer (1 cm2 in area) consisting of a 500 μm thick highly doped silicon substrate, 200 nm thick SiO2 layer as insulating material. The dielectric layer has been passivated by means of HMDS self-assembled monolayer via vapor treatment34. SuMBD has been performed in a supersonic jet of He carrier gas with stagnation pressure value of 2000 mbar, resulting in kinetic energy values of approximately 17 eV per molecule. Each deposition has been performed with substrates kept at room temperature, while the deposition rate was finely tuned heating the organic powder with temperatures between 203° C and 220° C, corresponding to deposition rates ranging from 0.01 nm/min to 0.22 nm/min. The deposition rate was measured by thickness monitor and after calibrated from the film thickness measurements.

### Characterization

Morphological characterization in air has been performed ex-situ by a Park Instruments Xe-100 AFM equipped with Silicon-doped cantilevers, in true-non-contact mode. Statistical analysis of AFM images was accomplished by Gwyddion 2.48 (http://gwyddion.net/). Samples were stored in dry air and in dark.

### Theoretical approach

The height profile and the correlation functions are theoretically evaluated by using a non-linear stochastic continuum equation solved through a mean-field time-dependent self-consistent scheme. All the temporal fluctuations of the non-linear terms are retained, its spatial fluctuations are correctly approximated. The self-consistency is imposed in the momentum space by using the power spectral density function. Very accurate and long dynamics are obtained by means of this approach.

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Zhang, X. N. et al. Evidence for a layer-dependent ehrlich-schwobel barrier in organic thin film growth. Physical Review Letters 103, 136101 (2009).

26. 26.

Kowarik, S. et al. Real-time observation of structural and orientational transitions during growth of organic thin films. Physical Review Letters 96, 125504 (2006).

27. 27.

Yim, S., Kim, K. I. & Jones, T. S. Growth morphology of perylene-3,4,9,10-tetracarboxylic dianhydride (ptcda) thin films: Influence of intermolecular interactions and step-edge barriers. Journal of Physical Chemistry C 111, 10993–10997 (2007).

28. 28.

Jones, B. A., Facchetti, A., Wasielewski, M. R. & Marks, T. J. Tuning orbital energetics in arylene diimide semiconductors. materials design for ambient stability of n-type charge transport. Journal of the American Chemical Society 129, 15259–15278 (2007).

29. 29.

Molinari, A. S., Alves, H., Chen, Z., Facchetti, A. & Morpurgo, A. F. High electron mobility in vacuum and ambient for pdif-cn2 single-crystal transistors. Journal of the American Chemical Society 131, 2462 (2009).

30. 30.

Barra, M. et al. Very low bias stress in n-type organic single-crystal transistors. Applied Physics Letters 100, 4 (2012).

31. 31.

Chiarella, F., Barra, M., Ricciotti, L., Aloisio, A. & Cassinese, A. Morphology, electrical performance and potentiometry of pdif-cn2 thin-film transistors on hmds-treated and bare silicon dioxide. Electronics 3, 76–86 (2014).

32. 32.

Perroni, C. A., Gargiulo, F., Nocera, A., Marigliano Ramaglia, V. & Cataudella, V. The effects of different electron-phonon couplings on the spectral and transport properties of small molecule single-crystal organic semiconductors. Electronics 3, 165–189 (2014).

33. 33.

Chiarella, F. et al. High mobility n-type organic thin-film transistors deposited at room temperature by supersonic molecular beam deposition. Applied Physics Letters 104, 143302 (2014).

34. 34.

Barra, M. et al. Addressing the use of pdif-cn2 molecules in the development of n-type organic field-effect transistors for biosensing applications. Biochimica Et Biophysica Acta-General Subjects 1830, 4365–4373 (2013).

35. 35.

Iannotta, S. & Toccoli, T. Supersonic molecular beam growth of thin films of organic materials: A novel approach to controlling the structure, morphology, and functional properties. Journal of Polymer Science Part B-Polymer Physics 41, 2501–2521 (2003).

36. 36.

Wu, Y. et al. Controlling the early stages of pentacene growth by supersonic molecular beam deposition. Physical Review Letters 98, 076601 (2007).

37. 37.

Killampalli, A. S., Schroeder, T. W. & Engstrom, J. R. Nucleation of pentacene on silicon dioxide at hyperthermal energies. Applied Physics Letters 87, 3 (2005).

38. 38.

Ratsch, C. & Venables, J. A. Nucleation theory and the early stages of thin film growth. Journal of Vacuum Science & Technology A 21, S96–S109 (2003).

39. 39.

Chiarella, F. et al. Spontaneous wetting dynamics in perylene diimide n-type thin films deposited at room temperature by supersonic molecular beam. Journal of Physical Chemistry C 120, 26076–26082 (2016).

40. 40.

Mullins, W. W. The statistical self-similarity hypothesis in grain-growth and particle coarsening. Journal of Applied Physics 59, 1341–1349 (1986).

41. 41.

Colle, R., Grosso, G., Cassinese, A. & Centore, R. Structural, electronic and vibrational properties of n,n′-1h,1h-perfluorobutyl dicyanoperylenecarboxydiimide (pdi-fcn2) crystal. Journal of Chemical Physics 139, 114507 (2013).

42. 42.

Weitz, R. T. et al. Organic n-channel transistors based on core-cyanated perylene carboxylic diimide derivatives. Journal of the American Chemical Society 130, 4637–4645 (2008).

43. 43.

Menichetti, G., Colle, R., Gatti, C. & Grosso, G. Heterointerface electronic states and charge transport of crystalline n,n′-1h,1 h′-perfluorobutil dicyanoperylene diimide in field-effect transistor configuration. Journal of Physical Chemistry C 120, 12083–12091 (2016).

44. 44.

Barabasi, A.-L. & Stanley, H. E. Fractal Concepts in Surface Growth. (Cambridge University Press, 1995).

45. 45.

Wen, J.-M. et al. Diffusion of large 2-dimentional AG cluster on AG(100). Phys. Rev. Lett. 73, 2591–2594 (1994).

46. 46.

Pai W.W. et al. Island Diffusion and Coarsening on Metal (100) Surfaces. Phys. Rev. Lett. 79, 3210–3213 (1997).

47. 47.

Zhao, Y. P., Yang, H. N., Wang, G. C. & Lu, T. M. Diffraction from diffusion-barrier-induced mound structures in epitaxial growth fronts. Physical Review B 57, 1922–1934 (1998).

48. 48.

Buzio, R. et al. Subnanometer resolution and enhanced friction contrast at the surface of perylene diimide pdi8-cn2 thin films in ambient conditions. Langmuir 34, 3207–3214 (2018).

49. 49.

