diff --git a/samples/texts/3727487/page_1.md b/samples/texts/3727487/page_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..b09813ecb4ad7f1c21c61aabfd0bab946de2321b
--- /dev/null
+++ b/samples/texts/3727487/page_1.md
@@ -0,0 +1,24 @@
+Majorana Fermions in Semiconductor Nanowires
+
+Tudor D. Stanescu,¹ Roman M. Lutchyn,² and S. Das Sarma³
+
+¹Department of Physics, West Virginia University, Morgantown, WV 26506
+
+²Station Q, Microsoft Research, Santa Barbara, CA 93106-6105
+
+³Condensed Matter Theory Center, Department of Physics,
+University of Maryland, College Park, MD 20742
+
+(Dated: November 17, 2011)
+
+We study multiband semiconducting nanowires proximity-coupled with an s-wave superconductor and calculate the topological phase diagram as a function of the chemical potential and magnetic field. The non-trivial topological state corresponds to a superconducting phase supporting an odd number of pairs of Majorana modes localized at the ends of the wire, whereas the non-topological state corresponds to a superconducting phase with no Majoranas or with an even number of pairs of Majorana modes. Our key finding is that multiband occupancy not only lifts the stringent constraint of one-dimensionality, but also allows having higher carrier density in the nanowire. Consequently, multiband nanowires are better-suited for stabilizing the topological superconducting phase and for observing the Majorana physics. We present a detailed study of the parameter space for multiband semiconductor nanowires focusing on understanding the key experimental conditions required for the realization and detection of Majorana fermions in solid-state systems. We include various sources of disorder and characterize their effects on the stability of the topological phase. Finally, we calculate the local density of states as well as the differential tunneling conductance as functions of external parameters and predict the experimental signatures that would establish the existence of emergent Majorana zero-energy modes in solid-state systems.
+
+PACS numbers:
+
+# I. INTRODUCTION
+
+The search for Majorana fermions has become an active and exciting pursuit in condensed matter physics [1-4]. Majorana fermions, particles which are their own antiparticles, were originally envisioned by E. Majorana in 1937 [5] in the context of particle physics (i.e., the physics of neutrinos). However, the current search for Majorana particles is mostly taking place in condensed matter systems [6, 7] where Majorana quasi-particles appear in electronic systems as a result of fractionalization, and as emergent modes occupying non-local zero energy states. The non-locality of these modes provides the ability to exchange and manipulate fractionalized quasiparticles and leads to non-Abelian braiding statistics [8-14]. Hence, in addition to being of paramount importance for fundamental physics, this property of the Majoranas places them at the heart of topological quantum computing schemes [13, 15-29]. We mention that solid-state systems, where the Majorana mode emerges as a zero energy state of an effective (but realistic) low-energy Hamiltonian, enable the realization of the Majorana operator itself, not just of the Majorana particle. Consequently, Majorana physics in solid-state systems is in fact much more subtle than originally envisioned by E. Majorana in 1937. For example, in condensed matter systems the non-local non-Abelian topological nature of the Majorana modes that are of interest to us is a purely emergent property.
+
+About ten years ago, Read and Green [9] discovered that Majorana zero-energy modes can appear quite naturally in 2D chiral p-wave superconductors where these
+
+quasiparticles, localized at the vortex cores, correspond to an equal superposition of a particle and a hole. A year later, Kitaev [11] introduced a very simple toy model for a 1D Majorana quantum wire with localized Majorana zero-energy modes at the ends. Both these proposals involve spinless p-wave superconductors where one can explicitly demonstrate the existence of Majorana zero-energy modes by solving the corresponding mean field Hamiltonian. Recently, several groups [30, 31] suggested a way to engineer spinless p-wave superconductors in the laboratory using a combination of strong spin-orbit coupling and superconducting proximity effect, thus opening the possibility of realizing Majorana fermions in solid-state systems to the experimental field. The basic idea of the semiconductor/superconductor proposal [31] is that the interplay of spin-orbit interaction, s-wave superconductivity and Zeeman spin splitting could, in principle, lead to a topological superconducting phase with localized zero-energy Majorana modes in the semiconductor. Since then, there have been many proposals for realizing solid-state Majoranas in various superconducting heterostructures [30-45]. Among them, the most promising ones involve quasi-1D semiconductor nanowires with strong spin-orbit interaction proximity-coupled with an s-wave superconductor [37, 38, 40]. The main advantage of this proposal is its simplicity: it does not require any specialized new materials but rather involves a conventional semiconductor with strong Rashba coupling such as InAs or InSb, a conventional superconductor such as Al or Nb, and an in-plane magnetic field. High quality semiconductor nanowires can be epitaxially grown (see, for example, Ref. [46] for InAs and Ref. [47] for InSb) and are known to have a large spin-orbit interaction strength
\ No newline at end of file
diff --git a/samples/texts/3727487/page_10.md b/samples/texts/3727487/page_10.md
new file mode 100644
index 0000000000000000000000000000000000000000..1df6e46eb866b560fac17e24dddb42eb1970f493
--- /dev/null
+++ b/samples/texts/3727487/page_10.md
@@ -0,0 +1,9 @@
+FIG. 6: (Color online) Upper panel: Low-energy spectra for weak SM-SC coupling ($\bar{\gamma} = 0.25\Delta_0$) in the vicinity of the “sweet spot” ($\mu = 14.5E_{\alpha}$, $\Gamma = 15.3E_{\alpha}$). The yellow circles correspond to inhomogeneous coupling with $\theta = 0.8$ and takes into account dynamical effects, the black squares are obtained by neglecting dynamical effects, i.e., $(Z^{1/2})_{n_y n'_y} = \delta_{n_y n'_y}$, and the red diamonds are for a homogeneous coupling. Lower panel: Same as in the upper panel for $\bar{\gamma} = \Delta_0$ near the “sweet spot” ($\mu = 14.5E_{\alpha}$, $\Gamma = 24.2E_{\alpha}$). Note that inclusion of dynamical effects through $Z^{1/2}$ renormalizes the energy scales, while inhomogeneous coupling play a critical role in establishing a finite gap near the “sweet spot”. The location of a given “sweet spot” depends on the coupling strength.
+
+FIG. 7: (Color online) Sequence of low-energy spectra obtained for four different values of the Zeeman field separated by points with a vanishing mini-gap. The spectra with an odd number of pairs of zero-energy modes (N) characterize topological SC phases, while those with an even N correspond to trivial SC phases. Note the overall decrease of the mini-gap with the Zeeman field. The system is characterized by the following parameters: $\mu = 30E_{\alpha}$, $\bar{\gamma} = 0.25\Delta_0$, and $\theta = 0.8$.
+
+spectrum for a system with a fixed chemical potential, $\mu = 30E_{\alpha}$, and different values of $\Gamma$, one from each of four intervals separated by points characterized by a vanishing gap. The results are shown in Fig. 7. The distinctive feature of the spectra in Fig. 7 is their number of zero-energy modes. In the presence of a very weak Zeeman field, the SC phase has trivial topology and no zero-energy modes. Increasing $\Gamma$ generates a transition to a topological SC phase characterized by one pair of Majorana bound states. The further increase of the applied Zeeman field produces alternating topologically
+
+trivial and nontrivial SC phases with increasing number of zero-energy bound states. Topological SC phases are characterized by an odd number of pairs of Majorana bound states, while trivial SC phases have an even number of pairs.
+
+This type of sequence of alternating SC phases is independent of the chemical potential or the strength of the SM-SC coupling. As discussed above, in the weak-coupling limit $\bar{\gamma} \to 0$ this behavior can be directly related to the number of sub-bands of an infinite wire that cross the chemical potential: an odd (even) number corresponds to a nontrivial (trivial) topological superconductor. At the phase boundary between two SC phases with different topologies, two sub-bands become degenerate at $k_x = 0$. In addition, at certain special values of the control parameters $\Gamma$ and $\mu$ two different phase boundaries intersect leading to multicritical points. We will call these special crossing points x-points. The “sweet spots” mentioned above are examples of such x-points. The key question is how the phase boundaries evolve when we turn on the SM-SC coupling and, in particular, what the physics is in the vicinity of the x-points, see also Ref. [52]. To obtain the global phase diagram in the $\Gamma - \mu$ plane,
\ No newline at end of file
diff --git a/samples/texts/3727487/page_11.md b/samples/texts/3727487/page_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..a8804596d04b693f8d46cb6393168b9e6e385afe
--- /dev/null
+++ b/samples/texts/3727487/page_11.md
@@ -0,0 +1,9 @@
+FIG. 8: (Color online) Phase diagram of the multiband nanowire as function of the Zeeman field $\Gamma$ and the chemical potential $\mu$. The quasi-particle gap $\Delta_{\infty}^*$ of the effective low-energy Hamiltonian (26) vanishes at the phase boundaries. Superconducting phases characterized by an odd (even) number of pairs of zero-energy Majorana modes are topologically nontrivial (trivial). The coupling of the SM nanowire to the s-wave SC is characterized by $\bar{\gamma} = 0.25\Delta_0$ and $\theta = 0.8$. The inhomogeneous coupling induces off-diagonal pairing $\Delta_{n_y n'_y}$ which removes the x-points creating regions with stable non-trivial (near the “sweet spots”) or trivial SC.
+
+we determine the parameters that satisfy the condition $\Delta_{\infty}^{*}(\Gamma, \mu) = 0$, i.e., we identify the phase boundaries for transitions between topologically trivial and non-trivial phases by imposing the condition of a vanishing quasi-particle gap. We show below that this approach is consistent with the calculation of the topological index $M$ (Majorana number) which distinguishes topologically trivial and non-trivial SC phases [11, 40]. The results are shown in Fig.8 for weak coupling ($\bar{\gamma} = 0.25\Delta_0$) and in Fig.9 for intermediate coupling ($\bar{\gamma} = \Delta_0$).
+
+For $\bar{\gamma} = 0.25\Delta_0$ (see Fig. 8), the only significant difference as compared to the weak-coupling picture presented above is the disappearance of the phase boundary crossings at the x-points. Instead, the region in the vicinity of these points is occupied by the phase that is robust against variations of the chemical potential. Near the “sweet spots”, this phase is the nontrivial topological SC. We note that the disappearance of phase boundary crossings is a direct result of the off-diagonal pairing induced by a non-uniform SM-SC coupling. Uniform tunneling ($\theta = 0$) does not eliminate the x-points, independent of the coupling strength, but pushes them to higher values of the Zeeman field as $\bar{\gamma}$ increases. Also, we note that the characteristic width of the stable phase in a given avoided
+
+FIG. 9: (Color online) Phase diagram of the multiband nanowire at intermediate coupling. The SM-SC coupling is $\bar{\gamma} = \Delta_0$, while the other parameters are the same as in Fig. 8. Note that the “sweet spots” of the topological phase characterized by $N=1$ are significantly expanded as compared with the weak-coupling case shown in Fig. 8. As a result, tuning the Zeeman field in the vicinity of $\Gamma = 30E_{\alpha}$ allows for huge variations of the chemical potential without crossing a phase boundary, i.e., without closing the gap.
+
+crossing region is controlled by a specific off-diagonal component $\Delta_{n_y, n'_y}$. For example, the “sweet spots” inside the phase characterized by $N=1$ (see Fig. 8) are controlled by the dominant matrix elements $\Delta_{n_y, n_y+1}$ (see Fig. 3), while the avoided crossings within the $N=2$ topologically trivial phase are controlled by matrix elements $\Delta_{n_y, n_y+2}$, that are typically smaller. Hence, we expect the strongest effect within the topological phase characterized by one pair of Majorana bound states. As this phase also requires relatively low Zeeman fields, it is the experimentally relevant phase for realizing and observing Majorana fermions. At intermediate couplings, the $N=1$ phase is pushed to slightly higher fields (see Fig. 9). However, as a result of the effective phase space of the “sweet spots” expanding significantly, this regime presents the remarkable possibility of being able to vary the chemical potential over energy scales of the order 10meV without crossing a phase boundary. As we will show below, this feature has major experimental implications in the sense that the elusive Majorana mode is most likely to be experimentally realized in the laboratory in this particular physically realistic parameter regime. We note that this interesting parameter regime exists for the multiband situation.
\ No newline at end of file
diff --git a/samples/texts/3727487/page_12.md b/samples/texts/3727487/page_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..595619fb9d7d9a1ada2a0576c9c41dfde40a2798
--- /dev/null
+++ b/samples/texts/3727487/page_12.md
@@ -0,0 +1,11 @@
+FIG. 10: (Color online) Dependence of the gap on the Zeeman field for a fixed chemical potential. The upper panel corresponds to weak SM-SC coupling with $\bar{\gamma} = 0.25\Delta_0$ and $\theta = 0.8$, while the lower panel is obtained for an intermediate coupling with $\bar{\gamma} = \Delta_0$ and $\theta = 0.8$. Within the red (darker gray) regions superconductivity is topologically trivial, while in the yellow (light gray) regions the system is topologically nontrivial. These curves correspond to horizontal cuts in the phase diagrams shown in figures 8 and 9, respectively, through a sweet spot of the phase $N=1$. Note that in the limit $\theta \to 0$, i.e., for uniform SM-SC coupling, the width of the lower field topologically nontrivial region shrinks to zero for both coupling strengths.
+
+**C. Dependence of the gap on the Zeeman field and the chemical potential**
+
+The phase diagram provides information about the topological nature of the phase characterized by given sets of control parameters ($\Gamma, \mu$). However, we are ultimately interested in the stability of the Majorana bound states that occur at the ends of a nanowire within the topologically nontrivial phase. As the Majorana zero-energy modes are protected by the quasi-particle gap, knowing the size of the gap and the dependence of $\Delta_{\infty}^*$ on the control parameters is critical. To address this issue, we determine the dependence of the gap on both the Zeeman field at fixed chemical potential and on $\mu$ at fixed $\Gamma$. The results are shown in figures 10 and 11. In general, the gap is non-zero everywhere except at points corresponding to phase boundaries. In the vicinity of a point $(\Gamma_c, \mu_c)$ with $\Delta^*(\Gamma_c, \mu_c) = 0$ the minimum gap in an infinitely long wire occurs at $k_x = 0$. The dependence of this minimum on the Zeeman field is approximately linear in the deviation from the critical field, $|\Gamma - \Gamma_c|$ (see Fig. 10). This generalizes the single-band results
+
+FIG. 11: (Color online) Dependence of the quasi-particle gap on the chemical potential for a fixed Zeeman field. The upper panel corresponds to weak coupling, $\bar{\gamma} = 0.25\Delta_0$, and $\Gamma = 25E_\alpha$, while the lower panel is obtained at intermediate coupling, $\bar{\gamma} = \Delta_0$, and $\Gamma = 32E_\alpha$. The curves represent vertical cuts through the $N=1$ topological phase in the phase diagrams shown in figures 8 and 9. At intermediate coupling (lower panel) the gap is finite over the entire chemical potential range, making the topological SC phase in a system with average chemical potential $\bar{\mu} \approx 60E_\alpha$ is robust against variations of the chemical potential of the order $\delta\mu = 5$meV.
+
+shown in Fig. 2. Note that outside this linear regime, the minimum gap occurs at finite wave vectors. Finally, we note that, at intermediate couplings, the Zeeman field can be tuned so that the gap remains finite over a large range of chemical potentials (Fig. 11, bottom panel). Such regimes are extremely stable against fluctuations of the chemical potential produced by disorder or other perturbations, as we will show explicitly in the next section. We emphasize that the critical ingredients for realizing this regime are: i) the off-diagonal pairing obtained by a non-uniform SM-SC coupling, and ii) an effective average coupling $\bar{\gamma}$ of the order of the bare SC gap $\Delta_0$. Note that the coupling strength $\bar{\gamma} \propto \tilde{t}^2/\Lambda L_z^3$ can be controlled by either modifying the tunneling $\tilde{t}$, or by changing the width $L_z$ of the nanowire in the direction perpendicular to the interface with the superconductor.
+
+As we mentioned above, the value of the gap in a finite system is, in general, smaller than the minimum gap in a system with the same parameters but no ends, e.g., with periodic boundary conditions. We emphasize that this is not a finite size effect, but is due to the appearance of in-gap states that are localized near the ends of the wire. In the topologically non-trivial phase, the characteristic length scale for these states is the same
\ No newline at end of file
diff --git a/samples/texts/3727487/page_13.md b/samples/texts/3727487/page_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..b42729168bb4884903275c128c38114247e10254
--- /dev/null
+++ b/samples/texts/3727487/page_13.md
@@ -0,0 +1,52 @@
+FIG. 12: (Color online) Top panel: Spectrum of a system characterized by $\mu = 54.5E_\alpha$, $\Gamma = 35E_\alpha$, $\bar{\gamma} = \Delta_0$, and $\theta = 0.8$. The states with $n = \pm 1$ are Majorana zero-modes, $n = 2, 3$ correspond to states localized near the ends of the wire, and states with $n \ge 4$ are extended states. The mini-gap is $\Delta^* = E_2$ and the minimum quasi-particle (QP) gap is $\Delta_\infty^* = E_4$. The finite-energy localized states can be viewed as precursors of the extra-pair of Majorana bound states that obtains for $\Gamma > 40E_\alpha$. Bottom panel: Dependence of the quasi-particle (QP) gap $\Delta_\infty^*$ and mini-gap $\Delta^*$ on the Zeeman field for a system with the same parameters as in Fig. 10. Note that $\Delta^* = \Delta_\infty^*$ in the vicinity of phase transition points. The quasi-particle gap $\Delta_\infty^*$ has the same values as in Fig. 10.
+
+as that of the Majorana zero modes and, for wires with
+lengths larger than this scale, their energy is indepen-
+dent of $L_x$. To illustrate this behavior, we show in Fig.
+12 (top panel) the spectrum of a system characterized by
+$\mu = 54.5E_\alpha$, $\Gamma = 35E_\alpha$, $\bar{\gamma} = \Delta_0$, and $\theta = 0.8$. We no-
+tice a pair of zero-energy Majorana modes and a number
+of states with almost the same energy that are extended
+over the entire length of the wire, as we checked explic-
+itly. In addition, we notice pair of states with energy
+within the bulk gap. An analysis of the position depen-
+dence of $|\psi(x)|^2$ reveals that these states are localized
+near the ends of the wire. The minimum energy of the
+extended states is equal to the quasi-particle gap $\Delta_\infty^*$,
+while the lowest energy of the bound states is equal to
+the mini-gap $\Delta^*$. We can understand these bound states
+as precursors of the extra-pair of Majorana zero-modes
+that characterize the topologically trivial phase that ob-
+tains for $\Gamma > 40E_\alpha$. Increasing the magnetic field pushes
+down the energy of the bound states until it vanishes at
+the transition, when $\Delta^* = \Delta_\infty^* = 0$. On the other side
+
+of the transition, an extra-pair of localized states will be
+characterized by zero energy (see Fig. 7). The evolu-
+tion of the mini-gap with the Zeeman field is shown in
+the lower panel of Fig. 12. Note that in the vicinity of
+the transition points $\Delta^* = \Delta_\infty^*$, while deep inside the
+topological phase $\Delta^* < \Delta_\infty^*$. Similar behavior can be
+observed throughout the phase diagram, including the
+topologically trivial phases.
+
+D. Phase diagram for an effective three–band model using the topological invariant.
+
+In this section we consider an effective three-band toy
+model which allows one to qualitatively understand sev-
+eral features observed in the detailed numerical simula-
+tions discussed in the previous sections.
+
+The topological phase diagram for the multiband nanowire can be obtained analytically using topological arguments due to Kitaev [11]. Namely, the superconducting phase hosting Majorana fermions has an odd fermion parity whereas the non-topological phase has even fermion parity. Thus, the phase diagram can be obtained by calculating the Z₂ topological index M (Majorana number) defined as
+
+$$
+\mathcal{M} = \operatorname{sgn}[PfB(0)]\operatorname{sgn}[PfB(\pi/a)] = \pm 1. \quad (31)
+$$
+
+The change of $\mathcal{M}$ signals the transition between trivial ($\mathcal{M} = 1$) and non-trivial phases ($\mathcal{M} = -1$). The anti-symmetric matrix $B$ in Eq.(31) represents the Hamiltonian of the system in the Majorana basis [11] and can be constructed by using the virtue of particle-hole symmetry [37, 61]. Specifically, the particle-hole symmetry of the BdG Hamiltonian is defined as
+
+$$
+\Theta H_{\text{BdG}}(p) \Theta^{-1} = -H_{\text{BdG}}(-p), \quad (32)
+$$
+
+where $\Theta = UK$ is an anti-unitary operator with $U$ and $K$ representing unitary transformation and complex conjugation, respectively. One can check that the matrix $B(P)=H_{\text{BdG}}(P)U$ calculated at the particle-hole invariant points where $H_{\text{BdG}}(P) = H_{\text{BdG}}(-P)$ is indeed anti-symmetric $B^T(P) = -B(P)$. In 1D there are two such points: $P = 0, \frac{\pi}{a}$ with $\frac{\pi}{a}$ being the momentum at the end of the Brillouin zone and $a$ being the lattice spacing. The function Pf in Eq. (31) denotes Pfaffian of the anti-symmetric matrix $B$. In the continuum approximation, where the lattice spacing $a \to 0$ and $P = \pi/a \to \infty$, the value of sgn[$PfB(\pi/a)$]=+1. Thus, the topological phase boundary given by the change in the topological index is determined by $PfB(0)$, and, thus, is accompanied by the gap closing at zero momentum. Note that the topological reconstruction of the spectrum is always accompanied by closing of the bulk gap [9] since $\det H_{\text{BdG}} = PfB^2$. Our approach of calculating the TP invariant relies on translational symmetry. Recently, the expression for the TP invariant was generalized to spatially inhomogeneous case [62, 63].
\ No newline at end of file
diff --git a/samples/texts/3727487/page_14.md b/samples/texts/3727487/page_14.md
new file mode 100644
index 0000000000000000000000000000000000000000..01fe73271a1375f47d95157e1f269341ac47d20b
--- /dev/null
+++ b/samples/texts/3727487/page_14.md
@@ -0,0 +1,19 @@
+We now calculate $PfB(0)$ for a simplified three-subband model and compare the phase diagram with the numerical one presented in the previous section. To make progress we assume that $\Delta_{i,i} = \Delta$, $\Delta_{i,i+1} = \Delta'$ and
+
+consider only diagonal in the subband index spin-orbit coupling terms in Eq. (4). With these approximations, $PfB(0)$ becomes
+
+$$ PfB(0) = (\delta E_{12}^2 (-V_x^2 + \Delta^2 + (\delta E_{13} - \mu)^2) (V_x^2 - \Delta^2 - \mu^2) - \delta E_{13}^2 (V_x^2 - (\Delta - \Delta')^2 - \mu^2) (V_x^2 - (\Delta + \Delta')^2 - \mu^2) \\ + (V_x^2 - \Delta^2 - \mu^2) (V_x^4 + (\Delta^2 - 2\Delta'^2)^2 + 2(\Delta^2 + 2\Delta'^2)\mu^2 + \mu^4 - 2V_x^2(\Delta^2 + 2\Delta'^2 + \mu^2)) \\ + 2\delta E_{13}\mu (V_x^4 + \Delta^4 - \Delta^2\Delta'^2 + 2\Delta'^4 + (2\Delta^2 + 3\Delta'^2)\mu^2 + \mu^4 - V_x^2(2\Delta^2 + 3\Delta'^2 + 2\mu^2)) \\ + 2\delta E_{12} (\delta E_{13}\mu (-V_x^2 + \Delta^2 + \Delta'^2 + \mu^2) + \mu(-V_x^2 + \Delta^2 + \mu^2)(-V_x^2 + \Delta^2 + 2\Delta'^2 + \mu^2) \\ + \delta E_{13}((V_x^2 - \Delta^2)\Delta'^2 + (2V_x^2 - 2\Delta^2 - 3\Delta'^2)\mu^2 - 2\mu^4)). \quad (33) $$
+
+Here $\delta E_{12}(3)$ represents the energy difference between first and second (third) subbands due to the transverse confinement. The superconducting gaps are related to the nominal bulk gap $\Delta_0$ via relations $\Delta = \gamma\Delta_0/(\gamma+\Delta_0)$ and $\Delta' = 0.25\gamma\Delta_0/(\gamma+\Delta_0)$ which take into account the dependence of the induced parameters on tunneling strength. Here we have chosen a reasonable ratio of $\Delta'/\Delta = 0.25$. Similarly, all other energies are renormalized in the following way $E \to E_{\gamma/\Delta_0}$ as explained in the previous sections.
+
+The phase diagram showing a sequence of topological phase transitions for the three sub-band toy model is shown in Fig. 13. The panels (a)-(c) represent the phase diagram with no interband mixing terms (i.e. $\Delta' = 0$). One can clearly see crossings in the phase diagram which represent the “sweet spots”. One can also notice the effect of the renormalization due to SM-SC tunneling as we increase $\gamma$ - the superconducting and normal terms are rescaled in a different way as explained above. In the weak-coupling regime $\gamma \ll \Delta_0$, the normal terms are not significantly renormalized since $\frac{\Delta_0}{\gamma+\Delta_0} \approx 1$ whereas induced pairing is small $\Delta \approx \gamma$ and is entirely determined by the normal state level broadening $\gamma$. On the other hand, in the strong-coupling regime $\gamma \gg \Delta_0$, the normal terms are decreasing with $\gamma$ because $\frac{\Delta_0}{\gamma+\Delta_0} \approx \Delta_0/\gamma \ll 1$ whereas $\Delta$ saturates at $\Delta_0$. Thus, strong SM-SC tunneling leads to important quantitative effect which should be taken into account in a realistic model for the proximity effect.
+
+The panels (d)-(f) represent the phase diagram with finite interband mixing terms $\Delta' \neq 0$. Here we find qualitative agreement with the numerical results presented in the previous sections, compare Figs. 8 and 9 with 13 (d) and (e). Different renormalization of the normal and SC terms has a two-fold effect on the phase diagram: a) with the increase of tunneling the topological phase is effectively “pushed” towards higher magnetic fields; b) even small interband pairing term opens a large gap at the sweet spot leading to extended vertical topological regions. This insensitivity of the topological phase
+
+FIG. 13: (Color online) Phase diagram of the multiband nanowire at different SM-SC tunneling strengths $\gamma$ and interband pairing $\Delta_{12} = \Delta_{23} = \Delta'$ calculated analytically for the effective three-band model. (a)-(c) correspond to $\gamma = 5, 20, 40$, respectively, and superconducting gaps $\Delta/E_\alpha = 4$ and $\Delta'/E_\alpha = 0$. (d)-(f) correspond to $\gamma = 5, 20, 40$, respectively, and superconducting gaps $\Delta/E_\alpha = 4$ and $\Delta'/E_\alpha = 1$. We used here $\delta E_{12}/E_\alpha = 30$ and $\delta E_{13}/E_\alpha = 80$.
+
+against chemical potential fluctuations can be exploited for the protection against disorder in the multisubband occupied nanowires with no such situation arising in the single channel case.
+
+### E. Phase diagram in the presence of a transverse external potential.
+
+A key feature of the phase diagrams shown above is represented by the hot spots. In the presence of intersub-band pairing, the topologically nontrivial phases expand in the vicinity of these hot spots, and, for cer-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_15.md b/samples/texts/3727487/page_15.md
new file mode 100644
index 0000000000000000000000000000000000000000..f5ebd809d5e8581c2559cb09bd2a17c22b904319
--- /dev/null
+++ b/samples/texts/3727487/page_15.md
@@ -0,0 +1,15 @@
+tain values of the Zeeman field, become stable over a wide range of values for the chemical potential. As we show in the subsequent sections, this property is critical for stabilizing the topological superconducting phase and, ultimately, for realizing and observing the Majorana zero modes. The necessary ingredient for obtaining this expansion of the hot spots is a non-vanishing inter-sub-band pairing, which is obtained by a non-uniform SM-SC coupling characterized by a strong position dependence along the direction transverse to the wire (see Fig. 3). The natural question is whether this effect can be obtained by breaking the symmetry in the transverse direction using an external field, e.g., generated by a gate potential, instead of a non-uniform coupling. This would constitute an alternative to engineering non-uniform SM-SC interfaces that would be much simpler to implement and would allow better control. To investigate this possibility, we consider an external potential that varies linearly across the nanowire,
+
+$$V_{\text{ext}}(y) = V_0(2y/Ly - 1), \quad (34)$$
+
+where $V_0$ is the amplitude of the transverse external potential. The matrix elements for the external potential are strictly off-diagonal and couple strongly the neighboring sub-bands, while other contributions are at least one order of magnitude smaller. Consequently, to a first approximation we have, $\langle n_y | V_{\text{ext}} | n'_y \rangle \approx 0.4V_0\delta_{n'_y, n_y \pm 1}$. Even in the presence of this inter-band coupling, the Pfaffian $Pf(B(0))$ for the effective three-band model can be determined analytically and it is given by an expression that generalizes Eq. (33). The corresponding phase diagrams for both uniform and non-uniform SM-SC couplings and different values of the external potential are shown in Fig. 14.
+
+The key conclusion of this calculation is that inter-sub-band mixing due to an external transverse potential does not lead to an expansion of the topological phase in the vicinity of the sweet spots, but rather determine a shift in their position. This is illustrated by diagrams (a-c) in Fig. 14, which correspond to homogeneous SM-SC coupling, i.e., $\theta=0$, and different values of the external field. Notice that, to a first approximation, the effect of the transverse potential is equivalent to increasing the inter-sub-band spacing $\delta E_{n_y n'_y}$. If the transverse potential is applied across a wire that is non-uniformly coupled to the SC, in addition to shifting the position of the sweet spots, it reduces the stability of the topological phase in their vicinity, as illustrated in panels (d-f) for $\theta \approx 0.4$. In addition, as a consequence of effectively increasing the inter-sub-band spacing, the regions between successive sweet spots occupied by the topological SC phase expands with increasing the transverse potential, as shown in panels (c) and (f). Note that, in the presence of non-uniform SM-SC coupling, the phase diagrams for $V_0 > 0$ and $V_0 < 0$ (not shown in Fig. 14 are slightly different. The strong dependence of the phase boundaries on the transverse potential, especially in the vicinity of the sweet spots, could be used experimentally for driving the
+
+FIG. 14: (Color online) Phase diagram of a multi-band wire in a transverse external field. The strength of the SM-SC coupling is $\bar{\gamma} = \Delta_0$, with uniform coupling ($\theta = 0$, i.e., $\Delta' = 0$) for the top panels and non-uniform tunneling ($\theta \approx 0.4$, $\Delta' = \Delta/4$) corresponding to the bottom panels. The amplitude of the transverse potential is: $V_0 = 0$ for panels (a) and (d), $V_0 = 50E_\alpha$ for (b) and (e), and $V_0 = 100E_\alpha$ for (c) and (f). Note that the topologically nontrivial phase has a vanishing width at the sweet spots in the absence of inter-sub-band pairing (a-c) for all values of the external potential. For non-uniform coupling (d-e), the transverse potential reduces the width of the topological phase near the sweet spots.
+
+system across the topological phase transition by tuning a gate potential, instead of changing the magnetic field.
+
+## IV. EFFECT OF DISORDER ON THE TOPOLOGICAL SUPERCONDUCTING PHASE
+
+In this section we consider the effect of disorder on the stability of the topological superconducting phase harming Majorana fermions. In a realistic system, disorder comes in various ways that affect the topological phase very differently. In this paper we consider three types of disorder: impurities in the s-wave superconductor, short- and long-range disorder in the semiconductor wire, and random nonuniform coupling between the semiconductor wire and the superconductor, which mimics a rough interface and the imprecision in engineering inhomogeneous y-dependent couplings. We begin by considering short-range impurities in the bulk superconductor, followed by the consideration of the other two types of disorder which are both more complex to treat theoretically and more detrimental to the existence of the Majorana modes.
\ No newline at end of file
diff --git a/samples/texts/3727487/page_16.md b/samples/texts/3727487/page_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..02e8a16e7d4690e5f10e8fec4932963e90b4a148
--- /dev/null
+++ b/samples/texts/3727487/page_16.md
@@ -0,0 +1,27 @@
+FIG. 15: (Color online) Diagrammatic perturbation theory in the tunneling between semiconductor and superconductor. Disorder-averaging is performed at each order in tunneling $t$. The thick solid line represents disorder-averaged Green's function in the superconductor $\tilde{G}(p, \omega)$. The last diagram corresponds to irreducible contributions, as far as disorder averaging is concerned.
+
+### A. Short-range impurities in the bulk superconductor.
+
+In order to understand the effect of non-magnetic impurities on the induced superconductivity in the semiconductor, we first review the results on the proximity effect for the infinite planar interface presented in the previous section. The basic idea is that the presence of short-range non-magnetic impurities in the metal modifies the bulk Green's function, which then is used to derive the appropriate superconducting proximity effect.
+
+We begin our discussion by considering the perturbation theory in the tunneling $t$, which is justified in the limit of low interface transparency. The lowest order contributions of the diagrammatic expansion in $t$ are shown in Fig. 15. One can notice that the self-energy at the second order in $t$ is determined by the disorder-averaged Green's function in the superconductor. Since typical s-wave superconductors are disordered (i.e. $\tau\Delta_0 \ll 1$ with $\tau$ being momentum relaxation time), the effect of impurity scattering is non-perturbative. In general, this yields a non-trivial problem because of the self-consistency condition which now has to be solved in the presence of disorder. The problem, however, can be substantially simplified if we neglect the effect of the in-plane magnetic field on the s-wave superconductivity. This condition can be justified due to the vast difference of the g-factors in the superconductor and semiconductor. We also assume here that the superconductor is thin enough so that we can neglect orbital effects. At this level of approximation, the problem reduces to understanding the effect of non-magnetic impurities on the bulk s-wave superconductivity, which has been investigated long time ago by Abrikosov and Gor'kov [64]. The main result of Abrikosov-Gor'kov's theory is that dis-
+
+order does not affect s-wave superconductivity, i.e., the superconducting gap is not suppressed by non-magnetic impurities, in agreement with Anderson's arguments invoking time-reversal symmetry. Specifically, the disorder at the single-particle Green's function level merely leads to the renormalization of the parameters $\tilde{\Delta}_0 = \eta_\omega \Delta_0$ and $\tilde{\omega} = \eta_\omega \Delta_0$:
+
+$$ \bar{G}(p, \omega) = \frac{\tilde{\omega}\tau_0 + \xi_p\tau_3 + \tilde{\Delta}_0\tau_x}{\tilde{\Delta}_0^2 + \xi_p^2 - \tilde{\omega}^2}, \quad (35) $$
+
+$$ \eta_{\omega} = 1 + \frac{1}{2\pi\sqrt{\tilde{\Delta}_{0}^{2} - \omega^{2}}}. \quad (36) $$
+
+Here bar represents disorder-averaged Green's function and the disorder strength is parameterized by the impurity scattering time $\tau$ defined as $1/\tau = \nu(\varepsilon_F)n_i u_0^2$ with $\nu(\varepsilon_F)$, $n_i$ and $u_0$ being the density of states at the Fermi level, the impurity concentration, and the scattering potential, respectively. Thus, at this level of the perturbation theory, the proximity effect can be included using the formalism developed for the clean case, see Eq. (13). By integrating out the degrees of freedom corresponding to the superconductor, one obtains the interface self-energy:
+
+$$ \Sigma(\omega) = |\tilde{t}|^2 \nu(\epsilon_F) \int d\epsilon \bar{G}(\epsilon, \omega) \quad (37) $$
+
+$$ = -|\tilde{t}|^2 \nu(\epsilon_F) \left[ \frac{\tilde{\omega}\tau_0 + \tilde{\Delta}_0\tau_x}{\sqrt{\tilde{\Delta}_0^2 - \tilde{\omega}^2}} + \frac{\zeta}{1 - \zeta^2} \tau_z \right] . \quad (38) $$
+
+Finally, upon substituting the expressions for $\tilde{\omega}$ and $\tilde{\Delta}_0$, we find that proximity-induced superconductivity is not affected by disorder in the s-wave superconductor
+
+$$ \Sigma(\omega) = -|\tilde{t}|^2 \nu(\epsilon_F) \left[ \frac{\omega\tau_0 + \Delta_0\tau_x}{\sqrt{\Delta_0^2 - \omega^2}} + \frac{\zeta}{1 - \zeta^2} \tau_z \right] . \quad (39) $$
+
+Moreover, one can see that, unlike impurities in the semiconductor, the disorder in the superconductor does not lead to momentum relaxation in the semiconductor at this order of the perturbation theory.
+
+As the interface transparency is increased, one needs to consider higher-order terms in tunneling. These terms involve reducible and irreducible contributions, see Fig. 15. The former depend only on the disorder-averaged Green's function, and, in a sense, are easy to take into account, whereas the latter involve non-trivial higher-order correlation functions (i.e. diffusons and Cooperons) and lead to momentum relaxation in the semiconductor. In this work, we consider only reducible contributions and neglect higher order correlation functions with respect to disorder-averaging which, one can show, are much smaller than the reducible ones [65]. Therefore, our minimal treatment of the disorder in the superconductor is justified and we believe that the superconducting disorder is irrelevant for the topological superconductivity. However, the disorder in the semiconductor (and at the interface) is relevant, as we discuss next.
\ No newline at end of file
diff --git a/samples/texts/3727487/page_17.md b/samples/texts/3727487/page_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..b26c8449b6f75e6b0b64910d46ff43313a9f154a
--- /dev/null
+++ b/samples/texts/3727487/page_17.md
@@ -0,0 +1,17 @@
+This conclusion holds as long as we neglect the effect of external magnetic field on the s-wave superconductivity, which is allowed as long as the field is not too large to destroy the superconducting state. This is realistic due to the large difference between the g-factors in superconductors ($g_{SC} \sim 1$) and semiconductors ($g_{SM} \sim 10-70$), allowing one to always find a parameter regime where the magnetic field opens a large spin gap in the spectrum without destroying superconductivity. Also, note that our conclusion regarding the short-range impurities in the superconductor is valid when the strength of the disorder is much larger than the superconducting gap: $\tau\Delta_0 \gg 1$ but is small enough not to affect the density of states. We have restricted our analysis to the experimentally relevant regime $\gamma \lesssim \Delta_0$. In the opposite limit $\gamma \gg \Delta_0$, electrons spend most of the time in the s-wave superconductor where their dynamics is not governed by the helical Hamiltonian required to have a topological phase and, thus, this limit is not experimentally desirable, see discussion in Sec. II B 1.
+
+We emphasize that our finding that short-range impurities in the bulk superconductor do not affect the topological superconducting phase emerging in the semiconductor is quite general since the electrons in the nanowire, being spatially separated from the bulk superconductor, simply do not interact directly with the short-range disorder in the bulk superconductor. Our use of the short-range impurity model to characterize the disorder in the bulk superconductor is justified, since strong metallic screening inside the bulk superconductor would render all bare long-range disorder into screened short-range one. Thus, as long as the applied magnetic field does not adversely affect the s-wave superconductivity, our conclusion regarding the validity of the Anderson theorem (i.e., no adverse effect from non-magnetic impurities) to the whole superconductor-semiconductor heterostructure system applies. In this context, we mention the recent theoretical analysis of Ref. [66], where it was explicitly established that, in the structure we are considering, the proximity-induced superconducting pair potential remains s-wave both in the topological and in the non-topological phase, in spite of the non-vanishing Zeeman field. The existence of the s-wave pairing potential, even in the presence of a parallel magnetic field (provided it is not too large), is the key reason for the short-range disorder in the superconductor not having any effect on the topological phase.
+
+## B. Disorder in the semiconductor nanowire
+
+In sharp contrast to disorder in the superconductor, disorder in the nanowire can have significant adverse effects on the stability of the topological SC phase. There are several different potential sources of disorder in the semiconductor. We focus on two sources that are the most relevant experimentally: random variations of the width of the SM nanowire and random potentials cre-
+
+ated by charged impurities. The first type of disorder is generally long-ranged, while the second type can be either long- or short-ranged. How to account for the effects of disorder depends crucially on the type of physical quantity that one is interested in. Here we focus on the low-energy spectrum, which carries information about the stability of the Majorana bound states, and on thermodynamic quantities such as the local density of states, which are experimentally relevant. The effect of disorder on these quantities is sample-dependent. We emphasize that averaging over different disorder realizations is equivalent in this case with sample averaging. As the goal is to observe stable Majorana fermions in a given nanowire, we investigate here the spectrum of the system for a given disorder realization and focus on establishing the general parameter regimes (e.g., amplitudes and length scales of the disorder potential) consistent with realistic experimental conditions that ensure the stability of the topological SC phase. More specifically, we study several different disorder realizations characterized by a given set of parameters and extract the generic features associated with that type of disorder.
+
+For a single channel Majorana nanowire, one can obtain some analytical results by performing disorder averaging [67–69]. Specifically, for a model of the spinless p-wave superconductor it has been shown [67, 68] that disorder drives the transition into non-topological phase when impurity scattering rate becomes comparable with the induced superconducting gap. In more realistic spinful models involving semiconductor nanowires, the physics is richer and depends on the strength of the magnetic field. We refer the reader to Ref. [69] for more details. The generalization of these results to the case of disordered multi-band semiconductor nanowires is an interesting open problem.
+
+### 1. Semiconductor nanowires with random edges
+
+The dimensions of the nanowire in the transverse direction satisfy the relation $L_y \gg L_z$. The small thickness $L_z$ is critical for the effectiveness of the superconducting proximity effect, as the SM-SC effective coupling $\bar{\gamma}$ scales, approximately, as $1/L_z^3$. On the other hand, the much larger width $L_y$ is required by the multichannel condition. Atomic-scale variations of $L_z$ generate huge local potential variations (of the order $500E_\alpha$) that would effectively cut the wire in several disconnected pieces. Topological SC phases may exist inside each of these pieces, but the Majorana states will be localized at the boundaries separating different segments and, in general, tunneling between them will be nonzero. To realize a single pair of Majorana zero-energy states localized at the ends of the wire, $L_z$ should be uniform along the system. (We mention in passing that modern MBE growth is consistent with very small variations in $L_z$ as necessary for the realization for the Majorana.)
+
+Engineering a long wire with constant width $L_y$ may
\ No newline at end of file
diff --git a/samples/texts/3727487/page_18.md b/samples/texts/3727487/page_18.md
new file mode 100644
index 0000000000000000000000000000000000000000..39543c9899c8f1bb2fc219df844d4c890395560b
--- /dev/null
+++ b/samples/texts/3727487/page_18.md
@@ -0,0 +1,15 @@
+FIG. 16: (Color online) Broadening of the sub-bands in a nanowire with random edges. For $n_y > 2$ the broadening becomes comparable with the inter-band gap in the presence of fluctuations $\Delta L_y$ representing a few percent of the width $L_y$, i.e., the sub-bands lose their identity.
+
+be, on the other hand, extremely challenging and less relevant for the stability of the topological SC phase. We assume that $L_y$ varies along the wire randomly in atomic-size steps that extend along hundreds of lattice sites in the x-direction, resulting in a function $L_y(x)$ that varies over length scales larger than the width of the wire. These fluctuations generate local variations of the bare sub-band energies given by Eq. (3) of the order of
+
+$$ \Delta\epsilon_{n_y} = -2t_0 \left[ \cos\left(\frac{n_y\pi a}{L_y + \Delta L_y}\right) - \cos\left(\frac{n_y\pi a}{L_y}\right) \right], \quad (40) $$
+
+where $L_y$ is the average width of the nanowire, $\Delta L_y$ the value of the local variation, and $a$ the lattice spacing. Note that different $n_y$ sub-bands are shifted differently, i.e., the random edge is not equivalent to long-range chemical potential fluctuations. To provide a quantitative measure of this effect, we show in Fig. 16 the evolution of the nanowire sub-bands with the size of the fluctuations $\Delta L_y$. Note that the sub-bands lose their identity in the presence of fluctuations representing a few percent of the wire width, as the broadening becomes comparable with inter-sub-band gaps. The natural question is how this broadening affects the low-energy physics of the nanowire and, in particular, the topological SC phase. Intuition based on the weak-coupling picture would suggest that the topological phase might become unstable, as the parity of the number of sub-bands crossing the chemical potential becomes an ill defined quantity.
+
+To quantify the effect of random edges on the low-energy physics, we first parametrize this type of disorder.
+
+FIG. 17: (Color online) Profile of the variation of the nanowire width, $\Delta L_y$, as a function of position along the wire. The characteristic length scale for profile I (upper panel) is about 10% of the nanowire length $L_x$, while for profile II (lower panel) it is approximately 5%. Each profile generates a series of particular disorder realizations characterized by different values of the maximum amplitude $|\Delta L_y|_{max}$. Random edge profiles corresponding to a given characteristic length and having the same maximum amplitude have similar effect on the low-energy physics of the nanowire.
+
+We consider a nanowire of width $L_y(x) = L_y + \Delta L_y(x)$, where $\Delta L_y(x)$ is a random function characterized by a certain maximum amplitude and a characteristic wavelength. Two examples of random edge profiles are shown in Fig. 17. We note that the actual width of the wire varies in atomic steps, i.e., $|\Delta L_y|_{min} = a$ and we assume that the characteristic length scale of these variations is much larger than the atomic scale. For example, for a nanowire with $L_x = 5\mu m$, a random edge profile like in the upper panel of Fig. 17 and a maximum amplitude of 10%, the atomic edge steps extend over hundreds of unit cells. In the calculations we explore the effect of random edges with maximum amplitudes up to 10% of $L_y$ and various characteristic wavelengths.
+
+The mini-gap $\Delta^*$ is reduced by the presence of random edges. However, for control parameters corresponding to points in the phase diagram away from phase boundaries, the amplitude of the variations of $L_y$ required for a complete collapse of the gap is well above 10%. Examples of the spectra for systems with random edges are shown in figures 18 and 19. Based on a number of similar calculations for different disorder realizations and control parameters, we have established the following general conclusions:
\ No newline at end of file
diff --git a/samples/texts/3727487/page_19.md b/samples/texts/3727487/page_19.md
new file mode 100644
index 0000000000000000000000000000000000000000..b4109f9a917a44b1237b27abb08581a9678c917d
--- /dev/null
+++ b/samples/texts/3727487/page_19.md
@@ -0,0 +1,15 @@
+FIG. 18: (Color online) Spectrum of a weakly coupled nanowire ($\bar{\gamma} = 0.25\Delta_0$) with random edges in the topological phase with $N=1$ (see Fig. 8). In the presence of disorder the mini-gap $\Delta^*$ decreases, but remains finite. The calculation was done using the random edge profiles shown in Fig. 17 for maximum amplitudes of 2% and 10%. Longer range disorder (upper panel) has stronger effects than short range disorder (lower panel). Small amplitude variations of the nanowire width of the order 2% generate a reduction of $\Delta^*$ up to 30%, but further increasing the amplitude has a weak effect at low energies.
+
+i) The details of the low-energy spectrum of a disordered nanowire depend on the particular disorder realization. Nonetheless, different disorder profiles characterized by a given amplitude and having the same characteristic length scale are likely to generate similar values of the mini-gap $\Delta^*$, with the exception of a few “rare events”, which are characterized by significantly lower gap values. A calculation that involves averaging over disorder will capture these “rare events” and will predict a value of the gap much lower than the typical value. Such a calculation would be relevant for an extremely long wire, i.e., in the limit $L_x \to \infty$, or for systems with a very large density of states at the relevant energies (e.g., a metal). However, in a typical semiconductor wire the number of states that control the low-energy physics is of the order of 100. How the energies of these states are modified in the presence of disorder, depends on the specific details of the disorder profile. Hence, any experimentally relevant conclusion regarding the low-energy spectrum or the local density of states of a disordered nanowire should be based on calculations involving specific disorder realizations. A direct consequence of these
+
+FIG. 19: (Color online) Spectrum of a nanowire with random edges at intermediate coupling, $\bar{\gamma} = \Delta_0$. The control parameters correspond to a point in the phase diagram inside the topological phase with $N=1$ (see Fig. 9). The random edges are given by the profiles in Fig. 17.
+
+considerations is that nominally identical samples with the same average disorder (e.g. same mobility) may have very different Majorana minigaps since they are likely to have different disorder configurations. The distribution of the minigaps was recently studied in Ref. [70].
+
+ii) Long range disorder has a stronger effect than short range disorder. This is general characteristic of disordered nanowires, regardless of the source of disorder, and will be studied in more detail in the next section. We note that the effects of variations of wire width $\Delta L_y(x)$ at atomic length scales are negligible.
+
+iii) Intermediate coupling $\bar{\gamma} \sim \Delta_0$ represents the optimal SM-SC coupling regime. The large gap that characterizes the topological $N=1$ phase in this regime is robust against fluctuations of the wire width of the order $\pm 10\%$ for any set of parameters that are not in the immediate vicinity of a phase boundary. Most importantly, this condition can be satisfied for specific values of the Zeeman field (e.g., of the order $30E_\alpha$ for the parameters corresponding to the phase diagram in Fig. 9) over a large range of chemical potentials.
+
+## 2. Nanowires with charged impurities
+
+A major source of disorder in the nanowire consists of charged impurities. Because the carrier density in the SM
\ No newline at end of file
diff --git a/samples/texts/3727487/page_2.md b/samples/texts/3727487/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..9c6532f47f75a47924183cc0b996d142ac42e0db
--- /dev/null
+++ b/samples/texts/3727487/page_2.md
@@ -0,0 +1,17 @@
+$\alpha$ as well as large Lande g-factor ($g_{\text{InAs}} \sim 10 - 25$ [48] and $g_{\text{InSb}} \sim 20 - 70$ [47]). Furthermore, these materials are known to form interfaces that are highly transparent for electrons, allowing one to induce a large superconducting gap $\Delta$ [49–51]. Thus, semiconductor nanowires show great promise for realizing and observing Majorana particles [6, 7]. It is important to emphasize that in the superconductor-semiconductor heterostructures the Majorana mode is constructed or engineered to exist as a zero-energy state, and as such, it should be experimentally observable in the laboratory under the right conditions.
+
+In a strictly 1D nanowire in contact with a superconductor, the condition for driving the system into a topological superconducting phase [37, 38] is $|V_x| > \sqrt{\Delta^2 + \mu^2}$, where $V_x$ is the Zeeman splitting due to the in-plane magnetic field, $\mu$ is the chemical potential and $\Delta$ is the proximity-induced superconducting gap. Thus, the key to the experimental realization of Majorana fermions in this system is the ability to satisfy a certain set of requirements that ensures the stability of the Majorana bound states. The challenging task here is the ability to suppress effects of disorder, control chemical potential fluctuations as well as fluctuations of other parameters. Given that realizing single channel (or one subband) nanowire is quite challenging, it is natural to consider semiconductor nanowires in the regime of multi-subband occupancy. It was shown in Ref. [40] that this is a promising route and the existence of Majorana fermions does not require strict one dimensionality. In fact, the stability of the topological superconducting phase is enhanced in multiband nanowires due to the presence of “sweet spots” (multicritical points in the topological phase diagram, see Ref. [52] for details) in the phase diagram where the system is most robust against chemical potential fluctuations [40]. In this paper, we expand on these ideas and explore effects of various perturbations such as disorder in the superconductor and semiconductor, fluctuations in the tunneling matrix elements, etc. on the stability of the topological phase. Our goal is to identify parameter regimes favorable for the exploration of the Majorana physics in the laboratory. We therefore use realistic physical models and realistic values of the parameters throughout this work so that our theoretical results are of direct relevance to experiments looking for Majorana modes in nanowires.
+
+The paper is organized as follows. We begin in Sec. II by introducing a tight-binding model for the semiconductor nanowires, and derive the superconducting proximity effect. We show that electron tunneling between semiconductor and superconductor leads to important renormalization of the parameters in the semiconductor. In Sec. III, we calculate the low-energy spectrum in the regime of multiband occupancy and identify the topological phase diagram. In Sec. IV, we study disorder effects on the stability of the topological phase. We consider several sources of disorder: short-range impurities in the superconductor, short-range and long-range impurities
+
+in the semiconductor as well as fluctuations in the tunneling matrix elements across the interface. In Sec. V, we present results for experimentally observable quantities (e.g., local density of states and tunneling conductance) calculated using realistic assumptions. Finally, we conclude in Sec. VI with the summary of our main results.
+
+## II. TIGHT-BINDING MODEL FOR SEMICONDUCTOR NANOWIRES
+
+### A. Spin-orbit interaction and Zeeman terms
+
+Single-channel semiconductor (SM) nanowires have been recently proposed [37, 38] as a possible platform for realizing and observing Majorana physics in solid state systems. Obtaining strictly one-dimensional (1D) nanowires would raise significant practical challenges [49, 50], but this requirement can be relaxed to a less stringent quasi-1D condition corresponding to multiband occupancy [40]. The physical system proposed for studying Majorana physics consists of a strongly spin-orbit interacting semiconductor, e.g., InAs and InSb, proximity-coupled to an s-wave superconductor (SC). The quasi-1D SM nanowire is strongly confined in the $\hat{z}$ direction, so that only the lowest corresponding sub-band is occupied, while the weaker confinement in the $\hat{y}$ direction is consistent with a few occupied sub-bands. Consequently, the linear dimensions of the rectangular nanowire satisfy the relation $L_z \ll L_y \ll L_x$ which is the usual physical situation in realistic semiconductor nanowires. The low-energy physics of the SM nanowire is described by the Hamiltonian
+
+$$ H_{\text{nw}} = H_0 + H_{\text{SOI}} = \sum_{i,j,\sigma} t_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} - \mu \sum_{i,\sigma} c_{i\sigma}^{\dagger} c_{i\sigma} \\ + \frac{i\alpha}{2} \sum_{i,\delta} \left[ c_{i+\delta_x}^{\dagger} \hat{\sigma}_y c_i - c_{i+\delta_y}^{\dagger} \hat{\sigma}_x c_i + \text{h.c.} \right], \quad (1) $$
+
+where $H_0$ includes the first two terms and describes hopping on a simple cubic lattice with lattice constant $a$ and the last term represents the Rashba spin-orbit interaction (SOI). We include only nearest-neighbor hopping with $t_{ii+\delta} = -t_0$, where $\delta$ are the nearest-neighbor position vectors. In Eq. (1), $c_i^\dagger$ represents a spinor $c_i^\dagger = (c_{i\uparrow}^\dagger, c_{i\downarrow}^\dagger)$ with $c_{i\sigma}^\dagger$ being electron creation operators with spin $\sigma$, $\mu$ is the chemical potential, $\alpha$ is the Rashba coupling constant, and $\hat{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ are Pauli matrices. In the long wavelength limit, $k \to 0$, our model reduces to an effective mass Hamiltonian with $t_0 = \hbar^2 a^{-2}/2m^*$ and Rashba spin-orbit coupling $\alpha_R(k_y\hat{\sigma}_x - k_x\hat{\sigma}_y)$, where $\alpha_R = \alpha a$. In the numerical calculations we use a set of parameters consistent with the properties of InAs, $a = 5.3Å$, $m^* = 0.04m_0$, and $\alpha_R = 0.1$ eV·Å. The position within the cubic lattice is described by $\mathbf{i} = (i_x, i_y, i_z)$ with $1 \le i_x(y,z) \le N_x(y,z)$. In the calculations we have used $N_z = 10$, $N_y = 250$, and $N_x$ between $10^4$ and $2 \cdot 10^4$,
\ No newline at end of file
diff --git a/samples/texts/3727487/page_20.md b/samples/texts/3727487/page_20.md
new file mode 100644
index 0000000000000000000000000000000000000000..2d746136841eb6083f8ca20495c8cd12c9b6c1c9
--- /dev/null
+++ b/samples/texts/3727487/page_20.md
@@ -0,0 +1,13 @@
+nanowire is small, charged impurities are not effectively screened and their presence can potentially have significant effects. The nature of these charged impurities, the values of their effective charge and their exact locations depend on the details of the nanowire engineering process and will have to be determined by future experimental studies. Here, we are interested in the fundamental question regarding the stability of the topological SC phase. In particular we address the following question: Is it possible to realize stable zero-energy Majorana modes in a nanowire with charged impurities within a realistic scenario? To answer this general question, we focus on four key aspects of the problem: a) The screening of charged impurities by the electron gas in the SM nanowire, b) The dependence of the low-energy physics on the concentration of impurities, c) The dependence of the low-energy spectrum on the Zeeman field, and d) The effect of long range random potentials.
