text:
\section{Device structure and basic characterization}

We show the schematic of the device structure in Fig. \ref{fig1}a.
By using a double gate structure, we can control the electric field and the chemical potential of the device independently.
Few-layer graphene serves as transparent gates,
top and bottom hBN serve as gate insulators, and
middle hBN (monolayer) serves as a tunnel barrier.
The crystal axis of the {${\rm MoSe_2}$}\ layers are aligned to be close to 0 degree
using the tear-and-stack technique\cite{Kim2016}.
Both {${\rm MoSe_2}$}\ layers are grounded via few-layer graphene contacts.
Fig. \ref{fig1}d shows the optical microscope image of the device.
Fig. \ref{fig1}b is a schematic image of a dipolar exciton formed by coherent coupling of
inter-layer exciton (IX) and intra-layer exciton (X) via hole tunneling.
Fig. \ref{fig1}c shows a schematic image of the electrons in a Moir\'e\ lattice
probed by intra-layer exciton in low electron density regime.

Figure~\ref{fig1}e shows a spatial map of total photoluminescence (PL) from the device.
Here, both top and bottom gate voltages are kept at zero Volts.

We observe PL from regions with monolayer {${\rm MoSe_2}$}, but not from bilayer {${\rm MoSe_2}$},
where two {${\rm MoSe_2}$}\ flakes are in direct contact
(the area around the point indicated by the white arrow in Fig. \ref{fig1}e).
On the other hand, the {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ area shows bright PL.
This indicates that the heterostructure becomes a direct band gap system
owing to the reduction of the inter-layer hybridization of the valence bands at the $\Gamma$ point\cite{Zhang2014}, due to the presence of monolayer hBN.
Typical PL spectra of the monolayer {${\rm MoSe_2}$}\ and the {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ area
are shown in the inset of Fig. \ref{fig1}e: there are pronounced intra-layer exciton luminescence peaks in both areas. Even though the heterostructure is fabricated from a single {${\rm MoSe_2}$}\ layer using tear-and-stack technique, we observe two distinct intra-layer exciton peaks in the {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ region. We observe that this strain-induced energy difference~\cite{He2013, Conley2013} between the PL from the top and bottom layers varies across the the sample (see Supplementary Information S2).

\section{Coherent inter-layer hole tunneling and dipolar excitons}

We first analyze the electric field ($E_z$) dependence of the elementary optical excitations of the {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ section in the absence of itinerant electrons or holes. To this end, we scan the top and bottom gate voltages (along L4 indicated in Fig. \ref{fig3}c) together to change $E_z$
while keeping the homo-bilayer system in the charge neutral regime. The PL spectrum we thus obtain is depicted in Fig. \ref{fig2}a: using the top ($V_{\rm tg}$) and bottom ($V_{\rm bg}$) gate voltage dependence, we determine that the PL spectra around 1.632\,eV and 1.640\,eV stem from intra-layer exciton in top and bottom layer ($\rm X_{top}$ and $\rm X_{bot}$), respectively. For high values of $|E_z|$ depicted in the top and bottom parts of the color-coded PL spectrum, we observe PL lines with a strong $E_z$ dependence: we identify these PL lines as originating from inter-layer excitons with a large dipole moment leading to a sizeable Stark shift.

The spectra for positive (negative) $V_{\rm tg}$ regime corresponds to the inter-layer exciton $\rm IX_{\uparrow}$ ($\rm IX_{\downarrow}$) which has a hole in the bottom (top) layer and an electron in the top (bottom) layer. The associated dipole-moment of the inter-layer exciton changes its polarity for $V_{\rm tg}$~$\sim 0$. By extrapolating the $\rm IX_{\uparrow}$  and $\rm IX_{\downarrow}$ PL lines and finding their crossing point, we estimate the energy difference between the inter- and the intra-layer exciton resonances at $E_z = 0$, which allows us to determine their binding energy difference to be $\sim$ 100\,meV.

Figure~\ref{fig2}b shows the differential reflectance ($\Delta R/R_0$) spectrum obtained for the same range of gate voltage scan as that of Fig. \ref{fig2}a. Here, $\Delta R / R_0 \equiv (R - R_0)/R_0$, with $R$ and $R_0$ denoting the reflectance signal from the {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ region, and background reflectance, respectively.
In accordance with the PL data (Fig. \ref{fig2}a), we find $\rm X_{top}$ and $\rm X_{bot}$ resonances around 1.632\,eV and 1.640\,eV, respectively. Moreover, for $V_{\rm tg}$ $\gtrsim$ 7.5\,V ($V_{\rm tg}$ $\lesssim$ -7.5\,V), we observe $\rm IX_{\uparrow}$ ($\rm IX_{\downarrow}$) hybridizing exclusively with $\rm X_{top}$ ($\rm X_{bot}$). Figures~\ref{fig2}c and \ref{fig2}d show the magnified plots of the regions highlighted with blue and green dashed lines in Fig.~\ref{fig2}b, confirming avoided crossing of an intra-layer exciton line with multiple inter-layer excitons. We first note that the observation of a sizeable reflection signal from $\rm IX_{\uparrow}$ away from the avoided crossing suggests that it is possible to resonantly excite long-lived inter-layer excitons in these structures. The hybridization of $\rm IX_{\uparrow}$ lines with $\rm X_{top}$, together with the lack of an avoided crossing with $\rm X_{bot}$ in Fig.~\ref{fig2}c, unequivocally shows that avoided crossings originate exclusively from coherent hole tunneling schematically shown in Fig. \ref{fig2}e.
Our observation, proving that the hole tunnel coupling is much larger than that of the electron, is consistent with the band alignment expected from first principle band-structure calculations\cite{Ozcelik2016}. This conclusion is also confirmed by the data depicted in Fig. \ref{fig2}d, where avoided crossing originates from coherent-hole-tunneling induced hybridization of $\rm IX_{\downarrow}$ and $\rm X_{bot}$
schematically shown in Fig. \ref{fig2}f.

One of the most remarkable features of the spectra depicted in Figs.~\ref{fig2}c~and~\ref{fig2}d is the existence of multiple avoided crossings associated with three inter-layer exciton resonances separated in energy by $\sim 3$\,meV. This inter-layer exciton fine-structure demonstrates the existence of a Moir\'e\ superlattice~\cite{Yu2017, Wu2018, Ruiz-Tijerina2019, Alexeev2019, Seyler2019, Tran2019, Jin2019}, originating from a small twist angle between the two {${\rm MoSe_2}$}\ layers. The presence of an hBN tunnel barrier strongly suppresses the strength of the associated Moir\'e\ potential, rendering it sizeable only for the inter-layer excitons~\cite{Yu2017}.

\section{Charge configuration detection by exciton-polaron spectroscopy}

It is well established that presence of itinerant charges drastically alters the optical excitation spectrum~\cite{Xu2014}. Recent theoretical and experimental work established that the modified spectrum originates from dynamical screening of excitons by electrons or holes~\cite{Sidler2017, Efimkin2017}, that lead to the formation of a lower energy attractive polaron (AP) branch. Concurrently, the exciton resonance evolves into a repulsive polaron (RP) (see Supplementary Information S1). The particularly strong sensitivity of the RP resonance energy to changes in electron density renders it an ideal spectroscopic tool for monitoring, or sensing,
the electron density $n$ in the same layer~\cite{Back2017, Smolenski2019}.
The strain induced resonance energy difference between $\rm X_{top}$ and $\rm X_{bot}$, ensuring different energies for the corresponding $\rm RP_{top}$ and $\rm RP_{bot}$, together with the much weaker sensitivity of $\rm RP_{top}$ ($\rm RP_{bot}$) on $n$ in the bottom (top) layer, allows us to determine the charging configuration of the two layers simultaneously.
Since we are predominantly interested in the low carrier density regime
where the quasi-particle (bare-exciton) weight of the RP resonance is
close to unity, we will refer to it as the exciton resonance.

Figures~\ref{fig3}a and \ref{fig3}b show the gate voltage dependence of $\Delta R/R_0$
at $E = 1.632\,{\rm eV}$ and $E = 1.640\,{\rm eV}$, which correspond to
the top ($\rm X_{top}$) and bottom ($\rm X_{bot}$) intra-layer exciton resonance energy
in the charge neutral regime, respectively. The inset to these figures show a line cut through the dispersive neutral exciton reflection spectrum, indicating the exciton energies at which we monitor $\Delta R/R_0$.
Since a small increase of $n$ from $\sim 0$ to $1 \times 10^{11}$\,cm$^{-2}$ results in a change of $\Delta R/R_0$ from $\sim -1$ to $\gtrsim 0$, the blue areas in Figs.~\ref{fig3}a and \ref{fig3}b  correspond to the charge neutral regime of each layer. The red and white areas in turn, correspond to the electron or hole doped regime of each layer. This all-optical determination of the electrically resolved charge map of the bilayer provides an invaluable tool for monitoring the bulk properties of 2D materials which is not directly accessible in transport measurements.

To enhance the sensitivity of the charge map to the transition between the undoped and doped regimes and to visualize the charge configuration of both layers at the same time, we first evaluate the derivative of $\Delta R/R_0$ with respect to energy at $E = 1.632\,\rm eV$ and $E = 1.640\,\rm eV$, and then overlay $d (\Delta R/R_0)/d E$ obtained for both layers. The resulting charge map, depicted in Fig.~\ref{fig3}c, is closely reminiscent of the charging plateaus used to characterize gate-defined quantum dots~\cite{Hanson2007}. Since $d (\Delta R/R_0)/d E$ is only sensitive to changes in $n$, the blue regions in Fig.~\ref{fig3}c correspond to the regime where the charge configuration changes, allowing us to clearly separate the regions (t,b) where the top or bottom layer is neutral (t=i or b=i), electron doped (t=n or b=n) or hole doped (t=p or b=p).

