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new file mode 100644 index 0000000000000000000000000000000000000000..99a5e0138f49452c077423a4052c340715a283b7 --- /dev/null +++ b/samples/pdfs/7827780.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fc9dab7c8205d0d975f843ea0aa4dc016d7780d939864311b076982b416e589d +size 755417 diff --git a/samples/pdfs/7856234.pdf b/samples/pdfs/7856234.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0a6c6671ae6be535b1b913b2b2060f3d6c4791e1 --- /dev/null +++ b/samples/pdfs/7856234.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fd88d200dce87dd871d82b0190b6e453776350f325fb63708c4a0fe43a4d0ca0 +size 795806 diff --git a/samples/texts/3332461/page_1.md b/samples/texts/3332461/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..57a56ea5907caa73e0c0e6ad9181cb9af0a49290 --- /dev/null +++ b/samples/texts/3332461/page_1.md @@ -0,0 +1,9 @@ +# Symmetry And Spectroscopy K V Reddy + +As recognized, adventure as competently as experience approximately lesson, amusement, as without difficulty as arrangement can be gotten by just checking out a book **symmetry and spectroscopy k v reddy** then it is not directly done, you could say yes even more regarding this life, vis--vis the world. + +We have the funds for you this proper as well as easy pretension to acquire those all. We pay for symmetry and spectroscopy k v reddy and numerous books collections from fictions to scientific research in any way. in the midst of them is this symmetry and spectroscopy k v reddy that can be your partner. + +If you're looking for an easy to use source of free books online, Authorama definitely fits the bill. All of the books offered here are classic, well-written literature, easy to find and simple to read. + +Symmetric and asymmetric stretching | Spectroscopy | Organic chemistry | Khan Academy \ No newline at end of file diff --git a/samples/texts/3332461/page_5.md b/samples/texts/3332461/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..8b5debf86f6340d306a461eafe5003d073a6f778 --- /dev/null +++ b/samples/texts/3332461/page_5.md @@ -0,0 +1,14 @@ +Group Theory in Spectroscopy e19 Example 3. Group of Non-singular Matrices All non-singular n ×n matrices4 with matrix multiplication as the operation form a group. Let us look at this now. Multiplication of a non-singular matrix A (i.e., detA = 0) by a non- singular matrix B gives a non-singular matrix C = AB, because detC = detAdetB = 0. The unit element is the unit matrix 1, and the ... + +**Group theory - ETH Z** + +Vibrational Spectroscopy (IR, Raman) Vibrational spectroscopy +Vibrational spectroscopy is an energy sensitive method. It is based on +periodic changes of dipolmoments (IR) or polarizabilities (Raman) +caused by molecular vibrations of molecules or groups of atoms and the +combined discrete energy transitions and changes of frequen-cies +during ... + +CHAPTER 5 - SYMMETRY AND VIBRATIONAL SPECTROSCOPY 5.1 ... + +Symmetry & IR Spectroscopy. One of the most importance applications of IR spectroscopy is structural assignment of the molecule depending on the relationship between the molecule and observed IR absorption bands. Every molecule is corresponding to one particular symmetry point group. Then we can predict which point group the molecule is ... \ No newline at end of file diff --git a/samples/texts/3332461/page_6.md b/samples/texts/3332461/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..f627794eb42c7fd7f9b5ea6be9cdce75cd5656dc --- /dev/null +++ b/samples/texts/3332461/page_6.md @@ -0,0 +1,15 @@ +**Download Symmetry And Spectroscopy K V Reddy PDF | pdf ...