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The next paragraph below has been taken from the formal short biography I usually send to IEEE papers. I hope to be far more interesting now since I am involved in many Bachelor's and Master's theses that use Arduino, NodeMCU and STM32L4 microcontrollers to make low-cost gadgets. I'd like to be considered as maker and STEM wielder. My research interests are veering to a new and interesting topic: energy harvesting for IoT applications. Let's see how all this ends.
Guillermo Robles was born in Madrid, Spain, in 1969. He received the M.Sc. and Ph.D. degrees in electronic engineering from the Comillas University of Madrid, Spain, in 1993 and 2002, respectively. In 2002, he joined the Department of Electrical Engineering in the Carlos III University of Madrid, Spain, where he has been an Associate Professor since 2009. He is also with the High-Voltage Research and Tests Laboratory (LINEALT). He has coauthored more than 70 papers in international journals and conferences. His research interests include the design of sensors, instrumentation and measurement techniques for high frequency currents and UHF signals, particularly due to partial discharges in noisy environments. He is currently involved in the development of algorithms for the localization of radio-frequency sources, the determination of time differences of arrival to the array of antennas and the optimization of the antenna layout to improve the localization accuracy.


Title: Software applications in Civil Engineering.
Now a days software applications in Civil Engineering are being used at all levels . In view of that Sri Raghuram Bhat delivered major softwares like Etabs,Rivet Architecture,STAAD Pro usage and utilities to enrich the practical knowledge of student and faculty.The technical lecture was very useful for final year students.


text:
\section{Introduction}

Strong fluctuations of eigenfunctions are the characteristic feature of the Anderson metal-insulator transition, which can be quantified by a set of inverse participation ratios, $P_q = \int d^d r |\psi^{2q}({ \bf r})|$ \cite{AMIT_Review}, where $\psi({ \bf r})$ denotes an eigenfunction for a given configuration of disorder. This is nothing but the $(q-1)^{th}$ moment of the probability density $|\psi^{2}({ \bf r})|$, described by $P_q=\int d^d r |\psi^{2(q-1)}({ \bf r})| |\psi^{2}({ \bf r})|$. A disorder average of the inverse participation ratio follows the scaling behavior of $\langle P_q\rangle \propto L^{-d(q-1)}$ in a metallic phase, where the eigenfunction at the Fermi energy is extended. Here, $L$ is the size of a system and $d$ is its dimension. On the other hand, the eigenfunction is localized in an Anderson insulating state, and the disorder average of the inverse participation ratio becomes independent of the system size, given by $\langle P_q\rangle \propto L^0$. In the vicinity of the Anderson metal-insulator transition, the inverse participation ratio shows an anomalous scaling behavior with respect to the system size $L$, given by $\langle P_q\rangle \propto L^{-\tau_q}$ with $\tau_q=D_q(q-1)$, where $D_q \not= d$ is the fractal dimension for each moment. If one replaces the probability density of an eigenfunction with an order parameter, he/she can calculate quantum mechanical averages for multiple moments of the order parameter. It turns out that such higher moments do not show critical scaling behaviors in conventional continuous quantum phase transitions. On the other hand, all the moments of eigenfunctions give rise to fractal behaviors in the vicinity of the Anderson metal-insulator transition, referred to as multifractality and regarded to be an essential feature of the Anderson metal-insulator transition \cite{Comments_Multifractality}.

Nature of the eigenfunction multifractality has been discussed both intensively and extensively in the vicinity of the Anderson metal-insulator transition. Analytical calculations based on the nonlinear $\sigma-$model field theory, which describes effective interactions between diffusions and Cooperons, turn out to be consistent with essentially exact numerical studies for fractal dimensions \cite{Multifractality_NLsM_vs_Numerics}. A natural question would be on the role of electron correlations in the multifractality of the Anderson metal-insulator transition. Recently, tunneling experiments on Ga$_{1-x}$Mn$_{x}$As have shown that the nature of eigenfunction multifractal correlations in the vicinity of the metal-insulator transition differs from that without electron correlations \cite{Multifractality_Interaction_Experiment}, suggesting that not only the eigenfunction multifractality survives electron interactions but also its nature gets modified. Motivated from such tunneling measurements, the fractal dimension of $D_{q}$ has been evaluated not only based on the nonlinear $\sigma-$model approach in the presence of Coulomb interaction \cite{Multifractality_Interaction_NLsM}, but also based on a numerical study, where the Coulomb interaction is treated within the Hartree-Fock approximation, but disorder effects are taken into account exactly \cite{Multifractality_Interaction_Numerics}. However, the nature of eigenfunction multifractality in the presence of Coulomb interactions is not still well understood, being under current debates.

We reexamine the effect of Coulomb interaction on the multifractality of Anderson metal-insulator transition, resorting to the Hartree-Fock approximation for the Coulomb interaction, where disorder effects are carried out exactly. Our major technical innovation is to utilize the Ewald-sum technique \cite{Ewald_Sum_Technique1,Ewald_Sum_Technique2,Ewald_Sum_Technique3}, which allows us to introduce the long-range nature of the Coulomb interaction into Hartree-Fock self-consistent equations of order parameters rather accurately. Based on this improved numerical technique, we evaluate the multifractal scaling exponent $\alpha_{q}$, given by the Legendre transformation of the fractal dimension $\tau_{q}$ discussed above, which characterizes the eigenfunction multifractal nature near the metal-insulator transition. Here, we focus on a characteristic disorder strength slightly below a critical value of the Anderson metal-insulator transition in three dimensions, above which all quantum states become localized. As a result, electrons at the Fermi energy remain delocalized to show diffusive dynamics in the absence of electron correlations. On the other hand, electrons at high energies become localized, where the density of states are much smaller than that at the Fermi energy and disorder potentials are not screened sufficiently due to the lack of the density of states. The characteristic energy for the Anderson localization is called the mobility edge \cite{Anderson_Localization_Review}. The multifractal nature of the mobility edge has been well understood in the absence of Coulomb interaction as discussed above.

Introducing the Coulomb interaction into the diffusive metallic phase at the characteristic disorder strength below the Anderson localization, the density of states at the Fermi energy evolves to be suppressed due to the Altshuler-Aronov correction \cite{Altshuer_Aronov_Correction} in the case of weak Coulomb interactions. Increasing the Coulomb interaction further, the density of states at the Fermi energy vanishes to show the Efros-Shklovskii pseudogap feature \cite{Coulomb_Gap_Review}. Our numerical analysis confirms the emergence of a mobility edge in the vicinity of the Coulomb-gap formation due to the suppression of the density of states in addition to the high-energy mobility edge involved with the Anderson localization without electron correlations. The emergence of the mobility edge near the Fermi energy seems to be consistent with the observation of the recent tunneling experiment \cite{Multifractality_Interaction_Experiment} although this measurement does not identify the mobility edge at a high energy. We find that the multifractal exponent at the high-energy mobility edge remains to be almost identical to that in the absence of Coulomb interactions. On the other hand, we reveal that the multifractal exponent near the Fermi energy is more enhanced than that at the high-energy mobility edge, suspected to result from interaction effects. However, both the multifractal exponents do not change even if the strength of the Coulomb interaction varies. We also show that the multifractality singular spectrum can be classified into two categories, confirming the appearance of two types of mobility edges.

Before going further, we would like to introduce two recent studies investigating the role of Coulomb interactions in the Anderson metal-insulator transition \cite{Coulomb_Disorder_DFT_I,Coulomb_Disorder_DFT_II}. Although these two studies are based on the density functional theory approximation, which differs from that of the present study, both papers pointed out that the Coulomb interaction changes the universality class of the Anderson metal-insulator transition, resorting to the multifractal analysis. In particular, Ref. \cite{Coulomb_Disorder_DFT_II} suggested a possible resolution of the long-standing ``exponent puzzle" due to the interplay between conduction and impurity states.

\section{Hartree-Fock approximation and Ewald summation technique}

We start from a disordered Hubbard Hamiltonian of spinless fermions on a three-dimensional cubic lattice of the size $L^{3}$, given by
\begin{equation}
  H=\sum_{\langle ij\rangle }(-t_{ij}+\varepsilon_i \delta_{ij})c_i^\dagger c_j+\frac{1}{2}\sum_{ij}U_{ij}\delta n_i\delta n_j .
  \label{modelHam}
\end{equation}
Here, the Coulomb interaction of $U_{ij}=\frac{e^2}{\kappa r_{ij}}$ is taken into account, where $\delta n_i=n_i-K$ is the fluctuation of the electron occupation $n_i$ around the mean value $K$. $e^{2} / \kappa$ with a dielectric constant $\kappa$ is referred to as the strength of the Coulomb interaction, denoted by $U$. $t_{ij}$ is a parameter for nearest neighbor hopping, set to be $t=1$ as the unit of energy. Onsite energies $\varepsilon_i$ are random, independently and uniformly distributed in $\varepsilon_i \in \left[ -W,W\right]$. The chemical potential $\mu$ can be renormalized by interactions, but chosen so as to keep the average density $K = 1/2$. For non-interacting particles at $U=0$, the Anderson metal-insulator transition occurs at a critical disorder strength $W_c=8.25$ \cite{AMIT_Critical_Disorder_Strength}, where the mobility edge comes from a high energy to the Fermi energy, localizing all quantum states of electrons.