We show typical gate voltage dependence of $\Delta R/R_0$ in Fig. \ref{fig3}d and
\ref{fig3}e which are obtained when the two gate voltages are scanned in a coordinated manner along the lines L1 and L2, indicated in Fig. \ref{fig3}a and \ref{fig3}b, respectively. In both plots, we confirm the emergence of the AP resonance and the associated blue shift of the exciton / RP energy around the charge configuration transition points, confirming the assignment obtained from $d (\Delta R/R_0)/d E$ in~Fig.~\ref{fig3}c.

In stark contrast to the case of monolayer {${\rm MoSe_2}$}\ (Supplementary Information S1), we find that the top (bottom) gate  dependence obtained by fixing the bottom (top) gate voltage, is not monotonic because of the screening of the applied gate voltage when one of the two layers is already doped.
The observed responsivity to the applied gate voltages is consistent with the interpretation that the lower (higher) energy exciton resonance at $E = 1.632\,\rm eV$ ($E = 1.640\,\rm eV$) is $\rm X_{top}$ ($\rm X_{bot}$).
For example in Fig. \ref{fig2}b, by fixing $V_{\rm tg}$ = 0\,V and sweeping $V_{\rm bg}$ from negative to positive, we find drastic change of $\Delta R/R_0$ around $V_{\rm bg} \simeq$ 2\,V.
On the other hand, by fixing $V_{\rm bg}$ = 0\,V and sweeping $V_{\rm tg}$ from negative to positive, we find much less change of $\Delta R/R_0$.
From this asymmetry of the gate dependence, we can confirm that
the resonance in Fig. \ref{fig2}b is originating from $\rm X_{bot}$.

Figure~\ref{fig3}f shows the gate voltage scan along L3, indicated in Fig. \ref{fig3}c, where we fixed $V_{\rm bg}$ at $4\,\rm{V}$ and scanned $V_{\rm tg}$. By sweeping $V_{\rm tg}$ from negative to positive, we find that the bottom layer gets electron doped around $V_{\rm tg} = -3\,\rm{V}$, and then gets depleted by increasing $V_{\rm tg}$ further.
This observation shows that electrons are transferred from the bottom layer to the top layer whilst  electrons are introduced into the top layer and  $V_{\rm bg}$ is kept unchanged. This counter-intuitive dependence shows up as the curving of the lines separating the charge configurations (n,i) and (n,n) in Fig. \ref{fig3}c. Similar inter-layer charge transfer behavior was previously observed in transport experiments in bilayer semiconductor systems \cite{Eisenstein1994, Larentis2014, Fallahazad2016} and was attributed to the negative compressibility. To the best of our knowledge, our experiments provide the first observation of negative compressibility, arising from dominance of intra-layer exchange interactions over kinetic energy, using optical spectroscopy.

\section{Interaction induced incompressible states}

The results we present in Sec.~II establish the existence of a Moir\'e\ superlattice for inter-layer excitons. On the one hand, the underlying periodic modulation of the electronic bands should lead to Moir\'e\ subbands for electrons (holes) in the conduction (valence) band. On the other hand, the absence of coherent electron tunneling indicates that the resulting subbands in the top and bottom layers do not hybridize. Taking into account the relatively strong conduction-band spin-orbit coupling in MoSe$_2$, the homobilayer system we are  investigating realizes a rather unique system exhibiting flat bands with layer and valley-spin degree of freedom; while the degeneracy associated with the former can be tuned using a perpendicular electric field ($E_z$), the latter can be controlled using a magnetic field ($B_z$). Moreover, our observation of negative compressibility (Sec.~III) indicates that the electron-electron interaction energy scale dominates over kinetic energy even at relatively high electron densities ($n \simeq 1 \times 10^{12}$\,cm$^{-2}$) where several Moir\'e\ bands in one layer are occupied. In this section, we explore electron correlation effects in the more interesting regime of low carrier densities by zooming in to the low-$n$ section of the charging map (Fig.~3c) where the (i,i), (i,n), (n,i) and (n,n) regions coalesce. The high sensitivity of the exciton/RP resonance energy, as well as the AP oscillator strength, to changes in electron density once again forms the backbone of our investigation.

Figures \ref{fig4}a and \ref{fig4}b show the gate voltage dependence of differential reflectance close to the $\rm X_{top}$ and $\rm X_{bot}$ resonance energy for $n=0$ at $E_{\rm X_{top}}^0 = 1.6320$\,eV and $E_{\rm X_{bot}}^0 = 1.6402$\,eV, respectively. In these maps, a shift of the exciton resonance energy due to a change in $n$ is detected as a modification
of the exciton reflectance. The specific choice of $E_{\rm X_{top}}^0$ and $E_{\rm X_{bot}}^0$, indicated by the magenta and cyan points in the insets of Fig. \ref{fig4}a and \ref{fig4}b, maximizes the sensitivity to $n$. Instead of showing the reflectance map as a function of the top and bottom gate voltages, we now choose the horizontal and vertical voltage axes to be given by $V_E = 0.5 V_{\rm tg} - 0.5 V_{\rm bg}$ and $V_{\mu} = (7/15) V_{\rm tg} + (8/15) V_{\rm bg}$.
With this choice, vertical ($V_{\mu}$ axis) and horizontal ($V_{E}$ axis) cuts through the reflectance map leave $E_z$ and $\mu$ unchanged respectively, where $\mu$ denotes chemical potential.

Figures~\ref{fig4}a and \ref{fig4}b show a periodic modulation of the RP differential reflectance as a function of $V_{\mu}$, particularly in the low $n$ regime. Moreover, the modulation of the top and bottom layer reflectance are correlated and symmetric with respect to the $V_E = -1$\,V axis, indicating that for this value of $V_E$, the energy detuning of
the top and bottom $\rm MoSe_2$ layers is zero.
To gain further insight into the structure of correlated changes, we first determine the excitonic resonance energy for the top and bottom layers ($E_{\rm X_{top}}$ and $E_{\rm X_{bot}}$) by fitting the reflectance spectrum with two dispersive Lorentzian lineshapes (see Supplementary Information S3) and then plot the derivative of $E_{\rm X_{top}}$ and $E_{\rm X_{bot}}$ with respect to $V_{\mu}$ in
Figs. \ref{fig4}c and \ref{fig4}d. The resulting map shows a remarkable checkerboard pattern that is complementary for the top and bottom layers. Since the blue shift of $\rm X_{top}$ and $\rm X_{bot}$ resonance while increasing $V_{\mu}$
(positive $dE_{\rm X_{top}}/dV_{\mu}$ and $dE_{\rm X_{bot}}/dV_{\mu}$)
corresponds to filling of electrons in the top and bottom layer respectively,
the complementary checkerboard like pattern indicates a layer by layer filling of electrons.
Note that a similar diagram has been reported in layer resolved capacitance measurement of Landau lavels in bilayer graphene\cite{Hunt2017}.

The observed periodicity in Fig.~\ref{fig4} evidences the existence of Moir\'e subbands for electrons.
In anticipation of the subsequent discussion,
we define a layer filling factor $\nu_{\rm L}$
(L = "top" or "bot" indicating top or bottom layer)
so that $\nu_{\rm L} = 1/2$ corresponds to 1 electron per Moir\'e
unit cell of a single layer, and a total filling factor $\nu$ as
$\nu = \nu_{\rm top} + \nu_{\rm bot}$.
From a capacitive model of our device,
we determine that $\nu = 1/2$ coincides with a remarkably
low electron density of $n = 2 \times 10^{11} {\rm cm^{-2}}$.
At this low electron density, $r_s$ parameter, which describes the ratio of interaction energy to kinetic energy, is estimated to be $r_s \simeq 14$.
The density periodicity corresponds to a Moir\'e\ superlattice latice constant of $\lambda_{\rm Moir\acute{e}} = 24\, {\rm nm}$ by assuming a triangular superlattice.
We indicate the values of $\nu$ corresponding to
$\nu$ = 1/2, 1, 3/2, 2 with blue dashed lines in Fig. \ref{fig4}c.

Figures~\ref{fig5}a and \ref{fig5}b show the $V_E$ dependence of the differential reflectance spectrum
for fixed $V_{\mu}$ where $\nu$ = 1/2 ($n = 2 \times 10^{11} {\rm\,cm^{-2}}$)
and $\nu$ = 1 ($n = 4 \times 10^{11}\,{\rm cm^{-2}}$), respectively. In Fig. \ref{fig5}a, we find an abrupt shift of exciton energy together with complete oscillator strength transfer between $\rm AP_{top}$ and $\rm AP_{bot}$. This shows that the electrons are completely and abruptly transferred from one layer to the other layer upon changing $E_z$. Figs. \ref{fig4}c and \ref{fig4}e show the extracted $\rm X_{bot}$ and $\rm X_{top}$ energies
around $\nu$ = 1/2. Remarkably, the abrupt jump in excitonic resonance is pronounced at $\nu$ = 1/2,
and smeared out for both lower ($\nu$ = 0.35) and higher filling factors ($\nu$ = 0.65).
These measurements show that abrupt transfer of practically all of the $\sim 1500$ electrons within the region we monitor optically is linked to the emergence of an interaction-induced incompressible state
in the lowest Moir\'e\ subband at $\nu$ = 1/2 filling. As the filling factor is increased (decreased) towards $\nu =1/2$, the electronic system shows an ever stronger layer-paramagnetism, due to enhanced role of interactions, but otherwise exhibits a continuous inter-layer transfer of electrons as a function of $E_z$ that would be expected from a compressible state. Close to $\nu=1/2$, there is a phase transition to an incompressible state that can be accommodated either in the top or bottom layer (see Fig. \ref{fig5}g).