Vandenbrouck, F., Valignat, M. P. & Cazabat, A. M. Thin nematic films: Metastability and spinodal dewetting. Physical Review Letters 82, 2693–2696 (1999).

## Acknowledgements

This work was funded by the Italian MIUR through Progetto Premiale 2012 “EOS: organic electronics for advanced research instrumentation”.

## Author information

Authors

### Contributions

F. Chiar., F. Chian., M.B., G.M.D. and A.C. conducted the experiments and analyzed the results. C.A.P. and V.C. performed the theoretical model and the simulations. All authors reviewed the manuscript.

### Corresponding author

Correspondence to Fabio Chiarella.

## 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

Chiarella, F., Perroni, C.A., Chianese, F. et al. Post-Deposition Wetting and Instabilities in Organic Thin Films by Supersonic Molecular Beam Deposition. Sci Rep 8, 12015 (2018). https://doi.org/10.1038/s41598-018-30567-7

• Accepted:

• Published:

• ### Perylene-Diimide Molecules with Cyano Functionalization for Electron-Transporting Transistors

• Mario Barra
• , Fabio Chiarella
• , Federico Chianese
• , Ruggero Vaglio
•  & Antonio Cassinese

Electronics (2019)


text:
# An electron (m(electron) = 9. 11 10-31 kg) emits photons of green light when it transitions between n 5 and

###### Question:

An electron (m(electron) = 9. 11 10-31 kg) emits photons of green light when it transitions between n 5 and n = 2 levels_ The wavelength of green light is 550 nm: What is the energy of the photon of green light in: (a) Joules (give answer to 2 dp times xl0-19 ) (b) electronVolts (2 dp)? [Assume: h 6. 63 10 -34 = 108 , 9(electron) 1. 6 10-19 C 10-19 Joules ev

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text:
# Why simple paper planes fly better than most “realistic” paper plane models?

Just a A4 paper sheet folded into a plane seems to fly decently. But a model of actual (big) plane made out of paper is only good at falling...

What is different for paper planes? Are there "modern and improved" designs of paper planes?

-
do you have pics of the ones you compared? often paper planes have a very thin hull and wings while model aircraft have much thicker hull and wings, often heavier as well – ratchet freak Jun 28 '14 at 20:48
Something like if one tries a Google Image Search of "paper plane model". – Anonymous Jun 28 '14 at 21:37
I suspect the answer will have something to do with Reynolds numbers. When you scale down your 747 to A4 size, you are not scaling down the viscosity of the fluid in which the object flies. – RedGrittyBrick Jun 28 '14 at 22:21
The article also mentions them. – Anonymous Jun 28 '14 at 22:37

The reason is really the size! Air molecules have the same size if they flow around a small paper airplane or a big real airplane, but since their relative size is so much smaller when they flow around the real airplane, friction effects are much smaller for the big airplane. When comparing airplanes of different sizes, a dimensionless number first formulated by Osborne Reynolds is most helpful. It is basically the ratio of inertial to viscous forces in the airflow, and the formula is $$Re = \frac{l \cdot v}{\nu}$$ Nomenclature:
$Re\:\:\:$ Reynolds number
$l\:\:\:\:\:\:$ characteristic length, for example wing chord
$v\:\:\:\:\:$ airspeed
$\nu\:\:\:\:\:$ kinematic viscosity of air

Flows with the same Reynolds number behave identically, but your scaled-down aircraft flies at a very different Reynolds number than the original (both speed and characteristic length are a lot lower), so the airflow around it will be very different. For it, the air looks more like a syrup when that around the original is like flowing water! Therefore, lift creation needs different means at this scale (wings with more chord, for example), and the paper airplane is better suited to cope with this particular flow condition.

The next difference is wing loading: The paper airplane is very light; its mass per area is low. Therefore, it can fly at a slower speed than your scaled-down real airplane, and it will be easier for you to launch the paper airplane properly into the air. The scaled-down airplane needs to be thrown harder, but still straight, and this might be another reason for it falling down quickly.

-
How will a giant paper (or other material if paper is too weak) plane fly? Will it also sink like a stone? – Anonymous Jun 28 '14 at 23:36
With a bit of cheating, it can be made to work. Look at cardboard, which is essentially glued paper. Smart construction can reinforce paper to be strong enough. You should be able to pull off a 2 meter glider. – MSalters Jun 28 '14 at 23:54
@Anonymous: Increasing size will help, even if the shape of the plane stays identical. But you definitely need something stronger than paper. The plane will fly, but will need more thrust (if engines are installed) or fly less far when launched compared to a plane with conventional shape due to its lower aerodynamic efficiency. – Peter Kämpf Jun 29 '14 at 5:37
@MSalters: A 2 meter glider is essentially a model aircraft, and they can be made similar to the real ones. But optimization for this scale will still lead to bigger and stubbier wings and airfoils with less camber and thickness, for example. – Peter Kämpf Jun 29 '14 at 5:40
and at the 2 meter scale you will be using fiberglass or if you have deep pockets carbon – ratchet freak Jun 29 '14 at 14:06


text:
# In an equilateral triangle ABC if AD ⊥ BC, then

Question:

In an equilateral triangle ABC if AD ⊥ BC, then

(a) $5 \mathrm{AB}^{2}=4 \mathrm{AD}^{2}$

(b) $3 \mathrm{AB}^{2}=4 \mathrm{AD}^{2}$

(c) $4 \mathrm{AB}^{2}=3 \mathrm{AD}^{2}$

(d) $2 \mathrm{AB}^{2}=3 \mathrm{AD}^{2}$

Solution:

$\triangle \mathrm{ABC}$ is an equilateral triangle and $\mathrm{AD} \perp \mathrm{BC}$.

In ∆ABD, applying Pythagoras theorem, we get

$\mathrm{AB}^{2}=\mathrm{AD}^{2}+\mathrm{BD}^{2}$

$\mathrm{AB}^{2}=\mathrm{AD}^{2}+\left(\frac{1}{2} \mathrm{BC}\right)^{2}\left(\because \mathrm{BD}=\frac{1}{2} \mathrm{BC}\right)$

$\mathrm{AB}^{2}=\mathrm{AD}^{2}+\left(\frac{1}{2} \mathrm{AB}\right)^{2}(\because \mathrm{AB}=\mathrm{BC})$

$\mathrm{AB}^{2}=\mathrm{AD}^{2}+\frac{1}{4} \mathrm{AB}^{2}$

$3 \mathrm{AB}^{2}=4 \mathrm{AD}^{2}$

We got the result as $(b)$.

Leave a comment


text:
Yesterday i wrote that you can use __($string,$domain) and _e($string,$domain) to internationalize a string with WordPress. I forgot to mention that if you use sprintf you can handle strings with variables too. An example:

echo sprintf(__('There are %d monkeys in the %s'), $domain),$number, \$location);


text:
# rotating vs lscape vs pdflscape

I'm asking this looking for the ultimate answer. What are the main differences between the three of them? When should I use anyone in particular?