We attack this problem numerically based on the Hartree-Fock approximation. Following Ref. \cite{Multifractality_Interaction_Numerics}, we write down an effective single-particle model with self-consistent onsite energies and hopping amplitudes
\begin{equation}
    H_{HF}=\sum_i \tilde{V}_i c_i^\dagger c_i -\sum_{ij} \tilde{t}_{ij} c_i^\dagger c_j +h.c.
    \label{H_nonint}
\end{equation}
Here, the self-consistent onsite potential energy and the self-consistent hopping kinetic energy are given by
\begin{eqnarray}
  \tilde{V}_i&=&\varepsilon_i+\sum_{j}\frac{U}{\bf |r_i-r_j |} \left[\langle c_j^\dagger c_j\rangle-K\right]-\mu , \label{selfconsol1} \\
  \tilde{t}_{ij}&=&t_{ij}+\frac{U}{\bf |{r_i-r_j}|}\langle c_j^\dagger c_i\rangle , \label{selfconsol2}
\end{eqnarray}
respectively, where $\langle ... \rangle$ denotes an ensemble average for a given disorder configuration. The effective onsite energy $\tilde{V}_j$ is renormalized by the interaction-induced Hartree term, which leads to ``correlated" on-site energies. The effective hopping parameter $\tilde{t}_{ij}$ is renormalized by the Fock term, where long-range hopping processes are generated by the Coulomb interaction. Here, we carry out exact diagonalization on a cubic three-dimensional lattice of the linear size $L=10$ $\sim$ $L = 24$.

A set of parameters to be determined self-consistently contains the ensemble average of all-range hopping $\langle c_i^\dagger c_j\rangle$ including the local density $\langle c_i^\dagger c_i\rangle$. In order to find a self-consistent solution, we begin with a random initial guess for all parameters, which should satisfy the condition
\begin{equation}
  \sum_i \langle n_i \rangle = \mathcal{N}_e,
\end{equation}
where the number of particles $\mathcal{N}_e=N/2$ is fixed at half filling in our simulation. Based on this initial condition, we diagonalize the effective Hamiltonian and find eigenfunctions $\psi_m({\bf r})$ and eigenvalues $\varepsilon_m$ for a given disorder configuration. Then, we obtain the ensemble average of all-range hopping $\langle c_i^\dagger c_j\rangle$ and the local density $\langle c_i^\dagger c_i\rangle$, resorting to
\begin{eqnarray}
  \langle n_i\rangle &=&\sum_m |\psi_m({\bf r}_i)|^2 f(\varepsilon_m), \\
  \langle c_i^\dagger c_j\rangle&=& \sum_m \psi_m^*({\bf r}_i) \psi_m({\bf r}_j)  f(\varepsilon_m),
\end{eqnarray}
regarded to be defining equations, where $f(\varepsilon_m)$ is the Fermi-Dirac distribution function with the chemical potential $\mu$. The chemical potential is adjusted to assure the system at half filling, determined by
\begin{equation}
  \mathcal{N}_e= \sum_m f(\varepsilon_m).
\end{equation}
Inserting these parameters into both Hartree-Fock self-consistent Eqs.~(\ref{selfconsol1}) and ~(\ref{selfconsol2}), we obtain an updated set of parameters, $\tilde{V}_{i}$ and $\tilde{t}_{ij}$, where $\tilde{t}_{ij}$ becomes long ranged. These renormalized parameters are introduced into the effective Hartree-Fock Hamiltonian Eq. (\ref{H_nonint}). We diagonalize the updated Hartree-Fock Hamiltonian and perform this iteration procedure until the output set of parameters converges within $10^{-4}$ of uncertainty. After convergence, the final set of eigenfunctions $\psi_m({\bf r})$ and eigenvalues $\varepsilon_m$ is used to compute physical quantities such as density of states and correlation functions. The result is then averaged over an ensemble of various disorder realizations, which are randomly selected by the rectangular distribution of a disordered potential $\varepsilon_i \in \left[ -W,W\right]$.

An essential point in solving these Hartree-Fock self-consistent equations is how to deal with the long-range nature of Coulomb interactions. In order to clarify the role of long-range interactions in the Hartree-Fock approximation, we implement the Ewald summation technique, where the Hartree term is split into two parts: a real-space portion based on a short-range interaction potential whose pairwise sum converges quickly and a long-range portion based on a slowly-varying interaction potential whose pairwise sum converges relatively quickly in a reciprocal space \cite{Ewald_Sum_Technique1,Ewald_Sum_Technique2,Ewald_Sum_Technique3}. The optimal implementation of the Ewald technique allows us to resolve the long-standing issue of the ill-convergence of the long-range potential and the multiplicity of Hartree-Fock solutions near the Anderson-Mott transition, which has been also reported in a recent Hartree-Fock numerical study \cite{Multifractality_Interaction_Numerics}. We refer all details, certainly important, to Appendix.

Based on the Ewald summation technique, the typical number of iterations required for a convergent solution for one realization of disorder is $\sim 30$. The bottleneck process in the overall computing steps is the $ \sim L^9$ operations to calculate the effective hopping matrix with the size $L^3 \times L^3$, which is performed in parallel using MPI. For a single disorder realization, the total time at $L=18 (L=24)$ is of the order of 50 hours (60 days) $/(\#$cores) for each parameter set of interaction and disorder strengths. For $50 (10)$ different disorder realizations, the total time at $L=18 (L=24)$ is of the order of 10 days (2 months) if $9(12)$ cores for parallel computing are used.
\vspace{-0.3cm}

\section{Result and Discussion}
\subsection{Density of states}
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.325]{dos_L18_log_linear.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Density of states for various interaction parameters of $0.3 \leq U \leq 0.8$ at $W = 7$ for a cubic lattice with the size $N=L^3=18^3$. Here, the interaction strength $U$ is given by the Coulomb interaction $V_{C}(|\bm{r}_{i} - \bm{r}_{j}|) = \frac{U}{|\bm{r}_{i} - \bm{r}_{j}|}$ and the critical disorder strength is $W_{c} \sim 8.25$ for the Anderson metal-insulator transition in three dimensions. The left panel displays a conventional plot for the density of states, which reproduces the Altshuler-Aronov correction in a metallic state and the Coulomb-gap feature in an insulating phase, clarified in the log-log plot of the right panel \cite{IRcutoff_mismatch}. An essential point is that the scaling behavior of $\rho(\omega) \sim |\omega|^{1/2}$, the Altshuler-Aronov correction in three dimensions, continues to dominate, approaching the metal-insulator transition from the metallic phase. This suggests that the critical interaction strength $U_{c}$ for the metal-insulator transition is estimated to be $0.3 < U_{c} < 0.5$ at $W = 7$, where the $\rho(\omega) \sim |\omega|^{1/2}$ scaling behavior turns into $\rho(\omega) \sim |\omega|^{2}$. This quantum critical regime shrinks in the insulating state and the Coulomb-gap scaling of $\rho(\omega) \sim |\omega|^{2}$ dominates in this region.}
\label{dos1}
\end{figure}

We show the density of states $\rho(\omega)$ in Fig. 1. Here, the interaction strength $U$ in the Coulomb potential $V_{C}(|\bm{r}_{i} - \bm{r}_{j}|) = \frac{U}{|\bm{r}_{i} - \bm{r}_{j}|}$ is varied from $U = 0.3$ to $U = 0.8$, and the disorder strength is fixed to be $W = 7$ below the critical disorder strength $W_{c} \sim 8.25$ \cite{AMIT_Critical_Disorder_Strength} for the Anderson metal-insulator transition in three dimensions. When the interaction strength is less than a critical value $U_{c}$, the density of states remains to be finite at the Fermi energy, but gets suppressed due to interaction corrections. This suppression is referred to as the Altshuler-Aronov correction, where diffusive electrons acquire strong renormalization effects even in the Hartree-Fock level \cite{Altshuer_Aronov_Correction}. The right panel confirms the typical suppression behavior of $\rho(\omega) - \rho_{0} \sim |\omega|^{1/2}$ in thee dimensions, where $\rho_{0}$ is the suppressed density of states at the Fermi energy \cite{IRcutoff_mismatch}. An interesting point is that the frequency scaling behavior of the Altshuler-Aronov type correction persists up around the critical point of a metal-insulator transition. This scaling behavior near the metal-insulator transition should be distinguished from the Altshuler-Aronov correction in the weak coupling approach, given by small corrections in the density of states. Here, the density of states changes more than two times, which seems to be beyond the weak coupling approach. This determines one critical exponent of the McMillan-Shklovskii scaling theory \cite{McMillan_Shklovskii_scaling_theory}. We point out that the critical exponent of $\rho(\omega) - \rho_{0} \sim |\omega|^{1/2}$ is consistent with that of the recent numerical study \cite{Multifractality_Interaction_Numerics}.
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.3]{alpha_q1q2_U0_W7.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Multifractal scaling exponents ($\alpha_1$ and $\alpha_2$) for various system sizes in the non-interacting case ($U=0$) with a disorder strength $W=7$ slightly below the critical disorder strength $W_{c} \sim 8.25$ of the Anderson metal-insulator transition. Here, $L$ denotes the size of a system and $b$ represents the size of a block, where the ratio of $(L/b)^{3} = 6^{3}$ is fixed. Enhancing the size of a system, the multifractal scaling exponent increases (decreases) in the metallic (insulating) region, where the energy is less (larger) than the mobility edge, i.e., $E < E_{m}$ ($E > E_{m}$). It does not depend on the system size at the mobility edge, identified with the critical energy for the Anderson-metal insulator transition. Based on this physics, we can determine not only the multifractal scaling exponents as $\alpha_{1} = 1.76$ and $\alpha_{2} = 1.13$ but also the position of the mobility edge as $E_{m} = 8.45$, all of which are consistent with previous studies \cite{AMIT_Review}.}
\vspace{-0.5cm}
\label{alpha_U0_W7}
\end{figure}