Figure~\ref{fig5}b shows that for $\nu =1$, the polaron reflectance spectrum as a function of $E_z$ is characterized by 3 plateau-like regions. We attribute the abrupt jumps in the excitonic resonance energy to the transition from $(\nu_{\rm top}, \nu_{\rm bot})$ = (0, 1), through (1/2, 1/2), to (1, 0) configurations (see Fig. \ref{fig5}h).
This explanation is confirmed by the corresponding changes in the oscillator strength of the AP resonances of the top and bottom layers. In (1, 0) and (0, 1) configurations, we measure a reflectance signal either from $\rm AP_{top}$ or $\rm AP_{bot}$, consistent with full layer polarization. In the (1/2, 1/2) configuration, we find the oscillator strength of $\rm AP_{top}$ and $\rm AP_{bot}$ to be identical and equal to half the value obtained under (1,0) for $\rm AP_{top}$.
The extracted excitonic resonance energy $\rm X_{bot}$ and $\rm X_{top}$ around $\nu = 1$
is shown in Figs.~\ref{fig5}d and \ref{fig5}f, respectively.
The plateau structure of the (1/2, 1/2) state with abrupt jumps in $n$ for $V_E = -1.2$\,V and $V_E = -0.7$\,V is clearly visible at $\nu = 1$, but is smeared out for both lower ($\nu = 0.85$) and higher fillings ($\nu = 1.15$).
We conclude that as the total $n$ is changed away from $\nu = 1$, both top and bottom layers become
compressible, showing smooth changes in layers occupancy as a function of small $E_z$ values ($-1.1\,{\rm V} < V_E < 0.8$\,V).
The emergence of the stabilized (1/2, 1/2) plateau at $\nu = 1$ strongly suggests that there is mutual stabilization of the incompressible electronic state due to the inter-layer electron-electron interactions. If the $\nu_{\rm top} = 1/2$ or  $\nu_{\rm bot} = 1/2$ had been incompressible for $\nu = 0.85$, we would have observed corresponding plateaus in the the excitonic resonance reflection map. The reflectance data for higher fillings ($\nu = 3/2$ and $\nu = 2$)
are shown in the supplementary information (Fig. S5): in stark contrast to the (1/2, 1/2) configuration at $\nu = 1$, a plateau at the (1, 1) electron configuration is missing at $\nu = 2$ filling, indicating that the state with the corresponding integer fillings is not sufficiently stabilized by the inter-layer interactions.

Finally, we emphasize that our identification of $n = 2 \times 10^{11}\,{\rm cm^{-2}}$ yielding half-filling of a single-layer Moir\'e\ subband is supported by our measurements at $B_z = 7$\,T: In Fig.~S6 we observe that the plateau structure observed for $\nu=2$ under full valley polarization of electrons is identical to that observed for $ B_z=0$\,T, even though the total number of electronic states per Moir\'e\ subband is halved due to giant valley-spin susceptibility of electrons in MoSe$_2$\cite{Back2017}. This observation shows that the incompressibility is determined by filling of each Moir\'e\ site by a single electron, irrespective of its degeneracy.

\section{Discussion}
The experiments we describe in Sec.~IV demonstrate the existence of Mott-like incompressible electronic states for half-filling of the lowest Moir\'e\ subband. Unlike prior reports\cite{Tran2019, Alexeev2019, Seyler2019, Jin2019}, our experiments are carried out for long Moir\'e\ superlattice latice constant of $\lambda_{\rm Moir\acute{e}} = 24\, {\rm nm}$ and $r_s$ parameter of $r_s  \simeq 14$. The weakness of the Moir\'e\ potential stemming from the hBN layer separating the two MoSe$_2$ layers in turn ensures that the on-site interaction strength is comparable to, or possibly larger than, the depth of the Moir\'e\ potential. In this sense, the homobilayer system realizes a rather unique regime of the Fermi-Hubbard model where some of the expectations such as an antiferromagnetic ground-state need not be applicable.

In addition to establishing twisted TMD homo-bilayers as a promising system for investigating Mott-Wigner physics\cite{Imada1998, Camjayi2008, Zarenia2017, Wu2018} originating from strong electronic correlations, our experiments open up new avenues for exploring interactions between dipolar excitons and electrons confined to flat bands. In particular, the structure we analyzed could be used to realize and study Bose-Fermi mixtures consisting of degenerate electrons strongly interacting with an exciton condensate generated by resonant laser excitation. The phase diagram of such a mixutre is currently not fully understood~\cite{Ludwig2011} but is expected to provide a rich playground for many-body physics, including but not limited to exciton-mediated superconductivity~\cite{Little1964, Ginzburg1970, Laussy2010, Cotlet2016}.

{\it Note added} After completion of this work, we became aware of two
manuscripts, arXiv:1910.08673 and arXiv:1910.09047, reporting similar results in a different material system.

\newpage

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{Fig1.pdf}
\caption{{\bf Device structure and the basic characteristics.}
{\bf a} Schematic image of the device structure.
$V_{\rm tg}$ ($V_{\rm bg}$) is the applied voltage to the top (bottom) gate.
{\bf b, c} Schematic image of coupled inter- and intra-layer exciton ({\bf b})
and electrons in the Moir\'e\ lattice probed by the intra-layer exciton ({\bf c}).
Purple planes correspond to {${\rm MoSe_2}$}\ layers, and dashed lines indicate a Moir\'e\ unit cell.
The pink (light blue) circles with $+$ ($-$) sign indicate holes (electrons) forming excitons, and the electron-hole pair enclosed by the red (yellow) ellipse indicates intra-layer (inter-layer) exciton.
The green double arrow in {\bf b} indicates tunnel coupling of holes
through the monolayer hBN barrier.
The light green circles in {\bf c} indicate electrons filling the Moir\'e\ lattice.
{\bf d} Optical microscope image of the device.
The border of each flake is highlighted with dashed lines,
and the material is indicated in the gray box
with the corresponding color. (The abbreviation "Gr" stands for Graphene.)
{\bf e} Spatial map of the integrated photoluminescence from 1.59eV to 1.65eV.
The blue and yellow dashed lines indicate the boundary of
the area of monolayer {${\rm MoSe_2}$}\ and {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}, respectively.
The inset shows representative PL spectra of monolayer {${\rm MoSe_2}$}\ and {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$}\ measured at
the positions indicated with the blue and the yellow stars in the main figure,
respectively.
} \label{fig1}
\end{figure*}

\newpage

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{Fig2.pdf}
\caption{{\bf Electric field dependence of photoluminescence and differential reflectance at charge neutrality.}
Gate dependence of photoluminescence {\bf a} and differential reflectance {\bf b} of $\rm MoSe_2/hBN/MoSe_2$.
Top and bottom gate voltages are scanned together to tune the electric field
at a constant chemical potential
(scanned along the dashed line L4 shown in Fig. \ref{fig3}c).
The intensity of the photoluminescnece {\bf a} is shown in log scale.
{\bf c}, {\bf d} Magnified plots of {\bf b}.
The corresponding area of {\bf c} and {\bf d} is indicated by the blue and green dashed rectangulars
in {\bf b}, respectively.
{\bf e}, {\bf f} Schematic of the energy bands and the exciton energy alignment
under electric fields at the hole resonant conditions of excitons.
} \label{fig2}
\end{figure*}

\newpage

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{Fig3.pdf}
\caption{{\bf Gate depedndence of differential reflectance spectrum of $\bf MoSe_2/hBN/MoSe_2$.}
{\bf a, b} Two gates dependence maps of differential reflectance
around top ({\bf a}) and bottom ({\bf b}) intra-layer exciton resonances
($E = 1.632 \rm eV$ and $E = 1.640 \rm eV$, respectively).
The insets of {\bf a} and {\bf b} show the differential reflectance spectrum
at $(V_{\rm tg}, V_{\rm bg}) = (0{\rm V}, 0{\rm V})$
(indicated with the white stars in the maps).
The magenta and cyan dots in the insets indicate the points
where $E = 1.632 \rm eV$ and $E = 1.640 \rm eV$, respectively.
{\bf c} Charge configuration diagram obtained by derivative of the differential reflectance spectrum
with respect to energy (sum of the derivatives at $E = 1.632 \rm eV$ and $E = 1.640 \rm eV$).
The charge configuration for each layer is indicated by p, i, n
which correspond to hole doped, neutral, electron doped, respectively,
and shown in the order of (top, bottom).
{\bf d} - {\bf f} Gate dependence of differential reflectance along the dashed lines L1 ({\bf d}), L2 ({\bf e}), and L3 ({\bf f})
shown in {\bf a}, {\bf b} and {\bf c}.
Magenta and cyan dashed lines indicate the top ($E = 1.632 \rm eV$) and bottom ($E = 1.640 \rm eV$)
exciton resonance energies, respectively.
$\rm AP_{L}^{C}$ and $\rm RP_{L}^{C}$
stand for intra-layer attractive and repulsive polarons,
where $\rm L = "top"\ or\ "bot"$ stands for top or bottom layer,
and $\rm C = +\ or\ -$ stands for hole or electron as Fermi sea carriers.
Charge configuration is written in green
together with the green dashed lines indicating the charge configuration transition
point.
} \label{fig3}
\end{figure*}