** + +Symmetry and Spectroscopy of Molecules by Reddy, K. Veera and a great selection of related books, art and collectibles available now at AbeBooks.com. + +**Symmetry And Spectroscopy K V Reddy - Edsa.com | pdf Book ...** + +Download Download Symmetry And Spectroscopy K V Reddy PDF book pdf free download link or read online here in PDF. Read online Download Symmetry And Spectroscopy K V Reddy PDF book pdf free download link book now. All books are in clear copy here, and all files are secure so don't worry about it. + +**Symmetry and Spectroscopy: An Introduction to Vibrational ...** + +Download Symmetry And Spectroscopy K V Reddy - edsa.com book pdf free download link or read online here in PDF. Read online Symmetry And Spectroscopy K V Reddy - edsa.com book pdf free download link book now. All books are in clear copy here, and all files are secure so don't worry about it. + +## CHAPTER 4: SYMMETRY AND GROUP THEORY + +The optical properties of a semiconductor can be de?ned as any property that involves ... The ?eld of optical spectroscopy is a very \ No newline at end of file diff --git a/samples/texts/3332461/page_7.md b/samples/texts/3332461/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..e52a4df102a6dbdb729b4442c94b4827373b5505 --- /dev/null +++ b/samples/texts/3332461/page_7.md @@ -0,0 +1,25 @@ +important area of science and technology since ... symmetry of the +space groups is also essential in determining the structure of the +energy + +Vibrational spectroscopy Vibrational Spectroscopy (IR, Raman) + +- Student and teacher friendly book with concepts of symmetry built +layer by layer leaving no room for confusion. - Expertly discusses +group theory, structure, bonding and spectroscopy of molecules. - The +style and pedagogical pattern of the book have developed from the +author's 25 years experience in teaching UG/PG courses and workshops. + +**Chapter 7 - Symmetry and Spectroscopy - Molecular ...** + +The Structure And Symmetry Of Fullerene Molecules Are Presented In +Some Detail For The First Time As A Class Room Example. The Background +Provided For Non-Mathematical Chemistry Students In Chapters 4 And 5 +Is Very Useful For The Advanced Aspects Of Group Theory. ... Symmetry +and Spectroscopy of Molecules K. Veera Reddy No preview available ... + +CHAPTER 36 OPTICAL PROPERTIES OF SEMICONDUCTORS + +In the following, we shall refer some molecules to the different +symmetry groups using a notation due to Schoenflies, 4 usual in +Theoretical Chemistry and Spectroscopy, giving the symmetry operations \ No newline at end of file diff --git a/samples/texts/3474975/page_13.md b/samples/texts/3474975/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..ccba809291b9961f45af03e01bc0490ce94420a5 --- /dev/null +++ b/samples/texts/3474975/page_13.md @@ -0,0 +1,31 @@ +objective $J_i : \mathcal{X}_i \times \mathcal{U}_i \rightarrow \mathbb{R}$ that is assumed to be in harmony with the team objectives. The myopic cost can be parameterized as a function of the coordination variable. This can be done by using the relationship $\mathbf{u}_i = f_i^\dagger(\mathbf{x}_i, \vartheta)$, for each $\vartheta \in \Theta_i(\mathbf{x}_i)$. The function + +$$\phi_i(\mathbf{x}_i, \vartheta) = J_i(\mathbf{x}_i, f_i^\dagger(\mathbf{x}_i, \vartheta)), \quad (2)$$ + +is a representation of the local myopic cost $J_i(\mathbf{x}_i, \mathbf{u}_i)$. Under the restriction that for each $\vartheta \in \Theta_i$ there is a unique $\mathbf{u}_i \in \mathcal{U}_i$, $\bigcup_{\vartheta \in \Theta_i(\mathbf{x}_i)} \phi_i(\mathbf{x}_i, \vartheta) = \bigcup_{\mathbf{u}_i \in \mathcal{U}_i(\mathbf{x}_i)} J_i(\mathbf{x}_i, \mathbf{u}_i)$. If this is not the case, it follows that $\bigcup_{\vartheta \in \Theta_i(\mathbf{x}_i)} f_i^\dagger(\mathbf{x}_i, \vartheta)$ may only be a proper subset of $\mathcal{U}_i$, and $\phi_i(\mathbf{x}_i, \cdot)$ an approximation of $J_i(\mathbf{x}_i, \cdot)$. The function + +$$\phi_i : \mathcal{X}_i \times \Theta_i(\mathbf{x}_i) \rightarrow \mathbb{R}$$ + +given by Equation (2) is called the *coordination function* of the *i*-th vehicle. For a given situation state $\mathbf{x}_i$, the coordination function parameterizes the myopic cost of the *i*-th vehicle versus the coordination variable. + +In this chapter, the cooperation problems of interest can be posed as a minimization of a team objective function, where the team objective is a function of the myopic objective functions. Let $\mathcal{J}_T : \mathbb{R}^N \rightarrow \mathbb{R}$ be the team objective function, then the cooperative control problem is to find influence variables $\mathbf{u}_1, \dots, \mathbf{u}_N$ that solve the following optimization problem: + +$$ (\mathbf{u}_1, \dots, \mathbf{u}_N) = \arg \min_{\mathcal{U}_1 \times \dots \times \mathcal{U}_N} \mathcal{J}_T (J_1(\mathbf{x}_1, \mathbf{u}_1), \dots, J_N(\mathbf{x}_N, \mathbf{u}_N)), \quad (3) $$ + +subject to + +$$ f_i(\mathbf{x}_i, \mathbf{u}_i) = f_j(\mathbf{x}_j, \mathbf{u}_j), \quad \forall i, j \in \{1, \dots, N\}. \quad (4) $$ + +This optimization problem will clearly pose computational problems as the number of vehicles increase, and for large states and influence dimensions. + +Using coordination variables and coordination functions, a decomposition of the optimization problem of Equations (3) and (4) that captures the information essential for cooperation can be posed: + +$$\theta = \arg \min_{\vartheta \in \Theta_i(\mathbf{x}_i)} \mathcal{J}_T (\phi_1(\vartheta), \dots, \phi_N(\vartheta)).$$ + +Once a team optimal value for the coordination variable is found, individual vehicle decisions can be found by solving for the influence variable from the relationship + +$$\mathbf{u}_i = f_i^\dagger(\mathbf{x}_i, \theta).$$ + +This two-level decomposition process significantly reduces the computation and communication loads. + +Coordination variables and functions have been applied successfully to UAV cooperative timing missions [21, 20] and UAV cooperative reconnaissance problems [4]. An illustrative example is given in the next section. \ No newline at end of file diff --git a/samples/texts/4730718/page_10.md b/samples/texts/4730718/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..0034b17ed0fda53046ebecde3517c58a98ddb092 --- /dev/null +++ b/samples/texts/4730718/page_10.md @@ -0,0 +1,37 @@ +time behavior of the Riemann problem and the Riemann problem with structure [26] for a system of balance laws proposed by Ruggeri and coworkers [27–29] – following an idea of Liu [30]. According to this conjecture, the solutions of both Riemann problems with and without structure, for large time, instead to converge to the corresponding Riemann problem of the equilibrium sub-system (i.e combination of shock and rarefaction waves), converge to solutions that represent a combination of shock structures (with and without sub-shocks) of the full system and rarefactions waves of the equilibrium sub-system. + +In particular, if the Riemann initial data correspond to a shock family $\mathcal{S}$ of the equilibrium sub-system, for large time, the solution of the Riemann problem of the full system converges to the corresponding shock structure. This means that, for the numerical study of the shock structure, instead of using a solver of ODE, which is not useful when a discontinuity (sub-shock) appears, Riemann solvers (e.g. [31]) can be used and if we wait enough