When the interaction parameter exceeds the critical value, the density of states at the Fermi energy vanishes, given by $\rho(\omega) \sim |\omega|^{2}$ and identified with the Efros-Shklovskii pseudogap \cite{Coulomb_Gap_Review}. This Coulomb gap feature starts to appear around $U = 0.5$ and becomes almost completed around $U = 0.8$, clarified by the right panel. Interestingly, the McMillan-Shklovskii scaling coexists with this Coulomb-gap scaling at $U = 0.5$. This evolution implies that the critical value of the interaction parameter is around $0.3 < U_{c} < 0.5$, which identifies the metal-insulator transition, where the density of states vanishes at the Fermi energy.
\vspace{-0.6cm}
\subsection{Multifractal analysis at the mobility edge of the Anderson model in the absence of electron correlations}
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{cc}
    \includegraphics[scale=0.3]{alpha_q2_U0_W7W8.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Multifractal scaling exponent $\alpha_2$ for two disorder strengths $W=7$ and $W=8$. Near the critical disorder strength ($W=8 \sim W_c = 8.25$), the  mobility edge, where the scale-invariance is realized, gets broader ($7.6 < E < 8.8$). The multifractal scaling exponent does not depend on the disorder strength $W$ at the mobility edge.}
\vspace{-0.3cm}
\label{alpha_U0_W7_W8}
\end{figure}
In order to find the fractal dimension $\tau_{q} = D_{q} (q-1)$ numerically, it is more convenient to introduce a coarse graining box with a volume of $b^{d}$, where the unit of an eigenfunction intensity is given by \cite{f_alpha}
\begin{equation}
  \mu_{k}(b) \equiv \sum_{j \in \mbox{box}_{k}} |\psi_{j}^{2}| .
\label{boxprob}
\end{equation}
Here, $j$ is a lattice site and box$_{k}$ is the coarse graining box with an effective index $k$. Then, moments of eigenfunction intensities are naturally introduced in the following way
\begin{equation}
  P_{q}(b) \equiv \sum_{k} [\mu_{k}(b)]^{q} .
\end{equation}
Accordingly, the fractal dimension can be defined as
\begin{equation}
  \tau_q = \lim_{\lambda\rightarrow 0}\frac{\ln \langle P_q \rangle}{\ln \lambda} ,
  \label{tauq}
\end{equation}
where $\lambda=b/L$.

The disorder average of the inverse participation ratio can be reformulated with the introduction of the distribution function $\mathcal{P}(|\psi^{2}|)$ for eigenfunction intensities, given by \cite{AMIT_Review}
\begin{equation}
  \mathcal{P}(|\psi^{2}|) \sim \frac{1}{|\psi^{2}|} L^{-d + f(- \ln |\psi^{2}| / \ln L)} ,
  \label{Distribution_Function_Alpha}
\end{equation}
where the function of $f(- \ln |\psi^{2}| / \ln L)$ defines the distribution function, referred to as the multifractal singularity spectrum. Actually, the disorder average of the moments $|\psi^{2q}|$ is expressed as
\begin{eqnarray}
  \langle P_q \rangle \equiv L^{d} \langle |\psi^{2q}| \rangle &=& L^{d} \int d |\psi^{2}| |\psi^{2 q}| \mathcal{P}(|\psi^{2}|) \nonumber \\ & \sim & \int d \alpha L^{- q \alpha + f(\alpha)} ,
  \label{Averaged_IPR}
\end{eqnarray}
where $\alpha = - \ln |\psi^{2}| / \ln L$ was introduced. Taking into account the limit of large $L$, this integral can be performed in the saddle-point approximation, resulting in
\begin{equation}
  \tau_{q} = q \alpha_{q} - f(\alpha_{q})
\label{tau_q2}
\end{equation}
with $q = \frac{d f(\alpha_{q})}{d \alpha_{q}}$ and $\alpha_{q} = \frac{d \tau_{q}}{d q}$. Here, we focus on the multifractal scaling exponent $\alpha_{q}$, which results from the Legendre transformation of the fractal dimension $\tau_{q}$ \cite{f_alpha}. Investigating the scaling behavior of such exponents with respect to the system size, we can determine the mobility edge and the multifractal exponent reliably.
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{cc}
    \includegraphics[scale=0.3]{alpha_q1_U0p8_W7_sgl.eps} &
    \includegraphics[scale=0.3]{alpha_q2_U0p8_W7_sgl.eps} \\
    \vspace{-0.3cm}\\
    \includegraphics[scale=0.3]{alpha_qm0p5_U0p8_W7_sgl.eps} &
    \includegraphics[scale=0.3]{alpha_qm1p5_U0p8_W7_sgl.eps}
    \vspace{-0.cm}
    \end{tabular}
\caption{Multifractal scaling exponents of $\alpha_1$, $\alpha_2$, $\alpha_{-0.5}$, and $\alpha_{-1.5}$ in the presence of the Coulomb interaction with $U=0.8$ at the disorder strength $W=7$. All these figures confirm the emergence of two types of mobility edges, one of which occurs near the Fermi energy and the other of which appears at a high energy, where the density of states at the Fermi energy shows the Coulomb-gap feature. It turns out that the multifractal scaling exponents at the high-energy mobility edge remain identical to those in the absence of Coulomb interactions. On the other hand, the low-energy mobility edge results from electron correlations, and the multifractal scaling exponents at the low-energy mobility edge differ from those at the high-energy mobility edge.}
\label{alpha_qvar}
\end{center}
\vspace{-0.5cm}
\end{figure}

\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{cc}
    \includegraphics[scale=0.3]{alpha_q2_U0p6_W7_sgl.eps} &
    \includegraphics[scale=0.3]{alpha_q2_U0p7_W7_sgl.eps} \nonumber \\
    \vspace{-0.3cm}\\
    \includegraphics[scale=0.3]{alpha_q2_U0p8_W7_sgl_wo24.eps} &
    \includegraphics[scale=0.3]{alpha_q2_U0p9_W7_sgl.eps} \nonumber \\
    \end{tabular}
    \vspace{-0.cm}
\caption{Multifractal scaling exponent $\alpha_2$ for various interaction parameters of $U=0.6$, $U = 0.7$, $U=0.8$, and $U = 0.9$ at the disorder strength $W=7$. The multifractal scaling exponent not only at the high-energy mobility edge but also at the low-energy mobility edge does not depend on the Coulomb interaction. This result seems to be consistent with a non-linear $\sigma-$model study although the value itself differs from our numerical value \cite{Multifractality_Interaction_NLsM}.}
\label{alpha_Uvar}
\end{center}
\vspace{-0.3cm}
\end{figure}

First, we show multifractal scaling exponents of $\alpha_1$ and $\alpha_2$ for various system sizes in the non-interacting case ($U=0$) with a disorder strength $W=7$ slightly below the critical disorder strength $W_{c} \sim 8.25$ of the Anderson metal-insulator transition, given by Fig.~\ref{alpha_U0_W7}. The horizontal axis is the energy scale, where the zero point is defined as the center of a band. The vertical axis corresponds to the multifractal scaling dimension $\alpha_q$. These critical exponents are evaluated for three different system sizes of $L = 12$, $L = 18$, and $L = 24$ with a fixed value of $\lambda=1/6$. It is clear that there exists a crossing point, denoted by $E_{m} = 8.45$, referred to as the mobility edge, where the multifractal scaling exponent $\alpha_q$ exhibits scale-invariance irrespective of the system size $L$ and the size of a coarse graining box. In the region of $E<E_m$, the multifractal scaling exponent $\alpha_q$ increases as the system size $L$ grows, regarded to be a characteristic feature of an electronic wave function extended over a space. In the clean limit, i.e., the absence of impurity scattering ($W=0$), $\alpha_q$ is proportional to the value of the spatial dimension of a system, given by $\alpha_q = d(q-1)$. See Fig.~\ref{alpha_U0_W7_W8}. In the presence of disorder scattering ($W \neq 0$), $\alpha_q$ shows a non-linear $q$ dependence, where $\alpha_q/(q-1)$ is always less than the spatial dimensionality $d$. In the region of $E>E_m$, the multifractal scaling exponent $\alpha_q$ decreases as $L$ increases, which indicates that the electronic wave function is confined within a finite volume.

Next, we discuss the evolution of the multifractal scaling exponent $\alpha_{2}$ with respect of the disorder strength $W$, shown in Fig.~\ref{alpha_U0_W7_W8}. The multifractal scaling exponent $\alpha_{2}$ decreases with increasing disorder strength $W$, which indicates the progress of Anderson localization near the transition point $W_c=8.25$. Near the critical disorder strength ($W=8\sim W_c=8.25$), the mobility edge is extended in a broad range of the energy scale ($7.6<E<8.8$) and the multifractal scaling exponent $\alpha_2$ of all electrons approaches $\alpha_2 \approx 1.13$. We also point out that the multifractal scaling exponent does not depend on the disorder strength $W$ at the mobility edge.
\vspace{-0.3cm}
\subsection{Emergence of two types of mobility edges and their multifractal scaling exponents in the presence of Coulomb interactions}
Now, we discuss various multifractal scaling exponents of $\alpha_1$, $\alpha_2$, $\alpha_{-0.5}$ and $\alpha_{-1.5}$ for the interacting case $U=0.8$ and the disorder strength $W=7$, shown in Fig.~\ref{alpha_qvar}. In addition to the mobility edge at $E_{m}^{UV}=8.91$ near the UV cutoff, an additional mobility edge appears near the band center at $E_{m}^{IR}=0.12$. The energy $E_{m}^{IR}=0.12$ corresponds to the crossover point above which the $\sim E^{2}$ behavior of the Coulomb gap in the density of states switches to the Altshuer-Aronov behavior $\sim \sqrt{E}$. See Fig.~\ref{dos1}. The multifractal scaling exponents at $E_{m}^{UV}=8.91$ near the UV cutoff turn out to be identical to the ones at the mobility edge at $E_m=8.45$ in the non-interacting case. See Fig.~\ref{alpha_U0_W7}. On the other hand, the multifractal scaling exponents at $E_{m}^{IR}=0.12$ near the Fermi energy become rather modified than the non-interacting ones at $E_{m}^{UV}=8.91$, enhanced by the factor of $1.24$ and $1.32$ for $\alpha_1$ and $\alpha_2$, respectively, and reduced by the factor of $1.24$ and $1.22$ for $\alpha_{-1/2}$ and $\alpha_{-3/2}$, respectively.