\newpage

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{Fig4.pdf}
\caption{{\bf Gate dependence of intra-layer exciton resonances in low electron density regime.}
{\bf a, b} Gate dependence maps of differential reflectance
around top ({\bf a}) and bottom ({\bf b}) intra-layer exciton resonances
($E = 1.6320 \rm eV$ and $E = 1.6402 \rm eV$, respectively).  The insets of {\bf a} and {\bf b} show the differential reflectance spectrum
at $(V_{E}, V_{\mu}) = (-1{\rm V}, 0.175{\rm V})$ (indicated with the white stars in the maps).
{\bf c, d} Gate dependence maps of top ({\bf c}) and bottom ({\bf d}) intra-layer exciton resonance energy differentiated by $V_{\mu}$.
} \label{fig4}
\end{figure*}

\newpage

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{Fig5.pdf}
\caption{{\bf Electric field dependence of differential reflectance spectrum in low electron density regime.}
{\bf a, b} Electric field ($V_E$) dependence of differential reflectance spectrum
for each fixed $V_{\mu}$ at $\nu = 1/2$ and $\nu = 1$.
Charging configuration of top and bottom layer is indicated by
$(\nu_{\rm top}, \nu_{\rm bot})$ in green.
{\bf c, d}  $V_E$ dependence of $\rm X_{bot}$ resonance energy around each total filling of $\nu = 1/2$ ({\bf c}), $\nu = 1$ ({\bf d}).
{\bf e, f}  $V_E$ dependence of $\rm X_{top}$ resonance energy around each total filling of $\nu = 1/2$ ({\bf e}), $\nu = 1$ ({\bf f}). In {\bf c} - {\bf f}, the cyan curves are without offset and other curves are displaced by 0.5meV and 1.0meV.
{\bf g, h} Schematic picture of charge configuration with density of states
at filling of $\nu$ = 1/2 ({\bf g}) and $\nu = 1$ ({\bf h}).
} \label{fig5}
\end{figure*}

\newpage

\section*{METHODS}
All {${\rm MoSe_2}$}, graphene, and hBN flakes are obtained by mechanical exfoliation of bulk crystals.
The flakes are assembled together using the dry transfer technique\cite{Wang2013} in an Ar filled glove box.
The crystal axis of top and bottom {${\rm MoSe_2}$}\ layers are aligned to be close to 0 degree
using tear-and-stack technique\cite{Kim2016}.
The metal electrodes to graphene layers are formed by Ti/Au (5\,nm/145\,nm).
The contact to the bottom graphene gate is formed by Cr/Au (3\,nm/147\,nm)
using the one-dimensional contact technique\cite{Wang2013}
by etching the hBN layer with reactive ion etching using $\rm CHF_3/O_2$ as mixture gas.

The photoluminescence measurements were performed using a HeNe laser (633nm) as an excitation source.
The reflectance measurements were performed using a single mode fiber coupled broadband LED
with a center wavelength of 760nm and a bandwidth of 20nm.
In both photoluminescence and reflectance measurements, we used a long working distance apochromatic
objective lens with NA = 0.65 (attocube LT-APO/LWD/VISIR/0.65).
All optical spectroscopy measurements have been performed at cryogenic temperature ($T \sim 4 {\rm K}$).

\bibliographystyle{naturemag.bst}

\section{Differential reflectance and Photoluminescence of monolayer {$\bf MoSe_2$}}
Here we show gate dependence of differential reflectance and photoluminescence (PL) of monolayer {${\rm MoSe_2}$}.
Fig. \ref{figS1}a shows the gate dependence of the differential reflectance at $E = 1.644 {\rm eV}$.
This energy corresponds to the minimum of the exciton reflection spectrum, as shown in the inset of Fig. \ref{figS1}a. In contrast to the case of $\rm MoSe_2/hBN/MoSe_2$ (Figs. 2a and 2b in the main text),
the dependences on top gate and bottom gate are similar.
The blue area where we have absorption from the exciton resonance is where {${\rm MoSe_2}$}\ is neutral.
In Fig. \ref{figS1}b,
we plot the gate dependence of the differential reflectance spectrum along L0 in Fig. \ref{figS1}a.
When {${\rm MoSe_2}$}\ is doped, the exciton resonance around $E = 1.644 {\rm eV}$ becomes the repulsive polaron and blue shifts. This is accompanied by the emergence of the attractive polaron\cite{Sidler2017} resonance around $E = 1.619 {\rm eV}$. Due to this blue shift of the exciton resonance
together with the dispersive shape of the exciton resonance as shown in the inset of Fig. \ref{figS1}a,
we can detect the changes in the charge configuration as we show in Fig. \ref{figS1}a.
In Fig. \ref{figS1}c, we plot gate dependence of the PL spectrum along L0 in Fig. \ref{figS1}a.
We observe that exciton PL around $E = 1.644 {\rm eV}$ is prominent in the neutral regime,
while attractive polaron PL around $E = 1.619 {\rm eV}$ dominates in the electron and hole doped regimes.

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{FigS1.pdf}
\caption{{\bf Gate dependence of differential reflectance spectrum and PL of $\bf MoSe_2$.}
{\bf a} Two gates dependence map of differential reflectance
around exciton resosnace ($E = 1.644 \rm eV$).
The charge configuration is indicated by p, i, n
which correspond to hole doped, neutral, electron doped, respectively.
The inset shows the differential reflectance spectrum at $(V_{\rm tg}, V_{\rm bg}) = (0{\rm V}, 0{\rm V})$
(indicated with the white star in the map).
The magenta dot in the inset indicate the point where $E = 1.644 \rm eV$.
{\bf b} Gate dependence of differential reflectance along the red dashed line L0 shown in {\bf a}.
Magenta dashed line indicates the exciton resonance energy ($E = 1.644 \rm eV$).
{\bf c} Gate dependence of PL along the red dashed line L0 shown in {\bf a}.
}
\label{figS1}
\end{figure*}

\newpage

\section{Differential reflectance and Photoluminescence of $\bf MoSe_2/hBN/MoSe_2$ at a different spot}
In the main text, we show the spectroscopy data of $\rm MoSe_2/hBN/MoSe_2$ at a specific spot.
Here we show  charge configuration detection and inter- and intra-layer exciton coupling at a different spot in the $\rm MoSe_2/hBN/MoSe_2$ section of the sample.

Fig. \ref{figS2}a and \ref{figS2}b show gate dependence the differential reflectance
at $E = 1.636 {\rm eV}$ and $E = 1.641 {\rm eV}$, respectively.
These energies correspond to minima of the exciton resonances in the charge neutral regime
(see the insets of Fig. \ref{figS2}a and \ref{figS2}b).
As we discussed in the main text and also in section S1,
carrier doping induces the blue shift of the exciton resonances
due to the formation of repulsive polaron,
which results in the abrupt enhancement of the differential reflectance
in Fig. \ref{figS2}a and \ref{figS2}b.
From the gate dependence,
we assign the exciton resonances at $E = 1.636 {\rm eV}$ and $E = 1.641 {\rm eV}$
to the bottom and the top intra-layer excitons, respectively.
Note that the energy order of top and bottom intra-layer excitons is opposite to
the result shown in the main text.
This is most likely caused by the inhomogeneity of the strain across the sample.
To enhance the border of the transition and visualize the charge configuration of both layers at the same time,
we calculated the derivative of the differential reflectance signal
with respect to energy at $E = 1.636 \rm eV$ and $E = 1.641 \rm eV$,
and added them together (Fig. \ref{figS2}c).
In Fig. \ref{figS2}d, e, and f, we show the gate dependence of the differential reflectance spectrum
along L'1, L'2 and L'3 (see Figs. \ref{figS2}a, \ref{figS2}b and \ref{figS2}c), respectively.
The results are similar to the results we show in the main text including the signature of negative compressibility. The only qualitative difference with respect to the data in the main text is the opposite energy order
of the top and the bottom intra-layer exciton resonances.

\begin{figure*}[h!]
	\centering
	\includegraphics[width=1.0\textwidth]{FigS2.pdf}
	\caption{{\bf Gate dependence of differential reflectance spectrum of $\bf MoSe_2/hBN/MoSe_2$ at a different spot.}
		{\bf a, b} Two gates dependence maps of differential reflectance
		around bottom ({\bf a}) and top ({\bf b}) intra-layer exciton resosnaces
		($E = 1.636 \rm eV$ and $E = 1.641 \rm eV$, respectively).
		The insets of {\bf a} and {\bf b} show the differential reflectance spectrum
		at $(V_{\rm tg}, V_{\rm bg}) = (0{\rm V}, 0{\rm V})$
		(indicated with the white stars in the maps).
		The magenta and cyan dots in the insets indicate the points
		where $E = 1.636 \rm eV$ and $E = 1.641 \rm eV$, respectively.
		{\bf c} Charge configuration diagram obtained by derivative of the differential reflectance spectrum
		with respect to energy (sum of the derivatives at $E = 1.636 \rm eV$ and $E = 1.641 \rm eV$).
		The charge configuration for each layer is indicated by p, i, n
		which correspond to hole doped, neutral, electron doped, respectively,
		and shown in the order of (top, bottom).
		{\bf d} - {\bf f} Gate dependence of differential reflectance along the red dashed lines L'1 ({\bf d}), L'2 ({\bf e}), and L'3 ({\bf f})
		shown in {\bf a}, {\bf b} and {\bf c}.
		Magenta and cyan dashed lines indicate the bottom ($E = 1.636 \rm eV$) and top ($E = 1.641 \rm eV$)
		exciton resonance energies, respectively.
	}
	\label{figS2}
\end{figure*}

We also show the electric field dependence of PL (Fig. \ref{figS3}a)
and differential reflectance (Fig. \ref{figS3}b)
in the charge neutrality regime at this spot.
For both measurements, the top and bottom gates are scanned along L'4 indicated in Fig. \ref{figS2}c
to control the electric field $E_z$ while keeping the system at charge neutrality.
In Fig. \ref{figS3}a, there are inter-layer exciton PL lines
which show Stark shift in the high $E_z$ regime
(top and botton side of the figure).
In Fig. \ref{figS3}b, we also observe differential reflection from the inter-layer excitons, exhibiting strong Stark shift, for high $E_z$. Fig. \ref{figS3}c and \ref{figS3}d show the magnified plots
of the differential reflection for high $|E_z|$ where we observe the inter-layer exciton reflection,
and we find avoided crossing structure in both plots. In Fig. \ref{figS3}c, we find the avoided crossings
between the top intra-layer exciton ($E\sim 1.641 \rm eV$) and the inter-layer excitons
 with an electron in the top and a hole in the bottom layer,
evidencing the coupling of these states via coherent hole tunneling (Fig. \ref{figS3}e).
In Fig. \ref{figS3}d, we find the avoided crossings
between the bottom intra-layer exciton ($E\sim 1.636 \rm eV$) and the inter-layer excitons
with the opposite dipole, once again confirming that the coherent coupling of these states via hole tunneling (Fig. \ref{figS3}f).

