| i = 6 | -(λi2 + g12g22/a1/3)e-q | -2e-qλa1/6/(g12g2) | (λi2 + g12g22/a1/3)e-q | (2e-qλa1/6/(g12g2)) e-qλa1/6(g12+g22) / (g12+g22) e-qλa-6/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30/7/30//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//7//8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999\\end{table}
+
diff --git a/samples/texts/3953766/page_7.md b/samples/texts/3953766/page_7.md
new file mode 100644
index 0000000000000000000000000000000000000000..ea9b176b1a2c800fc378dab1523aa44b602d9901
--- /dev/null
+++ b/samples/texts/3953766/page_7.md
@@ -0,0 +1,21 @@
+**Case 3.** First, the director is initially aligned in the second direction, such that $g_1 = a^{-1/6}$ and $g_2 = a^{1/3}$. The asymptotic limit of large stiffness ratio $\beta = \mu_f/\mu_s$ gives
+
+$$ \lambda_{\text{cr}} = a^{-1/6} - \frac{a^{-7/18}}{2} \left(\frac{3}{\beta}\right)^{2/3} + \mathcal{O}\left(\beta^{-4/3}\right), \quad (63) $$
+
+$$ q_{\text{cr}} = a^{-1/9} \left( \frac{3}{\beta} \right)^{1/3} + \mathcal{O} \left( \beta^{-1} \right). \qquad (64) $$
+
+Figure 6: Case 3 (LCE film, with director aligned in the second direction, on elastic substrate): The critical stretch ratio $0 < \lambda_{\text{cr}} < \min\{a^{1/12}, a^{1/3}\}$ and critical wave number $q_{\text{cr}}$ as functions of relative stiffness ratio $\beta = \mu_f/\mu_s$ when $g_1 = a^{-1/6}$ and $g_2 = a^{1/3}$. For different magnitudes of the nematic parameter $a$, the solid lines represent critical values, while the dashed lines show the value $\min\{a^{1/12}, a^{1/3}\}$, respectively.
+
+For finite values of $0 < q < \infty$ (see Appendix C for $q \to \infty$), Figure 6 displays the critical stretch ratios $\lambda_{\text{cr}}$ and critical wave numbers $q_{\text{cr}}$ as functions of the stiffness ratio $\beta$:
+
+* When $a < 1$, wrinkles parallel to the director form at the critical stretch ratio $0 < \lambda_{\text{cr}} < a^{1/3}$;
+
+* When $a > 1$, wrinkles parallel to the director are obtained at the critical stretch ratio $0 < \lambda_{\text{cr}} < a^{1/12}$.
+
+At constant temperature, the discussion is similar to that of Case 1. Namely, if the prolate LCE film is in its natural state prior to being attached to the elastic substrate, then decreasing the value of $\lambda < a^{-1/6}$ will not cause shear striping in the film.
+
+**Case 4.** Second, the director is initially aligned in the first direction, such that $g_1 = a^{1/3}$ and $g_2 = a^{-1/6}$. In the asymptotic limit of large stiffness ratio $\beta = \mu_f/\mu_s$, we find
+
+$$ \lambda_{\text{cr}} = a^{1/3} - \frac{a^{4/9} (a^{1/2} + 1)}{2^{5/3}} \left(\frac{3}{\beta}\right)^{2/3} + O\left(\beta^{-4/3}\right), \quad (65) $$
+
+$$ q_{\text{cr}} = \left( \frac{a^{2/3} + a^{1/6}}{2} \right)^{1/3} \left( \frac{3}{\beta} \right)^{1/3} + O\left( \beta^{-1} \right). \quad (66) $$
\ No newline at end of file
diff --git a/samples/texts/3953766/page_9.md b/samples/texts/3953766/page_9.md
new file mode 100644
index 0000000000000000000000000000000000000000..2c52251f0d4e29928a26bc69fa3cf705e0c98568
--- /dev/null
+++ b/samples/texts/3953766/page_9.md
@@ -0,0 +1,39 @@
+where $\xi''$ and $\xi^{iv}$ are the derivatives of order 2 and 4, respectively, of the normal deflection $\xi$ with
+respect to $x_1$, $S$ is the plane stress component in the first direction of the uniformly deformed film,
+and $f_s$ is the normal pressure exerted in the third direction by the (infinitely deep) substrate.
+
+Taking the wrinkling solution [12,24]
+
+$$ \xi = \xi_0 \cos(qx_1), \quad f_s = -2\mu_s q\xi = -2\mu_s q\xi_0 \cos(qx_1), \qquad (68) $$
+
+yields
+
+$$ S = -\frac{\mu_f}{3}q^2h^2 - \frac{2\mu_s}{qh}. \qquad (69) $$
+
+The critical wave number, which minimises the above stress, and the corresponding critical stress then take the form (see also [1])
+
+$$ q_{cr} = \frac{1}{h} \left( \frac{3}{\beta} \right)^{1/3}, \quad S_{cr} = -\mu_f \left( \frac{3}{\beta} \right)^{2/3}, \qquad (70) $$
+
+where $\beta = \mu_f/\mu_s$, as before. We recover here the scaling with respect to $\beta$ given in [53] based on a LCE plate theory. Equation (67) and the critical values given by (70) are valid for both elastic and LCE films. The difference between these two types of films lies in the material constitutive law, and hence, in the formula for the stress $S_{cr}$ in terms of the deformation (see [63] for details). Specifically, in each of the four cases described in the previous section, the critical plane stress $S_{cr}$ and the critical stretch ratio $\lambda_{cr}$ are related as follows:
+
+**Cases 1 and 2.** The film is made of an isotropic elastic material, hence,
+
+$$ S_{cr} = 2\mu_f (\lambda_{cr} - a^{-1/6}) \qquad (71) $$
+
+and, by (70),
+
+$$ \lambda_{cr} = a^{-1/6} - \frac{1}{2} \left(\frac{3}{\beta}\right)^{2/3}. \qquad (72) $$
+
+**Case 3 and 4.** The film is made of a monodomain liquid crystal elastomer, hence,
+
+$$ S_{cr} = 2\mu_f (\lambda_{cr} - g_1) \qquad (73) $$
+
+and, by (70),
+
+$$ \lambda_{cr} = g_1 - \frac{1}{2} \left( \frac{3}{\beta} \right)^{2/3}. \qquad (74) $$
+
+In each case, the critical stretch value for the plate model approximates well the corresponding value for the 3D models in the asymptotic limit of large stiffness ratio, $\beta \to \infty$. We conclude that the plate model is applicable when the film is much stiffer than the substrate.
+
+# 7 Conclusion
+
+We conducted a linear stability analysis for the onset of wrinkling in nonlinear two-layer systems formed from a hyperelastic film on a liquid crystal elastomer substrate, or a liquid crystal elastomer film on a hyperelastic substrate. We assumed that the hyperelastic material is described by a neo-Hookean strain-energy function, while the LCE is characterised by the neoclassical strain density. We also assumed that the substrate is infinitely deep and that the nematic director is uniformly aligned either parallel with or orthogonal to the biaxial forces causing wrinkling. We note that bilayer systems where the substrate is of arbitrary thickness can also be analysed by the same method if the original 8 × 8 algebraic system, which we derived, is used instead of its reduced 6 × 6 form corresponding to the
\ No newline at end of file
diff --git a/samples/texts/5029325/page_10.md b/samples/texts/5029325/page_10.md
new file mode 100644
index 0000000000000000000000000000000000000000..4d808b76c6a408a168ff2ecf0d87f54dd0f131a2
--- /dev/null
+++ b/samples/texts/5029325/page_10.md
@@ -0,0 +1,19 @@
+## 1.4 Open Problems
+
+In this work we prove that there exist unsatisfiable k-CNF formulas in $n$ variables that require $\delta$-regular Resolution refutations of size at least $2^{(1-\epsilon)n}$, where $k = \tilde{O}(\epsilon^{-4})$ and where $\delta = \tilde{O}(\epsilon^{-4})$. Hence a natural question is whether it is possible to improve the dependency of $\delta$ and $k$ on $\epsilon$.
+
+More generally, we have some proof systems stronger than $\delta$-regular Resolution, such as Resolution itself, Polynomial Calculus + Resolution, RES($k$), Cutting Planes, for which we know that there are some unsatisfiable CNFs in $n$ variables which require refutations of size $2^{\Omega(n)}$. Are those proof systems consistent with SETH?
+
+## 2 Preliminaries
+
+A *literal* is either a variable $x$ or its negation $\neg x$. A *clause* $C$ is a disjunction of literals and by its *width* we mean the number of literals appearing in $C$ and we denote this by $|C|$. A *Conjunctive Normal Form (CNF)* formula is a conjunction of a set of clauses.
+
+Given a boolean function $f$ on a set of variables $X$, a *partial assignment* is a function $\rho : X \to \{0, 1, *\}$. We call *domain* of $\rho$, dom($\rho$) the set $\rho^{-1}(\{0, 1\})$. The restriction of $f$ to $\rho$ denoted by $f|_{\rho}$ is a function on $\rho^{-1}(*)$ obtained from $f$ by fixing the value of all variables in $\rho^{-1}(0) \cup \rho^{-1}(1)$ according to $\rho$. We write $\rho \subseteq \sigma$ if for all $x \in X$, $\rho(x) \neq *$ implies $\sigma(x) = \rho(x)$. For a partial assignment $\rho$ for which $\rho(x) = *,$ by $\rho \cup \{(x, b)\}$ we denote a partial assignment $\rho'$ such that for all $y \neq x$, $\rho'(y) = \rho(y)$ and $\rho'(x) = b$. Given a (partial) assignment $\rho$ and a subset $B \subseteq X$, $\rho|_B$ is a partial assignment defined only on the variables in $B$ such that for all $x \in B$, $\rho|_B(x) = \rho(x)$.