+
+a. *Screening of charged impurities.* We start by considering a single charge $q$ inside or in the vicinity of the nanowire. For concreteness we assume that the charge is positioned near the middle of the wire and one lattice spacing away from its surface, i.e., for a wire that occupies the volume defined by $0 \le x_j \le L_j$, with $j \in \{x, y, z\}$, the position of the impurity is given by $(x_{imp}, y_{imp}, z_{imp}) = (L_x/2, L_y/2, -a)$. This corresponds, for example, to a charged impurity localized at the interface between the SM and the SC. We consider the extreme case $q = \pm e$, where $e$ is the elementary charge, although in practice it is likely that the effective charge is only a fraction of this value due to screening by electrons in the SC. We neglect the screening due to the presence of the superconductor, which may significantly reduce the effective potential created by the charge. A simple estimate of the screening effects due to the electrons in the semiconductor within the Thomas-Fermi approximation is highly inaccurate due to the low carrier density. We checked this property explicitly by calculating numerically the carrier density induced by a given effective potential $V(r)$, e.g., a screened Coulomb potential. We find that the relationship between the induced local carrier density $\delta n(r)$ and the local effective potential is highly non-linear. In addition, the density is characterized by strong Friedel-type oscillations (see Fig. 20). Hence, solving quantitatively the screening problem for the nanowire would require a self-consistent calculation that includes electron-electron interactions at the Hartree-Fock level. This calculation is beyond the scope of the present study and will be addressed elsewhere. Here we address a more limited question: What is the characteristic length scale over which the external charge $q$ is screened? We define this length scale $\lambda$ as the characteristic length of the volume that contains 63% (i.e., a fraction equal to $1 - 1/e$) of the induced charge. Specifically, the potential created by the charge $q$ at a point $r$ inside the semiconductor is
+
+$$V(\mathbf{r} - \mathbf{r}_{\text{imp}}) = \frac{-eq}{4\pi\epsilon_0\epsilon_r|\mathbf{r} - \mathbf{r}_{\text{imp}}|}, \quad (41)$$
+
+FIG. 20: (Color online) Induced carrier density as a function of distance from the impurity. The two curves, corresponding to different external sets of parameters for the nanowire, have been shifted for clarity. The upper curve corresponds to a system with a single occupied sub-band and is characterized by $\lambda \approx 45a$ (i.e., 63% of the induced charge is in a disk of radius $45a$), while the lower curve is for a system with three occupied sub-bands and has $\lambda \approx 30a$. Note that the effective impurity potentials that generate these density profiles are given by Eq. (42) with the corresponding values of $\lambda$.
+
+where $\mathbf{r}_{\text{imp}} = (x_{\text{imp}}, y_{\text{imp}}, z_{\text{imp}})$ is the position of the impurity and $\epsilon_r$ is the relative dielectric constant of the SM. For InAs $\epsilon_r = 14.6$. Next, we take into account the fact that the nanowire is extremely thin in the z direction ($L_z \approx 10a$), hence the wave function profile along this direction is very little affected by the presence of the impurity. On the other hand, the induced charge has a strong dependence on x (the direction along the wire) and y (the transverse direction). As mentioned above, the impurity potential is screened by the induced charge outside a region with a characteristic length scale $\lambda$ that contains a fraction equal to $1-1/e$ of the induced charge. We assume that the screened potential is qualitatively described by an expression of the form
+
+$$V_s = V(\mathbf{r} - \mathbf{r}_{\text{imp}}) \exp \left[ - \frac{\sqrt{(x-x_{\text{imp}})^2 + (y-y_{\text{imp}})^2}}{\lambda} \right]. \quad (42)$$
+
+The parameter $\lambda$ from Eq. (42) is determined self-consistently by imposing the condition that 63% of the induced charge be in a disk of radius $\lambda$ and thickness $L_z$ centered at $(x_{\text{imp}}, y_{\text{imp}}, L_z/2)$. The results for two different sets of parameters are shown in Fig. 20. As expected, nanowires with multiple occupied sub-bands provide a more effective screening, which is reflected in a lower value of the screening length $\lambda$. We emphasize that the present approach is not fully self-consistent and is therefore only of qualitative validity. An effective screened potential that includes exactly the contribution of the induced charge with details (e.g., oscillatory components) that are not captured by Eq. (42) should give results very similar to what we obtain here. Nonetheless, we expect these details to have a weak effect on
\ No newline at end of file
diff --git a/samples/texts/3727487/page_21.md b/samples/texts/3727487/page_21.md
new file mode 100644
index 0000000000000000000000000000000000000000..d4452126f8d2acf1a7faf77246b6373bb35d3a8b
--- /dev/null
+++ b/samples/texts/3727487/page_21.md
@@ -0,0 +1,13 @@
+FIG. 21: (Color online) Top panel: Comparison between the low-energy spectrum of a clean nanowire (yellow squares) and the spectrum of a nanowire with a charged impurity with $q = e$ (red circles). Note that the two spectra are almost on top of each other. Bottom panel: Electron wave function amplitudes for two low-energy states (marked by arrows in the top panel) in the presence of the charged impurity. The Majorana zero modes ($n = \pm 1$, not shown) and the in-gap modes ($n = \pm 2, \pm 3$, see Fig. 12 and the corresponding discussion) are localized near the ends of the wire and are not affected by the presence of the impurity. The low-energy bulk states $|n| > 4$ that are extended in a clean nanowire become localized near the impurity.
+
+the final results. We also note that the self-consistent calculation of the screening length is done for a non-superconducting nanowire, then the effective impurity potential $V_s$ is added to the total nanowire Hamiltonian before including the proximity effects due to the SM-SC coupling.
+
+What is the effect of the charged impurity on the low-energy spectrum of the superconducting nanowire? The effective potential in the vicinity of the impurity is extremely large, e.g., $V_s(L_x/2, L_y/2, L_z/2) \approx 640$ meV (i.e., $\approx 12000E_\alpha$), much larger than any other relevant energy scale in the problem, hence one would naively expect significant effects. However, the low-energy physics of the SM nanowire is controlled by single particle states with low wave numbers and the matrix elements of the impurity potential between these state are relatively small. In other words, the impurity will strongly affect the spectrum at intermediate and high energies, but will have a relatively small effect at low energies. To illustrate this property, we show in Fig. 21 a comparison between the
+
+FIG. 22: (Color online) Specific disorder realizations in nanowires with charged impurities. The dots represent the locations of the impurities. Each impurity has a charge $q = \pm e$ and is positioned one lattice spacing away from the surface of the nanowire along the z direction. The linear impurity densities $n_{\text{imp}}$ are: $1\mu\text{m}^{-1}$ (A), $2\mu\text{m}^{-1}$ (B), $4\mu\text{m}^{-1}$ (C), and $8\mu\text{m}^{-1}$ (D).
+
+low-energy spectra of a clean nanowire and of a nanowire with a charged impurity. We note that the presence of the external charge induces the formation of localized states in the vicinity of the impurity (see Fig. 21).
+
+Charge impurities in one-dimensional quantum wires produce weakly long-range disorder since the Coulomb potential decays as $\ln|qa|$ in the momentum space for $q \to 0$ with $a$ being the short-distance cut-off associated with the transverse dimensions of the nanowire [71]. This is to be contrasted with the much stronger $q^{-1}(q^{-2})$ long wavelength divergence of the bare Coulomb disorder in two (three) dimensions. The weakly long-range nature of 1D Coulomb potential suggests that any regularization of the long-range disorder would be a reasonable approximation in spite of the fact that Thomas-Fermi screening itself is weak in 1D. In particular, the presence of interband scattering in the multisubband situation would essentially lead to effective 2D screening in the system, which should suffice to regularize the singular Coulomb disorder. At very low densities, where the nanowire is strictly in 1D limit with only the lowest subband occupied, weak screening will lead to the formation of the inhomogeneous electron puddles in the system around the charge impurities due to the failure of screening. This situation is detrimental to the Majorana formation and must be avoided. It is clear that higher density and multisubband occupancy would be favorable for the experimental realization of the Majorana modes in the semiconductor nanowires.
+
+**b. Charged impurity disorder.** The next question that we address concerns the dependence of the low energy spectrum of a nanowire with charged disorder on the concentration of impurities. We emphasize that the details of the low-energy physics depend on the specific
\ No newline at end of file
diff --git a/samples/texts/3727487/page_22.md b/samples/texts/3727487/page_22.md
new file mode 100644
index 0000000000000000000000000000000000000000..8bb3392da2ced75c3f7a04ad035bf767b46cfd41
--- /dev/null
+++ b/samples/texts/3727487/page_22.md
@@ -0,0 +1,15 @@
+disorder realization. In particular, the low-energy states are localized in the vicinity of the impurities (see Fig. 21) and their energies depend on the specifics of the real-space disorder configuration. Hence, as discussed above, calculations of single-particle quantities, e.g., the local density of states, should involve specific disorder realizations, rather than disorder averaging. In Fig. 22 we show four different specific disorder realizations, corresponding to linear impurity densities ranging from $1\mu m^{-1}$ to $8\mu m^{-1}$, which are reasonably realistic impurity densities ($\sim 10^{15} cm^{-3}$) in high quality semiconductor structures. The impurities carry charge $q=e$ and are positioned at a distance of one lattice constant away from the SM surface. The effect of these impurities is incorporated through an impurity potential of the form
+
+$$V_{\text{imp}}(\mathbf{r}) = \sum_j V_s(\mathbf{r}, \mathbf{r}_j), \quad (43)$$
+
+where $V_s$ is given by Eq. (42) and $\mathbf{r}_j$ are the impurity position vectors. We note that in Eq. (43) the screened potential is characterized by a screening length $\lambda$ determined as described above for a single impurity. This approximation does not take into account the impact of the dependence of the effective potential associated with a given charged impurity on the location of the charge and on the presence of other impurities. It also neglects the effect of screening by the SC itself which should strongly suppress the effective disorder.
+
+The low-energy spectra of a nanowire with random charged impurities distributed as in Fig. 22 are shown in Fig. 23 for two sets of control parameters. The general trend is that the mini-gap decreases with increasing impurity concentration. However, for a given concentration $n_{\text{imp}}$ the exact value of the mini-gap depends on the specific disorder realization. As mentioned above, averaging over disorder includes rare configurations characterized by small mini-gaps, hence the averaged density of states is characterized at low energies by a small weight that does not correspond to any physical state in a typical disorder realization. The signature of a topological SC phase with $N=1$ (i.e., one pair of Majorana fermions) that distinguishes it from the trivial SC phase with $N=0$ is the presence of zero-energy quasiparticles separated by a finite gap from all other excitations. One key conclusion of our calculations is that the mini-gap that protects the topological SC phase remains finite for a significant range of realistic impurity concentrations.
+
+c. Dependence of the low-energy spectrum on the Zeeman field. As shown in Fig. 23, for certain high impurity concentrations (e.g., configuration D in fig. 22) it is possible that, in addition to the Majorana zero mode, another low-energy state has energy close to zero and is separated by a gap from the rest of the spectrum. How can one distinguish experimentally this state from a topological superconductor characterized by a finite mini-gap, on one hand, and from a trivial superconductor with $N=2$, on the other? As the low-energy spectrum has a strong dependence on $\Gamma$, the key is to vary the Zeeman
+
+FIG. 23: (Color online) Low-energy spectra of a superconducting nanowire with charged disorder for two different sets of control parameters and four disorder realizations. The symbols correspond to the specific disorder realizations shown in Fig. 22. Note that: i) The mini-gap $\Delta^*$ finite for impurity concentrations $n_{\text{imp}} \le 4\mu m^{-1}$ and collapses for $n_{\text{imp}} = 8\mu m^{-1}$, and ii) There is an overall tendency of the low-energy features to move at lower energies when increasing the impurity concentration, but the exact value of the mini-gap depends on the specific disorder realization, e.g., in the lower panel $\Delta^*(n_{\text{imp}} = 4) > \Delta^*(n_{\text{imp}} = 2)$.
+
+field. Tuning $\Gamma$ may push the system into the topological SC phase and the energy of the additional state will increase. On the contrary, it is possible that varying the Zeeman field will lead to the appearance of more low-energy excitations. This is the characteristic signature of the transition zone between phases with different topologies. To address this problem more systematically, let us consider a nanowire with the same parameters as in the bottom panel of Fig. 10: $\mu = 54.5E_\alpha$, $\Gamma = \Delta_0$, and $\theta = 0.8$. The dependence of the quasiparticle gap on $\Gamma$ for the infinite clean system is shown in Fig. 10, and the dependence of the mini-gap on $\Gamma$ for a finite wire is shown in Fig. 12. The transition between the trivial SC phase with $N=0$ and the topological phase with $N=1$ is clearly marked by the vanishing of the gap at $\Gamma \approx 21.5E_\alpha$.
+
+Let us now add disorder and follow the evolution of the lowest three energy levels with the Zeeman field. The results for two different values of the impurity concentra-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_23.md b/samples/texts/3727487/page_23.md
new file mode 100644
index 0000000000000000000000000000000000000000..4d668c624d54a6f2faab8de7794c938b9d1a3589
--- /dev/null
+++ b/samples/texts/3727487/page_23.md
@@ -0,0 +1,17 @@
+FIG. 24: (Color online) Dependence of the lowest three en- ergies $E_n$ ($n = 1, 2, 3$) on the Zeeman field for a disordered nanowire with $n_{\text{imp}} = 4\mu m^{-1}$ (top) and $n_{\text{imp}} = 7\mu m^{-1}$ (bot- tom). The lowest energy state is characterized by a gap $\Delta_1 = E_1$ (red/dark gray region) that vanishes for $\Gamma > 21.6E_\alpha$, i.e., when the system enters the topological SC phase. The gap be- tween the first and the second levels, $\Delta_2 = E_2 - E_1$ (yellow), becomes the mini-gap in the topological phase. The tran- sition zone is characterized by a high density of low-energy modes and expands with increasing the impurity concentra- tion.
+
+tion are shown in Fig. 24. First, we note that the main features are similar with those observed in a clean system: at low values of the Zeeman field the system is in a triv- ial SC phase characterized by finite gap to all excitations, while for $\Gamma$ above a certain critical value $E_c \approx 22E_\alpha$ the system has a Majorana zero mode separated by a mini- gap from the rest of the spectrum. The major difference from the clean case consists of the transition zone, which extends over a finite range of values of the Zeeman field and is characterized by multiple low-energy excitations. This transition zone extends with increasing impurity concentration. Assuming that we probe the low-energy properties of the system with a certain finite resolution, e.g., $\Delta E = 0.2E_\alpha$, the topological phase can be unam- biguously distinguished from the trivial SC provided the mini-gap is larger than the energy resolution. The trivial $N=0$ phase will be characterized by a finite gap and no zero-energy excitation, while the topological phase will have the characteristic zero-mode separated from the fi-
+
+FIG. 25: (Color online) Low-energy spectra of a nanowire with long-range disorder. The system in characterized by $\mu = 54.5E_\alpha$ and $\gamma = \Delta_0$ and the impurity potential is given by Eq. (44) with a profile $v_{\text{imp}}$ as in Fig. 17 (lower panel) and different amplitudes $U_0$. The parameters $U_0$ and $\Gamma$ are expressed in units of $E_\alpha$. For $\Gamma = 29$ the mini-gap collapses for $U_0 \ge 60$, while for a Zeeman field $\Gamma_{32}$ it survives up to an amplitude $U_0 \approx 100$.
+
+nite energy excitations by the mini-gap. In between the two phases there will be a transition zone characterized, within our finite energy resolution, by a continuum spec- trum. Starting with low values of the Zeeman field, i.e., deep inside the $N=0$ phase, by increasing $\Gamma$ one first reaches the transition zone, the topological $N=1$ phase. We emphasize that for $\Delta^* < \Delta E$ the topological phase becomes indistinguishable from the transition zone. In addition, at large impurity concentrations the mini-gap will collapse completely.
+
+d. Long range disorder potential. What is the effect of a long-range disorder potential on the stability of the topological SC phase? Are there qualitative differences from the short-range case discussed above? We are not interested here in the possible source of such long range disorder, but rather in identifying the magnitude of the amplitude of the random potential that would destroy the topological phase.
+
+Let us consider a random potential with a character-
+istic length scale $\lambda > L_y$. Neglecting the dependence of
+the potential on $y$ and $z$, we have
+
+$$V_{\text{imp}}(\mathbf{r}) = U_0 v_{\text{imp}}(x/L_x), \quad (44)$$
+
+where $U_0$ is the amplitude of the potential and $-1 \le v_{\text{imp}} \le 1$ is a random profile. For concreteness we consider the profile shown in the lower panel of Fig. 17 and a nanowire with $L_x = 5\mu m$ and $L_y = 0.12\mu m$. These parameters correspond to $\lambda \approx 2L_y$. The corresponding low-energy spectra for different values of the disorder amplitude are shown in Fig. 25. A striking feature is represented by the significant difference between the critical disorder amplitudes at which the mini-gap col-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_24.md b/samples/texts/3727487/page_24.md
new file mode 100644
index 0000000000000000000000000000000000000000..63dec48a535d7be8d123650b1ed14c41710cc1a7
--- /dev/null
+++ b/samples/texts/3727487/page_24.md
@@ -0,0 +1,15 @@
+lapses for the two slightly different values of the Zeeman field. Note that both values are near the “center” of the topological $N = 1$ phase, as one can see, for example, by inspecting the bottom panel of Fig. 12. What sets the scale for the critical amplitude? A hint can be obtained from the phase diagram shown in Fig. 9. In essence, in the presence of a smoothly varying random potential topological superconductivity is stable as long as the local chemical potential at any point within the nanowire has values within the topological phase. The permissible range of variation for the local chemical potential depends strongly on the Zeeman field. From the phase diagram in Fig. 9 and the chemical potential dependence in Fig. 11 it is clear that $\Gamma \approx 32E_\alpha$ represents a value of the Zeeman field that allows for large amplitude chemical potential fluctuations.
+
+We conclude that long range disorder can be treated as local chemical potential fluctuations. We emphasize that this is not the case for short range disorder. The topological phase is stable against chemical potential fluctuations up to a maximum amplitude that depends on the Zeeman field, on the average chemical potential, and on the SM-SC coupling. Single band occupancy necessarily limits the critical amplitude due to the close proximity of the phase boundary. Multi-band systems avoid this problem and provide a more efficient screening for the short range potentials created by charged impurities. In addition, a strongly non-uniform coupling between the nanowire and the s-wave SC (with $\theta \sim 1$) with a coupling strength in the intermediate regime ($\gamma \sim \Delta_0$) provides the optimal shape of the phase diagram (see Figures 9 and 12).
+
+### C. Disorder at the semiconductor-superconductor interface
+
+The inhomogeneous random coupling at the semiconductor-superconductor interface is another significant source of disorder. In principle, the tunneling matrix elements $\tilde{t}(i_x, i_y)$ between the nanowire and the s-wave SC are characterized by random real space variations due to inhomogeneities in the tunneling barrier. For example, realizing a nonuniform coupling $\tilde{t}(i_y)$, which is critical for generating off-diagonal pairing and for stabilizing the topological phase near the sweet spots, may require the growth at the interface of an insulating layer with variable thickness across the wire. Any growth imperfection will translate into variations of $\tilde{t}$. While ultimately the details of these variations will have to be determined by a careful experimental study of the interface, it is reasonable to assume that a typical interface is characterized by atomic size variations with a characteristic length scale of a few lattice spacing, as well as longer range inhomogeneities with characteristic length scales comparable to the width of the wire, $L_y$, or larger. The short range inhomogeneities could be generated by impurities or by point defects present at the interface, while longer range inhomogeneities could
+
+be due to extended defects. In the absence of a detailed microscopic description of the SM-SC interface, it is difficult to estimate the amplitude of these fluctuations. Here we consider a phenomenological model of the interface and we assume that the tunneling matrix is given by
+
+$$ \tilde{t}(i_x, i_y) = \tilde{t}(i_y) + \Delta\tilde{t}(i_x, i_y), \quad (45) $$
+
+where $\tilde{t}(i_y)$ is the smooth component of the nonuniform coupling, e.g., the interface transparency with the profile shown in Fig. 3, and $\Delta\tilde{t}(i_x, i_y)$ represents the random component. To model the short range disorder, we coarse grain the interface in square patches of side length $l$ and assume that $\Delta\tilde{t}$ is uniform within a patch, but varies randomly from patch to patch with an amplitude $(\Delta\tilde{t})_{\text{max}}$, i.e., with $-(\Delta\tilde{t})_{\text{max}} \le \Delta\tilde{t}(i_x, i_y) \le (\Delta\tilde{t})_{\text{max}}$. In the numerical calculations we considered patches of sizes $l = 20a$ and $l = 60a \approx Ly/4$ and an amplitude $(\Delta\tilde{t})_{\text{max}} = 0.25\tilde{t}(0)$, where $\tilde{t}(0)$ is the maximum value of the smooth nonuniform tunneling component shown in Fig. 3. We note that these are extremely large fluctuations of $\tilde{t}(i_x, i_y)$, larger than the minimum value of the smooth tunneling component, $\tilde{t}(Ly) = 0.2\tilde{t}(0)$, which result in significant variations of the local effective coupling $\gamma(i_x, i_y) \propto |\tilde{t}(i_x, i_y)|^2$. Examples of short-range random couplings within the patch model are shown in Fig. 26.
+
+Before we present the results of the numerical calculations, we would like to emphasize the specific way that interface disorder enters the effective Hamiltonian. While the effect of charged impurities can be included through a random potential, variations in the SM-SC coupling generate randomness in the effective SC order parameter, $\Delta_{n_y n'_y}$, as well as fluctuations of the renormalization matrix $Z^{1/2}$. From equations (25) and (27) we notice that the short range fluctuations of $\tilde{t}$ and, implicitly, the short range fluctuations of $\gamma$, are significantly reduced when taking the matrix elements with the eigenstates $|n_y\rangle$. As we are mainly interested in systems with only a few occupied sub-bands, integration over $y$ effectively averages out fluctuations with characteristic length scales $l \ll L_y$. As an illustration of this property, we consider the case of random coupling with $l = 60a$ and $(\Delta\tilde{t})_{\text{max}} = 0.25\tilde{t}(0)$ for a nanowire with $\mu = 54.5E_\alpha$, $\Gamma = 32E_\alpha$, and $\bar{\gamma} = \Delta_0$. In spite of the relatively large length scale of the fluctuation, $l = Ly/4$, the variations of the induced gap $\Delta_{11}(x)$ along the wire are only of the order of 10% of the average value. The dependence of the induced gap $\Delta_{11}$ on the position along the wire is shown in the lower panel of Fig. 20. The amplitude of the $\Delta_{11}$ fluctuations is further reduced if we consider shorter range coupling fluctuations. A similar behavior characterizes the renormalization matrix $Z^{1/2}$. In addition, as the low-energy properties of the system are determined by single particle states with small wave vectors $k_x$, we expect a further reduction of the effect of short range fluctuations as a result of the integration over $x$. In particular, if the clean, infinite wire is characterized by a maximum Fermi wave vector
\ No newline at end of file
diff --git a/samples/texts/3727487/page_25.md b/samples/texts/3727487/page_25.md
new file mode 100644
index 0000000000000000000000000000000000000000..5e4513a783a47c0405f7b6269202989d968d5c9d
--- /dev/null
+++ b/samples/texts/3727487/page_25.md
@@ -0,0 +1,13 @@
+FIG. 26: (Color online) Random coupling at the SM-SC interface. The coupling $\tilde{t}$ contains a random component $\Delta\tilde{t}$ that is constant within patches of length $l$, but takes a random value within each patch. The patch sizes are $l = 20a$ (top panel) and $l = 60a$ (middle panel). The strength of $\Delta\tilde{t}$ within the patches is color coded. Note that only part of the interface is shown, as the typical length of a nanowire used in the calculations is of the order $10^4 a$. Bottom: Dependence of the induced gap $\Delta_{11}$ on the position along the wire for $\Gamma = 32E_\alpha$, $\mu = 54.5E_\alpha$, $\bar{\gamma} = \Delta_0$, $\theta = 0.8$, and a random coupling with $l = 60a$ and an amplitude $(\Delta\tilde{t})_{max} = 0.25\tilde{t}$. The huge local variations of the coupling are strongly reduced by the integration over $y$ (see main text) and generate fluctuations of the order of 10% in $\Delta_{11}(x)$.
+
+$k_F$, we expect the low-energy physics to be insensitive to random variations of the SM-SC coupling with characteristic length-scales $l < 1/k_F$.
+
+The effects of a random SM-SC coupling on the low-energy spectrum of the nanowire are illustrated in Fig. 27. Remarkably, short range fluctuations with amplitudes up to 25% of the average coupling at $y=0$ (i.e., the maximum value of the coupling in a nonuniform profile - see Fig. 3) do not destroy the topological SC phase. The relatively weak effect of these strong fluctuations is due to the implicit averaging involved in the calculation of the matrix elements between single particle states with low wave vectors. Increasing the characteristic length $l$ makes this type of disorder more effective, as evident from the upper panel of Fig. 27. Hence the natural question: what is the effect of long-range SM-SC coupling fluctuations? We consider a smooth variation of $\tilde{t}$ along the wire with a profile as shown in the upper panel of Fig. 27. An amplitude of these fluctuations equal to 25% of $\tilde{t}(y=0)$ results in the collapse of the gap (see Fig. 27, lower panel). However, the topological phase is robust
+
+FIG. 27: (Color online) Spectra of a nanowire with random SM-SC coupling. Upper panel: Nanowire with short range fluctuations of the SM-SC coupling. The random coupling is considered within the patch model described in the text and corresponds to the distributions shown in Fig. 26. Lower panel: Nanowire with long-range SM-SC coupling fluctuations. The variations of $\tilde{t}$ along the wire have a profile as shown in the upper panel of Fig. 17 and an amplitude $\Delta\tilde{t}$. Note that for $\Delta\tilde{t} = 0.25\tilde{t}(y=0)$ the gap collapses.
+
+against long range coupling fluctuations with amplitudes smaller than 20%. We note that long range variations of the coupling strength could result from the engineering process of the nonuniform interface. Limiting the amplitude of these fluctuations below a certain limit of about 10–15% of the maximum coupling strength should be a priority of the experimental effort for realizing a topological SC state using semiconductor nanowires.
+
+V. EXPERIMENTAL SIGNATURES OF MAJORANA BOUND STATES
+
+Probing unambiguously the presence of Majorana bound states in the superconducting nanowire represents a critical task. In this section we show that local spectral measurements provide a simple and effective tool for accomplishing this task. We focus on the local density of states (LDOS), which could be measured using, for example, scanning tunneling spectroscopy (STS), and on the differential conductance associated with tunneling into the ends of the wire. We establish that these mea-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_26.md b/samples/texts/3727487/page_26.md
new file mode 100644
index 0000000000000000000000000000000000000000..6ae4918fb81b51001ca46b9f63859f130991109d
--- /dev/null
+++ b/samples/texts/3727487/page_26.md
@@ -0,0 +1,19 @@
+FIG. 28: (Color online) Typical spectra for a trivial SC with $N = 0$ (red squares), a topological SC with $N = 1$ (orange diamonds), and a trivial SC with $N = 2$ (black circles). The system is characterized by $\mu = 54.5E_\alpha$, $\theta = 0.8$, and $\bar{\gamma} = \Delta_0$. The finite energy in-gap states (for $\text{Gamma} = 15E_\alpha$ and $\Gamma = 45E_\alpha$) together with the Majorana zero-modes are localized near the ends of the wire (see also Fig. 12), while the rest of the states extend throughout the entire system. The corresponding LDOS is shown in Fig. 29.
+
+surements should suffice in establishing the existence of
+the zero-energy Majorana edge modes in semiconductor
+nanowires.
+
+A. Local density of states in superconducting nanowires
+
+In the previous section we have shown that disorder generates low-energy states and reduces the mini-gap. Nonetheless, a small mini-gap does not implicitly mean that the Majorana bound state cannot be resolved in a spectroscopic measurement. The key observation is that the undesirable low-energy states are generally localized near impurities and defects (see, for example, Fig. 21). A local measurement could easily distinguish between a zero-energy state localized at the end of the wire and a low-energy state localized somewhere inside the wire. To clarify the question regarding the real space distribution of the low-energy spectral weight, we calculate the local density of states (LDOS) for several relevant regimes and compare the LDOS of a clean ideal system with that of a disordered realistic nanowire.
+
+We start with a clean nanowire with three occupied sub-bands ($\mu = 54.5E_\alpha$) coupled non-uniformly to an
+
+s-wave superconductor. The coupling parameter is characterized by a non-homogeneity factor $\theta = 0.8$ and an intermediate coupling strength $\bar{\gamma} = \Delta_0$, where $\Delta_0$ is the SC order parameter of the superconductor. In the absence of a Zeeman field ($\Gamma = 0$), the nanowire is a trivial superconductor. Increasing the Zeeman field $\Gamma$ above a critical value induces a transition from the trivial SC phase with $N = 0$ (no Majorana modes) to a topological SC phase with $N = 1$ (one pair of Majorana modes). Further increasing $\Gamma$ drives the system through a series of alternating phases with trivial ($N$ even) and non-trivial ($N$ odd) topologies (see the phase diagram in Fig. 9). Of major practical interest are the first two phases ($N = 0$ and $N = 1$), as stronger Zeeman fields involve smaller gaps (see Fig. 10) or may destroy superconductivity altogether. Typical spectra from the first three phases ($N = 0, 1, 2$) are shown in Fig. 28 and the corresponding LDOS is shown in Fig. 29.
+
+The main conclusion suggested by the results shown in Fig. 29 is that clear-cut evidence for the existence of the Majorana zero modes can be obtained by driving the system from a trivial SC phase with $N = 0$ to a topological SC state with $N = 1$ by tuning the Zeeman field. In the trivial SC phase there is a well-defined gap for all excitations, including states localized near the ends of the wire. By contrast, the topological SC phase is characterized by sharp zero-energy peaks localized near the ends of the wire and separated from all other excitations (including possible localized in-gap states) by a well-defined mini-gap.
+
+Is it possible to clearly distinguish the two phases with different topologies in the presence of disorder? The answer is provided by the results shown in Fig. 30 for a nanowire with charged impurities. In contrast with the clean case, all the low-energy states are strongly localized. Nonetheless, the signature features of the two phases (the gaps and the zero-energy peaks) are preserved. At this point we emphasize two critical properties: i) The features illustrated in Fig. 30 are generic, i.e., they do not depend on the type or the source of disorder. Similar LDOS can be generated using any other significant type of disorder discussed in the previous section, or combinations of different types of disorder. ii) Observing a zero-energy peak at a certain value of the Zeeman field does not by itself prove the realization of a topological SC phase. The trivial SC state with $N = 2$ may also have a zero-energy peak separated from all other excitations by a mini-gap. To clearly identify the $N = 1$ phase one must measure the LDOS as a function of the Zeeman field starting from $\Gamma = 0$, i.e., from the trivial SC phase with $N = 0$. Continuously increasing $\Gamma$ will generate a transition from a phase characterized by a well-defined gap to a phase with strong zero-energy peaks localized near the ends of the wire. But what is the signature of the transition?
+
+Figure 31 shows the spectrum and the LDOS of a system with a Zeeman field $\Gamma = 21.5E_\alpha$. In a clean wire this corresponds to the transition between the $N = 0$ and
\ No newline at end of file
diff --git a/samples/texts/3727487/page_27.md b/samples/texts/3727487/page_27.md
new file mode 100644
index 0000000000000000000000000000000000000000..9a128572ad80ec8ccca203412ccfe1c6d7f9a4fa
--- /dev/null
+++ b/samples/texts/3727487/page_27.md
@@ -0,0 +1,11 @@
+FIG. 29: (Color online) LDOS for a clean nanowire in three different phases: trivial SC phase with $N = 0$ ($\Gamma = 15E_{\alpha}$, top), topological SC phase with $N = 1$ ($\Gamma = 25E_{\alpha}$, middle), and trivial SC phase with $N = 2$ ($\Gamma = 45E_{\alpha}$, bottom). The corresponding spectra are shown in Fig. 28. Notice the finite energy in-gap states localized near the ends of the wire (top and bottom) and the zero-energy Majorana modes (middle and bottom). The weight of the zero-energy modes in the $N = 2$ phase (bottom) is twice the weight of the Majorana modes in the topological SC phase with $N = 1$ (middle). However, the clearest distinction can be made between the $N = 0$ and the $N = 1$ phases. The LDOS is integrated over the transverse coordinates *y* and *z*.
+
+FIG. 30: (Color online) LDOS for a nanowire with charged impurities. The top picture corresponds to a trivial SC state with $N = 1$, while the bottom picture is for a system with $N = 1$. The linear impurity density is $n_{imp} = 7/\mu m$. Notice that all the low-energy states are strongly localized, but the clear-cut distinction between the two phases holds.
+
+$N = 1$ phases, which is marked by the vanishing of the quasi-particle gap, as shown in figures 10 and 12. The LDOS is characterized by a distribution of the spectral weight over the entire low-energy range of interest. This property holds at any position along the wire. Adding disorder induces localization, but does not change this key property. In fact, based on the analysis of the results shown in Fig. 24, we know that in disordered systems this
+
+type of critical behavior will characterize a finite range of Zeeman fields. Observing the transition between the topologically trivial and nontrivial phases, which is characterized by the closing of the gap and by a spectral weight distributed over a wide energy range, is the final ingredient necessary for unambiguously identifying the Majorana bound states using LDOS measurements.
+
+## B. Tunneling differential conductance
+
+An ideal type of measurement that exploits the properties identified in the previous subsection consists of tunneling into the ends of the wire and measuring the differential conductance [72–76]. To a first approximation, $dI/dV$ is proportional to the local density of states at the end of the wire, so the general discussion presented above should apply. Here, we focus on certain specific aspects
\ No newline at end of file
diff --git a/samples/texts/3727487/page_28.md b/samples/texts/3727487/page_28.md
new file mode 100644
index 0000000000000000000000000000000000000000..08182906d4fe747f1c03376eb5697f530ac0eabb
--- /dev/null
+++ b/samples/texts/3727487/page_28.md
@@ -0,0 +1,10 @@
+FIG. 31: (Color online) Energy spectrum and LDOS in the vicinity a topological phase transition. The top panel shows the spectra of a system with $\Gamma = 21.5E_α$ without disorder (blue diamonds) and in the presence of charged impurities ($n_{imp} = 0.7/\mu m$, orange squares). Note the absence of a gap. The corresponding LDOS are shown in the middle (clean system) and bottom (disordered wire) panels. The spectral weight is distributed over the entire energy range, including at positions near the ends of the wire.
+
+of a tunneling experiment, e.g., the specific form of the tunneling matrix elements and the role of finite temperature, that may limit the applicability of our conclusions. We find that values of the parameters consistent with an unambiguous identification of the Majorana bound state in the semiconductor nanowire are well within a realistic parameter regime.
+
+The tunneling current between a metallic tip and the nanowire can be evaluated within the Keldysh nonequilibrium formalism [77]. In terms of real space Green's functions we have
+
+$$ I = \frac{e}{h} \int d\omega \text{ReTr} \left\{ \left[1 - G_0^R(\omega)\Gamma^R(\omega - eV)\right]^{-1} \left[(1 - 2f_{\omega-eV})G_0^R(\omega)\Gamma^R(\omega - eV)\right. \right. \\ 
+\left. \left. + 2(f_{\omega-eV} - f_\omega)G_0^R\Gamma^A(\omega - eV) - (1 - 2f_\omega)G_0^A(\omega)\Gamma^A(\omega - eV)\right] \left[1 - G_0^A(\omega)\Gamma^A(\omega - eV)\right]^{-1} \right\}, \quad (46) $$
+
+where $V$ is the bias voltage applied between the tip and the nanowire and $f_\omega = 1/(e^{\beta\omega} + 1)$ is the Fermi-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_29.md b/samples/texts/3727487/page_29.md
new file mode 100644
index 0000000000000000000000000000000000000000..76225c4492574abfe3d1deee53d028407cc92554
--- /dev/null
+++ b/samples/texts/3727487/page_29.md
@@ -0,0 +1,27 @@
+FIG. 32: (Color online) Differential conductance for tunneling into the end of a superconducting nanowire. The curves correspond to different values of the Zeeman field ranging from $\Gamma = 11E_\alpha$ (bottom) to $\Gamma = 36E_\alpha$ (top) in steps of $E_\alpha$. The curves were shifted vertically for clarity. The trivial SC phase ($\Gamma < 21E_\alpha$) is characterized by a gap that vanishes in the critical region ($\Gamma \approx 21E_\alpha$). The signature of the topological phase is the zero-energy peak resulting from tunneling into the Majorana mode. The differential conductance was calculated at a temperature $T \approx 50mK$ for a disordered wire with a linear density of charged impurities $n_{imp} = 7/\mu m$.
+
+Dirac distribution function corresponding to a temperature $k_b T = \beta^{-1}$. The retarded (advanced) Green's function for the nanowire has the expression
+
+$$G_0^{R(A)}(\mathbf{r}, \mathbf{r}', \omega) = \sum_n \left\{ \frac{u_n^*(\mathbf{r})u_n(\mathbf{r}')}{\omega - E_n \pm i\eta} + \frac{v_n^*(\mathbf{r})v_n(\mathbf{r}')}{\omega + E_n \pm i\eta} \right\}, \quad (47)$$
+
+where $u_n$ and $v_n$ are the particle and hole components of the wave function corresponding to the energy $E_n$. The wave functions and the energies are obtained by diagonalizing the effective BdG Hamiltonian for the nanowire, including the contribution from disorder, as described in the previous sections. The matrices $\Gamma^{R(A)}$ contain information about the tip and the tip–nanowire coupling. Specifically, we have
+
+$$\Gamma^{R(A)}(\mathbf{r}, \mathbf{r}', \omega) = \gamma_{\mathbf{r},\mathbf{r}'} \int dx \frac{\nu(x)}{\omega - x \pm i\eta}, \quad (48)$$
+
+where $\nu(x)$ represents the density of states of the metallic
+
+tip and $\gamma_{\mathbf{r},\mathbf{r}}$ depends on the tunneling matrix elements between the tip and the wire. We note that in Eq. (46) the trace is taken over the position vectors. We consider a tunneling model in which the amplitude of the tunneling matrix elements vary exponentially with the distance from the metallic tip. Specifically, we have $\gamma_{\mathbf{r},\mathbf{r}}\gamma_0\theta_r\theta_{r'}$, where $\gamma_0$ gives the overall strength of the tip–nanowire coupling and the position–dependent factor is
+
+$$\theta_r = e^{-\frac{1}{2}\xi[\sqrt{(x-x_{tip})^2+(y-y_{tip})^2+(z-z_{tip})^2-x_{tip}}],} \quad (49)$$
+
+with $(x_{tip}, y_{tip}, z_{tip})$ being the position vector for the tip and $\xi$ a characteristic length scale associated with the exponential decay of the tip–wire coupling. In the numerical calculations we take $\xi = 0.4a$ and $(x_{tip}, y_{tip}, z_{tip}) = (-3a, L_y/2, L_z/2)$, i.e., the tip is at a distance equal with three lattice spacings away from the end of the wire. With these choices, the differential conductance becomes
+
+$$\frac{dI}{dV} \propto - \sum_n [f'(E_n - eV)|\langle u_n | \theta \rangle|^2 + f'(-E_n - eV)|\langle v_n | \theta \rangle|^2], \quad (50)$$
+
+where the matrix elements $\langle u_n | \theta \rangle$ and $\langle v_n | \theta \rangle$ involve summations over the lattice sites of the nanowire system and provide the amplitudes for tunneling into specific states. Finite temperature effects are incorporated through the derivatives of the Fermi–Dirac function, $f'$.
+
+The dependence of the tunneling differential conductance on the bias voltage for a superconducting nanowire with disorder is shown in Fig. 32. Different curves correspond to different values of the Zeeman field between $\Gamma = 11E_\alpha$ (bottom) and $\Gamma = 36E_\alpha$ (top) and are shifted vertically for clarity. The temperature used in the calculation is $50mK$, a value that can be easily reached experimentally. Lower temperature values will generate sharper features, but the overall picture remains qualitatively the same. Note that the closing of the gap in the critical region between the trivial SC phase and the topological SC phase can be clearly observed. In this region $dI/dV$ has features over the entire low-energy range, as discussed in the previous subsection. The Majorana bound state at $\Gamma > 22E_\alpha$ is clearly marked by a sharp peak at $V=0$, separated by a gap from other finite energy features. We conclude that measuring the tunneling differential conductance can provide a clear and unambiguous probe for Majorana bound states in semiconductor nanowires.
+
+## VI. CONCLUSIONS
+
+In conclusion, we have developed a comprehensive theory for the realization and the observation of the emergent non-Abelian Majorana mode in semiconductor (e.g., InAs, InSb) nanowires proximity coupled to an ordinary s-wave superconductor (e.g., Al, Nb) in the presence of a Zeeman splitting induced by an external magnetic field.
\ No newline at end of file
diff --git a/samples/texts/3727487/page_3.md b/samples/texts/3727487/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..fd9b9db945d914b665bee88a29b4202f5ff13c01
--- /dev/null
+++ b/samples/texts/3727487/page_3.md
@@ -0,0 +1,35 @@
+which corresponds to a nanowire with dimensions $L_z = 5$ nm, $L_y = 130$ nm, and $L_x$ between 5 µm and 10 µm.
+
+A brute force diagonalization of Hamiltonian (1) on a lattice containing more than $10^7$ sites would be numerically very expensive. More importantly, the relevant energy scales in the problem are of the order of a few meV allowing to construct the low-energy model as we show below and significantly reduce the Hilbert space. The largest energy scale is given by the gap between the lowest sub-bands, which, for example, for the first and second subband and the parameters used in our calculation is $\Delta E_{sb} \approx 1.6$ meV. Consequently, we are only interested in the low-energy eigenstates of the Hamiltonian (1). To obtain these states, we take advantage of the fact that the eigenproblem for $H_0$ can be solved analytically and notice that $H_{SOI}$ can be treated as a small perturbation. Explicitly, the eigenstates of $H_0$ are
+
+$$\psi_{n\sigma}(\mathbf{i}) = \prod_{\lambda=1}^{3} \sqrt{\frac{2}{N_{\lambda}+1}} \sin \frac{\pi n_{\lambda} i_{\lambda}}{N_{\lambda}+1} \chi_{\sigma}, \quad (2)$$
+
+where $\mathbf{n} = (n_x, n_y, n_z)$ with $1 \le n_\lambda \le N_\lambda$, and $\chi_\sigma$ is an eigenstate of the $\hat{\sigma}_z$ spin operator. The corresponding eigenvalues are
+
+$$\epsilon_n = -2t_0 \left( \cos \frac{\pi n_x}{N_x + 1} + \cos \frac{\pi n_y}{N_y + 1} + \cos \frac{\pi n_z}{N_z + 1} - 3 \right) - \mu, \quad (3)$$
+
+where the chemical potential in the semiconductor $\mu$ is calculated from the bottom of the band. We project the quantum problem into the low-energy subspace spanned by the eigenstates $\psi_{n\sigma}$ with energies below a certain cutoff value, $\epsilon_n < \epsilon_{max}$, where the cutoff energy $\epsilon_{max}$ is typically of the order 15meV, i.e., one order of magnitude larger than the inter sub-band spacing. The number of states in this low-energy basis is of the order $10^3$ and thus the numerical complexity of the problem is significantly reduced. The matrix elements of the SOI Hamiltonian are
+
+$$\langle \psi_{n\sigma} | H_{\text{SOI}} | \psi_{n'\sigma'} \rangle = \alpha \delta_{n_z n'_z} \left\{ \frac{1 - (-1)^{n_x+n'_x}}{N_x + 1} (i\hat{\sigma}_y)_{\sigma\sigma'} \times \frac{\sin \frac{\pi n_x}{N_x+1} \sin \frac{\pi n'_x}{N_x+1}}{\cos \frac{\pi n_x}{N_x+1} - \cos \frac{\pi n'_x}{N_x+1}} \delta_{n_y n'_y} - [x \leftrightarrow y] \right\}, (4)$$
+
+where the second term in the parentheses is obtained from the first term by exchanging the x and y indices.
+
+To realize nontrivial topological states that support Majorana modes, it is necessary that an odd number of sub-bands be occupied. [40] In the simplest case of a single sub-band nanowire, this can be achieved with the help of a Zeeman field (i.e. a spin splitting)
+
+$$H_{\text{Zeeman}} = \Gamma \sum_{i,\sigma,\sigma'} c_{i\sigma}^{\dagger} (\hat{\sigma}_x)_{\sigma\sigma'} c_{i\sigma'}, \quad (5)$$
+
+which opens a gap at small momenta and removes one of the helicities that characterize the spectrum of a Rashba-coupled electron system. In the multi-band nanowires
+
+the situation is more complicated (see below) but the Zeeman term is essential to avoid fermion doubling. The Zeeman term can be obtained using an external magnetic field applied along the $\hat{x}$ axis, $\Gamma = g\mu_B B_x/2$. When the chemical potential lies within one of the Zeeman gaps at $\mathbf{k}=0$, the condition for odd sub-band occupation is satisfied and thus fermion doubling is avoided, which allows for the existence of Majorana modes at the ends of the nanowire. Note that in SM with a large g-factor, e.g., $g_{InAs} \sim 10$ and $g_{InSb} \sim 50$, relatively small in-plane magnetic fields can open a sizable gap without significantly perturbing superconductivity. This is a crucial ingredient of the present proposal which is particularly important in the context of the effect of disorder on the topological phase as we discuss in Sec. IV. For example, in InAs a magnetic field $B_x \sim 1$T corresponds to $\Gamma \sim 1$meV. Finally, we note that in the basis given by Eq. (2) the Zeeman term has the simple form
+
+$$\langle \psi_{n\sigma} | H_{\text{Zeeman}} | \psi_{n'\sigma'} \rangle = \Gamma \delta_{nn'} \delta_{\bar{\sigma}\sigma'}, \quad (6)$$
+
+where $\bar{\sigma} = -\sigma$.
+
+## B. Proximity-induced superconductivity
+
+In addition to spin-orbit interaction and Zeeman spin splitting, the only other physical ingredient necessary for creating the Majorana mode is the ordinary s-wave superconductivity which can be induced in the semiconductor by the proximity effect, through coupling to an s-wave superconductor (SC). A model of the full system that supports the Majorana modes contains, in addition to the nanowire and Zeeman terms, Eqns. (1) and (5), respectively, the Hamiltonian for the superconductor, $H_{SC}$, and a term describing the nanowire-SC tunneling, $H_{nw-SC}$. We note that, to account quantitatively for the superconductivity induced in the nanowire, one should also include possible electron-phonon and electron-electron interactions within the SM itself, $H_{int}$. These interactions may enhance or inhibit the induced effect, depending on the details of the SM material [53, 54]. In this paper we do not take into account effect of interactions in the semiconductor on the proximity effect and use a simple model for the proximity effect using tunneling Hamiltonian approach [55] which is appropriate for the sample geometry considered here (thin semiconductor lying on top of the superconductor). The effects of interactions on the topological superconducting phase were recently considered in Refs. [52, 56–58]. In addition to affecting the proximity-induced SC gap, the repulsive Coulomb interactions among the SM electrons lead to an effective enhancement of the Zeeman splitting which might be favorable for inducing topological superconductivity [58].
+
+Thus, the total Hamiltonian for our model of semiconductor/superconductor heterostructure is given by
+
+$$H_{\text{tot}} = H_{\text{nw}} + H_{\text{int}} + H_{\text{Zeeman}} + H_{\text{SC}} + H_{\text{nw-SC}}, \quad (7)$$
\ No newline at end of file
diff --git a/samples/texts/3727487/page_30.md b/samples/texts/3727487/page_30.md
new file mode 100644
index 0000000000000000000000000000000000000000..6d5b2f5c56da12aa057d8f3c38853af7c9736154
--- /dev/null
+++ b/samples/texts/3727487/page_30.md
@@ -0,0 +1,11 @@
+The importance of our work lies in the thorough investigation of the experimental parameter space, which is required in order to predict the optimal parameter regime to search for the Majorana mode in nanowires. Since the number of possible physical parameters in the problem is large (e.g., electron density in the nanowire, geometric size of the wire, chemical potential, the strength of spin-orbit coupling, the superconducting gap, the hopping matrix elements between the semiconductor and the superconductor, the strengths of various disorder in the semiconductor, or in the superconductor, or at the interface between them), theoretical guidance, as provided in this work, is highly desirable for the success of the experimental search for the Majorana fermion in solid-state systems. Aside from the obvious conclusions (e.g., strong spin-orbit coupling and large Lande g-factor in the semiconductor, large superconducting gap in the superconductor, and low disorder everywhere stabilize the topological phase), we have discovered several unexpected results. In particular, we find somewhat surprisingly that the strict one-dimensional limit with purely one-subband occupancy for the nanowire, as originally envisioned by Kitaev [11] and later used by many researchers [14, 37, 38], is not only unnecessary, but is in fact detrimental to creating Majorana modes. In the presence of disorder, the optimal system should have a few (3-5) occupied subbands in the nanowire for the creation of the Majorana modes with maximal stability. This is, of course, great news from the experimental perspective, because fabricating strictly 1D semiconductor nanowires with pure one subband occupancy is a challenging task. Another important result of our analysis is the relative immunity of the Majorana modes to the presence of disorder in the system. The most dangerous disorder mechanisms are due to charged impurity centers in the semiconductor and to inhomogeneous hopping across the semiconductor-superconductor interface. Our work suggests that the optimal nanowires for observing the Majorana mode should not only have as little charge impurity disorder as possible in the semiconductor, but they should also have a thin insulating layer separating the semiconductor and superconductor (so that the tunneling across the interface is not too strong), as well as some non-uniformity in the tunneling amplitude between the semiconductor and the superconductor across the width (but not the length) of the nanowire. Our detailed numerical calculations establish that the zero-energy Majorana modes should clearly show up in experiments, even in the presence of considerable disorder in the nanowires. In addition, we establish that the disorder in the superconductor has little effect on the Majorana mode in the nanowire. Another salient aspect of our work is the detailed calculation of the expected tunneling spectroscopy spectra for observing the Majorana mode in the nanowire using realistic physical parameters.
+
+Our results establish that the predicted topological quantum phase transition between the trivial phase with no Majorana mode to the topological phase with a well-defined zero energy Majorana bound state should be clearly observable as a striking zero-bias anomaly in the tunneling current when the Zeeman splitting is tuned through the quantum critical point separating the two phases. More importantly, we calculate realistic tunneling spectra in the presence of uncontrolled spurious bound states in the system which are invariably present in real samples due to the localized random impurities in the semiconductor environment, clearly showing how to discern the topological features associated with the Majorana bound states from the background of contributions due to trivial bound states caused by impurities in the system.
+
+Our work emphasizes the tunneling measurements, which would directly establish the existence of a robust zero energy mode in the system, providing the necessary condition for the existence of the Majorana fermion. What we have done here is to develop a detailed theory for the existence of a topological phase in the semiconductor-superconductor heterostructure, by taking into account essentially all of the relevant physical effects. Once the presence of a robust zero energy mode is established, hence the necessary condition for the existence of the Majorana is realized, one must move on to establish the sufficient condition, which would obviously be a harder task. Several ideas for establishing definitively the existence of the Majorana mode (and its non-Abelian braiding statistics nature) have already been suggested in the literature, including experiments involving the fractional Josephson effect [11, 14, 33, 37, 78], the quantized differential conductance [76, 79], and Majorana interferometry [80–82].
+
+Our work establishes the realistic likelihood of the existence within laboratory conditions of the non-Abelian Majorana zero-energy mode in spin-orbit interacting semiconductor nanowires proximity-coupled to ordinary superconductors. We also establish that tunneling spectroscopy is one of the easiest techniques to directly observe the elusive Majorana in realistic solid-state systems. Greater challenges, such as carrying out topological quantum computation, lie ahead once the laboratory existence of the Majorana mode is established experimentally.
+
+## Acknowledgments
+
+We would like to thank Leonid Glazman, Matthew Fisher and Chetan Nayak for discussions. This work is supported by the DARPA QuEST, JQI-NSF-PFC, Microsoft Q (SDS) and WVU startup funds (TS).
\ No newline at end of file
diff --git a/samples/texts/3727487/page_31.md b/samples/texts/3727487/page_31.md
new file mode 100644
index 0000000000000000000000000000000000000000..14a868cf48eff53c6f7685dade7dda402f9c292a
--- /dev/null
+++ b/samples/texts/3727487/page_31.md
@@ -0,0 +1,179 @@
+[1] F. Wilczek, NATURE PHYSICS 5, 614 (2009).
+
+[2] A. Stern, Nature **464**, 187 (2010).
+
+[3] M. Franz, Physics **3**, 24 (2010).
+
+[4] C. Nayak, Nature **464**, 693 (2010).
+
+[5] E. Majorana, Il Nuovo Cimento (1924-1942) **14**, 171 (1937).
+
+[6] B. G. Levi, Physics Today **64**, 20 (2011).
+
+[7] R. F. Service, Science **332**, 193 (2011).
+
+[8] C. Nayak and F. Wilczek, Nucl. Phys. B **479**, 529 (1996).
+
+[9] N. Read and D. Green, Phys. Rev. B **61**, 10267 (2000).
+
+[10] D. A. Ivanov, Phys. Rev. Lett. **86**, 268 (2001).
+
+[11] A. Y. Kitaev, Physics-Uspekhi **44**, 131 (2001), cond-mat/0010440.
+
+[12] A. Stern, F. von Oppen, and E. Mariani, Phys. Rev. B **70**, 205338 (2004).
+
+[13] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. **80**, 1083 (2008), arXiv:0707.1889.
+
+[14] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nature Physics **7**, 412 (2011), 1006.4395.
+
+[15] A. Y. Kitaev, Ann. Phys.(N.Y.) **303**, 2 (2003).
+
+[16] S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. **94**, 166802 (2005).
+
+[17] A. Stern and B. I. Halperin, Phys. Rev. Lett. **96**, 016802 (2006), cond-mat/0508447.
+
+[18] P. Bonderson, A. Kitaev, and K. Shtengel, Phys. Rev. Lett. **96**, 016803 (2006), cond-mat/0508616.
+
+[19] S. Bravyi, Phys. Rev. A **73**, 042313 (2006).
+
+[20] S. Tewari, C. Zhang, S. Das Sarma, C. Nayak, and D.-H. Lee, Phys. Rev. Lett. **100**, 027001 (2008).
+
+[21] P. Bonderson, D. J. Clarke, C. Nayak, and K. Shtengel, Phys. Rev. Lett. **104**, 180505 (2010), arXiv:0911.2691.
+
+[22] F. Hassler, A. R. Akhmerov, C.-Y. Hou, and C. W. J. Beenakker, New Journal of Physics **12**, 125002 (2010).
+
+[23] J. D. Sau, S. Tewari, and S. Das Sarma (2010), arXiv:1007.4204.
+
+[24] P. Bonderson and R. M. Lutchyn, Phys. Rev. Lett. **106**, 130505 (2011).
+
+[25] D. J. Clarke, J. D. Sau, and S. Tewari, ArXiv e-prints (2010), 1012.0296.
+
+[26] K. Flensberg, Phys. Rev. Lett. **106**, 090503 (2011).
+
+[27] F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, ArXiv e-prints (2011), 1105.0315.
+
+[28] L. Jiang, C. L. Kane, and J. Preskill, Phys. Rev. Lett. **106**, 130504 (2011).
+
+[29] O. Zilberberg, B. Braunecker, and D. Loss, Phys. Rev. A **77**, 012327 (2008)
+
+[30] L. Fu and C. L. Kane, Phys. Rev. Lett. **100**, 096407 (2008), arXiv:0707.1692.
+
+[31] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. **104**, 040502 (2010).
+
+[32] S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B **73**, 220502 (2006).
+
+[33] L. Fu and C. L. Kane, Phys. Rev. B **79**, 161408 (2009).
+
+[34] M. Wimmer, A. R. Akhmerov, M. V. Medvedyeva, J. Tworzydlo, and C. W. J. Beenakker, Phys. Rev. Lett. **105**, 046803 (2010).
+
+[35] P. A. Lee, ArXiv e-prints (2009), 0907.2681.
+
+[36] J. Alicea, Phys. Rev. B **81**, 125318 (2010).
+
+[37] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.
+Lett. **105**, 077001 (2010), arXiv:1002.4033.
+
+[38] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev.
+Lett.
+**105**, 177002 (2010).
+
+[39] J.D.Sau,S.Tewari,R.M.Lutchyn,T.D.Stanescu,and
+S.Das Sarma,Phys.Rev.B **82**, 214509(2010).
+
+[40] R.M.Lutchyn,T.D.Stanescu,and S.Das Sarma,
+Phys.
+Rev.Lett.
+**106**, 127001(2011).
+
+[41] L.Mao,M.Gong,E.Dumitrescu,S.Tewari,and
+C.Zhang,ArXiv e-prints (2011), 1105.3483.
+
+[42] A.C.Potter and P.A.Lee,Physical Review Letters **105**,
+227003 (2010), 1007.4569.
+
+[43] A.C.Potter and P.A.Lee,Phys.Rev.B **83**, 094525
+(2011), 1011.6371.
+
+[44] X.-L Qi,T.L.Hughes,and S.-C.Zhang,Phys.Rev.B
+**82**, 184516 (2010).
+
+[45] J.Linder,Y.Tanaka,T.Yokoyama,A.Sudbo,and N.
+Nagaosa,Phys.Rev.Lett.
+**104**, 067001 (2010)
+
+[46] S.Nadj-Perge,S.M.Frolov,E.P.A.M.Bakkers,and
+L.P.Kouwenhoven,NATURE **468**, 1084 (2010).
+
+[47] H.A.Nilsson,P.Caroff,C.Thelander,M.Larsson,J.B.
+Wagner,L.-E.Wernersson,L.Samuelson,and H.Q.Xu,
+Nano Letters **9**, 3151 (2009).
+
+[48] V.Aleshkin,V.Gavrilenko,A.Ikonnikov,S.Krish-
+topenko,Y.Sadofyev,and K.Spirin,Semiconductors **42**,
+828 (2008), ISSN 1063-7826.
+
+[49] Y.-J.Doh,J.van Dam,A.Roest,E.Bakkers,L.Kouwen-
+hoven,and S.D.Franceschi,Science **309**, 272 (2005).