All of the results are fully consistent with the results we show in the main text,
even though the energy order of top and bottom intra-layer excitons is opposite.
Note that even the fine structure of the inter-layer exciton in Fig. \ref{figS3}c and Fig. \ref{figS3}d
is similar to the result which we show in the main text (Fig. 2c and 2d). This observation indicates that the origin of the fine structure is intrinsic to the system  and is consistent with Moir\'e\ potential picture. In particular, it shows that multiple inter-exciton lines do not stem from random spatial inhomogeneities in the sample.

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{FigS3.pdf}
\caption{{\bf Electric field dependence of PL and differential reflectance at charge neutrality.}
Gate dependence of PL ({\bf a}) and differential reflectance ({\bf b}) of $\rm MoSe_2/hBN/MoSe_2$
at the different spot along the dashed line L'4 shown in Fig. \ref{figS2}c.
Top and bottom gate voltages are scanned together to tune the electric field but to fix the chemical potential.
The intensity of the PL ({\bf a}) is shown in log scale.
{\bf c, d} Magnified plots of {\bf b}.
The corresponding area of {\bf c} and {\bf d} is indicated by the blue and green dashed rectangulars
in {\bf b}, respectively.
{\bf e, f} Schematic image of the energy band and the exciton energy alignment under electric fields
when the resonant conditions are achieved.
{\bf e} and {\bf f} correspond to the situation of {\bf c} and {\bf d}, respectively.
}
\label{figS3}
\end{figure*}

\newpage

\section{Detail of reflectance spectrum fitting and additional spectroscopy data in the low electron density regime}

We define the differential reflectance as $\Delta R/R_0 \equiv (R - R_0)/R_0$ where $R_0$ is the background reflectance, measured in a region of the heterostructure without {${\rm MoSe_2}$} \space flakes, and $R$ is the experimentally measured reflectance signal. In this work we are interested in the reflectance from {${\rm MoSe_2}$}/hBN/{${\rm MoSe_2}$} \space layers. In order to account for reflections and losses from multiple layers of the heterostructure, we assumed a Lorentzian lineshape for the excitons $\chi(E)\propto \frac{\gamma}{E-E_0+i\gamma/2}$ and effectively describe the total reflectance by ${
\rm Im}[\chi(E)\exp(i\alpha)]$ \cite{Smolenski2019}. Here, $\alpha$ is a phase shift that depends in energy and gate voltage. The total reflectance signal is given by

\begin{equation}
\Delta R/R_0 = A\cos(\alpha)\frac{\gamma^2/2}{(E-E_0)^2+\gamma^2/4}-A\sin(\alpha)\frac{\gamma (E-E_0)}{(E-E_0)^2+\gamma^2/4} + C ,
\label{eqS1}
\end{equation}
where we introduced the parameter $C$ to capture the broad background signal.

In Fig. \ref{figS4}a and \ref{figS4}b we show once again the gate voltage dependent maps of differential reflectance around top and bottom excitons used in Fig. 4a and 4b of the main text (where the gate voltages axes are $V_E = (7/15) V_{\rm tg} + (8/15) V_{\rm bg}$
and $V_{\mu} = 0.5 V_{\rm tg} - 0.5 V_{\rm bg}$). We used eq. \ref{eqS1} to fit the data and plotted the exciton resonance energies as function of gate voltages in Fig. \ref{figS4}c and \ref{figS4}d. The structures observed in Fig. \ref{figS4}a and \ref{figS4}b are well reproduced in Fig. \ref{figS4}c and \ref{figS4}d, respectively.
Figures \ref{figS4}e and \ref{figS4}f (and Fig. 4c and 4d in the main text) shows the derivative of the fitted exciton resonance from Fig. \ref{figS4}c and \ref{figS4}d with respect to $V_\mu$. The complementary checkerboard patterns are discussed in the main text, confirming the interpretation of layer by layer filling of electrons with changing chemical potential.
Periodic structures also appear in the integrated PL intensity of top and bottom layer attractive polaron (Fig. \ref{figS4}g and \ref{figS4}h, respectively), which is enhanced at the half integer filling of $\nu = $1/2, 1, 3/2 and 2. We indicate the charge configuration with ($\nu_{\rm top},\nu_{\rm bot}$), with $\nu_{\rm top} + \nu_{\rm bot} = \nu$.

Fig. \ref{figS5}f - g shows additional filling factors, complementing Fig. 5 of the main text; here, we plot $V_E$ dependence of differential reflectance spectrum for fixed $V_{\mu}$ where $\nu = $0, 1/2, 1, 3/2, and 2 ($\nu = 1/2$ and $\nu = 1$ are presented in Fig 5. in the main text).
The $V_{\mu}$ (or $\nu$) values corresponding to Fig. \ref{figS5}f - g are indicated by a blue dotted lines
shown in Fig. \ref{figS5}a - e, showing $dE_{\rm X_{bot}}/dV_{\mu}$ (Fig. \ref{figS4}f).
For $\nu = 0$ (Fig. \ref{figS5}f), both layers are neutral and therefore $\rm X_{top}$ and $\rm X_{bot}$ resonances do not shift. In stark contrast to (1/2, 1/2) state at $\nu = 1$ filling, at $\nu = 2$ filling the (1, 1) state is missing, indicating that the state of integer filling in both layers is not sufficiently stabilized by the interactions.

\begin{figure*}[h!]
	\centering
	\includegraphics[width=0.65\textwidth]{FigS4.pdf}
	\caption{{\bf Spectroscopy data of low electron density regime.}
		{\bf a, b} Gate dependence maps of differential reflectance
		around top ({\bf a}) and bottom ({\bf b}) intra-layer exciton resonances
		($E = 1.6320 \rm eV$ and $E = 1.6402 \rm eV$, respectively).  The insets of ({\bf a}) and ({\bf b}) show the differential reflectance spectrum
		at $(V_{E}, V_{\mu}) = (-1{\rm V}, 0.175{\rm V})$ (indicated with the white stars in the maps).
		{\bf c, d} Gate dependence maps of top ({\bf c}) and bottom ({\bf d}) intra-layer exciton resonance energy extracted from fitting of differential reflectance data.
		{\bf e, f} Gate dependence maps of top ({\bf e}) and bottom ({\bf f}) intra-layer exciton resonance energy differentiated by $V_{\mu}$.
		{\bf g, h} Gate dependence maps of top ({\bf g}) and bottom ({\bf h}) attractive polaron PL integrated over 1.6000 to 1.6083eV and 1.6125 to 1.6167eV, respectively.
	}
	\label{figS4}
\end{figure*}

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{FigS5.pdf}
\caption{{\bf Differential reflectance spectrum in low electron density regime.}
{\bf a} - {\bf e} Diagrams indicating electron filling of bottom layer (Fig. S4f).
{\bf f} - {\bf j} $V_E$ dependence of differential reflectance spectrum
for each fixed $V_{\mu}$ indicated by the blue broken lines in {\bf a} to {\bf e}.
Charging configuration of top and bottom layer is indicated by
$(\nu_{\rm top}, \nu_{\rm bot})$.
Green broken lines indicate the transition point of charging configuration,
which are also indicated in {\bf a} to {\bf e}.
}
\label{figS5}
\end{figure*}

\newpage

\section{Differential reflectance spectrum in the low electron density regime under a perpendicular magnetic field}
Here we show $V_E$ dependence of
the differential reflectance spectrum in the low electron density regime
(at $\nu = 2$) obtained under perpendicular magnetic fleld ($B_z$).
Figs. \ref{figS6}b and \ref{figS6}c show the differential reflectance specturm obtained at $B_z = 7{\rm T}$ for $\sigma_{-}$ and $\sigma_{+}$ circular polarization, respectively.
In Fig. \ref{figS6}b, attractive polaron (AP) signal is clearly visible, but is missing in Fig. \ref{figS6}c.
$\sigma_{+}$ ($\sigma_{-}$) circularly polarized exciton-polaron is formed between K (-K) valley exciton and -K (K) valley Fermi sea electrons \cite{Back2017, Smolenski2019}.
The full $\sigma_{-}$ polarization of AP reflectance demonstrates that
Fermi sea electrons are fully K-valley polarized at $B_z = 7{\rm T}$, $\nu = 2$.