+
+Resolution [7,24] is a proof system for refuting unsatisfiable CNF formulas. The only inference rule in Resolution is given as follows
+
+$$ \frac{C \lor x, D \lor \neg x}{C \lor D}, $$
+
+where $C$ and $D$ are clauses and $x$ is a variable. We say that $x$ is *resolved* and $C \lor D$ is called the *resolvant* of $C \lor x$ and $D \lor \neg x$. A *Resolution derivation* of a clause *D* from a CNF $\varphi$ is a sequence $\Pi = (C_1, ..., C_\tau)$ of clauses such that $C_\tau = D$ and each $C_i$ is either an *axiom*, that is a clause from $\varphi$, or it is derived by applying the Resolution rule on some clause $C_j$ and $C_{j'}$ such that $j, j' < i$. We will denote this by $\Pi : \varphi \vdash D$. When defining subsystems of Resolution we consider hardcoded in the sequence of clauses $\Pi$ also a function providing from which previous clauses a clause in $\Pi$ is inferred or if it is a clause from $\varphi$. Having at hand such function then a Resolution derivation $\Pi$ is given a structure of a DAG and hence we can talk of *paths* in the derivation intending paths in the DAG associated to the derivation. If $\varphi$ is an unsatisfiable formula, a *Resolution refutation* of $\varphi$ is a derivation of $\bot$, the empty clause, from $\varphi$. Resolution is *sound and complete*, that is we can derive $\bot$ from a CNF formula if and only if it is unsatisfiable.
+
+A $\delta$-regular *Resolution derivation* of a clause *D* from a formula $\varphi$ in $n$ variables is a Resolution derivation in which along any derivation path at most a fraction of $\delta$ variables are resolved multiple times. Hence a 0-regular Resolution refutation is just
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diff --git a/samples/texts/5029325/page_11.md b/samples/texts/5029325/page_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..6ee320341949dc9d2804e9d8f3aafaa2d27d6aad
--- /dev/null
+++ b/samples/texts/5029325/page_11.md
@@ -0,0 +1,33 @@
+a standard regular refutation and a 1-regular Resolution refutation is one without any constraint.
+
+The *size* of a Resolution derivation is the number of clauses appearing in it. We denote the minimum size of a derivation of *D* from $\varphi$ by **size**($\varphi \vdash D$). We also denote the minimum size of a $\delta$-regular derivation of *D* from $\varphi$ by **size**$_{\delta}$ ($\varphi \vdash D$). Similarly we define the *width* of a derivation to be the width of the largest clause appearing in it. We denote the minimum width of a derivation of *D* from $\varphi$ by **width**($\varphi \vdash D$).
+
+### 3 A Game View of Resolution
+
+In this section we present a common framework, Definition 3.1, for the games described by Atserias and Dalmau [1] and Pudlák [22] and then we recall the characterizations of width and size in Resolution.
+
+**Definition 3.1 (Game**(*φ*, *R*)**).** Given an unsatisfiable CNF formula *φ* in *n* variables and a set of partial assignments *R* containing the empty assignment, we define a game, **Game**(*φ*, *R*), between two players **Prover** (he) and **Delayer** (she).
+
+At each step *i* of the game a partial assignment αᵢ ∈ R is maintained (α₀ is the empty partial assignment), then at step *i* + 1 the following moves take place:
+
+1. **Prover** picks some variable *x* ∉ dom(αᵢ).
+
+2. **Delayer** then has to answer *x* = *b* for some bit *b* ∈ {0, 1}.
+
+3. Prover set αᵢ₊₁ ∈ R such that αᵢ₊₁ ⊆ αᵢ ∪ {⟨x, b⟩}.
+
+If at any point in the game αᵢ falsifies φ then Prover wins; otherwise we say that Delayer wins. As customary, we say that Prover has a winning strategy for the game if for any strategy of Delayer, he can play so that he wins the game. Otherwise we say that Delayer has a winning strategy.
+
+If in each run of the game Prover can query at most a fraction of δ variables, we call the corresponding game **Game**δ(φ, R).
+
+For a suitable choice of R the Game(φ, R) is exactly the one used by Atserias and Dalmau [1] to characterise the minimal width of Resolution refutations of φ. In particular in [1] the following result is shown (rephrased here with the notations we just set up).
+
+**Theorem 3.2 (Atserias and Dalmau [1]).** Let φ be an unsatisfiable CNF formula and let R be the set of all possible partial assignments with a domain of size strictly less than w. The following are equivalent
+
+1. Prover has a winning strategy for Game(φ, R);
+
+2. $width(\varphi \vdash \bot) < w.$
+
+Due to this equivalence, for this particular choice of R, we will denote Game(φ, R) by width-Game(φ, w).
+
+The next result is essentially due to Pudlák [22]: he shows that we can also characterize the minimal size of Resolution refutations of φ in terms of these games. From a Resolution refutation Π we can construct a winning strategy for Prover with a set R
\ No newline at end of file
diff --git a/samples/texts/5029325/page_12.md b/samples/texts/5029325/page_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..f211338898dabfcaa5ca900b630e6aaca48d6900
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@@ -0,0 +1,43 @@
+of the same size of Π and vice versa. Moreover a play of the $\mathbf{\Gamma}_{\delta}(\varphi, \mathcal{R})$ corresponds to a path in Π and, if Π is $\delta$-regular, in each run the set of variables **Prover** is going to query many times is at most a $\delta$ fraction of the total number of variables.
+
+**Theorem 3.3** Let $\varphi$ be an unsatisfiable CNF and let $\delta$ be any real in the interval $[0, 1]$. The following are equivalent
+
+1. there exists a set of partial assignments $\mathcal{R}$ such that $|\mathcal{R}| \le s$ for which **Prover** has a winning strategy for $\mathbf{\Gamma}_{\delta}(\varphi, \mathcal{R})$;
+
+2. $\text{size}_{\delta}(\varphi \vdash \bot) \le s$.
+
+Notice that this Theorem states an equivalence but in what follows we will only
+use the fact that (2) implies (1).
+
+4 Games and Xorifications
+
+Given a CNF formula $\varphi$ on the variables $x_1, \dots, x_n$, we define the $\ell$-xorification of $\varphi$
+as follows: it is a formula on the new variables $y_i^j$, where $1 \le i \le n$ and $1 \le j \le \ell$
+and it is obtained by replacing each $x_i$ with $y_i^1 \oplus \dots \oplus y_i^\ell$ and expanding the formula
+as a CNF formula. We denote the obtained CNF formula by $\varphi[\oplus^\ell]$ and note that if $\varphi$ is
+a $k$-CNF in $n$ variables, then $\varphi[\oplus^\ell]$ is a $k\ell$-CNF in $n\ell$ variables. Due to this notation
+we will refer to the variables of $\varphi$ as the $x$-variables and to the variables of $\varphi[\oplus^\ell]$ as
+the $y$-variables. Moreover we say that all the $y$-variables $y_i^1, \dots, y_i^\ell$ form a block of
+variables corresponding to the $x$-variable $x_i$. We say that a partial assignment over the
+$y$-variables fixes a value for a $x$-variable $x_i$ if it assigns all the $y$-variables in the block
+corresponding to $x_i$.
+
+**Restated Theorem 1.2** Let $\varphi$ be an unsatisfiable CNF formula in $n$ variables and $w$, $\delta$ and $\ell$ be parameters. If the width to refute $\varphi$ is Resolution is at least $w$ then the size to refute $\varphi[\oplus^\ell]$ in $\delta$-regular Resolution is at least $2^{(1-\epsilon)w\ell}$, where $\epsilon = \frac{1}{\ell}\log(\frac{e^3\ell n}{w}) + \frac{\delta n}{w}\log\frac{e^3\ell}{\delta}$.
+
+*Proof* For each partial assignment $\alpha$ over the $y$-variables there is naturally associated
+a partial assignment $\alpha'$ over the $x$-variables, defined as follows
+
+$$
+\alpha'(x_i) = \begin{cases}
+\alpha(y_i^1) \oplus \cdots \oplus \alpha_r(y_i^\ell) & \text{if } \forall j = 1, \ldots, \ell, y_i^j \in \operatorname{dom}(\alpha), \\
+* & \text{otherwise.}
+\end{cases}
+$$
+
+By Theorem 3.3, it is enough to show that if **Prover** wins **Game**$_{\delta}(\varphi[\oplus^\ell], \mathcal{R})$ then
+
+$$
+|\mathcal{R}| \geq 2^{w(\ell - \log(\frac{e^3\ell n}{w}) - \frac{\delta\ell n}{w} \log \frac{e^3\ell}{\delta})}.
+$$
+
+So suppose Prover wins Game$_{\delta}(\varphi[\oplus^{\ell}], R)$ for some set of partial assignments $R$. Since width$(\varphi \vdash \bot) \ge w$, by Theorem 3.2, there is a winning strategy $\sigma$ for Delayer in the game width-Game$(\varphi, w)$.