I know there is this question: Should I prefer {pdflscape} over [pdftex]{lscape}?, but it doesn't address the issue with the rotating package.

Also, there is this question: Rotating text by -90 degrees , but once gain: no reference about the rotating package.

Furthermore, I'm trying to get a figure (packages: graphicx, float, subcaption), and later on a Gantt diagram (packages: tikz, pgfgantt) on landscape layout.

Which one of the three should I use? What are the advantages of one over the other? (I have already gone through this question: Rotate Gantt Chart Figure, but the only answer is with lscape).

UPDATE: I ran some tests and it seems that when using a longtable environment pdflscape modifies all the pages along the table, whereas [pdftex]{lscape} only modifies the last one.

Code:

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{array}
\usepackage{longtable}
\usepackage{pdflscape}
%\usepackage[pdftex]{lscape}  % Uncomment if you want to see the output with [pdftex]{lscape}

\begin{landscape}
\begin{longtable}{l p{0.8\linewidth}}

WEFAPS          & \hspace{5mm} Wearable Fall Assessment \& Prediction System\\
WHO                 &   \hspace{5mm} World Health Organization\\
GDP                 & \hspace{5mm} Gross Domestic Product\\
PPP                 & \hspace{5mm} Purchasing Power Parities\\
MEMs                &   \hspace{5mm} Micro-Electromechanics\\
BDN                 & \hspace{5mm} Bayesian Dynamic Networks\\
ADL                 & \hspace{5mm} Activities of Daily Life\\
STST                &   \hspace{5mm} Sit-To-Stand Test\\
PPA                 &   \hspace{5mm} Physiological Profile Assessment\\
TUGT                &   \hspace{5mm} Timed Up-and-Go Test\\
DFRAC               & \hspace{5mm} Demura's Fall Risk Assessment Chart\\
STRATIFY        & \hspace{5mm} Saint Thomas's Risk Assessment Tool in Falling Elderly Inpatients\\
USUST               & \hspace{5mm} Unstructured and Unsupervised Test\\
WEFAPS          & \hspace{5mm} Wearable Fall Assessment \& Prediction System\\
WHO                 &   \hspace{5mm} World Health Organization\\
GDP                 & \hspace{5mm} Gross Domestic Product\\
PPP                 & \hspace{5mm} Purchasing Power Parities\\
MEMs                &   \hspace{5mm} Micro-Electromechanics\\
BDN                 & \hspace{5mm} Bayesian Dynamic Networks\\
ADL                 & \hspace{5mm} Activities of Daily Life\\
STST                &   \hspace{5mm} Sit-To-Stand Test\\
PPA                 &   \hspace{5mm} Physiological Profile Assessment\\
TUGT                &   \hspace{5mm} Timed Up-and-Go Test\\
DFRAC               & \hspace{5mm} Demura's Fall Risk Assessment Chart\\
STRATIFY        & \hspace{5mm} Saint Thomas's Risk Assessment Tool in Falling Elderly Inpatients\\
USUST               & \hspace{5mm} Unstructured and Unsupervised Test\\
WEFAPS          & \hspace{5mm} Wearable Fall Assessment \& Prediction System\\
WHO                 &   \hspace{5mm} World Health Organization\\
GDP                 & \hspace{5mm} Gross Domestic Product\\
PPP                 & \hspace{5mm} Purchasing Power Parities\\
MEMs                &   \hspace{5mm} Micro-Electromechanics\\
BDN                 & \hspace{5mm} Bayesian Dynamic Networks\\
ADL                 & \hspace{5mm} Activities of Daily Life\\
STST                &   \hspace{5mm} Sit-To-Stand Test\\
PPA                 &   \hspace{5mm} Physiological Profile Assessment\\
TUGT                &   \hspace{5mm} Timed Up-and-Go Test\\
DFRAC               & \hspace{5mm} Demura's Fall Risk Assessment Chart\\
STRATIFY        & \hspace{5mm} Saint Thomas's Risk Assessment Tool in Falling Elderly Inpatients\\
USUST               & \hspace{5mm} Unstructured and Unsupervised Test\\
WEFAPS          & \hspace{5mm} Wearable Fall Assessment \& Prediction System\\
WHO                 &   \hspace{5mm} World Health Organization\\
GDP                 & \hspace{5mm} Gross Domestic Product\\
PPP                 & \hspace{5mm} Purchasing Power Parities\\
MEMs                &   \hspace{5mm} Micro-Electromechanics\\
BDN                 & \hspace{5mm} Bayesian Dynamic Networks\\
ADL                 & \hspace{5mm} Activities of Daily Life\\
STST                &   \hspace{5mm} Sit-To-Stand Test\\
PPA                 &   \hspace{5mm} Physiological Profile Assessment\\
TUGT                &   \hspace{5mm} Timed Up-and-Go Test\\
DFRAC               & \hspace{5mm} Demura's Fall Risk Assessment Chart\\
STRATIFY        & \hspace{5mm} Saint Thomas's Risk Assessment Tool in Falling Elderly Inpatients\\
USUST               & \hspace{5mm} Unstructured and Unsupervised Test\\
WEFAPS          & \hspace{5mm} Wearable Fall Assessment \& Prediction System\\
WHO                 &   \hspace{5mm} World Health Organization\\
GDP                 & \hspace{5mm} Gross Domestic Product\\
PPP                 & \hspace{5mm} Purchasing Power Parities\\
MEMs                &   \hspace{5mm} Micro-Electromechanics\\
BDN                 & \hspace{5mm} Bayesian Dynamic Networks\\
ADL                 & \hspace{5mm} Activities of Daily Life\\
STST                &   \hspace{5mm} Sit-To-Stand Test\\
PPA                 &   \hspace{5mm} Physiological Profile Assessment\\
TUGT                &   \hspace{5mm} Timed Up-and-Go Test\\
DFRAC               & \hspace{5mm} Demura's Fall Risk Assessment Chart\\
STRATIFY        & \hspace{5mm} Saint Thomas's Risk Assessment Tool in Falling Elderly Inpatients\\
USUST               & \hspace{5mm} Unstructured and Unsupervised Test\\

\end{longtable}
\end{landscape}
\end{document}

• is it possible to add a MWE demonstrating the update ? code speaks better :) – texenthusiast May 1 '13 at 23:02

lscape and pdflscape is the same, the first for using latex the second for pdflatex. If your image needs more or less the total \textheigth as width then use the landscape environment, defined by these packages. If the image is much more smaller and doesn't need the whole page then use \rotatebox or the environment turn or \hvfloat from the package of the same name if you also need a caption and, of course, no page break.