Remarkably, even though Fermi sea electrons are fully valley polarized and the degeneracy of the Moir\'e\ subbands is reduced by a factor of 2, the structure of the spectrum remains essentially the same as the case of $B_z = 0{\rm T}$ (Fig. \ref{figS6}a). Our experiments therefore show that it is the number of electrons in Moir\'e\ unit cell that determines the structure, rather than the degeneracy of the quantum number. This observation is fully consistent with what we would expect from incompressible states induced by electron-electron interactions, where the degeneracy of Moir\'e\ subbands is lifted by strong interactions.

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth]{FigS6.pdf}
\caption{{\bf Differential reflectance spectrum under perpendicular magnetic field.}
{\bf a} - {\bf c} $V_E$ dependence of differential reflectance spectrum
at $\nu = 2$ ($V_{\mu} = 1.627{\rm V})$.
{\bf a} is the reflectance spectrum at $B_z = 0{\rm T}$.
{\bf b} and {\bf c} are the reflectance spectrum at $B_z = 7{\rm T}$ in $\sigma_-$ and $\sigma_+$ circular polarization, respectively.
}
\label{figS6}
\end{figure*}

\newpage

\bibliographystyle{naturemag.bst}

redpajama_set_name: RedPajamaArXiv


What Does the Federal Government Shutdown Mean for Real Estate?
The federal government's shutdown stretched into its 12th day today as President Donald Trump keeps pushing for $5 billion to fund his border wall while Democrats still refuse to allocate the cash. Despite a bipartisan meeting set with congressional leaders and a Republican Senate-floated compromise to give $2.5 billion to border security, Trump rejected the… Read More What Does the Federal Government Shutdown Mean for Real Estate?


Friends of Wexford General Hospital on behalf of the people of Co. Wexford has purchased much needed equipment for use in Theatre and ICU at Wexford General Hospital.
Following the recommendations of a project team formed at Wexford Gerneal Hospital, Friends of Wexford General Hospital donated €25,000 to the Emergency Department. The aim of the project team was to improve patient flow through the Emergency Department.
Friends contribute to Colorectal Screening at Wexford General Hospital.


Scholastic Responsibility: The well-being of the Chapter, the Fraternity, and the entire College Fraternity System depends upon achieving and maintaining a high degree of scholarship. It is the duty of every member to give his best efforts to his studies, to enhance the academic record of his Chapter.
Character Responsibility: To the college administration and the public, the moral conduct and personal behavior of each member affects the total Chapter image. This makes it imperative to act all times, according to the highest standards of integrity, propriety and good taste.
Leadership Responsibility: Since the continued existence of the Chapter requires regular succession of leaders, it is the obligation of every member to prepare for positions of service and authority - not seeking any office but being ready to accept when needed, and to serve to the best of his ability.
Obedience Responsibility: Policies and regulations are established by the college and the fraternity for the good of all. It is incumbent upon every member to comply, with all such rules and practices - including those relating to hazing, drinking and social affairs. Every member shall attend all Chapter meetings and official functions unless excused from doing so.
Financial Responsibility: Each member shall promptly meet all bills presented to him for Fraternity expenses in conformity with Chapter requirements; he shall avoid incurring debts with other members or with local merchants which are not paid when due. Members shall reside in and eat all meals at the Chapter house when such facilities are available.
Alpha Delta Gamma believes in the common brotherhood of mankind. As a part of Alpha Delta Gamma, Brothers must be responsible to their fellow man. Such responsibility means neither that we should interfere in the lives of others nor that we must be accountable for others. Rather, it means that we should be available to help others and responsive to the needs of others. We do not seek only to do great things for our fellow man, for often it is the small act of brotherhood which serves him best. We do not require acknowledgement for our aid to others, for our reward is in doing for others, not in receiving praise for our deeds. By striving to be of service to our fellow man and our Brothers, we will better mankind.
We enjoy the rights and privileges of Alpha Delta Gamma. We now have responsibility, in turn, to give to the Fraternity. For more than being a part of Alpha Delta Gamma, we are, in fact, Alpha Delta Gamma itself. Within the Fraternity itself, there are three levels of responsibility. We are responsible for full participation in the activities of our Chapter. In addition to taking part and enjoying the various events sponsored by the Chapter, there are certain things, such as attending Chapter meetings, prompt payment of dues, serving on committees, and accepting appointed offices or running for elected offices to help administer the Chapter, which should be automatic to you.
As an active or an Alumnus, there is a second level of responsibility within the Fraternity. Although we belong to a single Chapter, we are also seeking brotherhood in the total fraternity - Alpha Delta Gamma National Fraternity. Our responsibility to the National Chapter is not merely learning about the national organization and history. Rather we must all take a pro-active role by participating in national activities - attend Convention or seminars, serve on National committees, or support the Educational Foundation. By doing this, we will be able to strengthen the Fraternity and derive full advantage of being a member of a national Fraternity.
The Alpha Delta Gamma man's responsibility is constant. Alpha Delta Gamma does not exist in a vacuum, isolated from the needs of the world. To the contrary, we are devoted to "the world of men around us." The end of all responsibility is, of course, that end for which the Fraternity labors - Ad Dei Gloriam. The Fraternity is a Christian organization dedicated to serving God by serving man. The Alpha Delta Gamma man, then, must unite the Fraternity in this purpose and serve man so that the Fraternity, through its members, will indeed work for the glory of God.
We must also maintain a reasonable balance of responsibility so that one area is not stressed at the expense of another. To aid in our development, the Fraternity encourages us to adopt what we refer to as the "sacrament of the moment." This means simply doing the proper thing at the proper time - whether it be studying, playing or praying. By adhering to what we are supposed to be doing at all times, we will truly become the responsible man our Fraternity expects of us.


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Athlete swimmers from a group of colleges in Upstate New York may be at a severe disadvantage at this month's NCAA championship in North Carolina thanks to a 2016 executive order signed by Governor Andrew Cuomo in response to the southern state's transgender "bathroom ban" that was repealed two years ago. Over a dozen students from across three schools under the State University of New York (SUNY) system qualified for the Division III Swimming Championship in Greensboro, North Carolina, but under the current order from Albany, which bans publically funded travel to the Tar Heel State, the team will have to stay over an hour away from the competition site in neighboring Virginia.
A conservative group that filed a complaint with the FEC alleging Rep. Alexandria Ocasio-Cortez's campaign may have illegally funneled thousands of dollars through an allied PAC to boyfriend Riley Roberts on Thursday lodged a fresh complaint with the Office of Congressional Ethics.?
Google-owned YouTube is dealing with an exodus of some of its major advertisers in the wake of disturbing claims that pedophiles have not only been infiltrating the comments section of childrens videos, seemingly without consequence, until now, but that YouTube's own technology may have been facilitating this disturbing trend.
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text:
"Line 3 and 5 closed." Collaps of infrastructure and Suicidal trends
March 1, 2006 sigg3
That's the message roaming the loudspeakers when I was thrown off the train at Majorstua, one stop from my destination, the University. The cause? Too much snow. The people running things here have never conceived of snow as an actual weather condition, no, it's hypothetical. That's why they only have hypothetical means of making sure the routes go as they're supposed to. When fronting tons of snow, hypothetical means don't go further than a shrug and a pat on the back.
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We were going up this steep hill. There was four minutes left to my class would officially begin. I got my pace up and walked into the howling wind with a rather defying attitude. I thought of Alive and pondered on who of the two girls I'd have to eat first.
I got there ten minutes late.
When the lousy class was over, I ducked out from familiar faces, lit my cigarette and hurried down towards the trains. We'd gotten off kind of early and there was at least a hypothetical possibility of me reaching the first one going East. But as hypothesis go, this time the real world announced itself as uncompromising.
The schedule showed trains coming at 8 and 11 minutes past. It was 9 minutes past. "Still running a little late" I cursed to myself. You see, I take pride in ensuring that society runs the way I want to. My pride had completely vanished by 15 minutes past when the loudspeaker announced "an alternative schedule" for the day. What he was saying was that for people riding lines 3 and 5, which essentially was me, the trains would go to Majorstua but not further.
Cheers, mate!
I was getting awfully hungry, and I started to look around to see which of the other would-be passagers to eat. There was this fat woman in a fur coat, and I imagined that if you pulled off the coat, there would be some tender meat underneath I could sink my teeth in. And I could simply pull on her fur coat to protect myself from the agonizing cold, like Luke and Han Solo did at the Battle of Hoth.
Eventually the message started to sink in.
Despite the fact of us standing here there would be no train.
I leaped, or tumbled, into action and started walking. Again.
It was kind of nice walking through the semi sub-urban community that us frontier men and women had to tangle. At least when I walked, it didn't feel like I was going to have to amputate both of my legs after all. And I thought of the silly sods still waiting for the train. Stupid and worthless. It warmed my heart.
When I finally got to Majorstua they had come to terms with the fact that people wanted an explanation for why the trip they had paid for wasn't going to happen. The loudspeaker said that in addition to the snow there had been an incident, he actually used the word event, at the station west of the University. They also call it a personal tragedy. Personal tragedy my ass. It's a public tragedy! Why can't suicidal maniacs at least have the decency to just hang themselves? Why do they have to be all dramatic? It's just dying for chrissakes, it's not like your stopping the universe and time itself, so please leave those of us who's not going to die today alone. Yes, yes, we understand that your life was screwed, but you're not really doing much to deal with it now, are you?
This is collective transportation, okay?
Not singular or for yourself alone. That means we're in this together, and when you absolutely have to go and kill yourself using collective transportation, you're bringing down the infrastructure. You're making sure people don't get to their job interviews, that the happy fiancé never reaches the church in time, and that the pregnant chick never gets to the hospital – making a mess in some poor Pakistani's cab. What's wrong with a good old-fashioned hanging yourself, my boy? Now that's a way to go! It's not unoriginal, it's a classic! And it doesn't ruin the plans of those of us who are actually trying to make a better life for ourselves. Just do it.
But I finally got my ride homewards, without having to take a taxi. There's this "guarantee" that if the train is late, you can take a cab and send a receipt to the train company, who will look into your story. I don't need a year-long process telling me that they are insured against weather and other acts of gods. Besides I can't really prove that I lost anything of financial value. The only thing I lost was my lunch, adding to my growing appetite of destruction, and I lost my patience and I lost a few pounds running through the snow-covered outback of westside Suburbany. Not bad for a Wednesday. Can't wait till my next class tomorrow.
6 − = five