Fig.~\ref{alpha_Uvar} shows the multifractal scaling exponent $\alpha_2$ for four different interaction strengths $U$ and the fixed disorder strength $W=7$. The mobility edge close to the high energy cutoff remains the same as the non-interacting case. On the other hand, the mobility edge near the Fermi energy changes its position with increasing $U$ such as $E_{m}^{IR} = 0.08,~0.11,~0.12,~0.18$ for $U=0.6,~0.7,~0.8,~0.9$, respectively. The value of the critical exponent $\alpha_2$ at $E_{m}^{IR}$, however, does not depend on the strength of $U$ itself. See Fig. \ref{alpha2}.
\begin{figure}
\begin{center}
  \vspace{0.2cm}
  \begin{tabular}{cc}
    \includegraphics[scale=0.25]{alpha2.eps}
    \vspace{-0.3cm}
    \end{tabular}
\end{center}
\caption{Multifractal scaling exponent $\alpha_2$ at the high- and the low-energy mobility edge for various interaction strengths of $0.6<U<1.1$, denoted as $\alpha_{2,UV}$ and $\alpha_{2,IR}$, respectively. The multifractal scaling exponent not only at the high-energy mobility edge but also at the low-energy mobility edge does not depend on the Coulomb interaction.}
\vspace{-0.3cm}
\label{alpha2}
\end{figure}

We also calculate the multifractality singular spectrum for both noninteracting $U=0$ and interacting $U = 0.8$ cases at the disorder strength $W=7$, shown in Fig.~\ref{mult_spct}. This multifractality singular spectrum contains the information of the scale invariance, and thus it does not depend on the size of a system \cite{AMIT_Review}. The left panel displays that the multifractality singular spectrum collapses into a single curve, regardless of the system size, when electron correlations are turned off. The right panel shows that the multifractality singular spectrum can be classified into two categories, corresponding to the high-energy and low-energy mobility edges, respectively. The green single curve represents the multifractality singular spectrum at the high-energy mobility edge, essentially the same as that of the left panel, and the magenta single curve does it at the low-energy mobility edge, distinguished from the noninteracting multifractal spectrum.
\begin{figure}
\begin{center}
  \vspace{-0.1cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.3]{fspct_W7_U0_U0p8.eps}
    \vspace{-0.3cm}
    \end{tabular}
\end{center}
\caption{Multifractality singular spectrum for both noninteracting $U=0$ and interacting $U = 0.8$ cases at the disorder strength $W=7$. Regardless of the system size, the multifractality singular spectrum collapses into a single curve in the noninteracting case. On the other hand, the singularity spectrum becomes classified into two categories, which suggests the existence of two types of mobility edges.}
\vspace{-0.3cm}
\label{mult_spct}
\end{figure}
\vspace{-0.3cm}
\subsection{Comparison with recent analytical and numerical studies}
It is necessary to compare our numerical results with recent analytical and numerical studies. Although we focus on $\alpha_{2}$ in the present study, we also find $\tau_{2} \approx 1.7$, resorting to Eq.~(\ref{tau_q2}), where a typical value has been considered. A recent nonlinear $\sigma-$model study \cite{Multifractality_Interaction_NLsM} investigated the scaling behavior of moments of the local density of states for the unitary ensemble in the presence of Coulomb interactions, given by $[\rho(\omega,\bm{r}) / \langle \rho(\omega) \rangle]^{q}$. Based on the $\epsilon-$expansion near the lower critical dimension with $d = 2 + \epsilon$, this study found an anomalous fractal exponent $\Delta_{q} = - \frac{q(q-1) \epsilon}{4} \Big\{ 1 + \Big( 1 - A - \frac{\pi^{2}}{12} \Big) \epsilon \Big\} + \mathcal{O}(\epsilon^{3})$ up to the two-loop order, where the anomalous fractal exponent is given by $\Delta_{q} = \tau_{q} - d (q-1)$. Here, $A \approx 1.64$ is a positive numerical constant, which appears in the renormalization group equation for the inverse of the dimensionless conductance. Actually, this analytic study reported $\tau_{2} \approx 2.5$, which deviates from a recent numerical study \cite{Multifractality_Interaction_Numerics}. On the other hand, the nonlinear $\sigma-$model field theory gives rise to $\tau_{2} \approx 1.6$ up to the four-loop level in the absence of electron correlations, consistent with numerical results \cite{Multifractality_Interaction_NLsM}.

A recent Hartree-Fock numerical study reported $\tau_{2} = 1.57 \pm 0.05$ for the system size of $L = 10$ while $\tau_{2} = 1.34 \pm 0.05$ in the absence of electron interactions \cite{Multifractality_Interaction_Numerics}. This value is slightly smaller than the present typical value $\tau_{2} \approx 1.7$. Generally speaking, the fractal dimension increases in the presence of electron correlations, implying more sparse distributions of moments of eigenfunctions. This enhancement results from the fact that electron interactions give rise to linearly superposed states of fractal eigenfunctions in the absence of interactions, which weakens the multifractal nature of the Anderson metal-insulator transition. The Ewald summation technique seems to take into account long-ranged Coulomb interactions more strongly.
\section{Summary and Discussion}
\vspace{-0.3cm}

In summary, we investigated the role of Coulomb interactions in the nature of eigenfunction multifractality of an Anderson metal-insulator transition, based on the Hartree-Fock approximation and the Ewald summation technique. As a result, we showed that two types of mobility edges appear near the Fermi energy and at a high energy, respectively, where the low-energy mobility edge results from Coulomb interactions while the high-energy one is nothing but the mobility edge of the Anderson localization transition without electron correlations. Indeed, not only multifractal scaling exponents but also the multifractal singularity spectrum confirms the existence of two kinds of mobility edges: Their values differ from those of the Anderson metal-insulator transition and the singularity spectrum collapses into two types of curves, implying two kinds of scale-invariance, which depends on the energy scale. We speculate that this novel nature of the eigenfunction multifractality would serve as valuable information for possible instabilities near a metal-insulator transition in the presence of Coulomb interactions \cite{Kondo_Anderson_Transition,BCS_Anderson_Transition,Stoner_Anderson_Transition}.

Before closing, we would like to point out that our Hartree-Fock self-consistent equations with the Ewald summation technique do not take into account screening of the Coulomb interaction. In particular, the Coulomb interaction should be screened by particle-hole excitations near a Fermi surface in a metallic phase, described by the random phase approximation (RPA). Here, the RPA correction can be taken into account in a fashion of real space, given by matrix products to describe a convolution integral. Even if such corrections are not introduced into the self-consistent equations, order parameters protect the correct physics of the Altshuler-Aronov correction in a metallic state. In other words, the exchange hopping order parameter, which becomes long ranged potentially by Coulomb interactions, remains short ranged in a metallic phase, keeping the Altshuler-Aronov correction described by the Hartree-Fock approximation in the presence of disorder scattering. In an insulating phase, the Coulomb interaction itself persists, well described by the Ewald summation technique. However, the absence of the RPA correction may be dangerous in the vicinity of the metal-insulator transition because the screening effect can cause anomalous scaling behavior for the Coulomb interaction instead of the $\sim 1/ r$ potential. If this is the case, the present calculations would have uncertainties for multifractal scaling exponents. However, we emphasize that the existence of two kinds of mobility edges will not be affected by this approximation scheme, where the low energy mobility edge occurs after the formation of the Coulomb gap, i.e., in the insulating state.
\section{Acknowledgement}
This study was supported by the Ministry of Education, Science, and Technology (No. NRF-2015R1C1A1A01051629 and No. 2011-0030046) of the National Research Foundation of Korea (NRF). Computing resources were provided by the NSF via grant MRI: Acquisition of Conflux, A Novel Platform for Data-Driven Computational Physics (Tech. Monitor: Ed Walker). We appreciate helpful discussions with S. Kettemann, X. Wan, R. Narayanan, and V. Dobrosavljevic.
\section{Appendix}
\subsection{Ewald summation} \label{Ewald}
Consider $N$ charged particles subjected to the periodic boundary condition,
\begin{equation}
\rho{({\bf r})}=\rho({\bf r}+{\bf n} L),
\label{pbc}
\end{equation}
where ${\bf n}=n_1\hat{x}+n_2\hat{y}+n_3\hat{z}$ with arbitrary integers $n_1$, $n_2$ and $n_3$.
The total Coulomb interaction energy includes interactions between real and image charges in periodic supercells, given by
\begin{equation}
  E=\frac{1}{4\pi\varepsilon_0}\frac{1}{2}\sum_{\bf n}\sum_{i=1}^N{\sum_{j=1}^N}^\prime\frac{q_i q_j}{|{\bf r}_{ij}+{\bf n} L|},
\label{LR_int}
  \end{equation}
where ${\bf {r}}_{ij}+{\bf n}L$  is the distance between the two point charges $q_i$ and $q_j$ located in two separate supercells. The $\prime$ symbol means that the term $j=i$ is excluded, if and only if ${\bf n=0}$.