You do not need all those \hspace:

%\usepackage{pdflscape}
\usepackage{lscape}
[ ... ]
\begin{landscape}
\begin{longtable}{l @{\hspace{5mm}} p{0.8\linewidth}}

WEFAPS          &  Wearable Fall Assessment \& Prediction System\\

and viewing the pdf output with lscape is identical to the one with pdflscape

• I was running some tests and I saw that pdflscape rotates all pages from a longtable, whereas [pdftex]lscape only the last one – Mario S. E. May 1 '13 at 21:17
• textheigth-->text height – texenthusiast May 1 '13 at 21:23
• @MarioS.E.: I suppose you viewed the dvi output. View only the pdf one: latex->dvips->ps2pdf. And you do not need the [pdftex] option. – user2478 May 2 '13 at 6:52
• Why did't you suggest the package hvfloat ;-) – Marco Daniel May 2 '13 at 7:11
• @MarioS.E.: Again: if you run pdflatex then pdflscape and if you run latex then lscape – user2478 May 2 '13 at 10:08


text:
Are you between 5 and 25? If so, enter the Wicked Young Writer Awards NOW - click here for details...

# On Necessary and Sufficient Conditions for Lp-estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates

## Pascal Auscher

Part of the Memoirs of the American Mathematical Society Series

## Synopsis

On Necessary and Sufficient Conditions for Lp-estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates by Pascal Auscher

This memoir focuses on $Lp$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. The author introduces four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $Lp$ estimates. It appears that the case $p2$ which is new. The author thus recovers in a unified and coherent way many $Lp$ estimates and gives further applications. The key tools from harmonic analysis are two criteria for $Lp$ boundedness, one for $p2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.

## Book Info

#### Publication date

15th February 2007

#### Author

Pascal Auscher

More books by Pascal Auscher
Author 'Like for Like'
recommendations

#### Publisher

American Mathematical Society

Paperback
75 pages

#### Categories

Functional analysis & transforms
Differential calculus & equations

#### ISBN

9780821839416

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text:
OpenStudy (anonymous):

Help? If k, n, x, and y are positive numbers satisfying x^(-4/3) = k^-2 and y^(4/3) = n^2, what is (xy)^(-2/3) in terms of n and k? A. 1/(nk) B. (n/k) C. (k/n) D. nk E. 1

2 years ago
OpenStudy (retireed):

I hate negative exponents so I get rid of then. How would you do this?

2 years ago
OpenStudy (retireed):

What is X to the minus 1 X^-1 $X ^{-1}$

2 years ago
OpenStudy (retireed):

Without the negative exponent.

2 years ago
OpenStudy (retireed):

Sorry that does not help me like I thought it would.

2 years ago


text:
Graph containing all trees? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:39:26Z http://mathoverflow.net/feeds/question/54193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54193/graph-containing-all-trees Graph containing all trees? Bill Bradley 2011-02-03T13:16:08Z 2011-02-07T14:07:29Z
Consider graphs on $n$ nodes. I am trying to find a graph $G$ that contains all $n$-node trees as sub-graphs but contains as few edges as possible. The complete graph $K_n$ suffices, but can we get by with fewer edges? Maybe $O(n)$ edges?

(This problem arose in the context of circuit design, where edges in $G$ correspond to wires in a chip layout.)
 http://mathoverflow.net/questions/54193/graph-containing-all-trees/54196#54196 Answer by Flo Pfender for Graph containing all trees? Flo Pfender 2011-02-03T13:37:36Z 2011-02-03T13:37:36Z
Well, $O(n)$ will not do, as you need 1 vertex of degree $n-1$, a total of two vertices of degree $\ge \frac{n}{2}$, etc., to embedd all the trees with few vertices of high degree and only leaves otherwise. That should get you $cn\log n$ edges as a lower bound.

My guess is that this should be about right asymptotically, as you should be able to take a random graph with average degree $c\Delta\log n$ to get (all?) trees of max degree $\Delta$ (see http://arxiv.org/abs/1007.2326), and add some large degree vertices to get all other trees from this.
 http://mathoverflow.net/questions/54193/graph-containing-all-trees/54199#54199 Answer by Manor Mendel for Graph containing all trees? Manor Mendel 2011-02-03T14:41:53Z 2011-02-03T14:41:53Z
The standard term is "universal graphs", I think. See http://en.wikipedia.org/wiki/Universal_graph According to that entry the answer is indeed $O(n \log n)$.
 http://mathoverflow.net/questions/54193/graph-containing-all-trees/54211#54211 Answer by David Eppstein for Graph containing all trees? David Eppstein 2011-02-03T16:40:08Z 2011-02-03T16:40:08Z
See Chung and Graham, On Universal Graphs for Spanning Trees. They prove that the number of edges is $\Theta(n\log n)$.
 http://mathoverflow.net/questions/54193/graph-containing-all-trees/54636#54636 Answer by Mohsen for Graph containing all trees? Mohsen 2011-02-07T14:07:29Z 2011-02-07T14:07:29Z
see representation number of graph by ring

it is useful.


text:
# D2Editor:registerFilter

registerFilter is a function that is part of D2Editor (2D Editor).

This routine registers a table of handler functions that can filter events before they are sent to the 2D editor widget, or unregisters if nil is passed.

The handlerTable is a table of event handler functions. Any event function described within the handlerTable can be filtered by a function in the handler table.

In the example code below, if the user presses Tab in the text editor ed, the tabKey filter function moves the focus to text editor ed2. Events charIn and arrowKey simply report which key was pressed and then allow the event to pass on through to the text editor.

This has been introduced in TI-Nspire OS 3.2 (Changes).

## Syntax

D2Editor.registerFilter(editor, [handlerTable])
( or editor:registerFilter([handlerTable]) )

Parameter Type Description
handlerTable
table table of event functions (like enterKey, tabKey...)

The handlerTable argument is optional, since not passing an argument (nil) is the way to reset (turn off) the 2DEditor's event filtering.

## Example

-- Create an editor
ed = D2Editor.newRichText()

-- Register filters for events
ed:registerFilter {
tabKey = function() ed2:setFocus() return true end,
charIn = function(ch) print(ch) return false end
arrowKey = function(key) print(key) return false end
}

## Good to know

• The event handler functions have to be defined "on their own", like "enterKey", "tabKey" etc, and not like the usual definitions "on.enterKey", "on.tabKey" (this is because in that case, you're defining methods of the "on" table, the global events)
• The event handler functions in the registerFilter table have to return either true or false : true meaning that no further action will be taken for that event, while false means that the default action for the event will still be executed.


text:
## Tuesday, 15 April 2008

### Relativity Confusion

I've just been working through some relativity questions, and most of the stuff I can just about manage, when prompted by the solutions. Indeed, figuring out which Lorentz transformation to use, and the rough procedure to solve a problem, aren't actually too bad, but today I'm having problems with basic mathematics.