+
+[50] J.A.van Dam,Y.V.Nazarov,E.P.A.M.Bakkers,
+S.D.Franceschi,and L.P.Kouwenhoven,Nature **442**,
+667 (2006).
+
+[51] A.Chrestin,T.Matsuyama,and U.Merkt,Phys.Rev.
+B **55**, 8457 (1997).
+
+[52] R.M.Lutchyn and M.P.A.Fisher,ArXiv e-prints
+(2011), 1104.2358.
+
+[53] P.G.de Gennes,Rev.Mod.Phys.
+**36**, 225 (1964).
+
+[54] A.S.Alexandrov and V.V.Kabanov,Phys.Rev.B
+**78**,
+132510 (2008).
+
+[55] W.L.Mcmillan,Phys.Rev.
+**175**, 537 (1968).
+
+[56] S.Gangadharaiah,B.Braunecker,P.Simon,and D.Loss,
+ArXiv e-prints (2011), 1101.0094.
+
+[57] E.Sela,A.Altland,and A.Rosch,ArXiv e-prints
+(2011),
+1103.4969.
+
+[58] E.M.Stoudenmire,J.Alicea,O.A.Starykh,and
+M.P.A.Fisher,ArXiv e-prints
+(2011), 1104.5493.
+
+[59] J.D.Sau,S.Tewari,R.M.Lutchyn,T.D.Stanescu,and
+S.Das Sarma,Phys.Rev.B **82**, 214509 (2010).
+
+[60] T.D.Stanescu,J.D.Sau,R.M.Lutchyn,and
+S.Das Sarma,Phys.Rev.B **81**, 241310 (2010).
+
+[61] P.Ghosh,J.D.Sau,S.Tewari,and S.Das Sarma,
+Phys.
+Rev.B **82**, 184525 (2010).
+
+[62] A.R.Akhmerov,J.P.Dahlhaus,F.Hassler,M.Wimmer,
+and C.W.J.Beenakker,Phys.Rev.Lett.
+**106**, 057001
+(2011).
+
+[63] C.W.J.Beenakker,J.P.Dahlhaus,M.Wimmer,and
+A.R.Akhmerov,Phys.Rev.B
+**83**, 085413 (2011).
+
+[64] A.A.Abrikosov and L.P.Gor'kov,Soviet Phys.JETP
+**12**,
+**1243** (1961).
+
+[65] A.C.Potter and P.A.Lee,Phys.Rev.B
+84, 059906(E)
+(2011);R.M.Lutchyn,T.Stanescu and S.Das Sarma,
+arXiv:1110.5643 (2011).
+
+[66] S.Tewari,T.D.Stanescu,J.D.Sau,and S.D.Sarma,
+New Journal of Physics
+**13**, 065004 (2011).
\ No newline at end of file
diff --git a/samples/texts/3727487/page_32.md b/samples/texts/3727487/page_32.md
new file mode 100644
index 0000000000000000000000000000000000000000..1a2fd6c87ea67cb8163ff791992d609958ebbefb
--- /dev/null
+++ b/samples/texts/3727487/page_32.md
@@ -0,0 +1,35 @@
+[67] O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B 63, 224204 (2001).
+
+[68] I. A. Gruzberg, N. Read, and S. Vishveshwara, Phys. Rev. B 71, 245124 (2005).
+
+[69] P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, ArXiv e-prints (2011), 1103.2746.
+
+[70] P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, ArXiv e-prints (2011), 1104.1531.
+
+[71] S. Das Sarma and W.-Y. Lai, Phys. Rev. B 32, 1401 (1985).
+
+[72] M. Yamashiro, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 56, 7847 (1997).
+
+[73] C. J. Bolech and E. Demler, Phys. Rev. Lett. 98, 237002 (2007).
+
+[74] S. Tewari, C. Zhang, S. Das Sarma, C. Nayak, and D.-H. Lee, Phys. Rev. Lett. 100, 027001 (2008).
+
+[75] J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker,
+
+Phys. Rev. Lett. 101, 120403 (2008)
+
+[76] K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. 103, 237001 (2009)
+
+[77] C. Berthod and T. Giamarchi, ArXiv e-prints (2011), 1102.3895.
+
+[78] H. J. Kwon, K. Sengupta, and V. M. Yakovenko, Eur. Phys. J. B 37, 349 (2003).
+
+[79] M. Wimmer, A.R. Akhmerov, J.P. Dahlhaus, C.W.J. Beenakker, New J. Phys. 13, 053016 (2011)
+
+[80] L. Fu and C. L. Kane , Phys. Rev. Lett. 102, 216403 (2009)
+
+[81] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,
+Phys. Rev. Lett. 102, 216404 (2009)
+
+[82] J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. B
+84, 085109 (2011)
\ No newline at end of file
diff --git a/samples/texts/3727487/page_4.md b/samples/texts/3727487/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..940223bde77a9c9ec22e121cc208b6f8ffa04713
--- /dev/null
+++ b/samples/texts/3727487/page_4.md
@@ -0,0 +1,93 @@
+where the tunneling term reads
+
+$$
+H_{\text{nw-SC}} = \sum_{i,j,\sigma} [\tilde{t}_{i,j} c_{i\sigma}^{\dagger} a_{j\sigma} + \text{h.c.}], \quad (8)
+$$
+
+with $c_{i\sigma}$ and $a_{j\sigma}$ being electron destruction operators
+acting within the SM and SC, respectively. We assume
+that the matrix elements $\tilde{t}_{i,j}$ couple the sites of the SC
+located at the interface, $\mathbf{j} = (\mathbf{r}_{\parallel}, z_{\text{interface}})/a$, where $\mathbf{r}_{\parallel}$ is position vector in a plane parallel to the interface and
+$z_{\text{interface}}$ is the coordinate of the interface layer in a slab geometry, to the first layer of the semiconductor wire,
+$\mathbf{i} = (i_x, i_y, 1)$.
+
+The proximity effect can be now derived by integrating
+out superconducting degrees of freedom in Eq. (7) and
+considering the resulting effective low-energy theory for
+the SM. To identify the form of the effective low-energy
+Hamiltonian, we consider first the case of a single-band
+SM coupled to an s-wave SC through an infinite planar
+interface, then we address the specific issues related to
+multiband nanowires.
+
+1. Infinite planar interface
+
+Within the tunneling Hamiltonian approach, the
+proximity-effect induced by an s-wave SC can be cap-
+tured by integrating out the superconducting degrees of
+freedom and calculating the surface self-energy due to
+the exchange of electrons between SC and SM [59]. We
+briefly review this approach here and use these results
+later when discussing the disordered s-wave superconduc-
+tors. The interface self-energy is given by
+
+$$
+\Sigma_{\sigma\sigma'}(\mathbf{r}, \mathbf{r}', \omega) = \mathrm{Tr}_{\mathbf{r}_1, \mathbf{r}_2} \tilde{t}(\mathbf{r}, \mathbf{r}_1) G_{\sigma\sigma'}(\mathbf{r}_1, \mathbf{r}_2, \omega) \tilde{t}^\dagger(\mathbf{r}_2, \mathbf{r}'), \quad (9)
+$$
+
+where $\tilde{t}(\mathbf{r}, \mathbf{r}_1)$ is the matrix describing tunneling between semiconductor and superconductor. To illustrate the basic physics, we use here the simplest form for tunneling matrix elements $\tilde{t}(\mathbf{r}, \mathbf{r}_1) = \tilde{t}\delta(z)\delta(z_1)\delta(\mathbf{r}^\| - \mathbf{r}_1^\|)$ with $\mathbf{r}^\|$ and $z$ denoting in-plane and out-of-plane coordinates. After some algebra, the surface self-energy $\Sigma(\mathbf{r}-\mathbf{r}', \omega)$ is given by
+
+$$
+\Sigma(\mathbf{r} - \mathbf{r}', \omega) = |\tilde{t}|^2 \delta(z) \delta(z') \int \frac{d^3\mathbf{p}}{(2\pi)^3} e^{i\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}')} G(\mathbf{p}, \omega), \quad (10)
+$$
+
+and finally becomes in the momentum space
+
+$$
+\Sigma(p^{\parallel}, \omega) &= |\tilde{t}|^2 \int \frac{dp_z}{2\pi} G(p, \omega) \tag{11} \\
+&= |\tilde{t}|^2 \int_{-\Lambda}^{\Lambda} d\epsilon \int \frac{dp_z}{2\pi} \delta(\epsilon - \xi_p) G(\epsilon, \omega). \tag{12}
+$$
+
+where $\Lambda$ is half bandwidth. The density of states
+$\nu(\varepsilon, \mathbf{p}^{\parallel}) = \int \frac{dp_z}{2\pi} \delta(\varepsilon - \xi_{\mathbf{p}})$ is usually a weakly-dependent
+function of momenta and energy $\nu(\varepsilon_F, \mathbf{p}^{\parallel}) \approx \nu(\varepsilon_F) =$
+
+$2\sqrt{1-\zeta^2}/\Lambda$ with $\zeta = (\Lambda-\varepsilon_F)/\Lambda$ and $\varepsilon_F$ being the Fermi level in the superconductor. With these approximations, the surface self-energy becomes
+
+$$
+\Sigma(\omega) = |\tilde{t}|^2 \nu(\epsilon_F) \int d\varepsilon G(\varepsilon, \omega) \quad (13)
+$$
+
+$$
+= -|\tilde{t}|^2\nu(\epsilon_F) \left[ \frac{\omega\tau_0 + \Delta_0\tau_x}{\sqrt{\Delta_0^2 - \omega^2}} + \frac{\zeta}{1 - \zeta^2\tau_z} \right] . \quad (14)
+$$
+
+In the homogeneous case the last term in Eq. (13) rep-
+resents a shift of the chemical potential and can be ne-
+glected as the chemical potential should be determined
+self-consistently by solving the appropriate equation for
+the fixed total electron density.
+
+We can now include the surface self-energy Σ(ω) into
+the SM Hamiltonian and study the effective low-energy
+model for the semiconductor. This can be done by inves-
+tigating the poles of the SM Green’s function
+
+$$
+G^{-1}(\mathbf{k}, \omega) = \omega \left( 1 + \frac{\gamma}{\sqrt{\Delta_0^2 - \omega^2}} \right) - \Gamma \hat{\sigma}_z \quad (15) \\
+- [\xi_k + \alpha_R (k_y \hat{\sigma}_x - k_x \hat{\sigma}_y)] \tau_z - \frac{\gamma \Delta_0}{\sqrt{\Delta_0^2 - \omega^2}} \hat{\tau}_x,
+$$
+
+where $\xi_k = h^2 k_x^2 / (2m^* - m)$, $\alpha_R$ is the Rashba coupling, $\Gamma$ the strength of a Zeeman field oriented perpendicular to the interface, $\Delta_0$ is the value of the superconducting gap inside the SC, and $\gamma$ is the effective SM-SC coupling. In the calculations, in addition to the values specified in Sec. II A, i.e., $m^* = 0.04m_0$ and $\alpha_R = 0.1\text{eV}\cdot\text{Å}$, we have $\Delta_0 = 1\text{meV}$. The effecting coupling $\gamma = \tilde{t}^2 |\psi(i_z = 1)|^2 \nu(\epsilon_F)$ depends on the transparency of the interface, $\tilde{t}$, the amplitude of the SM wave function at the interface, $\psi(i_z = 1)$, and the local density of states of the non-superconducting metal at the interface, which can be expressed in terms of the half-bandwidth $\Lambda$ and the Fermi energy $\epsilon_F$ of the metal.[59] Note that the Green's function (15) is written in the Nambu spinor basis ($u_\uparrow, u_\downarrow, v_\downarrow, -v_\uparrow)^T$ using the Pauli matrices $\hat{\tau}_\lambda$ and $\sigma_\lambda$ that correspond to the Nambu and spin spaces, respectively. The identity matrices $\tau_0$ and $\sigma_0$ are omitted for simplicity.
+
+Explicit comparison between the effective theory de-
+scribed by Eq. (15) and microscopic tight-binding
+calculations [59] has shown remarkable agreement. A sim-
+ilar effective description has proven extremely accurate in
+describing the proximity effect induced at a topological
+insulator – superconductor interface.[60] Can the low–
+energy physics contained in Eq. (15) be captured by
+an effective Hamiltonian description? To address this
+question, we determine the poles of the Green’s function
+at frequencies within the superconducting gap, $\omega < \Delta_0$,
+i.e., we solve the Bogoliubov–de Gennes (BdG) equation
\ No newline at end of file
diff --git a/samples/texts/3727487/page_5.md b/samples/texts/3727487/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..dd2f601f0b519634e43ae7b1f5344478fc164022
--- /dev/null
+++ b/samples/texts/3727487/page_5.md
@@ -0,0 +1,56 @@
+$$
+\det[G^{-1}] = 0. \text{ Explicitly, we have}
+$$
+
+$$
+\begin{equation} \tag{16}
+\begin{split}
+\omega^2 \left( 1 + \frac{\gamma}{\sqrt{\Delta_0^2 - \omega^2}} \right)^2 &= \xi_k^2 + \lambda_k^2 + \Gamma^2 + \frac{\gamma^2 \Delta_0^2}{\Delta_0^2 - \omega^2} \\
+&\quad - 2\sqrt{\xi_k^2 (\lambda_k^2 + \Gamma^2) + \frac{\gamma^2 \Delta_0^2 \Gamma^2}{\Delta_0^2 - \omega^2}}.
+\end{split}
+\end{equation}
+$$
+
+Note that we have considered only the lowest energy
+mode. In Eq. (16) dynamical effects are generated
+by the frequency dependence of the proximity–induced
+terms containing the expression $\gamma/\sqrt{\Delta_0^2 - \omega^2}$, with a rel-
+ative magnitude that depends on the SM-SC coupling
+strength. In general, we distinguish a weak-coupling
+regime characterized by $\gamma \ll \Delta_0$ and a strong coupling
+regime, $\gamma \gg \Delta_0$. In the weak coupling regime we expect
+negligible dynamical effects at all energies that are not
+very close to the gap edge, $\omega = \Delta_0$. Neglecting the fre-
+quency dependence in the proximity–induced terms, the
+solution of Eq. (15) becomes
+
+$$
+E_k = Z \sqrt{\xi_k^2 + \lambda_k^2 + \Gamma^2 + \gamma^2 - 2\sqrt{\xi_k^2(\lambda_k^2 + \Gamma^2) + \gamma^2\Gamma^2}}, \quad (17)
+$$
+
+where $Z = (1+\gamma/\Delta_0)^{-1} < 1$ is the quiasparticle residue
+at zero energy.
+
+To evaluate the dynamical effects, we compare the BdG spectrum given by Eq. (17) with the full solution of Eq. (16). The results are shown in Fig. 1 for a weak cou- pling regime characterized by $\gamma = 0.25$meV (top panel) and at large coupling, $\gamma = 2$meV (bottom panel). Notice that, even for effective couplings larger than the gap, e.g., $\gamma = 2\Delta_0$, Eq. (17) represents a very good approximation of the low energy spectrum. What about couplings that are much larger than $\Delta_0$? To answer this question, let us consider the dependence of the low-energy spectrum on the coupling constant $\gamma$, and the Zeeman field $\Gamma$. For $\Gamma = 0$ the spectrum is gaped and the minimum of the gap is located at $k=0$ (see Fig. 1). Applying a Zeeman field reduces this minimum gap continuously and at the critical value $\Gamma_c = \sqrt{\gamma^2 + \mu^2}$ the spectrum becomes gap- less (see Fig. 1). For $\Gamma > \Gamma_c$ a gap opens again with a minimum at $k=0$ for $\Gamma \ge \Gamma_c$ and at a finite wave vector for large values of the Zeeman field. The dependence of the minimum gap on the Zeeman field is shown in Fig. 2 for three different values of the SM-SC coupling. The vanishing of the gap at the critical point $\Gamma = \Gamma_c$ marks a quantum phase transition from a normal SC at low Zee- man fields to a topological SC ( when $\Gamma > \Gamma_c$).[37, 59] The change in the location of the quasiparticle gap from $k=0$ to a finite wave vector is marked in Fig. 2 by a discontinuity in the slope at $\Gamma^* \ge \Gamma_c$. The optimal value of the excitation gap in the topological phase is obtained for $\Gamma \approx \Gamma^*$. This optimal value depends weakly on the ef- fective coupling $\gamma$, but varies strongly with the spin-orbit coupling.
+
+From the above analysis we conclude that the strong-
+coupling regime characterized by $\gamma \gg \Delta_0$ is not ex-
+perimentally desirable, as it would require extremely
+
+FIG. 1: (Color online) Top: Low-energy BdG spectrum of a semiconductor with proximity-induced superconductivity. The effective SM-SC coupling is $\gamma = 0.25$meV and the chemical potential is $\mu = 0$. The induced gap vanishes at $k = 0$ in the presence of a Zeeman field $\Gamma = \sqrt{\gamma^2 + \mu^2} = 0.25$meV. The full lines are obtained by solving Eq. (16), while the symbols represent the spectrum given by Eq. (17). Bottom: Low-energy BdG spectrum for $\gamma = 2$meV, $\mu = 0$ and three different values of the Zeeman field. The filled area corresponding to energies above $\Delta_0 = 1$meV represents the SC continuum. Note that for energies $E_k < \Delta_0/2$, Eq. (17) represents a very good approximation of the BdG spectrum even in the strong-coupling regime.
+
+high magnetic fields to reach the topologically nontrivial
+phase, i.e., Γ > √γ² + μ² ≫ Δ₀. In addition, it would
+be difficult to tune the chemical potential and drive the
+system into a topological superconducting phase for a
+large SM-SC coupling. Also, as follows from Eq. (17), the
+quasiparticle excitation spectrum decays with increasing
+γ/Δ₀ which leads to the reduced stability of the topo-
+logical phase against thermal fluctuations. Hence, an ex-
+perimentally useful interface should be characterized by
+an effective coupling γ of order Δ₀ or less, i.e., in the
+intermediate to weak-coupling regime. As shown above,
+in these regimes the BdG spectrum is accurately approx-
+imated by Eq. (17). Consequently, we can model the
+low-energy spin-orbit coupled semiconductor with prox-
+imity induced superconductivity using an effective tight-
\ No newline at end of file
diff --git a/samples/texts/3727487/page_6.md b/samples/texts/3727487/page_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..31a07d675ad72c8ff9347b8a8d1b496f6deb5b71
--- /dev/null
+++ b/samples/texts/3727487/page_6.md
@@ -0,0 +1,39 @@
+FIG. 2: (Color online) Dependence of the minimum quasiparticle excitation gap in the BdG spectrum given by Eq. (16) on the Zeeman field $\Gamma$ for different SM-SC couplings. The chemical potential is $\mu = 0$ and the Rashba coefficient is $\alpha_r = 0.15\text{eV}\cdot\text{Å}$ for the curve represented by small (green) circles and $\alpha_r = 0.1\text{eV}\cdot\text{Å}$ for the other three curves. The system becomes gapless at $\Gamma_c = \sqrt{\gamma^2 + \mu^2}$. The superconducting state with $\Gamma < \Gamma_c$ is topologically trivial, while for $\Gamma > \Gamma_c$ one has a topological superconductor that supports Majorana bound states. Note that the optimal quasiparticle gap for the topological SC has a weak dependence on the SM-SC coupling but varies strongly with the strength of the spin-orbit coupling.
+
+binding model given by the Hamiltonian
+
+$$H_{\text{eff}} = H_{\text{SM}} + H_{\text{Zeeman}} + H_{\Delta}, \quad (18)$$
+
+where the semiconductor term $H_{\text{SM}}$ has the same form as the Hamiltonian $H_{\text{nw}}$ for the nanowire given by Eq. (1) but with $N_x \to \infty$ and $N_y \to \infty$ and the Zeeman term $H_{\text{Zeeman}}$ is given by Eq. (5) with $\hat{\sigma}_x \to \hat{\sigma}_z$. In addition, all the energy scales are renormalized by a factor $Z = (1+\gamma/\Delta_0)^{-1} < 1$, i.e., $t_0 \to Zt_0$, $\alpha \to Z\alpha$, etc. The physical meaning of the factor $Z$ written as $Z = \gamma^{-1}/(\gamma^{-1} + \Delta_0^{-1})$ is intuitively clear - it corresponds to a probability to find an electron in the semiconductor. The induced superconductivity is induced by the effective pairing term
+
+$$H_{\Delta} = \sum_{i} \left( \Delta c_{i\uparrow}^{\dagger} c_{i\downarrow}^{\dagger} + \text{h.c.} \right), \quad (19)$$
+
+with an effective SC order parameter $\Delta = \gamma\Delta_0/(\gamma + \Delta_0)$. With these choices, $H_{\text{eff}}$ given by Eq. (18) has the same low energy spectrum as the one described by Eq. (17).
+
+## 2. The multiband case
+
+To account for the specific aspects that characterize the proximity effect in finite-size systems, we return to the details of deriving the effective low-energy Green's
+
+function description, e.g., Eq. (15), starting with the microscopic Hamiltonian (7). After integrating out the SC degrees of freedom, the effective SM Green's function acquires a self-energy
+
+$$\Sigma_{\text{SC}}(\mathbf{n}, \mathbf{n}') = \sum_{i_x, i_y, j_x, j_y} \psi_{\mathbf{n}}(i_x, i_y, 1) \tilde{t}(i_x, i_y) \times G_{\text{SC}}(\omega, \mathbf{r}_{\parallel}, z_{\text{interface}}) \tilde{t}(j_x, j_y) \psi_{\mathbf{n}'}(j_x, j_y, 1), \quad (20)$$
+
+where $\psi_n(i_x, i_y, 1)$ are the orbital components of the eigenstates described by Eq. (2) and $G_{\text{SC}}(\omega, r_\parallel, z_{\text{interface}})$ is the SC Green's function, both evaluated at the interface. In Eq. (20) $r_\parallel = (i_x - j_x, i_y - j_y)$ and $\tilde{t}(i_x, i_y)$ are matrix elements that couple two sites with in-plane coordinates $(i_x, i_y)$ across the interface. In the discussion of the planar interface we implicitly assumed translational invariance for the SM-SC coupling, i.e., $\tilde{t}(i_x, i_y) = t$. Here, we consider position-dependent couplings and argue that engineering interfaces with a transparency that varies across the wire, i.e., $\tilde{t}(i_x, i_y) = \tilde{t}(i_y)$, generates off-diagonal components of the effective SC order parameter that help stabilize the Majorana modes.[40] We note that variations of the coupling matrix element along the wire, i.e., in the x direction, act as an effective disorder potential. We will address this issue below in Sec.IV C.
+
+The SC Green's function integrated over momenta can be written explicitly as (see Eq. (13))
+
+$$G_{\text{SC}} = -\nu(\varepsilon_F) \left[ \frac{\omega\tau_0 + \Delta_0\tau_x}{\sqrt{\Delta_0^2 - \omega^2}} + \frac{\zeta}{1-\zeta^2}\tau_z \right], \quad (21)$$
+
+where $\zeta = (\Lambda - \varepsilon_F)/\Lambda$ with $\Lambda$ being the half-bandwidth, $\varepsilon_F$ being the Fermi energy and $\Delta_0 = 1\text{meV}$ being the s-wave SC gap. In the numerical calculations we have $\varepsilon_F = \Lambda/2$, i.e., $\zeta = 0.5$. Using the expression of the SC Green's function given by Eq. (21), the self-energy (20) becomes
+
+$$\Sigma_{\text{SC}}(\mathbf{n}, \mathbf{n}') = -\gamma_{n_y n'_y} \left[ \frac{\omega + \Delta_0 \hat{\tau}_x}{\sqrt{\Delta_0^2 - \omega^2}} + \frac{\zeta \hat{\tau}_z}{\sqrt{1 - \zeta^2}} \right] \delta_{n_x n'_x}, \quad (22)$$
+
+with the implicit assumption that the wire is very thin, $L_z \ll L_y$, and $n_z = n'_z = 1$. The coupling matrix in Eq. (22) is
+
+$$\begin{align}
+\gamma_{n_y n'_y} &= \langle n_y | \gamma | n'_y \rangle \tag{23}\\
+&= \frac{2}{N_y + 1} \sum_{i_y} \gamma(i_y) \sin \left[ \frac{n_y i_y \pi}{N_y + 1} \right] \sin \left[ \frac{n'_y i'_y \pi}{N_y + 1} \right], \nonumber \\
+\gamma(i_y) &= \frac{4 \sqrt{1 - \zeta^2} \sin^2 \left[ \frac{\pi}{N_z+1} \right]}{(N_z+1)\Lambda} t^2(i_y). \tag{24}
+\end{align}$$
+
+We note that for position-independent SM-SC couplings, $\tilde{t}(i_y) = t$, the matrix $\gamma$ is proportional to the unit matrix and the effective low-energy Hamiltonian can be obtained along the lines of Sec. IIB 1. However, for non-uniform couplings, $\gamma_{n_y n'_y}$ acquires off-diagonal elements
\ No newline at end of file
diff --git a/samples/texts/3727487/page_7.md b/samples/texts/3727487/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..f6be2bfd621b05b9c3f0a07ad8d214a34a367ed1
--- /dev/null
+++ b/samples/texts/3727487/page_7.md
@@ -0,0 +1,27 @@
+FIG. 3: (Color online) Upper panel: Non-uniform SM-SC coupling $\tilde{t}(y) = \tilde{t}_0[1 - \theta p(y)]$ with a smooth profile $p(y)$ and $\theta = [\tilde{t}(0) - \tilde{t}(L_y)]/\tilde{t}(0) = 0.8$. Lower panel: Dependence of the induced gap on the non-uniformity of the coupling across the wire. The inhomogeneous proximity effect induces inter-sub-band pairing with $\Delta_{n_y n_y \pm 1} \approx \Delta_{n_y n_y}/2$ for $\theta > 0.7$.
+
+that generates normal and anomalous inter-sub-band terms in the effective Hamiltonian via the self-energy (22). The relative magnitude of the off-diagonal terms depends on the non-homogeneity of the SM-SC coupling. To quantify this property, we consider a fixed profile $p(y)$ with the property $p(0) = 0$ and $p(L_y) = 1$ and the position-dependent tunneling $\tilde{t}(y) = \tilde{t}_0[1 - \theta p(y)]$, where $0 \le \theta \le 1$ is a parameter that measures the degree of non-uniformity of the coupling. Shown in Fig 3 (upper panel) is $\tilde{t}(y)$ for $\theta = 0.8$. As we show below, the non-uniform coupling induces an effective pairing $\Delta_{n_y n'_y} = \langle n_y | \gamma \Delta_0 / (\gamma + \Delta_0) | n'_y \rangle$, where $\gamma$ is given by Eq. (24). The dependence of $\Delta_{n_y n'_y}$ on the coupling asymmetry parameter $\theta$ is shown in Fig 3 (lower panel). For uniform tunneling ($\theta = 0$) the off-diagonal elements vanish and the diagonal elements become equal. By contrast, strongly non-homogeneous tunneling ($\theta \to 1$) generate off-diagonal terms $\Delta_{n_y n'_y}$ that reach about 50% of the diagonal contributions for neighboring sub-bands, $n'_y = n_y \pm 1$, and are much smaller for $|n'_y - n_y| > 1$.
+
+The low-energy effective Hamiltonian for the nanowire can be derived following the scheme described in Sec. II B 1. As before, at low energies ($\omega \ll \Delta_0$) we can
+
+neglect the frequency dependence of the dynamically-generated terms, i.e., $1/\sqrt{\Delta_0^2 - \omega^2} \approx 1/\Delta_0$. However, due to the inter-sub-band coupling induced by non-homogeneous proximity effect, one cannot simply renormalize the energy by a factor Z. This is due to the fact that the Green's function for the proximity-coupled nanowire, $(G^{-1})_{nn'} = \omega \delta_{nn'} - (H_{nw} + H_{Zeeman})_{nn'} - \Sigma_{SC}(n, n')$, contains a frequency-dependent term $\omega(1 + \gamma_{n_y n'_y} / \Delta_0)$ that is not proportional to the unit matrix. However, we notice that we can define a matrix $Z^{1/2}$ with the property $Z^{1/2} G^{-1} Z^{1/2} = \omega - H_{eff}$. Explicitly, we have
+
+$$ (Z^{1/2})_{n_y n'_y} = \left\langle n_y \left| \sqrt{\frac{\Delta_0}{\Delta_0 + \gamma}} \right| n'_y \right\rangle, \quad (25) $$
+
+where $\gamma(i_y)$ is given by Eq. (24) and $\langle i_y | n_y \rangle = \sqrt{2/(N_y + 1)} \sin[i_y n_y \pi / (N_y + 1)]$. Because $\det[Z^{1/2}] > 0$, the renormalized Green's function satisfies the same BdG equation as the original Green's function, i.e., $\det[\omega - H_{eff}] = \det[G^{-1}] = 0$. We conclude that the low-energy physics of a nanowire proximity-coupled to an s-wave SC can be described by an effective Hamiltonian $H_{eff}$ that can be conveniently characterized by its matrix elements in the Nambu basis $||n\sigma\rangle\rangle = (\psi_{n\sigma}, -\sigma\psi^\dagger_{n\sigma})^T$ provided by the eigenstates of $H_0$ given by Eq. (2). Considering only the lowest band, i.e., **n** = (n_x, n_y, 1), we can write explicitly
+
+$$ 
+\begin{align}
+\langle\langle n\sigma || H_{\text{eff}} || n'\sigma' \rangle\rangle &= (26) \\
+(Z^{1/2})_{n_y m_y} \langle m\sigma | H_{\text{nw}} \hat{\tau}_z + H_{\text{Zeeman}} | m'\sigma' \rangle (Z^{1/2})_{m'_y n'_y} &\nonumber \\
+&\quad - \delta_{n_x n'_x} \delta_{\sigma\sigma'} \frac{\zeta \Delta_{n_y n'_y}}{\sqrt{1-\zeta^2}} \hat{\tau}_z - \delta_{n_x n'_x} \delta_{\bar{\sigma}\bar{\sigma}'} \Delta_{n_y n'_y} \hat{\tau}_x,
+\end{align}
+$$
+
+where $\bar{\sigma} = -\sigma$, **m** = (n_x, m_y, 1), $m' = (n'_x, m'_y, 1)$, and summation over the repeating indices $m_y, m'_y$ is implied. In Eq. (26) the matrix $Z^{1/2}$ is given by Eq. (25), the Hamiltonian for the nanowire is $H_{nw} = H_0 + H_{SOI}$, and the matrix elements of $H_0, H_{SOI}$, and $H_{Zeeman}$ are given by equations (3), (4), and (6), respectively. We note that for a homogeneous SM-SC interface the normal contribution proportional to $\Delta_{n_y n'_y}$ becomes diagonal and can be absorbed in the chemical potential, but in general it generates inter-sub-band mixing. These induced off-diagonal terms can be significant in the strong-coupling limit. Finally, the effective SC order parameter is
+
+$$ \Delta_{n_y n'_y} = \left\langle n_y \left| \frac{\gamma \Delta_0}{\Delta_0 + \gamma} \right| n'_y \right\rangle, \quad (27) $$
+
+where $\gamma$ is given by Eq. (24) and $|n_y\rangle$ has the same significance as in Eq. (25).
+
+The effective low-energy BdG Hamiltonian given by Eq. (26) is the main result of this section. In the remainder of this work we will study the low-energy physics of a nanowire with proximity-induced superconductivity by diagonalizing numerically this effective Hamiltonian (26).
\ No newline at end of file
diff --git a/samples/texts/3727487/page_8.md b/samples/texts/3727487/page_8.md
new file mode 100644
index 0000000000000000000000000000000000000000..464b432aabcf0f614f7b782594e887df522624a9
--- /dev/null
+++ b/samples/texts/3727487/page_8.md
@@ -0,0 +1,60 @@
+III. LOW-ENERGY SPECTRUM, MAJORANA
+BOUND STATES, AND PHASE DIAGRAM
+
+A. General properties of the BdG spectrum
+
+The effective BdG Hamiltonian (26) can be written as
+
+$$
+H_{\text{eff}} = \tilde{H}_{\text{nw}} \hat{\tau}_z + \tilde{H}_{\text{Zeeman}} + \tilde{H}_{\Delta} \hat{\tau}_x, \quad (28)
+$$
+
+where $\tilde{H}_{\text{nw}}$ and $\tilde{H}_{\text{Zeeman}}$ are renormalized nanowire and Zeeman Hamiltonians, respectively, and $\tilde{H}_{\Delta}$ is the effective pairing with matrix elements $-\delta_{n_x n'_x} \delta_{\sigma\sigma'} \Delta_{n_y n'_y}$. We note that the normal contribution proportional to $\Delta_{n_y n'_y}$ from Eq. (26) is included in $\tilde{H}_{\text{nw}}$. As mentioned above, for homogeneous SM-SC coupling the only effect of this term is to generate an overall shift of the energy. For convenience, we eliminate this shift by adding a term $\delta_{n_x n'_x} \delta_{\sigma\sigma'} \tilde{\Delta}\zeta / \sqrt{1 - \zeta^2} \hat{\tau}_z$ to the effective Hamiltonian (26), where
+
+$$
+\bar{\Delta} = \frac{\Delta_0}{N_y} \sum_{i_y} \frac{\gamma(i_y)}{\Delta_0 + \gamma(i_y)} \quad (29)
+$$
+
+is the “average” effective pairing. Note that $\Delta_{n_y n_y} \rightarrow \bar{\Delta}$ for $n_y \gg 1$ and the diagonal contribution to the energy-shifting term is partially canceled even in the non-homogeneous case. Finally, to be able to compare results corresponding to various degrees of inhomogeneity in the coupling, i.e., different $\theta$ parameters, we define the average coupling strength as
+
+$$
+\bar{\gamma} = \frac{1}{N_y} \sum_{i_y} \gamma(i_y). \tag{30}
+$$
+
+To obtain better insight into the properties of the BdG Hamiltonian, we start with a non-superconducting system described by the Hamiltonian $\tilde{H}_{\text{nw}} + \tilde{H}_{\text{Zeeman}}$ and $L_x \to \infty$, i.e., an infinitely long renormalized nanowire placed into an effective magnetic field. The spectrum of the renormalized wire is shown in Fig. 4 for a chemical potential $\mu/E_\alpha = 5$ and a Zeeman field $\Gamma/E_\alpha \approx 15$, where $E_\alpha = m^* \alpha_R \approx 0.6K$ is the characteristic spin-orbit coupling energy. Here and below we systematically use $E_\alpha$ as energy unit. The bare nanowire spectrum is renormalized due to a weak inhomogeneous coupling with a profile given in Fig. 3, $\theta = 0.8$, and $\bar{\gamma} = 0.25\Delta_0$. For $\Gamma = 0$ the sub-bands with a given value of $n_y$ are double degenerate at $k_x = 0$, but this degeneracy is removed by the Zeeman field. However, for special values of $\Gamma$, sub-bands corresponding to different values of $n_y$ may become degenerate at $k_x = 0$ (see Fig. 4). If the chemical potential has a value such that the degeneracy point occurs at zero energy, ($\Gamma, \mu$) represent a so-called “sweet spot”.[40] Adding superconductivity, will now open a gap in the spectrum near $E=0$. In the weak-coupling limit, one can determine the topological nature of the induced
+
+FIG. 4: (Color online) Spectrum of an infinite non-superconducting wire in the presence of a Zeeman field. The energies are obtained by diagonalizing the Hamiltonian $\tilde{H}_{\text{nw}} + \tilde{H}_{\text{Zeeman}}$ for $\mu = 5E_\alpha$ and $\Gamma \approx 15E_\alpha$ and are measured relative to the chemical potential. Due to the presence of the Zeeman field, each $n_y$ band is split into two sub-bands marked $\{n_y-\}$ and $\{n_y+\}$. Note that, for this value of $\Gamma$, the sub-bands $\{1+\}$ and $\{2-\}$ are degenerate at $k_x = 0$.
+
+superconductivity by simply counting the number of sub-
+bands crossed by the chemical potential: an odd number
+corresponds to a topologically non-trivial SC, while an
+even number results in a standard superconductor.[40]
+Within this simplified picture, the “sweet spots” repre-
+sent critical points. As will be shown below, the prop-
+erties of the system in the interesting parameter regimes
+near the “sweet spots” are determined by the effective
+inter-band pairing, i.e., by the non-homogeneity of the
+SM-SC proximity effect.
+
+Next, we diagonalize numerically the full effective
+Hamiltonian (26) for a finite wire using the same set of
+control parameters, $\mu/E_\alpha = 5$ and $\Gamma/E_\alpha \approx 15$. The cor-
+responding low-energy spectrum is shown in the upper
+panel of Fig. 5. The eigenstates are labeled by an inte-
+ger number $n$ that has the same sign as the corresponding
+eigenvalue $E_n$. The spectrum is characterized by a gap
+$\Delta^* \approx 1.8E_\alpha$ and a pair of zero-energy Majorana bound
+states. To prove the localized nature of these states, we
+calculate the wave function amplitude and show that the
+Majorana modes are localized near the ends of the wire
+(Fig. 5 lower panel). By contrast, finite energy states ex-
+tend over the entire system. The oscillations of the wave
+function amplitudes is associated with the Fermi momen-
+tum $k_F \approx 0.02/a$, as shown in Fig. 4. We note that
+amplitudes shown in Fig. 5 represent the particle com-
+ponent of the BdG wave-functions, i.e., $|u_{n\uparrow}|^2 + |u_{n\downarrow}|^2$.
+For a finite-energy state, e.g., $n=2$ the corresponding
+total spectral weight is about 1/2, with the other half
+coming from from $n=-2$. The Majorana modes have
+a total weight of one, which corresponds to one physi-
+cal particle, but this weight is spatially separated into
\ No newline at end of file
diff --git a/samples/texts/3727487/page_9.md b/samples/texts/3727487/page_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..355f3d0f2c821980c9794a44812ee6816289e8e4
--- /dev/null
+++ b/samples/texts/3727487/page_9.md
@@ -0,0 +1,13 @@
+two contributions localized near the ends of the wire. Removing the Majorana pair would require overlapping the two components, which cannot be done by local perturbations. This is, of course, the topological immunity of the Majorana modes, which is crucial for topological quantum computation. The characteristic length scale for the localized modes is controlled by the minimum value of the quasi-particle gap in a wire with no ends, e.g., with periodic boundary conditions, $\Delta_{\infty}^*$. As will be shown below (see Sec III C), in a finite wire it is possible that bound states localized near the ends of the system have energies within the gap. When in-gap states are present, the lowest-energy localized state sets the value of the mini-gap, $\Delta^* < \Delta_{\infty}^*$. We emphasize that for a set of control parameters $(\Gamma, \mu)$ corresponding to a non-vanishing minimum quasi-particle gap $\Delta_{\infty}^*$, the mini-gap $\Delta^*$ is always nonzero. The characteristic length scale for the localized modes diverges in the limit $\Delta^* \to 0$. Hence, the topological phase is protected as long as the quasi-particle gap remains finite. Consequently, to determine the stability of the Majorana bound states, our key task is to determine the dependence of $\Delta_{\infty}^*$ on various physical parameters and experimentally-relevant perturbations, e.g., chemical potential, Zeeman field, SM-SC coupling, and charged impurity and coupling-induced disorder.
+
+At this point in our analysis it is important to clarify the role played by the parameters that incorporate the SM-SC proximity effect into the low-energy effective theory. In particular, we address the following question: how does the low-energy spectrum depend on the strength of the SM-SC coupling (i.e., on $\bar{\gamma}$), on the non-homogeneity of the coupling ($\theta$), and on the dynamical effects included in the effective description ($Z^{1/2}$)? The non-homogeneity of the coupling is responsible for generating inter sub-band pairing in Eq. (27). These off-diagonal contributions play a minor role away from the “sweet spots”, where they generate a small quantitative change of the quasi-particle gap. However, in the vicinity of the “sweet spots” $\Delta_{\infty}^*$ vanishes in the absence of inter sub-band pairing and the non-homogeneity of the coupling (i.e., $\theta > 0$) becomes crucial. As expected, inclusion of dynamical effects through $Z^{1/2}$ renormalizes the energy scales. Without these effects, the mini-gap would increase monotonically with the coupling strength, but dynamical effects limit its maximum value. The optimal quasi-particle gap obtains in the intermediate coupling regime $\bar{\gamma} \sim \Delta_0$. Further increase of the coupling leads to a decrease of the gap. To illustrate the features described above, we show in Fig. 6 low-energy spectra in the vicinity of the “sweet spot” ($\mu = 14.5E_{\alpha}$, $\Gamma = 15.3E_{\alpha}$) at weak coupling ($\bar{\gamma} = 0.25\Delta_0$, top panel) and intermediate coupling ($\bar{\gamma} = \Delta_0$, bottom panel). Note that in the absence of inter-band pairing, i.e., for homogeneous SM-SC coupling, the gap near the “sweet spot” collapses. Also, inclusion of dynamical effects at intermediate and strong coupling is the key for obtaining the correct energy scales. Finally, the location of the “sweet spots” in the $\Gamma - \mu$ plane depends on the coupling strength and,
+
+FIG. 5: (Color online) Upper panel: BdG energy spectrum for a finite superconducting wire obtained by numerical diagonalization of $H_{\text{eff}}$. The parameters $\mu$ and $\Gamma$ are the same as in Fig. 4 and $n$ labels the eigenvalues of $H_{\text{eff}}$ staring with the lowest energy and has the same sign as $E_n$. The in-gap states are Majorana zero-energy modes. Lower panel: The particle-component of the wave-function amplitude for the lowest energy states. The Majorana modes ($n=1$) are localized at the ends of the wire, while the finite energy states extend over the entire wire.
+
+more generally, the location of phase boundaries depends on the strength of the SC proximity effect.
+
+## B. Phase diagram for multi-band superconducting nanowires
+
+Topological superconductivity and, implicitly, the Majorana bound states are protected by the quasi-particle gap $\Delta_{\infty}^*$, as discussed above. The vanishing of $\Delta_{\infty}^*$ signals a transition between topologically nontrivial and topologically trivial superconductivity. In a multi-band system, such transitions can be caused, for example, by varying the Zeeman field while maintaining a fixed value of the chemical potential. The vanishing of $\Delta_{\infty}^*$ at certain specific values of $\Gamma$ reveals a sequence of alternating SC phases with trivial and nontrivial topological properties. A natural question is whether different topologically nontrivial (or trivial) phases have exactly the same low-energy properties. While topologically identical, these phases may have some distinct features, at least in certain parameter regimes.
+
+To address this question, we calculate the low-energy
\ No newline at end of file
diff --git a/samples/texts/4477302/page_1.md b/samples/texts/4477302/page_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..c948ce0f7114afbb1fc0db16bd87fb39af470960
--- /dev/null
+++ b/samples/texts/4477302/page_1.md
@@ -0,0 +1,36 @@
+Analyse complexe/Complex Analysis
+
+(Analyse mathématique/Mathematical Analysis)
+
+Effective Cartan-Tanaka connections
+for $\mathscr{C}^6$-smooth strongly pseudoconvex hypersurfaces $M^3 \subset \mathbb{C}^2$
+
+Mansour Aghasi$^{a}$, Joël Merker$^{b}$, Masoud Sabzevari$^{a}$
+
+$^{a}$Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, IRAN
+
+$^{b}$Département de Mathématiques d'Orsay, Bâtiment 425, Faculté des Sciences, F-91405 Orsay Cedex, FRANCE
+
+Received ****; accepted after revision ++++
+
+Presented by
+
+Abstract
+
+Explicit Cartan-Tanaka curvatures, the vanishing of which characterizes sphericity, are provided in terms of the 6-th order jet of a graphing function for a $\mathscr{C}^6$ strongly pseudoconvex hypersurface $M^3 \subset \mathbb{C}^2$. To cite this article:
+
+Résumé
+
+Connections de Cartan-Tanaka effectives pour les hypersurfaces strictement pseudoconvexes $M^3 \subset \mathbb{C}^2$ de classe $\mathscr{C}^6$. Des courbures de Cartan-Tanaka explicites, dont l'annulation identique caractérise la sphéricité, sont fournies en termes du jet d'ordre 6 d'une fonction graphante pour une hypersurface $M^3 \subset \mathbb{C}^2$ de classe $\mathscr{C}^6$ strictement pseudoconvexe. Pour citer cet article :
+
+# 1. Cartan connection, curvature function and second cohomology of Lie algebras
+
+**Definition** 1.1 ([12,4]) Let $G$ be a real Lie group with a closed subgroup $H$, and let $\mathfrak{g}$ and $\mathfrak{h}$ be the corresponding real Lie algebras. A Cartan geometry of type $(G,H)$ on a $\mathscr{C}^\infty$ manifold $M$ is a principal $H$-bundle $\pi: \mathcal{P} \to M$ together with a $\mathfrak{g}$-valued 'Cartan connection' 1-form $\omega: T\mathcal{P} \to \mathfrak{g}$ satisfying:
+
+(i) $\omega_p: T_p\mathcal{P} \to \mathfrak{g}$ is an isomorphism at every $p \in \mathcal{G}$;
+
+(ii) if $R_h(p) := ph$ is the right translation on $\mathcal{G}$ by an $h \in H$, then $R_h^*\omega = \text{Ad}(h^{-1}) \circ \omega$;
+
+(iii) $\omega(H^\dagger) = h$ for every $h \in \mathfrak{h}$, where $H^\dagger|_p := \frac{d}{dt}\big|_0(R_{\exp(th)}(p))$ is the left-invariant vector field on $\mathcal{G}$ associated to $h$.
+
+*Email addresses: m.aghasi@cc.iut.ac.ir (Mansour Aghasi), merker@math.u-psud.fr (Joël Merker), sabzevari@math.iut.ac.ir (Masoud Sabzevari).*
\ No newline at end of file
diff --git a/samples/texts/4477302/page_2.md b/samples/texts/4477302/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..207a790a1e016d3999b9ab91414337bed0819770
--- /dev/null
+++ b/samples/texts/4477302/page_2.md
@@ -0,0 +1,37 @@
+Since the associated curvature 2-form $\Omega(X, Y) := d\omega(X, Y) + [\omega(X), \omega(Y)]_g$, with $X, Y \in \Gamma(T\mathcal{P})$, vanishes if either $X$ or $Y$ is vertical ([12]), $\Omega$ is fully represented by the curvature function $\kappa \in \mathcal{C}^\infty(\mathcal{P}, \Lambda^2(\mathfrak{g}^*/\mathfrak{h}^*) \otimes \mathfrak{g})$ which sends a point $p \in \mathcal{P}$ to the map $\kappa(p): (\mathfrak{g}/\mathfrak{h}) \wedge (\mathfrak{g}/\mathfrak{h}) \to \mathfrak{g}$ defined by:
+
+$$ (x' \mod \mathfrak{h}) \wedge (x'' \mod \mathfrak{h}) \longmapsto -\Omega_p(\omega_p^{-1}(x'), \omega_p^{-1}(x'')) = -[x', x'']_g + \omega_p([\hat{X}'', \hat{X}']) $$
+
+where $\hat{X} := \omega^{-1}(X)$ is the constant field on $\mathcal{P}$ associated to an $x \in g$. Denote then $r := \dim_{\mathbb{R}} g$, $n := \dim_{\mathbb{R}} (\mathfrak{g}/\mathfrak{h})$ whence $n-r = \dim_{\mathbb{R}} h$ and suppose $r \ge 2$, $n \ge 1$, $n-r \ge 1$ so that $g, g/h$ and $h$ are all nonzero. Picking an adapted basis $(x_k)_{1 \le k \le r}$ with $g = \text{Span}_{\mathbb{R}}(x_1, \dots, x_n, x_{n+1}, \dots, x_r)$ and $h = \text{Span}_{\mathbb{R}}(x_{n+1}, \dots, x_r)$, one may expand the curvature function by means of $\mathbb{R}$-valued components:
+
+$$ \kappa(p) = \sum_{1 \le i_1 < i_2 \le n} \sum_{k=1}^{r} \kappa_{i_1, i_2}^{k}(p) x_{i_1}^{*} \wedge x_{i_2}^{*} \otimes x_k. $$
+
+**Lemma 1.2** ([2]) For any field $Y^\dagger = \frac{d}{dt}|_0 R_{\exp(ty)}$ on $\mathcal{P}$ associated to an arbitrary $y \in h$, one has:
+
+$$ (Y^\dagger \kappa)(p)(x', x'') = -[y, \kappa(p)(x', x'')]_g + \kappa(p)([y, x']_g, x'') + \kappa(p)(x', [y, x'']_g). \quad \square $$
+
+For any $k \in \mathbb{N}$, consider $k$-cochains $\mathcal{C}^k := \Lambda^k(\mathfrak{g}^*/\mathfrak{h}^*) \otimes g$ and define the differential $\partial^k : \mathcal{C}^k \to \mathcal{C}^{k+1}$ by:
+
+$$ (\partial^k \Phi)(z_0, z_1, \dots, z_k) := \sum_{i=0}^{k} (-1)^i [z_i, \Phi(z_0, \dots, \hat{z}_i, \dots, z_k)]_g + \sum_{0 \le i < j \le k} (-1)^{i+j} \Phi([z_i, z_j]_g, z_0, \dots, \hat{z}_i, \dots, \hat{z}_j, \dots, z_k). $$
+
+Especially for $k=2$, the cohomology spaces $\mathcal{H}^k := \ker(\partial^k)/\im(\partial^{k-1})$ encode deformations of Lie algebras and are useful for Cartan connections ([13,4,6]), cf. [1] for an algorithm using Gröbner bases.
+
+**Lemma 1.3** (Bianchi identity [4,6,2]) For any three $x', x'', x''' \in g$, one has at every point $p \in \mathcal{P}$:
+
+$$ 0 = (\partial^2 \kappa)(p)(x', x'', x''') + \sum_{\text{cycl}} \kappa(p)(\kappa(p)(x', x''), x''') + \sum_{\text{cycl}} (\hat{X}'(\kappa))(p)(x'', x'''). \quad \square $$
+
+When $g = g_{-\mu} \oplus \cdots \oplus g_{-1} \oplus g_0 \oplus g_1 \oplus \cdots \oplus g_\nu$ is graded as in Tanaka's theory, with $[g_{\lambda_1}, g_{\lambda_2}]_g \subset g_{\lambda_1+\lambda_2}$ for any $\lambda_1, \lambda_2 \in \mathbb{Z}$ and when $h = g_0 \oplus g_1 \oplus \cdots \oplus g_\nu$, cochains enjoy a natural grading, the second cohomology is graded too: $\mathcal{H}^2 = \bigoplus_{h \in \mathbb{Z}} \mathcal{H}^2_{[h]}$, and the graded Bianchi identities ([6,2]):
+
+$$ \partial_{[h]}^2 (\kappa_{[h]}) (x', x'', x''') = -\sum_{\text{cycl}} \sum_{h'=1}^{h-1} (\kappa_{[h-h']} (\text{proj}_{g/\mathfrak{h}} (\kappa_{[h']}(x', x'')), x''')) - \sum_{\text{cycl}} (\hat{X}'\kappa_{[h+x'|]}) (x'', x''') $$
+
+show that the lowest order nonvanishing curvature must be $\partial$-closed, and more generally, any homogeneous curvature component is determined by the lower components up to a $\partial$-closed component.
+
+## 2. Geometry-preserving deformations of the Heisenberg sphere $\mathbb{H}^3 \subset \mathbb{C}^2$
+
+Let now $\mathcal{M}^3 \subset \mathbb{C}^2$ be a local strongly pseudoconvex $\mathcal{C}^6$-smooth real 3-dimensional hypersurface, represented in coordinates $(z, w) = (x + iy, u + iv)$ as the graph (for background, see [7,8,9,10,11]):
+
+$$ v = \varphi(x, y, u) = x^2 + y^2 + O(3) $$
+
+of a certain real-valued $\mathcal{C}^6$ function $\varphi$ defined in a neighborhood of the origin in $\mathbb{R}^3$. Its complex tangent bundle $\mathcal{T}^c M = \text{Re } T^{0,1} M$ is generated by the two vector fields:
+
+$$ H_1 := \frac{\partial}{\partial x} + (\frac{\varphi_y - \varphi_x}{1+\varphi_u^2}) \frac{\partial}{\partial u} \quad \text{and} \quad H_2 := \frac{\partial}{\partial y} + (\frac{-\varphi_x - \varphi_y}{1+\varphi_u^2}) \frac{\partial}{\partial u}, $$
\ No newline at end of file
diff --git a/samples/texts/4477302/page_3.md b/samples/texts/4477302/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..74a95c3671e3a2c04f3b35b81a0a040016db1d10
--- /dev/null
+++ b/samples/texts/4477302/page_3.md
@@ -0,0 +1,72 @@
+which make a frame joint with the third, Levi form-type Lie-bracket:
+
+$$
+T := \frac{1}{4} [H_1, H_2] = \left( \frac{1}{4} \frac{1}{(1+\varphi_u^2)^2} \left\{ -\varphi_{xx} - \varphi_{yy} - 2\varphi_y \varphi_{xu} - \varphi_x^2 \varphi_{uu} + 2\varphi_x \varphi_{yu} - \varphi_y^2 \varphi_{uu} + \right. \right. \\
+\left. \left. + 2\varphi_y \varphi_u \varphi_{yu} + 2\varphi_x \varphi_u \varphi_{xu} - \varphi_u^2 \varphi_{xx} - \varphi_u^2 \varphi_{yy} \right\} \right) \frac{\partial}{\partial u}.
+$$
+
+Such $M^3$'s are geometry-preserving deformations of the Heisenberg sphere $\mathbb{H}^3$: $v = x^2 + y^2$. It is known
+that the Lie algebra $\mathfrak{ho}(\mathbb{H}^3) := \{\mathbf{X} = Z(z, w) \frac{\partial}{\partial z} + W(z, w) \frac{\partial}{\partial w} : \mathbf{X} + \mathbf{X}^\text{ tangent to } \mathbb{H}^3\}$ of infinitesimal CR
+automorphisms of the Heisenberg sphere $\mathbb{H}^3$ in $\mathbb{C}^2$ is 8-dimensional and generated by:
+
+$$
+\begin{align*}
+T &:= \partial_w, &
+H_1 &:= \partial_z + 2iz \partial_w, &
+H_2 &:= i \partial_z + 2z \partial_w, &
+D &:= z \partial_z + 2w \partial_w, &
+R &:= iz \partial_z, \\
+l_1 &:= (w + 2iz^2) \partial_z + 2izw \partial_w, &
+l_2 &:= (iw + 2z^2) \partial_z + 2zw \partial_w &
+J &:= zw \partial_z + w^2 \partial_w.
+\end{align*}
+$$
+
+Recently, Ezhov, McLaughlin and Schmalz published in the Notices of the AMS an expository article
+[6] in which they reconstruct — within Tanaka’s framework and assuming that M is $\mathscr{C}^\omega$ — Cartan’s
+connection ([5]) valued the eight-dimensional abstract real Lie algebra:
+
+$$
+g := R t \oplus R h_1 \oplus R h_2 \oplus R d \oplus R r \oplus R i_1 \oplus R i_2 \oplus R j \quad (\text{with } h := R d \oplus R r \oplus R i_1 \oplus R i_2 \oplus R j)
+$$
+
+spanned by some eight abstract vectors enjoying the same commutator table as T, . . . , J. A natural Tanaka grading is: $g_{-2} = R t; g_{-1} = R h_1 \oplus R h_2; g_0 = R d \oplus R r; g_1 = R i_1 \oplus R i_2; g_2 = R j$. By performing the above choice {$H_1, H_2, T$} of an initial frame for TM which is explicit in terms of the graphing function $\varphi(x, y, u)$, we deviate from the initial normalization made in [6] (with a more geometric-minded approach), since our computational objective is to provide a Cartan-Tanaka connection all elements of which are completely effective in terms of $\varphi(x, y, u)$ — assuming only $\mathcal{C}^6$-smoothness of M.
+
+Call $\Upsilon$ the numerator of $T = \frac{1}{4}[H_1, H_2] = \frac{1}{4}\frac{\Upsilon}{\Delta^2}\frac{\partial}{\partial u}$, allow the two notational coincidences: $x_1 \equiv x$, $x_2 \equiv y$; introduce the two length-three brackets:
+
+$$
+[H_i, T] = \frac{1}{4} [H_i, [H_1, H_2]] =: \Phi_i T \qquad (i=1,2),
+$$
+
+which are both multiples of *T* by means of two functions $\Phi_i := \frac{A_i}{\Delta^2 \Upsilon}$; lastly, introduce furthermore the
+$H_k$-iterated derivatives of the functions $\Phi_i$ up to order 3, where $i, k_1, k_2, k_3 = 1, 2:$
+
+$$
+H_{k_1}(\Phi_i) = \frac{A_{i,k_1}}{\Delta^4 \Upsilon^2}, \quad H_{k_2}(H_{k_1}(\Phi_i)) = \frac{A_{i,k_1,k_2}}{\Delta^6 \Upsilon^3}, \quad H_{k_3}(H_{k_2}(H_{k_1}(\Phi_i))) = \frac{A_{i,k_1,k_2,k_3}}{\Delta^8 \Upsilon^4}.