  In the Ewald technique, the long-range interaction in Eq.~(\ref{LR_int}) is split into two parts; a short-range interaction potential whose pairwise sum readily converges in real space and a long-range portion based on a slowly-varying interaction potential whose pairwise sum converges relatively quickly in reciprocal space \cite{Ewald_Sum_Technique1,Ewald_Sum_Technique2,Ewald_Sum_Technique3}.

The original charge distribution $\rho({\bf r})$ can be split into two terms,
\begin{eqnarray}
  \rho_i({\bf r})&=&\rho^S_i({\bf r})+\rho^L_i({\bf r}),\\
  \rho^S_i({\bf r})&=&q_i \delta({\bf r-r_i})-q_i\mathcal{G}({\bf r-r_i}), \\
  \rho^L_i({\bf r})&=&q_i\mathcal{G}({\bf r-r_i}).
  \end{eqnarray}
where $\mathcal{G}({\bf r})$ is a Gaussian distribution,
\begin{equation}
  \mathcal{G}({\bf r})=\frac{1}{(2\pi\sigma^2)^{3/2}}{\rm exp}\left[ -\frac{|{\bf r}|^2}{2\sigma^2}\right].
\end{equation}
The potential field generated by a charge distribution of the Gaussian form is obtained as
\begin{eqnarray}
  \phi(r) &=&\frac{1}{4\pi\varepsilon_0 r} {\rm erf}(\frac{r}{\sqrt{2}\sigma}) ,
\end{eqnarray}
where ${\rm erf}(z)\equiv \frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2}dt$. Accordingly, the Coulomb potential is written as
\begin{eqnarray}
  \phi_i({\bf r})&=&\phi^S_i({\bf r})+\phi^L_i({\bf r}) , \\
  \phi^S_i({\bf r})&=&\frac{q_i}{4\pi\varepsilon_0}\frac{1}{|\bf r-r_i|}\left[{\rm erfc}\left(\frac{|{\bf r-r_i}|}{\sqrt{2}\sigma}\right)\right] ,
\label{phi_s}
\\
  \phi^L_i({\bf r})&=&\frac{q_i}{4\pi\varepsilon_0}\frac{1}{|\bf r-r_i|}\left[{\rm erf}\left(\frac{|{\bf r-r_i}|}{\sqrt{2}\sigma}\right)\right] .
  \label{phi_r}
  \end{eqnarray}
  Note that ${\rm erfc } (x)=1-{\rm erf}(x)$. Here, the Ewald parameter $\sigma$ is the cutoff length scale on which the short-range function $\frac{1}{r}{\rm erfc}(\frac{r}{\sqrt{2}\sigma})$ decays. As $\sigma$ decreases, more of the summation is performed in reciprocal space, whereas, setting $\sigma \sim L$, the Coulomb interaction is taken into account entirely in real space.

Using Eq.~(\ref{phi_s}) and Eq.~(\ref{phi_r}), the total Coulomb interaction energy in Eq.~(\ref{LR_int}) can be written as
\begin{eqnarray}
  E   &=&\frac{1}{4\pi\varepsilon_0}\frac{1}{2}\sum_{\bf n}\sum_{i=1}^N{\sum_{j=1}^N}^\prime\frac{q_i q_j}{{|\bf r_i-r_j}+{\bf n}L|}\times  \nonumber\\
 && \left[   {\rm erfc}     \left(   \frac{|{\bf r_i-r_j}+{\bf n}L|}{\sqrt{2}\sigma}   \right) +{\rm erf}\left(\frac{|{\bf r_i-r_j}+{\bf n}L|}{\sqrt{2}\sigma}\right)\right].\nonumber\\
\label{LR_int1}
\end{eqnarray}

  Now we define a cavity field $\phi_{\left[i \right]}({\bf r})$ as the potential field generated by all the ions plus their images, excluding the ion at $i$,
  \begin{equation}
 \phi_{\left[i \right]}({\bf r}) \equiv \phi({\bf r}) - \phi_{i }({\bf r}) =\frac{1}{4\pi\varepsilon_0}\sum_{\bf n}{\sum_{j=1}^{N}}^\prime \frac{q_j}{|{\bf r-r_j+n}{ L}|} .
\label{cavity}
 \end{equation}
  The $\prime$ symbol means that the term $j=i$ is excluded, if and only if ${\bf n=0}$. Using Eq.~(\ref{cavity}), the total Coulomb interaction energy in Eq.~(\ref{LR_int1}) can be written as
  \begin{eqnarray}
    E&=&\frac{1}{2} \sum_{i=1}^N q_i \phi^S_{\left[ i \right]}({\bf r}_i) +\frac{1}{2} \sum_{i=1}^N q_i \phi^L({\bf r}_i)-\frac{1}{2} \sum_{i=1}^N q_i \phi_i^L({\bf r}_i)\nonumber \\
    &=&E^S+E^L+E^{self} .
  \end{eqnarray}

With taking the $r \rightarrow 0$ limit,
\begin{eqnarray}
  \lim_{\bf r\rightarrow r_i}\phi^L_i({\bf r})&=&\frac{q_i}{4\pi\varepsilon_0}\lim_{r\rightarrow 0}\frac{1}{{r}}{\rm erf}\left(\frac{r}{\sqrt{2}\sigma}\right) ,
\label{phi_self}
\end{eqnarray}
we can easily obtain the self-energy term,
\begin{equation}
  E^{self}=\frac{1}{4\pi\sigma \varepsilon_0}\sqrt{\frac{1}{2\pi}}\frac{1}{\sigma}\sum_{i=1}^{N}q_i^2 .
\end{equation}

In order to handle the long-range portion $E^L$ in the reciprocal space, we make the Fourier transform of the total charge density
\begin{eqnarray}
  \rho^L({\bf r})&=&\sum_{\bf n}\sum_{j=1}^{N}q_j \mathcal{G}({\bf r-r_j+n}L) ,
  \end{eqnarray}
and obtain
\begin{eqnarray}
  \rho^{L}({\bf k})  &=&\mathcal{N}_{cell}\sum_{j=1}^{N} q_j e^{-i{\bf k}\cdot {\bf r_j}}e^{-\sigma^2k^2/2} ,
  \label{rho_k}
  \end{eqnarray}
where $\mathcal{N}_{cell}$ is the number of supercells. The Poisson's equation
\begin{equation}
  \nabla^2 \phi^L({\bf r})=-\frac{1}{\varepsilon_0}\rho^L({\bf r})
\end{equation}
can be Fourier-transformed into the reciprocal space, given by
\begin{equation}
  \phi^L({\bf k})=\frac{1}{\varepsilon_0}\frac{\rho^L({\bf k})}{k^2} .
  \label{poisson_k}
\end{equation}
As a result, Eq.~(\ref{rho_k}) and Eq.~(\ref{poisson_k}) give the potential field in the reciprocal space as follows
\begin{eqnarray}
  \phi^{L}({\bf k})&=&\mathcal{N}_{cell}\sum_{j=1}^{N}q_j  e^{-i{\bf k}\cdot {\bf r_j}}\frac{e^{-\sigma^2k^2/2}}{k^2}.
  \label{phi_k}
\end{eqnarray}
Applying the inverse Fourier transform, we get
\begin{eqnarray}
  \phi^{L}{(\bf r})&=&\frac{1}{V}\sum_{\bf k\neq 0} \phi^L({\bf k}) e^{i{\bf k}\cdot {\bf r}}  \nonumber\\
  &=&\frac{1}{v\varepsilon_0}\sum_{\bf k\neq 0} S({\bf k}) e^{i{\bf k}\cdot {\bf r}}\frac{e^{-\sigma^2k^2/2}}{k^2} ,
  \label{phi_L}
\end{eqnarray}
where $S({\bf k})=\sum_{j=1}^{N} q_j  e^{-i{\bf k}\cdot {\bf r_j}}$. Here, $v=\frac{V}{\mathcal{N}_{cell}}$ is the volume of a single supercell. The contribution to the ${\bf k=0}$ term is zero if the supercell is charge neutral, i.e. $\sum_i^N q_i=0.$

 In practice, we introduce an IR momentum cutoff $\varepsilon$ to neglect the small momentum contribution $|{\bf k}|<\varepsilon$ so that we can bypass the poor resolution at $|{\bf k}|<\varepsilon$ associated to the finite size of system $N=L^3$, i.e.
\begin{eqnarray}
  \phi^{L}({\bf r})  &=&\frac{1}{v\varepsilon_0}\sum_{|{\bf k}|>\varepsilon} S({\bf k}) e^{i{\bf k}\cdot {\bf r}}\frac{e^{-\sigma^2k^2/2}}{k^2}.
  \label{phi_L2}
\end{eqnarray}
Technically, the IR cutoff $\varepsilon>0$ can be regarded as an effective convergence factor which helps the summation in Eq.~(\ref{phi_L2}) absolutely convergent. Otherwise the pairwise sum of the long-range potential in Eq.~(\ref{phi_L}), which is conditionally convergent but not absolutely convergent, yields discrepant results depending on the sequence of the summation \cite{Ewald_Sum_Technique1,Ewald_Sum_Technique2}. We find that the ill-convergence of the long-range potential $\phi^L{(\bf r})$ can lead to multiplicity of Hartree-Fock solutions.