\ No newline at end of file
diff --git a/samples/texts/5029325/page_13.md b/samples/texts/5029325/page_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..3ff30f16704788835dd5d77ce571a7e1a2188fce
--- /dev/null
+++ b/samples/texts/5029325/page_13.md
@@ -0,0 +1,25 @@
+For each total assignment $\beta$ on the y-variables, consider a strategy $\sigma_\beta$ for **Delayer** in the game $\mathbf{Game}_\delta(\varphi[\oplus^\ell], \mathcal{R})$ as follows. Let $\alpha_r$ be the partial assignment on y-variables at stage $r$ of the game $\mathbf{Game}_\delta(\varphi[\oplus^\ell], \mathcal{R})$ and $y_i^j$ the variable queried by **Prover** at stage $r+1$. Then the strategy $\sigma_\beta$ for **Delayer** goes as follows:
+
+1. if there exists $j' \neq j$ such that $y_i^{j'} \notin \text{dom}(\alpha_r)$, set $y_i^j$ to $\beta(y_j^i)$;
+
+2. otherwise, if for all $j' \neq j$, $y_i^{j'} \in \text{dom}(\alpha_r)$, then look at the value $b \in \{0, 1\}$ the strategy $\sigma$ sets the variable $x_i$ when given the partial assignment $\alpha_r'$. Then set $y_i^j$ to $q \in \{0, 1\}$ such that
+
+$$q \oplus \bigoplus_{j' \neq j} \alpha_r (y_i^{j'}) = b.$$
+
+This can be done since $x_i \equiv y_i^1 \oplus \cdots \oplus y_i^\ell$ and the value of $x_i$ can be set freely to 0 or 1 appropriately even after all but one of $y_i^1, \dots, y_i^\ell$ have been set. Moreover, by induction on $r$, it is easy to see that the strategy $\sigma$ must provide an answer when challenged by **Prover** by any variable $x_i$ not on the record $\alpha_r$, hence $\sigma_\beta$ is well-defined.
+
+Moreover, it is easy to see that for each total assignment $\beta$ over the y-variables, $\sigma_\beta$ is a winning strategy for **Delayer** in the game width-Game$(\varphi[\oplus^\ell], w\ell)$. Since we are assuming that **Prover** has a winning strategy for $\mathbf{Game}_\delta(\varphi[\oplus^\ell], \mathcal{R})$, in particular, this means that for any $\beta$ he wins against the **Delayer**'s strategy $\sigma_\beta$. This means that for each total assignment $\beta$ over the y-variables, $\mathcal{R}$ must contain some partial assignment, denoted by $\rho_\beta$, with domain of size at least $w\ell$ and such that at least $w$ blocks of y-variables are completely fixed by $\rho_\beta$. Without loss of generality we assume that each $\rho_\beta$ fixes exactly $w$ blocks of y-variables, that is if $\rho_\beta$ is setting more y-variables we simply ignore some of the variables and only consider $w$ blocks. Our goal is to show that we have ‘many distinct’ such partial assignments $\rho_\beta$.
+
+Let $B \subseteq [n]$ denote a generic set of size $w$ and consider for each possible such $B$ the set $S_B$ of the total assignments $\beta$s such that $\rho_\beta$ is fixing all the $y_i^1, \dots, y_i^\ell$ corresponding to some $i$ in $B$. There are $2^{n\ell}$ possible total assignments $\beta$ and $\binom{n}{w}$ possible sets $B$, hence by the pigeonhole principle, there is a set $B^* \subseteq [n]$ of size $w$ such that
+
+$$|S_{B^*}| \geq \frac{2^{n\ell}}{\binom{n}{w}}. \quad (1)$$
+
+Let $S_{B^*}'$ be the set of partial assignments $\beta|_{B^*}$ where $\beta \in S_{B^*}$. We clearly have that
+
+$$|S_{B^*}| \le |S_{B^*}'| \cdot 2^{n\ell - |\overline{B^*}|} = |S_{B^*}'| \cdot 2^{n\ell - w\ell}.$$
+
+By Eq. (1), we get
+
+$$|S_{B^*}'| \geq \frac{2^{w\ell}}{\binom{n}{w}}. \quad (2)$$
+
+We have now that both $S_{B^*}'$ and $\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}$ consist of assignments with domain the y-variables $y_i^j$ such that $i \in B^*$ and $1 \le j \le l$. We show that $|\{\rho_\beta \in \mathcal{R}\}:
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diff --git a/samples/texts/5029325/page_2.md b/samples/texts/5029325/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..c46d053be98aa3f99240f5f9ce1f19c23667ab64
--- /dev/null
+++ b/samples/texts/5029325/page_2.md
@@ -0,0 +1,27 @@
+$β \in S_{B^*} |$ cannot be too small compared to $|S'_{B^*}|$, this will be, intuitively, due to the fact that the $β$s we start with are very different.
+
+Let $Z^\beta$ be the set of variables that Prover re-queried when playing against $\sigma_\beta$ and for any $i = 1, \dots, n$ let $Z_i^\beta = Z^\beta \cap \{y_i^1, \dots, y_i^\ell\}$. By hypothesis, Prover is allowed to re-query in each game at most a $\delta$ fraction of variables, hence $|Z^\beta| \le \delta\ln n$.
+
+When Delayer follows the strategy $\sigma_\beta$ and fixes all y-variables in a block corresponding to $x_i$, the assignment produced $\rho_\beta$ is within Hamming distance $|Z_i^\beta| + 1$ from $\beta$ in that block. This means that for each $\beta \in S_{B^*}$ and for each $i$, $\rho_\beta|_{\{y_1^{i_1}, \dots, y_\ell^{i_\ell}\}}$ has Hamming distance at most $|Z_i^\beta| + 1$ from some partial assignment in $S'_{B^*}$ restricted to $\{y_1^{i_1}, \dots, y_\ell^{i_\ell}\}$. Let $\mathcal{Z}$ be the set of all possible sets Z that are subsets of the y-variables of size $\delta\ell n$ and such that there exists $\beta \in S_{B^*}$ with $Z^\beta \subseteq Z$. For any $i=1, \dots, n$ let $Z_i = Z \cap \{y_i^1, \dots, y_i^\ell\}$. Then, by counting the variables where $\rho_\beta$ and an assignment in $S'_{B^*}$ could differ, we have that
+
+$$|S'_{B^*}| \le |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \sum_{Z \in \mathcal{Z}} \prod_{i \in B^*} 2^{|Z_i|+1} \binom{\ell}{|Z_i|+1}. \quad (3)$$
+
+Hence we have the following chain of inequalities
+
+$$|S'_{B^*}|^{\text{eq.}(3)} \le |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \sum_{Z \in \mathcal{Z}} \prod_{i \in B^*} 2^{|Z_i|+1} \binom{\ell}{|Z_i|+1} \quad (4)$$
+
+$$\le |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \sum_{Z \in \mathcal{Z}} \prod_{i \in B^*} \left( \frac{e^{2\ell}}{|Z_i|+1} \right)^{|Z_i|+1} \quad (5)$$
+
+$$\le |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \sum_{Z \in \mathcal{Z}} \left( \frac{\sum_{i \in B^*} e^{2\ell}}{\sum_{i \in B^*} (|Z_i|+1)} \right)^{\sum_{i \in B^*} (|Z_i|+1)} \quad (6)$$
+
+$$\le |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \binom{\ell n}{\delta \ell n} \cdot \left(\frac{\sum_{i \in B^*} e^{2\ell}}{w}\right)^{\delta \ell n + w} \quad (7)$$
+
+$$= |\{\rho_\beta \in \mathcal{R} : \beta \in S_{B^*}\}| \cdot \binom{\ell n}{\delta \ell n} \cdot (e^{2\ell})^{\delta \ell n + w} \quad (8)$$
+
+The inequality (6) follows from the weighted AM-GM inequality¹ and the inequality (7) follows from the fact that $w \le \sum_{i \in B^*} (|Z_i| + 1) \le \delta\ell n + w$. Putting all together we have that
+
+¹ The weighted Arithmetic Mean - Geometric Mean inequality says that given non-negative numbers $a_1, \dots, a_n$ and non-negative weights $w_1, \dots, w_n$ then
+
+$$\prod_i a_i^{w_i} \le \left( \frac{\sum_i w_i a_i}{w} \right)^w,$$
+
+where $w = \sum_i w_i$. We applied this inequality with $a_i = \frac{e^{2\ell}}{|Z_i|+1}$ and $w_i = |Z_i| + 1$.
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diff --git a/samples/texts/5029325/page_7.md b/samples/texts/5029325/page_7.md
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+## 1.1 Previous Work
+
+Exponential lower bounds supporting ETH have long been known for natural proof systems such as Resolution since mid 1980s, see e.g. [28]. These are $2^{\Omega(n)}$ lower bounds for k-CNF formulas on n variables and hence not strong enough to support SETH. Some thirteen years passed until the first lower bounds supporting SETH were shown. Pudlák and Impagliazzo [23] proved such lower bounds for tree-like Resolution via Prover-Delayer games. Another thirteen years later, Beck and Impagliazzo [5] obtained a very strong width lower bound which simplified and improved the result of [23] for tree-like Resolution and they were able to prove lower bounds supporting SETH for regular Resolution, a sub-system of Resolution. Beck and Impagliazzo in [5] showed that there are unsatisfiable k-CNF formulas in n variables requiring refutations of size at least $2^{n(1-\epsilon_k)}$ in regular Resolution. Their proof is an adaptation of a probabilistic technique from [4] and, from an high level, it can be seen as a variation of the bottleneck counting of Haken in [16]. In their argument a rule is given which maps assignments to particular clauses of the proof, at which a significant amount of 'work' is done.
+
+Prior to this work, the strongest proof system with lower bounds supporting SETH was regular Resolution [5]. This work proves lower bounds supporting SETH in a subsystem of Resolution stronger than regular Resolution and moreover it gives a different, conceptually simpler, game-theoretic proof of the fact that SETH is consistent with regular Resolution.