The solutions given are in quite a brief format, where huge swathes of steps have been missed out. Often, it's just a few lines of algebra which is fairly straightforward, but today I don't seem to be able to "see" these, and have to work through them, blind.

One particular bit, which I've tried for the best part of 15 minutes, and have now given up on temporarily, is convincing myself that
\frac{1-v/c}{\sqrt{1-v^2/c^2}} = \sqrt{\frac{1-v/c}{1+v/c}}

This is a fairly fundamental step in calculating Relativistic Doppler shifts, since:
\nu^\prime = \nu\sqrt{\frac{1-v/c}{1+v/c}}

If anyone can tell me how those two things are equal, I'd love to know! Of course, maybe tomorrow it'll be clear. Today is not a good day to revise, but I have to do it anyway.

#### 1 comment:

1. (1-v/c)/sqrt(1-v^2/c^2)=(1-v/c)/sqrt((1-v/c)(1+v/c))=(1-v/c)^(1-1/2)/sqrt(1+v/c)=sqrt((1-v/c)/(1+v/c))
Where the identity 1-x^2=(1-x)(1+x) has been used ;)


text:
# Condition for algebra being sigma algebra

Suppose that we defined $\mathcal{M}_{\mu^{*}}=\left\{B:B \ \ \mu^{*}-\text{measurable}\right\}$ as the set of all $\mu^{*}$ measurable sets.A set is measurable with respect to outter measure $\mu^{*}$ if for every $A\in\mathcal{P}(X)$ we have $\mu^{*}(A)=\mu^{*}(A\cap B)+\mu^{*}(A\cap B^{c})$ then $B\in\mathcal{M}_{\mu^{*}}$

Outter measure is defined as

$\mu^{*}:\mathcal{P}(X)\rightarrow (0,\infty]$

$A\subset B \Rightarrow \mu^{*}(A)\leq \mu^{*}(B)$

for every $(A_{n})_{n\mathbb{N}}$ , $\mu^{*}(\cup_{n=1}^{\infty}A_{n})\leq \sum_{n=1}^{\infty}\mu^{*}(A_{n})$

$\mu^{*}(\varnothing )=0$

I proved that $M_{\mu^{*}}$ is algebra and based on this result I found the following proposition:

An algebra is sigma-algebra if for every $(A_{n})_{n\mathbb{N}}$ (where $A_{n}$ disjoint pairwise) ,$\cup_{n=1}^{\infty}A_{n}$ belongs in algebra.

It seems rational that we want to have the infinite countable union of $A_{n}$ in algebra because that's the definition of sigma algebra.But I'm confused on how to prove this proposition.

• – BCLC Mar 19 '18 at 6:10

Let $\mathcal A$ be an algebra that is closed under the formation of countable unions of sets in $\mathcal A$ that are mutually disjoint.
For $n=1,2,\dots$ let $B_n\in\mathcal A$.
Then $A_1:=B_1\in\mathcal A$ and for every $n\geq2$ also $A_n:=B_n\cap\left(\bigcup_{k=1}^{n-1}B_k\right)^{\complement}\in\mathcal A$.
This because $\mathcal A$ is closed under the formation of finite unions and complements.
Observe that the sets $A_n$ are mutually disjoint so that: $$\bigcup_{n=1}^{\infty}B_n=\bigcup_{n=1}^{\infty}A_n\in\mathcal A$$
So $\mathcal A$ is an algebra closed under the formation of countable unions, hence is a $\sigma$-algebra.


text:
# Definitions

• A subsequence may not be contiguous, e.g. [1, 1, 1] is a subsequence of [1, 2, 1, 2, 1].
• An equal subsequence is a subsequence in which every element is equal.
• The longest equal subsequence may not be unique, e.g. [1, 1] and [2, 2] are both longest equal subsequences of [2, 1, 1, 2].

# Input

A non-empty list of positive integers in one of the format below:

• as the native implementation of an array of positive integers in your language
• as a string of newline-separated integers in decimal
• as a string of newline-separated integers in unary
• any other reasonable formats

# Output

All of the longest equal subsequences in any order in one of the formats below:

• as a 2D nested array in your language (if the input is an array)
• as a flattened array with the equal elements being contiguous
• any other reasonable format

# Scoring

Although we are looking for something long, the code used should be as short as possible in terms of number of bytes, since this is

# Testcases

Inputs:

[1, 2, 3]
[1, 2, 2, 1]
[1, 2, 3, 2, 1]
[1, 2, 1, 2, 3, 4, 1]

Outputs:

[[1], [2], [3]]
[[1, 1], [2, 2]]
[[1, 1], [2, 2]]
[[1, 1, 1]]

Note that for the outputs above, any order is valid.

A flattened array is also valid, as long as the equal elements are contiguous.

• It would be simpler to talk of “most frequent elements” IMO: subsequences are used when the order is important, but here, every permutation of the input has the same set of allowed correct outputs. – ShreevatsaR May 1 '17 at 7:50
• @ShreevatsaR Sorry, I've edited the question. – Leaky Nun May 1 '17 at 7:51
• Does a flat list work for output? E.g. 1 2 3, 1 1 2 2, 1 1 2 2, 1 1 1? – Conor O'Brien May 1 '17 at 14:26
• @ConorO'Brien saying yes would invalidate most of the answers here... – Leaky Nun May 1 '17 at 14:29
• @LeakyNun As in, is it an acceptable alternative? – Conor O'Brien May 1 '17 at 14:30

# Jelly, 5 bytes

ĠLÐṀị

Try it online!

### How it works

ĠLÐṀị  Main link. Argument: A (array)

Ġ      Group; partition the indices of A by their corresponding values.
LÐṀ   Select all index arrays with maximal length.
ị  Unindex; retrieve the items of A at the specified indices.
• I thought Jelly doesn't have a maximum quick... – Leaky Nun May 1 '17 at 3:31
• It's technically a maximal quick, but yes, it does. – Dennis May 1 '17 at 3:31

# Brachylog, 7 bytes

⊇ᶠ=ˢlᵍh

Try it online!

## Explanation

⊇ᶠ=ˢlᵍh
⊇ᶠ        Find all subsequences
=ˢ      Keeping only those for which all elements are equal
lᵍ    Group by length
h   Take the first group

's natural order generates the longest subsequences first, so those are the ones that end up in the first group.