Using the results in Eq.~(\ref{phi_self}) and Eq.~(\ref{phi_L2}), the cavity potential field generated by the surrounding electrical charges can be written as
  \begin{eqnarray}
    \phi_{\left[i \right]}({\bf r_i})&=& \phi^S_{\left[i \right]}({\bf r_i})+\phi^L({\bf r_i})-\phi^{self}({\bf r_i})\nonumber \\
    &=&\frac{1}{4\pi\varepsilon_0}\sum_{\bf n}{\sum_{j=1}^{N}}^\prime q_j\frac{ {\rm erfc}\left(\frac{|{\bf r_i-r_j+n} L|}{\sqrt{2}\sigma} \right)}{|{\bf r_i-r_j+n} L|}\nonumber \\
    &+&\frac{1}{v\varepsilon_0}\sum_{|{\bf k}|>\varepsilon }\sum_{j=1}^{N} q_j  e^{i{\bf k}\cdot {\bf (r_i-r_j)}}\frac{e^{-\sigma^2k^2/2}}{k^2}\nonumber\\
    &-& \frac{q_i}{4\pi\varepsilon_0} \sqrt{\frac{2}{\pi}}\frac{1}{\sigma} .
    \label{tot_pot}
  \end{eqnarray}
Accordingly, the Hartree potential in Eq.~(\ref{selfconsol1}) is written as
\begin{eqnarray}
  \tilde{V}_i&=&\varepsilon_i-\mu +U\sum_{\bf n}{\sum_{j=1}^{N}}^\prime \delta n_j\frac{ {\rm erfc}\left(\frac{|{\bf r_i-r_j+n} L|}{\sqrt{2}\sigma} \right)}{|{\bf r_i-r_j+n} L|}\nonumber\\
  &+&\frac{4\pi U}{v}\sum_{|{\bf k}|>\varepsilon}\sum_{j=1}^{N} \delta n_j  e^{-i{\bf k}\cdot {\bf (r_i-r_j)}}\frac{e^{-\sigma^2k^2/2}}{k^2}-U\delta n_i \sqrt{\frac{2}{\pi}}\frac{1}{\sigma},\nonumber\\
  \label{tot_hartree_pot}
\end{eqnarray}
where $\delta n_i=n_i-K$ is the fluctuation of the electron occupation $n_i$ around the mean value $K$.
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.31]{dos_sizedep1.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Density of states at the interaction strength $U=0.5$ and the disorder strength $W=7$ for three different sizes of systems $N=L^3=16^3,~18^3,~24^3$ with the control parameters of $\sigma=2$ and $\varepsilon=0.8$.}
\label{dos5}
\vspace{-0.4cm}
\end{figure}

In thermodynamic limit, $N\rightarrow \infty$ and $\varepsilon \rightarrow 0$, the last two terms in Eq.~(\ref{tot_hartree_pot}) perfectly compensate each other to yield the correct power-law feature of the density of states near the Fermi level. In the presence of a finite size effect, however, the contribution of self-energy $\phi^{self}({\bf r_i})$ dominates the long-range term $\phi^L({\bf r_i})$ to open a hard gap in the density of states near zero frequency.

The parameters $\sigma$ and $\varepsilon$ in Eq.~(\ref{tot_hartree_pot}), therefore, are optimized to fulfill the two requirements. First, it should give a unique solution which is absolutely convergent, i.e., independent of the size of a system. Second, the solution should exhibit the correct power-law behavior of the density of states.

Fig.~\ref{dos5} shows the density of states evaluated at the interaction strength $U=0.5$ and the disorder strength $W=7$ for three different sizes of systems $L=16,~18$, and $24$. With $\sigma=2$ and $\varepsilon=0.8$, the density of states of these three different sizes of systems collapse into a single curve showing an insulating behavior ($\sim E^2$). The small energy region ($E<0.03$) subjected to an exponential decay is due to the mismatch between the self-energy and the long-range potential as mentioned before.
\begin{figure}
\begin{center}
  \vspace{0.cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.31]{dos_sizedep2.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Density of states at the interaction strength $U=0.3$ and the disorder strength $W=7$ for three different sizes of systems $N=L^3=14^3,~18^3,~24^3$  with the control parameters of $\sigma=3,~\varepsilon=0.4$ and $\sigma=4,~\varepsilon=1$.}
\label{dos6}
\vspace{-0.4cm}
\end{figure}

Fig.~\ref{dos6} shows the density of states at $U=0.3$ and $W=7$ for the size of systems $L=14,~18,~24$, where two sets of the control parameters $\sigma=3,~\varepsilon=0.4$ and $\sigma=4,~\varepsilon=1$ are considered. At present, it is not conclusive that $U=0.3$ is metallic. These three curves show the same Altshuler-Aronov behavior at the energy range $0.01<E<0.1$ but deviate from each other at $E<0.01$. If these three lines exhibit the same $\sim E^2$ below $E<0.01$ with an alternative set of parameters $\sigma$ and $\varepsilon$, the $U=0.3$ case can also correspond to an insulating phase. We find that, within the current Ewald scheme, it is more difficult to get convergence among systems with different sizes in a metallic phase.

Fig.~\ref{IRedge} shows the low-energy mobility edge $E_{m}^{IR}$ for various interaction strengths. For $U < 0.6$, the mobility edge has an order of magnitude $10^{-2}$ and is susceptible to numerical uncertainty attributed to finite system size and the number of disorder realizations. The linear extrapolation estimates that $E_m^{IR}$ goes to zero around $U \sim 0.32$, indicating the possibility of the metal-insulator transition at $U\sim 0.3$. It is desirable to perform multifractal finite-size scaling analysis \cite{AMIT_Critical_Disorder_Strength,MFM_Numerics}, which permits the systematic analysis of the interacting mobility edge near the Fermi level.
\subsection{Long-range hopping matrix element}
  In the Hartree-Fock approximation, the hopping matrix element is self-consistently determined such as
  \begin{equation}
    \tilde{t}_{ij}=t_{ij}+\frac{U}{\bf |{r_i-r_j}|}\langle c_j^\dagger c_i\rangle
  \end{equation}
  and the effective hopping Hamiltonian is
  \begin{equation}
    \hat{H}_{hop}=\sum_{i}^{N}\sum_{j}^{N}\tilde{t}_{ij}  ({c_i}^\dagger c_j+{c_j}^\dagger c_i).
  \end{equation}
\begin{figure}
\begin{center}
  \vspace{-0.3cm}
  \begin{tabular}{c}
    \includegraphics[scale=0.25]{IR_edge.eps}
    \end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Low-energy mobility edge $E_{m}^{IR}$ for various interaction strengths $0.5< U < 1.2$ obtained from the multifractal scaling analysis. The dashed line is linear extrapolation.}
\label{IRedge}
\vspace{-0.3cm}
\end{figure}
    Now consider $N$ charged particles subjected to the periodic boundary condition in Eq.~(\ref{pbc}). Including the effect of hopping between the image charges in the periodic supercells, the hopping Hamiltonian can be written as
  \begin{eqnarray}
    \hat{H}_{hop}&=&\sum^{\mathcal{N}_{cell}}_{\bf n}\sum^{\mathcal{N}_{cell}}_{\bf m} \sum_{i}^{N}\sum_{j}^{N}\tilde{t}^{\bf (nm)}_{ij}  ({c^{\bf n}_i}^\dagger c^{\bf m}_j+{c^{\bf m}_j}^\dagger c^{\bf n}_i) .
    \end{eqnarray}
Here $\mathcal{N}_{cell}$ is the total number of cells in the supercell structure. In this work, we keep the hopping matrix elements within each supercell neglecting the intercell matrix elements,
   \begin{eqnarray}
    \hat{H}_{hop}    &\approx&\sum_{\bf n}\sum_{ij}\tilde{t}^{(\bf nn)}_{ij}({c^{\bf n}_i}^\dagger c^{\bf n}_j+{c^{\bf n}_j}^\dagger c^{\bf n}_i).
   \end{eqnarray}
   This approximation is valid if the range of electron hopping is shorter than the linear size of a cell $L$, which is easily fulfilled in the metallic or the insulating phase where the electron hopping remains short ranged. In the vicinity of the critical region, however, the hopping can also have a long-range nature and the approximation is valid only when the cell-size is large enough to cover the range of hopping.