+
+## 1.2 Results
+
+In this work we consider proof systems that are intermediate between regular Resolution and Resolution. The *Resolution* proof system [7, 24] is a proof system for refuting unsatisfiable CNF formulas. A Resolution refutation of a CNF formula $\varphi$ is a sequence of clauses ending with the empty clause such that each clause is either a clause from $\varphi$ or it is derived from previous clauses in the sequence according to the following inference rule:
+
+$$ \frac{C \lor x, \quad D \lor \neg x}{C \lor D}, $$
+
+where C and D are clauses and x is a variable. Clearly a Resolution refutation can be annotated with directed edges keeping track of the applications of the inference rule, in particular each clause in the sequence will have either 0 or 2 predecessors according to if it is a clause from $\varphi$ or an inferred clause. The resulting directed graph is a DAG and the sequence of clauses in the Resolution refutation is a topological ordering of it. Notice that given a resolution Refutation the DAG we can associate is not uniquely determined. Anyway, when defining subsystems of Resolution based on restrictions on the DAG structures allowed for a proof, it is customary to consider a Resolution refutation directly as a DAG with vertices labeled with clauses. This is the case for *tree-like* Resolution where are allowed as valid Resolution refutations only the ones with a tree-like structure.
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diff --git a/samples/texts/5029325/page_8.md b/samples/texts/5029325/page_8.md
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index 0000000000000000000000000000000000000000..433fef45aae46b0f122cedd371c883701f477966
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+A $\delta$-regular Resolution refutation of a formula $\varphi$ is a Resolution derivation in which along any path of the refutation DAG at most a fraction $\delta$ of the variables of $\varphi$ are resolved multiple times. Hence a 0-regular Resolution refutation is just a standard regular Resolution refutation, that is a Resolution refutation where no variable is resolved multiple times along any path. A 1-regular Resolution refutation is just one without any constraint. A more formal definition of Resolution and $\delta$-regular Resolution is given in Sect. 2 together with all the other necessary preliminaries.
+
+The main result of this work is the following.
+
+**Theorem 1.1 (Main theorem)** For any large $n$ and $k$, there exists an unsatisfiable $k$-CNF formula $\psi$ on $n' \ge n$ variables such that any $\delta$-regular Resolution refutation of $\psi$ requires size at least $2^{(1-\epsilon_k)n}$ where both $\epsilon_k$ and $\delta$ are $\tilde{O}(k^{-1/4})$.
+
+We recall that the width of a Resolution refutation is the number of literals in the largest clause appearing in the refutation. The way we prove this result is via a strong width lower bound, that is a lower bound of the form $(1-\epsilon_k)n$, with $\epsilon_k \to 0$, relative to Resolution refutations of some particular $k$-CNFs in $n$ variables. Width lower bounds of this form were proved in [5] and improved in the asymptotic in [8] (see Theorem 4.1 for the precise statement we are going to use).
+
+The reader could be tempted to think that once we are given a strong Resolution width of the form above then a size lower bound as in Theorem 1.1 will follow by the standard relation between width and size by Ben-Sasson and Wigderson [6]. In [6] the authors showed that if a formula requires refutations of large width, it also requires refutations with many clauses. More precisely they showed that if a $k$-CNF formula can only have Resolution refutations of width at least $W$, then it requires Resolution size at least $2^{(W-k)^2/16n}$, where $n$ is the number of variables. The constant loss in the exponent is the reason why we do not immediately get $2^{(1-\epsilon_k)n}$ size lower bounds from strong width lower bounds. However, the result in [6] holds for any $k$-CNF without any particular assumption. If the formula is structured in some sense, for instance if it is a *xorification*, we show we can avoid this loss (in a subsystem of Resolution).
+
+The $\ell$-xorification of a CNF formula $\varphi$ in $n$ variables is a new CNF formula in $n\ell$ variables obtained substituting each variable $x_i$ in $\varphi$ with $\ell$ new variables $y_i^1 \oplus \cdots \oplus y_i^\ell$ and then expanding again as a CNF. We denote the CNF resulting from such operation $\varphi[\oplus^\ell]$. Our main technical result, Theorem 1.2 informally states that if a $k$-CNF $\varphi$ requires width $w$ to be refuted in Resolution, then any $\delta$-regular Resolution refutation of $\varphi[\oplus^\ell]$ requires size $2^{(1-\epsilon)nw}$, where $\epsilon$ is a function of $k$, $\ell$, $\delta$ and $w$. More precisely we prove the following:
+
+**Theorem 1.2** Let $\varphi$ be an unsatisfiable CNF formula in $n$ variables and $w$, $\delta$ and $\ell$ be parameters. If the width to refute $\varphi$ is Resolution is at least $w$ then the size to refute $\varphi[\oplus^\ell]$ in $\delta$-regular Resolution is at least $2^{(1-\epsilon)w\ell}$, where $\epsilon = \frac{1}{\ell} \log(\frac{e^3\ell n}{w}) + \frac{\delta n}{w} \log \frac{e^3\ell}{\delta}$.
+
+Once we have proved this result then Theorem 1.1 follows just by carefully tuning the parameters.
+
+The way we prove Theorem 1.2 relies on two known games characterizations: the *Pudlák game* characterizing Resolution size [22] and the *Atserias-Dalmau game* characterizing Resolution width [1]. In Sect. 3 we give a precise account for both games in a common setting and terminology, see Definition 3.1. We conclude this
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+introductory part giving some intuition behind those games and the proof of Theorem 1.2.
+
+In the *Pudlák game*, informally, we have two players, *Prover* and *Delayer*, that play on some formula $\varphi$. *Prover* has the objective of showing that the formula $\varphi$ is unsatisfiable by querying variables. *Delayer* on the other hand wants to play as long as possible before the formula is falsified while answering to the queries *Prover* asks her. The size of Resolution proofs of $\varphi$ is then characterized as the minimal number of records, i.e. partial assignments, *Prover* has to consider in a winning strategy [22]. If we force the *Prover* in each game to re-query a fraction of at most $\delta$ variables from $\varphi$ then the minimal number of records such *Prover* has to consider in a winning strategy characterize the size in $\delta$-regular Resolution. This is the content of Theorem 3.3 which we leave without proof since it is a trivial observation over the result in [22].
+
+Hence to prove a Resolution (or a $\delta$-regular Resolution) size lower bound we show that, in order to win, **Prover** must keep a large number of records and we can do that by producing a lot of sufficiently different strategies for **Delayer**. **Prover** must win against each of them, hence in his winning strategy he must have a lot of distinct records, since the strategies of **Delayer** are sufficiently different. In the literature this is done essentially by making **Prover** play against a **Delayer** that plays accordingly to a random strategy [12,22]. Then the size lower bound, that is a lower bound on the number of records that **Prover** must have in a winning strategy, is obtained by probabilistic arguments. This may very likely lead to some loss in the constants that we need to avoid to prove a SETH lower bound for Resolution size so we choose another way: we play the Pudlák game over a xorified formula $\varphi[\oplus^\ell]$.
+
+The construction of multiple strategies for Delayer relies on the characterization
+of Resolution width as a game [1] where again we have a Prover and a Delayer,
+the goal of the prover is to falsify the formula φ but he can use assignments with a
+bounded number of variables. At a very high level, a winning strategy for Delayer in
+the width game on φ gives rise to a multitude of strategies for Delayer on the Pudlák
+game on φ[⊕^l]. The new strategies act differently from each other on φ[⊕^l], but in
+a sense they all act the same as the original strategy for Delayer in the width game
+on the original formula φ. The size lower bound then follows by a counting argument
+exploiting the combinatorial properties of the xorified formula in such a way that the
+number of Delayer strategies, for the Pudlák game played on φ[⊕^l], does indeed
+hugely amplify.
+
+Notice that Theorem 1.2 does not depend on the particular formula we choose to
+apply it but, to get the result in Theorem 1.1, we need to apply it to a particular formula
+φ for which we have strong width lower bounds, such formulas are provided by [5]
+and, with better asymptotic, by [8].
+
+## 1.3 Outline of Paper
+
+In the next Section we give some preliminaries and notations about Resolution and δ-regular Resolution. Section 3 contains the common framework for *Pudlák games* [22] characterizing *size* in Resolution and the *Atserias and Dalmau games* [1] characterizing *width* in Resolution. Section 4 contains the core results of this work (Theorem 1.1 and Theorem 1.2).
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diff --git a/samples/texts/7104097/page_1.md b/samples/texts/7104097/page_1.md
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+# The Range 1 Query (R1Q) Problem*
+
+Michael A. Bender¹², Rezaul Chowdhury¹, Pramod Ganapathi¹,
+Samuel McCauley¹, and Yuan Tang³
+
+¹ Department of Computer Science, Stony Brook University, Stony Brook, NY, USA
+{bender, rezaul, pganapathi, smccauley}cs.stonybrook.edu
+
+² Tokutek, Inc.
+
+³ Software School, Fudan University, Shanghai, China
+yuantang@csail.mit.edu
+
+**Abstract.** We define the *range 1 query* (R1Q) problem as follows. Given a *d*-dimensional ($d \ge 1$) input bit matrix $A$, preprocess $A$ so that for any given region $\mathcal{R}$ of $A$, one can efficiently answer queries asking if $\mathcal{R}$ contains a 1 or not. We consider both orthogonal and non-orthogonal shapes for $\mathcal{R}$ including rectangles, axis-parallel right-triangles, certain types of polygons, and spheres. We provide space-efficient deterministic and randomized algorithms with constant query times (in constant dimensions) for solving the problem in the word RAM model. The space usage in bits is sublinear, linear, or near linear in the size of $A$, depending on the algorithm.