+$$
+
+**Proposition 2.1** ([2]) All the numerators appearing above are explicitly given by:
+
+$$
+\begin{align*}
+A_i &:= \Delta^2 \Upsilon x_i + \Delta(-2\Delta x_i \Upsilon + \Lambda_i \Upsilon_u - \Upsilon \Lambda_{i,u}) - \Lambda_i \Upsilon \Delta_u, \\
+A_{i,k_1} &:= \Delta^2(\Upsilon A_{i,x_{k_1}} - \Upsilon x_{k_1} A_i) + \Delta(-2\Delta x_{k_1} \Upsilon A_i + \Upsilon \Lambda_{k_1} A_{i,u} - \Upsilon_u \Lambda_{k_1} A_i) - 2\Delta_u \Upsilon \Lambda_{k_1} A_i, \\
+A_{i,k_1,k_2} &:= \Delta^2(\Upsilon A_{i,k_1,x_{k_2}} - 2\Upsilon x_{k_2} A_{i,k_1}) + \Delta(-3\Delta x_{k_2} \Upsilon A_{i,k_1} + \Upsilon \Lambda_{k_2} A_{i,k_1,u} - 2\Upsilon_u \Lambda_{k_2} A_{i,k_1}) - 3\Delta_u \Upsilon \Lambda_{k_2} A_{i,k_1}, \\
+A_{i,k_1,k_2,k_3} &:= \Delta^2(\Upsilon A_{i,k_1,k_2,x_{k_3}} - \Upsilon x_{k_3} A_{i,k_1,k_2}) + \Delta(-6\Delta x_{k_3} \Upsilon A_{i,k_1,k_2} + \Upsilon \Lambda_{k_3} A_{i,k_1,k_2,u} - 3\Upsilon_u \Lambda_{k_3} A_{i,k_1,k_2}) - 6\Delta_u \Upsilon \Lambda_{k_3} A_{i,k_1,k_2}.
+\end{align*}
+$$
+
+Furthermore, these iterated derivatives identically satisfy $H_2(\Phi_1) = H_1(\Phi_2)$ and:
+
+$$
+\begin{align*}
+0 &\equiv -H_1(H_2(H_1(\Phi_2))) + 2H_2(H_1(H_1(\Phi_2))) - H_2(H_2(H_1(\Phi_1))) - \Phi_2 H_1(H_2(\Phi_1)) + \Phi_2 H_2(H_1(\Phi_1)), \\
+0 &\equiv -H_2(H_1(H_1(\Phi_2))) + 2H_1(H_2(H_1(\Phi_2))) - H_1(H_1(H_2(\Phi_2))) - \Phi_1 H_2(H_1(\Phi_2)) + \Phi_1 H_1(H_2(\Phi_2)), \\
+0 &\equiv -H_1(H_1(H_1(\Phi_2))) + 2H_1(H_2(H_1(\Phi_1))) - H_2(H_1(H_1(\Phi_1))) + \Phi_1 H_1(H_1(\Phi_2)) - \Phi_1 H_2(H_1(\Phi_1)), \\
+0 &\equiv -H_2(H_2(H_1(\Phi_2))) + 2H_2(H_1(H_2(\Phi_2))) - H_1(H_2(H_2(\Phi_2))) + \Phi_2 H_2(H_1(\Phi_2)) - \Phi_2 H_1(H_2(\Phi_2)).
+\end{align*}
+$$
+
+**3. Explicit Cartan-Tanaka connection**
+
+**Theorem 3.1** ([2]) Associated to such an $M^3 \subset \mathbb{C}^2$, there is a unique $g$-valued Cartan connection which is normal and regular in the sense of Tanaka. Its curvature function reduces to:
\ No newline at end of file
diff --git a/samples/texts/4477302/page_4.md b/samples/texts/4477302/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..970ee4d880df0acf6d3613d022257e3769525312
--- /dev/null
+++ b/samples/texts/4477302/page_4.md
@@ -0,0 +1,67 @@
+$$
+\begin{align*}
+\kappa(p) ={}& \kappa_{i_1}^{h_1 t}(p) h_1^* \wedge t^* \otimes i_1 + \kappa_{i_2}^{h_1 t}(p) h_1^* \wedge t^* \otimes i_2 + \kappa_{i_1}^{h_2 t}(p) h_2^* \wedge t^* \otimes i_1 \\
+& + \kappa_{i_2}^{h_2 t}(p) h_2^* \wedge t^* \otimes i_2 + \kappa_j^{h_1 t}(p) h_1^* \wedge t^* \otimes j + \kappa_j^{h_2 t}(p) h_2^* \wedge t^* \otimes j,
+\end{align*}
+$$
+
+where the two main curvature coefficients, having homogeneity 4, are of the form:
+
+$$
+\begin{align*}
+\kappa_{i_1}^{h_1 t}(p) &= -\Delta_1 c^4 - 2\Delta_4 c^3 d - 2\Delta_4 cd^3 + \Delta_1 d^4 \quad \text{and} \quad \\
+\kappa_{i_2}^{h_1 t}(p) &= -\Delta_4 c^4 + 2\Delta_1 c^3 d + 2\Delta_1 cd^3 + \Delta_4 d^4,
+\end{align*}
+$$
+
+in which the two functions $\Delta_1$ and $\Delta_4$ of only the three variables $(x,y,u)$ are explicitly given by:
+
+$$
+\begin{align*}
+\Delta_1 &= \frac{1}{384} [H_1(H_1(H_1(\Phi_1))) - H_2(H_2(H_2(\Phi_2)))) + 11H_1(H_2(H_1(\Phi_2))) - 11H_2(H_1(H_2(\Phi_1))) + 6\Phi_2 H_2(H_1(\Phi_1)) - 6\Phi_1 H_1(H_2(\Phi_2)) \\
+&\quad - 3\Phi_2 H_1(H_1(\Phi_2)) + 3\Phi_1 H_2(H_2(\Phi_1)) - 3\Phi_1 H_1(H_1(\Phi_1)) + 3\Phi_2 H_2(H_2(\Phi_2)) - 2\Phi_1 H_1(\Phi_1) + 2\Phi_2 H_2(\Phi_2) \\
+&\quad - 2(\Phi_2)^2 H_1(\Phi_1) + 2(\Phi_1)^2 H_2(\Phi_2) - 2(\Phi_2)^2 H_2(\Phi_2) + 2(\Phi_1)^2 H_1(\Phi_1)], \\
+\Delta_4 &= \frac{1}{384} [-3H_2(H_1(H_2(\Phi_2))) - 3H_1(H_2(H_1(\Phi_1))) + 5H_1(H_2(H_2(\Phi_2))) + 5H_2(H_1(H_1(\Phi_1))) \\
+&\quad + 4\Phi_1 H_1(H_1(\Phi_2)) + 4\Phi_2 H_2(H_1(\Phi_2)) - 3\Phi_2 H_1(H_1(\Phi_1)) - 3\Phi_1 H_2(H_2(\Phi_2)) \\
+&\quad - 7\Phi_2 H_1(H_2(\Phi_2)) - 7\Phi_1 H_2(H_1(\Phi_1)) - 2H_1(\Phi_1)H_1(\Phi_2) - 2H_2(\Phi_2)H_2(\Phi_1) \\
+&\quad + 4\Phi_1\Phi_2 H_1(\Phi_1) + 4\Phi_1\Phi_2 H_2(\Phi_2)],
+\end{align*}
+$$
+
+and where the remaining four secondary curvature coefficients are given by:
+
+$$
+\kappa_{i_{1}}^{h_{2}t} = \kappa_{i_{2}}^{h_{1}t}, \quad \kappa_{i_{2}}^{h_{2}t} = -\kappa_{i_{1}}^{h_{1}t}, \quad \kappa_{j}^{h_{1}t} = \hat{H}_{1}(\kappa_{i_{2}}^{h_{2}t}) - \hat{H}_{2}(\kappa_{i_{2}}^{h_{1}t}), \quad \kappa_{j}^{h_{2}t} = -\hat{H}_{1}(\kappa_{i_{1}}^{h_{2}t}) + \hat{H}_{2}(\kappa_{i_{1}}^{h_{1}t}).
+$$
+
+Corollary 3.2 A $\mathcal{C}^6$-smooth strongly pseudoconvex local hypersurface $M^3 \subset \mathbb{C}^2$ is biholomorphic to $\mathbb{H}^3$, namely is spherical, if and only if $0 \equiv \Delta_1 \equiv \Delta_4$, identically as functions of $(x,y,u)$.
+
+**Acknowledgements.** One year ago, Gerd Schmalz kindly provided us with a pdf copy of the accepted version of [6], and this was of great help during the preparation of the presently announced memoir [2].
+
+References
+
+[1] M. Aghasi, B. Alizadeh, J. Merker, M. Sabzevari, A Gröbner-bases algorithm for the computation of the cohomology of Lie (super) algebras, arxiv.org/abs/1104.5300, 23 pp.
+
+[2] M. Aghasi, J. Merker, M. Sabzevari, Effective Cartan-Tanaka connections on $\mathcal{C}^6$ strongly pseudoconvex hypersurfaces $M^3 \subset \mathbb{C}^2$, arxiv.org/abs/1104.1509, 113 pp.
+
+[3] V. K. Beloshapka, V. Ezhov, G. Schmalz, Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russian J. Mathematical Physics 14 (2007) 121–133.
+
+[4] A. Čap, H. Schichl, Parabolic geometries and canonical Cartan connections, Hokkaido Math. J. 29 (2000) 453–505.
+
+[5] É. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes I, Ann. Math. Pures Appl. 4 (1932) 17–90; II, Ann. Scuola Norm. Sup. Pisa 2 (1932) 333–354.
+
+[6] V. Ezhov, B. McLaughlin, G. Schmalz, From Cartan to Tanaka: getting real in the complex world, Notices of AMS 58 (2011) 20–27
+
+[7] J. Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets, Bull. Soc. Math. France 129 (2001) 547–591.
+
+[8] J. Merker, On the local geometry of generic submanifolds of $\mathbb{C}^n$ and the analytic reflection principle, J. Math. Sciences (N. Y.) 125 (2005) 751–824.
+
+[9] J. Merker, Lie symmetries and CR geometry, J. Math. Sciences (N. Y.) 154 (2008) 817–922.
+
+[10] J. Merker, Nonrigid spherical real analytic hypersurfaces in $\mathbb{C}^2$, Complex Variables and Elliptic Equations, **55** (2010), no. 12, 1155–1182.
+
+[11] J. Merker, E. Porten, Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities, Int. Math. Research Surveys 2006 Article ID 28295 287 pages.
+
+[12] R. W. Sharpe, Differential Geometry. Cartan's generalization of Klein's Erlangen program, Springer, Berlin, 1997.
+
+[13] N. Tanaka, On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. 10 (1970) 1–82.
\ No newline at end of file
diff --git a/samples/texts/4807413/page_1.md b/samples/texts/4807413/page_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..02acf0b06dd50be2b3cdc88511806e111356ceaf
--- /dev/null
+++ b/samples/texts/4807413/page_1.md
@@ -0,0 +1,13 @@
+Topology Proceedings
+
+**Web:** http://topology.auburn.edu/tp/
+
+**Mail:** Topology Proceedings
+Department of Mathematics & Statistics
+Auburn University, Alabama 36849, USA
+
+**E-mail:** topolog@auburn.edu
+
+**ISSN:** 0146-4124
+
+COPYRIGHT © by Topology Proceedings. All rights reserved.
\ No newline at end of file
diff --git a/samples/texts/4807413/page_2.md b/samples/texts/4807413/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..bbf0c3d7b2ea5287376b0772258a0604be0180a0
--- /dev/null
+++ b/samples/texts/4807413/page_2.md
@@ -0,0 +1,10 @@
+ON FINITE PRODUCTS OF MENGER SPACES
+AND 2-HOMOGENEITY
+
+DENNIS J. GARITY
+
+**ABSTRACT.** G. S. Ungar has shown that homogeneous metric continua that are 2-homogeneous are locally connected. K. Kuperberg, W. Kuperberg and W. R. R. Transue gave examples of homogeneous metric continua that were locally connected but were not 2-homogeneous. In this paper, examples are produced that show that adding the additional requirement of local n-connectivity is not enough to produce a converse to Ungar's theorem. For every positive integer n, a homogeneous metric continua of dimension (n+1) that is locally (n-1) connected is produced. These spaces are shown not to be 2-homogenous. The examples are produced by taking products of the universal Menger n-dimensional space with $S^1$. Other examples are produced by taking finite products of Menger spaces. An analysis of Čech homology properties of Menger spaces is needed in the examples.
+
+# 1. INTRODUCTION
+
+G. S. Ungar has shown that homogeneous metric continua having the stronger homogeneity property of being 2-homogeneous are necessarily locally connected [Un]. This result leads to the question of whether imposing local connectivity or local n-connectivity conditions on homogeneous continua would imply that they possessed stronger homogeneity properties such as 2-homogeneity. In 1980, K. Kuperberg, W. Kuperberg and W. R. R. Transue showed that $\mu_1 \times \mu_1$ and $\mu_1 \times S^1$ were not 2-homogeneous [KKT]. Here, $\mu_1$ is the universal curve, defined
\ No newline at end of file
diff --git a/samples/texts/4807413/page_3.md b/samples/texts/4807413/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..9be29f2a66c4b1cad55946754642b907718f688e
--- /dev/null
+++ b/samples/texts/4807413/page_3.md
@@ -0,0 +1,11 @@
+below. Their results gave examples of homogeneous spaces that were locally connected, but not 2-homogeneous. They also ask whether finite or countable products of $\mu_1$ with itself are 2-homogeneous.
+
+The results in this paper were first presented in a talk at the Workshop in Geometric Topology in Colorado Springs in June, 1992. A summary of the results appeared in the Proceedings of that conference [Ga]. Since that time, it has been pointed out that arguments of J. Kennedy Phelps [Ke1], [Ke2] that showed that $\mu_1 \times X$, where $X$ is an arbitrary continuum, is not 2-homogeneous, can be generalized to obtain the same result for the other Menger spaces. K. Kuperberg, W. Kuperberg and W. R. R. Transue also have a more recent paper on 2-homogeneity [KKT2].
+
+The techniques used in this paper obtaining similar results are different enough to be of general interest. In this paper, we are able to show that imposing higher local connectivity conditions on metric homogeneous continua does not lead to a converse to Ungar's Theorem. For each positive integer $n$, we give examples of homogeneous $(n+2)$-dimensional metric continua that are locally $n$-connected, but are not 2-homogeneous. We also show that finite products with each factor a Menger space are not 2-homogeneous. It remains open whether there is a homogeneous metric continuum that is $n$-connected for all $n$, and is not 2-homogeneous.
+
+The results in [KKT] depend on certain one-dimensional facts from Curtis and Fort [CF] that do not generalize directly to higher dimensions. In this paper, we replace the one-dimensional arguments with higher dimensional Čech homology arguments that allow us to generalize the results in [KKT]. As a special case, we are able to show that $\mu_m \times \mu_n$ is not 2-homogeneous for all values of $m$ and $n$ where $\max\{m,n\} \ge 1$.
+
+Section 2 contains the necessary definitions. Section 3 contains results on the Čech homology of Menger spaces. The main results are in Section 4.
+
+The author would like to thank John J. Walsh and Krystyna
\ No newline at end of file
diff --git a/samples/texts/4807413/page_4.md b/samples/texts/4807413/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..f2f78f703c516907ad9b027dbf756c9a15253321
--- /dev/null
+++ b/samples/texts/4807413/page_4.md
@@ -0,0 +1,19 @@
+M. Kuperberg for numerous helpful conversations.
+
+## 2. DEFINITIONS
+
+All spaces under consideration are separable metric spaces. We begin with definitions of the Menger Spaces. These spaces were originally defined by Menger in 1932 [Mg]. An inductive definition is as follows. Let $M_n^0$ be $I^{2n+1} \subset R^{2n+1}$, where $I = [0, 1]$. Inductively assume that $M_n^k$ is a union of $(2n+1)$-dimensional cells with sides of length $(1/3)^k$. Subdivide each cell in $M_n^k$ into $3^{2n+1}$ smaller cells by subdividing each side in thirds. Then $M_n^{k+1}$ is the union of all of those smaller cells that intersect the $n$-skeleton of $M_n^k$. The Menger $n$-dimensional space, $\mu_n$, is then defined as $\bigcap_{i=0}^{\infty} M_n^i$. The space $\mu_0$ is the standard middle thirds Cantor set and the space $\mu_1$ is the universal curve characterized by R. D. Anderson [An1], [An2].
+
+In 1984, M. Bestvina characterized all the remaining Menger Spaces [Be]. The following theorem gives Bestvina's characterization.
+
+**Theorem 2.1.** [Be] *The Menger universal $n$-dimensional space $\mu_n$ is the unique space satisfying the following conditions:*
+
+(1) $\mu_n$ is a compact $n$-dimensional metric space.
+
+(2) $\mu_n$ is locally $(n-1)$-connected ($LC^{n-1}$).
+
+(3) $\mu_n$ is $(n-1)$-connected ($C^{n-1}$).
+
+(4) $\mu_n$ satisfies the Disjoint $n$-cells Property ($DD^nP$).
+
+**Definition 2.2.** A space $X$ is *k*-connected if every map of $S^j, 0 \le j \le k$ into $X$ extends to a map of $B^{j+1}$ into $X$. A space $X$ is *locally k*-connected if for each point $p \in X$ and for each neighborhood $U$ of $p$, there exists a neighborhood $V$ of $p$ so that each map of $S^j, 0 \le j \le k$ into $V$ extends to a map of $B^{j+1}$ into $U$. A space $X$ satisfies the *Disjoint k-Cells Property* if for each $\epsilon > 0$ and for each pair of maps $f_1$ and $f_2$ from $I^k$ into $X$, there are maps $g_1$ and $g_2$ from $I^k$ into $X$ with $g_1(I^k) \cap g_2(I^k) = \emptyset$ and $d(g_i, f_i) < \epsilon$.
\ No newline at end of file
diff --git a/samples/texts/4807413/page_5.md b/samples/texts/4807413/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..c4e842e6f7684fe0d58de9ebcbb8b859fbe2f8b9
--- /dev/null
+++ b/samples/texts/4807413/page_5.md
@@ -0,0 +1,11 @@
+Note that the spaces $M_n^i$ used in the construction of the $n$-dimensional Menger space $\mu_n$ are both $C^{n-1}$ and $LC^{n-1}$. The conditions in Theorem 2.1 yield the result that $\mu_n$ is a universal $n$-dimensional separable metric space, i.e. $\mu_n$ is $n$-dimensional and contains a homeomorphic copy of every separable metric $n$-dimensional space. The details are given in [Be]. We are interested in the homogeneity properties of Menger Spaces and of products of Menger Spaces. The relevant definitions are provided next.
+
+**Definition 2.3.** A space $X$ is *homogeneous* if and only if for each pair of points $p$ and $q$ in $X$, there is a homeomorphism $h : X \to X$ with the property that $h(p) = q$. A space $X$ is *$n$-homogeneous* if for each pair of $n$-point subsets of $X$, $A$ and $B$, there is a homeomorphism $h : X \to X$ with the property that $h(A) = B$. A space $X$ is *countable dense homogeneous* if and only if for each pair of countable dense subsets $A$ and $B$ of $X$, there is a homeomorphism $h : X \to X$ with the property that $h(A) = B$.
+
+It is well known that the Cantor set $(\mu_0)$, the Hilbert Cube, and all manifolds satisfy these types of homogeneity. R. D. Anderson established that $\mu_1$ also satisfies these types of homogeneity [An1, An2]. M. Bestvina established the analogous results for the higher dimensional Menger spaces.
+
+**Theorem 2.4.** [Be, pg. 73]. Each Menger Space $\mu_n$ is $k$-homogeneous for each $k$, and is countable dense homogeneous.
+
+### 3. ČECH HOMOLOGY OF MENGER SPACES
+
+We need some preliminary results about embeddings and maps of $S^n$ into $\mu_n$. For these computations, we use singular and Čech homology with coefficients in the rationals. See [ES] for the properties of Čech homology. A map $f$ from $S^n$ into a space $X$ is said to be essential with respect to $n$-th Čech
\ No newline at end of file
diff --git a/samples/texts/4807413/page_6.md b/samples/texts/4807413/page_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..66d0be5388272877dfc089849c21a079f1ed6948
--- /dev/null
+++ b/samples/texts/4807413/page_6.md
@@ -0,0 +1,23 @@
+homology if the induced homomorphism on n-th Čech homology groups is nontrivial. The map is said to be essential with respect to homotopy if it is not homotopic to a constant map.
+
+**Lemma 3.1.** For each $\mu_n$, the following results hold:
+
+(1) Any embedding of $S^n$ into $\mu_n$ is essential both with respect to homotopy and with respect to n-th Čech homology.
+
+(2) If $f_1$ and $f_2$ are any maps from $S^n$ into $\mu_n$ that are essential with respect to n-th Čech homology, and if $f_1$ and $f_2$ have disjoint images, then $f_1$ and $f_2$ are not homotopic.
+
+*Proof:* Consider the exact Čech homology sequence of the pair $(\mu_n, E)$ where $E$ is either $e(S^n)$ or $f_1(S^n) \cup f_1(S^n)$. Since $\mu_n$ is $n$-dimensional, it follows that $\tilde{H}_{n+1}(\mu_n, E)$ is trivial and thus the inclusion induced homomorphism $\tilde{H}_n(E) \to \tilde{H}_n(\mu_n)$ is a monomorphism. It follows that if $E$ is $e(S^n)$, $\tilde{H}_n(E)$ is non-trivial and thus the map $e$ is essential with respect to $n$-th Čech homology. It follows from this that the map is essential with respect to homotopy.
+
+If $E$ is $f_1(S^n) \cup f_1(S^n)$, then $\tilde{H}_n(E) \simeq \tilde{H}_n(E_1) \oplus \tilde{H}_n(E_2)$
+where $E_i = f_i(S^n)$. The hypotheses imply that each $\tilde{H}_n(E_i)$ is
+nontrivial. If $f_1$ and $f_2$ were homotopic, they would induce the
+same homomorphism on Čech homology, contradicting the fact
+that the inclusion induced homomorphism $\tilde{H}_n(E) \to \tilde{H}_n(\mu_n)$
+is a monomorphism.
+
+**Lemma 3.2.** For each $\epsilon > 0$ and for each point $p \in \mu_n$, there is an embedding $f : S^n \to \mu_n$ with image contained in the $\epsilon$ neighborhood of p.
+
+*Proof:* This follows from the construction of $\mu_n$ since each point in $\mu_n$ has arbitrarily small neighborhoods that are homeomorphic to $\mu_n$ and since $\mu_n$ itself contains embedded copies of $S^n$. $\square$
+
+**Theorem 3.3.** Let X be a compact subspace of some ANR Z.
+Fix $n \ge 1$. Let $Y = \mu_n \times X$. Let $p$ and $q$ be projections from
\ No newline at end of file
diff --git a/samples/texts/4807413/page_7.md b/samples/texts/4807413/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..0f5a02b79e3b29dfdf8aeb917629871b29686fe5
--- /dev/null
+++ b/samples/texts/4807413/page_7.md
@@ -0,0 +1,15 @@
+$Y$ onto $\mu_n$ and $X$ respectively. If $f : S^k \to Y, 1 \le k \le n$ is a map that is essential with respect to $k$-th Čech homology, then either $p_0 f$ or $q_0 f$ is essential with respect to $k$-th Čech homology. If $k < n$, then $q_0 f$ is essential with respect to $k$-th Čech homology.
+
+*Proof:* We write the homology class of a cycle $z$ as $[z]$. Express $X$ as a nested intersection of compact neighborhoods $X_i$ in $Z$. The space $\mu_n$ is a nested intersection of the compact polyhedra $M_n^i$ defined above. So the space $Y$ is a nested intersection of the compact neighborhoods $Y_i = M_n^i \times X_i$ in $Z \times R^{2n+1}$. The Čech homology of $Y$ may be thus computed as the inverse limit of the singular homology of the spaces $Y_i$. Let $p_i$ and $q_i$ be projections from $Y_i$ onto $M_n^i$ and $X_i$ respectively.
+
+The spaces $M_n^i$ have trivial singular homology in dimensions 1 through $(k-1)$. So the Künneth and Eilenberg-Zilber theorems for singular homology (see [Mu]) imply that if $\alpha_i = [c]$ is a nontrivial element of the $k$-th singular homology of $Y_i$, either $p_{i*}(\alpha_i)$ or $q_{i*}(\alpha_i)$ is nontrivial.
+
+To see this, note that since we are using rational coefficients, the Künneth and Eilenberg Zilber Theorems provide isomorphisms
+
+$$ H_k(Y_i) \xrightarrow{u} H_k(S(M_n^i) \otimes S(X_i)) \\ \int \theta \\ (H_k(M_n^i) \otimes H_0(X_i)) \oplus (H_0(M_n^i) \otimes H_k(X_i)) $$
+
+Here
+
+$$ u([c]) = \left[ \sum_j p_{i*}(cF_j) \otimes q_{i*}(cB_{k-j}) \right] $$
+
+where $F_j$ and $B_{k-j}$ are the front $j$-face and back $k-j$ face operators. Also, $\theta([a] \otimes [b]) = [a \otimes b]$. An inverse isomor-
\ No newline at end of file
diff --git a/samples/texts/5140397/page_1.md b/samples/texts/5140397/page_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..c3c377e077b6ec5297caa47e18a518aac14a6861
--- /dev/null
+++ b/samples/texts/5140397/page_1.md
@@ -0,0 +1,25 @@
+First detection of threshold crossing events under intermittent sensing
+
+Aanjaneya Kumar,∗ Aniket Zodage,† and M. S. Santhanam‡
+
+Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India.
+
+(Dated: June 8, 2021)
+
+The time of the first occurrence of a threshold crossing event in a stochastic process, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional 'sensor' monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the *first detection time*, which denotes the time when the sensor detects the threshold crossing event for the first time. In this work, a birth-death process monitored by a random intermittent sensor is studied, for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to, and is obtainable from, the first passage time distribution. Our analytical results display an excellent agreement with simulations. Further, this framework is demonstrated in several applications – the SIS compartmental and logistic models, and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.
+
+In many situations, the time taken for an observable to reach a pre-determined threshold for the first time, called the first passage time (FPT), is of immense interest and carries practical value. The first occurrence of breakdown of an engineered structure, triggering of bio-chemical reactions, or a stock market index reaching a specific value are all examples that can be posed as first passage problems (FPP). This class of problems has been extensively studied in statistical physics for many decades [1–3], and has a large number of applications in many areas of physical sciences [1, 4–6], engineering [3, 7–9], finance [10–12] and biology [13–15]. In its simplest formulation, the FPP is solved under an implicit assumption of perfect detection conditions – the notional sensor, which monitors the first occurrence of a threshold crossing event, is active at all the times.
+
+In practical applications, there is an energy cost associated with an always-on sensor, and the threshold crossing events in processes of interest are thus monitored by intermittent sensors that are not active at all times. A principal example is the wireless sensor networks widely deployed to monitor rare events at remote locations and operate under tight energy constraints [16, 17]. These sensors are typically not always active in order to optimise power consumption, and event sensing under such conditions can be modeled as an FPP [18–20]. Intermittent sensing has been extensively deployed in industrial and military environments [21] to detect events and is even thought to be the future trend for Internet-of-Things and wireless monitoring technologies [22]. In several bio-chemical processes too, the sensors make stochastic transitions between active and inactive states [23]. A relevant example is of the heat shock response in a cell due to environmental stresses [24, 25], where the HSF
+
+family of proteins, which upregulate heat shock proteins under heat stress, can perform upregulation only when present in their trimeric state, while they are inactive in their monomeric state.
+
+Motivated by these phenomena, in this paper, the standard FPP for threshold crossing is expanded to include intermittent sensing by an independent sensor that stochastically switches between active and inactive
+
+FIG. 1. (a) The schematic shows the first passage time $T_f$ for a stochastic process to reach a threshold $X^*$, and the first detection time $T_d$ of the threshold crossing event under intermittent sensing. The red (green) shaded region indicates if the sensor is inactive (active). The sensor can detect events only if it is active. (b) A birth-death process models the underlying process, (c) a sensor switching between active and inactive states. (d) The composite process with $N = 5$ and $m = 4$. The sensor can detect the absorbing states (yellow).
+
+* kumar.aanjaneya@students.iiserpune.ac.in
+
+† aniket.zodage@students.iiserpune.ac.in
+
+‡ santh@iiserpune.ac.in
\ No newline at end of file
diff --git a/samples/texts/5140397/page_10.md b/samples/texts/5140397/page_10.md
new file mode 100644
index 0000000000000000000000000000000000000000..91be557d51fbe7c7eb36c2515e8c40e32b5a81f3
--- /dev/null
+++ b/samples/texts/5140397/page_10.md
@@ -0,0 +1,80 @@
+where
+
+$$
+\tilde{L}(s) = \frac{1 - \tilde{F}_s(m|n_0)}{s} \tag{S23}
+$$
+
+$$
+\tilde{g}_1(s) = \frac{\beta}{\alpha + \beta} \tilde{F}_s(m|n_0) - \frac{\beta}{\alpha + \beta} \tilde{F}_{s+\alpha+\beta}(m|n_0) \quad (S24)
+$$
+
+$$
+\tilde{g}_2(s) = \tilde{F}_{s+\alpha}(m-1|m) \tag{S25}
+$$
+
+$$
+\tilde{g}_3(s) = \frac{\beta}{\alpha + \beta} \tilde{F}_s(m|m-1) + \frac{\alpha}{\alpha + \beta} \tilde{F}_{s+\alpha+\beta}(m|m-1) \quad (S26)
+$$
+
+$$
+\tilde{g}_4(s) = \frac{1 - \tilde{F}_s(m|m-1)}{s} \tag{S27}
+$$
+
+$$
+\tilde{g}_5(s) = \frac{1 - \tilde{F}_{s+\alpha}(m-1|m)}{s+\alpha}. \qquad (S28)
+$$
+
+Because of several repetitive factors in each term, we identify both the series as geometric series
+
+$$
+\tilde{S}_s(a, n_0) = \tilde{L}(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_4(s)[1 + \tilde{g}_2(s)\tilde{g}_3(s) + \tilde{g}_2(s)^2\tilde{g}_3(s)^2 + \dots] + \tilde{g}_1(s)\tilde{g}_5(s)[1 + \tilde{g}_2(s)\tilde{g}_3(s) + \tilde{g}_2(s)^2\tilde{g}_3(s)^2 + \dots] \tag{S29}
+$$
+
+both of which can be individually summed to give
+
+$$
+\begin{equation}
+\tilde{S}_s(a, n_0) = \tilde{L}(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_4(s) \left[ \frac{1}{1-\tilde{g}_2(s)\tilde{g}_3(s)} \right] + \tilde{g}_1(s)\tilde{g}_5(s) \left[ \frac{1}{1-\tilde{g}_2(s)\tilde{g}_3(s)} \right]. \tag{S30}
+\end{equation}
+$$
+
+This allows us to write the Laplace transform of the survival probability as
+
+$$
+\tilde{S}_s(a, n_0) = \tilde{L}(s) + \frac{\tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_4(s) + \tilde{g}_1(s)\tilde{g}_5(s)}{1 - \tilde{g}_2(s)\tilde{g}_3(s)}. \quad (S31)
+$$
+
+A term by term interpretation of the above equation is possible, as done for the Laplace transformed FDTD in the
+previous section.
+
+In above analysis, we considered an initial state in which the sensor is active and the birth death process is in state $n_0(< m)$. If the initial conditions is such that the sensor is inactive, a similar analysis can be performed. But now the initial state of the birth-death process can be any number between 0 to N. We summarise the results of such analysis below.
+
+1. $n_0 < m$:
+
+$$
+\tilde{S}_s(i, n_0) = \tilde{L}(s) + \frac{\tilde{g}_6(s)\tilde{g}_2(s)\tilde{g}_4(s) + \tilde{g}_6(s)\tilde{g}_5(s)}{1 - \tilde{g}_2(s)\tilde{g}_3(s)} \quad (S32)
+$$
+
+with
+
+$$
+g_6(t) = F_t(m|n_0) p_t(i|i) \quad \text{and} \quad \tilde{g}_6(s) = \frac{\beta}{\alpha + \beta} \tilde{F}_s(m|n_0) + \frac{\alpha}{\alpha + \beta} \tilde{F}_{s+\alpha+\beta}(m|n_0) \quad (\text{S33})
+$$
+
+$g_6(t)$ is the probability that starting with in the state $n_0$ with inactive sensor, the birth death process reaches the state $m$ for the first time at time $t$ and the sensor is inactive at $t$. $g_6(t)$ accounts for the probability of the initial part of the survival trajectories, which involve crossing of threshold state by the birth death process. Only this is the difference from initial active target case.
+
+2. $n_0 \ge m$
+
+$$
+\tilde{S}_s(i, n_0) = \tilde{L}_1(s) + \frac{\tilde{g}_7(s)\tilde{g}_3(s)\tilde{g}_5(s) + \tilde{g}_7(s)\tilde{g}_4(s)}{1 - \tilde{g}_2(s)\tilde{g}_3(s)} \quad (S34)
+$$
+
+with
+
+$$
+L_1(t) = e^{-\alpha t} \int_t^\infty dt' F_{t'}(m-1|n_0) \quad \text{and} \quad \tilde{L}_1(s) = \frac{1 - \tilde{F}_{s+\alpha}(m-1|n_0)}{s+\alpha} \tag{S35}
+$$
+
+$$
+g_7(t) = e^{-\alpha t} F_t(m-1|n_0) \quad \text{and} \quad \tilde{g}_7(s) = \tilde{F}_{s+\alpha}(m-1|n_0) \tag{S36}
+$$
\ No newline at end of file
diff --git a/samples/texts/5140397/page_11.md b/samples/texts/5140397/page_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..f1782dc4bcf8725da1a0fa39bec43ea08bf7ad02
--- /dev/null
+++ b/samples/texts/5140397/page_11.md
@@ -0,0 +1,35 @@
+$L_1(t)$ is the probability that starting in the state $n_0 \ge m$ with inactive sensor, the birth death process remains above the state $m-1$ and the sensor is inactive throughout this time $t$. $g_7(t)$ is the probability that starting in the state $n_0$ with the inactive sensor, the birth death process reaches the state $m-1$ for the first time at time $t$ and the sensor is inactive throughout this time. Similar to $g_6(t)$, probability $g_7(t)$ accounts for the initial part of the trajectories, which involve crossing of threshold state by the underlying birth death process.
+
+### C. Splitting Probabilities
+
+A key feature of this modeling approach is that when the threshold crossing event is detected, the state of the birth death process is a random number which can take values lying in the set $\{m, m+1, \dots, N\}$. A quantity of interest in such a scenario is the splitting probability $H_t(\zeta)$ that gives us the probability that the birth death process is in state $\zeta$ while the threshold crossing event is detected. Using insights from the calculation of the survival probability, it is easy to obtain $H_t(\zeta)$.
+
+It turns out that $\zeta = m$ is a special case. Detection of the threshold crossing event at the state $m$ can happen with the last jump of the birth death process being – (1) from the state $m-1$ to $m$ where sensor can be active or inactive at the time of the jump or (2) from $m+1$ to $m$ where the sensor has to be inactive at the time of the jump (the becoming active after the jump while the birth death process is till in the state $m$). Whereas for $\zeta > m$, sensor has to be inactive at the time of the last jump, irrespective of the type of jump ($\zeta + 1 \text{ to } \zeta$ or $\zeta - 1 \text{ to } \zeta$). We consider $\zeta = m$ and $\zeta > m$ cases separately.
+
+We define a random variable $\tau_{m,m-1}$ which denotes the first time when the birth-death process reaches the state $(m-1)$ starting from the state $m$.
+
+First we consider the case $k=m$. The trajectories leading to detection of the threshold crossing event with the birth death process in the state $m$ can be grouped in two parts – (I) trajectories with the last jump of the birth death process at time $t$ from $m-1$ to $m$, (II) trajectories with the last jump of the birth death process from $m+1$ to $m$ at time $t$ and trajectories with the last jump of the birth death process from $m-1$ to $m$ but at some time $s < t$.
+
+The possible trajectories in the part I are:
+
+* The birth death process reaches state $m$ for the first time at time $t$ and the sensor is active at the same time.
+
+* The birth death process reaches state $m$ for the first time at time $t_1$ but the sensor is inactive. Then birth death process remains in states greater than $m$ till time $t_2$ and the sensor is inactive throughout. At time $t_2$, the birth death process reaches the state $m-1$. Then birth death process reaches the state $m$ at time $t$ for the first time after $t_2$ and the sensor is active at the same time.
+
+Summing up the probabilities of aforementioned trajectories gives
+
+$$ 
+\begin{aligned}
+H_t^{(1)}(m) &= F_t(m|n_0)p_t(a|i) \\
+&\quad + \int_0^t \int_0^{t_2} dt_1 F_{t_1}(m|n_0)p_{t_1}(i|a)F_{t_2-t_1}(m-1|m)e^{-\alpha(t_2-t_1)}F_{t-t_2}(m|m-1)p_{t-t_2}(a|i) \\
+&\quad + \int_0^t \int_0^{t_4} \int_0^{t_3} \int_0^{t_2} dt_4 \cdot dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0)p_{t_1}(i|a)F_{t_2-t_1}(m-1|m)e^{-\alpha(t_2-t_1)} \\
+&\quad F_{t_3-t_2}(m|m-1)p_{t_3-t_2}(i|i)F_{t_4-t_3}(m-1|m)e^{-\alpha(t_4-t_3)}F_{t-t_4}(m|m-1)p_{t-t_4}(a|i) + \dots
+\end{aligned} 
+\quad (\text{S37})
+$$
+
+The possible trajectories in the second part are:
+
+* The birth death process reaches the state $m$ for the first time at time $t_1$ and the sensor is inactive. Then birth death process remains in states greater than $m-1$ till time $t$ and the sensor is inactive throughout. At time $t$, the birth death process is in the state $m$ and at the same time sensor becomes active for the first time after $t_1$.
+
+* The birth death process reaches the state $m$ for the first time at time $t_1$ and the sensor is inactive. Then birth death process remains in states greater than $m-1$ for some time and sensor is inactive throughout. At time $t_2$, the birth death process reaches the state $m-1$ for the first time after $t_1$. Then the birth death process reaches state $m$ at time $t_3$ for the first time after $t_2$ and the sensor is inactive. Then birth death process remains in states greater than $m-1$ till time $t$ and the sensor is inactive throughout. At time $t$, the birth death process is in state $m$ and at the same time sensor becomes active for the first time after $t_3$.
\ No newline at end of file
diff --git a/samples/texts/5140397/page_12.md b/samples/texts/5140397/page_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..9fb1be752992b5939c535203142d22ceaf4d45fa
--- /dev/null
+++ b/samples/texts/5140397/page_12.md
@@ -0,0 +1,82 @@
+Summing up the probabilities of the aforementioned trajectories gives
+
+$$
+\begin{align}
+H_t^{(\mathrm{II})}(m) = & \int_0^t dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) P_{t-t_1}(m|m, \tau_{m,m-1} > t - t_1) \alpha e^{-\alpha(t-t_1)} \nonumber \\
+& + \int_0^t \int_0^{t_3} \int_0^{t_2} dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0) p_{t_1}(i|i) F_{t_2-t_1}(m-1|m) e^{-\alpha(t_2-t_1)} F_{t_3-t_2}(m|m-1) \nonumber \\
+& p_{t_3-t_2}(i|i) P_{t-t_3}(m|m, \tau_{m,m-1} > t - t_3) \alpha e^{-\alpha t - t_3} + \dots \tag{S38}
+\end{align}
+$$
+
+Then splitting probability for the detection at *m* can be written as $H_t(m) = H_t^{(I)}(m) + H_t^{(II)}(m)$. For the brevity, we define the following quantities
+
+$$
+h_1(t) = F_t(m|n_0) p_t(a|a) \tag{S39}
+$$
+
+$$
+h_2(t) = F_t(m|m-1) p_t(a|i) \quad (S40)
+$$
+
+$$
+h_3(k, m, t) = \alpha e^{-\alpha t} P_t(k|m, \tau_{m,m-1} > t) \quad \text{For } k = m, m+1, m+2, \dots, N \tag{S41}
+$$
+
+We take the Laplace transform of $H_t(m)$, which gives
+
+$$
+\begin{align}
+\tilde{H}_s(m) &= \tilde{h}_1(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{h}_2(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_3(s)\tilde{g}_2(s)\tilde{h}_2(s) + \dots \tag{S42} \\
+&\quad + \tilde{g}_1(s)\tilde{h}_3(m,m,s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_3(s)\tilde{h}_3(m,m,s) + \dots \tag{S43}
+\end{align}
+$$
+
+where
+
+$$
+\tilde{h}_{1}(s)=\frac{\alpha}{\beta+\alpha} \tilde{F}_{s}\left(m | n_{0}\right)+\frac{\beta}{\beta+\alpha} \tilde{F}_{s+\alpha+\beta}\left(m | n_{0}\right)
+$$
+
+$$
+\tilde{h}_2(s) = \frac{\alpha}{\beta + \alpha} \tilde{F}_s(m|m-1) + \frac{\alpha}{\beta + \alpha} \tilde{F}_{s+\alpha+\beta}(m|m-1) \quad (S45)
+$$
+
+$$
+\tilde{h}_3(m, m, s) = \alpha \overline{\mathcal{L}C}_{s+\alpha}(m) \qquad (\text{S46})
+$$
+
+where $\mathcal{L}C_t(k) = P_t(k|m, \tau_{m,m-1} > t)$ For $k = m, m+1, m+2, \dots, N$
+
+Factoring out the common terms, a geometric series can be identified.
+
+$$
+\begin{equation}
+\begin{split}
+\tilde{H}_s(m) = {}& \tilde{h}_1(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{h}_2(s) [1 + \tilde{g}_2(s)\tilde{g}_3(s) + \tilde{g}_2(s)^2\tilde{g}_3(s)^2 + \cdots] \\
+& + \tilde{g}_1(s)\tilde{h}_3(m,m,s) [1 + \tilde{g}_2(s)\tilde{g}_3(s) + \tilde{g}_2(s)^2\tilde{g}_3(s)^2 + \cdots]
+\end{split}
+\tag{S47}
+\end{equation}
+$$
+
+Both the geometric series can be summed up to give
+
+$$
+\hat{H}_s(m) = \hat{h}_1(s) + \frac{\tilde{g}_1(s)\tilde{g}_2(s)\tilde{h}_2(s) + \tilde{g}_1(s)\tilde{h}_3(m,m,s)}{1 - \tilde{g}_2(s)\tilde{g}_3(s)}. \quad (S48)
+$$
+
+Now we consider the case where $k = m+r$, where $r = 1, 2, 3, \dots, N-m$. It is easy to see that $H_t(m+r)$ will be given by
+
+$$
+\begin{align}
+H_t(m+r) = & \int_0^t dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) P_{t-t_1}(m+r|m, \tau_{m,m-1} > t - t_1) \alpha e^{-\alpha t - t_1} \nonumber \\
+& + \int_0^t \int_0^{t_3} \int_0^{t_2} dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0) P_{t_1}(i|i) F_{t_2-t_1}(m-1|m) e^{-\alpha(t_2-t_1)} F_{t_3-t_2}(m|m-1) \nonumber \\
+& p_{t_3-t_2}(i|i) P_{t-t_3}(m+r|m, \tau_{m,m-1} > t - t_3) \alpha e^{-\alpha t - t_3} + \dots \tag{S49}
+\end{align}
+$$
+
+Following similar steps as in the case of $k = m$, we are led to the Laplace transform of $H_t(m+r)$ as
+
+$$
+\hat{H}_s(m+r) = \frac{\tilde{g}_1(s)\tilde{h}_3(m+r, m, s)}{1 - \tilde{g}_2(s)\tilde{g}_3(s)} \quad (S50)
+$$
\ No newline at end of file
diff --git a/samples/texts/5140397/page_13.md b/samples/texts/5140397/page_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..bb4465cbde4d17000b706b323c97734390e9a29f
--- /dev/null
+++ b/samples/texts/5140397/page_13.md
@@ -0,0 +1,27 @@
+**D. SIS Model**
+
+The susceptible-infected-susceptible (SIS) model of disease propagation is a stochastic model that describes the spread of a disease in a population, which we consider to be well-mixed. The population consists of two types of individuals, those that are susceptible to the infection, and those that are currently infected. The rate at which the disease is transmitted between individuals is $\lambda$, and the recovery rate for each infected individual is $\mu$. If $j$ represents the number of infected individuals, there are $j(N-j)$ pairwise contacts between infected and susceptible people in a well-mixed population. Each of the $j$ infected individuals recover at rate $\mu$. The rates of increase and decrease of the number of infected individuals is given by
+
+$$W_+(j) = \lambda j(N-j), \quad W_-(j) = \mu j. \qquad (S51)$$
+
+The parameter values chosen to obtain the curve for Fig. 5 from the main text are $N = 15, m = 7, n_0 = 1, \lambda = 0.1$, and $\mu = 0.4$.
+
+**E. Logistic Model**
+
+The stochastic version of the logistic model describes the dynamics of a population of mortal agents, who can reproduce. A constant rate $B$ is assumed for each agent, which means that in a small time interval $dt$, each individual gives birth to a new individual with probability $Bdt$. For each agent, there is constant death rate (set to 1), when the population size is low. However, for larger population sizes, the death rate increases by an amount which is quadratic in the size of the population. In the birth-death formulation, the transition rates are
+
+$$W_+(j) = Bj, \quad W_-(j) = j + Kj^2/N, \qquad (S52)$$
+
+where $K$ determines the strength of influence of competition towards the death rates. The parameter values chosen to obtain the curve for Fig. 5 from the main text are $N = 15, m = 8, n_0 = 4, B = 1.5, K = 0.1, \alpha = \beta = 1$.
+
+**F. Stochastic Resets**
+
+In the birth-death process with stochastic resets, apart from the simple birth-death dynamics, with a rate $r$, the underlying process can be reset to the state 0. This dynamics is reminiscent of fluctuating dynamics that undergo burst-like relaxations. Furthermore, one can also consider the process, where the reset happens to a dormant state, at rate $r$. When this reset happens, the underlying process spends some refractory time in this dormant state, and resumes its birth-death dynamics at a rate $g$.
+
+The parameter values chosen to obtain the curve for Fig. 5 from the main text are $N = 10, m = 5, n_0 = 0, r = \alpha = \beta = 1$ while the birth-death rates were chosen to be the same as the ones for the curves in Fig. 2. In the case where refractory period is also considered, we choose $g = 1$.
+
+**G. Derivation of the first passage time distribution conditioned on first detection at $T_d$**
+
+We define $F_{T_f}(m|n_0, \sigma_0, T_d)$ to be the density that the first passage to threshold $m$ happens at time $T_f$, conditioned on the fact that the first detection of the threshold crossing event happens at $T_d$, and that the underlying process starts from a state $n_0$ whereas the sensor starts from state $\sigma_0$. As mentioned in the main text, for $T_f > T_d$, $F_{T_f}(m|n_0, \sigma_0, T_d) = 0$. For $T_f < T_d$, we are interested in the trajectories that reach the threshold $m$ for the first time at $T_f$, but go undetected, and eventually, the threshold crossing event is detected at time $T_d$. We can break each such trajectory in two parts: the evolution up to time $T_f$, and the evolution up to time $T_d$, starting from time $T_f$. The first part of each trajectory is a *first passage trajectory*, i.e. one which reaches the threshold for the first time at time $T_f$. We can immediately also conclude that at time $t = T_f$, the state of the underlying birth death process must be equal to the threshold, and the state of the sensor must have been inactive. Furthermore, the second part of each trajectory is a *first detection trajectory*, that starts from an initial state such that the birth death process is at the threshold and the sensor is inactive, and subsequently, the first detection of the threshold crossing event happens at time $T_d$. Putting this together, we have
+
+$$F_{T_f}(m|n_0, \sigma_0, T_d) = F_{T_f}(m|n_0) \cdot p_{T_f}(i|\sigma_0) \frac{D_{T_d-T_f}(m,i)}{D_{T_d}(n_0, \sigma_0)} \qquad (S53)$$
\ No newline at end of file
diff --git a/samples/texts/5140397/page_14.md b/samples/texts/5140397/page_14.md
new file mode 100644
index 0000000000000000000000000000000000000000..d60beb46af1da8a51d54a043c3da29394dc610e8
--- /dev/null
+++ b/samples/texts/5140397/page_14.md
@@ -0,0 +1,3 @@
+where the denominator $D_{T_d}(n_0, \sigma_0)$ is due to the fact that we are looking at the subset of trajectories, which are conditioned to undergo the first detection event at time $T_d$.
+
+A similar argument enables us to see that for $T_f = T_d$, the probability $F_{T_f}(m|n_0, \sigma_0, T_d = T_f) = \frac{F_{T_d}(m|n_0)}{D_{T_d}(n_0, \sigma_0)}$.
\ No newline at end of file
diff --git a/samples/texts/5140397/page_2.md b/samples/texts/5140397/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..e0a3d3c827d1dbcdb65b2fd0de8b6ecef12ca943
--- /dev/null
+++ b/samples/texts/5140397/page_2.md
@@ -0,0 +1,31 @@
+states. The central quantity of interest is the first detection time distribution (FDTD) of the threshold crossing event by the sensor. This can be thought of as first detection of an extreme event [26–28] or a general threshold activated process [29–31] under intermittent sensing. A schematic of the processes are displayed in Fig. 1(a), where a stochastic process $X(t)$ is evolving in time, while a sensor, which monitors whether the stochastic process has crossed a pre-defined threshold $X^*$ or not, switches between inactive (red background) and active (green) states. A threshold crossing event is detected only if $X(t) > X^*$ and the sensor is active at time $t$. Both the first passage time for threshold crossing and first detection time are marked in Fig. 1(a). This process is distinct from the FPP extensively studied in the context of intermittent search problems [32–35] or intermittent target problems [36–39]. In the intermittent search/target problems, the process is terminated only when the searcher is at the target when the searcher/target is “active”. In contrast, in the threshold crossing under intermittent sensing, the detection of the crossing event can happen in any state greater than the threshold, provided the sensor is active (Fig. (1)).
+
+We model the underlying process of interest as a Markovian, continuous-time birth-death process (BDP) (Fig. 1(b)), which has previously been used to study a variety of processes [40–44]. The state of the BDP can be interpreted as the stress or damage accumulated over time, or any other physical quantity where threshold crossing is of prime interest. The BDP is defined on the state space $\mathcal{S} \in \{0, 1, 2, \dots, N\}$ with its dynamics governed by the rates $W_+(j)$ and $W_-(j)$ for transitioning from state $j$ to states $j+1$ and $j-1$ respectively, where $j \in \mathcal{S}$, with $W_+(N) = W_-(0) = 0$. The probability of finding the BDP in state $n \in \mathcal{S}$ at time $t$, given that the system started from state $n_0 \in \mathcal{S}$ initially, is given by the propagator $P_t(n|n_0)$, and its Laplace transform is denoted by $\tilde{P}_s(n|n_0)$. A well studied quantity of the BDP is the first passage time distribution (FPTD), denoted by $F_t(m|n_0)$, which is probability density that the BDP reaches state $m$ for the first time, at time $t$, given that it started from state $n_0$. Through the renewal formula [1], the Laplace transform of the FPTD is given by $\tilde{F}_s(m|n_0) = \frac{\tilde{P}_s(m|n_0)}{\tilde{P}_s(m|m)}$, for $n_0 \neq m$. This analysis assumes perfect detection, i.e., as soon as the threshold $m$ is reached, the event is detected.
+
+To study the case of imperfect detection, an intermittent sensor is assumed and is modeled by a two-state Markov process (Fig. 1(c)). The sensor dynamics is assumed to be independent of the underlying BDP whose threshold crossing will be monitored. The sensor can be found in any of the two states $\Omega = \{a, i\}$, with $a$ and $i$ denoting, respectively, active and inactive states. The sensor switches from state $i$ to $a$ at rate $\alpha$, and from $a$ to $i$ at $\beta$. For $\sigma_0, \sigma \in \Omega$, we define $p_t(\sigma|\sigma_0)$ to be the probability that the sensor is in state $\sigma$ at time $t$, given that it was in state $\sigma_0$ initially. The Markov diagram of
+
+the composite process, i.e., the underlying process and the sensor combined, is shown in Fig. 1(d) for the special case of $N=5$, and threshold $m=4$. The composite process is another continuous time Markov process on the state space $\mathcal{S} = \mathcal{S} \times \Omega$. The key object of interest is the statistics of first detection time (FDT) of a threshold crossing event, defined as the first time when the composite process is found in any of the states $(n,a)$ such that $n \ge m$, where $m$ is the pre-defined threshold. Thus, the composite process has a total of $N-m+1$ absorbing states. We denote the FDTD by $D_t(n_0, \sigma_0)$, which is the probability density for the first detection event to happen at time $t$, given that the initial condition for the composite process was $(n_0, \sigma_0)$.
+
+In the analysis that follows, the knowledge of $P_t(n, n_0)$ for the BDP is assumed. This is known exactly for a variety of examples [42, 45, 46]. Furthermore, the propagator can be obtained, in principle, for any BDP governed by an $N \times N$ tridiagonal Markovian transition matrix $\mathbb{W}$ as $P_t(n|n_0) = \langle n | e^{\mathbb{W}t} | n_0 \rangle$ where $\langle l | = (0 \ 0 \ 0 \ \dots \ 0 \ 1 \ 0 \ \dots \ 0)$ denotes a row vector with 1 as its $l^{th}$ element, with 0 elsewhere.
+
+To obtain the FDT statistics, we note that the FDTD satisfies the following equations:
+
+$$D_t(n_0, \sigma_0) = f_1(t) + \int_0^t f_2(t') D_{t-t'}(m, i) dt', \quad (1)$$
+
+$$D_t(m, i) = f_3(t) + \int_0^t e^{-\alpha t'} F_{t'}(m-1|m) D_{t-t'}(m-1, i) dt' \quad (2)$$
+
+where $n_0 < m$ and the following functions are defined:
+
+$$f_1(t) = F_t(m|n_0)p_t(a|\sigma_0), \quad f_2(t) = F_t(m|n_0)p_t(i|\sigma_0),$$
+
+$$f_3(t) = \alpha e^{-\alpha t} \int_t^\infty F_{t'}(m-1|m)dt'. \quad (3)$$
+
+We obtain $D_t(m-1, i)$ from Eq. (1) in terms of $D_t(m,i)$, and taking a Laplace transform of Eqs. 1 and 2, we can write
+
+$$\tilde{D}_s(n_0, \sigma_0) = \tilde{f}_1(s) + \frac{\tilde{f}_2(s)(\tilde{f}_3(s) + \tilde{f}_4(s)\tilde{F}_{s+\alpha}(m-1|m))}{1 - \tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)}. \quad (4)$$
+
+where we define:
+
+$$f_4(t) = F_t(m|m-1)p_t(a|i), \quad f_5(t) = F_t(m|m-1)p_t(i|i). \quad (5)$$
+
+Equation 4 is our central result, that asserts that the FDTD can be obtained in terms of the FPTD alone. An alternate derivation of this result is presented in SI. The first term on the RHS, $f_1(t)$ denotes trajectories where the FPT and FDT coincide, whereas the second term accounts for all trajectories where the first passage event goes unnoticed, and detection happens at a later time. In the limit of $\beta \to 0^+$, then $f_2(t) \to 0$, and it leads to $D_t(n_0, \sigma_0) = f_1(t)$. This is consistent with the
\ No newline at end of file
diff --git a/samples/texts/5140397/page_3.md b/samples/texts/5140397/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..b61bb8aedf76a4584701c2eab51892c0fcb919c3
--- /dev/null
+++ b/samples/texts/5140397/page_3.md
@@ -0,0 +1,17 @@
+FIG. 2. The FDTD for the BDP with stochastic switching (solid lines), with $n_0 = 0, N = 20, \lambda = 0.1$ and $k = 1$, shown for threshold values $m = 3, 5, 7$ and $\alpha = \beta = 1$. The symbols are from simulations. The dashed lines are FDTD if the sensor is not intermittent and is always on.
+
+expectation that in the $\beta \to 0^+$ limit deactivation of the sensor is extremely unlikely and renders the composite process equivalent to a simple BDP.
+
+To illustrate these results, consider a BDP with transition rates $W_+(j) = \lambda(N-j)$ and $W_-(j) = kj$, with $j \in \{0, 1, 2, \dots, N\}$. These rates were previously used to model threshold crossing processes [30] in the context of triggering of biochemical reactions [29]. Figure 2 shows the FDTD for this process with parameters $n_0 = 0, N = 20, \lambda = 0.1$ and $k = 1$, for threshold values $m = 3, 5$ and $7$, and $(\alpha, \beta) = (1, 1), (1, 10)$ and $(10, 1)$. This figure shows analytical results (solid lines) for which inverse Laplace transform of Eq. 4 has been numerically performed, and simulations (open triangles) were generated by performing $10^7$ realizations of the stochastic process. An excellent agreement is observed between the analytical result and the simulations. For comparison, the case of sensor being always-on is also shown (as dashed lines) and it effectively corresponds to the FPTD. If the sensor is initially active, then for $t \ll \frac{1}{\beta}$, the FDTD matches with FPTD. This is because, in this limit, the threshold is crossed earlier than the typical timescale for inactivation of the sensor. Starting from around $t \approx \frac{1}{\beta}$, the FDTD deviates from the FPTD, a feature captured by the analytical result. For $t \gg \frac{1}{\beta}$, both the FDTD and FPTD show an exponential tail, with different decay rates governed by the mean FDT and FPT respectively.