redpajama_set_name: RedPajamaCommonCrawl


text:
Microsoft/
The new Mac vs. PC war is all about touch
By Tom Warren / @tomwarren
Oct 27, 2016, 9:05 PM UTC |
It's been brewing for a while, but the Mac vs. PC war is back. Apple and Microsoft have shared various battles over many years, in an ongoing rift over the direction of computing, but this week has marked a switch in both of their war strategies. Microsoft is now going after Apple's core customers, and the two are split over the role of touch and input interfaces for PCs.
The fundamental differences and disagreements between the pair of tech giants can be tracked back to the hardware and software products they both produce. Apple separates its tablets and phones with the iOS operating system, and its range of Macs running macOS. Microsoft on the other hand believes in having one operating system, Windows 10, to rule them all and work across phones, PCs, tablets, and even devices like the Xbox One and HoloLens. The strategies are remarkably different.
Microsoft and Apple went head-to-head this week
This week, both Microsoft and Apple held events back-to-back, providing us with a rare opportunity to compare their latest takes on modern computing all in less than 24 hours. Microsoft spent the first part of its keynote demonstrating new Windows 10 features, a new Paint app, and its VR strategy, before the big news of a Surface Studio all-in-one PC. Apple spent the first part of its keynote highlighting pictures from its iPhone 7, and launching a new TV app for its Apple TV, before moving onto the unveiling of its new MacBook Pro lineup.
Apple doesn't believe in touchscreen Macs, but it did add a touchscreen of sorts to its MacBook Pro, just not where you'd expect to find one: the Touch Bar, an OLED touchscreen, replaces the function keys at the top of the keyboard. It includes Touch ID and is easily accessible, customizable, and has the relevant controls displayed dynamically in each app. If you're in Photoshop you'll get tools for manipulating images, and in iTunes you'll get all the music controls you need.
It's too early to judge whether Apple's Touch Bar will be beneficial or just a gimmick, but it's an entirely different approach from just adding a touchscreen to a MacBook. Apple is expecting developers to create contextually aware controls for their apps, and for MacBook users to learn these new controls. There's a clear value here and an interesting take on touch input, and Apple isn't shying away from ripping out old function keys and replacing them with something entirely new.
Apple's Touch Bar has many challenges it needs to meet
The challenges are clear, too. Apple has a lot to prove here, and the company's demos showed controls that could have easily been achieved from using existing keyboard shortcuts or just the trackpad. Apple highlighted emoji input at one point, which seemed like an unusual aside for a laptop that's designed for professionals who care about the speeds and feeds, not gimmicks. Apple enlarged the trackpad on its MacBook Pro, too, but it barely noted the change or explained why bigger is necessarily better. It now looks like a miniature Wacom tablet, but Apple didn't do something wild like getting the Pencil working with it. Touch Bar aside, it was a cautious refresh to the MacBook Pro line, with modern additions instead of big bets.
Microsoft's approach to its Surface Studio was the type you would normally expect from Apple. The sleek product videos, the focus on power, and a new theme of focusing on creatives. A new Surface Dial forms the input mechanism for the Surface Studio, and the whole device is geared toward professionals and creators. It's the type of market that Apple typically dominates, but Microsoft is trying everything to get these users to switch to Windows.
Microsoft still needs to convince everyone that touch PCs are the future
Microsoft has a lot to prove, too. The Surface Studio is a niche product, and the Surface Dial will require developers to build contextual menus into their apps to really make use of the new hardware. Apple is betting on a touch strip interaction model, and Microsoft wants you to wear headsets and create 3D objects, or place crazy puck-like devices on a Surface Studio display. It's tough to compare a laptop directly to a desktop, but it further highlights Apple's resistance to touchscreens on Macs, and Microsoft's new focus on stealing some of Apple's usual customers. The two tech giants are split by fundamental disagreements about the new methods to provide input to modern PCs.
There's a lot of work for both companies to achieve what they've demonstrated this week. This war is far from over, and it's impossible to predict a winner just yet. Apple might not like touchscreens on Macs, but the company has shown it's willing to compete with Microsoft's Surface devices with an iPad Pro. Now Microsoft is showing it's willing to go the extra mile to try and tempt a niche market of professionals over to the Windows world. If you thought the Mac vs. PC days were long gone, think again. They're just about to jump back to life, and customers will elect the winner.
Microsoft Surface Studio PC first look
More from Apple
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Apple's Studio Display costs a slightly less ridiculous $1,299 at Amazon
Quick fixes: how to transfer iPhone videos to a Windows PC when it keeps disconnecting


text:
136th Birth Anniversary of Don Tomas Remigio, co-founder of the KKK CO-FOUNDING the Katipunan (Kataastaasan, Kagalang-galangan na Katipunan ng mga Anak ng Bayan) and helping to make it the most powerful weapon of his downtrodden countrymen was the greatest contribution to the Filipino struggle for freedom of Don Tomas Remigio. One of the intrepid Filipino patriots who founded the Katipunan on July 7, 1892, he established the first Katipunan branch – Balangay Trozo – in his town of Tondo, Manila. Inspired by this example, fellow Katipuneros followed and established Katipunan branches in many places. Thus, when the colonial authorities discovered the KKK, they were astounded to know that it already had some 30,000 armed adherents.
Born in Barrio Masambong, Sampaloc, Manila, on March 7, 1867, Don Tomas' education and upbringing led him to join patriotic organizations. He first affiliated with the reformist group of Marcelo H. del Pilar and Dr. Jose P. Rizal. In this group, Don Tomas was a zealous member in spreading the reformists' works which helped, in no small measure, in awakening the socio-political consciousness of the Filipino people.
The outbreak of the 1896 Revolution saw Don Tomas active in the battlefield. With the rank of Captain of infantry, he served throughout the revolution against Spain and against the United States in the Filipino-American War.
The end of the war found him zealously carrying on the task of advancing the cause of his country and fellow-Filipinos. Also a good writer, Don Tomas devoted his remaining years to writing on themes that promoted the socio-political consciousness of his countrymen. Luha ng Ina, Ang Isang Haka ni Rizal, Malaya, and Hweteng were some of his wellknown works. Don Tomas Remigio died on August 1, 1916.
On January 6, 1953, the City Council of Manila renamed Tondo Elementary School to Don Tomas Remigio Elementary School in recognition of his many contributions and sacrifices to the Filipino nation.
Encouraging the Filipino youth to protect the Philippine freedom is one laudable way of preserving the memory of Don Tomas Remigio and those of his "Generation of 1896," the Filipino people's paladins of Philippine freedom.
Source: The Manila Bulletin Online March 3, 2003.


text:
Àtal fou un cap dels agrians en temps d'Alexandre el Gran. Va dirigir als agrians que combatien al costat d'Alexandre en les batalles d'Issos (333 aC) i Arbela (331 aC).