+
+**Keywords:** R1Q, range query, range emptiness, randomized, rectangular, orthogonal, non-orthogonal, triangular, polygonal, circular, spherical.
+
+## 1 Introduction
+
+Range searching is one of the fundamental problems in computational geometry [1,12]. It arises in application areas including geographical information systems, computer graphics, computer aided design, spatial databases, and time series databases. Range searching encompasses different types of problems, such as range counting, range reporting, emptiness queries, and optimization queries.
+
+The *range 1 query* (R1Q) problem is defined as follows. Given a *d*-dimensional ($d \ge 1$) input bit matrix $A$ (consisting of 0's and 1's), preprocess $A$ so that one can efficiently answer queries asking if any given range $\mathcal{R}$ of $A$ is empty (does not contain a 1) or not, denoted by $R1Q_A(\mathcal{R})$ or simply $R1Q(\mathcal{R})$. In 2-D, the range $\mathcal{R}$ can be a rectangle, a right triangle, a polygon or a circle.
+
+In this paper, we investigate solutions in the word RAM model sharing the following characteristics. First of all, we want queries to run in constant time, even for $d \ge 2$ dimensions. Second, we are interested in solutions that have space linear or sublinear in the number of bits in the input grid. Note that while our sublinear bounds are parameterized by the number of 1s in the grid, this is still larger
+
+* Rezaul Chowdhury & Pramod Ganapathi are supported in part by NSF grant CCF-1162196. Michael A. Bender & Samuel McCauley are supported in part by NSF grants IIS-1247726, CCF-1217708, CCF-1114809, and CCF-0937822.
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diff --git a/samples/texts/7104097/page_11.md b/samples/texts/7104097/page_11.md
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+We first show that for any given $p \in [0, \log N]$ at most $2\gamma$ fraction of the queries of size $2^p$ can return incorrect answers. Consider any two consecutive blocks of size $2^p$, say, blocks $i \in [0, \frac{N}{2^p} - 1)$ and $i+1$. Exactly $2^p$ different queries of size $2^p$ will cross the boundary between these two blocks. The answer to each of these queries will depend on the estimates of $R_p[i]$ and $L_p[i+1]$ obtained from the CM sketches. Under our construction the estimates are $\hat{R}_p[i] \le (1+\gamma)R_p[i] \le R_p[i] + \gamma \cdot 2^p$ and $\hat{L}_p[i+1] \le (1+\gamma)L_p[i+1] \le L_p[i+1] + \gamma \cdot 2^p$. Hence, at most $\gamma \cdot 2^p$ of those $2^p$ queries will produce incorrect results due to the error in estimating $R_p[i]$, and at most $\gamma \cdot 2^p$ more because of the error in estimating $L_p[i+1]$. Thus with probability at least $(1-\delta)^2$, at most $2\gamma$ fraction of those $2^p$ queries will return wrong results. Recall from Section 2.1 that we answer given queries by decomposing the query range into two overlapping query ranges. Hence, with probability at least $(1-\delta)^4 \ge 1-4\delta$, at most $2\gamma+2\gamma = 4\gamma$ fraction of all queries can produce wrong answers.
+
+**Theorem 4.** Given a 1-D bit array of length *N* containing *N*₁ nonzero entries, and two parameters γ ∈ (0, 1/4) and δ ∈ (0, 1/4), and an integer constant *c* > 1, one can construct a data structure occupying O(N₁ log³ N log₁+γ (1/δ) + N¹/c log log N) bits (and discard the input array) to answer each R1Q in O(log c/δ) worst-case time such that with probability at least 1 − 4δ at most 4γ fraction of all query results will be wrong.
+
+**Sampling Based Algorithm.** Suppose we are allowed to use only $\mathcal{O}(s)$ bits of space (in addition to the input array $A$), and $s = \Omega(\log_2 N)$. We are also given two constants $\varepsilon \in (0, 1)$ and $\delta \in (0, 1)$. We build $L_p$ and $R_p$ arrays for each $p \in [\log \frac{N}{s} + \log \log N, \log N]$, and an MSB lookup table to support constant time MSB queries for integers in $[1, s/\log N]$. Consider the query $R_1 Q_A(i, j)$. If $j-i+1 \le w$, we answer the query correctly in constant time by reading at most 2 words from $A$ and using bit shifts. If $j-i+1 \ge 2^{\log \frac{N}{s} + \log \log N} = \frac{N \log N}{s}$, we use the $L_p$ and $R_p$ arrays to correctly answer the query in constant time. If $w < j-i+1 < \frac{N \log N}{s}$, we sample $\lceil \frac{1}{\epsilon} \ln (\frac{1}{\delta}) \rceil$ entries uniformly at random from $A[i...j]$, and return their bitwise OR. It is easy to show that the $L_p$ and $R_p$ tables use $\mathcal{O}(s)$ bits in total, and the MSB table uses $o(s)$ bits of space. The query time is clearly $\mathcal{O}(\frac{1}{\epsilon} \ln (\frac{1}{\delta}))$.
+
+**Error Bound.** If at least an $\epsilon$ fraction of the entries in $A[i...j]$ are nonzero then the probability that a sample of size $\lceil \frac{1}{\epsilon} \ln(\frac{1}{\delta}) \rceil$ chosen uniformly at random from the range will pick at least one nonzero entry is $\ge 1-(1-\epsilon)^{\frac{1}{\epsilon} \ln(\frac{1}{\delta})} \approx 1-\delta$.
+
+**Theorem 5.** Given a 1-D bit array of length *N*, a space bound *s* = *Ω*(log *N*), and two parameters *ε* ∈ (0,1) and *δ* ∈ (0,1), one can construct a data structure occupying only O(*s*) bits of space (in addition to the input array) that in O(1/ε ln(1/δ)) time can answer each R1QA(i,j) correctly with probability at least 1 − δ provided at least an ε fraction of the entries in A[i...j] are nonzero. If j−i+1 ≤ *w* or j−i+1 ≥ N log N/s, the query result is always correct.
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+**Fig. 4.** Black grid points contain 1's and white grid points contain 0's. Polygon in (a) satisfies Prop. 1. Polygons in (b) and (c) do not satisfy Prop. 1. Still, R1Q can be answered for (c).
+
+in constant time). Observe that we can answer R1Q(*ABC*) if we can answer R1Q for triangles *ADG* and *CEH*, and the rectangle *BDFE*. R1Q for the rectangle can be answered using our deterministic algorithm described in Section 2.2. R1Q for a right triangle of a particular orientation with height or base length equal to a power of two can be answered in constant time. This is done by checking whether the point stored (from preprocessing) with the appropriate endpoint of the hypotenuse for that specific orientation is inside the triangle or not.
+
+**Theorem 6.** Given a 2-D bit matrix of size $N = \sqrt{N} \times \sqrt{N}$ containing $N_0$ zero bits, one can construct a data structure occupying $O(N \log N + N_0 \log^2 N)$ bits in $O(N^{1.5})$ time (and discard the input matrix) to answer each axis-aligned right triangular R1Q with the three vertices on the grid points in $O(1)$ time.
+
+## 3.2 Polygonal R1Q
+
+Consider a simple polygon with its vertices on grid points satisfying the following.
+
+*Property 1.* For every two adjacent vertices $(a, b)$ and $(c, d)$, one of the two right triangles with the third vertex being either $(a, d)$ or $(c, b)$ is completely inside the polygon.
+
+It can be shown that such a polygon can be decomposed into a set of possibly overlapping right triangles and rectangles with only grid points as vertices that completely covers the polygon (see Fig. 4(a)). Examples of polygons that do not satisfy the constraint are given in Fig. 4(b,c), but we can still answer R1Q for the polygon in (c). A simple polygon with $k$ vertices satisfying property 1 can be decomposed into $O(k)$ right triangles and rectangles and hence can be answered in $O(k)$ time.
+
+## 3.3 Spherical R1Q
+
+The spherical R1Q problem is defined as follows. Given a $d$-dimensional ($d \ge 2$) input bit matrix $A$, preprocess $A$ such that given any grid point $p$ in $A$ and a radius $r \in \mathbb{R}^{+}$, find efficiently if there exists a 1 in the $d$-sphere centered at $p$ of radius $r$. Here, we present the algorithm for 2-D. The approach can be extended to higher dimensions.
+
+A nearest 1-bit of a grid point $p$ is called a *nearest neighbor* (NN) of $p$. We preprocess $A$ by computing and compressing the NNs of all grid points in $A$
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diff --git a/samples/texts/7104097/page_3.md b/samples/texts/7104097/page_3.md
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+(only one NN per grid point). We then answer a spherical R1Q by checking
+whether a NN of the given center point is inside the circle of given radius.
+
+**Preprocessing.** We store the locations of the NNs of the grid points of $A$ in a temporary NN matrix that occupies $\mathcal{O}(N \log N)$ bits, but can be compressed to occupy $\mathcal{O}(N\sqrt{\log N})$ bits as follows.
+
+We divide the grid into $\sqrt{\frac{\log N}{6}} \times \sqrt{\frac{\log N}{6}}$ blocks. We store the NN position for all points on the boundary of each block. The interior points will be replaced with arrows ($\rightarrow, \leftarrow, \uparrow, \downarrow$) and bullets ($\bullet$) as follows. If a grid point $p$ contains a 1 then $p$ is replaced with a $\bullet$. An arrow at a grid point gives the direction of its NN. If we follow the arrows from any interior point, we end up in either a boundary point or an interior point containing a 1. For any given block the matrix created as above will be called a *symbol matrix* representing the block.