+
+The mean FDT can be computed as
+
+$$ \langle T_d \rangle = \langle T_{n_0, \sigma_0} \rangle = -\left. \frac{d}{ds} \tilde{D}_s(n_0, \sigma_0) \right|_{s=0} . \quad (6) $$
+
+As $t \to \infty$, the FDTD decays as $\frac{1}{\langle T_d \rangle} e^{-t/\langle T_d \rangle}$. As shown in Fig. 3(a) for a threshold of $m = 5$ and for several pairs $(\alpha, \beta)$, the exact FDTD agrees with the asymptotic result for $t \gg 1$. Further, we define $\kappa = \beta/\alpha$, which is the
+
+FIG. 3. (a) Asymptotic FDTD for $t \gg 1$ with $m = 5$ for $(\alpha, \beta) = (1, 1), (0.1, 1), (1, 0.1), (10, 1)$, and $(1, 10)$. Other parameters are the same as in Fig. 2. (b) Mean detection time as a function of $\alpha$, where along each curve $\frac{\beta}{\alpha}$ is held constant.
+
+fraction of time the sensor spends in the inactive state. If $\kappa \ll 1$, then the sensor is active nearly all the time. The mean FDT $\langle T_d \rangle$ is plotted as a function of $\alpha$ for constant value of $\kappa$ in Fig. 3(b). As this figure reveals, the $\langle T_d \rangle \approx \langle T_f \rangle$ for small and large values of $\alpha$. For intermediate values of $\alpha$, the mean first passage and detection times can differ from one another by several orders of magnitude depending on the value of $\kappa$ – larger $\kappa$ leads to larger $\langle T_d \rangle$. This has a surprising outcome for event detections. Physically this implies that even if $\kappa \gg 1$, where the sensor spends most of its time in the inactive state, the detection can happen at timescales comparable to $\langle T_f \rangle$ as long as the time scales of the sensor switching is much faster than the intrinsic time scale of the underlying process. This effectively renders the switching to have little effect on the detection times. Though seemingly counterintuitive, a similar result was also noted in a different scenario of diffusing particles searching for intermittent target [36].
+
+In general, the detection of the threshold crossing event does not necessarily happen at state $m$ of the underlying process, but can happen at any state $\zeta \in \{m, m+1, \dots, N\}$. Thus, a natural question arises – what are the splitting probabilities $H_\zeta$ that the event is detected in state $\zeta$? By performing an analysis similar to the FDTD (full calculation in SI) the density $H_t(\zeta)$ of the threshold crossing event being detected at $\zeta$ at time $t$ can be obtained. For brevity, the following quantities are defined:
\ No newline at end of file
diff --git a/samples/texts/5140397/page_4.md b/samples/texts/5140397/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..fa2d01299dfd4dabcca91513fcdabbd3b1d8cd63
--- /dev/null
+++ b/samples/texts/5140397/page_4.md
@@ -0,0 +1,27 @@
+FIG. 4. Splitting probability for states of the underlying process (lines), for $\lambda = 1$, $k = 1$, $N = 10$, $n_0 = 0$, $m = 4$ for three different pairs of $(\alpha, \beta) = (1, 1)$, $(10, 1)$, and $(1, 10)$. Symbols are from simulations. (Inset) Splitting probability density as a function of time, for $(\alpha, \beta) = (1, 1)$.
+
+$$h_1(t) = F(m-1|m;t)e^{-\alpha t},$$
+
+$$h_2(k,m,t) = \alpha e^{-\alpha t} P_t(k|m, \tau_{m,m-1} > t), \quad (7)$$
+
+for $k \in \{m, m+1, \dots, N\}$, and $\tau_{m,m-1}$ is the first time when the underlying process visits the state $(m-1)$ starting from the state $m$. Summing over trajectories that lead to the threshold crossing event at $\zeta = m$, the Laplace transform of splitting probability density is
+
+$$\tilde{H}_s(m) = \tilde{f}_1(s) + \frac{\tilde{f}_2(s)(\tilde{h}_1(s)\tilde{f}_4(s) + \tilde{h}_2(m,m,s))}{1 - \tilde{h}_1(s)\tilde{f}_5(s)}.$$
+
+and for $\zeta = m+r$, where $r \in \{1, 2, \dots, N-m\}$, we obtain
+
+$$\tilde{H}_s(m+r) = \frac{\tilde{f}_2(s)\tilde{h}_2(m+r,m,s)}{1-\tilde{h}_1(s)\tilde{f}_5(s)}. \quad (8)$$
+
+The splitting probability $H_\zeta$ can thus be obtained as $H_\zeta = \int_0^\infty dt H_t(\zeta)$, and is equal to the $s \to 0$ limit of $\tilde{H}_s(\zeta)$. Figure 4 shows the splitting probabilities $H_\zeta$ for different values of $\zeta$, for the parameter values $\lambda = 1$, $k=1$, $N=10$, $n_0=0$, $m=4$ for three different pairs of $(\alpha, \beta) - (1, 1)$, $(10, 1)$ and $(1, 10)$. This demonstrates an excellent agreement between the analytical and simulations results. Furthermore, the inset in Fig. 4 shows the analytically computed splitting probability density for $(\alpha, \beta) = (1, 1)$.
+
+If the first detection of a threshold crossing event happens at time $T_d$, can we infer the first passage time $T_f$? This has practical value as it estimates the first occurrence time for an event that possibly went undetected. In sensors that detect abnormal voltage fluctuations (with potential for damage), $T_f$ corresponds to the time until
+
+FIG. 5. FDTD for various processes – Logistic model (violet), SIS epidemiological model (green), BDP with stochastic RESETS (red) and that with a refractory period (blue) – showing an excellent agreement with numerical simulations of these processes (circles). Details about parameter values and additional derivations are provided in SI.
+
+when the device being monitored was fully functional, and $T_d$ denotes the time when sensor detects the large fluctuation. Let $F_{T_f}(m|n_0, \sigma_0, T_d)$ be the density that the first passage to threshold $m$ happens at time $T_f$, conditioned on the fact that the first detection of the threshold crossing event happens at $T_d$, and that the underlying process starts from state $n_0$ and the sensor starts from $\sigma_0$. Clearly, for $T_f > T_d$, $F_{T_f}(m|n_0, \sigma_0, T_d) = 0$. It is easy to see that for $T_f = T_d$, the probability density is $\frac{F_{T_d}(m|n_0)}{D_{T_d}(n_0, \sigma_0)}$. For $T_f < T_d$, the exact expression is
+
+$$F_{T_f}(m|n_0, \sigma_0, T_d) = F_{T_f}(m|n_0) \cdot p_{T_f}(i|\sigma_0) \frac{D_{T_d-T_f}(m,i)}{D_{T_d}(n_0, \sigma_0)}.$$
+
+This shows that the FPTD conditioned on detection at a specific time $F_{T_f}(m|n_0, \sigma_0, T_d)$ can be expressed explicitly as the unconditioned FPTD $F_{T_f}(m|n_0)$, multiplied by additional tilting factors which ensure that the threshold crossing event is detected exactly at $T_d$, after it goes undetected at $T_f$ (See SI).
+
+**Applications:** These results can be applied to threshold crossing events in processes with other absorbing states. Important examples are models of population dynamics, and compartmental models for disease propagation. These models can estimate the time taken for the size of a population, or the infected case load to cross a threshold, and such models contain an absorbing state where the size of the population, or number of infected individuals goes to zero. During a pandemic, while the dynamics of the number of infected individuals follows a continuous time BDP with disease-dependent rates, they are intermittently reported in specific time windows. Thus the formalism developed in this work has practical relevance as well. In Fig. 5, the analytical FDTD (details in SI) for the SIS model and the logistic model are shown along with simulation results.
\ No newline at end of file
diff --git a/samples/texts/5140397/page_5.md b/samples/texts/5140397/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..be4d007ebee2b24eed4e2220cf58ef1eff9df527
--- /dev/null
+++ b/samples/texts/5140397/page_5.md
@@ -0,0 +1,59 @@
+These results can further be extended to processes with stochastic resetting [47–52], in which some observable such as the accumulated stress or damage can undergo burst-like relaxations. Recently, this process has received considerable research attention with extensive applications that include population dynamics under stochastic catastrophes [53–55] to the dynamics of queues subject to intermittent failure. In Fig. 5, the FDT under intermittent sensing is shown for two cases: a BDP with simple resets, and with resets that include a refractory period [56]. In both cases, analytical and simulations results are in agreement.
+
+In this Letter, the general problem of threshold crossing under intermittent sensing is studied using the versatile BDP, and a sensor that stochastically switches between active and inactive states. A general relation between the FDTD, under intermittent sensing, and the
+
+FPTD is obtained. The central result is that the first detection time can be obtained from the first passage times, which is known for a wide range of problems. The validity of these results is demonstrated in a variety of applications including the SIS model, logistic model, and BDP with resetting. The splitting densities, arising due to intermittency of the sensor, can be obtained by this approach. Finally, the practically relevant problem of inferring the FPTD from the FDTD is solved.
+
+*Acknowledgements.* AK gratefully acknowledges helpful discussions with Basila Moochickal Assainar and Gaurav Joshi, and the Prime Minister's Research Fellowship of the Government of India for financial support. AZ acknowledges support of the INSPIRE fellowship. MSS acknowledges the support of MATRICS grant from SERB, Govt. of India.
+
+[1] S. Redner, *A Guide to First-Passage Processes* (Cambridge University Press, 2001).
+
+[2] R. Metzler, G. Oshanin, and S. Redner, *First-passage Phenomena and Their Applications* (World Scientific, 2014).
+
+[3] D. S. Grebenkov, D. Holcman, and R. Metzler, Preface: new trends in first-passage methods and applications in the life sciences and engineering, Journal of Physics A: Mathematical and Theoretical **53**, 190301 (2020).
+
+[4] A. Auffinger, M. Damron, and J. Hanson, *50 years of first-passage percolation*, Vol. 68 (American Mathematical Soc., 2017).
+
+[5] G. H. Weiss, First passage time problems in chemical physics, Adv. Chem. Phys **13**, 1 (1967).
+
+[6] A. Szabo, K. Schulten, and Z. Schulten, First passage time approach to diffusion controlled reactions, The Journal of chemical physics **72**, 4350 (1980).
+
+[7] G. A. Whitmore, First-Passage-Time Models for Duration Data: Regression Structures and Competing Risks, Journal of the Royal Statistical Society. Series D (The Statistician) **35**, 207 (1986).
+
+[8] J. W. Shipley and M. C. Bernard, The First Passage Time Problem for Simple Structural Systems, Journal of Applied Mechanics **39** (1972).
+
+[9] H. Vanvinckenroye, T. Andrianne, and V. Denoël, First passage time as an analysis tool in experimental wind engineering, Journal of Wind Engineering and Industrial Aerodynamics **177**, 366 (2018).
+
+[10] R. Chicheportiche and J.-P. Bouchaud, Some applications of first-passage ideas to finance, in *First-Passage Phenomena and Their Applications* (World Scientific, 2014) Chap. Chapter 1, pp. 447–476.
+
+[11] D. Zhang and R. V. Melnik, First passage time for multivariate jump-diffusion processes in finance and other areas of applications, Applied Stochastic Models in Business and Industry **25**, 565 (2009).
+
+[12] R. Chicheportiche and J.-P. Bouchaud, Some applications of first-passage ideas to finance, in *First-passage Phenomena And Their Applications* (World Scientific, 2014) pp. 447–476.
+
+[13] T. Chou and M. R. DOrsogna, First passage problems
+
+in biology, in *First-Passage Phenomena and Their Applications* (World Scientific, 2014) Chap. Chapter 1, pp. 306–345.
+
+[14] S. Iyer-Biswas and A. Zilman, First-Passage Processes in Cellular Biology, in *Advances in Chemical Physics* (2016) pp. 261–306.
+
+[15] Y. Zhang and O. K. Dudko, First-Passage Processes in the Genome, Annual Review of Biophysics **45**, 117 (2016).
+
+[16] M. S. Obaidat and S. Misra, *Principles of Wireless Sensor Networks* (Cambridge University Press, 2014).
+
+[17] W. Dargie and C. Poellabauer, *Fundamentals of wireless sensor networks: theory and practice* (John Wiley & Sons, 2010).
+
+[18] H. Inaltekin, C. R. Tavoularis, and S. B. Wicker, Event detection time for mobile sensor networks using first passage processes, in *IEEE GLOBECOM 2007 - IEEE Global Telecommunications Conference* (2007) pp. 1174–1179.
+
+[19] C. Hsin and M. Liu, Randomly duty-cycled wireless sensor networks: Dynamics of coverage, IEEE Transactions on Wireless Communications **5**, 3182 (2006).
+
+[20] D. Song, C. Kim, and J. Yi, On the time to search for an intermittent signal source under a limited sensing range, IEEE Transactions on Robotics **27**, 313 (2011).
+
+[21] P. C. Hew, An intermittent sensor versus a target that emits glimpses as a homogeneous poisson process, Journal of the Operational Research Society **71**, 433 (2020).
+
+[22] J. Hester and J. Sorber, The future of sensing is batteryless, intermittent, and awesome, in *Proceedings of the 15th ACM Conference on Embedded Network Sensor Systems*, SenSys ’17 (Association for Computing Machinery, New York, NY, USA, 2017).
+
+[23] P. C. Bressloff, Stochastic switching in biology: from genotype to phenotype, Journal of Physics A: Mathematical and Theoretical **50**, 133001 (2017).
+
+[24] J. Li, J. Labbadia, and R. I. Morimoto, Rethinking HSF1 in Stress, Development, and Organismal Health, Trends in Cell Biology **27** (2017).
+
+[25] P. K. Sorger, Heat shock factor and the heat shock response, Cell **65**, 363 (1991).
\ No newline at end of file
diff --git a/samples/texts/5140397/page_6.md b/samples/texts/5140397/page_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..90b57eed394afb2312e452d4ee870e6ab8b4d568
--- /dev/null
+++ b/samples/texts/5140397/page_6.md
@@ -0,0 +1,63 @@
+[26] V. Kishore, M. S. Santhanam, and R. E. Amritkar, Extreme Events on Complex Networks, Physical Review Letters **106**, 188701 (2011).
+
+[27] N. Malik and U. Ozturk, Rare events in complex systems: Understanding and prediction, Chaos: An Interdisciplinary Journal of Nonlinear Science **30**, 090401 (2020).
+
+[28] A. Kumar, S. Kulkarni, and M. S. Santhanam, Extreme events in stochastic transport on networks, Chaos: An Interdisciplinary Journal of Nonlinear Science **30**, 043111 (2020).
+
+[29] D. S. Grebenkov, First passage times for multiple particles with reversible target-binding kinetics, The Journal of Chemical Physics **147**, 134112 (2017).
+
+[30] S. D. Lawley and J. B. Madrid, First passage time distribution of multiple impatient particles with reversible binding, The Journal of Chemical Physics **150**, 214113 (2019).
+
+[31] K. Dao Duc and D. Holcman, Threshold activation for stochastic chemical reactions in microdomains, Physical Review E **81**, 041107 (2010).
+
+[32] O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Optimal search strategies for hidden targets, Phys. Rev. Lett. **94**, 198101 (2005).
+
+[33] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. **83**, 81 (2011).
+
+[34] J. M. Newby and P. C. Bressloff, Directed intermittent search for a hidden target on a dendritic tree, Phys. Rev. E **80**, 021913 (2009).
+
+[35] P. Bressloff and J. Newby, Directed intermittent search for hidden targets, New Journal of Physics **11**, 023033 (2009).
+
+[36] G. Mercado-Vásquez and D. Boyer, First hitting times to intermittent targets, Phys. Rev. Lett. **123**, 250603 (2019).
+
+[37] P. C. Bressloff, Diffusive search for a stochastically-gated target with resetting, Journal of Physics A: Mathematical and Theoretical **53**, 425001 (2020).
+
+[38] Y. Scher and S. Reuveni, A Unified Approach to Gated Reactions on Networks, arXiv:2101.05124 [cond-mat, physics:physics] (2021).
+
+[39] G. Mercado-Vásquez and D. Boyer, First hitting times between a run-and-tumble particle and a stochastically gated target, Physical Review E **103**, 042139 (2021).
+
+[40] S. Azaele, S. Suweis, J. Grilli, I. Volkov, J. R. Banavar, and A. Maritan, Statistical mechanics of ecological systems: Neutral theory and beyond, Reviews of Modern Physics **88**, 035003 (2016).
+
+[41] A. S. Novozhilov, G. P. Karev, and E. V. Koonin, Biological applications of the theory of birth-and-death processes, Briefings in Bioinformatics **7**, 70 (2006).
+
+[42] F. W. Crawford and M. A. Suchard, Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution, Journal of Mathematical Biology **65**, 553 (2012).
+
+[43] J. Freidenfelds, Capacity Expansion when Demand Is a Birth-Death Random Process, Operations Research **28**, 712 (1980).
+
+[44] L. S. T. Ho, J. Xu, F. W. Crawford, V. N. Minin, and M. A. Suchard, Birth/birth-death processes and their computable transition probabilities with biological applications, Journal of Mathematical Biology **76**, 911 (2018).
+
+[45] P. R. Parthasarathy and R. Sudhesh, Exact transient solu-
+
+tion of a state-dependent birth-death process, Journal of Applied Mathematics and Stochastic Analysis **2006**, e97073 (2006), publisher: Hindawi.
+
+[46] R. Sasaki, Exactly solvable birth and death processes, Journal of Mathematical Physics **50**, 103509 (2009), publisher: American Institute of Physics.
+
+[47] M. R. Evans and S. N. Majumdar, Diffusion with Stochastic Resetting, Physical Review Letters **106**, 160601 (2011).
+
+[48] S. Reuveni, Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal, Physical Review Letters **116**, 170601 (2016).
+
+[49] A. Pal and S. Reuveni, First Passage under Restart, Physical Review Letters **118**, 030603 (2017).
+
+[50] M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, Journal of Physics A: Mathematical and Theoretical **53**, 193001 (2020).
+
+[51] A. Pal, L. Kusmierz, and S. Reuveni, Search with home returns provides advantage under high uncertainty, Physical Review Research **2**, 043174 (2020).
+
+[52] A. Pal and V. V. Prasad, First passage under stochastic resetting in an interval, Physical Review E **99**, 032123 (2019).
+
+[53] A. Di Crescenzo, V. Giorno, A. G. Nobile, and L. M. Ricciardi, A note on birth–death processes with catastrophes, Statistics & Probability Letters **78**, 2248 (2008).
+
+[54] A. Di Crescenzo, V. Giorno, A. Nobile, and L. Ricciardi, On the M/M/1 Queue with Catastrophes and Its Continuous Approximation, Queueing Systems **43**, 329 (2003).
+
+[55] B. K. Kumar and D. Arivudainambi, Transient solution of an M/M/1 queue with catastrophes, Computers & Mathematics with Applications **40**, 1233 (2000).
+
+[56] M. R. Evans and S. N. Majumdar, Effects of refractory period on stochastic resetting, Journal of Physics A: Mathematical and Theoretical **52**, 01LT01 (2018).
\ No newline at end of file
diff --git a/samples/texts/5140397/page_7.md b/samples/texts/5140397/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..35ca82889d9077b63e0b2c4c9fe3039276e4f3c6
--- /dev/null
+++ b/samples/texts/5140397/page_7.md
@@ -0,0 +1,37 @@
+# S1. SUPPLEMENTARY MATERIAL FOR “THRESHOLD CROSSING PROCESSES UNDER INTERMITTENT SENSING”
+
+This Supplemental Material provides additional discussions and mathematical derivations which support the results described in the Letter and provides details of the various examples used to demonstrate the validity and applicability of the results.
+
+## A. Interpreting the FDTD formula
+
+We noted in Eq. 4, that the first detection time distribution can be obtained from the first passage time distributions alone, which can be computed through several standard results. In Laplace space, the first detection time is expressed as
+
+$$ \tilde{D}_s(n_0, \sigma_0) = \underbrace{\tilde{f}_1(s)}_{\text{First detection at first passage}} + \frac{\underbrace{\tilde{f}_2(s)(\tilde{f}_3(s) + \tilde{f}_4(s)\tilde{F}_{s+\alpha}(m-1|m))}_{\substack{\text{First detection happening} \\ \text{strictly after first passage event}}} - \underbrace{1 - \tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)}_{\text{First detection not happening after first passage event}}, \quad (S1) $$
+
+where we define:
+
+$$ f_1(t) = F_t(m|n_0)p_t(a|\sigma_0), \quad (S2) $$
+
+$$ f_2(t) = F_t(m|n_0)p_t(i|\sigma_0), \quad (S3) $$
+
+$$ f_3(t) = \alpha e^{-\alpha t} \int_t^\infty F_{t'}(m-1|m)dt' \quad (S4) $$
+
+$$ f_4(t) = F_t(m|m-1)p_t(a|i) \quad (S5) $$
+
+$$ f_5(t) = F_t(m|m-1)p_t(i|i). \quad (S6) $$
+
+The second term on the RHS can be understood if expressed as the following:
+
+$$ \frac{\tilde{f}_2(s)(\tilde{f}_3(s) + \tilde{f}_4(s)\tilde{F}_{s+\alpha}(m-1|m))}{1 - \tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)} = \underbrace{\tilde{f}_2(s)}_{\substack{\text{Factor I:} \\ \text{First passage} \\ \text{while sensor} \\ \text{is inactive}}} \cdot \frac{1}{\underbrace{1 - \tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)}_{\substack{\text{Factor II: Accounts for the} \\ \text{number of undetected threshold} \\ \text{crossings before first detection}}}} \cdot \underbrace{\left(\tilde{f}_3(s) + \tilde{f}_4(s)\tilde{F}_{s+\alpha}(m-1|m)\right)}_{\substack{\text{Factor III: Ensures detection} \\ \text{of threshold crossing event}}}, \quad (S7) $$
+
+Factor II is the sum of the following geometric series in Laplace space, which accounts for the number of threshold crossings that go undetected before eventual detection:
+
+$$ \frac{1}{1 - \tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)} = 1 + (\tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m)) + (\tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m))^2 + (\tilde{f}_5(s)\tilde{F}_{s+\alpha}(m-1|m))^3 + \dots \quad (S8) $$
+
+Factor III consists of two different terms:
+
+1. $\tilde{f}_3(s)$: since the last undetected threshold crossing, the birth-death process stays above the threshold, and at time $t$, the sensor becomes active, and thus the threshold crossing event is detected.
+
+2. $\tilde{f}_4(s)\tilde{F}_{s+\alpha}(m-1|m)$: since the last undetected threshold crossing, the birth-death process stays above the threshold for some time and remains undetected. It then comes below the threshold, and finally, the birth-death processes reaches the threshold at time $t$ when the sensor is active.
+
+Both the terms add up to give two different ways of detection of the threshold crossing event, without any undetected transitions from the state $m-1$ to $m$. Overall, our central result computes the sum of probabilities of all trajectories where first detection of the threshold crossing event occurs at time $t$, and the sum can be expressed completely in terms of the first passage probabilities without any imperfect sensing.
\ No newline at end of file
diff --git a/samples/texts/5140397/page_8.md b/samples/texts/5140397/page_8.md
new file mode 100644
index 0000000000000000000000000000000000000000..383c0b4f9a3c24d7de0b271adc895d0281979466
--- /dev/null
+++ b/samples/texts/5140397/page_8.md
@@ -0,0 +1,41 @@
+## B. Alternate derivation: Computing the survival probability for a birth-death process under intermittent sensing
+
+In this section, we provide a step by step derivation of the first detection time statistics. Given that the dynamics of the birth death process and the sensor are independent of each other, it is instructive to first understand the dynamics of the sensor. The switching between the two states, active (a) and inactive (i), is modelled by a two-state Markov process with rates $\alpha$ for activation and $\beta$ for deactivation. For $\sigma, \sigma_0 \in \{a, i\}$, we define $p_t(\sigma|\sigma_0)$ as the probability that the sensor is in state $\sigma$ at time $t$, given that it was in state $\sigma_0$ initially. It is easy to obtain the following set of equations that govern the dynamics of the sensor
+
+$$p_t(a|a) = \frac{\alpha}{\alpha + \beta} + \frac{\beta}{\alpha + \beta} e^{-(\alpha+\beta)t} \quad (S9)$$
+
+$$p_t(i|a) = \frac{\beta}{\alpha + \beta} - \frac{\beta}{\alpha + \beta} e^{-(\alpha+\beta)t} \quad (S10)$$
+
+$$p_t(a|i) = \frac{\alpha}{\alpha + \beta} - \frac{\alpha}{\alpha + \beta} e^{-(\alpha+\beta)t} \quad (S11)$$
+
+$$p_t(i|i) = \frac{\beta}{\alpha + \beta} + \frac{\alpha}{\alpha + \beta} e^{-(\alpha+\beta)t} \quad (S12)$$
+
+Now, we shift our discussion to the computation of the first detection time distribution when the sensor is intermittent. To do this, we will adopt the approach of computing the survival probability $S_t(\sigma_0, n_0)$ which gives us the probability that the threshold crossing event has not been detected until time $t$, given that, initially, the sensor was in state $\sigma_0$ and the birth death process was in state $n_0$. The Laplace transform of the FDTD and survival probability are related as $\tilde{D}_s(n_0, \sigma_0) = 1 - s\tilde{S}_s(n_0, \sigma_0)$.
+
+Let us consider the case $\sigma_0 = a$. We break up $S_t(a, n_0)$ in two exclusive but exhaustive parts, and write them separately. The first part contains the possible surviving trajectories of our process such that, at time $t$, the underlying birth death process is in states $n < m$, where $m$ denotes the threshold. The second part enlists the surviving trajectories in which, at time $t$, the underlying birth death process is in states $n \ge m$. Needless to say, in the surviving trajectories, whenever the underlying birth death process is in states $n \ge m$, the sensor must be inactive.
+
+The possible trajectories in the first part are:
+
+* The birth death process does not reach the threshold (state $m$) until time $t$.
+
+* The birth death process reaches state $m$ for the first time at time $t_1$ but the sensor is inactive at time $t_1$. The birth death process remains in states $\ge m$ for some time and the sensor remains inactive throughout. Then at time $t_2$, for the first time, the birth death process reaches state $m-1$ and then always stays below $m$ until time $t$.
+
+...
+
+We can sum up the probabilities of such trajectories as the following
+
+$$ 
+\begin{aligned}
+S_t^{(1)}(a, n_0) &= \int_t^\infty dt' F_{t'}(m|n_0) \\
+&\quad + \int_0^t \int_0^{t_2} \int_{t-t_2}^\infty dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) F_{t_2-t_1}(m-1|m) e^{-\alpha(t_2-t_1)} F_{t_3}(m|m-1) \\
+&\quad + \int_0^t \int_0^{t_4} \int_0^{t_3} \int_0^{t_2} \int_{t-t_4}^\infty dt_5 \cdot dt_4 \cdot dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) F_{t_2-t_1}(m-1|m) e^{-\alpha(t_2-t_1)} \\
+&\quad F_{t_3-t_2}(m|m-1) p_{t_3-t_2}(i|i) F_{t_4-t_3}(m-1|m) e^{-\alpha(t_4-t_3)} F_{t_5}(m|m-1) + \dots
+\end{aligned}
+\quad (S13)
+$$
+
+Similarly, the possible trajectories in the second part are:
+
+* The birth death process reaches state $m$ for the first time at time $t_1$ and the sensor is inactive at $t_1$. Then, the birth death process remains in states greater than or equal to $m$ till time $t$ and the sensor too, remains inactive till time $t$.
+
+* The birth death process reaches state $m$ for the first time at time $t_1$ but the sensor is inactive at time $t_1$. The birth death process remains in states greater than equal to $m$ for some time and the sensor remains inactive throughout that time. Then at time $t_2$, for the first time, the birth death process comes below $m$ and stays
\ No newline at end of file
diff --git a/samples/texts/5140397/page_9.md b/samples/texts/5140397/page_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..da30918281d62072487bf5628e50f8685d93b169
--- /dev/null
+++ b/samples/texts/5140397/page_9.md
@@ -0,0 +1,44 @@
+below *m* until time *t*₃. Next, at *t*₃, the birth death process reaches state *m* for the first time after *t*₂ (while the sensor is inactive). Subsequently, the birth death process, till time *t*, remains in states higher than *m* and sensor remains inactive.
+
+...
+
+Summing up the probabilities of the aforementioned trajectories, we get
+
+$$
+\begin{align}
+S_t^{(\mathrm{II})}(a, n_0) = & \int_0^t \int_{t-t_1}^\infty dt_2 \cdot dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) e^{-\alpha(t-t_1)} F_{t_2}(m-1|m) \nonumber \\
+& + \int_0^t \int_0^{t_3} \int_0^{t_2} \int_{t-t_3}^\infty dt_4 \cdot dt_3 \cdot dt_2 \cdot dt_1 F_{t_1}(m|n_0) p_{t_1}(i|a) F_{t_2-t_1}(m-1|m) e^{-\alpha(t_2-t_1)} \nonumber \\
+& F_{t_3-t_2}(m|m-1) p_{t_3-t_2}(i|i) e^{-\alpha(t-t_3)} F_{t_4}(m-1|m) + \dots \tag{S14}
+\end{align}
+$$
+
+We can get the total survival probability as $S_t(a, n_0) = S_t^{(I)}(a, n_0) + S_t^{(II)}(a, n_0)$. For brevity, it is convenient to define the following quantities
+
+$$
+\begin{align}
+L(t) &= \int_t^\infty dt' F_{t'}(m|n_0) \tag{S15} \\
+g_1(t) &= F_t(m|n_0) p_t(i|a) \tag{S16} \\
+g_2(t) &= F_t(m-1|m) e^{-\alpha t} \tag{S17} \\
+g_3(t) &= F_t(m|m-1) p_t(i|i) \tag{S18} \\
+g_4(t) &= \int_t^\infty dt' F_{t'}(m|m-1) \tag{S19} \\
+g_5(t) &= e^{-\alpha t} \int_t^\infty dt' F_{t'}(m-1|m) \tag{S20}
+\end{align}
+$$
+
+Eqs. (S15-S20) denote probability density functions for the following relevant quantities which have the following interpretations: (i) $L(t)$ denotes the probability that the birth death process does not reach $m$ till time $t$, given that initially it was in state $n_0 < m$; (ii) $g_1(t)$ is the probability that the birth death process reaches state $m$ for the first time at $t$, starting from an initial state $n_0$ and the sensor being inactive at time $t$ starting from an active state; (iii) $g_2(t)$ denotes the probability that the birth death process comes down to $m-1$ for the first time at $t$ starting from the state $m$ and the sensor remains inactive throughout the time 0 to $t$, when the birth death process was in states $\ge m$; (iv) $g_3(t)$ denotes the probability that, starting from the state $m-1$, the birth death process reaches the state $m$ for the first time at $t$ and that the sensor is inactive at $t$ starting from the inactive state, (v) $g_4(t)$ is the probability that the birth death process does not reach the state $m$ till time $t$ starting from the state $m-1$, and finally (vi) $g_5(t)$ represents the probability that the birth death process remains in the states greater than or equal to $m$ till time $t$ with starting in the state $m$, and that the sensor remains inactive throughout this time.
+
+With the above definitions in mind, summing Eqs. (S13) and (S14), we can write the probability that the threshold crossing event is not detected upto time *t*, with the state of the birth death process being *n*₀ and the sensor being active initially as
+
+$$
+\begin{align}
+S_t(a, n_0) = L(t) &+ \int_0^t \int_0^{t_2} dt_2 dt_1 g_1(t_1) g_2(t_2 - t_1) g_4(t - t_2) + \nonumber \\
+& \int_0^t \int_0^{t_4} \int_0^{t_3} \int_0^{t_2} dt_4 dt_3 dt_2 dt_1 g_1(t_1) g_2(t_2 - t_1) g_3(t_3 - t_2) g_2(t_4 - t_3) g_4(t - t_4) + \dots \nonumber \\
+& + \int_0^t dt_1 g_1(t_1) g_5(t - t_1) + \int_0^t \int_0^{t_3} \int_0^{t_2} dt_3 dt_2 dt_1 g_1(t_1) g_2(t_2 - t_1) g_3(t_3 - t_2) g_5(t - t_3) + \dots \tag{S21}
+\end{align}
+$$
+
+The fact that all the terms form a convolution in time, the analysis is much simpler in Laplace space. We obtain the Laplace transformed survival probability as a sum of two infinite series
+
+$$
+\tilde{S}_s(a, n_0) = \tilde{L}(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_4(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_3(s)\tilde{g}_2(s)\tilde{g}_4(s) + \cdots + \tilde{g}_1(s)\tilde{g}_5(s) + \tilde{g}_1(s)\tilde{g}_2(s)\tilde{g}_3(s)\tilde{g}_5(s) + \cdots \quad (\text{S22})
+$$
\ No newline at end of file
diff --git a/samples/texts/6573214/page_1.md b/samples/texts/6573214/page_1.md
new file mode 100644
index 0000000000000000000000000000000000000000..666f9403bb2f1e22347e4f8317c5e2abcaced482
--- /dev/null
+++ b/samples/texts/6573214/page_1.md
@@ -0,0 +1,20 @@
+# THE WEIERSTRASS ROOT FINDER IS NOT GENERALLY CONVERGENT
+
+BERNHARD REINKE, DIERK SCHLEICHER, AND MICHAEL STOLL
+
+**ABSTRACT.** Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root finding method of Weierstrass, a root finder that tries to approximate all roots of a given polynomial in parallel (in the Jacobi version, i.e., with parallel updates). This method has a good reputation for finding all roots in practice except in obvious cases of symmetry, but very little is known about its global dynamics and convergence properties.
+
+We show that the Weierstrass method, like the well known Newton method, is not generally convergent: there are open sets of polynomials $p$ of every degree $d \ge 3$ such that the dynamics of the Weierstrass method applied to $p$ exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to $Z^3 + Z + 175$ have attracting cycles of period 4. Here, period 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 or 3 that attract open sets of starting points.
+
+We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree $d \ge 3$ there are orbits that are defined for all iterates but converge to $\infty$; this is a problem that does not occur for Newton's method.
+
+Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra.
+
+## 1 Introduction
+
+Finding roots of polynomials is one of the fundamental tasks in mathematics that is highly relevant for the theory of many fields, as well as for numerous practical applications. Since the work of Ruffini-Abel, it is clear that in general the roots cannot be found by finite radical extensions, so numerical approximation methods are required. One may find it surprising that, despite age and relevance of this problem, no clear algorithm is known that has a well-developed theory and works well in practice.
+
+There are “algorithms” (in the sense of heuristics) that seem to work in practice fast and reliably, among them the Weierstrass and Ehrlich-Aberth methods: these are both iterations in as many variables as the number of roots to be found, and are supposed to converge to a vector of roots under iteration. They are known to converge quadratically resp. cubically near the roots (at least when all roots are simple), but have essentially no known global theory. Then there are algorithms such as Pan’s [Pan02] that have excellent theoretical complexity (optimal up to log-factors), but they cannot be used in practice because of their lack of stability.
+
+2010 Mathematics Subject Classification. 65H04, 37F80, 37N30, 68W30.
+Key words and phrases. Weierstrass, Durand-Kerner, root-finding methods, attracting cycle, escaping orbits.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_10.md b/samples/texts/6573214/page_10.md
new file mode 100644
index 0000000000000000000000000000000000000000..bdcde5815f27b661d2f747b84bbec2068ec2ba52
--- /dev/null
+++ b/samples/texts/6573214/page_10.md
@@ -0,0 +1,37 @@
+For polynomials with multiple roots, the local dynamics are more complicated. It is not even true that a neighborhood of the vector containing the roots converges to the roots; see for instance the case of $Z \mapsto Z^3$ discussed in Section 3.2, and e.g. [HM96] for a more detailed discussion.
+
+## 3.1 Properties of the Weierstrass method
+
+We state some elementary and well known properties of $W_p$ that will be important to us.
+
+**Lemma 3.3 (Simple properties of the Weierstrass method).**
+
+(1) Let $p \in P'_d$ and $T \in \text{Aff}(\mathbb{C})$. Then $W_{Tp}$ is conformally conjugate to $W_p$ by $T$, i.e., $W_{Tp} = T \circ W_p \circ T^{-1}$, where the action of $T$ on $\mathbb{C}^d$ is component-wise.
+
+(2) For each $p \in P'_d$, $W_p$ is equivariant with respect to the natural action of the symmetric group $S_d$ on $\mathbb{C}^d$ by permuting the coordinates: if $\sigma \in S_d$, then $W_p(\sigma z) = \sigma W_p(z)$.
+
+*Proof.* (1) Writing $p = \prod_{k=1}^d (Z - \alpha_k)$ in (3.5), we see that the relation is unchanged when we replace $\alpha_k, z_k, z'_k$ and $Z$ by their images under $T$. Undoing the transformation on $Z$ then gives a valid equation between polynomials, which is equivalent to $W_{Tp}(TZ) = TW_p(Z)$, or $W_{Tp} = T \circ W_p \circ T^{-1}$, where the action of affine transformations on $\mathbb{C}^d$ is coordinate-wise.
+
+(2) This is clear. $\square$
+
+By the first property we can use the same parameter space $P_d$ for the Weierstrass iteration on polynomials of degree $d$ as we did for Newton's method.
+
+Equation (3.5) leads to a simple proof of the following useful property.
+
+**Lemma 3.4 (Invariant hyperplane).** Let $p = Z^d - aZ^{d-1} + \dots$. Then the sum of the entries of $W_p(\underline{z})$ is $a$, for all $\underline{z} \in \mathbb{C}^d \setminus \Delta$.
+
+This was already observed in [Wei91, Paragraph 22].
+
+*Proof.* Comparing coefficients of $Z^{d-1}$ in (3.5), we see that
+
+$$ \sum_{k=1}^{d} (z'_{k} - z_{k}) = - \sum_{k=1}^{d} z_{k} + a, $$
+
+which gives the claim. $\square$
+
+This means that the dynamics is effectively only $(d-1)$-dimensional and takes place on the hyperplane $z_1 + \dots + z_d = a$. As mentioned earlier, we can restrict to centered polynomials, i.e., $a=0$.
+
+**Lemma 3.5 (Degree reduction if root is present).** Fix $k \in \{1, \dots, d\}$. If $z_k$ is a root of $p$ and $\underline{z} \in \mathbb{C}^d \setminus \Delta$, then $W_p(\underline{z})_k = z_k$, and the dynamics on the remaining entries is that of the Weierstrass method for $p(Z)/(Z-z_k)$.
+
+*Proof.* Clear from the definition. $\square$
+
+**Lemma 3.6 (Weierstrass in degree 2 is Newton).** If $p$ has degree 2, then the dynamics of $W_p$ reduces to Newton's method for $p$. In particular, for $p$ with distinct roots, $W_p$ restricted to the invariant hyperplane (which is a line in this case) is conjugate to the squaring map $z \mapsto z^2$, which has no attracting cycles that are not fixed points.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_11.md b/samples/texts/6573214/page_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..fabbe1e6b7adf0a8b5395b3330096a1c09096d6b
--- /dev/null
+++ b/samples/texts/6573214/page_11.md
@@ -0,0 +1,30 @@
+*Proof.* By Lemma 3.3, we can assume that $p(Z) = Z^2 - 1$ if $p$ has distinct roots. By Lemma 3.4, all iterates after the initial vector will have the form $(z, -z)$. It is then easy to check that $W_p(z, -z) = (w, -w)$ with $w = z - (z^2 - 1)/(2z) = N_p(z)$. The last claim follows from Lemma 2.3.
+
+If $p$ has a double root, then $N_p$ and $W_p$ are conjugate to $N_{Z^2}$ and $W_{Z^2}$, respectively; again, $W_{Z^2}$ agrees with $N_{Z^2}$ when restricted to the invariant line. $\square$
+
+When $p$ is linear, then $N_p$ and $W_p$ both find the unique root immediately by definition. So this lemma tells us that interesting behavior in the Weierstrass method can occur only when $d \ge 3$.
+
+When looking for periodic orbits under $W_p$, Lemma 3.5 tells us that we can assume that no entry of $\tilde{z}$ is a root of $p$, since otherwise we can reduce to a case of lower degree. However, this very observation allows us to promote counterexamples of low degrees to higher degrees. To do this, we need the following lemma.
+
+**Lemma 3.7 (Lifting to higher degrees).** Let $p$ be a monic polynomial of degree $d$ and let $\alpha \in \mathbb{C}$. Set $\tilde{p}(Z) = (Z - \alpha)p(Z)$.
+
+(1) For a point $\underline{z} = (z_1, \dots, z_d, \alpha)$ with pairwise distinct entries, the Jacobi matrix
+$$\mathbf{D}(W_{\tilde{p}})|_{\underline{z}}$$ has the form
+
+$$\mathbf{D}(W_{\tilde{p}})|_{\underline{z}} = \begin{pmatrix} \mathbf{D}(W_p)|_{\underline{z}'} & *_{d \times 1} \\ 0_{1 \times d} & \lambda \end{pmatrix}$$
+
+with $\underline{z}' = (z_1, \dots, z_d)$ and
+
+$$\lambda = 1 - \frac{p(\alpha)}{\prod_{j=1}^{d} (\alpha - z_j)}.$$
+
+(2) If $q \in \mathbb{C}^d$ is a periodic point of $W_p$ of period $n$ such that all eigenvalues of $\mathbf{D}(W_p^{\alpha n})|_q$ have absolute values strictly less than 1, then for $|\alpha|$ sufficiently large, $\tilde{q} := (q, \alpha) \in \mathbb{C}^{d+1}$ is a periodic point of $W_{\tilde{p}}$ of period $n$ such that all eigenvalues of $\mathbf{D}(W_{\tilde{p}}^{\alpha n})|_{\tilde{q}}$ have absolute values strictly less than 1.
+
+*Proof.* The first claim results from an easy computation.
+
+Now assume that $q \in \mathbb{C}^d$ is a periodic point of $W_p$ of period $n$. By Lemma 3.5, $\tilde{q} = (q, \alpha)$ is a periodic point of $W_{\tilde{p}}$ of period $n$. We obtain an analogous formula relating the derivatives of $W_p^{\alpha n}$ at $q$ and $W_{\tilde{p}}^{\alpha n}$ at $\tilde{q}$, with the product $\lambda_0 \cdots \lambda_{n-1}$ replacing $\lambda$, where $\lambda_m$ arises from $(W_p^{\alpha m}(q), \alpha)$. In particular, the eigenvalues of $\mathbf{D}(W_{\tilde{p}}^{\alpha n})|_{\tilde{q}}$ are those of $\mathbf{D}(W_p^{\alpha n})|_q$ together with $\lambda_0 \cdots \lambda_{n-1}$. As $|\alpha| \to \infty$, we see that $\lambda_m \to 0$ for all $0 \le m < n$, and the claim follows. $\square$
+
+## 3.2 The dynamics of $W_{Z^3}$
+
+In this section we prove some results on the dynamics of the Weierstrass iteration in the simple case when $p(Z) = Z^3$. By Lemma 3.4, we can restrict consideration to the hyperplane $H = \{z_1 + z_2 + z_3 = 0\}$. We will show that all starting points in $H$ outside a set of measure zero converge to the unique root vector $(0, 0, 0)$, but that there are uncountably many orbits that converge to infinity, even arbitrarily close to $(0, 0, 0)$.
+
+From Lemma 3.3 (1) and since the unique root 0 of $Z^3$ is invariant under scaling, it follows that $W_{Z^3}(\lambda\underline{z}) = \lambda W_{Z^3}(\underline{z})$, so $W_{Z^3}$ induces a rational map $\varphi: \mathbb{P}H \to \mathbb{P}H$, where
\ No newline at end of file
diff --git a/samples/texts/6573214/page_12.md b/samples/texts/6573214/page_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..507006cdb7919ab3978573b36a88348b25ddbab5
--- /dev/null
+++ b/samples/texts/6573214/page_12.md
@@ -0,0 +1,40 @@
+$PH \simeq P^1$ is the complex projective line obtained by considering the nonzero points of $H$
+up to scaling.
+
+Writing a nonzero point in $H$ up to scaling in the form $(1, z, -1-z)$ (the missing scalar multiples of $(0, 1, -1)$ correspond to the limit case $z = \infty$), we find that
+
+$$
+(3.6) \qquad W_{Z^3}(1, z, -1-z) = s(z)(1, \varphi(z), -1-\varphi(z))
+$$
+
+with
+
+$$
+(3.7) \qquad s(z) = \frac{1-z-z^2}{2-z-z^2} \quad \text{and} \quad \varphi(z) = \frac{z(2+z)(1+z-z^2)}{(1+2z)(1-z-z^2)}.
+$$
+
+We can say something about the dynamics of $\varphi$.
+
+**Lemma 3.8 (Dynamics of $\varphi$).** The map $\varphi$ has two attracting fixed points at $\omega$ and $\omega^2$, where $\omega = e^{2\pi i/3}$ is a primitive cube root of unity. Let $z \in \mathbb{C}$. If $\operatorname{Im}(z) > 0$, then $\varphi^{\operatorname{on}}(z)$ converges to $\omega$ as $n \to \infty$, and if $\operatorname{Im}(z) < 0$, then $\varphi^{\operatorname{on}}(z)$ converges to $\omega^2$ as $n \to \infty$. The real line is forward and backward invariant under $\varphi$.
+
+*Proof*. Conjugating $\varphi$ by the Möbius transformation $z \mapsto (\omega^2 z - \omega^2)/(z - \omega)$, we obtain
+
+$$
+f(z) = z \frac{2z^3 + 1}{z^3 + 2}.
+$$
+
+This map $f$ is the product of $z$ with the composition of $z \mapsto z^3$ by $z \mapsto (2z+1)/(z+2)$; the latter is an automorphism of the open unit disk (and also of the complement of the closed unit disk in the Riemann sphere). Therefore, $|f(z)| < |z|$ when $0 < |z| < 1$ and $|f(z)| > |z|$ for $|z| > 1$ (in other words, $f$ is a Blaschke product with a fixed point at $z=0$). This implies that the open unit disk is attracted to the fixed point 0 of $f$, while the complement of the closed unit disk is attracted to $\infty$; the unit circle is forward and backward invariant (and maps to itself as a covering map with degree 4). Translating back to $\varphi$, this gives the result. $\square$
+
+From this, we can deduce the following statement on the global dynamics of $W_{Z^3}$.
+
+**Theorem 3.9 (Convergence of $W_{Z^3}$).** If $\underline{z} \in H$ is not a scalar multiple of a vector with real entries, then $W_{Z^3}^{on}(\underline{z})$ converges to the zero vector. The convergence is linear with rate of convergence 2/3.
+
+Note that the rate of convergence is the same as that for $N_{Z^3}$.
+
+*Proof.* Let $\underline{z} \in H$ be such that $\underline{z}$ is not a scalar multiple of a vector with real entries. In particular, $\underline{z}$ is not the zero vector. By symmetry, we can assume that the first entry is nonzero; then $\underline{z} = z_1(1, z, -1-z)$ with $z = z_2/z_1 \in C\backslash\mathbb{R}$. We then have that $\varphi^{\circ n}(z) \neq \infty$ for all $n \ge 0$ by Lemma **3.8**, and
+
+$$
+W_{Z^3}^{\circ n}(\underline{z}) = z_1 \prod_{k=0}^{n-1} s(\varphi^{\circ k}(z)) \cdot (1, \varphi^{\circ n}(z), -1 - \varphi^{\circ n}(z)).
+$$
+
+By Lemma **3.8** again, $\varphi^{\circ n}(z)$ converges to $\omega$ or $\omega^2$. Since $s(\omega) = s(\omega^2) = 2/3$, the factor in front will linearly converge to zero with rate of convergence 2/3, whereas the vector will converge to $(1, \omega, \omega^2)$ (if Im$(z) > 0$) or to $(1, \omega^2, \omega)$ (if Im$(z) < 0$). $\square$
\ No newline at end of file
diff --git a/samples/texts/6573214/page_13.md b/samples/texts/6573214/page_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..6ea545b86fbbc295ecc4af2acf6dc5ee05159c13
--- /dev/null
+++ b/samples/texts/6573214/page_13.md
@@ -0,0 +1,29 @@
+We have seen that orbits of starting points in $H$ that are not scalar multiples of real vectors converge to zero, whereas there are real vectors in $H$ whose orbit tends to infinity; see Section 3.3 below. There are also starting points whose orbits cease to be defined after finitely many steps; this occurs if and only if some iterate is a multiple of $(1, 1-2)$ or one of its permutations. On the other hand, there are many real starting points in $H$ whose orbits tend to zero (one example is obtained by replacing $\alpha$ with $-\alpha$ in the proof of Theorem 3.10 below). In fact, we expect that almost all real starting points have this property.
+
+### 3.3 Escaping points
+
+In this section we prove Theorem B: the Weierstrass iteration $W_p$ has escaping points for all polynomials of degree $d \ge 3$ with distinct roots.
+
+We first continue our study of the cubic case, $d=3$. As observed earlier, we can always assume that our polynomial $p$ is centered, i.e., has the form $p = Z^3 + aZ + b$. Then the image of $W_p$ is contained in the plane $H = \{z_1 + z_2 + z_3 = 0\}$, so it is sufficient to consider the induced map $H \to H$. We identify $H$ with $\mathbb{C}^2$ by projecting to the first two coordinates. We can then extend $W_p$ to a rational map $\mathbb{P}^2 \to \mathbb{P}^2$, which is given by the following triple of quartic polynomials, as a simple computation shows.
+
+$$ 
+\begin{aligned}
+(z_1:z_2:z_0) & \mapsto (z_1(z_1-z_2)(z_1+2z_2)(2z_1+z_2) - (z_1+2z_2)(z_1^3 + az_1z_0^2 + bz_0^3)) \\
+(3.8) \qquad & : z_2(z_1-z_2)(z_1+2z_2)(2z_1+z_2) + (2z_1+z_2)(z_2^3 + az_2z_0^2 + bz_0^3) \\
+& : z_0(z_1-z_2)(z_1+2z_2)(2z_1+z_2))
+\end{aligned}
+$$
+
+Here the line at infinity is given by $z_0 = 0$; it is forward invariant, and the induced dynamics on this projective line is given by the rational map $\varphi$ from Section 3.2.
+
+**Theorem 3.10 (Escaping orbits for cubic polynomials).** For every cubic polynomial $p$, there are starting points $\underline{z} \in \mathbb{C}^3$ such that the iteration sequence $(W_p^{\text{on}}(\underline{z}))$ exists for all times and converges component-wise to infinity. The set of escaping points contains a holomorphic curve.
+
+*Proof.* Let
+
+$$ \alpha = -\sqrt{\frac{5+\sqrt{21}}{2}}. $$
+
+Then one can check that the point $q_0 = (1:\alpha:0)$ on the line at infinity is 2-periodic for the extension of $W_p$ to $\mathbb{P}^2$. We consider $q_0$ as a fixed point of the second iterate of this extension. Its multiplier matrix has eigenvalues
+
+$$ 12\alpha^3 - 72\alpha + 43 = 43 + 12\sqrt{7} > 1 \quad \text{and} \quad \frac{-\alpha^3 + 6\alpha + 4}{2} = 2 - \frac{1}{2}\sqrt{7} \in (0,1). $$
+
+The eigenspace for the first of these eigenvalues is tangential to the line at infinity, whereas the eigenspace for the second eigenvalue points away from the line. So the point $q_0$ has a stable manifold (see [PdM82, Ch. 2, Section 6] for the general theory) that meets the (complex) line at infinity locally only at $q_0$ and is a holomorphic curve by [Hub05, Cor. 8]. In particular, all points $q \in H$ that lie on the stable manifold and are sufficiently close to $q_0$ will converge in $\mathbb{P}^2$ to the 2-cycle that $q_0$ is part of. Since the points of this 2-cycle are on the line at infinity (and different from $(1:0:0)$, $(0:1:0)$, $(1:-1:0)$, which are the points corresponding to the lines $z_1=0$, $z_2=0$ and $z_3=-z_1-z_2=0$), the claim follows. □
\ No newline at end of file
diff --git a/samples/texts/6573214/page_14.md b/samples/texts/6573214/page_14.md
new file mode 100644
index 0000000000000000000000000000000000000000..a74e5d0edb5b7a7a73c65b3444dee53b1ff33ae1
--- /dev/null
+++ b/samples/texts/6573214/page_14.md
@@ -0,0 +1,19 @@
+**Remark 3.11.** When $p = Z^3$, the stable manifold of $q_0$ is the complex line joining it to $(0:0:1)$. So in this case, every scalar multiple of $(1,\alpha, -1-\alpha) \in H$ escapes to infinity.
+
+Now Theorem B follows from Theorem 3.10 in the following way. Write $p = p_1p_2$ with $p_1$ of degree 3 and $p_2$ with simple roots. By Theorem 3.10 there is a vector $q_1 \in \mathbb{C}^3$ that escapes to infinity under $W_{p_1}$. Now set $q = (q_1, q_2)$, where $q_2 \in \mathbb{C}^{d-3}$ has the roots of $p_2$ (in some order) as entries. Then iterating $W_p$ on $q$ has the effect of fixing the last $d-3$ coordinates, whereas the effect on the first three is that of $W_{p_1}$; see Lemma 3.5. In particular, the first three coordinates of the vectors in the orbit of $q$ under $W_p$ tend to infinity. Note that this result covers a slightly larger set of polynomials than those with simple roots: the cubic factor $p_1$ is arbitrary, so $p$ can have a multiple root of order at most 4 or two double roots.
+
+Taking iterated preimages under $W_p$ of the curve to infinity whose existence we have shown in Theorem 3.10 above, we obtain countably infinitely many (complex) curves to infinity full of escaping points. Here we restrict to iterated preimage curves ending in an iterated preimage of the point $q_0$ (notation as in the proof above) that is on the line at infinity. Two of the immediate preimage curves end at the origin, which is a point of indeterminacy for the rational map (3.8) induced by $W_p$. There are very likely other escaping points, but we expect the set of escaping points to be of measure zero within $H$.
+
+# 4 Algebraic description of periodic orbits
+
+Since we will be using methods from Computer Algebra to obtain a proof of the Theorem A, we now discuss how we can describe the periodic points of $W_p$ of any given period algebraically. We begin with a description of $W_p$ itself.
+
+## 4.1 Algebraic description of $W_p$
+
+For the purpose of studying periodic orbits under $W_p$ algebraically as $p$ varies, equation (3.5) is preferable to (3.1), since it is a polynomial equation involving the entries of $\underline{z}$ and $\underline{z}'$ and the coefficients of $p$, rather than an equation involving rational functions. The following result shows that we do not get extraneous solutions by doing so, in the sense that all solutions we find that involve points in $\Delta$ arise as degenerations of “honest” solutions living outside $\Delta$.
+
+**Proposition 4.1** (Polynomial equation describing iteration). *Fix* $p \in P_d'$. *The algebraic variety in* $\mathbb{C}^d \times \mathbb{C}^d$ *described by equation (3.5)* *is the Zariski closure of the graph of* $W_p$ *(*which is contained in*$ (\mathbb{C}^d \setminus \Delta) \times \mathbb{C}^d$*).
+
+*Proof.* Let $V_p$ denote the variety in question. Equation (3.5) corresponds to $d$ equations in the $2d$ coordinates of $\underline{z}$ and $\underline{z}'$, so each irreducible component of $V_p$ must have dimension at least $d$. We have to show that no irreducible component is contained in $\Delta \times \mathbb{C}^d$. We do this by showing that $\dim(V_p \cap (\Delta \times \mathbb{C}^d)) < d$.
+
+Assume that $\underline{z} \in \Delta$. We first consider the simplest case that $z_1 = z_2$, but $z_2, \dots, z_d$ are distinct. Substituting $Z \leftarrow z_1$ in (3.5), we obtain that $p(z_1) = 0$, so that $z_1$ must be a root of $p$. The subset of $\Delta$ consisting of $\underline{z}$ with this property has dimension $d-2$. Substituting $Z \leftarrow z_k$ with $k \ge 3$, we see that $z_k'$ is uniquely determined by $\underline{z}$ (it is still given by (3.1)). On the other hand, taking the derivative with respect to $Z$ on both sides and then substituting $Z \leftarrow z_1$, we see that $z_1' + z_2'$ is uniquely determined, so the
\ No newline at end of file
diff --git a/samples/texts/6573214/page_15.md b/samples/texts/6573214/page_15.md
new file mode 100644
index 0000000000000000000000000000000000000000..e830ec71f547bf538c40ca3ca5b2ef8da742351b
--- /dev/null
+++ b/samples/texts/6573214/page_15.md
@@ -0,0 +1,23 @@
+fiber above $\tilde{z}$ of the projection of $V_p$ to the first factor has dimension 1. So the part of
+$V_p \cap (\Delta \times \mathbb{C}^d)$ lying above points $\tilde{z}$ with only one double entry has dimension $d-1$.
+
+In general, we see by similar considerations (taking higher derivatives as necessary) that when $\tilde{z}$ has entries of multiplicities $m_1, \dots, m_l$ (with $m_1 + \dots + m_l = d$ and some $m_j \ge 2$), then these entries must be roots of $p$ of multiplicities (at least) $m_1 - 1, \dots, m_l - 1$, and the fiber of $V_p$ above $\tilde{z}$ is a linear space of dimension $(m_1 - 1) + \dots + (m_l - 1) = d - l$. On the other hand, the set of $\tilde{z}$ of this type has dimension $\#\{j : m_j = 1\} < l$, so the dimension of the corresponding subset of $V_p$ is $< d$.