redpajama_set_name: RedPajamaArXiv


The moment has come when the great deceivers since time immemorial are now using our language against us. Such would not be possible but for our own laxness in demanding of ourselves and others precision and lucidity with this evolutionary gift for the advancement of civilization over primality. As rust never sleeps, neither do those who would set us against one another for their purposes of vindicating their claims of the foregone conclusion that we are failures to the designs of the divine. It matters little why they set themselves up as champions for the demise of the architect of existence. It may be they themselves are deceived. It may be they garner perverse pleasure in causing suffering and confusion. It may be they are possessed of the same vision held by demons of wanton destruction, who in their own realization that they cannot create, must then destroy to fabricate an imagined dominance over an existence wherein their part is to serve.
What matters to us is, our existence is in peril because we allowed complacency, comfort and convenience to distract us from the substantial and edifying fruits of true education. We allowed our schools and universities to become mere employment training centers to serve the economic appetite. In so doing we failed ourselves as a species and opened ourselves up to relentless erosion by the willfully ignorant who have always been among us. We allowed personal preferences for carnal gratification override our duty to vigilance. Now, we must extricate ourselves or pay the ultimate price just as the mountains succumb to become sands - just as the mighty tree is laid low by the constant intrusions of time.
The hour is dire. The years of gentle remonstrance and kindly correction were replaced with wise warnings and foretellings of misfortune. Yet, the race toward opulence and self-gratification surged, and the enemy is at last upon us. Armed with intentional equivocation, they aim to rewrite knowledge itself. They operate under the guise of making truth at last free from subjective distortions driven by conspirators. Yet, they claim the advancement of evolution is an assault upon the tossed aside, time-honored ways of old. They claim to advance ourselves is to become the ones who are wantonly destroying the past.
You will not defeat them meeting force with force. The best to be hoped for there is a tragically diminishing stalemate. They can only be defeated with establishing unerring and precise truth. The language can no longer be used as a flexible source of permissiveness, for it is our only weapon. It is the only tool which can decisively vindicate or invalidate any point of view, or perspective. In this regard, responsibility to our historical account as an unimpeachable narrative, despite how this may reflect upon one group, or another, must be achieved as this is the ground upon which the conflict will be fought.
All sides wishing victory by defeating an imagined opponent have thus far, and for centuries, advocated manipulation of these two things to gain decisive advantage - the language and the narrative. To be a culture of integrity these two elements of said culture must be impervious to manipulation by anyone, agenda or vested interests will out. The only way this can be achieved is for that society to agree it must be done, and then take the steps to ensure what was agreed to is animated. This is the will of the people, not the will of some people over that of others.
Though all involved may believe this issue is between themselves, the major concern should be: How does the divine see this? I assure all, the divine does not see the inhabitants of earth as many different people with honest issues of conflict with one another. The divine sees all the people of the world as children of the divine. What we see as insurmountable, the divine does not. The conflict we bring upon ourselves in the end damages and destroys our children; their lives and the conditions of their futures. Of all things we do to displease the divine, this numbers as the worst. Forbearance may be expected with regard to this, but I assure all, the divine will not be interested in our rationales and excuses. The divine will, however, have it well in mind how many children our self-inflicted confusion has cost.


text:
Named the first headliner of Leopolis Jazz Fest 2021
Rhythmic beating of the clock in the town hall, melodic calls of trams, voices and music coming from street cafes, echoes of footsteps in cozy courtyards ... Lviv always sounds like jazz, Lviv is jazz!
So it is not surprising that the city of Leva annually hosts the legendary jazz festival Leopolis Jazz Fest (until 2017 - Alfa Jazz Fest), recognized by the famous British publication The Guardian as one of the best jazz festivals in Europe. And every time it's not just an event, it's a celebration of music, a special atmosphere and incredible meetings with the taste of coffee in the middle of summer.
The singer also recorded a welcome address to Ukrainians, in which he wished everyone to stay healthy and meet him at the festival in Lviv.
One of the headliners of Leopolis Jazz Fest 2021 will be the winner of four Grammy Awards, British singer Seal.
According to the organizers of the festival on the Facebook page, Seal will perform on June 25, 2021 on the Stage. E. Rosner.
"During his outstanding career, which spans more than two decades, London multiplatinum singer and songwriter Seal has focused on a creative aspiration: to express the full feeling of love through song," said the organizers of Leopolis Jazz Fest.
Known for his unique hoarse baritone and classic songwriting, Seal has been successful in many music genres. His soft timbre, which sounds in the hits "Crazy" and "Kiss from a Rose", quickly secured him the status of a respected singer.
Over the years, Seal has expanded its influence with a wealth of original music and emotional covers, as well as winning numerous Grammy Awards and selling more than 30 million albums worldwide.
Seal also recorded a welcome address to Ukrainians, in which he wished everyone to stay healthy and meet him at the anniversary festival in Lviv.
The singer is also well known as the ex-husband of German supermodel Heidi Klum thanks to their tumultuous romance and loud breakup (2004-2012). Long after the divorce, the singer wore an engagement ring, arguing that he still loves Heidi Klum.
Swiss Hotel Lviv will continue the jazz pleasure. Here you can relax after all the impressions and events, taste the dishes prepared by the chef and admire the incredible view of Lviv from the rooftop hotel Restaurant "Valentino Roof terrace".
Come to Lviv summer - where there is a holiday of life, where hospitality and joy, where coffee, love and jazz!

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to fix that OpenCL freezes? I'm trying to detect blinking pixels. I've written the code in C++ first but I realized, that a CPU is not suitable for it. So I found the OpenCL library. I've never used it before.
Besides, I haven't found a good documentation for OpenCL.
OpenCLHelper.cpp
#include <CL/cl.hpp>
#include <fstream>
#include <iostream>
#include <stdlib.h>

cl::Program CreateProgram(const std::string& fileName) {

    std::vector<cl::Platform> platforms;
    cl::Platform::get(&platforms);

    auto platform = platforms.front();
    std::vector<cl::Device> devices;
    platform.getDevices(CL_DEVICE_TYPE_GPU, &devices);

    auto device = devices.front();

    std::ifstream file(fileName);
    std::string src(std::istreambuf_iterator<char>(file), (std::istreambuf_iterator<char>()));

    cl::Program::Sources sources(1, std::make_pair(src.c_str(), src.length()+1));

    cl::Context context(device);

    cl::Program program(context, sources);

    std::cout << program.build("-cl-std=CL1.2") << std::endl;
    return program;
}

main.cpp (not the whole code of the file)
cl::Context context = program.getInfo<CL_PROGRAM_CONTEXT>();
vector<cl::Device> devices = context.getInfo<CL_CONTEXT_DEVICES>();
cl::Device device = devices.front();

unsigned char* shootFrame(unsigned char *data) {

    unsigned char* frequencyImage = new unsigned char[pixelsPerFrame];
    strcopy(data, frequencyImage);

        cl_int err = 0;
        cl::Buffer inBuf(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(unsigned char) * pixelsPerFrame * equalxFramesAtTheSameTime, lastFrames, &err);
        cout << err << endl;

        cl::Buffer outBuf(context, CL_MEM_WRITE_ONLY | CL_MEM_HOST_READ_ONLY, sizeof(unsigned char) * pixelsPerFrame, nullptr, &err);

        cl::Buffer var1(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(int), &isLightOn, &err);cout << err << endl;

//I creates variables, because I can't use #define in there
        int equalxFramesAtTheSameTime2 = equalxFramesAtTheSameTime;
        cl::Buffer var2(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(int), &equalxFramesAtTheSameTime2, &err);cout << err << endl;
        int thresholdPixel2 = thresholdPixel;
        cl::Buffer var3(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(int), &thresholdPixel2, &err);cout << err << endl;
        int ppf = pixelsPerFrame;
        cl::Buffer var4(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(int), &ppf, &err);cout << err << endl;

        cl::Buffer var5(context, CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR, sizeof(int), &currentFrameID, &err);cout << err << endl;

        cl::Kernel kernel(program, "ProcessImage"); cout << err << endl;

        err = kernel.setArg(0, var1); cout << err << endl;
        err = kernel.setArg(1, var2); cout << err << endl;
        err = kernel.setArg(2, var3); cout << err << endl;
        err = kernel.setArg(3, var4); cout << err << endl;
        err = kernel.setArg(4, var5); cout << err << endl;
        err = kernel.setArg(5, inBuf); cout << err << endl;
        err = kernel.setArg(6, outBuf); cout << err << endl;

        cl::CommandQueue queue(context, device);

        err = queue.enqueueNDRangeKernel(kernel, cl::NullRange, cl::NDRange(pixelsPerFrame)); cout << err << endl;

        err = queue.enqueueReadBuffer(outBuf, CL_FALSE, 0, sizeof(unsigned char) * pixelsPerFrame, done); cout << err << endl;

        cout << "done: " << queue.finish() << endl;

        return getXYfromRawImage(done, frequencyImage, updown, leftright);
}

ProcessImage.cl
__kernel void ProcessImage(const int isLightOn, const int frameSize, const int thresholdPixel, const int pixelsPerFrame, const int currentFrameID, __global unsigned char* lastFrames, __global unsigned char* outData) {
    int isBegin = 1;

    bool mustBrightNow = !isLightOn;

    int lastPixel = 0;

    int isWrongPixel = 0;

    for (int i=currentFrameID; i<frameSize + currentFrameID; i++) {

        int i2 = i;

        if(i >= frameSize) {
            i2 = i2 - frameSize;
        }

        int id = (i2 * pixelsPerFrame) + get_global_id(0);

        if (isBegin == 1) {
            lastPixel = (int) lastFrames[ id ];
            isBegin = 0;

        } else {
            int currentPixel = (int) lastFrames[ id ];

            if (mustBrightNow == false) {
                if (currentPixel + thresholdPixel < lastPixel) {
                    mustBrightNow = true;
                } else {
                    isWrongPixel = 0; //It freezes when I write '1'
instead of '0'

                    break;
                }
            } else {
                if (currentPixel - thresholdPixel > lastPixel) {
                    mustBrightNow = false;
                } else {
                    isWrongPixel = 0; //Also it freezes when I write '1'. And it doesn't matter if is an integer or a boolean.

                    break;
                }
            }

            lastPixel = currentPixel;
        }

    }
    if (isWrongPixel == 0) {
        outData[get_global_id(0)] = (uchar) (0);
    } else {
        outData[get_global_id(0)] = (uchar) (1);
    }

}

On all cout I get 0. So "no" noticeable errors.
I know, that some parts of the code aren't optimized, but it should work.
If you want to now, the unsigned char* lastFrames is built like for every pixel a char (monochrome). So its size is 20 million (Width x Height x previous frames). So it has multiple frames and I can compare different frames in OpenCL.
So what could that be?

*

*Is there a problem, that not the main thread starts the kernel but
the pthread_t.

*And OpenCL freezes mostly at the 146th frame or randomly. Maybe a
mistake in memory?