• Oh hey, another Brachylogist. – Leaky Nun May 1 '17 at 6:51
• Somehow you and me must have missed each other repeatedly in the Brachylog chat; I've been using it for months, and was surprised to learn that apparently someone else besides Fatalize was too. – user62131 May 1 '17 at 6:53

# Pyth, 5 bytes

S.M/Q

Test suite

Explanation:

This is implicitly S.M/QZQ. .M is the maximal function, so .M/QZQ selects all elements where the value of /QZ, count the number of occurrences of the element in the input, is maximal. S then sorts the list so that identical elements are contiguous.

## bash, 66 bytes

sort|uniq -c|sort -rn|awk 'NR==1{a=$1}$1==a{for(i=a;i--;)print$2}' This seems like it should be way shorter, but I can't figure out how. sort # sort the input |uniq -c # group runs of identical lines and prefix with count |sort -rn # sort by count, with largest at top |awk ' # pipe to awk... NR==1{a=$1}         # on the first line, set the variable "a" to field 1
$1==a{ # on any line, if first field is a (max count)... for(i=a;i--;) # a times... print$2           # print the second field
}
'

Try it online!

Thanks to Leaky Nun for 3 bytes!

• 3 bytes off – Leaky Nun May 1 '17 at 3:50
• Consider updating your explanation – Leaky Nun May 1 '17 at 4:24

# Python 2, 68 63 bytes

lambda x:sorted(n for n in x if x.count(n)/max(map(x.count,x)))

Try it online!

• Would like to see an answer in Python 3 :p – Leaky Nun May 1 '17 at 5:32
• Porting this one is trivial: just replace print with return. – Dennis May 1 '17 at 5:33
• Oh, I thought Python 3 doesn't have map. – Leaky Nun May 1 '17 at 5:35
• It's a bit different in 3 (returns a generator and truncates longer iterables if there are more than two arguments), but it's there. – Dennis May 1 '17 at 5:36
• I thought Python had a built-in for this – Beta Decay May 1 '17 at 6:07

# Mathematica, 4231 25 bytes

Thanks @GregMartin for 5 bytes and @MartinEnder for another byte!

MaximalBy[Length]@*Gather

## Explanation

MaximalBy[Length]@*Gather  (*                       {1, 2, 3, 2, 1}       *)
Gather  (* Gather same numbers:  {{1, 1}, {2, 2}, {3}} *)
@*        (* Function composition                        *)
MaximalBy[Length]          (* Find longest:         {{1, 1}, {2, 2}}      *)
• You can save 5 bytes with Gather@#~MaximalBy~Length&. – Greg Martin May 1 '17 at 5:59
• @GregMartin and then MaximalBy[Length]@*Gather. – Martin Ender May 1 '17 at 14:20
• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:38

# Stacked, 5552 43 bytes

sorted rle toarr:[1#]map MAX@K[1#K=]YES rld

Try it online!

Works by run-length encoding the input, sorting by occurrences, keeping occurances for which the number of occurrences is maximal, and run length decoding. Outputs through a flat list, as is acceptable by the challenge.

# Actually, 23 bytes

;╗⌠;╜ck⌡M;♂NM╗⌠N╜=⌡░♂FS

Thanks to Leaky Nun for pointing out a one-byte improvement that really should've been obvious to me

-3 bytes from relaxed output format

Explanation:

;╗⌠;╜ck⌡M;♂NM╗⌠N╜=⌡░♂FS
;╗                        save a copy of the input to register 0
⌠;╜ck⌡M                 for each value in the input list:
;                        make a copy on the stack
╜c                      count the occurrences in the input list (from register 0)
k                     make a list: [value, count]
;♂N             make a copy, take last value of each list in the 2D list
M╗           store the maximum count in register 0
⌠N╜=⌡░     filter the other copy of the list of [value, count] lists:
N╜=         take items where the count equals the maximum count
♂FS  take first items (values) and sort them

# Python 2, 138 bytes

lambda l:[[x[0]]*x[1] for x in next(__import__('itertools').groupby(__import__('collections').Counter(l).most_common(),lambda x:x[1]))[1]]
• itertools is never the shortest :p – Leaky Nun May 1 '17 at 5:31
• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:40

# MATL, 10 bytes

3#XMg1bX"&

Try it online!

### Explanation

Similar to my Octave answer. Consider input [10, 20, 30, 20, 10] as an example.

3#XM   % Three-output version of mode function. Gives the first mode, the
% number of repetitions, and a cell array with all modes
% STACK: 10, 2, {10; 20}
g      % Convert from cell array to matrix
% STACK: 10, 2, [10; 20]
1      % Push 1
% STACK: 10, 2, [10; 20], 1
b      % Bubble up in the stack
% STACK: 10, [10; 20], 1, 2
X"     % Repeat those number of times vertically and horizontally
% STACK: 10, [10, 10; 20, 20]
&      % Specify that implicit display will show only the top of the stack.
% Since this is singleton cell array that contains a matrix, that
% matrix is directly displayed
• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:39
• @LeakyNun Thanks for letting me know – Luis Mendo May 1 '17 at 17:55
• It is my responsibility. – Leaky Nun May 1 '17 at 17:55

# Octave, 47 bytes

[~,b,c]=mode(input(0));disp([repmat(c,1,b){:}])

Try it online!

### Explanation

The second and third outputs of mode (obtained as [~,b,c]=mode(...)) respectively give the number of repetitions (b) and a column cell array (c) of the most repeated elements in the input (input(0)) . The cell array c is then repeated horizontally b times (repmat(c,1,b)), converted to a comma-separated list ({:}) and contatenated horizontally ([...]) to give a numeric matrix, which is displayed (disp(...)).

• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:39

# 05AB1E, 8 5 bytes

Outputs a flat list in order

.M¹Ã{

Uses the 05AB1E encoding. Try it online!

• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:39
• @LeakyNun Thanks for the notification :) – Adnan May 1 '17 at 14:49

# CJam, 22 bytes

{$ez~\__:e>f=.*\]ze~} This is an anonymous block (function) that takes the input from the top of the stack and repaces it with the output. The output is a flattened array withequal elements being contiguous. Try it online! ### Explanation Consider input [10 20 30 20 10 ] as an example. { e# Begin block e# STACK: [10 20 30 20 10]$    e#   Sort
e#   STACK: [10 10 20 20 30]
e   e#   Run-length encoding
e#   STACK: [[2 10] [2 20] [1 30]]
z    e#   Zip
e#   STACK: [[2 2 1] [10 20 30]]
~    e#   Dump array contents onto the stack
e#   STACK: [2 2 1] [10 20 30]
\    e#   Swap
e#   STACK: [10 20 30] [2 2 1]
__   e#   Duplicate twice
e#   STACK: [10 20 30] [2 2 1] [2 2 1] [2 2 1]
:e>  e#   Fold maximum over array. Gives the maximum of the array
e#   STACK: [10 20 30] [2 2 1] [2 2 1] 2
f=   e#   Map "is equal" with number (2) over the array ([2 2 1])
e#   STACK: [10 20 30] [2 2 1] [1 1 0]
.*   e#   Vectorized multiplication
e#   STACK: [10 20 30] [2 2 0]
\    e#   Swap
e#   STACK: [2 2 0] [10 20 30]
]    e#   Pack into array
e#   STACK: [[2 2 0] [10 20 30]]
z    e#   Zip
e#   STACK: [[2 10] [2 20] [0 30]]
e~   e#   Run-length decoding
e#   STACK: [10 10 20 20]
}      e# End block

# Perl 5, 58 bytes

sub{sort grep$x{$_}>$m,grep{$/=$x{$_}++;$m=$/if$m<$/;1}@_}

# APL (Dyalog), 22 bytes

Requires ⎕ML←3 which is default on many systems.