Referències

Prínceps

redpajama_set_name: RedPajamaWikipedia


text:
# Wed 16 Sep, 10:00
Radomlje 2 - 0 Rudar Velenje
Radomlje vs Rudar Velenje Summary
Home Win
Away Win
Home Lost
Away Lost
Radomlje 1 15 6 16 5 120% 480% 00% 1100% 00%
Rudar Velenje 9 7 6 7 9 150% 150% 116% 266% 133%
Radomlje vs Rudar Velenje head 2 head
16 Sep 10:00 10:00, 16 Sep fin
Rudar Velenje
27 May 10:55 10:55, 27 May fin
02 Apr 12:00 12:00, 02 Apr fin
25 Nov 11:00 11:00, 25 Nov fin
2. Division league table
1 Radomlje 2 1 0 1 3 2 3 4 4 0 0 13 3 12 6 5 0 1 16 5 15 11 2.70
2 NK Krka 4 2 2 0 10 4 8 2 2 0 0 5 0 6 6 4 2 0 15 4 14 11 2.50
3 NK Nafta 4 4 0 0 10 2 12 2 0 2 0 2 2 2 6 4 2 0 12 4 14 8 2.00
4 NK Dob 2 2 0 0 7 2 6 4 2 0 2 9 8 6 6 4 0 2 16 10 12 6 2.70
5 ND Bilje 3 2 0 1 4 6 6 3 2 0 1 6 3 6 6 4 0 2 10 9 12 1 1.70
6 NK Brezice 3 1 0 2 3 7 3 3 2 1 0 3 0 7 6 3 1 2 6 7 10 -1 1.00
7 Beltinci 3 1 0 2 7 5 3 3 2 0 1 5 5 6 6 3 0 3 12 10 9 2 2.00
8 Fuzinar 4 1 1 2 9 5 4 2 1 1 0 4 3 4 6 2 2 2 13 8 8 5 2.20
9 Rudar Velenje 3 1 0 2 2 3 3 3 1 1 1 5 6 4 6 2 1 3 7 9 7 -2 1.20
10 Primorje 4 2 1 1 5 4 7 2 0 0 2 1 7 0 6 2 1 3 6 11 7 -5 1.00
11 Krsko 3 1 1 1 5 7 4 3 1 0 2 2 5 3 6 2 1 3 7 12 7 -5 1.20
12 Triglav Kranj 2 1 0 1 4 2 3 4 1 0 3 2 5 3 6 2 0 4 6 7 6 -1 1.00
13 Smartno 1928 3 0 0 3 2 9 0 3 2 0 1 4 4 6 6 2 0 4 6 13 6 -7 1.00
14 Jadran Dekani 2 0 1 1 0 1 1 4 1 1 2 6 5 4 6 1 2 3 6 6 5 0 1.00
15 NK Drava PTUJ 3 0 0 3 3 8 0 3 1 0 2 5 7 3 6 1 0 5 8 15 3 -7 1.30
16 Brda 3 0 0 3 0 7 0 3 1 0 2 2 11 3 6 1 0 5 2 18 3 -16 0.30
Radomlje Last Games
- - - - - 0 0 - - - - - 0 0 + - - + + + 0 0 ? ? ? ? ? ? ? - 0 + - + - + + + +
Triglav Kranj
Krsko
Jadran Dekani
29 Aug 11:00 11:00, 29 Aug fin
NK Dob
NK Brezice
Smartno 1928
ND Bilje
10 Jun 11:30 11:30, 10 Jun fin
HT: 0-1 FT: 1-1 ET: 1-1 PN: 2-4
NK Nafta 2
Rudar Velenje Last Games
0 ? - - 0 ? 0 + + - - - 0 0 - ? ? ? ? ? ? ? ? ? ? ? + + - - - - - - - 0 - - 0 0 - + - - +
NK Brezice 1
Rudar Velenje 1
ND Bilje 1
NK Krka
22 Jul 12:00 12:00, 22 Jul fin
Radomlje Results
Rudar Velenje Results
Radomlje Goals
Rudar Velenje Goals
Radomlje Strength
calculated by the robot
Rudar Velenje Strength
Radomlje Goals (Last 30 games)
Total Scored: 52. Total Conceded: 49
Rudar Velenje Goals (Last 30 games)
Radomlje Scorers (Last 15 games)
Matic Ahacic 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2
Grega Marinsek 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2
Anze Pisek 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 2
Zan Nikolic 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2
Luka Cerar 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 2
Dusan Ivanov 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Anel Hajric 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Nerman Basic 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
David Laban 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Zan Poglajen 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Lan Gojak 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Sacha Varga 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
Luka Gucek 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
Jaka Stromajer 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
Rudar Velenje Scorers (Last 15 games)
Andrej Panadic 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Sandro Jovanovic 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Mateo Panadic 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
Vosnjak Amadej 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
Radomlje Bad Boy (Last 30 games)
Rudar Velenje Bad Boy (Last 30 games)
Unknown 23.08.2020
Game Prediction
calculated by the bots
Bot2
Radomlje 1.67 1.33 2.90 1.92 1.75
Rudar Velenje 1.50 1.33 0.86 0.73 1.13
Radomlje vs Rudar Velenje
The last meeting of these teams took place more than a month ago (27 May 2017) and ended with the draw with a score 0-0.
The most common result of matches between these teams is 2-0. 2 matches have ended with this result.
The distance between the teams in the league table is 8 points.
Radomlje is the leader of the tournament with 15 points. This is the 6th of 30 team game in the tournament.
The nearest pursuer is 1 point lower and has the opportunity to score 72 points additionally. The goal difference is 16:5. In additional, Radomlje has the best offensive record in the league. The Radomlje defensive line is one of the TOP-4 in the tournament.
Rudar Velenje takes the 9th position in the league table with 7 points. This is the 6th of 30 team game in the tournament.
The difference with the nearest team on top is 1 point (Rudar Velenje can score 72 points). The distance from the bottom league record is 4 points. The nearest pursuer has the same number of points with Rudar Velenje and can score 72 points. The goal difference is 7:9.
Radomlje Stats
Radomlje has won all of 5 recent matches.
The team scored 17 points оn its own field in 10 last matches, with the following results 5 wins, 2 draws and 3 losses. The difference between scored and conceded goals is 15-14. The average number of goals is 1.5. The average number of cards in 10 last matches is 2.7.
Leading players in the last 15 matches in all tournaments: Matic Ahacic (2), Grega Marinsek (2), Anze Pisek (2), Zan Nikolic (2), Luka Cerar (2), Dusan Ivanov (1), Anel Hajric (1).
Rudar Velenje Stats
The team scored 8 points when playing away in 10 last matches, with the following results 2 wins, 2 draws и 6 losses. The difference between scored and conceded goals is 8-21. The average number of goals is 0.8. The average number of cards in 10 last matches is 2.6.
Leading players in the last 15 matches in all tournaments: Andrej Panadic (1), Sandro Jovanovic (1), Mateo Panadic (1), Vosnjak Amadej (1).


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Service: A quick delivery and a very pleasing product bought in the sale.
Product: Really comfy padded socks . Wash well and a quick know delivery.


text:
Switzerland to pull plug on FM radio by 2023
Just 13% of the Swiss population still listened to radio on FM in 2020, a survey shows. © Keystone / Petra Orosz
The radio industry has reached an agreement on turning off its analogue FM transmitters, with the public broadcaster expected to make the definitive switch to digital in mid-2022. Private radio stations will follow suit by January 2023.
This content was published on January 4, 2021 - 17:06 January 4, 2021 - 17:06
Keystone-SDA/swissinfo.ch/gw
日本語 (ja) スイス、2023年までにFMラジオ放送を廃止
Português (pt) Suíça vai desligar rádio FM para sempre em 2023
The end of very high frequency radio waves (VHF-FM) will occur earlier than planned, the Federal Office of Communications (OFCOM) has said. Initially the authorities had announced the switch to digital audio broadcasting, or DAB, for the end of 2024.
A survey conducted in 2020 showed that just 13% of the Swiss population still listened to FM radio, OFCOM said. Digital radio use, on the other hand, increased by 22 percentage points between 2015 and 2020. DAB+ offers better sound quality and broader coverage areas and is already the most widely used transmission channel for radio services in the country, according to the communications authority.
Hear Swiss stories for the world on The Swiss Connection podcast
This content was published on Jan 19, 2021 Jan 19, 2021 If a friend abandoned you in a foreign country, how angry would you be? And how healthy is it to dwell on the past?
Last year OFCOM launched a CHF5 million ($5.67 million) multi-year public information campaign to prepare listeners for the switch to digital radio. The government created the legal framework for the end of FM radio back in 2017 – the same year Norway became the first country in the world to end national broadcasts on FM – and has provided financial support to radio broadcasters for making the transition to digital.
When the world got its news from shortwave radio
This content was published on Nov 3, 2019 Nov 3, 2019 What did swissinfo.ch sound like for the first seven decades of its existence? The short answer: a radio station.
When Switzerland broadcast Esperanto around Europe
This content was published on Sep 1, 2019 Sep 1, 2019 From 1946 to 1992 swissinfo.ch's predecessor broadcast programmes in Esperanto. Delving into the archives, we look at why – and why they stopped.
What does diversity mean for public service media?
This content was published on Feb 13, 2020 Feb 13, 2020 For World Radio Day, we joined forces with other public service media to discuss what role diversity has in the content we publish and in how we work.
TV and radio still more popular than online media
This content was published on Nov 21, 2018 Nov 21, 2018 Streaming services like Netflix and Spotify are increasingly popular in Switzerland, but television and radio remain the most popular media.


text:
Q: finding spectral radius of the jacobi iteration matrix I am using Matlab to find the spectral radius of the Jacobi iteration matrix where A=[4 2 1;1 3 1;1 1 4].
I can't seem to input the correct commands to get the size of the error after 5 iterations. Can someone help me?
Here are a list of commands that I put into Matlab so far:
A=[4 2 1;1 3 1;1 1 4]

A =

 4     2     1
 1     3     1
 1     1     4

 D=diagonal(diagonal(A));L=(A,-1);U=(A,1);

 b=([3 -1 4])

 x0j=zeros([0 0 0]);

 x=D\(-(U+L)*x0j+b);r=b-A*x    %Jacobi iteration.
------------------------------------------------------------------------------
 Error using  *
 Inputs must be 2-D, o enter code here r at least one input must be scalar.
 To compute element wise TIMES, use TIMES (.*) instead.

A: The spectral radius of a matrix is the maximum of the modulus of its eigenvalues. It can be simply computed using max(abs(eig(·))).
However, as others have noticed, your whole code seems pretty mixed up and not actually valid Matlab code, so your problem is not really to compute the spectral radius, is it? The algorithm is very straightforward and easy to implement:
% diagonal part of A and rest
D = diag(diag(A));
R = A - D;

% iteration matrix and offset
T = - inv(D) * R;
C = inv(D) * b;

% spectral radius condition
rho = max(abs(eig(T)));
if rho >= 1
    error('no convergence')
end

% initial guess
x = randn(size(b));

% iteration
while norm(A * x - b) > 1e-15
    x = T * x + C;
end

Note that I used inv(D) to directly follow the description in Wikipedia, but the inverse of a diagonal matrix can be easily computed using diag(1 ./ diag(D)).
I don't really see why one would need to separate R into an upper and lower part. I suppose it has to do with numerical efficiency, but then, Matlab is a very efficient high-level language for matrix computations already. So actually there is no need to implement the Jacobi algorithm in it explicitly when you can simply write A \ b – except for educational purposes I guess.

redpajama_set_name: RedPajamaStackExchange