+
+Two blocks are of the same type if they have the same symbol matrix. Each
+symbol can be represented using only three bits. Since each block has $(\log N)/6$
+symbols, there are $2^{\frac{3(\log N)}{6}} = \sqrt{N}$ possible block types. For each block type we
+create a *position matrix* that stores, for each grid point within a block, the pointer
+to its NN if the NN is an interior point, or a pointer to a boundary point if the
+NN is an exterior or boundary point. The boundary point will have stored its
+own NN position in the input array.
+
+We can now discard the original input matrix, and replace it with the follow-
+ing compressed representation. For each block in the input matrix we store its
+block type (i.e., a pointer to the corresponding block type) followed by the NN
+positions of its boundary points. For each block type we retain its position matrix.
+
+**Query Execution.** We can answer a spherical R1Q by checking whether the NN position of the center point is inside the query sphere. The approach of finding the NN position is as follows. We find the block to which the given point belongs and follow the pointer to its block type. We check the position stored at the given point in the position matrix. If it points to an internal point, then that point is the correct NN. If it points to a boundary point, we again follow the pointer stored at the boundary point to get the correct NN.
+
+**Theorem 7.** Given a 2-D bit array of size *N*, one can construct a data structure occupying $\mathcal{O}(N\sqrt{\log N})$ bits (and discard the input array) to answer each spherical R1Q in $\mathcal{O}(1)$ time.
+
+**Acknowledgments.** We like to thank Michael Biro, Dhruv Matani, Joseph S. B. Mitchell, and anonymous referees for insightful comments and suggestions.
+
+References
+
+1. Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. Contemporary Mathematics 223, 1–56 (1999)
+2. Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. Journal of Algorithms 57(2), 75–94 (2005)
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diff --git a/samples/texts/7104097/page_4.md b/samples/texts/7104097/page_4.md
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+3. Chazelle, B., Rosenberg, B.: Computing partial sums in multidimensional arrays. In: SoCG. pp. 131–139. ACM (1989)
+
+4. Cormode, G., Muthukrishnan, S.: An improved data stream summary: the count-min sketch and its applications. Journal of Algorithms 55(1), 58–75 (2005)
+
+5. De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational geometry. Springer (2008)
+
+6. Fischer, J.: Optimal succinctness for range minimum queries. In: LATIN, pp. 158–169. Springer (2010)
+
+7. Fischer, J., Heun, V., Stiihler, H.: Practical entropy-bounded schemes for o (1)-range minimum queries. In: Data Compression Conference. pp. 272–281. IEEE (2008)
+
+8. Golynski, A.: Optimal lower bounds for rank and select indexes. TCS 387(3), 348–359 (2007)
+
+9. González, R., Grabowski, S., Mäkinen, V., Navarro, G.: Practical implementation of rank and select queries. In: Poster Proc. WEA. pp. 27–38 (2005)
+
+10. Navarro, G., Nekrich, Y., Russo, L.: Space-efficient data-analysis queries on grids. TCS (2012)
+
+11. Overmars, M.H.: Efficient data structures for range searching on a grid. Journal of Algorithms 9(2), 254–275 (1988)
+
+12. Sharir, M., Shaul, H.: Semialgebraic range reporting and emptiness searching with applications. SIAM Journal on Computing 40(4), 1045–1074 (2011)
+
+13. Tang, Y., Chowdhury, R., Kuszmaul, B.C., Luk, C.K., Leiserson, C.E.: The Pochoir stencil compiler. In: SPAA. pp. 117–128. ACM (2011)
+
+14. Yao, A.C.: Space-time tradeoff for answering range queries. In: STOC. pp. 128–136. ACM (1982)
+
+15. Yuan, H., Atallah, M.J.: Data structures for range minimum queries in multidimensional arrays. In: SODA. pp. 150–160 (2010)
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+than the information-theoretic lower bounds. For our motivating applications, information-theoretically optimal space is less important than constant query times. Third, we are interested in grid inputs [10, 11], viewing the problem in terms of pixels/voxels rather than a set of spatial points. This grid perspective enables constant-time operations such as table lookup and hashing. Finally, we are interested in both orthogonal and nonorthogonal queries, and we require solutions that are concise enough to be implementable.
+
+**Previous Results.** The R1Q problem can be solved using data structures such as balanced binary search trees, kd-trees, quad trees, range trees, partition trees, and cutting trees (see [5]), which take the positions of the 1-bits as input. It can also be solved using a data structure of Overmars [11], which uses priority search trees, y-fast tries, and q-fast tries and takes the entire grid as input. However, in $d$-D ($d \ge 2$), in the worst case these data structures have a query time at least polylogarithmic and occupy a near-linear number of bits.
+
+The R1Q problem can also be solved via range partial sum [3, 14] and the range minimum query (RMQ) [2, 6, 7, 15] problems. Though several efficient algorithms have been developed to solve the problem in 1-D and 2-D, their generalizations to 3-D and higher dimensions occupying a linear number of bits are not known yet. Also, there is little work on space-efficient constant-time RMQ solutions for non-orthogonal ranges.
+
+The R1Q problem can also be solved using rank queries [8, 9]. Again, its generalization to 2-D and higher dimensions has not yet been studied.
+
+**Motivation.** We encountered the R1Q and R0Q (whether a range contains a 0) problems while trying to optimize stencil computations in the *Pochoir* stencil compiler [13], where we had to answer octagonal R1Q and octagonal R0Q on a static 2-D property grid. Stencil computations have applications in physics, computational biology, computational finance, mechanical engineering, adaptive statistical design, weather forecasting, clinical medicine, image processing, quantum dynamics, oceanic circulation modeling, electromagnetics, multigrid solvers, and many other areas (see the references in [13]).
+
+In Fig. 1, we provide a simplified exposition of the problem encountered in Pochoir. There are two grids of the same size: a static property grid and a dynamic value grid. Each property grid cell is set to 1 if it satisfies property $\mathcal{P}$ and 0 otherwise. When Pochoir needs to update a range $\mathcal{R}$ in the value grid (see Alg. 1), its runtime system checks whether all or none of the points in $\mathcal{R}$ satisfy $\mathcal{P}$ in the property grid, and based on the query result it uses an appropriate precompiled optimized version of the original code (see Algs. 3, 4) to update the range in the value grid. To check if all points in $\mathcal{R}$ satisfy $\mathcal{P}$, Pochoir uses $R_0\mathcal{Q}(\mathcal{R})$, and to check if no points in $\mathcal{R}$ satisfy $\mathcal{P}$, it uses $R_1\mathcal{Q}(\mathcal{R})$.
+
+Pochoir needs time-, space-, and cache-efficient data structures to answer R1Q. It can also tolerate some false-positive errors. The solutions should achieve constant query time and work in all dimensions. Although it is worth trading off space to achieve constant query times, space is still a scarce resource.
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+**Fig. 1.** Examples of the procedures in Pochoir that make use of R1Q and R0Q.
+
+**Our Contributions.** We solve the R1Q problem for orthogonal and non-orthogonal ranges. Our major contributions as shown in Table 1 are as follows:
+
+1. [Orthogonal Deterministic.] We present a deterministic data structure to answer R1Q for orthogonal ranges in all dimensions and for any data distribution. It occupies linear space in bits and answers queries in constant time for any constant dimension.
+
+2. [Orthogonal Randomized.] We present randomized data structures to answer R1Q for orthogonal ranges. The structures occupy sublinear space in bits and provide a tradeoff between query time and error probability.
+
+3. [Non-Orthogonal Deterministic.] We present deterministic data structures to answer R1Q for non-orthogonal shapes such as axis-parallel right-triangles (for 2-D) and spheres (for all dimensions). The structures occupy near-linear space in bits and answer queries in constant time.
+
+We use techniques such as power hyperrectangles, power right-triangles, sketches, sampling, the four Russians trick, and compression in our data struc-
+tures. A careful combination of these techniques allows us to solve a large class of
+R1Q problems. Techniques such as power hyperrectangles, table lookup, and the
+four Russians trick are already common in RMQ-style operations, while sketches,
+power right-triangles, and compression are not.
+
+**Organization of the Paper.** Section 2 presents deterministic and randomized algorithms to answer orthogonal R1Qs on a grid in constant time for constant dimensions. Section 3 presents deterministic algorithms to answer non-orthogonal R1Qs on a grid, for axis-parallel right triangles, some polygons, and spheres.
+
+## 2 Orthogonal Range 1 Queries (R1Q)
+
+In this section, we present deterministic and randomized algorithms for answering
+orthogonal R1Qs in constant time and up to linear space.
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+**Query Execution.** To answer R1Q$_A(i, j)$, we consider two cases: (1) *Intra-word queries*: If $(i, j)$ lies inside one word, we answer R1Q using bit shifts. (2) *Inter-word queries*: If $(i, j)$ spans multiple words, then the query gets split into three subqueries: (a) R1Q from $i$ to the end of its word, (b) R1Q of the words between $i$'s and $j$'s word (both exclusive), and (c) R1Q from the start of $j$'s word to $j$.
+
+The answer to an inter-word query is 1 if and only if the R1Q for at least one of the three subqueries is 1. The first and third subqueries are intra-word queries and can be answered using bit shifts. Let the words containing indices $i$ and $j$ be $I$ and $J$, respectively. Then, the second subquery, denoted by $R1Q_{L_0}(I+1, J-1)$, is answered as follows. Using the MSB of $J-I-1$, we find the largest integer $p$ such that $2^p \le J - I - 1$. The query $R1Q_{L_0}(I+1, J-1)$ is then decomposed into the following two overlapping queries of size $2^p$ each: $R1Q_{L_0}(I+1, I+2^p)$ and $R1Q_{L_0}(J-2^p, J-1)$. If either of those two ranges contains a 1 then the answer to the original query will be 1, and 0 otherwise. We show below how to answer $R1Q_{L_0}(I+1, I+2^p)$. Query $R1Q_{L_0}(J-2^p, J-1)$ is answered similarly.