*And it only freezes when I write in isWrongPixel 1 or
true.

*And when I write 0 or false it will work, but I need a
boolean.

*And I haven't tried more than 500 frames in a run.

So what am I doing wrong ?
I know that my grammar is not completely correct..
Thanks in advance

A: Following on the discussion from comments.
Here's an example how to use Images:
#include <CL/cl.hpp>
#include <vector>

std::vector<cl::Platform> clPlatforms;
cl::Platform::get(&clPlatforms);

// TODO Set correctly
cl::Device chosenDevice;
bool first = true;
for (auto &&platform : clPlatforms) {
    std::vector<cl::Device> clDevices;
    platform.getDevices(CL_DEVICE_TYPE_ALL, &clDevices);

    if (first) { // REMOVE
        chosenDevice = clDevices[0];
        first = false;
    }

    std::cout << platform.getInfo<CL_PLATFORM_NAME>()<<'\n';
    for (auto &&device : clDevices) {
        std::cout << device.getInfo<CL_DEVICE_NAME>()<<'\n';
    }
}

cl::Context context{chosenDevice};
// Possible values
// https://www.khronos.org/registry/OpenCL/sdk/1.2/docs/man/xhtml/cl_image_format.html
cl::ImageFormat format{ CL_R, CL_UNSIGNED_INT8};

std::size_t imageWidth = 640;
std::size_t imageHeight = 480;
std::size_t numFrames = 128;
// Fill as sequences of rows for each 2D
std::uint8_t
    *input = new std::uint8_t[imageWidth * imageHeight * numFrames];

std::size_t i = 0;
for (std::size_t frameI = 0; frameI < numFrames; ++frameI)
    for (std::size_t y = 0; y < imageHeight; ++y)
        for (std::size_t x = 0; x < imageWidth; ++x)
            input[i++] = 0; // INIT

// Zeroes specify data format, see
// https://www.khronos.org/registry/OpenCL/sdk/1.2/docs/man/xhtml/clCreateImage3D.html
// Note that images cannot be both read and write
cl::Image3D
    inImage{context,
            CL_MEM_READ_ONLY | CL_MEM_HOST_NO_ACCESS | CL_MEM_COPY_HOST_PTR,
            format,
            imageWidth,
            imageHeight,
            numFrames,
            0,
            0,
            input,
            nullptr};
cl::Image2D outImage{context, CL_MEM_WRITE_ONLY | CL_MEM_HOST_READ_ONLY,
                     format, imageWidth, imageHeight};

std::string source = "PASTE SOURCE HERE";
cl::Program program(context, source);
program.build("-cl-std=CL1.2");

cl::Kernel kernel(program, "ProcessImage");
kernel.setArg(0, (int)0);
kernel.setArg(1, (int)numFrames);
int thresholdPixel = 10; // SET
kernel.setArg(2, (int)thresholdPixel);
kernel.setArg(3, (int)(imageWidth * imageHeight));
int currentFrameID = 12; // SET
kernel.setArg(4, (int)currentFrameID);
kernel.setArg(5, inImage);
kernel.setArg(6, outImage);
cl::CommandQueue queue(context, chosenDevice);
queue.enqueueNDRangeKernel(kernel, cl::NullRange,
                           cl::NDRange(imageWidth, imageHeight));

std::uint8_t *output = new std::uint8_t[imageWidth * imageHeight];
// See
// https://www.khronos.org/registry/OpenCL/sdk/1.0/docs/man/xhtml/clEnqueueReadImage.html
cl::size_t<3> region;
region[0] = imageWidth;
region[1] = imageHeight;
region[2] = 1;
//Might as well block if the next call would be clFinish anyway.
queue.enqueueReadImage(outImage, true, cl::size_t<3>(), region, 0, 0,
                       output);

The kernel source:
__kernel void ProcessImage(const int isLightOn, const int frameSize, const int thresholdPixel, const int pixelsPerFrame, const int currentFrameID, read_only image3d_t lastFrames, write_only image2d_t outData) {
    int isBegin = 1;
    bool mustBrightNow = !isLightOn;
    int lastPixel = 0;
    int isWrongPixel = 0;

    for (int i=currentFrameID; i<frameSize + currentFrameID; i++) {
        int i2 = i;

        if(i >= frameSize) {
            i2 = i2 - frameSize;
        }

        int pixValue = (int)read_imageui(lastFrames,(int4)(get_global_id(0),get_global_id(1),i2,0)).x;
        if (isBegin == 1) {
            lastPixel = pixValue;
            isBegin = 0;
        } else {
            int currentPixel = pixValue;

            if (mustBrightNow == false) {
                if (currentPixel + thresholdPixel < lastPixel) {
                    mustBrightNow = true;
                } else {
                    isWrongPixel = 1;
                    break;
                }
            } else {
                if (currentPixel - thresholdPixel > lastPixel) {
                    mustBrightNow = false;
                } else {
                    isWrongPixel = 1;
                    break;
                }
            }
            lastPixel = currentPixel;
        }
    }
    write_imageui(outData,(int2)(get_global_id(0),get_global_id(1)),(uint4)(isWrongPixel,0,0,0));
}

I was able to run this code both on my 1050TI, Intel 7700HQ and Intel 630HD without freezes, I hope you will to :)
I replaced buffers with images, and also made this task "2D" which is reflected in enqueueNDRangeKernel. The kernel does exactly the same thing, but it makes indexing images more natural. I was not sure if you are aware of multiple platforms, which might allow you to use your GPU. It should only need up-to-date drivers, nothing else for it to show up in platforms and devices. There's no need to create buffers for const variables, just use correct types in kernel.setArg template function.
So try it :)

redpajama_set_name: RedPajamaStackExchange


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The Milwaukee Bucks won't sign guard D.J. Stephens to a second 10-day deal, sources tell RealGM.
Stephens will scan his other options in the NBA and overseas.
Stephens appeared in three games with the Bucks, averaging 2.3 points and 1.7 rebounds in 4.9 minutes.


text:
Apple/
Mobile/
Apple will launch a new 4-inch iPhone early next year, says analyst
But he doesn't predict a plastic iPhone 6c
By James Vincent
Source Ming-Chi Kuo (Mac Rumors)
Nov 4, 2015, 9:41 AM UTC |
Apple could release a new 4-inch iPhone next year that "resembles an upgraded iPhone 5S," according to a report by respected KGI Securities analyst Ming-Chi Kuo. The device is targeted to be released in the first half of 2016, says Kuo, and will be powered by the A9 chip seen in the iPhone 6S and 6S Plus. However, unlike these larger models, the 4-inch handset will reportedly not offer the pressure sensitive 3D Touch feature, in order to differentiate the lineup.
"As there is still demand for a 4-inch iPhone, we believe Apple will upgrade this product line," writes Kuo in an investors' note reported by Mac Rumors. "Because the iPhone 5S is more popular than the iPhone 5C, we think Apple is likely to launch an upgraded iPhone 5S. We predict Apple will mass-produce this new 4-inch iPhone in 1H16 with metal casings. In order to make the current iOS 9 or next-generation iOS 10 run smoothly, Apple may adopt an A9 chip for this new phone."
Kuo's predictions suggest this is a small iPhone 6s, not a new iPhone 6c
Kuo's note about metal casings stresses that the small iPhone would not be a retread of the plastic and colorful iPhone 5c. Although there have been rumors about a 4-inch device since 2015, Kuo has a strong track record that makes him particularly convincing. Over recent years he correctly predicted the launch of the 4.7-inch and 5.5-inch iPhones, as well the two sizes of the Apple Watch, the new 12-inch MacBook, and pretty much the entire spec sheet for the iPhone 6S. However, he's not always right at the right time — he didn't think the 5.5-inch iPhone would arrive until later in 2015.
When it comes to the iPhone 7, Kuo's predictions are safe and not particularly stirring. According to Mac Rumors, he foresees an upgraded A10 chip for the iPhone 7 and 7 Plus, with the Plus model getting 3GB of RAM instead of 2GB to differentiate it as the top-end device. If Apple does launch a total of three devices next year, the iPhone lineup will look oddly similar to Sony's Xperia trio, although even Sony thinks a 4-inch smartphone is too small for today's market.
Verge Video: iPhone 6s review
More from Apple
The best smartphone you can buy for under $500
The best phone to buy right now
Quick fixes: how to transfer iPhone videos to a Windows PC when it keeps disconnecting


We have an external entity that we consume through an extension. This entity has a few computed columns that make use of sql functions and can't be moved to OutSystems for obvious reasons.
"The column "*************" cannot be modified because it is either a computed column or is the result of a UNION operator."
Mark the attributes as Ignore and Outsystems doesn't show the attributes in consumer modules anymore. Downside is that we can't use those attributes anymore. I though of importing that entity twice as read-only and write-only but Integration studio won't allow for it.
Tried changing the Auto Generated attribute on the extension entity to let the platform know that it can't modify it. But when I update the reference to the entity it marks the IsMandatory property on those attributes as true by default. I can see in the change log that IsMandatory has changed from "Yes" to "No" but when I refresh the reference nothing changes and on re-opening the dependencies again the Refresh option is still there.
This obviously looks like a bug in the service studio to me but I am also not sure if it'll help me in fixing the actual issue which is to keep the computed columns without OutSystems trying to update those.
I was suggested that in this case, a workaround would be to use SQL instead of the Entity Actions.
Thanks Eduardo. Appreciate the response. I'm not totally against it but at the current stage we don't want to go to Advanced SQL.
I have logged a support ticket to check if there's another way of doing it or working around the Service Studio bug. I'll update here once we have a resolution.