+
+So we have seen that $V_p \cap (\Delta \times \mathbb{C}^d)$ is a finite union of algebraic sets of dimension $< d$;
+therefore it cannot contain an irreducible component of $V_p$. $\square$
+
+**Remark 4.2.** As in the proof above, we will usually think of (3.5) as a system of $d$ equations that are obtained by comparing the coefficients of the various powers of $Z$ on both sides. Note that the equation for the coefficient of $Z^j$ is of degree $d-j$ in $z_1, \dots, z_d, z'_1, \dots, z'_d$. So the total system has degree $d!$.
+
+## 4.2 Periodic points
+
+We use equation (3.5) to obtain a system of equations representing periodic points. Fix the degree $d$ and the period $n$. We consider $nd$ variables, grouped into $n$ vectors $\underline{z}^{(k)} = (z_1^{(k)}, \ldots, z_d^{(k)})$, for $0 \le k < n$, which we think of as representing an $n$-cycle $\underline{z}^{(0)}, \underline{z}^{(1)} = W_p(\underline{z}^{(0)}), \ldots, \underline{z}^{(n-1)} = W_p(\underline{z}^{(n-2)}), \underline{z}^{(0)} = W_p(\underline{z}^{(n-1)})$. We therefore define the scheme $\mathcal{P}'_d(n) \subset \mathcal{P}'_d \times \mathbb{C}^{nd}$ by collecting the equations arising from comparing coefficients on both sides of (3.5), where we replace $(\underline{z}, \underline{z}')$ successively by $(\underline{z}^{(0)}, \underline{z}^{(1)}), (\underline{z}^{(1)}, \underline{z}^{(2)}), \ldots, (\underline{z}^{(n-1)}, \underline{z}^{(0)})$; $p$ runs through the monic degree $d$ polynomials in $\mathcal{P}'_d$. This encodes that $\underline{z}^{(0)} \mapsto \underline{z}^{(1)} \mapsto \ldots \mapsto \underline{z}^{(n-1)} \mapsto \underline{z}^{(0)}$ under $W_p$. We then take $\mathcal{P}_d(n)$ to be the quotient of $\mathcal{P}'_d(n)$ by the group of affine transformations on $\mathbb{C}$, acting via
+
+$$T \cdot (p, z_1^{(0)}, \dots, z_d^{(n-1)}) = (Tp, Tz_1^{(0)}, \dots, Tz_d^{(n-1)}).$$
+
+We expect the fibers of the projection $\mathcal{P}_d(n) \to \mathcal{P}_d$ to be finite, i.e., that for each polynomial $p$, there are only finitely many points of period $n$ under $W_p$. The following lemma gives a criterion for when this is the case.
+
+**Lemma 4.3** (Criterion for finiteness of $n$-periodic points). Let $\mathcal{P}_d^{(0)}(n) \subset \mathbb{C}^{nd}$ be the fiber of $\mathcal{P}_d'(n)$ above $p = Z^d$. The projection $\mathcal{P}_d'(n) \to \mathcal{P}_d'$ is finite if and only if $\mathcal{P}_d^{(0)}(n) = \{\emptyset\}$.
+
+*Proof.* We first note that since the unique root 0 of $Z^d$ is fixed by scaling, the same is true for $\mathcal{P}_d^{(0)}(n)$ under simultaneous scaling of the coordinates. So $\mathcal{P}_d^{(0)}(n) = \{\emptyset\}$ is equivalent to $\mathcal{P}_d^{(0)}(n)$ being zero-dimensional. In particular, if $\mathcal{P}_d^{(0)}(n) \neq \{\emptyset\}$, then the projection is not finite, since the fiber above $Z^d$ has positive dimension. This proves one direction of the claimed equivalence.
+
+Now assume that the projection is not finite, so there is some $p \in \mathcal{P}_d$ such that the fiber $\mathcal{P}_d^{(p)}(n)$ above $p$ has positive dimension. Let $\bar{\mathcal{P}}_d^{(p)}(n) \subset \mathbb{P}^{nd}$ denote the projective scheme obtained by homogenizing the equations defining $\mathcal{P}_d'(n)$ and specializing to $p$. Then $\mathcal{P}_d^{(p)}(n)$ meets the hyperplane at infinity of $\mathbb{P}^{nd}$. But the intersection of $\mathcal{P}_d^{(p)}(n)$ with the hyperplane at infinity is exactly the image of $\mathcal{P}_d^{(0)}(n)$ under the projection $\mathbb{C}^{nd} \setminus \{\emptyset\} \to \mathbb{P}^{nd-1}$. So this image is non-empty, which implies that $\mathcal{P}_d^{(0)}(n)$ contains non-zero points. This shows the other direction. $\square$
\ No newline at end of file
diff --git a/samples/texts/6573214/page_16.md b/samples/texts/6573214/page_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..9bb890295ef9f87590c7459ac1533cc63342fd72
--- /dev/null
+++ b/samples/texts/6573214/page_16.md
@@ -0,0 +1,23 @@
+We can test the condition "$\mathcal{P}_d^{(0)}(n) = \{0\}$" with a Computer Algebra System by setting up the ideal that is generated by the equations defining $\mathcal{P}_d^{(0)}(n)$, together with $z_1^{(0)} - 1$ (for symmetry reasons, if there is some nonzero point, then there is one with $z_1^{(0)} \neq 0$, and by scaling, we can assume that $z_1^{(0)} = 1$). Then we compute a Groebner basis for this ideal. The condition is satisfied if and only if this Groebner basis contains 1. We did this for $d=3$ and small values of $n$.
+
+**Lemma 4.4** (Finiteness of $n$-periodic points). For every cubic polynomial $p$ with at least two distinct roots, there are only finitely many points of period $n \le 8$ under $W_p$. For cubic polynomials with a triple root, the statement holds for all $n \le 8$ except $n=6$.
+
+*Proof.* The claim follows for $n \in \{1, 2, 3, 4, 5, 7, 8\}$ from Lemma 4.3 and a computation as described above. For $n=6$, we find that $\mathcal{P}_3^{(0)}(6)$ consists of six lines through the origin (plus the origin with high multiplicity). These six lines correspond to 6-cycles of rotation type (see Section 5 below for the definition). By an explicit computation (see also Proposition 5.8), we check that the fiber above any polynomial with at least two distinct roots of the scheme describing 6-periodic points of rotation type is finite. For the remaining components of $\mathcal{P}_3'(6)$, we find that the corresponding part of $\mathcal{P}_3^{(0)}(6)$ has the origin as its only point; we can then conclude as in the proof of Lemma 4.3 that there are only finitely many 6-periodic points not of rotation type for all cubic polynomials. $\square$
+
+**Remark 4.5.** We expect that for cubic polynomials without a triple root, the statement of Lemma 4.4 holds for all $n$. For cubic polynomials with a triple root, we expect that the 6-periodic points of rotation type are the only exceptions, i.e., that there are no points of exact order $n \ge 2$ except the 6-periodic points of rotation type described in the proof above.
+
+We do not venture to formulate a conjecture for polynomials of degrees higher than 3. We did verify the criterion of Lemma 4.3 also for $d=4$ and $n=1,2,3$, however; beyond that, the computations become infeasible.
+
+There is a simple argument that shows that periodic points of any order always exist.
+
+**Lemma 4.6** (Existence of periodic points). Fix a monic polynomial $p$ of degree $d \ge 2$ with distinct roots. Then $W_p$ has periodic points of all periods $n \ge 1$.
+
+*Proof.* For $n=1$, all the vectors consisting of the roots of $p$ in some order are fixed points. So we fix now some $n \ge 2$. Write $p(Z) = \prod_{j=1}^d (Z - \alpha_j)$. Let $\omega$ be a primitive $(2^n-1)$-th root of unity. Then $\omega$ has exact period $n$ under the squaring map $z \mapsto z^2$, so by Lemma 3.6, there is a point $(z_1, z_2)$ of exact order $n$ for the Weierstrass map associated to $(Z - \alpha_1)(Z - \alpha_2)$. By Lemma 3.5, the point
+
+$$ \underline{z} = (z_1, z_2, \alpha_3, \dots, \alpha_d) $$
+
+then has exact period $n$ under $W_p$. $\square$
+
+One might ask whether there are always periodic points of all periods that do not fix any coordinate (or even, for which all coordinates have the same period $n$).
+
+We are interested in attracting periodic points, i.e., points $q \in \mathbb{C}^d$ with the property that there is a period $n \ge 2$ and a neighborhood $U$ of $q$ in $\mathbb{C}^d$ so that $W_p^{\text{omn}}(z) \to q$ as $m \to \infty$ for all $z \in U$. Consider the linearization $\mathbf{D}(W_p^{\text{on}})_q$ of the first return map at the point $q$. We call this the multiplier matrix of $q$. Local fixed point theory relates the
\ No newline at end of file
diff --git a/samples/texts/6573214/page_17.md b/samples/texts/6573214/page_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..dfa912e784acce7c2502c50d19e699e42f065c1e
--- /dev/null
+++ b/samples/texts/6573214/page_17.md
@@ -0,0 +1,15 @@
+topological property of being attracting to an algebraic property of this matrix, as in the following statement, which is a consequence of the fact that a differentiable map is locally well-approximated by its derivative.
+
+**Lemma 4.7 (Attracting fixed point).** The fixed point $q$ of a differentiable map $W: \mathbb{C}^d \to \mathbb{C}^d$ is attracting if all eigenvalues of $\mathbf{D}(W)|_q$ have absolute values strictly less than 1. It cannot be attracting unless all eigenvalues have absolute values at most 1.
+
+In the context of points of period $n$, we consider $W = W_p^{(n)}$. The lemma then tells us that $q$ can only be attracting when all eigenvalues of its multiplier matrix have absolute value at most 1. Equivalently, the characteristic polynomial of the multiplier matrix has all its roots in the closed complex unit disk. The set of monic polynomials of degree $d$ with this property forms a compact subset $\mathcal{A}_d$ of $\mathcal{P}'_d$.
+
+In the following, we will always assume that we pick a representative in the affine equivalence class of the polynomial in question that is centered, i.e., with vanishing sum of roots. Then the dynamics of $W_p$ takes place in the linear hyperplane $H$ given by $z_1 + \dots + z_d = 0$, and we get $\mathcal{P}_d(n) \subset \mathcal{P}_d \times H^n$. We can identify $\mathcal{P}_d(n)$ with its image in $\mathcal{P}_d \times H$ obtained by projection to the first two factors, $(p, \underline{z}^{(0)}, \dots, \underline{z}^{(n-1)}) \mapsto (p, \underline{z}^{(0)})$. Then the points of $\mathcal{P}_d(n)$ are represented by pairs $(p, q)$, where $p$ is a centered polynomial and $q \in H$ satisfies $W_p^{(n)}(q) = q$. Since we restrict to $H$, the multiplier matrix of any periodic point $q$ is of size $\dim H = d-1$.
+
+To study whether the $n$-cycles parameterized by $\mathcal{P}_d(n)$ can be attracting, we would like to associate to each such point $(p, q)$ the $d-1$ eigenvalues of the multiplier matrix of $q$ (the eigenvalues do not change under affine conjugation, so this gives a well-defined map). However, there is no natural order on these eigenvalues. To capture them as an unordered $(d-1)$-tuple, we express the eigenvalues instead through their elementary symmetric functions and hence through the characteristic polynomial of the multiplier matrix. In this way, we obtain an algebraic morphism (and therefore a holomorphic map) $\mu_{d,n}: \mathcal{P}_d(n) \to \mathcal{P}'_{d-1}$, in much the same way as in the context of Newton's method. Here we think of $\mathcal{P}'_{d-1}$ as the space of coefficient vectors of the characteristic polynomials.
+
+Our goal is now to find out if the image of $\mu_{d,n}$ meets $\mathcal{A}_{d-1}$, the set of polynomials all of whose roots are in the closed unit disk.
+
+Since we expect that $\mathcal{P}_d(n)$ is a finite-degree covering of $\mathcal{P}_d$, it should in particular have dimension $\dim \mathcal{P}_d = d-2$. This would imply that the image of $\mu_{d,n}$ has dimension at most $d-2$ (and we expect it to be exactly $d-2$), so it is contained in a proper algebraic subvariety of $\mathcal{P}'_{d-1}$. Each irreducible component of $\mathcal{P}_d(n)$ will map to an irreducible component of this subvariety. Such a subvariety of codimension at least 1 does not have to intersect a given bounded subset like $\mathcal{A}_{d-1}$. This is a marked difference compared to the situation with Newton's method, where the corresponding multiplier map is surjective, and so examples of attracting $n$-cycles can easily be found.
+
+So our strategy will be to get as good control as we can on the varieties $\mathcal{P}_d(n)$ (or suitable components of them), find the Zariski closure $X$ of their image under $\mu_{d,n}$ and then check if $X$ meets $\mathcal{A}_{d-1}$. If it does not, then clearly no stable $n$-cycle can exist on the component of $\mathcal{P}_d(n)$ that we are considering. If it does, then we check that it also meets the open subset of $\mathcal{A}_{d-1}$ consisting of polynomials with all roots in the open unit disk; then the intersection will contain a relative open subset of $X$ and so it will contain points in the image and such that the corresponding polynomial $p$ has distinct roots.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_18.md b/samples/texts/6573214/page_18.md
new file mode 100644
index 0000000000000000000000000000000000000000..7d72c5f521b63a4eec88e4a20bf74b97c4b8e0e0
--- /dev/null
+++ b/samples/texts/6573214/page_18.md
@@ -0,0 +1,51 @@
+5 Cycles for cubic polynomials
+
+We will now restrict consideration to cubic polynomials $p$. Using affine transformations, we can assume that $p(Z) = Z^3 + Z + t$ with some $t \in \mathbb{C}$. This choice of parameterization excludes only (the affine equivalence classes of) $Z^3 - 1$ (which corresponds to $t \to \infty$) and the degenerate case $Z^3$. The induced map to the true parameter space $\mathcal{P}_3$ is a double cover identifying $t$ and $-t$. We will abuse notation slightly in the following by writing $\mathcal{P}_3(n)$ for what is really the pull-back of the true $\mathcal{P}_3(n)$ to the $t$-line via the parameterization we use here. As mentioned earlier, for such centered polynomials, the dynamics restricts to the plane $H = \{z_1 + z_2 + z_3 = 0\}$.
+
+Let $\sigma_k(\underline{z})$ denote the $k$-th elementary symmetric polynomial in the entries of $\underline{z}$. We introduce the quantities
+
+$$w_2(\underline{z}) = \sigma_2(\underline{z}) - 1 \quad \text{and} \quad w_3(\underline{z}) = \sigma_3(\underline{z}) + t.$$
+
+(We shift by the elementary symmetric polynomials in the roots of $p$ to move the image
+of the fixed points to $(0,0)$.) The map $\mathbb{C}^3 \supset H \to \mathbb{C}^2$ given by $(w_2, w_3)$ has degree 6.
+
+By the second property in Lemma **3.3**, $W_p$ induces a map $\widetilde{W}_p$ on (a subset of) $\mathbb{C}^2$ such
+that
+
+$$ (w_2(W_p(\underline{z})), w_3(W_p(\underline{z}))) = \widetilde{W}_p(w_2(\underline{z}), w_3(\underline{z})) $$
+
+for all $\underline{z} \in (\mathbb{C}^3 \setminus \Delta) \cap H$.
+
+**Lemma 5.1.** $\widetilde{W}_p$ is given by
+
+$$ 
+\begin{aligned}
+\widetilde{W}_p(w_2, w_3) = \frac{1}{\delta} & (w_2^2 + 2w_2^3 - 3w_3^2 - 9tw_2w_3 + w_2^4 + 6w_2w_3^2, \\
+& 4w_2w_3 + 3tw_2^2 + 4w_2^2w_3 + 2tw_2^3 - 9tw_2^2 + w_2^3w_3 + 8w_3^3),
+\end{aligned}
+$$
+
+where
+
+$$ \delta = 4(1 + w_2)^3 + 27(t - w_3)^2. $$
+
+*Proof.* Routine calculation with a Computer Algebra System. $\square$
+
+Note that this explicit expression shows the quadratic convergence to (0,0) when $p$ has
+distinct roots, which is equivalent to $4 + 27t^2 \neq 0$.
+
+Now suppose we have an *n*-cycle ($\underline{z}^{(0)}, \underline{z}^{(1)}, \dots, \underline{z}^{(n-1)}$) under $W_p$. It will be attracting
+only if all eigenvalues of the multiplier matrix $\mathbf{D}(W_p^{\text{on}})|_{\underline{z}^{(0)}}$ have absolute value at most 1.
+Concretely, we consider the map $\mu_{3,n}: \mathcal{P}_3(n) \to \mathcal{P}'_2$ as discussed in Section **4.2**. The
+characteristic polynomial will have the form $Z^2 + c_1Z + c_0$ with $c_0, c_1 \in \mathbb{C}$, and we
+know from the discussion in Section **4.2** that $c_0$ and $c_1$ must satisfy an algebraic relation,
+i.e., the points $(c_0, c_1)$ lie on some plane algebraic curve as we run through all possible
+characteristic polynomials.
+
+We can also consider the image of this *n*-cycle under $(w_2, w_3)$, as the map $\mu_{3,n}$ factors
+through the $(w_2, w_3)$-plane. Assuming that *n* is the minimal period of the cycle, the
+image cycle can have minimal period *n*, *n*/2 or *n*/3. The second possibility occurs when
+*n* = 2*k* is even and $W_p^{\text{ok}}$ acts as a transposition on the vectors in the cycle. In this case,
+we say that the cycle is of *transposition type*. The last possibility occurs when *n* = 3*k* is
+divisible by 3 and $W_p^{\text{ok}}$ acts as a cyclic shift on the vectors in the cycle. In this case, we
+say that the cycle has *rotation type*. We can then equivalently look at the characteristic
\ No newline at end of file
diff --git a/samples/texts/6573214/page_19.md b/samples/texts/6573214/page_19.md
new file mode 100644
index 0000000000000000000000000000000000000000..24efbfea5654987ea80ca58a65cd95b27613d573
--- /dev/null
+++ b/samples/texts/6573214/page_19.md
@@ -0,0 +1,50 @@
+polynomial of $D(\tilde{W}_p^{(n)})_{|(w_2,w_3)(z)}$ (or with $k$ in place of $n$ in the transposition or rotation type cases).
+
+We will need a criterion that we can use to show that the two relevant eigenvalues can
+never simultaneously be in the unit disk, in cases when the relation between $c_0$ and $c_1$ is
+somewhat involved. The following lemma provides one such criterion.
+
+**Lemma 5.2.** Let $P(\lambda, \mu) \in \mathbb{C}[\lambda, \mu]$ be a polynomial. Fix a half-line $\ell$ emanating from the origin and some $N \in \mathbb{Z}_{>0}$. Let $B$ be the sum of the absolute values of the coefficients of the two partial derivatives of $P$. If for all $j, k \in \{0, 1, \dots, N-1\}$, the distance from $P(e^{2\pi i j/N}, e^{2\pi i k/N})$ to $\ell$ exceeds $\pi B/N$, then $P(\lambda, \mu) = 0$ has no solutions in $\mathbb{C}^2$ with $|\lambda|, |\mu| \le 1$.
+
+*Proof.* We first show that the assumptions imply that the image of $P$ on the torus $\mathbb{S}^1 \times \mathbb{S}^1$
+is contained in the slit plane $\mathbb{C}\setminus\ell$. So consider $(u, v) \in [0, 1]^2$ and pick $(j, k) \in \{0, \dots, N\}^2$
+so that $|u-j/N|, |v-k/N| \le 1/(2N)$. Note that the sum of the absolute values of the
+partial derivatives of $(u, v) \mapsto F(u, v) := P(e^{2\pi i u}, e^{2\pi i v})$ for $u, v \in \mathbb{R}$ is bounded by $2\pi B$.
+This shows that
+
+$$
+|P(e^{2\pi i u}, e^{2\pi i v}) - P(e^{2\pi i j/N}, e^{2\pi i k/N})| \leq \frac{1}{2N} \|F_u\|_{\infty} + \frac{1}{2N} \|F_v\|_{\infty} \leq \frac{1}{2N} \cdot 2\pi B = \pi B/N.
+$$
+
+Since the distance of $P(e^{2\pi i j/N}, e^{2\pi i k/N})$ from $\ell$ is by assumption larger than $\pi B/N$, it
+follows that $P(e^{2\pi i u}, e^{2\pi i v}) \notin \ell$.
+
+We now assume that there is a solution with $|\lambda|, |\mu| \le 1$, so that the curve defined by $P$
+in $\mathbb{C}^2$ meets the unit bi-disk. Since the curve is unbounded, by continuity there will be a
+solution with $|\lambda| = 1$ and $|\mu| \le 1$ or $|\mu| = 1$ and $|\lambda| \le 1$. By symmetry, we can assume
+the former. By the argument principle, the closed curve $\gamma: [0, 1] \ni s \mapsto P(\lambda, e^{2\pi i s})$ has to
+pass through the origin or wind around it at least once. However, since the assumptions
+imply that the image of $\gamma$ is contained in the slit plane $\mathbb{C} \setminus \ell$, which does not contain the
+origin and is simply connected, we obtain a contradiction. $\square$
+
+The general procedure for obtaining the results given below is as follows.
+
+1. Set up equations for the variety $\mathcal{P}_3(n)$ or parts of it using (3.5).
+
+2. Set up the map $\mu_{3,n}$ as a map to the projective plane given by the coefficients of the characteristic polynomial of the multiplier matrix.
+
+3. Use the Groebner Basis machinery of a Computer Algebra System like Magma [BCP97] or Singular [DGPS19] to find the equation of the image curve.
+
+4. Either find a point on the image curve corresponding to a characteristic polynomial with both roots in the unit disk, or show using Lemma 5.2 that no such points exist.
+
+The available machinery can also be used to obtain additional information on the com-
+ponents of the curves $\mathcal{P}_3(n)$, for example smoothness or the (geometric) genus.
+
+Since the map $\mu_{3,n}$ is given by fairly involved rational functions when $n$ is not very small,
+Step 3 above may not necessarily be feasible as stated. In this case, we can instead sample
+some algebraic points on the variety considered (e.g., by specializing the parameter $t$ to
+a rational value and then determining the solutions of the resulting zero-dimensional
+system) and consider their images under $\mu_{3,n}$. Given enough of these image points, we
+can fit a curve of lowest possible degree through them (this is just linear algebra). We
+can then check that this curve is correct by constructing a generic point on the original
+variety and checking that its image lies indeed on the curve.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_2.md b/samples/texts/6573214/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..8c618e52cb82b9f48a755d61d614252e7ea6049b
--- /dev/null
+++ b/samples/texts/6573214/page_2.md
@@ -0,0 +1,19 @@
+An interesting method is Newton's, which may well be the best-known method; it approximates one root at a time. This is a simple method that is stable and converges quadratically near simple roots, so it is often used to polish approximate roots. However, it is an iterated rational map, so it is "chaotic" on its Julia set, and its global dynamics is hard to describe. In particular, it is well known to be not generally convergent: there are open sets of polynomials and open sets of starting points on which the Newton dynamics does not converge to any root, but rather to an attracting periodic orbit ("an attracting cycle") of period 2 or higher. Its use has thus often been discouraged. However, in recent years quite some theory has been developed about its global dynamics and its expected (rather efficient) speed of convergence. At the same time, it has been used in practice successfully to find all roots of polynomials of degree exceeding $10^9$ in remarkable speed. Some of these results are described in Section 2. Therefore, Newton's method stands out as one that at the same time has good theory and performs well in practice.
+
+The focus of our work is on the Weierstrass iteration method [Wei91], also known as the Durand-Kerner-method [Dur60, Ker66]. For this method, we are not aware of any global theory of its dynamics, but it is well known that in practice it usually finds all roots of a complex polynomial (except in the presence of obvious symmetries: for instance, when the polynomial is real but some of its roots are not, then any purely real vector of starting points cannot converge to the roots because the method respects complex conjugation).
+
+Our first result says that this observation does not hold in general.
+
+**Theorem A** (The Weierstrass method is not generally convergent).
+
+(1) There is an open set of polynomials $p$ of every degree $d \ge 3$ such that the (partially defined) Weierstrass iteration $W_p: \mathbb{C}^d \to \mathbb{C}^d$ associated to $p$ has attracting cycles of period 4. In particular, the Weierstrass method is not generally convergent for polynomials of degree at least 3.
+
+(2) Period 4 is minimal with this property: for every cubic polynomial $p$ the associated Weierstrass iteration $W_p: \mathbb{C}^3 \to \mathbb{C}^3$ associated to $p$ cannot have an attracting cycle of period 2 or 3.
+
+This theorem answers in the affirmative a question asked by Steve Smale: he expected
+the existence of attracting cycles in the 1990’s, if not earlier, in analogy to the Newton
+dynamics (Victor Pan, personal communication).
+
+Following McMullen [McM87], we say that a root-finding method in one variable is *generally convergent* if, for an open dense set of polynomials of fixed degree, there is an open dense set of starting points in $\mathbb{C}$ that converge to one of the roots. To our knowledge, the only known way to establish failure of general convergence is to find a polynomial $p$ that, under the given iteration method, has an attracting periodic orbit (an “attracting cycle”) of period $n \ge 2$. This attracting cycle must attract a neighborhood of the cycle, and it would persist under small perturbations of $p$, so convergence to a root fails on an open set of starting points for an open set of polynomials. Therefore, our theorem establishes that the Weierstrass method is not generally convergent for polynomials of degrees 3 or higher. (Other ways of failure of general convergence are of course conceivable but have apparently never been observed).
+
+It is well known that the Weierstrass method has another problem: some orbits are not defined forever. The Weierstrass method $W_p: \mathbb{C}^d \to \mathbb{C}^d$ is not defined whenever two coordinates in $\mathbb{C}^d$ coincide; this problem may occur even after any number of iteration steps from a starting vector with distinct entries.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_20.md b/samples/texts/6573214/page_20.md
new file mode 100644
index 0000000000000000000000000000000000000000..60559a9df91f786d37ba37d761f599cbb5fcd4ee
--- /dev/null
+++ b/samples/texts/6573214/page_20.md
@@ -0,0 +1,29 @@
+In the following, we always tacitly assume that the vectors occurring in the cycles do not contain roots of $p$. Those that do can easily be described using Lemmas 3.5 and 3.6.
+
+The computations leading to the results given below have been done using the Magma Computer Algebra System [BCP97] and also in many cases independently with Singular [DGPS19]. A Magma script containing code that verifies most of the claims made is available at [Sto].
+
+## 5.1 Points of order 2
+
+We begin by considering 2-cycles. Note that a 2-cycle of transposition type fixes one component of the vector, which then must be a root of $p$. Since we have excluded cycles of this form (up to the obvious symmetries, there are three of them, one for each root), no 2-cycles of transposition type have to be considered.
+
+**Proposition 5.3.** The 2-cycles form a smooth irreducible curve of geometric genus 0; it maps with degree 12 to the t-line. So for each polynomial, there is (generically) one orbit of 2-cycles under the natural action of $S_3 \times C_2$, where the first factor permutes the vector entries and the second factor performs a cyclic shift along the cycle. The image in $(t, w_2, w_3)$-space is the curve
+
+$$w_2 = -3, \quad 27t^2 - 45tw_3 + 20w_3^2 - 20 = 0$$
+
+of genus 0. The characteristic polynomial $X^2 + c_1X + c_0$ of the multiplier matrix at a point on this curve satisfies the relation $c_0 + 2c_1 + 6 = 0$. In particular, no 2-cycle can be attracting.
+
+*Proof.* This follows the method outlined above. Note that when both eigenvalues have absolute values at most 1, we have $|c_0| \le 1$ and $|c_1| \le 2$. $\square$
+
+## 5.2 Points of order 3
+
+We begin by considering the 3-cycles of rotation type. They can be defined by (3.5) together with $(z'_1, z'_2, z'_3) = (z_2, z_3, z_1)$ (for one choice of the cyclic permutation involved). Their images under $(w_2, w_3)$ are fixed points of $\tilde{W}_p$.
+
+**Proposition 5.4.** The 3-cycles of rotation type form two smooth irreducible curves (as $t$ varies) of geometric genus 0, according to which of the possible two cyclic permutations results from the action of $W_p$; the map to the t-line is of degree 6 in both cases. The images of both curves in $(t, w_2, w_3)$-space agree; the image curve is given by the equations
+
+$$w_2 = -\frac{3}{2}, \quad 216t^2 - 360tw_3 + 152w_3^2 - 1 = 0,$$
+
+describing a curve of genus 0. The characteristic polynomial of the multiplier matrix at a point on this curve (as a fixed point under $\tilde{W}_p$) has the form $X^2 + 3X + a$ for some $a \in \mathbb{C}$. In particular, such a 3-cycle cannot be attracting.
+
+*Proof.* This again follows the procedure outlined above. The characteristic polynomials lie on the curve $c_1 = 3$. So the sum of the eigenvalues is $-3$, hence it is not possible that both eigenvalues are in the closed unit disk. $\square$
+
+Now we consider “general” 3-cycles, i.e., 3-cycles that are not of rotation type.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_21.md b/samples/texts/6573214/page_21.md
new file mode 100644
index 0000000000000000000000000000000000000000..e270996d64dff949d87d59e2a95bebab53d30b26
--- /dev/null
+++ b/samples/texts/6573214/page_21.md
@@ -0,0 +1,26 @@
+**Proposition 5.5.** The 3-cycles that are not of rotation type form two irreducible curves of geometric genus 19, which each map with degree 72 to the t-line and are interchanged by the action of any transposition in $S_3$. Each curve therefore contains 8 orbits of 3-cycles under the action of $A_3 \times C_3$, and there are in total 8 orbits under $S_3 \times C_3$, for each fixed $t$. The coefficients $(c_0, c_1)$ of the characteristic polynomial of the multiplier matrix at a point in such a 3-cycle give a point on a rational curve of degree 12 that can be parameterized as $(c_0(u)/c_2(u), c_1(u)/c_2(u))$, where
+
+$$ 
+\begin{align*}
+c_0(u) &= -9u^{12} - 162u^{11} - 693u^{10} + 1434u^9 + 11958u^8 - 32202u^7 - 182301u^6 \\
+&\quad + 578742u^5 + 2069910u^4 - 919718u^3 - 3065685u^2 + 892254u + 264295, \\
+c_1(u) &= u^{12} + 26u^{11} + 230u^{10} + 693u^9 - 3867u^8 - 5844u^7 + 123074u^6 \\
+&\quad - 38381u^5 - 1320149u^4 + 420552u^3 + 4310940u^2 - 4206447u + 1442574, \\
+c_2(u) &= -9u^{10} - 63u^9 + 301u^8 + 1126u^7 - 7693u^6 - 3641u^5 \\
+&\quad + 52375u^4 + 13526u^3 - 104463u^2 - 47919u + 20987.
+\end{align*}
+$$
+
+In particular, no such 3-cycle can be attracting.
+
+*Proof.* The computations get quite a bit more involved, so we give more details here. We work in 7-dimensional affine space over $\mathbb{Q}$ with coordinates $(t, x_0, y_0, x_1, y_1, x_2, y_2)$, where the three vectors in the cycle are $\tilde{z}^{(j)} = (x_j, y_j, -x_j - y_j)$ for $j = 0, 1, 2$. We first set up the scheme giving the cycle $\tilde{z}^{(0)} \mapsto \tilde{z}^{(1)} \mapsto \tilde{z}^{(2)} \mapsto \tilde{z}^{(0)}$ under $W_p$. Then we remove the subschemes corresponding to cycles that have a fixed component or to 3-cycles of rotation type. The resulting scheme is a curve mapping with degree 144 to the $t$-line. Its projection to the $(x_2, y_2)$-plane is a curve of degree 48, whose defining polynomial factors into two irreducibles of degree 24 each that are interchanged by $x_2 \leftrightarrow y_2$. Let $\mathbb{Q}$ denote one of the factors, considered as a bivariate polynomial. Since the projection is birational, this induces the splitting of the original curve into two components. We could compute the genus by working with the birationally equivalent plane curve given by $\mathbb{Q}(x, y) = 0$. It has 222 simple nodes (six of which are defined over $\mathbb{Q}(\sqrt{6})$; the remaining 214 are conjugate) and a pair of conjugate singularities defined over $\mathbb{Q}(\sqrt{-3})$ that each contribute 6 to the difference between arithmetic and geometric genus. We obtain
+
+$$ g = \frac{23 \cdot 22}{2} - 222 - 2 \cdot 6 = 19 $$
+
+as claimed.
+
+This is a case where we had to use the sampling-and-interpolation trick to determine the image curve of $\mu_{3,3}$ on one of the components.
+
+After showing that the image curve has geometric genus 0 (there is one point of multiplicity 4 at $(8, -9)$ that gives an adjustment of 8, and there are 47 further simple nodes, so we obtain $g = 11 \cdot 10/2 - 8 - 47 = 0$) and finding some smooth rational points on it, we computed a parameterization modulo some large prime that maps $0, 1, \infty$ to three specified rational points and lifted it to $\mathbb{Q}$. It is then easy to verify that we indeed obtain a parameterization of the curve over $\mathbb{Q}$. We then used Magma's (fairly new and contributed by the third author) `ImproveParametrization` command to simplify the resulting parameterization.
+
+Finally, we use Lemma **5.2** to show that there is no attracting 3-cycle (not of rotation type). We find the polynomial $P(\lambda, \mu) = 0$ that gives the relation between the eigenvalues $\lambda$ and $\mu$ (by substituting $(c_0, c_1) \leftarrow (\lambda\mu, -(\lambda+\mu))$ in the equation relating the coefficients
\ No newline at end of file
diff --git a/samples/texts/6573214/page_22.md b/samples/texts/6573214/page_22.md
new file mode 100644
index 0000000000000000000000000000000000000000..520a7b98becb393db4c3923829d0040f2cd61888
--- /dev/null
+++ b/samples/texts/6573214/page_22.md
@@ -0,0 +1,32 @@
+of the characteristic polynomial) and check that the criterion of Lemma 5.2 is satisfied when $\ell$ is the positive real axis and $N = 18$. $\square$
+
+## 5.3 Points of order 4
+
+Judging by the heavy lifting that was necessary to deal with case of general 3-cycles, looking at general $n$-cycles with $n \ge 4$ seems too daunting a task to attack with confidence along the lines described here. We can, however, consider cycles with extra symmetries. Here we look at 4-cycles of transposition type.
+
+**Proposition 5.6.** The 4-cycles of transposition type form three irreducible smooth curves of geometric genus 1, each of degree 24 over the t-line, that are permuted by a cyclic shift of the coordinates. The characteristic polynomial $X^2 + c_1X + c_0$ of the multiplier matrix at any associated point (considered as a point of order 2 under $\tilde{W}_p$) satisfies the relation
+
+$$ 
+\begin{aligned}
+& 34c_0^4c_1^2 + 169c_0^3c_1^3 - 675c_0^2c_1^4 - 2997c_0c_1^5 - 2187c_1^6 + 68c_0^5 + 984c_0^4c_1 + 3359c_0^3c_1^2 \\
+& \quad - 19182c_0^2c_1^3 - 88965c_0c_1^4 - 91584c_1^5 + 4254c_0^4 + 29059c_0^3c_1 - 93688c_0^2c_1^2 \\
+& \quad - 634050c_0c_1^3 - 809379c_1^4 + 76045c_0^2 + 60846c_0^2c_1 - 725626c_0c_1^2 - 1171592c_1^3 \\
+& \quad + 487003c_0^2 + 4167623c_0c_1 + 8653407c_1^2 + 5442895c_0 + 15506760c_1 - 35154225 = 0,
+\end{aligned}
+$$
+
+which describes a curve birationally equivalent to the elliptic curve over $\mathbb{Q}$ with Cremona label 15a4. In particular, there do exist values of the parameter $t$ such that there are attracting 4-cycles of transposition type. Such parameters can be found near
+
+$$ t \approx 177.68741192204597. $$
+
+*Proof.* We set up the variety describing 4-cycles of transposition type as a subscheme of 5-dimensional affine space with coordinates $t, x_0, y_0, x_1, y_1$, where $t$ is the parameter and the iteration satisfies
+
+$$ (x_0, y_0, -x_0 - y_0) \mapsto (x_1, y_1, -x_1 - y_1) \mapsto (y_0, x_0, -x_0 - y_0), $$
+
+and we remove the component consisting of cycles in which the last coordinate is fixed. This results in a smooth irreducible curve of degree 24 over the $t$-line that has genus 1. We find the image curve in the $(c_0, c_1)$-plane. We compute that the geometric genus of the image curve is 1 and find a smooth rational point on it. This allows us to identify the elliptic curve it is birational to. From the explicit equation, we find that there is a characteristic polynomial that has a double root near $-0.68916660883309$. This leads to the given value of $t$ (and its negative). $\square$
+
+**Remark 5.7.** The region in the $t$-plane consisting of parameter values for which an attracting 4-cycle of transposition type exists is a union of two components, mapped to each other by $t \mapsto -t$. Each of them is symmetric with respect to the real axis; the component containing values with positive real part is shown in Figure 2 in blue.
+
+One can verify numerically that as $t$ increases along the real axis beyond the boundary of this region, a symmetry-breaking bifurcation occurs, and we find an adjacent region where attracting general 4-cycles (i.e., not of transposition type) exist. This region is shown in green in Figure 2.
+
+In Figure 3 we show how these regions are located relative to the parameter space of cubic Newton maps, in terms of a parameterization that is more commonly used in this context. It is apparent that these regions in parameter space are quite small. In addition, the left part of Figure 2 shows that the basin of attraction of the attracting 4-cycles is
\ No newline at end of file
diff --git a/samples/texts/6573214/page_23.md b/samples/texts/6573214/page_23.md
new file mode 100644
index 0000000000000000000000000000000000000000..8ee8d657df5a12ed00e9310c6d522854dfeeb218
--- /dev/null
+++ b/samples/texts/6573214/page_23.md
@@ -0,0 +1,23 @@
+also quite small as a subset of the dynamical plane. It is therefore not very surprising
+that examples of polynomials for which the Weierstrass method exhibits attractive cycles
+had not been found previously by numerical methods.
+
+It is well known that the parameters $\lambda$ for which the Newton map has attracting cycles of period 2 or greater are organized in the form of little Mandelbrot sets, finitely many for each period, and that every parameter in the bifurcation locus (common boundary point of any two colors) contains, in every neighborhood, infinitely many such little Mandelbrot sets. In Figure 4 we compare with one of these regions in parameter space where attractive 4-cycles exist for Newton's method. This period 4 component ranges roughly from imaginary parts 0.62095 to 0.6272 along the imaginary axis, hence is of diameter about 0.00625; for comparison: the period 4 component for Weierstrass has imaginary parts between 0.88439 and 0.88589, hence diameter about 0.0016, which is roughly comparable (even though there is no uniform Euclidean scale across parameter space).
+
+**FIGURE 2.** **Left:** An approximation to the real section of the immediate basin of attraction around a periodic point of order 4 for $W_{Z^3+Z+t}$ with $t = 177.68741192204597$. The coordinates shown are $(z_1, z_2)$; the corresponding point is $(z_1, z_2, -z_1 - z_2)$. The coloring encodes the number of iteration steps necessary to get within distance $10^{-4}$ of the periodic point.
+
+**Right:** Parameter values $t \in \mathbb{C}$ for which there exists a stable 4-cycle of transposition type (left region, blue) or a stable 4-cycle without extra symmetry (right region, green). The components touch at a point where the multiplier matrix under $\tilde{W}_p^{o2}$ has eigenvalue $-1$.
+
+## 5.4 Points of order 6
+
+Finally, we consider 6-cycles of rotation type.
+
+**Proposition 5.8.** The 6-cycles of rotation type form two irreducible smooth curves of geometric genus 5, each of degree 24 over the t-line, that are permuted by a transposition of the coordinates. The characteristic polynomial $X^2 + c_1X + c_0$ of the multiplier matrix at any associated point (considered as a point of order 2 under $\tilde{W}_p$) satisfies a relation that specifies a curve of geometric genus 0 and degree 5. This curve can be parameterized as $(c_0(u)/c_2(u), c_1(u)/c_2(u))$, where
+
+$$c_0(u) = -36u^5 - 12u^4 + 60u^3 + 236u^2 + 260u - 4,$$
+
+$$c_1(u) = -4u^5 - 51u^4 - 90u^3 + 59u^2 + 42u + 5,$$
+
+$$c_2(u) = -9u^4 - 18u^3 + u^2 + 10u - 1.$$
+
+In particular, no such 6-cycle can be attracting.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_24.md b/samples/texts/6573214/page_24.md
new file mode 100644
index 0000000000000000000000000000000000000000..85878d2499814fd0819211b4c17a32747cfecdf0
--- /dev/null
+++ b/samples/texts/6573214/page_24.md
@@ -0,0 +1 @@
+FIGURE 3. The parameter space of cubic polynomials up to affine precomposition, parameterized as $p(Z) = (Z-1)(Z+\frac{1}{2}-\lambda)(Z+\frac{1}{2}+\lambda)$ with $\lambda \in \mathbb{C}$; shown is the complex $\lambda$-plane. This is a standard parameterization used to visualize Newton dynamics, which the picture illustrates: the three colors indicate to which of the three roots 1, $\frac{1}{2} + \lambda$, and $\frac{1}{2} - \lambda$ the free critical point 0 converges. A point is colored black when there is no convergence. The top picture shows a global view of parameter space, with two subsequent magnifications shown below (first left, then right). The regions shown on the right in Figure 2, which indicate parameter values for which an attractive 4-cycle exists for the Weierstrass iteration, are superimposed on the last magnification (shown in yellow and converted to the different parameterization used here).
\ No newline at end of file
diff --git a/samples/texts/6573214/page_25.md b/samples/texts/6573214/page_25.md
new file mode 100644
index 0000000000000000000000000000000000000000..2d763d491ec9011aa7842f92e7206c8b8b260b79
--- /dev/null
+++ b/samples/texts/6573214/page_25.md
@@ -0,0 +1,19 @@
+FIGURE 4. Sequence of close-ups towards one of the largest “little Mandelbrot sets” around attracting cycles of period 4 for the Newton iteration, starting with the first two pictures in Figure 3 (top and left); the square in the bottom of the latter shows the domain where the magnifications start that are shown here.
+
+*Proof*. We set up the variety describing 6-cycles of rotation type as a subscheme of 5-dimensional affine space with coordinates $t, x_0, y_0, x_1, y_1$, where $t$ is the parameter and the iteration satisfies
+
+$$ (x_0, y_0, -x_0 - y_0) \mapsto (x_1, y_1, -x_1 - y_1) \mapsto (y_0, -x_0 - y_0, x_0), $$
+
+and we remove components coming from 3-cycles of rotation type. This results in a smooth irreducible curve of degree 24 over the $t$-line that has genus 5. We find the image curve in the $(c_0, c_1)$-plane. Since the degree and the coefficient size are moderate, we can directly check that the curve has geometric genus 0 and then find a parameterization. We then use the explicit equation and Lemma 5.2 with $\ell$ the negative real axis and $N = 12$ to verify that no characteristic polynomial lying on the curve can have both roots in the unit disk. □
+
+## 5.5 Proof of Theorem A
+
+The results obtained in this section provide a proof of part (2) of Theorem A. Proposition 5.6 gives a proof of part (1) for the case $d=3$. To obtain the conclusion for all $d \ge 3$, we invoke Lemma 3.7.
+
+## References
+
+[BAS16] Todor Bilarev, Magnus Aspenberg, and Dierk Schleicher, *On the speed of convergence of Newton's method for complex polynomials*, Math. Comp. **85** (2016), no. 298, 693–705, DOI 10.1090/mcom/2985. ↑2
+
+[BLS13] Béla Bollobás, Malte Lackmann, and Dierk Schleicher, *A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method*, Math. Comp. **82** (2013), no. 281, 443–457, DOI 10.1090/S0025-5718-2012-02640-8. ↑2
+
+[BCP97] Wieb Bosma, John Cannon, and Catherine Playoust, *The Magma algebra system. I. The user language*, J. Symbolic Comput. **24** (1997), no. 3-4, 235–265, DOI 10.1006/jsco.1996.0125. Computational algebra and number theory (London, 1993). ↑1, 3, 5
\ No newline at end of file
diff --git a/samples/texts/6573214/page_26.md b/samples/texts/6573214/page_26.md
new file mode 100644
index 0000000000000000000000000000000000000000..334c88fecf6244388729893b97fc08df3b2e9755
--- /dev/null
+++ b/samples/texts/6573214/page_26.md
@@ -0,0 +1,43 @@
+[BT18] Paul Breiding and Sascha Timme, *Homotopy Continuation.jl: a package for homotopy continuation in Julia*, Mathematical software — ICMS 2018. 6th international conference (South Bend, IN, USA, July 24), Lecture Notes in Computer Science, vol. 10931, Springer, Cham, 2018, pp. 458–465. ↑1
+
+[BH03] Xavier Buff and Christian Henriksen, *On König's root-finding algorithms*, Nonlinearity **16** (2003), no. 3, 989–1015, DOI 10.1088/0951-7715/16/3/312. ↑2
+
+[DGPS19] Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann, SINGULAR 4-1-2 — A computer algebra system for polynomial computations, 2019. available at http://www.singular.uni-klu.de. ↑1, 3, 5
+
+[Doč62] Kiril Dočev, *A variant of Newton's method for the simultaneous approximation of all roots of an algebraic equation*, Fiz.-Mat. Spis. Bulgar. Akad. Nauk. **5**(**38**) (1962), 136–139 (Bulgarian). ↑3
+
+[Dur60] Émile Durand, *Solutions numériques des équations algébriques*. Tome I: Équations du type $F(x) = 0$; racines d'un polynôme*, Masson et Cle, Editeurs, Paris, 1960. ↑1
+
+[Erë89] Alexandre È. Erëmenko, *On the iteration of entire functions*, Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339–345. ↑1
+
+[HM96] T. E. Hull and R. Mathon, *The mathematical basis and a prototype implementation of a new polynomial rootfinder with quadratic convergence*, ACM Trans. Math. Software **22** (1996), no. 3, 261–280, DOI 10.1145/232826.232830. ↑3
+
+[Hub05] John Hubbard, *Parametrizing unstable and very unstable manifolds*, Mosc. Math. J. **5** (2005), no. 1, 105–124, DOI 10.17323/1609-4514-2005-5-1-105-124 (English, with English and Russian summaries). ↑3.3
+
+[HSS01] John Hubbard, Dierk Schleicher, and Scott Sutherland, *How to find all roots of complex polynomials by Newton's method*, Invent. Math. **146** (2001), no. 1, 1–33, DOI 10.1007/s002220100149. ↑2
+
+[Ker66] Immo O. Kerner, *Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen*, Numer. Math. **8** (1966), 290–294, DOI 10.1007/BF02162564. ↑1, 3
+
+[LMS15] Russell Lodge, Yauhen Mikulich, and Dierk Schleicher, *A classification of postcritically finite Newton maps*, October 9, 2015. arXiv preprint, https://arxiv.org/abs/1510.02771. ↑2
+
+[McM87] Curt McMullen, *Families of rational maps and iterative root-finding algorithms*, Ann. of Math. (2) **125** (1987), no. 3, 467–493, DOI 10.2307/1971408. ↑1, 2
+
+[McN02] John Michael McNamee, *A 2002 update of the supplementary bibliography on roots of polynomials*, J. Comput. Appl. Math. **142** (2002), no. 2, 433–434, DOI 10.1016/S0377-0427(01)00546-5. ↑1
+
+[McN07] ________, *Numerical methods for roots of polynomials. Part I*, Studies in Computational Mathematics, vol. 14, Elsevier B. V., Amsterdam, 2007. ↑1
+
+[PdM82] Jacob Palis Jr. and Welington de Melo, *Geometric theory of dynamical systems*, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. ↑3.3
+
+[Pan97] Victor Y. Pan, *Solving a polynomial equation: some history and recent progress*, SIAM Rev. **39** (1997), no. 2, 187–220, DOI 10.1137/S0036144595288554. ↑1
+
+[Pan02] ________, *Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding*, J. Symbolic Comput. **33** (2002), no. 5, 701–733, DOI 10.1006/jsco.2002.0531. Computer algebra (London, ON, 2001). ↑1
+
+[Pan21] ________, *New Progress in Polynomial Root-finding*, February 28, 2021. arXiv preprint, https://arxiv.org/abs/1805.12042. ↑1
+
+[RSS17] Marvin Randig, Dierk Schleicher, and Robin Stoll, *Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees*, December 31, 2017. arXiv preprint, https://arxiv.org/abs/1703.05847. ↑2
+
+[Rei20] Bernhard Reinke, *Diverging orbits for the Ehrlich-Aberth and the Weierstrass root finders*, November 20, 2020. arXiv preprint, https://arxiv.org/abs/2011.01660. ↑1
+
+[RRRS11] Günter Rottenfusser, Johannes Rückert, Lasse Rempe, and Dierk Schleicher, *Dynamic rays of bounded-type entire functions*, Ann. of Math. (2) **173** (2011), no. 1, 77–125, DOI 10.4007/annals.2011.173.1.3. ↑1
+
+[Sch16] Dierk Schleicher, *On the Efficient Global Dynamics of Newton's Method for Complex Polynomials*, October 8, 2016. arXiv preprint, https://arxiv.org/abs/1108.5773. ↑2
\ No newline at end of file
diff --git a/samples/texts/6573214/page_27.md b/samples/texts/6573214/page_27.md
new file mode 100644
index 0000000000000000000000000000000000000000..7b26ca1abdcb0e346690d2fb9e2624d90d5086bb
--- /dev/null
+++ b/samples/texts/6573214/page_27.md
@@ -0,0 +1,21 @@
+[SS17] Dierk Schleicher and Robin Stoll, *Newton's method in practice: Finding all roots of polynomials of degree one million efficiently*, Theoret. Comput. Sci. **681** (2017), 146–166, DOI 10.1016/j.tcs.2017.03.025. ↑2
+
+[SCR⁺²⁰] Sergey Shemyakov, Roman Chernov, Dzmitry Rumiantsau, Dierk Schleicher, Simon Schmitt, and Anton Shemyakov, *Finding polynomial roots by dynamical systems — a case study*, Discrete Contin. Dyn. Syst. **40** (2020), no. 12, 6945–6965, DOI 10.3934/dcds.2020261. ↑2
+
+[Sma85] Steve Smale, *On the efficiency of algorithms of analysis*, Bull. Amer. Math. Soc. (N.S.) **13** (1985), no. 2, 87–121, DOI 10.1090/S0273-0979-1985-15391-1. ↑2
+
+[Sto] Michael Stoll, *Magma code verifying the results in Section 5*. available at http://www.mathe2.uni-bayreuth.de/stoll/magma/index.html#Weierstrass. ↑5
+
+[Wei91] Karl Weierstraß, *Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Funktionen derselben Veränderlichen*, Sitzungsberichte der königlich preussischen Akademie der Wissenschaften zu Berlin (1891), 1085–1101. ↑1, 3, 3.1
+
+AIX-MARSEILLE UNIVERSITÉ, INSTITUT DE MATHÉMATIQUES DE MARSEILLE, 163 AVENUE DE LUMINY CASE 901, 13009 MARSEILLE, FRANCE.
+
+Email address: Bernhard.Reinke@univ-amu.fr
+
+AIX-MARSEILLE UNIVERSITÉ, INSTITUT DE MATHÉMATIQUES DE MARSEILLE, 163 AVENUE DE LUMINY CASE 901, 13009 MARSEILLE, FRANCE.
+
+Email address: Dierk.Schleicher@univ-amu.fr
+
+MATHEMATISCHES INSTITUT, UNIVERSITÄT BAYREUTH, 95440 BAYREUTH, GERMANY.
+
+Email address: Michael.Stoll@uni-bayreuth.de
\ No newline at end of file
diff --git a/samples/texts/6573214/page_3.md b/samples/texts/6573214/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..53271f20719fa48b3d40afd28f11969a3eb79f33
--- /dev/null
+++ b/samples/texts/6573214/page_3.md
@@ -0,0 +1,17 @@
+Our second main result establishes the existence of a very different kind of problem for the Weierstrass method that apparently was not known: there are orbits in $\mathbb{C}^d$ for which the iteration is always defined that converge to $\infty$ (in the sense that the orbit leaves every compact subset of $\mathbb{C}^d$). This problem exists (at least) for every polynomial of degree $d \ge 3$ that has only simple roots. In fact, we prove a slightly stronger result; see Section 3.3.
+
+**Theorem B** (The Weierstrass method has escaping points). For every polynomial $p$ of degree $d \ge 3$ with only simple roots, there are vectors in $\mathbb{C}^d$ whose orbits under $W_p$ tend to infinity. The set of escaping points contains a holomorphic curve.
+
+We have subsequently established the existence of similar escaping orbits also for the Ehrlich–Aberth-method, as well as for the Weierstrass method in which the components of the approximation vectors in $\mathbb{C}^d$ are updated immediately upon computation (Gauss-Seidel update scheme); see [Rei20]. In the present paper, we consider simultaneous updates of all components (Jacobi update scheme).
+
+It might be interesting to observe that this problem does not exist for Newton’s method: here, $\infty$ is a “repelling fixed point”, and all points sufficiently close to $\infty$ will always iterate closer toward the roots. For degenerate polynomials like $Z \mapsto Z^d$, all Newton orbits converge to the single root, while Weierstrass has escaping orbits even for $Z \mapsto Z^3$ (see Remark 3.11).
+
+We cannot resist stating an analogy to the dynamics of transcendental entire functions in one complex variable: all such functions have escaping points (points that converge to $\infty$ under iteration); see [Erë89]. Already Fatou observed that in many cases, the set of escaping points contains curves to $\infty$; in the 1980’s Eremenko raised the conjecture that all escaping points were on such curves to $\infty$. This conjecture was established for many classes of entire functions, and disproved in general, in [RRRS11]. It is plausible that the set of escaping points for $W_p$ has the following property: every escaping point can be joined with $\infty$ by a curve consisting of escaping points.
+
+There is a substantial body of literature on root finding in general, and on background on our methods in particular. In particular, there is an excellent survey by Pan [Pan97] about various known methods and their properties, with a recent update [Pan21]; let us also mention the surveys by McNamee [McN02, McN07], as well as the references in all these papers.
+
+## Structure of this paper
+
+In Section 2, we describe some background on Newton's method and its properties, in order to describe analogies and to build up some intuition. Basic properties of the Weierstrass method are then described in Section 3. In particular, we discuss escaping points for the Weierstrass method, starting with the simple polynomial $Z \mapsto Z^3$, and give a proof of Theorem B.
+
+In Section 4, we describe some algebraic properties of the Weierstrass method and its periodic points. In the final Section 5 we focus on the case of cubic polynomials, giving an explicit description of periodic points of low periods; in particular we give a proof of Theorem A.
\ No newline at end of file
diff --git a/samples/texts/6573214/page_4.md b/samples/texts/6573214/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..bfaf4ba4f98bd97bde7b33c25b4540ece0315f21
--- /dev/null
+++ b/samples/texts/6573214/page_4.md
@@ -0,0 +1,25 @@
+## Notation and conventions
+
+All our polynomials will be univariate and over the complex numbers, so we have polynomials $p \in \mathbb{C}[Z]$ (the indeterminate variable will usually be called $Z$). The associated Newton map is denoted $N_p$, the Weierstrass map $W_p$. In general, we denote the $n$-th iterate of a map $F$ by $F^{\circ n}$. When we want to highlight that a point $z \in \mathbb{C}^d$ is a vector, we write $\tilde{z}$ for $(z_1, \dots, z_d)$. The Jacobi matrix of a map $F$ at a point $\tilde{z}$ is denoted $\mathbf{D}(F)|_{\tilde{z}}$.
+
+A polynomial $p \in \mathbb{C}[Z]$ is *monic* if its leading coefficient equals 1; that means, if the roots of $p$ are $\alpha_1, \dots, \alpha_d$, that $p(Z) = \prod_j (Z - \alpha_j)$. It turns out that both for Newton and for Weierstrass, it is sufficient to consider monic polynomials.
+
+## Acknowledgments
+
+We gratefully acknowledge support by the European Research Council in the form of the ERC Advanced Grant HOLOGRAM.
+
+This research was inspired by discussions with Dario Bini and Victor Pan, and it completes work initiated jointly with Steffen Maass. Over the years, we also had useful discussions with various other colleagues including Erik Bedford, Xavier Buff, Lukas Geyer, John Hubbard, Sarah Koch, and Scott Sutherland. We are grateful for the discussions we had with them, as well as with other members of our research team.
+
+The algebraic computations described in this paper were done using Magma [BCP97] and Singular [DGPS19]; further numerical computations were done using HomotopyContinuation.jl [BT18].
+
+## 2 Newton's method and its properties
+
+Even though the main results in this paper are about the Weierstrass method, we provide a review of the Newton method in order to build up intuition and explain analogies, especially since some of these analogies were guiding us in our research. Interestingly, much more is known about the global dynamics of Newton's method than about the Weierstrass method.