time after the initial time, we obtain the shock-structure profile with or without sub-shocks. This strategy was adopted in several shock phenomena of ET [2]. In particular the conjecture was tested numerically for a Grad 13-moment system and a mixture of fluids [27, 32] and was verified in a simple 2 × 2 dissipative model considered by Mentrelli and Ruggeri [29] for which it is possible to calculate the shock structures of the full system and the rarefactions of the equilibrium subsystem analytically. + +We perform numerical calculations on the shock structure obtained after long time for the Riemann problem consisted with two equilibrium states $\mathbf{U}_0 = (\rho_0, 0, T_0, 0, 0, 0)^T$ and $\mathbf{U}_1 = (\rho_1, v_1, T_1, 0, 0, 0)^T$ satisfying the RH conditions for the system of the Euler equations (18). For the numerical calculations on the shock structure, the BGK model for the production terms is adopted and therefore the relaxation times $\tau_\Pi$, $\tau_S$ and $\tau_q$ are constant and have the same value $\tau_\Pi = \tau_S = \tau_q$. We developed and adopted the parallel numerical code written in C language on the basis of the Uniformly accurate Central Scheme of order 2 (UCS2) proposed by Liotta, Romano and Russo [33] for analyzing the hyperbolic balance laws with production term. + +Figure 2 shows typical examples of the mass density profile with $D = 7$. The Mach numbers are $M_0 = 1.5$ and $M_0 = 2$. The profile for $M_0 = 1.5$ is continuous and no sub-shock arises. In the profile for $M_0 = 2$, only one sub-shock, which corresponds to the fastest mode, appears. The present situation is similar to the ones obtained in the case of a rarefied monatomic gas [1, 6]. The singular point in the shock-structure solution becomes regular except for the maximum characteristic velocity. This result implies that the system of ET for rarefied polyatomic gas has the same property on the sub-shock formation and this property seems common for the systems satisfying the requirements of the ET theory. + +FIG. 3. (Case A) Dependence of the characteristic velocities in the perturbed state $\lambda_1$ on the shock speed $s$ for $u_0 = 1.15$. + +## IV. 2 × 2 HYPERBOLIC DISSIPATIVE SYSTEM + +### A. General form of 2 × 2 hyperbolic dissipative system + +Let us consider the following 2 × 2 dissipative hyperbolic system of balance laws proposed by Mentrelli and Ruggeri [29]: + +$$ +\begin{aligned} +\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \left( \frac{\partial K}{\partial u} \right) &= -\frac{1}{\tau} (u-v), \\ +\frac{\partial v}{\partial t} + \frac{\partial}{\partial x} \left( \frac{\partial K}{\partial v} \right) &= -\frac{1}{\tau} (v-u), +\end{aligned} +\quad (20) $$ + +or, alternatively, + +$$ +\begin{aligned} +& \frac{\partial}{\partial t}(u+v) + \frac{\partial}{\partial x}\left(\frac{\partial K}{\partial u} + \frac{\partial K}{\partial v}\right) = 0, \\ +& \frac{\partial u}{\partial t} + \frac{\partial}{\partial x}\left(\frac{\partial K}{\partial u}\right) = -\frac{1}{\tau}(u-v) +\end{aligned} +\quad (21) $$ + +for the unknown field $\mathbf{U} = (u, v)^T$, which is a function of space $x$ and time $t$. Here $K \equiv K(u, v)$ is an arbitrary smooth function in terms of the variables $u$ and $v$ and $\tau > 0$ represents a constant relaxation time. The equilibrium state is achieved when $u = v$. + +The system (20) (or, (21)) was proposed because this satisfies all the requirement of rational extended thermodynamics. In fact, the solution of the balance equations (20) (or, (21)) satisfies the following entropy inequality [34]: + +$$ \frac{\partial h}{\partial t} + \frac{\partial h^1}{\partial x} = \Sigma > 0, \quad (22) $$ \ No newline at end of file diff --git a/samples/texts/4730718/page_12.md b/samples/texts/4730718/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..e819ebbb7dc149d0bd20b9a890d04d12c8cdf432 --- /dev/null +++ b/samples/texts/4730718/page_12.md @@ -0,0 +1,116 @@ +($r^{(i)}$ represents the $i^{th}$ characteristic eigenvector of the +hyperbolic system (1)), it was proven that, for small ini- +tial data, smooth solutions exist for all times and con- +stant states are stable [37–40]. The K-condition (31) is +equivalent to [41]: + +$$ +\delta \mathbf{f}|_{E} \neq 0. +$$ + +In the present case, from (30)$_3$, we have + +$$ +\delta \mathbf{f}|_E \neq 0 \iff (\delta u - \delta v)|_E \neq 0. \quad (32) +$$ + +We need to consider the two possible cases separately: + +• If + +$$ +\left. \frac{\partial^2 K}{\partial u \partial v} \right|_E = 0, \tag{33} +$$ + +from (25) and (24), we have + +$$ +\lambda_E^{(1)} = \left. \frac{\partial^2 K}{\partial u^2} \right|_E, \quad (\delta u)|_E = 1, (\delta v)|_E = 0, +$$ + +$$ +\lambda_E^{(2)} = \left. \frac{\partial^2 K}{\partial v^2} \right|_E, \quad (\delta u)|_E = 0, (\delta v)|_E = 1 +$$ + +and (32) is automatically satisfied for both eigen- +vectors. + +• If + +$$ +\left. \frac{\partial^2 K}{\partial u \partial v} \right|_E \neq 0, +$$ + +from (24), we have + +$$ +(\delta u)|_E = - \left. \frac{\partial^2 K}{\partial u \partial v} \right|_E , \quad (\delta v)|_E = - \lambda_E + \left. \frac{\partial^2 K}{\partial u^2} \right|_E +$$ + +and therefore the K-condition (32) is satisfied when + +$$ +\lambda_E \neq \omega, \quad \text{with} \quad \omega = \left. \frac{\partial^2 K}{\partial u^2} \right|_E + \left. \frac{\partial^2 K}{\partial u \partial v} \right|_E . \quad (34) +$$ + +From (25), we have + +$$ +P_E(\omega) = \left. \frac{\partial^2 K}{\partial u \partial v} \right|_E \left( \frac{\partial^2 K}{\partial u^2} - \frac{\partial^2 K}{\partial v^2} \right) |_E . +$$ + +The K-condition (34) implies $P_E(\omega) \neq 0$ and there- +fore + +$$ +\left( \frac{\partial^2 K}{\partial u^2} - \frac{\partial^2 K}{\partial v^2} \right) |_{E} \neq 0. \quad (35) +$$ + +We notice that, if (35) holds, the equilibrium char- +acteristic velocities for the full system have the dif- +ferent values from the one for the equilibrium sub- +system and the inequalities in (28) and (29) become +strict. + +FIG. 5. (Case C) Dependence of the characteristic velocities in the perturbed state $\lambda_1$ on the shock speed $s$ for $u_0 = 0.3$. + +Therefore we can summarize as follows: + +**Statement 1** For any smooth function $K(u, v)$ such that + +$$ +\begin{align*} +& \left. \frac{\partial^2 K}{\partial u \partial v} \right|_{u=v} = 0, \\ +& \text{or} \\ +& \left. \left( \frac{\partial^2 K}{\partial u^2} - \frac{\partial^2 K}{\partial v^2} \right) \right|_{u=v} \neq 0, +\end{align*} +$$ + +and initial data sufficiently small, according with the the- +orems stated in [37–40], the system (20) has global smooth +solutions for all time. + +B. **2 × 2 system with** $K = u^4/12 + v^6/30$ + +In the paper [29], the system with $K = uv^2$ was stud- +ied. In this system, two characteristic velocities have the +different sign; one is positive and another one is nega- +tive. In order to discuss the sub-shock formation with +the slower shock velocity than the maximum characteris- +tic velocity, we need to construct a new system in which +both characteristic velocities are positive. In the present +paper, we adopt + +$$ +K = u^4/12 + v^6/30. +$$ + +In this case, we have the following balance equations: + +$$ +\begin{align} +\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \left( \frac{u^3}{3} \right) &= -\frac{u-v}{\tau}, \\ +\frac{\partial v}{\partial t} + \frac{\partial}{\partial x} \left( \frac{v^5}{5} \right) &= -\frac{v-u}{\tau}, +\end{align} +\tag{36} +$$ \ No newline at end of file diff --git a/samples/texts/4730718/page_13.md b/samples/texts/4730718/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..2b63064c7f718ea67dae0a9817681acbc818f3c0 --- /dev/null +++ b/samples/texts/4730718/page_13.md @@ -0,0 +1,133 @@ +or, alternatively, + +$$ +\begin{align} +\frac{\partial}{\partial t}(u+v) + \frac{\partial}{\partial x}\left(\frac{u^3}{3} + \frac{v^5}{5}\right) &= 0, \tag{37} \\ +\frac{\partial u}{\partial t} + \frac{\partial}{\partial x}\left(\frac{u^3}{3}\right) &= -\frac{u-v}{\tau}. \nonumber +\end{align} +$$ + +The solution of the balance equations (36) (or, (37)) satisfies the entropy inequality (22) where the entropy density $h$, the entropy flux $h^1$ and the entropy production density $\Sigma$ are, in the present case, given by + +$$ +\begin{align*} +h &= -\frac{1}{2}(u^2 + v^2), \\ +h^1 &= -\frac{u^4}{4} - \frac{v^6}{6}, \\ +\Sigma &= \frac{1}{\tau}(u - v)^2. +\end{align*} +$$ + +The characteristic velocities $\lambda$ are + +$$ +\lambda = u^2, v^4. \tag{38} +$$ + +We adopt the following notation: + +$\lambda^{(u)} = u^2$ and $\lambda^{(v)} = v^4.$ + +The equilibrium sub-system (26) becomes + +$$ +\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \left( \frac{u^3}{6} + \frac{u^5}{10} \right) = 0 \quad (39) +$$ + +and the characteristic velocity of the equilibrium sub- +system $\mu$ (27) becomes + +$$ +\mu = \frac{u^2 + u^4}{2}. \tag{40} +$$ + +From equations (38) and (40), it can be easily proven that +the sub-characteristic condition (29) holds. The Shizuta- +Kawashima condition is always satisfied because the con- +dition (33) holds. + +V. IDENTIFICATION OF POSSIBLE SUB-SHOCKS + +Let us consider the shock-structure solution of the sys- +tem (37). In the present case, (8) becomes: + +$$ +\begin{gather} +\frac{d}{dz} \left\{ -s(u+v) + \left(\frac{u^3}{3} + \frac{v^5}{5}\right) \right\} = 0, \tag{41} \\ +(-s+u^2) \frac{du}{dz} = \frac{v-u}{\tau}, \notag +\end{gather} +$$ + +with the following boundary conditions + +$$ +\lim_{z \to +\infty} (u, v) = (u_0, u_0), +$$ + +$$ +\lim_{z \to -\infty} (u, v) = (u_1, u_1). \tag{42} +$$ + +From (41)$_1$ we have : + +$$ +-s(u+v) + \left(\frac{u^3}{3} + \frac{v^5}{5}\right) = \text{const.} +$$ + +and by taking (42) into account, we obtain the Rankine- +Hugoniot conditions for the equilibrium subsystem (39): + +$$ +-2su_0 + \left( \frac{u_0^3}{3} + \frac{u_0^5}{5} \right) = -2su_1 + \left( \frac{u_1^3}{3} + \frac{u_1^5}{5} \right) = \text{const.} \quad (43) +$$ + +Therefore we have a relation between $s$, $u_0$ and $u_1$ given +by (43), and, by excluding the null shock $u_1 = u_0$, the +relation can be rewritten as: + +$$ +s = \frac{u_1^2 + u_0 u_1 + u_0^2}{6} + \frac{u_1^4 + u_0 u_1^3 + u_0^2 u_1^2 + u_0^3 u_1 + u_0^4}{10}. \quad (44) +$$ + +From the RH conditions and the expression of the char- +acteristic eigenvalue of the subsystem (40), we conclude +that the Lax condition [42] for the equilibrium subsystem +is satisfied when + +$\mu_0 < s < \mu_1$, provided $u_1 > u_0 > 0$. + +In the present case we have: + +$$ +\lambda_{0}^{(u)} = u_{0}^{2}, \quad \lambda_{0}^{(v)} = u_{0}^{4} +$$ + +$$ +\lambda_1^{(u)} = u_1^2, \quad \lambda_0^{(v)} = u_1^4. +$$ + +The first two are constants depending on $u_0$ and the other +two are functions of $s$ through the relation (44) that gives +$u_1$ as function of $s$ and $u_0$. + +We classify the RH curves into the following three dif- +ferent cases: Case. A: $u_0 > 1.06$; Case. B: $0.536 < u_0 < 