+
+Split $L_0$ into blocks of size $2^p$. Then, the range $R1Q_{L_0}(I+1, I+2^p)$ can be covered by one or two consecutive blocks. Let $I+1$ be in the $k$th block. If the range lies in one block, we find whether a 1 exists in that block by checking whether $L_p[k] < 2^p$ is true. If the range is split across two consecutive blocks, we find whether a 1 exists in at least one of the two blocks by checking whether at least one of $R_p[k] \le (k+1)2^p - I$ or $L_p[k+1] \le I + 2^p - (k+1)2^p$ is true.
+
+## 2.2 Deterministic d-D Algorithm
+
+For d-D ($d \ge 2$) R1Q, the input is a bit matrix $A$ of size $N = n \times n$, but the algorithm extends an algorithm for a 2-D matrix of size $N = n \times n$, but the algorithm extends to higher dimensions. For simplicity, we assume $n$ is a power of 2. The query $R1Q([i_1, j_1][i_2, j_2])$ asks if there exists a 1 in the submatrix $A[i_1 \dots j_1][i_2 \dots j_2]$.
+
+**Preprocessing.** For each $p, q \in [0, \log n]$, we partition $A$ into $\frac{n}{2^p} \times \frac{n}{2^q}$ blocks, each of size $2^p \times 2^q$ called a $(p, q)$-block. For each $(p, q)$ pair, we construct four tables of size $\frac{N}{2^{p+q}} \times \min(2^p, 2^q)$ each:
+
+(i) *TL**p,q*: if $p \le q$, *TL**p,q*[*i*,*j*][$k$] indicates that any rectangle of height $k \in [0, 2^p)$ starting from the top-left corner of the current block must have width at least *TL**p,q*[*i*,*j*][$k$] in order to include at least one 1-bit.
+
+(ii) *BL*, *TR*, *BR*: similar to *TL* but starts from the bottom-left, top-right and bottom-right corners, respectively.
+
+In all cases, a stored value of $\max(2^p, 2^q)$ indicates that the block has no 1.
+
+**Query Execution.** Given a query [*i*₁, *j*₁][*i*₂, *j*₂], we find the largest integers *p* and *q* such that $2^p \le j_1 - i_1 + 1$ and $2^q \le j_2 - i_2 + 1$. The original query range can then be decomposed into four overlapping (*p*, *q*)-blocks, which we call *power rectangles*, each with a corner at one of the four corners of the original rectangle, as in Fig. 2(a). If any of these four rectangles contains a 1, the answer to the original query will be 1, and 0 otherwise. We show below how to answer an R1Q for a power rectangle.
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+**Fig. 2.** Rectangles: (a) Query rectangle split into four possibly overlapping power rectangles. (b) Power rectangle divided into four regions by four split rectangles.
+
+We consider the partition of A into preprocessed $(p, q)$-blocks. It is easy to see that each of the four power rectangles of size $2^p \times 2^q$ will intersect at most four preprocessed $(p, q)$-blocks. We call each rectangle contained in both the power rectangle and a $(p, q)$-block a *split rectangle* (see Fig. 2(b)). The R1Q for a split rectangle can be answered using a table lookup, checking if the table values of the appropriate $(p, q)$-blocks are inside the power rectangle boundary, as shown in Fig. 2(b). The proof of the following theorem will be given in the full paper.
+
+**Theorem 2.** Given a $d-D$ input grid of size $N = n^d$, each orthogonal R1Q on the grid can be answered deterministically in $O(4^d d)$ time after preprocessing the grid in $\Theta(N)$ time using $O((d+1)!(2/\ln 2)^d N)$ bits of space. In 1-D, the space can be reduced to $O(N/\log N)$ bits.
+
+## 2.3 Randomized Algorithms
+
+In this section, we present randomized algorithms that build on the deterministic algorithms given in Sections 2.1 and 2.2. We describe the algorithms for one dimension only. Extensions to higher dimensions are straightforward.
+
+**Sketch Based Algorithms.** Our algorithms provide probabilistic guarantees based on the Count-Min (CM) sketch data structure proposed in [4]. Let $N_1$ be the number of 1-bits in the input bit array $A[0...N-1]$ for any data distribution. Then, the (preprocessing) time and space complexities depend on $N_1$ while the query time remains constant.
+
+A CM sketch with parameters $\epsilon \in (0, 1]$ and $\delta \in (0, 1)$ can store a summary of any given vector $\mathbf{a} = \langle a_0, a_1, \ldots, a_{n-1} \rangle$ with $a_i \ge 0$ in only $\lceil \frac{\epsilon}{\delta} \rceil [\ln \frac{1}{\delta}] \log ||\mathbf{a}||_1$ bits of space, where $||\mathbf{a}||_1$ (or $||\mathbf{a}||$) = $\sum_{i=0}^{n-1} a_i$, and can provide an estimate $\hat{a}_i$ of any $a_i$ with the following guarantees: $a_i \le \hat{a}_i$, and with probability at least $1-\delta$, $\hat{a}_i \le a_i + \epsilon ||\mathbf{a}||_1$. It uses $t = \lceil \ln \frac{1}{\delta} \rceil$ hash functions $h_1 \ldots h_t : \{0 \ldots n-1\} \rightarrow \{1 \ldots b\}$ chosen uniformly at random from a pairwise-independent family, where bucket size $b = \lceil \frac{\epsilon}{\delta} \rceil$. These hash functions are used to update a 2-D matrix $c[1:t][1:b]$ of bt counters initialized to 0. For each $i \in [0, n-1]$ and each $j \in [1,t]$ one then updates $c[j][h_j(i)]$ to $c[j][h_j(i)] + a_i$. After the updates, an estimate $\hat{a}_i$ for any given query point $a_i$ is obtained as $\min_{1 \le j \le t} c[j][h_j(i)]$.
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+# MORAVA K-THEOIRES OF CLASSIFYING SPACES OF COMPACT LIE GROUPS
+
+MASAKI KAMEKO
+
+**ABSTRACT.** This is the abstract for the talk I will give in the 54th Topology Symposium in August 6, 2007 at the University of Aizu. I will talk about the computational aspect of cohomology and generalized cohomology theories of classifying spaces of connected compact Lie groups.
+
+## 1. INTRODUCTION
+
+Let $p$ be a prime number. Let $G$ be a connected compact Lie group and let us denote by $BG$ its classifying space. We say $G$ has $p$-torsion if and only if $H_*(G; \mathbb{Z})$ has $p$-torsion. In the case $G$ has no $p$-torsion, the mod $p$ cohomology of $BG$ is well-known. However, in the case $G$ has $p$-torsion, the computation of the mod $p$ cohomology is not an easy task. I refer the reader for the book of Mimura and Toda [9] for the detailed account on the cohomology of classifying spaces of compact Lie groups.
+
+I will give the current state of computation of the mod $p$ cohomology theory, Brown-Peterson cohomology and Morava $K$-theories of classifying spaces of some connected compact Lie groups. The coefficient ring of Brown-Peterson cohomology is
+
+$$BP^* = \mathbb{Z}_{(p)}[v_1, v_2, \dots, v_n, \dots],$$
+
+where $\deg v_n = -2(p^n - 1)$ and the coefficient ring of the Morava $K$-theory $K(n)$ is
+
+$$K(n)^* = \mathbb{Z}/p[v_n, v_n^{-1}].$$
+
+I will describe the results of the joint work with Yagita [2] and some other results obtained after writing of [2]. One of the explicit computational results is as follows:
+
+**Theorem 1.1.** For $p = 2$, $G = G_2$, the Morava $K$-theory $K(n)^*(BG_2)$ of the classifying space of the exceptional Lie group $G_2$ is given by
+
+$$K(n)^* \otimes_{BP^*} (BG_2).$$
+
+As an $\text{gr } BP^*$-module, we have that
+
+$$\text{gr } BP^*(BG_2)$$
+
+is isomorphic to
+
+$$\text{gr } BP^*[[y_8, y_{12}, y_{14}]]/(y_{14}\rho_0, y_{14}\rho_{-2}, y_{14}\rho_{-6}) \bigoplus \text{gr } BP^*[[y_8, y_{12}, y_{14}]]\{w_4\},$$
+
+where the index indicates the degree.
+
+The author was partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) 19540105.
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+The computation of the Brown-Peterson cohomology above is due to Kono and
+Yagita in [6]. The computation of Morava K-theories is a new result. Kono
+and Yagita compute the Brown-Peterson cohomology of BSpin(n) for n ≤ 10 and
+BPU(4n+2), BSp(4n+2), too. Wilson compute the Brown-Peterson cohomology
+of BO(n) in [12] and Kono and Yagita compute their Morava K-theories in [6].
+Recently, Inoue and Yagita compute the Brown-Peterson cohomology and Morava
+K-theories of BSO(n) for n ≥ 2 in [1]. In the case p = 3, we have the following
+results for simply connected simple Lie groups.
+
+
+
+
+ | (p = 3) |
+ G |
+ BP |
+ Morava K |
+
+
+
+
+ |
+ F4 |
+ ○ |
+ ○ |
+
+
+ |
+ E6 |
+ ○ |
+ ?? |
+
+
+ |
+ E7 |
+ ○ |
+ ○ |
+
+
+ |
+ E8 |
+ ?? |
+ ?? |
+
+
+
+
+The computation of $p = 3$, $G = F_4$, $PU(3)$ is done in [6]. In the case $p = 3$, $G = E_6$, in [2], we computed the Brown-Peterson cohomology of $BE_6$, however, we could not conclude
+
+$$
+K(n)^*(BE_6) = K(n)^* \otimes_{BP} BP^*(BE_6).