+
+Newton's method is perhaps the most classical root finding method. One of its virtues is its simplicity: to find roots of a monic polynomial $p(Z) = \prod_j (Z - \alpha_j)$, update any approximation $z \in \mathbb{C}$ to a root by
+
+$$ (2.1) \qquad N_p(z) = z - \frac{p(z)}{p'(z)} = z - \left( \sum_j \frac{1}{z - \alpha_j} \right)^{-1} $$
+
+and hope that the new number is a better approximation to some root, at least after a few more iterations. Of course, as long as the roots of $p$ are not known, it is the expression in the middle of (2.1) that is used to evaluate the Newton iteration. The right hand side involving the roots cannot be computed, but it may be helpful in analyzing the properties of the Newton map. Since only the expression $p/p'$ enters into the Newton formula, there is no loss of generality in considering only monic polynomials.
+
+An important property of Newton's method is its compatibility with affine transformations. We denote the space of all monic polynomials of degree $d$ with complex coefficients by $\mathcal{P}'_d$; this is an affine space of dimension $d$. It can be identified with $\mathbb{C}^d$ by taking the coefficients of $Z^k$ for $k = 0, 1, \dots, d-1$ as coordinates. Alternatively, it can be seen as
\ No newline at end of file
diff --git a/samples/texts/6573214/page_5.md b/samples/texts/6573214/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..7689d2769c5c0ee22af53615425d1e32ccfaa0a2
--- /dev/null
+++ b/samples/texts/6573214/page_5.md
@@ -0,0 +1,25 @@
+the quotient $S_d \backslash \mathbb{C}^d$, where $\mathbb{C}^d$ parameterizes the $d$ roots and the symmetric group $S_d$ acts by permutation of the coordinates on $\mathbb{C}^d$. The group $\mathrm{Aff}(\mathbb{C})$ of affine transformations of $\mathbb{C}$ acts on $\mathcal{P}'_d$ via its action on the roots of the polynomials.
+
+**Lemma 2.1** (Newton’s method and affine transformations). If $p$ is a polynomial and $T: \mathbb{C} \to \mathbb{C}$, $z \mapsto \alpha z + \beta$, is an affine transformation, then
+
+$$N_{Tp} = T \circ N_p \circ T^{-1};$$
+
+i.e., the Newton dynamics for $p$ and $Tp$ are affinely conjugate via $T$.
+
+*Proof*. The defining equation (2.1) can be written as
+
+$$\frac{1}{z - N_p(z)} = \sum_j \frac{1}{z - \alpha_j},$$
+
+where the $\alpha_j$ are the roots of $p$. From this, the claim is obvious. $\square$
+
+The lemma above shows that the dynamics of $N_p$ is conjugate (and therefore essentially unchanged) if we replace $p$ by another polynomial in its orbit under $\mathrm{Aff}(\mathbb{C})$. So the true parameter space $\mathcal{P}_d$, i.e., the space of polynomial Newton maps up to affine conjugation, is the quotient of $\mathcal{P}'_d$ by the action of $\mathrm{Aff}(\mathbb{C})$. This quotient is not a nice space: the polynomials with a $d$-fold root have a one-dimensional stabilizer under $\mathrm{Aff}(\mathbb{C})$, whereas for all other polynomials, the stabilizer is finite. This implies that the closure of any point in $\mathcal{P}_d$ contains the point $\bullet$ representing the polynomials with $d$-fold roots. Removing this point, however, results in a reasonable space, which has complex dimension $\dim \mathcal{P}'_d - \dim \mathrm{Aff}(\mathbb{C}) = d-2$.
+
+There are two fairly natural ways to construct this space. We can use the action of $\mathrm{Aff}(\mathbb{C})$ to move two of the roots to 0 and 1. The remaining roots form a $(d-2)$-tuple of complex numbers specifying the polynomial. This representation is not unique, since we can re-order the roots (and then normalize the first two roots again). This gives an action of the symmetric group $S_d$, and we obtain $\mathcal{P}_d \setminus \{\bullet\} = S_d \backslash \mathbb{C}^{d-2}$. We can also use the translations in $\mathrm{Aff}(\mathbb{C})$ to make the polynomial *centered*. i.e., such that the sum of the roots is zero; equivalently, the coefficient of $Z^{d-1}$ vanishes. The set of such polynomials can be identified with $\mathbb{C}^{d-1}$. This leaves the action of $\mathbb{C}^\times$ by scaling the roots, which has the effect of scaling the coefficient of $Z^k$ by $\lambda^{d-k}$ (for $k=0, \dots, d-2$). Leaving out the origin of $\mathbb{C}^{d-1}$ (it corresponds to the “bad” polynomials), we obtain $\mathcal{P}_d \setminus \{\bullet\}$ as the quotient of $\mathbb{C}^{d-1} \setminus \{0\}$ by this $\mathbb{C}^\times$-action. The resulting space is a weighted projective space of dimension $d-2$ with weights $(2, 3, \dots, d)$.
+
+We now fix a period length $n$. Then the space
+
+$$\mathcal{P}_d(n) = \{(p, q) \in \mathcal{P}_d \times \mathbb{C} : q \text{ has period } n \text{ under } N_p\}$$
+
+is a finite-degree cover of $\mathcal{P}_d$; it particular, it also has dimension $d-2$. On $\mathcal{P}_d(n)$ we have the holomorphic map $\mu_{d,n}: (p,q) \mapsto (N_p^{\alpha_n})'(q)$ associating to each point $q$ of period $n$ its multiplier. It is a standard fact that the cycle consisting of $q$ and its iterates is *attracting* (i.e., there is an open neighborhood $U$ of $q$ such that for all $z \in U$, the sequence $(N_p^{\alpha_m}(z))_{m\ge 0}$ converges to $q$) if and only if $|\mu_{d,n}(p,q)| < 1$.
+
+A great virtue of Newton's method is its fast local convergence: close to a simple root, the convergence is quadratic, so the number of valid digits doubles in every iteration step. Therefore, Newton is often employed for "polishing" approximate roots (once the roots have been separated from each other). Yet another virtue is that it can be applied in a
\ No newline at end of file
diff --git a/samples/texts/6573214/page_6.md b/samples/texts/6573214/page_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..f72ef17d5beeb4c4b303090b1a81bff0f323110e
--- /dev/null
+++ b/samples/texts/6573214/page_6.md
@@ -0,0 +1,19 @@
+great variety of contexts, in many dimensions as well as for maps that are smooth but not analytic.
+
+However, Newton's method is not an algorithm but a heuristic: it is a formula that suggests a hopefully better approximation to any given initial point $z$. This formula says little about the properties of the global dynamics, which is an iterated rational map. As such, it has a Julia set with “chaotic” dynamics, and which may well have positive (planar Lebesgue) measure. Worse yet, Newton's method can have open sets of starting points that fail to converge to any root, but instead converge to periodic points of period 2 or higher. Therefore Newton's method fails to be generally convergent. The problems occur even in the simplest possible case: for the cubic polynomial $p(Z) = Z^3 - 2Z + 2$, the Newton method has an attracting 2-cycle, as illustrated in Figure 1. Steven Smale had observed this phenomenon, and he asked for a classification of such polynomials [Sma85, Problem 6 on p. 98]. Partially in response to this question, a complete classification of all (postcritically finite) Newton maps of arbitrary degrees was developed in [LMS15]; in particular, it implies the following result.
+
+**Proposition 2.2** (Polynomials with attracting periodic orbits). For every degree $d \ge 2$, the Newton map of a degree $d$ polynomial can have up to $d-2$ attracting periodic orbits that are not fixed points, and the periods can independently be arbitrary numbers $2$ or greater. This bound is sharp.
+
+This is a rather weak corollary of the general classification result of postcritically finite Newton maps, in which the dynamics can be prescribed with far greater precision. Here we give a heuristic explanation.
+
+The upper bound comes from a well-known fact in holomorphic dynamics. The Newton map $N_p$ of a polynomial $p$ with $d$ distinct roots (of possibly higher multiplicity) is a rational map of degree $d$, and as such it has $2d-2$ critical points. Each of the roots of $p$ is an attracting fixed point and must attract (at least) one of these critical points, so up to $d-2$ “free” critical points remain. Each attracting cycle of period at least 2 must attract one of these critical points; thus the bound.
+
+For the lower bound, to establish that up to $d-2$ cycles of period at least 2 can be made attracting, the fundamental observation is that the multipliers of these cycles form a map from $(d-2)$-dimensional parameter space to a $(d-2)$-dimensional space of multipliers, so under conditions of genericity one expects this map to have dense image. This will be not so for Weierstrass; see Section 3.
+
+Newton's method for polynomials of degree 1 is trivial: the Newton map is the constant map with value at the root. For degree 2, the dynamics is very simple as well; we note this here for later use.
+
+**Lemma 2.3** (Newton’s method for quadratic polynomials). *If* $p$ is a polynomial of degree 2 with distinct roots, then $N_p$ is conformally conjugate to the squaring map $z \mapsto z^2$ on the Riemann sphere. In particular, $N_p$ has periodic orbits of each exact period at least 2, none of which are attracting.
+
+*Proof*. By Lemma 2.1, we can take $p(Z) = Z^2 - 1$. Then
+
+$$N_p(z) = \frac{z^2 + 1}{2z} = T^{-1}(T(z)^2) \quad \text{with} \quad T(z) = \frac{z+1}{z-1}.$$
\ No newline at end of file
diff --git a/samples/texts/6573214/page_7.md b/samples/texts/6573214/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..14a9f15adc0b2272327ba191d654df2f4e68e28c
--- /dev/null
+++ b/samples/texts/6573214/page_7.md
@@ -0,0 +1,13 @@
+FIGURE 1. For every degree $d \ge 3$ and every period $m \ge 2$ there is a polynomial $p$ of degree $d$ so that $N_p$ has a periodic point of period $m$ that attracts a neighborhood of each of its points. **Left:** The Newton dynamics plane for $p(Z) = Z^3 - 2Z + 2$, where $N_p(z) = z - \frac{z^3 - 2z + 2}{3z^2 - 2} = \frac{2z^3 - 2}{3z^2 - 2}$ has an attracting 2-cycle $0 \mapsto 1 \mapsto 0$; its basin is shown in black. **Right:** Detail near center.
+
+Now fix $n$ and let $\omega$ be a primitive $(2^n - 1)$-th root of unity. Then $\omega$ has exact order $n$ under the squaring map, so $T^{-1}(\omega)$ has exact order $n$ under $N_p$. The multiplier of $\omega$ as a point of order $n$ is $2^n$, and this is the same as the multiplier of $T^{-1}(\omega)$ under $N_p$. $\square$
+
+For completeness, we might note that the Newton map for a quadratic polynomial with a double root is conformally conjugate to $z \mapsto z/2$.
+
+## Positive results about Newton's method
+
+Meanwhile, there is a substantial body of knowledge about the global dynamics of Newton's method, in stark contrast to the Weierstrass method. Here we mention some of the relevant results.
+
+For the Newton dynamics $N_p$, any particular orbit may or may not converge to a root. However, one can estimate that asymptotically at least a fraction of $1/(2\log 2) \approx 0.72$ of randomly chosen points in $\mathbb{C}$ will converge to some root (see [HSS01, Section 4]). More explicitly, for every degree $d$ there is a universal set $S_d$ of starting points that will find, for every polynomial $p$ of degree $d$, normalized so that all roots are in the unit disk, all the roots of $p$ under iteration of $N_p$. This set is universal in the sense that it depends only on $d$, and it may have cardinality as low as $1.1d(\log d)^2$ [HSS01]. If one accepts probabilistic results, then $cd(\log \log d)^2$ starting points are sufficient to find all roots with given probability, where $c$ depends only on this probability [BLS13]. Upper bounds on the complexity of Newton's method to find all roots with prescribed precision $\epsilon$ were established in [Sch16, BAS16]; they can be as good as $O(d^2(\log d)^4 + d \log |\log \epsilon|)$, which is close to optimal when the starting points are outside of a disk containing the roots.
+
+In addition to these strong theoretical results, Newton's method has also been used successfully in practice for finding all roots of polynomials of degrees exceeding $10^9$ [SS17, RSS17], and it is interesting to compare the experimental complexity between the Newton and Ehrlich-Aberth methods; see [SCR$^{+}$20]: depending on the efficiency how the
\ No newline at end of file
diff --git a/samples/texts/6573214/page_8.md b/samples/texts/6573214/page_8.md
new file mode 100644
index 0000000000000000000000000000000000000000..5788f3295f95d98ab22ecd456b4f0e958b748f9c
--- /dev/null
+++ b/samples/texts/6573214/page_8.md
@@ -0,0 +1,26 @@
+polynomials can be evaluated, and how the roots are located, one or the other method
+may be faster.
+
+Finally, we might mention that there are several other complex one-dimensional root finding iteration methods, including König's method; see [BH03]. However, there is a theorem by McMullen [McM87] that no one-dimensional root finding method can be generally convergent. It is natural to ask whether a similar result holds also for root finding methods in several variables.
+
+### 3 The Weierstrass method
+
+The *Weierstrass root finding method*, also known as the Durand–Kerner method, tries to approximate all *d* roots of a degree *d* polynomial simultaneously (unlike the Newton method, which approximates only one root at a time). Recall that $P_d'$ is the space of monic polynomials of degree *d*. Let $p \in P_d'$. Then the Weierstrass root finding method consists of iterating the (partially defined) map $W_p: \mathbb{C}^d \to \mathbb{C}^d$, $\underline{z} \mapsto \underline{z}'$, where the components $z_k'$ of $\underline{z}'$ are given in terms of those of $\underline{z}$ by
+
+$$ (3.1) \qquad z_k' = z_k - \frac{p(z_k)}{\prod_{j \neq k} (z_k - z_j)}. $$
+
+This map is defined for all $\underline{z} \in \mathbb{C}^d \setminus \Delta$, where $\Delta$ is the “big diagonal”
+
+$$ \Delta = \{\underline{z} \in \mathbb{C}^d : z_j = z_k \text{ for some } 1 \le j < k \le d\}. $$
+
+If *p* is not necessarily monic, then $W_p$ is defined to be the same as $W_{p/c}$, where *c* is the leading coefficient of *p*. It is therefore sufficient to consider only monic polynomials.
+
+The Weierstrass method converges on a non-empty open subset of $\mathbb{C}^d$ to a vector containing the $d$ roots in some order. It is well known that iteration of $W_p$ may land on $\Delta$ after any number of steps even when the starting point $\underline{z}$ is not in $\Delta$. Moreover, even when an orbit is defined forever it may fail to converge to roots: for instance, when a polynomial is real but its roots are not, then a vector of purely real initial points cannot converge to non-real solutions; similar arguments apply in the presence of other symmetries. More generally, different vectors of starting points may converge to the roots in different order, and the respective domains of convergence in $\mathbb{C}^d$ must have non-empty boundaries on which convergence cannot occur. The best possible outcome to hope for would be that convergence to roots occurs on an open dense subset of $\mathbb{C}^d$, ideally with complement of measure zero.
+
+Obviously, if $z_k$ is already a root, then the map has a fixed point in the $k$-th coordinate; all roots already found stabilize in the approximation vector (as long as they are all distinct).
+
+One heuristic interpretation of the Weierstrass method is as follows. Each of the $d$ component variables “thinks” that all other roots have already been found and tries to find its own value necessary to match the value of the polynomial at a single point. To make this precise, write again $p(Z) = \prod_k (Z - \alpha_k)$. Take a coordinate $k \in \{1, \dots, d\}$; if we assume that $z_j = \alpha_j$ for all $j \neq k$, then
+
+$$ (3.2) \qquad p(z_k) = (z_k - \alpha_k) \prod_{j \neq k} (z_k - z_j), $$
+
+and then the method simply “finds” the missing root $\alpha_k$ as the only unknown quantity in (3.2) to make the equation fit. This leads to the Weierstrass iteration formula (3.1).
\ No newline at end of file
diff --git a/samples/texts/6573214/page_9.md b/samples/texts/6573214/page_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..89ba28486f89688318683d541a4c2c3152738223
--- /dev/null
+++ b/samples/texts/6573214/page_9.md
@@ -0,0 +1,48 @@
+For the Weierstrass method, all $k$ variables make the same “assumption” and in general
+they are all wrong, but it turns out anyway that this leads to a reasonable approximation
+of the root vectors, at least sufficiently close to a true solution.
+
+We will now show that the Weierstrass method can be interpreted as a higher-dimensional
+Newton iteration. Consider the map
+
+$$F: \mathbb{C}^d \to \mathcal{P}'_d, \quad (z_1, \dots, z_d) \mapsto \prod_{k=1}^d (Z - z_k).$$
+
+Then the task of finding all the roots of $p$ is equivalent to finding some preimage of $p$
+under $F$. To solve this problem, we can employ Newton's method in $d$ dimensions. This
+leads to the iteration
+
+$$ (3.3) \qquad \underline{z} \mapsto \underline{z} - (\mathbf{D}(F)|_{\underline{z}})^{-1} (F(\underline{z}) - p), $$
+
+which is defined on the set of $\underline{z} \in \mathbb{C}^d$ where $\mathbf{D}(F)|_{\underline{z}}$ is invertible, which is the case if and
+only if $\underline{z} \notin \Delta$. (The "if" direction follows from the proof below; the "only if" direction is
+easy.)
+
+**Lemma 3.1** (Weierstrass method as higher-dimensional Newton). *The map given by (3.3) is* $W_p$.
+
+A particular reference for this is [Ker66], where the Weierstrass method is derived as a
+higher-dimensional Newton method.
+
+*Proof*. First note that the partial derivative of $F$ with respect to the $k$-th coordinate $z_k$
+is
+
+$$ \frac{\partial F}{\partial z_k}(\underline{z}) = - \prod_{j \neq k} (Z - z_j), $$
+
+where the expression on the right is a polynomial of degree less than $d$; we identify the
+space of such polynomials with $\mathbb{C}^d$. If we denote the right hand side of (3.3) by $\underline{z}'$, we
+can write (3.3) in the form
+
+$$ (3.4) \qquad \mathbf{D}(F)|_{\underline{z}}(\underline{z}' - \underline{z}) = p - F(\underline{z}). $$
+
+Written out, this gives
+
+$$ (3.5) \qquad \sum_{k=1}^{d} (z'_{k} - z_{k}) \prod_{j \neq k} (Z - z_{j}) = \prod_{k=1}^{d} (Z - z_{k}) - p. $$
+
+If we assume that the entries of $\underline{z}$ are distinct and, separately for each $m \in \{1, \dots, d\}$,
+we set $Z \leftarrow z_m$, the product on the right and most products on the left vanish and the
+remaining equation gives (3.1) (with $m$ in place of $k$). $\square$
+
+The following local convergence result is well known, see e.g. [Wei91, Doč62].
+
+**Lemma 3.2** (Local convergence of the Weierstrass method). For a polynomial $p$ with distinct roots, every vector consisting of the $d$ roots of $p$ has a neighborhood in $\mathbb{C}^d$ on which the Weierstrass method converges quadratically to this solution vector.
+
+*Proof.* This follows from the fact that $W_p$ is Newton's method applied to $F(\underline{z}) - p$. $\square$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_16.md b/samples/texts/7149586/page_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..c4dcbe7020084ee0b66d666c0b00975a7c509d33
--- /dev/null
+++ b/samples/texts/7149586/page_16.md
@@ -0,0 +1,42 @@
+algorithm,⁴⁶ and is available in the modeling system GAMS.⁵⁰ It should be noted that
+this code has a heuristic termination criterion for nonconvex problems. The code
+α-ECP implements the extended cutting plane method by Westerlund and
+Pettersson.⁴⁹ Codes that implement the branch and bound method include the code
+MINLP_BB⁴⁴ available in AMPL, and the program SBB which is also available in
+GAMS. Recently, the open source MINLP solver Bonmin,⁵¹ which is part of the
+COIN-OR project,⁵² implements an extension of the branch-and-cut
+outer-approximation algorithm that was proposed by Quesada and Grossmann,⁴⁸ as
+well as the branch and bound and outer-approximation method.
+
+**5.2. MINLP Reformulation**
+
+In order to improve the computational efficiency of solving the MINLP model
+(P1) with the above cited solvers, we present in this section a reformulation of (P1).
+
+The square root term in the objective function of (P1) can give rise to difficulties
+in the optimization procedure. When the DC in location *j* is not selected, both
+square root terms would take a value of zero, which leads to unbounded gradients in
+the NLP optimization and hence numerical difficulties. Thus, we reformulate the
+model in order to eliminate the square root terms. We first introduce two sets of
+non-negative continuous variables, Z1<sub>j</sub> and Z2<sub>j</sub>, to represent the square-root terms
+in the objective function:
+
+$$
+Z1_j^2 = \sum_{i \in I} \mu_i Y_{ij}, \quad \forall j \in J \tag{15}
+$$
+
+$$
+Z 2_j^2 = \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}, \quad \forall j \in J \qquad (16)
+$$
+
+$$
+Z1_j \ge 0, Z2_j \ge 0, \quad \forall j \in J \tag{17}
+$$
+
+Because the non-negative variables Z1_j and Z2_j are introduced in the
+objective function with positive coefficients, and this problem is a minimization
+problem, (15) and (16) can be further relaxed as the following inequalities,
+
+$$
+-Z1_j^2 + \sum_{i \in I} \mu_i Y_{ij} \le 0, \quad \forall j \in J \qquad (18)
+$$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_17.md b/samples/texts/7149586/page_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..d65b6df584f91ddfa3f6a0a8435c55a8a8fabc91
--- /dev/null
+++ b/samples/texts/7149586/page_17.md
@@ -0,0 +1,51 @@
+$$
+-Z 2_j^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij} \le 0, \quad \forall j \in J \tag{19}
+$$
+
+Thus, the reformulated model can then be expressed as the following MINLP problem denoted as (P2),
+
+$$
+\textbf{(P2)} \quad \text{Min:} \quad \sum_{j \in J} \left( f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j Z 1_j + q Z 2_j \right) \qquad (20)
+$$
+
+$$
+\text{s.t.} \quad \sum_{j \in J} Y_{ij} = 1, \quad \forall i \in I. \tag{8}
+$$
+
+$$
+Y_{ij} \le X_j, \quad \forall i \in I, \forall j \in J. \tag{9}
+$$
+
+$$
+-Z1_j^2 + \sum_{i \in I} \mu_i Y_{ij} \le 0, \quad \forall j \in J \tag{18}
+$$
+
+$$
+-Z2_j^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij} \le 0, \quad \forall j \in J \qquad (19)
+$$
+
+$$
+X_j \in \{0,1\}, \quad \forall j \in J
+\quad (10)
+$$
+
+$$
+Y_{ij} \ge 0, \quad \forall i \in I, \forall j \in J. \tag{14}
+$$
+
+$$
+Z1_j \ge 0, Z2_j \ge 0, \quad \forall j \in J
+\quad (17)
+$$
+
+(P2) can be trivially shown to be equivalent to (P1) but with linear objective function and quadratic terms in the constraints (18) and (19). As shown in Appendix A, the following property can be established for problem (P2).
+
+**Proposition 2.** The global optimal solution of problem (P2), or a local optimal solution with fixed 0-1 value for $X_j$, has all the continuous variables $Y_{ij}$ take on integer value (0 or 1).
+
+5.3. Heuristic Algorithm
+
+Since problem (P2) is a nonconvex problem, the solution is highly dependent of
+the starting point when using an MINLP solver that relies on convexity assumption.
+To obtain a “good” feasible starting point, we first relax the nonconvex nonlinear
+constraints (18) and (19) in (P2) by replacing the concave terms with their
+corresponding secants, which represent the convex envelopes⁵³ of these functions.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_19.md b/samples/texts/7149586/page_19.md
new file mode 100644
index 0000000000000000000000000000000000000000..c5bfe1d811f5ffa45270eb1f33ba6a3e77cf17f9
--- /dev/null
+++ b/samples/texts/7149586/page_19.md
@@ -0,0 +1,21 @@
+**Algorithm 1:** (Heuristic Algorithm)
+
+**Step 1:** Solve the MILP model (P3).
+
+**Step 2:** Use the optimal values of variables $X_j$ and $Y_{ij}$ obtained from Step 1 as the starting point, and solve problem (P2) with an MINLP solver that relies on convexity assumptions (such as DICOPT, SBB, $\alpha$-ECP, MINLP_BB, Bonmin, etc.) for obtaining a near-optimal solution.
+
+Note that if we solve problem (P2) with Algorithm 1 by using an MINLP solver that relies on convexity assumptions, the optimal solution may not be globally optimal. However, the optimal solution still has all the $Y_{ij}$ variables at integer values based on Proposition 1 (see Appendix A for details). Furthermore, the solution obtained by using heuristic Algorithm 1 for problem (P2) is also a feasible solution of problem (P1).
+
+## 5.4. A Lagrangean Relaxation Algorithm
+
+In order to obtain potentially better solutions, we propose a Lagrangean relaxation algorithm for obtaining global optimal or near global optimal solutions of model (P2).
+
+### 5.4.1. The Decomposition Procedure
+
+In the Lagrangean relaxation algorithm, we use a “spatial” decomposition scheme by dualizing the assignment constraints (8) in (P2) using the Lagrangean multipliers $\lambda_i$, which is similar to the works by Beasley²³ and Daskin et al.³³ As a result, we obtain the following relaxed problem (denoted by $\mathbf{P}(\boldsymbol{\lambda}))$,
+
+$$ (\mathbf{P}(\boldsymbol{\lambda})) \quad V = \text{Min:} \quad \sum_{j \in J} \left( f_j X_j + \sum_{i \in I} (\hat{d}_{ij} - \lambda_i) Y_{ij} + K_j Z_1 + qZ_2 \right) + \sum_{i \in I} \lambda_i \quad (23) $$
+
+$$ \text{s.t.} \quad Y_{ij} \le X_j, \quad \forall i \in I, \forall j \in J \quad (9) $$
+
+$$ -Z_1^2 + \sum_{i \in I} \mu_i Y_{ij} \le 0, \quad \forall j \in J \quad (18) $$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_2.md b/samples/texts/7149586/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..528989f0bf81662118f3ebc3629cdfb6afbeca06
--- /dev/null
+++ b/samples/texts/7149586/page_2.md
@@ -0,0 +1,7 @@
+# 1. Introduction
+
+Due to increasing pressure for remaining competitive in the global market place, an emerging challenge for the process industries has become how to manage the inventories at the enterprise level so as to reduce costs and improve the customer service.¹, ² A key challenge to achieve this goal is to integrate inventory management with supply chain network design decisions, so that decisions such as the number of inventory stocking locations and the associated amount of inventory can be determined simultaneously for lower costs and higher customer service level.
+
+Although supply chain network design problems and the inventory management problems have been studied extensively in recent years,³⁻⁸ most of the models consider inventory management and supply chain network design separately. On the other hand, there are related works on supply chain optimization that take into account the inventory costs, but consider inventory issues without detailed inventory management policies. In these models the safety stock level is given as a parameter, and usually treated as a lower bound of the total inventory level,⁹⁻¹¹ or considered as the inventory targets that would lead to some penalty costs if violated.¹²⁻¹⁴ This approach cannot optimize the safety stock levels, especially when considering demand uncertainty.¹⁵⁻¹⁷ Thus, it can only provide an approximation of the inventory cost, and therefore lead to suboptimal solutions. Jung et al.¹⁸ introduce a simulation-optimization framework to estimate the optimal safety stock levels, but the supply chain design decisions are not jointly optimized.
+
+Recently, Shen et al.¹⁹ proposed a joint location-inventory model that integrates supply chain network design model with inventory management under demand uncertainty. In their work, the management of working inventory and safety stock are taken into account besides the distribution center location decisions. To solve the resulting nonlinear integer programming problem, the authors simplified the model by assuming that the uncertain demand in each retailer has the same variance-to-mean ratio. Based on this assumption, they reformulated the model as a set-covering problem and solved it with a branch-and-price algorithm. The proposed algorithm performs well
\ No newline at end of file
diff --git a/samples/texts/7149586/page_21.md b/samples/texts/7149586/page_21.md
new file mode 100644
index 0000000000000000000000000000000000000000..6f124027ece1a6a333409de9de6430988dbe6053
--- /dev/null
+++ b/samples/texts/7149586/page_21.md
@@ -0,0 +1,22 @@
+$$
+V = \sum_{j \in J} V_j + \sum_{i \in I} \lambda_i . \tag{25}
+$$
+
+For each fixed value of the Lagrangean multipliers $\lambda_i$, we solve problem (P$_j(\lambda)$)
+by globally minimizing (24) for each candidate DC location j (e.g. using BARON).
+Then, based on (25), the optimal objective function value of problem (P$(\lambda)$) can be
+calculated for each fixed value of $\lambda_i$. Using a standard subgradient method$^{20, 21}$ to
+update the Lagrangean multiplier $\lambda_i$, the algorithm iterates until a preset optimality
+tolerance is reached.
+
+**5.4.2. Lagrangean Relaxation Subproblems**
+
+In each iteration with fixed values of the Lagrange multipliers $\lambda_i$, the design variables ($X_j$) are optimized separately in each subproblem ($\mathbf{P}_j(\boldsymbol{\lambda})$) in the aforementioned decomposition procedure. For each subproblem ($\mathbf{P}_j(\boldsymbol{\lambda})$), we can observe that the objective function value of ($\mathbf{P}_j(\boldsymbol{\lambda})$) is 0 if and only if $X_j = 0$ (i.e. we do not select DC $j$). In other words, there is a feasible solution that leads to the objective function value of subproblem ($\mathbf{P}_j(\boldsymbol{\lambda})$) equal to 0. Therefore, the globally minimum objective function value of subproblem ($\mathbf{P}_j(\boldsymbol{\lambda})$) should be less than or equal to zero. Given this observation, it is possible that under some value of $\lambda_i$ (such as $\lambda_i = 0$, $i \in I$) the optimal objective function values for all the subproblem ($\mathbf{P}_j(\boldsymbol{\lambda})$) are 0 (i.e., $X_j = 0$, $j \in J$, we do not select any DC). However, the original assignment constraint (8) implies a redundant constraint that at least one DC should be selected to meet the demands, i.e.
+
+$$
+\sum_{j \in J} X_j \geq 1. \tag{26}
+$$
+
+Once constraint (8) is relaxed, constraint (26) becomes “not redundant” and should be
+taken into account in the algorithm.²³,²⁶ To satisfy the constraint (26) in the Lagrangean
+relaxation procedure, we make the following modifications to the aforementioned step
\ No newline at end of file
diff --git a/samples/texts/7149586/page_23.md b/samples/texts/7149586/page_23.md
new file mode 100644
index 0000000000000000000000000000000000000000..1d6a0f432cfca33898a5b4357ce2fd1b86d3675b
--- /dev/null
+++ b/samples/texts/7149586/page_23.md
@@ -0,0 +1,28 @@
+objective function value of problem (PR<sub>j</sub>(λ)), which is a negative value. Therefore, it
+is optimal to have X<sub>j</sub> = 1 under this value of the Lagrange multiplier.
+
+On the other hand, if the minimum objective function value of problem (PR<sub>j</sub>(λ))
+is positive, it means that when X<sub>j</sub> = 1, the optimal objective function value of problem
+(P<sub>j</sub>(λ)) could not be negative. Thus the optimal objective function value of problem
+(P<sub>j</sub>(λ)) would be 0 when X<sub>j</sub> = 0 (because if X<sub>j</sub> = 1 the minimum objective function
+value would be positive, as given in the objective function value of problem (PR<sub>j</sub>(λ))).
+
+A possible extreme case is that the minimum objective function value of all the
+Lagrangean subproblems (PR<sub>j</sub>(λ)) are positive (for example, λ<sub>i</sub> = 0, i ∈ I). In this
+case, it means that the globally minimum objective function value of problem (P<sub>j</sub>(λ))
+are all 0, i.e. we do not select any DC. To satisfy the implied constraint (26) that we
+need to select at least one DC, we just install the DC j with smallest objective
+function value, though this value is positive. By using the relationship between problem
+(PR<sub>j</sub>(λ)) and (P<sub>j</sub>(λ)), we can solve (PR<sub>j</sub>(λ)) instead of (P<sub>j</sub>(λ)) for equivalent
+optimality.
+
+Therefore, the algorithm for solving the Lagrangean relaxation subproblems is as
+follows. For each fixed value of λᵢ, we solve PRⱼ(λ) for every candidate DC location
+j. Then select the DCs in candidate location j (i.e. let Xⱼ = 1), for which V̂ⱼ ≤ 0.
+For all the remaining DCs for which V̂ⱼ > 0, we do not select them and set Xⱼ = 0. On
+the other hand, if all the V̂ⱼ > 0, ∀j ∈ J, we select only one DC with the minimum V̂ⱼ,
+i.e. Xⱼ* = 1 for the j* such that V̂ⱼ* = minⱼ{Vⱼ}.
+
+By doing this at each iteration of the Lagrangean relaxation (for each value of the
+multiplier λᵢ), we ensure that the optimal solution always satisfies Σⱼ Xⱼ ≥ 1. Thus the
+globally optimal objective function of (P(λ)) can be recalculated as:
\ No newline at end of file
diff --git a/samples/texts/7149586/page_24.md b/samples/texts/7149586/page_24.md
new file mode 100644
index 0000000000000000000000000000000000000000..736136cfe155acc42c2f24634638e02083d82854
--- /dev/null
+++ b/samples/texts/7149586/page_24.md
@@ -0,0 +1,21 @@
+$$V = \sum_{j \in J} \hat{V}_j + \sum_{i \in I} \lambda_i . \quad (28)$$
+
+**5.4.3. Obtaining Feasible Solutions**
+
+As the original model (P2) has very few constraints, there are several methods to obtain a feasible solution for the problem.
+
+The initial feasible solution can be obtained with the following two methods:
+
+The first method is to select a DC, and then assign all the retailers to this DC, i.e. pick up a $j^* \in J$, and let $X_{j^*} = 1$, $Y_{ij^*} = 1, \forall i \in I$. This method provides a simple way for obtaining a feasible solution and the resulting objective function value provides a valid upper bound of the global optimal objective function value.
+
+The second method is to solve the problem (P2) with an MINLP solver, possibly using Algorithm 1 to obtain a near optimal solution, which is also a feasible solution of the original problem. This usually provides a “tighter” upper bound than the one obtained with the first method.
+
+To obtain a feasible solution during the iterations, we first fix the values of the design variables ($X_j$) at the optimal values of the Lagrangean relaxation subproblems, and then solve the original model (P2) with a nonlinear programming (NLP) solver (not necessarily a global solver). The optimal values of the assignment variables ($Y_{ij}$) in the Lagrangean relaxation subproblems are used as the initial values in the nonlinear optimization procedure. Nonlinear solvers such as MINOS, CONOPT, SNOPT, KNITRO, IPOPT can be used in this step.
+
+Note that solving (P2) with fixed values of $X_j$ variables using a local or global NLP solver guarantees that the feasible solutions generated in this step have all the $Y_{ij}$ variables at integer values based on Proposition 2 (see Appendix A for details).
+
+**5.4.4. The Solution Algorithm**
+
+To summarize, the solution algorithm is as follows:
+
+**Algorithm 2: (Lagrangean Relaxation Algorithm)**
\ No newline at end of file
diff --git a/samples/texts/7149586/page_25.md b/samples/texts/7149586/page_25.md
new file mode 100644
index 0000000000000000000000000000000000000000..1335efbabdf82e782219df251ed07556edd884be
--- /dev/null
+++ b/samples/texts/7149586/page_25.md
@@ -0,0 +1,21 @@
+**Step 1:** (Initialization) for the initial value of the multiplier $\lambda^1$ of constraint (8) use an arbitrary guess, or the multiplier values corresponding to a local optimum of the NLP relaxation of model (P2). Let the incumbent upper bound be $UB = +\infty$, lower bound be $LB = -\infty$ and iteration number be $t=1$. Set the step length parameter $\theta = 2$.
+
+**Step 2:** Solve the modified Lagrangean relaxation program ($PR_j(\lambda^t)$) with fixed Lagrangean multiplier vector $\lambda^t$ for all the $j$ using a global optimization solver (e.g. BARON). Denote the optimal objective function value as $\hat{V}_j(\lambda^t)$ and the optimal solutions as $\hat{Y}_{ij}(\lambda^t)$.
+
+If $\hat{V}_j(\lambda^t) > 0, \forall j \in J$, let $X_{j*}(\lambda^t) = 1$ for the $j^*$ such that $\hat{V}_{j*}(\lambda^t) = \min_{j \in J}\{\hat{V}_j(\lambda^t)\}$.
+Else, let $X_j(\lambda^t) = 1$ for all $j$ with $\hat{V}_j(\lambda^t) \le 0$, and $X_j(\lambda^t) = 0$ for all $j$ such that
+$\hat{V}_j(\lambda^t) > 0$. Calculate $V(\lambda^t) = \sum_{j \in J, X_j(\lambda^t)=1} \hat{V}_j + \sum_{i \in I} \lambda_i^t$.
+
+If $V(\lambda^t) > LB$, update the lower bound by setting $LB = V(\lambda^t)$.
+
+If more than 2 iterations of the subgradient procedure²⁰ are performed without an increment of $LB$, then halve the step length parameter by setting $\theta = \frac{\theta}{2}$.
+
+**Step 3:** Fixing the design variable values as $X_j = X_j(\lambda^t)$ and using $\hat{Y}_{ij}(\lambda^t)$ as the initial values of the assignment variables $Y_{ij}$, solve problem ($P(\lambda^t)$) in the reduced space with fixed $\lambda^t$ and $X_j(\lambda^t)$ using an NLP solver (local or global). Denote the optimal solution as $Y_{ij}(\lambda^t)$ and the optimal objective function value as $\bar{V}(\lambda^t)$.
+
+If $\bar{V}(\lambda^t) < UB$, update the upper bound by setting $UB = \bar{V}(\lambda^t)$.
+
+**Step 4:** Calculate the subgradient ($G_i$) using
+
+$$ G_i^t = 1 - \sum_{j \in J} \hat{Y}_{ij}(\lambda^t), \quad i \in I \qquad (29) $$
+
+Compute the step size $T$, ²⁰, ²¹
\ No newline at end of file
diff --git a/samples/texts/7149586/page_26.md b/samples/texts/7149586/page_26.md
new file mode 100644
index 0000000000000000000000000000000000000000..edc2bc27aed74bb996e4afee9586f419d36e4e0c
--- /dev/null
+++ b/samples/texts/7149586/page_26.md
@@ -0,0 +1,15 @@
+$$T^t = \frac{\theta \cdot (UB - LB)}{\sum_{i \in I} (G_i^t)^2} \quad (30)$$
+
+Update the multipliers:
+
+$$\lambda^{t+1} = \max\{0, \lambda^t + T^t \cdot G^t\} \quad (31)$$
+
+**Step 5:** If $gap = \frac{UB - LB}{UB} < tol$ (e.g. $10^{-5}$), or $\|\lambda^{t+1} - \lambda^t\|^2 < tol$ (e.g. $10^{-3}$) or the maximum number of iterations has been reached, set $UB$ as the optimal objective function value, and set $X_j(\lambda^t)$ and $Y_{ij}(\lambda^t)$ as the optimal solution.
+
+Else, increment $t$ as $t+1$, go to Step 2.
+
+We should note that the above algorithm is guaranteed to provide rigorous lower bounds in Step 2 since the subproblems are globally optimized. Also, the feasible solution generated in Step 3 has all the $Y_{ij}$ variables at integer values as we mentioned in Section 5.3.3. Thus, the solution obtained by using this algorithm for problem (P2) is also a feasible solution of problem (P1). Due to the duality gap, the above algorithm must be stopped after a finite number of iterations. As will be shown in the computational results, the duality gaps are quite small.
+
+## 6. Illustrative Example
+
+To illustrate the application of this model, we consider a small illustrative example for the supply chain of liquid oxygen (LOX) consisting of one plant, three potential DCs and six customers as given in Figure 5. The tri-echelon “plant-DC-customer” supply chain is similar to the “supplier-DC-retailer” network that we discussed before, and the joint supply chain design and inventory management model can be also used to minimize total network design, transportation and inventory costs.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_27.md b/samples/texts/7149586/page_27.md
new file mode 100644
index 0000000000000000000000000000000000000000..86e697bf2660906c45879e0f2f8336467b6fed51
--- /dev/null
+++ b/samples/texts/7149586/page_27.md
@@ -0,0 +1,20 @@
+Figure 5 LOX supply chain network superstructure for the illustrative example
+
+**6.1. Illustrative Example for Industrial Application**
+
+We first consider an instance to illustrate the application of the joint supply chain
+design and inventory management model. In this instance, the fixed cost to install a DC
+($f_j$) is $10,000/year, and the fixed cost for ordering from supplier ($F_j$) is $100/
+replenishment. The order lead time for all the DCs are 7 days, and we consider 97.5%
+service level, thus the associated service level parameter $z_{\alpha}$ is 1.96. We consider 365
+days in a year, and the annual inventory holding for LOX is $3.65/Liter, i.e. daily
+inventory holding cost is $0.01/Liter. Both weight parameters $\beta$ and $\theta$ are set to 1.
+The remaining data for demand uncertainty and transportation costs are given in Table
+B1, B2 and B3 in Appendix B.
+
+We solve model (P2) directly to obtain the global optimum by using the BARON
+solver with GAMS,⁵⁰ because the problem only includes 3 binary variables, 24
+continuous variables and 30 constraints. The resulting optimal supply chain is given in
+Figure 6. We can see that only two DCs are installed, and they both serve the nearest
+three customers. The optimal replenishment number for DC1 is around 44 times, and
+for DC3 is around 57 times. This means that 108,770 liters of LOX are shipped from the
\ No newline at end of file
diff --git a/samples/texts/7149586/page_28.md b/samples/texts/7149586/page_28.md
new file mode 100644
index 0000000000000000000000000000000000000000..1db7a91229f48a09a9962d5294fa597cb25fea90
--- /dev/null
+++ b/samples/texts/7149586/page_28.md
@@ -0,0 +1,7 @@
+plant to DC1 in 44 shipments, i.e. roughly one shipment every eight days, and 182,865 liters of LOX are shipped from the plant to DC3 with 57 shipments, i.e. roughly one shipment every 6 days. The yearly expected flows of the corresponding transportation links are given in Figure 6. The optimal total cost is $366,624.27/year, which includes $200,000/year for installing DC1 and DC3, $65,320.40/year for the transportation cost from the DCs to customers, $77,444.86/year for the transportation cost from the plant to DCs, $10,163.62/year for fixed ordering cost, $10,301.48/year for the cost of working inventory and $3,393.91/year for the cost of safety stocks in the two installed DCs. The major trade-off for this instance is between DC installation costs, transportation costs and inventory costs.
+
+**Figure 6** Optimal network structure for the LOX supply chain
+
+## 6.2. Illustrative Example for the key trade-offs
+
+To better illustrate the trade-offs in this problem, we consider different weighted parameters for the transportation and inventory cost. All the data for demand uncertainty and transportation costs are the same as the previous example and given in Tables B1-B3 in Appendix B. Other important model coefficients for instances discussed in this section are given in Table B4 in Appendix B. Note that to reveal the
\ No newline at end of file
diff --git a/samples/texts/7149586/page_29.md b/samples/texts/7149586/page_29.md
new file mode 100644
index 0000000000000000000000000000000000000000..7b3155582a51e8dfdac94b99d6d87e886865ea26
--- /dev/null
+++ b/samples/texts/7149586/page_29.md
@@ -0,0 +1,13 @@
+trade-offs and using different weighted parameters, the units of some parameters are removed for scaling purpose.
+
+**Table 1** Comparison result for the illustrative example
+
+<table><thead><tr><td>Transportation cost weight factor (β)</td><td>Inventory cost weight factor (θ)</td><td>Objective Function (Cost)</td><td>No. DCs</td><td>Network structure</td></tr></thead><tbody><tr><td>0.01</td><td>0.01</td><td>2260.26</td><td>2</td><td>Figure 6a</td></tr><tr><td>0.1</td><td>0.01</td><td>8122.93</td><td>3</td><td>Figure 6b</td></tr><tr><td>0.001</td><td>0.01</td><td>1099.25</td><td>1</td><td>Figure 6c</td></tr><tr><td>0.01</td><td>0.1</td><td>5359.18</td><td>1</td><td>Figure 6d</td></tr><tr><td>0.01</td><td>0.001</td><td>1341.04</td><td>3</td><td>Figure 6e</td></tr></tbody></table>
+
+GAMS/BARON is also used to solve model (P2) directly for obtaining global optimal solutions. We first consider a base case with the transportation cost and inventory cost weighted factor as $β = 0.01$, $θ = 0.01$, and then consider different values of weights.
+
+(a) $β = 0.01, θ = 0.01, Cost = 2260.26$ (base case)
+
+(b) $β = 0.1, θ = 0.01, Cost = 8122.93$
+
+(c) $β = 0.001, θ = 0.01, Cost = 1099.25$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_3.md b/samples/texts/7149586/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..af7bc5b27708927f3b94ef65f9c32cf4b402dfc0
--- /dev/null
+++ b/samples/texts/7149586/page_3.md
@@ -0,0 +1,12 @@
+for large scale problems. However, the assumption for identical variance-to-mean ratio
+might not provide a good approximation to real world problems because the demand
+uncertainties for each retailer may vary significantly. Thus, to allow the model to
+accommodate more general cases, an efficient algorithm is needed for the model
+without the simplifications.
+
+Lagrangean relaxation and Lagrangean decomposition methods are recognized as efficient tools for solving large-scale optimization problems with “special” structures. The Lagrangean relaxation and subgradient optimization are discussed by Fisher.²⁰, ²¹ Later, Guignard and Kim²² proposed the well-known Lagrangean decomposition method that yields stronger bounds than the Lagrangean relaxation algorithm. A large number of applications of Lagrangean-based algorithms for supply chain optimization and related problems have been reported in the past. Various Lagrangean-based heuristic algorithms for large scale facility location problems are discussed by Beasley.²³ Based on this work, Holmberg and Ling²⁴ proposed a novel Lagrangean heuristic method for location problems with staircase costs. Sridharan²⁵ implemented the Lagrangean relaxation method for the plant location problem with consideration of capacity issues. A Lagrangean relaxation and decomposition method for multiproduct tri-echelon supply chain design problem is proposed by Pirkul and Jayaraman.²⁶ Later, Klose²⁷ developed a relax-and-cut algorithm for the capacitated facility location problems. The proposed method yielded significant improvements in computational efficiency. van den Heever et al.²⁸ developed a Lagrangean heuristic method for the design and planning of offshore oil fields. A Lagrangean based temporal decomposition algorithm for supply chain planning was proposed by Jackson and Grossmann.¹² Recently, Neiro and Pinto²⁹ applied the Lagrangean based method to a petroleum supply chain planning model. The results showed that significant improvement in computational efficiency can be achieved by using Lagrangean decomposition.
+
+The objective of this work is to develop effective algorithms for large-scale joint
+supply chain network design and inventory management problem for a given product.
+This work relies on the integer nonlinear programming model proposed by Shen et al.¹⁹.
+We first reformulate the model as a mixed-integer nonlinear programming (MINLP)
\ No newline at end of file
diff --git a/samples/texts/7149586/page_30.md b/samples/texts/7149586/page_30.md
new file mode 100644
index 0000000000000000000000000000000000000000..b0af32b40f7e19bbd4285c9f15c4692a1f1b6238
--- /dev/null
+++ b/samples/texts/7149586/page_30.md
@@ -0,0 +1,9 @@
+(d) $\beta = 0.01, \theta = 0.1, \text{Cost} = 5359.18$
+
+(e) $\beta = 0.01, \theta = 0.001, \text{Cost} = 1341.04$
+
+**Figure 7** Optimal LOX supply chain network structure of the illustrative example for different transportation cost and inventory cost weighted parameters
+
+The results for different instances are given in Table 1, and their associated optimal supply chain network structures are given in Figure 7. We can see that in the base case, only two DCs are selected to install and they are connected to three retailers respectively. When we increase the transportation cost factor to $\beta = 0.1$, all the three DCs are installed and each of them serves two retailers (Figure 7b). If we decrease the transportation cost factor to $\beta = 0.001$, only DC 3 is selected to be installed and it serves all the retailers. Thus, the larger the weighted factor for transportation costs $\beta$, the more DCs are installed. On the other hand, when we fix the transportation cost factor $\beta = 0.01$, and consider different values of the inventory cost factor $\theta$, we can similarly find out from Figure 7d and 6e that the larger weighted factor for inventory costs, the fewer DCs are installed.
+
+Based on this analysis, we obtain a similar conclusion as Shen et al.¹⁹ that the more DCs are installed, the more transportation costs are potentially reduced, but less inventory cost are saved. The major reason of this performance is that from an inventory cost aspect, the more retailers are pooled to a DC, the more cost saving can be achieved, but from a transportation cost viewpoint, installing more DC to serve different retailers may reduce the total transportation cost. Thus, the trade-off between inventory and transportation costs is established and reflects on the number of DCs besides the tradeoffs for supply chain design costs and operation costs.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_31.md b/samples/texts/7149586/page_31.md
new file mode 100644
index 0000000000000000000000000000000000000000..01a3dd505720825f3f3a1b8cdd8a8b885b25450d
--- /dev/null
+++ b/samples/texts/7149586/page_31.md
@@ -0,0 +1,11 @@
+## 7. Computational Results
+
+In order to illustrate the applicability of the proposed solution strategies, we carry out computational experiments for instances with 33, 88 and 150 retailer locations with different weight parameters $\beta$ and $\theta$. In all cases, each retailer location is also a candidate DC location, i.e. there are as many candidate DC locations as the retailer locations for each instance. Note that in analogy to the industrial gases supply chain introduced in Section 6, the “retailers” correspond to the “customers”.
+
+The model sizes of instance for $n$ retailer locations (and $n$ candidate DC locations) are given in Table 2. This means that for the largest problem with 150 retailers and 150 candidate distribution centers, the original INLP problem (P0) includes 22,650 binary variables and 22,650 constraints, while the reformulated problem (P2) includes only 150 binary variables, 22,800 continuous variables and 22,950 constraints.
+
+**Table 2 Model statistics of instance for n retailer locations (and n candidate distribution center locations)**
+
+<table><thead><tr><th>Model</th><th>(P0)</th><th>(P1)</th><th>(P2)</th><th>(P3)</th><th>(PR<sub>j</sub>(λ))</th></tr></thead><tbody><tr><td>No. of discrete variables</td><td>n<sup>2</sup> + n</td><td>n</td><td>n</td><td>n</td><td>0</td></tr><tr><td>No. of continuous variables</td><td>0</td><td>n<sup>2</sup></td><td>n<sup>2</sup> + 2n</td><td>n<sup>2</sup> + 2n</td><td>n + 2</td></tr><tr><td>No. of constraints</td><td>n<sup>2</sup> + n</td><td>n<sup>2</sup> + n</td><td>n<sup>2</sup> + 3n</td><td>n<sup>2</sup> + 3n</td><td>2</td></tr></tbody></table>
+
+For every retailer $i$, we set the annual inventory holding cost $h=1$, the service level parameter $z_{\alpha} = 1.96$ (97.5% service level), the order lead time $L = 7$ days, the fixed order cost $F_i = 10$, the unit shipping cost (from supplier to retailer) $a_i = 5$, and the fixed shipping cost (from supplier to retailer) $g_i = 10$. Each retailer location
\ No newline at end of file
diff --git a/samples/texts/7149586/page_32.md b/samples/texts/7149586/page_32.md
new file mode 100644
index 0000000000000000000000000000000000000000..bb16a43ff630da1008d5909984da6054647c0f0b
--- /dev/null
+++ b/samples/texts/7149586/page_32.md
@@ -0,0 +1,7 @@
+represents a city in the U.S., with mean demand $\mu_i$ equal to the city population divided by 2000, based on the data from U.S. Census 2000.⁵⁴ The standard deviation-to-mean ratio of each customer demand $\sigma_i / \mu_i$ is generated uniformly on $U[0, 0.3]$, and the fixed DC installation cost $f_i$ is generated uniformly on $U[90, 110]$.
+
+All the instances are modeled with GAMS⁵⁰ and solved with solvers including DICOPT, SBB, Bonmin and BARON on an Intel 3.2 GHz machine with 512 MB RAM.
+
+Tables 3 shows the optimal objective function values for solving problems (P1) and (P2) directly and with Algorithm 1 by using different solvers, including the outer-approximation algorithm in solver DICOPT, the branch & bound method in solver SBB, and the global optimization solver BARON. We can see that for all the instances we considered, problems (P1) and (P2) cannot be solved directly by using MINLP solvers such as DICOPT and SBB. This is presumably due to the unbounded gradient when solving the NLP relaxation of (P1) and (P2). In contrast, with the proposed heuristic algorithm 1 which involves the solution of (P3) and (P2), we obtain “good quality” solutions by using all the solvers. It is interesting to note that for all the computational instances, the optimal solutions are found in the NLP relaxation step. This shows that the NLP relaxation of the MINLP problems (P1) and (P2) are quite effective in practice, although theoretically they are not guaranteed to yield integer solutions. Similar conclusions are also reported in Shen et al.,¹⁹ Cornuejols et al.,³⁴ Conn and Cornuejols.³⁵
+
+Table 4 shows the detailed computational times and the objective function values from the computational experiments using different algorithms and solvers to solve instances ranging from 33 to 150 retailer locations with different weight parameters. All the instances are solved with Algorithm 1 by using the global optimization solver BARON and MINLP solvers that rely on convexity assumptions (Bonmin, DICOPT, SBB) with default options, and the proposed Lagrangean heuristic algorithm (Algorithm 2) of which the Lagrangean subproblems (for lower bounds) are solved with the global optimization solver BARON and the feasibility subproblem (for upper
\ No newline at end of file
diff --git a/samples/texts/7149586/page_33.md b/samples/texts/7149586/page_33.md
new file mode 100644
index 0000000000000000000000000000000000000000..dcf5aba3c495615010a687eb9990c30077ec1dd4
--- /dev/null
+++ b/samples/texts/7149586/page_33.md
@@ -0,0 +1,7 @@
+bounds) are solved with the NLP solver CONOPT. For comparison purposes, all the instances are also solved with the global optimization solver BARON to obtain global optimal solutions, although BARON failed to terminate the search after more than 10 hours for large instances. By comparing the solution and the associated computational times, we can see that both Algorithms 1 and 2 can obtain good quality solutions that are equal to, or very close to the global optimum.
+
+Algorithm 1 requires much less computational time, and for the smaller instances the solutions obtained are quite close to the global optimum. Note that for all the instances where we can obtain an exact global optimum, the solutions from Algorithm 1 are within 5% of the global optimal solution.
+
+The Lagrangean relaxation algorithm requires longer computational times than Algorithm 1, but the quality of the solutions is significantly improved. The detailed computational results of the Lagrangean relaxation algorithm and the global optimization solver BARON are given in Table 5, where the solutions and computational times are compared. For all instances, the Lagrangean relaxation algorithm requires much shorter computational times than using BARON to obtain the same quality solutions. The instances that BARON can solve to global optimality in less than 10 hours, are solved more efficiently by using the Lagrangean based algorithm in shorter computational times and with 0% optimality gap. For the remaining large scale instances that BARON cannot close the gaps in 10 hours, the optimality gaps of the Lagrangean based algorithm are much smaller (usually less than 1.2%) than the optimality gaps by using BARON for 10 hours.
+
+The results show that good solutions without excessive computational times can be obtained with the proposed Algorithm 1, and near-global solutions can be obtained with the proposed Lagrangean relaxation method (Algorithm 2).