### Program: s/⍨(⌈/=⊢)≢¨s←⊂⍨(⍋⊃¨⊂)⎕

get numeric (evaluated) input

() tacit function
the indices of ascending items
⊃¨ each pick from
the entire array

⊂⍨ partition by cutting at its increases

s← store as s

≢¨ tally each

() tacit function
⌈/ the maximum (tally)
= equals
the argument (the tallies)

s/⍨ filter s with that

### Function: {s/⍨(⌈/=⊢)≢¨s←⊂⍨⍵[⍋⍵]}

{} anonymous function where argument is

⍵[⍋⍵] sort (lit. index with indices of ascending items)

⊂⍨ partition by cutting at its increases

s← store as s

≢¨ tally each

() tacit function
⌈/ the maximum (tally)
= equals
the argument (the tallies)

s/⍨ filter s with that Try it online!

• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:40

# PHP, 69 Bytes

<?print_r(preg_grep("#".max($r=array_count_values($_GET))."#",$r)); Online Version ## Output Format key = value , value = count Array ( [1] => 2 [2] => 2 ) ## PHP, 96 Bytes <?foreach($_GET as$v)$r[$m[]=count($l=preg_grep("#^{$v}$#",$_GET))][$v]=$l;print_r($r[max($m)]); Online Version ## Output Format 1D Key= value 2D Key = position in the input array for each value Array ( [1] => Array ( [0] => 1 [4] => 1 ) [2] => Array ( [1] => 2 [3] => 2 ) ) ## PHP, 97 Bytes <?foreach($_GET as$v)$r[count($l=preg_grep("#^{$v}$#",$_GET))][$v]=$l;ksort($r);print_r(end($r));
• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:40

## JavaScript (ES6), 84 83 bytes

Returns a sorted flattened array.

a=>a.sort().filter((_,i)=>b[i]==Math.min(...b),b=a.map(i=>a.filter(j=>i-j).length))

### Test cases

let f =

a=>a.sort().filter((_,i)=>b[i]==Math.min(...b),b=a.map(i=>a.filter(j=>i-j).length))

console.log(JSON.stringify(f([1, 2, 3])))
console.log(JSON.stringify(f([1, 2, 2, 1])))
console.log(JSON.stringify(f([1, 2, 3, 2, 1])))
console.log(JSON.stringify(f([1, 2, 1, 2, 3, 4, 1])))

• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:39
• @LeakyNun Thanks for the notification. – Arnauld May 1 '17 at 15:43

## CJam, 24 bytes

{$e_$W=0=\{0=1$=},e~\;} I wanted to do this in 05ab1e, but I gave up :P This is a block. Input and output are arrays on the stack. Try it online! Explanation: { e# Stack: | [1 2 3 2 1]$                     e# Sort:                 | [1 1 2 2 3]
e                   e# RLE encode:           | [[2 1] [2 2] [1 3]]
_$W= e# Copy elements: | [[2 1] [2 2] [1 3]] [2 1] 0= e# First element: | [[2 1] [2 2] [1 3]] 2 \ e# Swap: | 2 [[2 1] [2 2] [1 3]] {0=1$=},     e# Filter where x[0]==2: | 2 [[2 1] [2 2]]
e~   e# RLE decode:           | 2 [1 1 2 2]
\; e# Delete back:          | [1 1 2 2]
}
• This only works if the smallest integer belongs to the most common elements. You'll need \$W= instead of the first 0=. – Martin Ender May 1 '17 at 14:22
• I have added another acceptable alternative which might help you golf off some bytes. – Leaky Nun May 1 '17 at 14:39

## Clojure, 65 bytes

#(let[P partition-by C count](last(P C(sort-by C(P +(sort %))))))

Ungolfed:

(def f #(->> %
(sort-by      identity)   ; sort so that identical values are one after another, same as sort
(partition-by identity)   ; partition by identity (duh!)
(sort-by      count)      ; sort by item count
(partition-by count)      ; partition by item count
last))                    ; get the last partition

# C#, 145 Bytes

l=>{var t=Enumerable.Range(0,l.Max()+1).Select(i=>l.Count(a=>a==i));return t.Select((a,i)=>Enumerable.Repeat(i,a)).Where(d=>d.Count()==t.Max());}

This must be possible better as well, however I'm kind of stuck.

Explanation

l =>                                                   //Takes the list
{                                                      //...
var t = Enumerable.Range(0, l.Max() + 1)           //Makes a range till the count, so that the items together with their indices are double defined (i.e. the items are 0,1,2,3... and the indices are the same)
.Select(i =>                     //Takes the items
l.Count(a => a == i));       //And replaces them with the count of themselves in the list (so the item has the index with its old value and the count as it's actual value)
return t.Select((a, i) =>                          //Then it takes this list and selects the items together with the indices
Enumerable.Repeat(i, a))                       //Repeats them as often as they appeared in the list
.Where(d => d.Count() == t.Max());   //And just keeps those which appear the maximum amount of times
};                                                     //...

Probably a totally different approach would be much shorter, so the C# challenge is still open :)

# Ruby, 57 bytes

->a{a.reject{|i|a.map{|j|a.count j}.max>a.count(i)}.sort}

Try it online!


text:
# STEVE The two of us can go, Charlie. (he turns to look back at the car) It couldn’t be the meteor.

###### Question:

STEVE
The two of us can go, Charlie.
(he turns to look back at the car)
It couldn’t be the meteor. A meteor couldn’t do this.
He and Charlie exchange a look then they start to walk away from the group.
What do the stage directions tell the reader about Steve’s and Charlie’s point of view?
They do not want to leave their homes and walk into town.
They suspect something strange may be going on.
They know what caused the flash of light but do not want to say it.
They are angry at the group of people for making them leave.

### Dalton's Law: John puts 1250 PSI of Oxygen, 780 PSI of Nitrogen and 550 PSI of Helium in a SCUBA tank. What is the total pressure?