+$$
+
+It remains to be an open problem. In the case *p* = 5, we have the following result
+for simply connected simple Lie groups.
+
+$$
+\begin{array}{c|c|c|c}
+(p=5) & G & BP & \text{Morava } K \\
+\hline
+& E_8 & \bigcirc & \bigcirc \\
+\end{array}
+$$
+
+4. ADAMS SPECTRAL SEQUENCE
+
+In [6], they use the Atiyah-Hirzebruch spectral sequence in order to compute
+the Brown-Peterson cohomology. The Atiyah-Hirzebruch spectral sequence in [6]
+does not collapse at the $E_2$-level. In [2], we use the Adams spectral sequence in
+order to compute $P(n)$-cohomology and show that the assumption in Theorem 3.3
+holds. The Adams spectral sequence is an spectral sequence converging to $P(n)$
+cohomology whose $E_2$-term is given by
+
+$$
+\mathrm{Ext}_{\mathcal{A}}(H^*(P(n); \mathbb{Z}/p), H^*(BG; \mathbb{Z}/p)) = \mathrm{Ext}_{\mathcal{E}_n}(\mathbb{Z}/p, H^*(BG; \mathbb{Z}/p)),
+$$
+
+where $\mathcal{A}$ is the mod $p$ Steenrod algebra and $\mathcal{E}_n$ is the subalgebra generated by Milnor operations $Q_n$, $Q_{n+1}$, .... The $E_2$-term could be computed by taking the (co)homology of the (co)chain complex
+
+$$
+d_1 : grP(n)^*\hat{\otimes} H^*(BG; \mathbb{Z}/p) \rightarrow grP(n)^*\hat{\otimes} H^*(BG; \mathbb{Z}/p)
+$$
+
+where
+
+$$
+\operatorname{gr} P(n)^* = \mathbb{Z}/p[v_n, v_{n+1}, \dots, ]
+$$
+
+and
+
+$$
+d_1(v \otimes x) = \sum_{k=n}^{\infty} vv_k \otimes Q_k x.
+$$
+
+We compute this (co)chain complex. In this talk, we deal with the case $p = 2$,
+$G = G_2$, $E_6$ and give an outline of the computation.
+
+First, we deal with the case $p = 2$, $G = G_2$. Let us recall the mod 2 cohomology of the classifying space of the exceptional Lie group $G_2$. There is an non-toral elementary abelian 2-subgroup of rank 3, say $A$, in $G_2$. The induced homomorphism
+
+$$
+H^*(BG_2;\mathbb{Z}/2) \to H^*(BA;\mathbb{Z}/2)
+$$
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+is a monomorphism and the image of this induced homomorphism is the ring
+of invariants of general linear group $GL_3(\mathbb{F}_2)$ acting on the polynomial algebra
+$H^*(BA;\mathbb{Z}/2)$ in the usual manner. So, the mod 2 cohomology of $BG_2$ is the Dick-
+son invariants
+
+$$D = \mathbb{Z}/2[y_0, y_1, y_2]$$
+
+where $\deg y_k = 2^3 - 2^k$ and through this inclusion, we have the action of the Steenrod algebra, in particular, the action of Milnor operations on $H^*(BG_2; \mathbb{Z}/2)$. Let $R$ be the subalgebra of $D$ generated by $y_0^2, y_1^2$ and $y_2^2$. Then $D$ is a free $R$-module with the basis
+
+$$\{1, y_0, y_0y_1, y_0y_2, y_1, y_2, y_1y_2, y_0y_1y_2\}.$$
+
+Let us write $\beta_0 = y_0^2$, $\beta_1 = y_0y_1$, $\beta_2 = y_0y_2$ and $\beta_3 = y_0$. Then, we have
+
+$$D = R\{1, \beta_1, \beta_2, \beta_3, \frac{\beta_1\beta_2}{\beta_0}, \frac{\beta_1\beta_3}{\beta_0}, \frac{\beta_2\beta_3}{\beta_0}, \frac{\beta_1\beta_2\beta_3}{\beta_0}\}.$$
+
+**Proposition 4.1.** There exit $f_1, f_2, f_3 \in grP(n)^*\hat{\otimes}R$ such that
+
+$$d_1(\beta_1) = f_1\beta_0, \quad d_1(\beta_2) = f_2\beta_0, \quad d_1(\beta_3) = f_3\beta_0,$$
+
+where $\deg f_1 = 0$, $\deg f_2 = -2$, $\deg f_3 = -6$. Moreover, $f_1, f_2, f_3$ is a regular sequence in $grP(n)^*\hat{\otimes}R$.
+
+Since $R$ has no odd degree elements and since
+
+$$\deg d_1(\beta_1\beta_2\beta_3/\beta_0^2) = 4,$$
+
+the following proposition completes the proof of Theorem 1.1. The element
+
+$$d_{1}(\beta_{1}\beta_{2}\beta_{3}/\beta_{0}^{2})$$
+
+corresponds to $w_4$ in Theorem 1.1.
+
+**Proposition 4.2.** Let $R$ be a graded algebra over $\mathbb{Z}/2$ and suppose that $R$ is an integral domain. Let $f_1, \dots, f_n$ be elements in $R$. Let $z_0$ be an element in $R$ and consider $R$-submodules $C_k$ of
+
+$$R[z_0^{-1}] \otimes \Delta(\beta_1, \dots, \beta_n)$$
+
+generated by
+
+$$\frac{\beta_{i_1} \cdots \beta_{i_k}}{z_0^{k}},$$
+
+where $\Delta(\beta_1, \dots, \beta_n)$ is a simple system of generators, $1 \le i_1 < i_2 < \cdots < i_k \le n$, $k > 0$ and $C_0 = R$. Consider the differential $d$ given by $d(\beta_k) = z_0 f_k$ for $k = 1, \dots, n$ and by $d(xy) = d(x)y + xd(y)$. If $f_1, \dots, f_n$ is a regular sequence, then the homology of $(C, d)$ is given by
+
+$$H_{0}(C, d) = R/(z_{0}f_{1}, \dots, z_{0}f_{n})$$
+
+and
+
+$$H_k(C, d) = \{0\}$$
+
+for $k > 0$.
+
+Next, we deal with the case $p = 2$, $G = E_6$. The mod 2 cohomology $H^*(BE_6; \mathbb{Z}/2)$ is generated by $y_4, y_6, y_7, y_{10}, y_{18}, y_{32}, y_{48}$ and $y_{34}$ with the relations
+
+$$y_{7}y_{10} = 0, \quad y_{7}y_{18} = 0, \quad y_{7}y_{34} = 0, \quad y_{34}^{2} = y_{10}^{2}y_{48} + y_{18}^{2}y_{32} + \text{lower terms},$$
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+REFERENCES
+
+ANSYS Inc. (2014). ANSYS Fluent Version 15.0 User's Guide.
+
+AFSHIN E., MITRA J. (2012). Comparison of Mixture and VOF Models for Numerical Simulation of Air-entrainment in Skimming Flow over Stepped Spillways Journal of Science Direct. Procedia engineering Vol.28, pp 657 – 66.
+
+BENTALHA C., HABI M. (2015). Numerical Simulation of Air Entrainment for Flat-sloped Stepped Spillway, The Journal of Computational Multiphase Flows, Vol. 7, n°1, pp. 33-41.
+
+BENTALHA C., HABI M. (2017). Numerical simulation of water flow along stepped spillways with non uniform step heights, Larhyss Journal, n°31, pp. 115-129.
+
+BENTALHA C., HABI M. (2019). Numerical study of the flow field for moderate-slope stepped spillways. Larhyss Journal, n°37, pp.39-51.
+
+BOMBARDELLI FA, MEIRELES I, MATOS J. (2010). Laboratory measurements and multi-block numerical simulations of the mean flow and turbulence in the non-aerated skimming flow region of steep stepped spillways. Environmental Fluid Mechanics. Vol. 11. issue .3 pp.263-288.
+
+CHANSON H. (1997). Air Bubble Entrainment in Free-Surface Turbulent Shear Flows. Academic Press, London, UK, 401 pages ISBN 0-12-168110-6.
+
+CHANSON H. (2001) .The Hydraulics of Stepped Chutes and Spillways. Balkema Publisher,. Rotterdam, The Netherlands. ISBN 90 5809 352 2, 384 p.
+
+CHARLES E. RICE., KEM C.KADAVY. (1996). Model of A roller compacted concrete stepped spillway. Journal of Hydraulic Engineering, ASCE, Vol 122, n°6, pp.292-297.
+
+CHENG XIANGJU., CHEN YONGCAN., LUO LIN. (2006). Numerical simulation of air-water two-phase flow over stepped spillways. Science in China Series E, Technological Sciences, Vol 49, n°6 , pp.674-684
+
+CHEN Q., G.Q. DAI, H.W. LIU., (2002). Volume of Fluid Model for Turbulence Numerical Simulation of Stepped Spillway Over Flow. Journal of Hydraulic Engineering, ASCE Vol 128, n°7,pp. 683-688.
+
+CHINNARASRI C., KOSITGITTIWONG K., JULIEN PY. (2012). Model of flow over spillways by computational fluid dynamics. Proceedings of the Institution of Civil Engineers, Vol. 167, n° 3, pp. 164-175
+
+CHRISTODOULOU G.C. (1993). Energy dissipation on stepped spillways. Journal of Hydraulic Engineering, ASCE Vol 119, n°5 ,pp. 644-650.
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