\ No newline at end of file
diff --git a/samples/texts/7149586/page_34.md b/samples/texts/7149586/page_34.md
new file mode 100644
index 0000000000000000000000000000000000000000..e3aed8894c8a42e3578dc37e18332e4206206417
--- /dev/null
+++ b/samples/texts/7149586/page_34.md
@@ -0,0 +1,5 @@
+Table 3 Comparison of the optimal objective function values for solving different problems with different solvers and algorithms
+
+<table><thead><tr><th rowspan="2">No. Retailers</th><th rowspan="2">β</th><th rowspan="2">θ</th><th colspan="2">Solving MINLP problem (P1) Directly</th><th colspan="2">Solving MINLP problem (P2) Directly</th><th colspan="4">Algorithm 1 for MINLP Problem (P2)</th></tr><tr><th>DICOPT</th><th>SBB</th><th>DICOPT</th><th>SBB</th><th>MILP Relaxation (P3)</th><th>SBB</th><th>DICOPT</th><th>BARON</th></tr></thead><tbody><tr><td>33</td><td>0.001</td><td>0.1</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>398.64</td><td>398.64</td><td>398.64</td><td>398.64</td></tr><tr><td>33</td><td>0.001</td><td>0.5</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>580.46</td><td>580.46</td><td>580.46</td><td>580.46</td></tr><tr><td>33</td><td>0.005</td><td>0.1</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>890.44</td><td>1023.00</td><td>1023.00</td><td>1023.00</td></tr><tr><td>33</td><td>0.005</td><td>0.5</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>1072.25</td><td>1384.03</td><td>1384.03</td><td>1384.03</td></tr><tr><td>88</td><td>0.001</td><td>0.1</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>837.68</td><td>935.02</td><td>935.02</td><td>867.55*</td></tr><tr><td>88</td><td>0.001</td><td>0.5</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>1161.12</td><td>1386.28</td><td>1386.28</td><td>1295.02*</td></tr><tr><td>88</td><td>0.005</td><td>0.1</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>1956.30</td><td>2297.74</td><td>2297.74</td><td>2297.80*</td></tr><tr><td>88</td><td>0.005</td><td>0.5</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>2279.74</td><td>3082.19</td><td>3082.19</td><td>3022.67*</td></tr><tr><td>150</td><td>0.001</td><td>0.5</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>1674.08</td><td>2205.37</td><td>2205.37</td><td>1847.93*</td></tr><tr><td>150</td><td>0.005</td><td>0.1</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>Loc. Infeas.</td><td>3107.87</td><td>4069.09</td><td>4069.09</td><td>3689.71*</td></tr></tbody></table>
+
+* Suboptimal solution obtained with BARON for 10 hour limit. Detailed upper and lower bounds are reported in Table 5.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_35.md b/samples/texts/7149586/page_35.md
new file mode 100644
index 0000000000000000000000000000000000000000..f9be60e0b96aa572b7dacce5283fa779433651a3
--- /dev/null
+++ b/samples/texts/7149586/page_35.md
@@ -0,0 +1,7 @@
+**Table 4** Comparison of computational results for solving model (P2) with different algorithms and solvers
+
+<table><thead><tr><th rowspan="3">No.<br>Retailers</th><th rowspan="3">β<br>θ</th><th colspan="6">Algorithm 1</th><th colspan="2">Algorithm 2</th></tr><tr><th colspan="2">Bonmin<br>(Ipopt)</th><th colspan="2">DICOPT<br>(CONOPT)</th><th colspan="2">SBB<br>(CONOPT)</th><th colspan="2">BARON<br>(global optimum)</th></tr><tr><th>Obj. fun.</th><th>Time (s)</th><th>Obj. fun.</th><th>Time (s)</th><th>Obj. fun.</th><th>Time (s)</th><th>Obj. fun.</th><th>Time (s)</th><th>Obj. fun.</th><th>Time (s)</th></tr></thead><tbody><tr><td>33</td><td>0.001<br>0.1</td><td>398.64</td><td>10.125</td><td>398.64</td><td>0.22</td><td>398.64</td><td>0.20</td><td>398.64</td><td>53.31</td><td>398.64</td><td>15.8</td></tr><tr><td>33</td><td>0.001<br>0.2</td><td>457.61</td><td>366.97</td><td>457.61</td><td>0.22</td><td>457.61</td><td>0.25</td><td>457.61</td><td>54.12</td><td>457.61</td><td>16.8</td></tr><tr><td>33</td><td>0.001<br>0.5</td><td>580.46</td><td>496.281</td><td>580.46</td><td>0.23</td><td>580.46</td><td>0.22</td><td>580.46</td><td>74.27</td><td>580.46</td><td>17.9</td></tr><tr><td>33</td><td>0.001<br>1.0</td><td>728.21</td><td>227.828</td><td>728.21</td><td>0.17</td><td>728.21</td><td>0.17</td><td>728.21</td><td>39.14</td><td>728.21</td><td>15.85</td></tr><tr><td>33</td><td>0.001<br>5.0</td><td>1460.40</td><td>235.765</td><td>1460.40</td><td>0.18</td><td>1460.40</td><td>0.20</td><td>1460.40</td><td>93.22</td><td>1460.40</td><td>37.28</td></tr><tr><td>33</td><td>0.001<br>0.1</td><td>398.64</td><td>10.125</td><td>398.64</td><td>0.22</td><td>398.64</td><td>0.20</td><td>398.64</td><td>53.31</td><td>398.64</td><td>15.8</td></tr><tr><td>33</td><td>0.003<br>0.1</td><td>770.26</td><td>274.078</td><td>770.26</td><td>0.23</td><td>770.26</td><td>0.27</td><td>734.60</td><td>75.67</td><td>734.60</td><td>42.79</td></tr><tr><td>33</td><td>0.005<br>0.1</td><td>1023.00</td><td>244.641</td><td>1023.00</td><td>0.72</td><td>1023.00</td><td>0.72</td><td>1007.31*</td><td>&gt; 10 hr</td><td>1006.01</td><td>90.85</td></tr><tr><td>33</td><td>0.008<br>0.1</td><td>1248.59</td><td>333.953</td><td>1248.59</td><td>0.80</td><td>1248.59</td><td>0.80</td><td>1249.37*</td><td>&gt; 10 hr</td><td>1248.59</td><td>53.13</td></tr><tr><td>33</td><td>0.010<br>0.1</td><td>1418.57</td><td>201.610</td><td>1418.57</td><td>0.80</td><td>1418.57</td><td>0.76</td><td>1398.54*</td><td>&gt; 10 hr</td><td>1398.39</td><td>92.74</td></tr><tr><td>88</td><td>0.001<br>0.1</td><td>--**</td><td>--**</td><td>935.02</td><td>20.89</td><td>935.02</td><td>20.94</td><td>867.55*</td><td>&gt; 10 hr</td><td>867.55</td><td>356.1</td></tr><tr><td>88</td><td>0.001<br>0.5</td><td>--**</td><td>--**</td><td>1386.28</td><td>42.16</td><td>1386.28</td><td>38.50</td><td>1295.02*</td><td>&gt; 10 hr</td><td>1230.99</td><td>322.54</td></tr><tr><td>88</td><td>0.005<br>0.1</td><td>--**</td><td>--**</td><td>2297.74</td><td>42.11</td><td>2297.74</td><td>42.16</td><td>2297.80*</td><td>&gt; 10 hr</td><td>2284.06</td><td>840.28</td></tr><tr><td>88</td><td>0.005<br>0.5</td><td>--**</td><li>--**<li></li></li></li></table>
+
+* Suboptimal solution obtained with BARON for 10 hour limit Detailed upper and lower bounds are reported in Table 5.
+
+** Data not available due to solver failure..
\ No newline at end of file
diff --git a/samples/texts/7149586/page_36.md b/samples/texts/7149586/page_36.md
new file mode 100644
index 0000000000000000000000000000000000000000..3c6ea5a6cd1f510915d39016db58be423ff8de8b
--- /dev/null
+++ b/samples/texts/7149586/page_36.md
@@ -0,0 +1,7 @@
+**Table 5** Comparison of bounds by using the Lagrangean heuristic algorithm and global optimizer BARON
+
+<table><thead><tr><th rowspan="2">No.<br>Retailers</th><th rowspan="2">β</th><th rowspan="2">θ</th><th colspan="5">Lagrangean Relaxation (Algorithm 2)</th><th colspan="4">BARON (global optimum)</th></tr><tr><th>Upper<br>Bound</th><th>Lower<br>Bound</th><th>Gap</th><th>Iterations</th><th>Time (s)</th><th>Upper<br>Bound</th><th>Lower<br>Bound</th><th>Optimality<br>Gap</th><th>Time (s)</th></tr></thead><tbody><tr><td>33</td><td>0.001</td><td>0.1</td><td>398.64</td><td>398.64</td><td>0 %</td><td>10</td><td>15.8</td><td>398.64</td><td>398.64</td><td>0 %</td><td>53.31</td></tr><tr><td>33</td><td>0.001</td><td>0.2</td><td>457.61</td><td>457.61</td><td>0 %</td><td>6</td><td>16.8</td><td>457.61</td><td>457.61</td><td>0 %</td><td>54.12</td></tr><tr><td>33</td><td>0.001</td><td>0.5</td><td>580.46</td><td>580.46</td><td>0 %</td><td>6</td><td>17.9</td><td>580.46</td><td>580.46</td><td>0 %</td><td>74.27</td></tr><tr><td>33</td><td>0.001</td><td>1.0</td><td>728.21</td><td>728.21</td><td>0 %</td><td>6</td><td>15.85</td><td>728.21</td><td>728.21</td><td>0 %</td><td>39.14</td></tr><tr><td>33</td><td>0.001</td><td>5.0</td><td>1460.40</td><td>1460.40</td><td>0 %</td><td>13</td><td>37.28</td><td>1460.40</td><td>1460.40</td><td>0 %</td><td>93.22</td></tr><tr><td>33</td><td>0.001</td><td>0.1</td><td>398.64</td><td>398.64</td><td>0 %</td><td>10</td><td>15.80</td><td>398.64</td><td>398.64</td><td>0 %</td><td>53.31</td></tr><tr><td>33</td><td>0.003</td><td>0.1</td><td>734.60</td><td>734.60</td><td>0 %</td><td>16</td><td>42.79</td><td>734.60</td><td>734.60</td><td>0 %</td><td>75.67</td></tr><tr><td>33</td><td>0.005</td><td>0.1</td><td>1006.01</td><td>1004.53</td><td>0.147 %</td><td>32</td><td>90.85</td><td>1007.31*</td><td>965.29</td><td>4.353 %</td><td>36000</td></tr><tr><td>33</td><td>0.008</td><td>0.1</td><td>1248.59</td><td>1248.59</td><td>0 %</td><td>19</td><td>53.13</td><td>1249.37*</td><td>1215.12</td><td>2.819 %</td><td>36000</td></tr><tr><td>33</td><td>0.010</td><td>0.1</td><td>1398.39</td><td>1397.7</td><td>0.049 %</td><td>33</td><td>92.74</td><td>1398.54*</td><td>1364.82</td><td>2.471 %</td><td>36000</td></tr><tr><td>88</td><td>0.001</td><td>0.1</td><td>867.55</td><td>867.54</td><td>0.001 %</td><td>21</td><td>356.1</td><td>867.55*</td><td>837.68</td><td>3.566 %</td><td>36000</td></tr><tr><td>88</td><td>0.001</td><td>0.5</td><td>1230.99</td><td>1223.46</td><td>0.615 %</td><td>24</td><td>322.54</td><td>1295.02*</td><td>1165.15</td><td>11.146 %</td><td>36000</td></tr><tr><td>88</td><td>0.005</td><td>0.1</td><td>2284.06</td><td>2280.74</td><td>0.146 %</td><td>55</td><td>840.28</td><td>2297.80*</td><td>2075.51</td><td>10.710 %</td><td>36000</td></tr><tr><td rowspan="2">88<br/>150<br/>150</tbody></tr></tbody></table>
+
+* Suboptimal solution obtained with BARON for 10 hour limit.
+
+-36-
\ No newline at end of file
diff --git a/samples/texts/7149586/page_37.md b/samples/texts/7149586/page_37.md
new file mode 100644
index 0000000000000000000000000000000000000000..2fa7f3de8e42ba496503aaa16eff8c66b350df64
--- /dev/null
+++ b/samples/texts/7149586/page_37.md
@@ -0,0 +1 @@
+Figure 8 Network structure for the case of 33 retailers with $\beta=0.001$, $\theta=0.1$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_38.md b/samples/texts/7149586/page_38.md
new file mode 100644
index 0000000000000000000000000000000000000000..7262da465d63c8d2adfc66f44b108fbef4897ff6
--- /dev/null
+++ b/samples/texts/7149586/page_38.md
@@ -0,0 +1,5 @@
+(b) Computational time
+
+**Figure 9** Results with different $\theta$ values for the case of 33 retailers, $\beta=0.001$
+
+(a) Objective function value
\ No newline at end of file
diff --git a/samples/texts/7149586/page_39.md b/samples/texts/7149586/page_39.md
new file mode 100644
index 0000000000000000000000000000000000000000..85b1f350f25d5e3c6f2955408095a9725d7a0ea6
--- /dev/null
+++ b/samples/texts/7149586/page_39.md
@@ -0,0 +1,9 @@
+(b) Computational time
+
+**Figure 10** Results with different $\beta$ values for the case of 33 retailers, $\theta=0.1$
+
+The network structure of the 33 retailer case with $\beta = 0.001$, $\theta = 0.1$ is given in Figure 8. Figure 9 and Figure 10 show how the objective function values and computational times change as the weights for transportation costs ($\beta$) and inventory costs ($\theta$) change. We can see that large weights will lead to an increase of the objective function value for both cases. From Figure 9, we can see that global optimal solutions can be obtained by using either Algorithm 1 or Algorithm 2, but Algorithm 1 requires less computational time. From Figure 10, we can see that although Algorithm 1 with MINLP solvers that rely on convexity assumptions always converges more quickly, the optimal objective function values are often higher. Compared with the global optimizer BARON, the proposed Lagrangean relaxation algorithm can converge to the global optimum in much shorter times for 33 retailers, $\theta = 0.1$ and different $\beta$ values.
+
+# 8. Conclusion
+
+This paper has proposed two algorithms for solving the joint supply chain network design and stochastic inventory management model presented by Shen et al.¹⁹ The first algorithm is a heuristic method based on using MINLP optimization methods that rely
\ No newline at end of file
diff --git a/samples/texts/7149586/page_4.md b/samples/texts/7149586/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..c0f337e0303774df51d390e64c02f4e7a28eafd2
--- /dev/null
+++ b/samples/texts/7149586/page_4.md
@@ -0,0 +1,9 @@
+model, and then solve it with different solution approaches, including a proposed heuristic method that relies on initialization from convex relaxations and a Lagrangean relaxation algorithm. The results from the full-scale solution and those from the various solution strategies are then compared and analyzed.
+
+The rest of this paper is organized as follows. Some basic concepts of inventory management with risk pooling are discussed in Section 2. Section 3 presents the problem statement, while Section 4 provides a detailed description of the joint supply chain network design and inventory management model proposed by Shen et al.,¹⁹ stating and defining the objective function and constraints of the model. The solution strategies including the MINLP reformulation and the Lagrangean relaxation algorithm are proposed in Section 5. Two small illustrative examples on a supply chain for liquid oxygen (LOX) are given in Section 6. Section 7 presents the comparison between different solution strategies and the full scale model, along with an analysis of the solution quality. Finally, Section 8 concludes on the performance of the proposed algorithm and the overall results.
+
+## 2. Inventory Management Model with Risk Pooling
+
+In this section, we briefly review some inventory management models that are related to the problem addressed in this work. Detailed discussion about inventory management models are given by Zipkin.⁵
+
+Figure 1 shows the inventory profile in a distribution center (or any stocking facility) for a given product. As we can see, the inventory level decreases due to the customer demand, and increases when replenishments arrive. The reorder point is a specific inventory level. It means that each time when the inventory level goes down to the reorder point, a replenishment order will be placed. The time it requires from placing an order until the replenishment arrives at the distribution center is defined as the ordering lead time. Typically, the total inventory consists of two parts, working inventory and safety stock. The working inventory represents products that have been ordered from the supplier due to replenishment, but not yet shipped out of the distribution center to satisfy the demand. The safety stock is the inventory for buffering
\ No newline at end of file
diff --git a/samples/texts/7149586/page_40.md b/samples/texts/7149586/page_40.md
new file mode 100644
index 0000000000000000000000000000000000000000..2fefb6c76121881e86fd38080c806fdc04464ffb
--- /dev/null
+++ b/samples/texts/7149586/page_40.md
@@ -0,0 +1,13 @@
+on assuming convexity in the functions. Computational experiments show that this heuristic algorithm, which includes an initialization scheme, can obtain good quality solutions (typically within 5% of the global optimum). The second algorithm is a heuristic Lagrangean relaxation and decomposition algorithm for obtaining global or near-global optimal solutions. Although there are duality gaps due to the nonconvex nature of the model, extensive numerical examples suggest that the solutions obtained with this algorithm are typically within 1.2% of the global optimum. Moreover, the second algorithm requires much less computational effort than the global optimization solver BARON.
+
+This research can be extended to consider capacity constraints, as both the supplier(s) and the distribution centers are assumed to have infinite capacity in this model. It is likely that Proposition 1 will not hold for the joint capacitated facility location and inventory management models due to the additional capacitated constraints. However, the proposed algorithms can still be used to solve the relaxed problems when branching on the assignment variables.
+
+Another possible extension is to consider different lead times for the distribution centers. In this model the lead times from supplier(s) to all the distribution centers are assumed to be the same, so cost saving can be achieved by risk pooling. If the replenishment lead times of each distribution centers are different, pooling the customers may or may not save costs. It would be also interesting to see how the supply chain network structure and the associated inventory levels change, as the lead time at each distribution center changes, especially when addressing responsiveness issue in the supply chain network design.
+
+## Acknowledgment
+
+The authors acknowledge financial support from the Pennsylvania Infrastructure Technology Alliance (PITA) and the National Science Foundation under Grant No. DMI-0556090.
+
+## Appendix A: Properties of Model (P0)
+
+In this section, we will present some properties of the INLP model (P0). Especially, we show in Proposition 1 that the binary variables for assignment decisions ($Y_{ij}$) in
\ No newline at end of file
diff --git a/samples/texts/7149586/page_41.md b/samples/texts/7149586/page_41.md
new file mode 100644
index 0000000000000000000000000000000000000000..6d2fa4f6d6e30c850e926f14082b94dae16cc11b
--- /dev/null
+++ b/samples/texts/7149586/page_41.md
@@ -0,0 +1,39 @@
+the model (P0) can be relaxed as continuous variables while treating the $X_j$ variables as integer without changing the global optimal integer solution or a local optimal solution for fixed 0-1 values for $X_j$.
+
+Let us first consider the following relaxation problem (P1) with the assignment variables $Y_{ij}$ in (P0) relaxed as continuous variables as in constraint (14).
+
+$$
+\begin{align}
+\text{(P1)} \quad & \text{Min:} && \sum_{j \in J} \left( f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}} \right) \tag{13} \\
+& \text{s.t.} && \sum_{j \in J} Y_{ij} = 1, && \forall i \in I. \tag{8} \\
+& && Y_{ij} \le X_j, && \forall i \in I, \forall j \in J. \tag{9} \\
+& && X_j \in \{0,1\}, && \forall j \in J \tag{10} \\
+& && Y_{ij} \ge 0, && \forall i \in I, \forall j \in J. \tag{14}
+\end{align}
+$$
+
+We also consider problem (P1) in the reduced space where all the binary variables $X_j$ are fixed to be $X_j^* = 0$ or 1, $\forall j \in J$. We denote this problem as (AP1).
+
+$$
+\text{(AP1)} \quad \text{Min:} \quad \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}} \right) \quad (\text{A1})
+$$
+
+$$
+\text{s.t.} \quad \sum_{j \in J} Y_{ij} = 1, \qquad \forall i \in I. \hfill (\text{A2})
+$$
+
+$$
+Y_{ij} \le X_j^*, \quad \forall i \in I, \forall j \in J. \tag{A3}
+$$
+
+$$
+Y_{ij} \ge 0, \quad \forall i \in I, \forall j \in J. \tag{A4}
+$$
+
+Problem (P1) is an MINLP problem, and problem (AP1) is a nonlinear programming (NLP) problem with all the binary variables $X_j$ in (P1) fixed to certain values. Note that problem (AP1) has the following properties given in Lemma 1 and Lemma 2.
+
+**Lemma 1.** *Problem (AP1) is a concave minimization problem defined over a polyhedron.*
+
+**Proof:**
+
+It is trivial to see that all the constraints of problem (AP1) are linear, therefore the linear constraints correspond to a polyhedron.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_42.md b/samples/texts/7149586/page_42.md
new file mode 100644
index 0000000000000000000000000000000000000000..043016b3beb203b62aebe06db7e162b1a8e4c808
--- /dev/null
+++ b/samples/texts/7149586/page_42.md
@@ -0,0 +1,33 @@
+Next, we prove that the objective function given in (A1) is concave. Let us assume $\mathbf{Y}^1 = \{Y_{ij}^1 | i \in I, j \in J\}$ and $\mathbf{Y}^2 = \{Y_{ij}^2 | i \in I, j \in J\}$ are two feasible solutions satisfying the constraints of problem (AP1). Let $V^1$ and $V^2$ be the associated objective function values, we have:
+
+$$V^1 = \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij}^1 + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}^1} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}^1} \right) \quad (A5)$$
+
+$$V^2 = \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij}^2 + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}^2} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}^2} \right) \quad (A6)$$
+
+Let $0 \le t \le 1$, and $\mathbf{Y}^0 = t \cdot \mathbf{Y}^1 + (1-t) \cdot \mathbf{Y}^2 = \{Y_{ij}^0 = t \cdot Y_{ij}^1 + (1-t) \cdot Y_{ij}^2 | i \in I, j \in J\}$. Since all the constraints of problem (AP1) are linear, it is trivial to show $\mathbf{Y}^0$ is also a feasible solution of problem (AP1). Let $V^0$ be the associated objective function value.
+
+We then have,
+
+$$ 
+\begin{aligned}
+V^0 &= \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij}^0 + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}^0} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}^0} \right) \\
+&= \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} [t \cdot Y_{ij}^1 + (1-t) \cdot Y_{ij}^2] + K_j \sqrt{\sum_{i \in I} \mu_i [t \cdot Y_{ij}^1 + (1-t) \cdot Y_{ij}^2]} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 [t \cdot Y_{ij}^1 + (1-t) \cdot Y_{ij}^2]} \right)
+\end{aligned}
+\quad (A7) $$
+
+Thus we have:
+
+$$ 
+\begin{aligned}
+& V^0 - [t \cdot V^1 + (1-t) \cdot V^2] \\
+&= \sum_{j \in J} K_j \left( \sqrt{[t \cdot \sum_{i \in I} (\mu_i Y_{ij}^1) + (1-t) \cdot \sum_{i \in I} (\mu_i Y_{ij}^2)] - [t \cdot \sqrt{\sum_{i \in I} \mu_i Y_{ij}^1} + (1-t) \cdot \sqrt{\sum_{i \in I} \mu_i Y_{ij}^2}] } + q \sqrt{[t \cdot \sum_{i \in I} (\hat{\sigma}_i^2 Y_{ij}^1) + (1-t) \cdot \sum_{i \in I} (\hat{\sigma}_i^2 Y_{ij}^2)] - [t \cdot \sqrt{\sum_{i \in I} (\hat{\sigma}_i^2 Y_{ij}^1)} + (1-t) \cdot \sqrt{\sum_{i \in I} (\hat{\sigma}_i^2 Y_{ij}^2)}]} \right)
+\end{aligned}
+\quad (A8) $$
+
+Now consider the following function:
+
+$$f(t) = \sqrt{t \cdot a + (1-t) \cdot b} - [t \cdot a + (1-t) \cdot b] - [t \cdot b + (1-t) \cdot a]$$
+
+(A9)
+
+where $0 \le t \le 1$ and $a, b \ge 0$. Thus, we have:
\ No newline at end of file
diff --git a/samples/texts/7149586/page_43.md b/samples/texts/7149586/page_43.md
new file mode 100644
index 0000000000000000000000000000000000000000..79bc39d311b233530c02f46d3f21a48e54274aa7
--- /dev/null
+++ b/samples/texts/7149586/page_43.md
@@ -0,0 +1,26 @@
+$$
+\begin{align}
+f(t) &= \sqrt{t \cdot a + (1-t) \cdot b} - [t \cdot \sqrt{a} + (1-t) \cdot \sqrt{b}] = \frac{t \cdot a + (1-t) \cdot b - [t \cdot \sqrt{a} + (1-t) \cdot \sqrt{b}]^2}{\sqrt{t \cdot a + (1-t) \cdot b} + [t \cdot \sqrt{a} + (1-t) \cdot \sqrt{b}]} \nonumber \\
+&= \frac{t \cdot (1-t) \cdot (\sqrt{a} - \sqrt{b})^2}{\sqrt{t \cdot a + (1-t) \cdot b} + [t \cdot \sqrt{a} + (1-t) \cdot \sqrt{b}]} \ge 0 \tag{A10}
+\end{align}
+$$
+
+Comparing (A9), (A10) with each term in (A8), it follows that
+
+$$V^0 - [t \cdot V^1 + (1-t) \cdot V^2] \geq 0$$
+
+Therefore, the objective function of problem (AP1) is a concave function. ■
+
+**Lemma 2.** The $Y_{ij}$ variables take on integer values (0 or 1) for any local or global optimal solution of problem (API).
+
+**Proof:**
+
+Based on Lemma 1, we know that problem (AP1) corresponds to the minimization of a concave function over a polyhedron. As proved by Falk and Hoffmann,⁵⁵ any local or global optimal solution of this problem always lies on a vertex of the polyhedron.
+
+Furthermore, it is trivial to see the coefficient matrix of constraint (A2) is totally unimodular,⁵⁶ while constraints (A3) and (A4) only provide integer bounds of the $Y_{ij}$ variables. Thus, constraints (A2), (A3) and (A4) define an integral polyhedron, of which the extreme points are at the integer values (0 or 1) of the $Y_{ij}$ variables.⁵⁶ Therefore, all the $Y_{ij}$ variables equal to 0 or 1 for any local or global optimal solution of problem (AP1). ■
+
+**Proposition 1.** *The continuous variables* $Y_{ij}$ *yield 0-1 integer values when* (P0) *is globally optimized or locally optimized for fixed 0-1 value for* $X_j$.
+
+**Proof:**
+
+The MINLP problem (**P1**) is a relaxation of the INLP problem (**P0**) by treating all the $Y_{ij}$ as continuous variables, while problem (**AP1**) is an NLP subproblem of (**P1**) with a
\ No newline at end of file
diff --git a/samples/texts/7149586/page_44.md b/samples/texts/7149586/page_44.md
new file mode 100644
index 0000000000000000000000000000000000000000..8aefb315413adb7f70e451805e89fa40d3a07616
--- /dev/null
+++ b/samples/texts/7149586/page_44.md
@@ -0,0 +1,34 @@
+fixed value of the integer variables $X_j$. Based on Lemma 2, we can then conclude that
+when (P1) is globally optimized or locally optimized for fixed 0-1 values for $X_j$, all
+the $Y_{ij}$ variables take integer values (0 or 1).
+
+**Proposition 2.** The global optimal solution of problem (P2), or a local optimal solution with fixed 0-1 value for $X_j$, has all the continuous variables $Y_{ij}$ take on integer value (0 or 1).
+
+**Proof:**
+
+Similarly, we consider problem (P2) in the reduced space where all the binary
+variables $X_j$ are fixed to be $X_j^* = 0$ or 1, $\forall j \in J$. We denote this problem as (AP2).
+
+$$
+\begin{align}
+\text{(AP2) Min:} \quad & \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j Z_1 + qZ_2 \right) \tag{A11} \\
+\text{s.t.} \quad & \sum_{j \in J} Y_{ij} = 1, \qquad \forall i \in I. \tag{A2} \\
+& Y_{ij} \le X_j^*, \qquad \forall i \in I, \forall j \in J. \tag{A3} \\
+& Y_{ij} \ge 0, \qquad \forall i \in I, \forall j \in J \tag{A4} \\
+& -Z_1^2 + \sum_{i \in I} \mu_i Y_{ij} \le 0, \qquad \forall j \in J \tag{A12} \\
+& -Z_2^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij} \le 0, \qquad \forall j \in J \tag{A13} \\
+& Z_1 \ge 0, \qquad \forall j \in J \tag{A14} \\
+& Z_2 \ge 0, \qquad \forall j \in J \tag{A15}
+\end{align}
+$$
+
+We associate the Lagrange multiplier $\lambda_i$ with constraint (A2), $\mu_{ij}$ with constraint (A3), $t_{ij}$ with constraint (A4), $\rho_j$ with constraint (A12), $\gamma_j$ with constraint (A13), $\zeta_j$ with constraint (A14) and $\xi_j$ with constraint (A15). The Lagrange function for this problem is:
+
+$$
+\begin{equation}
+\begin{aligned}
+L = & \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j Z_1 + qZ_2 \right) + \sum_{i \in I} \lambda_i \left( 1 - \sum_{j \in J} Y_{ij} \right) + \sum_{i \in I} \sum_{j \in J} \mu_{ij} (Y_{ij} - X_j^*) \\
+& - \sum_{i \in I} \sum_{j \in J} t_{ij} Y_{ij} + \sum_{j \in J} \rho_j \left( -Z_1^2 + \sum_{i \in I} \mu_i Y_{ij} \right) + \sum_{j \in J} \gamma_j \left( -Z_2^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij} \right) - \sum_{j \in J} \zeta_j Z_1 - \sum_{j \in J} \xi_j Z_2
+\end{aligned}
+\end{equation}
+$$
\ No newline at end of file
diff --git a/samples/texts/7149586/page_45.md b/samples/texts/7149586/page_45.md
new file mode 100644
index 0000000000000000000000000000000000000000..6a96dbb698103b513dd4b089d6d9ff62ce911d2a
--- /dev/null
+++ b/samples/texts/7149586/page_45.md
@@ -0,0 +1,39 @@
+Then the KKT conditions for this problem are:
+
+$$ \frac{\partial L}{\partial Y_{ij}} = \hat{d}_{ij} - \lambda_i + \mu_{ij} - t_{ij} + \rho_j \mu_i + \gamma_j \hat{\sigma}_i^2 = 0, \forall i \in I, j \in J \quad (A16) $$
+
+$$ \frac{\partial L}{\partial Z1_j} = K_j - 2 \cdot \rho_j \cdot Z1_j - \zeta_j = 0, \forall j \in J \quad (A17) $$
+
+$$ \frac{\partial L}{\partial Z2_j} = q - 2 \cdot \gamma_j \cdot Z2_j - \xi_j = 0, \forall j \in J \quad (A18) $$
+
+$$ \mu_{ij}(Y_{ij} - X_j^*) = 0, \forall i \in I, \forall j \in J \quad (A19) $$
+
+$$ t_{ij}Y_{ij} = 0, \forall i \in I, \forall j \in J \quad (A20) $$
+
+$$ \rho_j(-Z1_j^2 + \sum_{i \in I} \mu_i Y_{ij}) = 0, \forall j \in J \quad (A21) $$
+
+$$ \gamma_j(-Z2_j^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}) = 0, \forall j \in J \quad (A22) $$
+
+$$ \zeta_j \cdot Z1_j = 0, \forall j \in J \quad (A23) $$
+
+$$ \xi_j \cdot Z2_j = 0, \forall j \in J \quad (A24) $$
+
+$$ \sum_{j \in J} Y_{ij} = 1, \quad \forall i \in I. \quad (A25) $$
+
+$$ Y_{ij} \le X_j^*, \quad \forall i \in I, \forall j \in J. \quad (A3) $$
+
+$$ Y_{ij} \ge 0, \quad \forall i \in I, \forall j \in J. \quad (A4) $$
+
+$$ -Z1_j^2 + \sum_{i \in I} \mu_i Y_{ij} \le 0, \quad \forall j \in J \quad (A12) $$
+
+$$ -Z2_j^2 + \sum_{i \in I} \hat{\sigma}_i^2 Y_{ij} \le 0, \quad \forall j \in J \quad (A13) $$
+
+$$ Z1_j \ge 0, \quad \forall j \in J \quad (A14) $$
+
+$$ Z2_j \ge 0, \quad \forall j \in J \quad (A15) $$
+
+$$ \lambda_i \ge 0, \mu_{ij} \ge 0, t_{ij} \ge 0, \forall i \in I, \forall j \in J. \quad (A25) $$
+
+$$ \rho_j \ge 0, \gamma_j \ge 0, \zeta_j \ge 0, \xi_j \ge 0, \forall j \in J. \quad (A26) $$
+
+On the other hand, for problem (AP1), if we similarly associate Lagrange multiplier $\lambda_i$ with constraint (A2), $\mu_{ij}$ with constraint (A3) and $t_{ij}$ with constraint (A4). The Lagrange function for problem (AP1) is:
\ No newline at end of file
diff --git a/samples/texts/7149586/page_46.md b/samples/texts/7149586/page_46.md
new file mode 100644
index 0000000000000000000000000000000000000000..216fdf7fdc4eba4f0256497657a5d05afe3314c9
--- /dev/null
+++ b/samples/texts/7149586/page_46.md
@@ -0,0 +1,32 @@
+$$L' = \sum_{j \in J} \left( f_j X_j^* + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + K_j \sqrt{\sum_{i \in I} \mu_i Y_{ij}} + q \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}} \right) + \sum_{i \in I} \lambda_i \left( 1 - \sum_{j \in J} Y_{ij} \right) + \sum_{i \in I} \sum_{j \in J} \mu_{ij} (Y_{ij} - X_j^*) - \sum_{i \in I} \sum_{j \in J} t_{ij} Y_{ij}$$
+
+Then the KKT conditions for this problem are:
+
+$$\frac{\partial L'}{\partial Y_{ij}} = \hat{d}_{ij} + \frac{K_j \mu_i}{2 \sqrt{\sum_{i \in I} \mu_i Y_{ij}}} + \frac{q \hat{\sigma}_i^2}{2 \sqrt{\sum_{i \in I} \hat{\sigma}_i^2 Y_{ij}}} - \lambda_i + \mu_{ij} - t_{ij} = 0, \quad \forall i \in I, j \in J \quad (A27)$$
+
+$$\mu_{ij}(Y_{ij} - X_j^*) = 0, \quad \forall i \in I, \forall j \in J \quad (A19)$$
+
+$$t_{ij}Y_{ij} = 0, \quad \forall i \in I, \forall j \in J \quad (A20)$$
+
+$$\sum_{j \in J} Y_{ij} = 1, \quad \forall i \in I. \quad (A2)$$
+
+$$Y_{ij} \le X_j^*, \quad \forall i \in I, \forall j \in J. \quad (A3)$$
+
+$$Y_{ij} \ge 0, \quad \forall i \in I, \forall j \in J. \quad (A4)$$
+
+$$\lambda_i \ge 0, \mu_{ij} \ge 0, t_{ij} \ge 0, \forall i \in I, \forall j \in J. \quad (A25)$$
+
+By substituting equations (A17), (A18), (A21), (A22), (A23) and (A24) into (A27),
+we can have
+
+$$\frac{\partial L'}{\partial Y_{ij}} = \hat{d}_{ij} + \frac{K_j \mu_i}{2\sqrt{\sum_{i\in I} \mu_i Y_{ij}}} + \frac{q\hat{\sigma}_i^2}{2\sqrt{\sum_{i\in I} (\hat{\sigma}_i^2 Y_{ij})}} - \lambda_i + \mu_{ij} - t_{ij} = \hat{d}_{ij} - \lambda_i + \mu_{ij} - t_{ij} + \rho_j \mu_i + \gamma_j \hat{\sigma}_i^2 = \frac{\partial L}{\partial Y_{ij}}$$
+
+which is the same as (A14).
+
+This then shows that the KKT conditions of problem (AP1) and (AP2) are equivalent,
+although due to the square root terms in (A23) problem (AP1) may have unbounded
+gradients. Because we know that all the optimal solutions of problem (AP1) have $Y_{ij}$
+take on integer values 0 or 1, we can conclude that $Y_{ij}$ variables also have integer
+values when (AP2) is (locally or globally) optimized. Similarly to Proposition 1, we
+can conclude that problem (P2) has all the $Y_{ij}$ take on integer values when it is 1
+globally optimized or locally optimized for fixed $X_j$.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_47.md b/samples/texts/7149586/page_47.md
new file mode 100644
index 0000000000000000000000000000000000000000..ed9ae36981f4653acea207e17d0e4f8dc6791845
--- /dev/null
+++ b/samples/texts/7149586/page_47.md
@@ -0,0 +1,17 @@
+# Appendix B: Data for the Illustrative Examples
+
+**Table B1** Parameters for demand uncertainty for the illustrative example
+
+<table><thead><tr><th></th><th>Mean demand μ<sub>i</sub> (Liters/day)</th><th>Standard Deviation σ<sub>i</sub> (Liters/day)</th></tr></thead><tbody><tr><td>Customer 1</td><td>95</td><td>30</td></tr><tr><td>Customer 2</td><td>157</td><td>50</td></tr><tr><td>Customer 3</td><td>46</td><td>25</td></tr><tr><td>Customer 4</td><td>234</td><td>80</td></tr><tr><td>Customer 5</td><td>75</td><td>25</td></tr><tr><td>Customer 6</td><td>192</td><td>80</td></tr></tbody></table>
+
+**Table B2** Parameters for unit transportation cost ($d_{ij}$) between DCs and customers ($/Liter)
+
+<table><thead><tr><th></th><th>DC 1</th><th>DC 2</th><th>DC 3</th></tr></thead><tbody><tr><td>Customer 1</td><td>0.04</td><td>2.00</td><td>2.88</td></tr><tr><td>Customer 2</td><td>0.08</td><td>1.36</td><td>1.32</td></tr><tr><td>Customer 3</td><td>0.36</td><td>0.08</td><td>1.04</td></tr><tr><td>Customer 4</td><td>0.88</td><td>0.10</td><td>0.52</td></tr><tr><td>Customer 5</td><td>1.52</td><td>1.80</td><td>0.12</td></tr><tr><td>Customer 6</td><td>3.36</td><td>2.28</td><td>0.08</td></tr></tbody></table>
+
+**Table B3** Parameters for shipping cost between Plant and DCs
+
+<table><thead><tr><th rowspan="2"></th><th>Fixed shipping cost from plant to DC ($g_j$)</th><th>Unit shipping cost ($a_j$)</th></tr><tr><th>($/shipment)</th><th>($/Liter)</th></tr></thead><tbody><tr><th scope="row">DC 1</th><td>13</td><td>0.24</td></tr><tr><th scope="row">DC 2</th><td>10</td><td>0.20</td></tr><tr><th scope="row">DC 3</th><td>14</td><td>0.28</td></tr></tbody></table>
+
+**Table B4** Model coefficients for the illustrative example in Section 6.2
+
+<table><tbody><tr><th scope="row">F<sub>j</sub></th><td>Fixed order cost per replenishment</td><td>10</td></tr><tr><th scope="row">f<sub>j</sub></th><td>Fixed cost to install a DC (annual)</td><td>100</td></tr><tr><th scope="row">z<sub>&alpha;</sub></th><td>Service level parameter</td><td>1.96</td></tr><tr><th scope="row">L</th><td>Order lead time (days)</td><td>7</td></tr><tr><th scope="row">h</th><td>Unit inventory holding cost (annual)</td><td>12</td></tr></tbody></table>
\ No newline at end of file
diff --git a/samples/texts/7149586/page_48.md b/samples/texts/7149586/page_48.md
new file mode 100644
index 0000000000000000000000000000000000000000..1a2dbcab42a39f34333bdbec7787f4d2190fb6a5
--- /dev/null
+++ b/samples/texts/7149586/page_48.md
@@ -0,0 +1,33 @@
+Reference
+
+1. Grossmann, I. E., Enterprise-wide Optimization: A New Frontier in Process Systems Engineering. *AIChE Journal* **2005**, *51*, 1846-1857.
+
+2. Chopra, S.; Meindl, P., *Supply Chain Management: Strategy, Planning and Operation*. Prentice Hall: Saddle River, NJ, 2003.
+
+3. Daskin, M. S., *Network and Discrete Location: Models, Algorithms, and Applications*. Wiley: New York, 1995.
+
+4. Owen, S. H.; Daskin, M. S., Strategic facility location: A review. *European Journal of Operations Research* **1998**, *111*, 423-447.
+
+5. Zipkin, P. H., *Foundations of Inventory Management*. McGraw-Hill: Boston, MA, 2000.
+
+6. Cachon, G. P.; Fisher, M., Supply Chain Inventory Management and the Value of Shared Information. *Management Science* **2000**, *46*, (8), 1032-1048.
+
+7. Kok, A. G.; Graves, S. C., *Supply Chain Management: Design, Coordination and Operation*. Elsevier: 2003.
+
+8. Tsiakis, P.; Shah, N.; Pantelides, C. C., Design of Multi-echelon Supply Chain Networks under Demand Uncertainty. *Industrial & Engineering Chemistry Research* **2001**, *40*, 3585-3604.
+
+9. Bok, J.-K.; Grossmann, I. E.; Park, S., Supply Chain Optimization in Continuous Flexible Process Networks. *Industrial & Engineering Chemistry Research* **2000**, *39*, (5), 1279-1290.
+
+10. Relvas, S.; Matos, H. A.; A.P.F.D., B.-P.; Fialho, J.; Pinheiro, A. S., Pipeline Scheduling and Inventory Management of a Multiproduct Distribution Oil System. *Industrial & Engineering Chemistry Research* **2006**, *45*, 7841-7855.
+
+11. Schulz, E. P.; Diaz, M. S.; Bandoni, J. A., Supply chain optimization of large-scale continuous process. *Computers & Chemical Engineering* **2006**, *29*, 1305-1316.
+
+12. Jackson, J. R.; Grossmann, I. E., Temporal decomposition scheme for nonlinear multisite production planning and distribution models. *Industrial & Engineering Chemistry Research* **2003**, *42*, (13), 3045-3055.
+
+13. Lim, M.-F.; Karimi, I. A., Resource-Constrained Scheduling of Parallel Production Lines Using Asynchronous Slots. *Industrial & Engineering Chemistry Research* **2003**, *42*, 6832-6842.
+
+14. Varma, V. A.; Reklaitis, G. V.; Blau, G. E.; Pekny, J. F., Enterprise-wide modeling & optimization – An overview of emerging research challenges and opportunities. *Computers & Chemical Engineering* **2007**, *31*, 692-711.
+
+15. Chen, C.-L.; Lee, W.-C., Multi-objective optimization of multi-echelon supply chain networks with uncertain product demands and prices. *Computers & Chemical Engineering* **2004**, *28*, 1131-1144.
+
+16. Gupta, A.; Maranas, C. D., Managing demand uncertainty in supply chain planning.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_49.md b/samples/texts/7149586/page_49.md
new file mode 100644
index 0000000000000000000000000000000000000000..0fa0d2bca01302dbcec1907ea63cf785d5888f68
--- /dev/null
+++ b/samples/texts/7149586/page_49.md
@@ -0,0 +1,35 @@
+Computers & Chemical Engineering **2003**, *27*, 1219-1227.
+
+17. Gupta, A.; Maranas, C. D.; McDonald, C. M., Mid-term supply chain planning under demand uncertainty: customer demand satisfaction and inventory management. *Computers & Chemical Engineering* **2000**, *24*, 2613-2621.
+
+18. Jung, J. Y.; Blau, G.; Pekny, J. F.; Reklaitis, G. V.; Eversdyk, D., Integrated safety stock management for multi-stage supply chains under production capacity constraints. In *Computers & Chemical Engineering*, Submitted: 2007.
+
+19. Shen, Z.-J. M.; Coullard, C.; Daskin, M. S., A joint location-inventory model. *Transportation Science* **2003**, *37*, 40-55.
+
+20. Fisher, M. L., The Lagrangean relaxation method for solving integer programming problems. *Management Science* **1981**, *27*, (1), 18.
+
+21. Fisher, M. L., An application oriented guide to Lagrangian relaxation. *Interfaces* **1985**, *15*, (2), 2-21.
+
+22. Guignard, M.; Kim, S., Lagrangean decomposition: A model yielding stronger Lagrangean bounds. *Mathematical Programming* **1987**, *39*, 215-228.
+
+23. Beasley, J. E., Lagrangean heuristics for location problems. *European Journal of Operations Research* **1993**, *65*, (3), 383-399.
+
+24. Holmberg, K.; Ling, J., A Lagrangean heuristic for the facility location problem with staircase costs. *European Journal of Operations Research* **1997**, *97*, (1), 63-74.
+
+25. Sridharan, R., The capacitated plant location problem. *European Journal of Operations Research* **1993**, *87*, 203-213.
+
+26. Pirkul, H.; Jayaraman, V., Production, transportation, and distribution planning in a multi-commodity tri-echelon system. *Transportation Science* **1996**, *30*, 291-302.
+
+27. Klose, A., An Lagrangean relax-and-cut approach for two-stage capacitated facility location problems. *Journal of the Operational Research Society* **2000**, *126*, 408-421.
+
+28. van den Heever, S. A.; Grossmann, I. E.; Vasantharajan, S., A Lagrangean decomposition heuristic for the design and planning of offshore hydrocarbon field infrastructures with complex economic objectives. *Industrial & Engineering Chemistry Research* **2001**, *40*, (13), 2857-2875.
+
+29. Neiro, S. M. S.; Pinto, J. M., A general modeling framework for the operational planning of petroleum supply chains. *Computers & Chemical Engineering* **2004**, *28*, 871-896.
+
+30. Axsater, S., Using the Deterministic EOQ Formula in Stochastic Inventory Control. *Management Science* **1996**, *42*, (6), 830-834.
+
+31. Zheng, Y. S., On properties of stochastic inventory systems. *Management Science* **1992**, *38*, 87-103.
+
+32. Eppen, G., Effects of centralization on expected costs in a multi-echelon newsboy problem. *Management Science* **1979**, *25*, (5), 498-501.
+
+33. Daskin, M. S.; Coullard, C.; Shen, Z.-J. M., An inventory-location model: formulation, solution algorithm and computational results. *Annals of Operations Research* **2002**, *110*, 83-106.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_5.md b/samples/texts/7149586/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..f42b34a178e451bc4e62378081e6312cad9f3113
--- /dev/null
+++ b/samples/texts/7149586/page_5.md
@@ -0,0 +1,7 @@
+the system against stockouts due to the uncertain demands during the ordering lead time.
+
+**Figure 1 Inventory profile changing with time**
+
+**Figure 2 Inventory profile for deterministic demand with (Q, r) policy**
+
+A popular inventory control policy widely used in practice is the order quantity/reorder point (Q, r) inventory policy. When using this policy, each time when inventory level depletes to reorder point r, a fixed order quantity Q will be placed for replenishment. When the demand is deterministic with consistent demand rate, the inventory profile is a series of identical square triangle as given in Figure 2. Each of
\ No newline at end of file
diff --git a/samples/texts/7149586/page_50.md b/samples/texts/7149586/page_50.md
new file mode 100644
index 0000000000000000000000000000000000000000..c816954fd5c0bcecb56761d74a54c2e9c3751c14
--- /dev/null
+++ b/samples/texts/7149586/page_50.md
@@ -0,0 +1,35 @@
+34. Cornuejols, G.; Nemhauser, G. L.; Wolsey, L. A., The uncapacitated facility location problem. In *Discrete Location Theory*, Mirchandani, P.; Francis, R., Eds. John Wiley and Son Inc.: New York, 1990; pp 119-171.
+
+35. Conn, A. R.; Cornuejols, G., A Projection Method for the Uncapacitated Facility Location Problem. *Mathematical Programming* **1990**, *46*, 273-298.
+
+36. Ryoo, H. S.; Sahinidis, N. V., A branch-and-reduce approach to global optimization. *Journal of Global Optimization* **1996**, *8*, (2), 107-138.
+
+37. Tawarmalani, M.; Sahinidis, N. V., Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. *Mathematical Programming* **2004**, *99*, 563-591.
+
+38. Adjiman, C. S.; Androulakis, I. P.; Floudas, C. A., Global optimization of mixed-integer nonlinear problems. *AIChE Journal* **2000**, *16*, (9), 1769-1797.
+
+39. Quesada, I.; Grossmann, I. E., A Global Optimization Algorithm for Linear Fractional and Bilinear Programs. *Journal of Global Optimization* **1995**, *6*, 39-76.
+
+40. Smith, E. M. B.; Pantelides, C. C., A symbolic reformulation/spatial branch and bound algorithm for the global optimization of nonconvex MINLPs. *Computers & Chemical Engineering* **1999**, *23*, 457-478.
+
+41. Kesavan, P.; Allgor, R. J.; Gatzke, E. P.; Barton, P. I., Outer Approximation Algorithms for Separable Nonconvex Mixed-Integer Nonlinear Programs. *Mathematical Programming* **2004**, *100*, (3), 517-535.
+
+42. Sahinidis, N. V., BARON: A general purpose global optimization software package. *Journal of Global Optimization* **1996**, *8*, (2), 201-205.
+
+43. Grossmann, I. E., Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. *Optimization and Engineering* **2002**, *3*, 227-252.
+
+44. Leyffer, S., Integrating SQP and branch and bound for mixed integer nonlinear programming. *Computational Optimization and Applications* **2001**, *18*, 295-309.
+
+45. Geoffrion, A. M., Generalized Benders Decomposition. *Journal of Optimization Theory and Applications* **1972**, *10*, (4), 237-260.
+
+46. Duran, M. A.; Grossmann, I. E., An outer-approximation algorithm for a class of mixed-integer nonlinear programs. *Mathematical Programming* **1986**, *36*, (3), 307.
+
+47. Viswanathan, J.; Grossmann, I. E., A combined penalty-function and outer-approximation method for MINLP optimization. *Computers & Chemical Engineering* **1990**, *14*, (7), 769-782.
+
+48. Quesada, I.; Grossmann, I. E., An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. *Computers & Chemical Engineering* **1992**, *16*, 937-947.
+
+49. Westerlund, T.; Pettersson, F., A cutting plane method for solving convex MINLP problems. *Computers & Chemical Engineering* **1995**, *19*, S131-S136.
+
+50. Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R., GAMS- A User's Manual. In GAMS Development Corp.: 1998.
+
+51. Bonami, P.; Waechter, A.; Biegler, L. T.; Conn, A. R.; Cornuéjols, G.; Grossmann, I. E.; Laird,
\ No newline at end of file
diff --git a/samples/texts/7149586/page_51.md b/samples/texts/7149586/page_51.md
new file mode 100644
index 0000000000000000000000000000000000000000..4db8d5a85c0ac6b264878c037764c6adee878818
--- /dev/null
+++ b/samples/texts/7149586/page_51.md
@@ -0,0 +1,11 @@
+C. D.; Lee, J.; Lodi, A.; Margot, F.; Sawaya, N., An Algorithmic Framework for Convex Mixed Integer Nonlinear Programs. In *Technical Report RC23771*, IBM Thomas J. Watson Research Center: Yorktown Heights, NY, 2005.
+
+52. http://www.coin-or.org
+
+53. Falk, J. E.; Soland, R. M., An Algorithm for separable nonconvex programming problems. *Management Science* **1969**, *15*, 550-569.
+
+54. http://www.census.gov/main/www/cen2000.html
+
+55. Falk, J. E.; Hoffman, K. R., A Successive Underestimation Method for Concave Minimization Problems. *Mathematics of Operations Research* **1976**, *1*, (3), 251-259.
+
+56. Nemhauser, G. L.; Wolsey, L. A., *Integer and Combinatorial Optimization*. John Wiley and Son Inc.: New York, NY, 1988.
\ No newline at end of file
diff --git a/samples/texts/7149586/page_6.md b/samples/texts/7149586/page_6.md
new file mode 100644
index 0000000000000000000000000000000000000000..7d07c22917bfda286a697e7228e8f4f3fd41f3db
--- /dev/null
+++ b/samples/texts/7149586/page_6.md
@@ -0,0 +1,22 @@
+these square triangles has the same height (the order quantity Q), and the same width
+denoted as the replenishment interval. The optimal order quantity and replenishment
+interval for this deterministic demand case can be determined by using an economic
+order quantity (EOQ) model, which takes into account the trade-off between fixed
+ordering costs, transportation costs and working inventory holding costs (EOQ model
+formulation for our model is given in equation (3) in Section 3.2.). Although the EOQ
+model uses the deterministic demands, it has proved to provide very good
+approximations for working inventory costs of systems with (Q, r) policy under
+demand uncertainty.³⁰,³¹ A common approach for the (Q, r) inventory model, as pointed
+out by Axsater,³⁰ is to first replace the stochastic demand with its mean value and then
+determine the optimal order quantity Q with the deterministic EOQ model, and finally
+find out the optimal reorder point under uncertain demand based on the order quantity.
+
+**Figure 3 Safety stock and service level under normally distributed demand**
+
+A distribution center under demand uncertainty may not always have sufficient
+stock to handle the changing demand. If the reorder point (inventory level) is less than
+the demand during the order lead time, stockout may happen. *Type I service level* is
+defined as the probability that the total inventory on hand is more than the demand (as
+shown Figure 3). If the demand is normally distributed with mean $\mu$ and standard
+deviation $\sigma$ and the ordering lead time is $L$, the optimal safety stock level to
+guarantee a service level $\alpha$ is $z_{\alpha}\sqrt{L\sigma}$, where $z_{\alpha}$ is a standard normal deviate such
\ No newline at end of file
diff --git a/samples/texts/7149586/page_7.md b/samples/texts/7149586/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..337362621c56d2a5365b0b0ad3c4e79ad674b137
--- /dev/null
+++ b/samples/texts/7149586/page_7.md
@@ -0,0 +1,13 @@
+that $Pr(z \le z_{\alpha}) = \alpha$. We should note that the acceptable practice in this field is to assume a normal distribution of the demand, although of course other distribution functions can be specified.
+
+To consider the total safety stock of an inventory system, Eppen³² proposed the “risk pooling effect”, which states that significant safety stock cost can be saved by grouping retailers. In particular, Eppen considered a single period problem with *N* retailers and one supplier. Each retailer *i* has normally distributed demand with mean $\mu_i$ and standard deviation $\sigma_i$, and the correlation coefficient of demand at retail *i* and *j* is $\rho_{ij}$. The order lead time from the supplier to all these retailers are the same and given as *L*. Eppen compared two operational modes of retailer supply chain: decentralized mode and centralized mode. In the decentralized mode, each retailer orders independently to minimize its own expected cost. Since in this mode the optimal safety stock in retailer *i* is $z_{\alpha}\sqrt{L}\sigma_i$, the total safety stock in the system is given by,
+
+$$z_{\alpha}\sqrt{L}\sum_{i=1}^{N}\sigma_{i}.$$
+
+In the centralized mode, all the retailers are considered as a whole and a single quantity is ordered for replenishment, so as to minimize the total expected cost of the entire system. Since in the centralized mode all the retailers are grouped, and the demand at each retailer follows a normal distribution $N(\mu_i, \sigma_i^2)$, the total uncertain demand of the entire system during the order lead time will also follow a normal distribution with mean $L\sum_{i=1}^{N}\mu_i$ and standard deviation $\sqrt{L\sum\sigma_i^2+2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\sigma_i\sigma_j\rho_{ij}}$. Therefore, the total safety stock of the distribution centers in the centralized mode is,
+
+$$z_{\alpha}\sqrt{L}\left[\sum_{i=1}^{N}\sigma_{i}^{2}+2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\sigma_{i}\sigma_{j}\rho_{ij}\right]$$
+
+Thus, if the demands of all the *N* retailers are independent, the optimal safety stock can be expressed by $z_{\alpha}\sqrt{L}\sum_{i=1}^{N}\sigma_{i}^{2}$, which is less than $z_{\alpha}\sqrt{L}\sum_{i=1}^{N}\sigma_{i}$. Eppen's simple model illustrates the potential saving in safety stock costs due to risk pooling.
+
+In summary, for an inventory system including multiple distribution centers operating with (Q, r) policy and Type I service level under demand uncertainty, the total
\ No newline at end of file
diff --git a/samples/texts/7149586/page_8.md b/samples/texts/7149586/page_8.md
new file mode 100644
index 0000000000000000000000000000000000000000..76d98b61a8719c468069ccde8b2c156d9fffaba5
--- /dev/null
+++ b/samples/texts/7149586/page_8.md
@@ -0,0 +1,9 @@
+inventory cost consists of working inventory costs and safety stock costs. The optimal working inventory costs can be estimated with a deterministic EOQ model, and the safety stock costs can be reduced by risk pooling.
+
+### 3. Problem Statement
+
+We assume we are given a supply chain consisting of one or more suppliers and a set of retailers $i \in I$ (this can also be customers or markets, for convenience, we denote as “retailer” in the rest part of this paper unless specified), together with a number candidate sites for distribution centers $j \in J$. For an example of the network structure, see Figure 4). The locations of the supplier(s), potential distribution centers and the retailers are known and the distances between them are given. The replenishment lead time $L$ of each distribution center is assumed to be the same for all the candidate distribution centers. This in turn means that the suppliers can be treated implicitly and lumped into one supplier. There is a fixed setup cost $f_j$ when each distribution center is installed. Each retailer $i$ has a normally distributed demand with mean $\mu_i$ and variance $\sigma_i^2$, which is independent of the other retailers’ demands (The model can be easily extended to consider correlated demands in the retailers by modifying the safety stock terms as discussed at the end of Section 2). Each distribution center can serve more than one retailer, but each retailer should be only assigned to exactly one distribution center to satisfy the demand. Linear transportation costs are incurred for shipments from supplier to distribution center $j$ with fixed cost $g_j$ and unit cost $a_j$ and from distribution center $j$ to retailer $i$ with unit cost $d_{ij}$.
+
+Most of the inventory in the network is held in the distribution centers where the inventory is managed with a $(Q, r)$ policy with type I service.⁵ Inventory costs are incurred at each distribution centers, and consist of both working inventory and safety stock. The retailers only maintain a very small amount of inventory whose costs are ignored.
+
+The problem is to determine how many distribution centers (DCs) to install, where
\ No newline at end of file
diff --git a/samples/texts/7149586/page_9.md b/samples/texts/7149586/page_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..2131a3aa85e60f229a2a70903b1b1468e05a7ed9
--- /dev/null
+++ b/samples/texts/7149586/page_9.md
@@ -0,0 +1,22 @@
+to locate them, which DCs to assign to each retailer, how often to reorder for
+replenishment at each DC, and what level of safety stock to maintain so as to minimize
+the total location, transportation, and inventory costs, while ensuring a specified level
+of service.
+
+**Figure 4 Supply chain network structure (three echelons)**
+
+# 4. Model Formulation
+
+The joint supply chain network design and inventory management model of Shen et al. ¹⁹ is used as the basis for the present work, in which we will not rely on the assumption that each customer has identical variance-to-mean ratio. The joint location-inventory model is a nonlinear integer program that deals with the supply chain network design for a given product, and considers its detailed inventory management. The definition of sets, parameters, and variables of the model are as follows:
+
+**Sets/Indices**
+
+$I$ Set of retailers indexed by $i$
+
+$J$ Set of candidate DC site indexed by $j$
+
+**Parameters**
+
+Fixed cost (annual) of locating a DC at candidate site $j$
+
+$f_j$
\ No newline at end of file