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### How is the cell membrane like a bag with tiny holes?

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### Write a verbal expression for 4(2x7)

Write a verbal expression for 4(2x7)...

### Need help im confused 15 points

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### What caused The Great Depression?

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### Gia measures the mass of her football. She uses four 100g weights and two 10g weights to balance the scale. What is the mass

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### Although early scientists attempted to classify people into races based strictly on static –categories, we now know that

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### Please Join It’s just for fun

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### What is the only island state in the us

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### The diagram shows a dilation of segment AB. Which statements are true about the dilation? Select all that apply.

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### 10. Chocolate malt balls used to sell for sixty-five cents a pound, but you see that the new price iseighty-one

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### I need help the area of the triangle is 24 square feet. Hat is the lengths of the base

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### Elige la respuesta correcta, que represente el siguiente problema: En mi rancho tenemos ovejas, mi abuela vendió 40 y ahora tenemos 120.

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text:
1 posts
Why does *Row has to be a uint8 instead of a uint32
Hi I was just hoping someone could please explain why *Row has to be a uint8 instead of a uint32 and why then *Pixel is a uint32. Thanks

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 uint8 *Row = Buffer->Memory + MinX * Buffer->BytesPerPixel + MinY * Buffer->Pitch for(int Y = MinY; Y < MaxY; Y++) { uint32 *Pixel = (uint32 *)Row; for(int X = MinX; X < MaxX; X++) { *Pixel++ = Color; } Row += Buffer->Pitch; }
504 posts
Why does *Row has to be a uint8 instead of a uint32
So that Row += Buffer->Pitch; will work.

pointer arithmetic will automatically multiply the size of the thing pointer ad with the offsets you add and subtract to it. uint8 is the same granularity in which Pitch is provided so it makes sense to use that.
Simon Anciaux
1204 posts
Why does *Row has to be a uint8 instead of a uint32
1 posts
Why does *Row has to be a uint8 instead of a uint32
 1 2 3 4 fake Buffer->Memory 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n

you can pretend that each element of the array above is the size uint8
so from the start of the array, 0, going 4 uint8's in gets you to 4
 1 [start here]0, 1, 2, 3, [end here]4

 1 uint8 *arraystart + 4 = 4

and from
 1 (uint8 *)(j's position) + 4 = n

since a uint32 is equal to 4 uint8's, it represents 4 elements in the array.
from the start of the array going 1 uint32 further gets you to 4

 1 (uint32 *)arraystart + 1 = 4

pitch = amount to get to the next row but under where the old position was
pitch = 8
row is a uint8 pointer type
 1 uint8 *Row
so from 0's position adding the pitch puts you at 8, at c it puts you to k.

if row were a uint32 * then adding pitch would add pitch * 4 = 32
and would go outside of the array above, the point being it's not working as a pitch then

with the uint32 *Pixel, it's the same logic, you're accessing 4 elements at a time.
 1 [0, 1, 2 , 3], [4, 5, 6, 7] ... [g, h, i, j] ...


text:
# Request : Make Message Board

Hi Everyone !

It is Request From my friend's to make thier Message Board's . Here Brilliant communication gap is an problem . So that when I want to talk to you guy's , I can talk .

Even Many of us are really wanted It from you guy's . Special Request :

@DEEPANSHU GUPTA

@RONAK AGARWAL

@Mvs Saketh

@Megh choksi

@Krishna sharma

@Pratik shastri

@Pranjal Jain

@Sudeep salgia

@Shwahank shukla

@Azhaghu Roopesh

@Satvik Pandey

@Raghav

And all other's .

Thanks ! It will be great for all lerner's Like me for learning from you. and asking doubt's , share views etc.

Note by Karan Shekhawat
6 years, 5 months ago

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I want to know some aspects of yours, would you please ?

1. At what level/position/status, you want to see yourself in the coming approx. 8 years ?

2. What is/are those thing(s) which hold you back from attaining your target(s) and what you do to overcome those hurdles ?

3. Your favorite teacher up to the day, and why ? @KARAN SHEKHAWAT

- 6 years, 5 months ago

Thank's Sir ! first of all sir I just curious to know why u r asking such quest. ?

1)- Sir I just wanted to complete my graduation in Good IIT's . with Meachanical branch andThen I want to do Specialisation in Energy field . This almost take 5-6 year's and after that I have no current Plan's.

2)- Sir It was only Hunger to get selected in good IIT , and my ambition which Hold's me , during this year . Also I alway's think that I have no option to do , so this alway's hold's me . But Yes Sometimes I feel very boared and alone , since my all friend's goe's to collage's and I am at home . It is Really very disappointing and depressing but Yet above method's help's me lot .

3)- I have Respect My all Coaching Teacher's and Teacher's upto 10th class . , Except my 11th - 12th class school teacher's since I was in govt. school so you know , what was condition's of us . No teacher will Taught us , they said come on his tution's . which is insane . And I Respect The " Anil Kumar Gupta " Sir who is Physic's HOD of bansal classes , jaipur . I was studied 11th class there , But then I move other institutes due to Far distance of house . But Still I Respect him from bottom of my heart . I can't tell about him in word's , But yet He is best , Beacuse He alway's enforce to study with understanding , Don't ever learn even single thing , it is useless . And Also He enforce to clear our doubt's at own . He say's first think yourself ,and if not able to solved then ask to him . and many other thing's ......

- 6 years, 5 months ago

I was asking you just like that, no specific reasons, just to know. All the best. If you mentally refuse to loose, you will always find a way to win. Keep it up. @KARAN SHEKHAWAT

- 6 years, 5 months ago

A nice idea from you :)

But I don't think that I am qualified enough to make a Message Board of my own .

Also my name is Roopesh , you have misspelt my name in both the Notes that you've posted .It doesn't matter if you leave out my surname , but please just correct my name's spelling

- 6 years, 5 months ago

Thanks for editing my name :)

- 6 years, 5 months ago

Yes ! I'am sorry for that !

- 6 years, 5 months ago

This is a good idea. Count me in if this happens! Although it might be a bit difficult for the people you requested to help you out. Board exam in 5 days and JEE in a month. But after that it will be possible, assuming we still do problems from here as a hobby.

- 6 years, 5 months ago

I liked your idea too. :) I would like to see a message board coming from other guys that you have mentioned. I don't want to compel you all for doing that because I know you all are too busy now a days. But I want to request you all to share a note about your experiences of JEE after the exam.

I don't think I am qualified enough to clear your doubts instead I would like to learn from you guys. :)

- 6 years, 5 months ago

I guess @Deepanshu Gupta has made his message Board and it has already garnered around 120 replies !!

- 6 years, 5 months ago