1$; +Case. C: $0 < u_0 < 0.536$ or $1 < u_0 < 1.06$. + +A. Case A + +If we choose the unperturbed state $\mathbf{U}_0 = (u_0, u_0)^T$ +with $u_0 > 1.06$, the relationship $\lambda_0^{(u)} < \lambda_0^{(v)}$ holds +and both characteristic velocities in the perturbed state +$\mathbf{U}_1 = (u_1, u_1)^T$, $\lambda_1^{(u)} < s$ and $\lambda_1^{(v)} > s$ never meet +the shock velocity $s$. Therefore the necessary conditions +(14) are violated and there exists only one possibility of +sub-shock formation when the shock velocity is larger +than the maximum characteristic velocity: $s > \lambda_0^{(v)}$. As +a typical example, we show the shock velocity depen- +dence of the characteristic velocities in the perturbed +state $\mathbf{U}_1 = (u_1, u_1)^T$ for $u_0 = 1.15$ in Figure 3. In this +case, $\mu_0 \approx 1.54$, $\lambda_0^{(1)} \approx 1.32$ and $\lambda_0^{(2)} \approx 1.75$. + +B. Case B + +If we choose $\mathbf{U}_0 = (u_0, u_0)^T$ with $0.536 < u_0 < 1$, the relationship $\lambda_0^{(u)} > \lambda_0^{(v)}$ holds. The characteristic velocity $\lambda_1^{(v)}$ in the perturbed state $\mathbf{U}_1 = (u_1, u_1)^T$ is equal \ No newline at end of file diff --git a/samples/texts/4730718/page_4.md b/samples/texts/4730718/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..83ae3544eac7dfedf3761deaef0a1d419ea2a0ee --- /dev/null +++ b/samples/texts/4730718/page_4.md @@ -0,0 +1,67 @@ +profile of the shock structure is continuous and no sub-shock exists like in Figures 6$_{1,2,4}$. If the two curve have two points in common like in Figure 6$_{5}$, we understand that a sub-shock appears. + +As a typical example of Case A, Figure 6 shows the numerical shock structure with or without a sub-shock for $u_1 = 1.2$ ($s = 1.64$) and for $u_1 = 1.3$ ($s = 1.89$). As was predicted, we have the continuous shock wave structure for $u_1 = 1.2$ and observe only one sub-shock for $u_1 = 1.3$. + +As a typical example of Case B, Figure 7 shows the numerical shock structure for $u_1 = 0.9$ ($s = 0.677$), for $u_1 = 0.935$ ($s = 0.717$) and for $u_1 = 0.95$ ($s = 0.735$). We see the continuous shock structure for $u_1 = 0.9$. It should be emphasized that we observe the sub-shock formation for $u_1 = 0.935$ which satisfies the RH conditions for the sub-shock and that this is clearly a counter example of the sub-shock slower than the maximum unperturbed characteristic velocity. We see also the multiple sub-shock for $u$ and $v$ for $u_1 = 0.95$. + +As a typical example of Case C, Figure 8 shows the numerical shock structure for $u_1 = 0.4$ ($s = 0.069$), for $u_1 = 0.55$ ($s = 0.11$) and for $u_1 = 0.65$ ($s = 0.15$). As was predicted, we see the continuous shock wave structure for $u_1 = 0.4$, the structure with one sub-shock for $u$ for $u_1 = 0.55$ and the formation of the multiple sub-shock for $u$ and $v$ for $u_1 = 0.65$. + +## VII. SUMMARY AND CONCLUDING REMARKS + +In this paper, first, we have shown that ET for a rarefied polyatomic gas with 14 independent variables does not predict the sub-shock formation with slower shock velocity than the maximum unperturbed characteristic + +velocity. Second, we have shown an example of the clear sub-shock formation with slower shock velocity than the maximum characteristic velocity by adopting a simple 2 × 2 hyperbolic dissipative system that satisfies all requirements of the ET theory. We have concluded that the requirements of the entropy principle, the convexity of the entropy and the Shizuta-Kawashima condition, are not enough to characterize the property on the sub-shock formation of ET. + +Therefore, if we conjecture that ET theories have this strange beautiful property such that the sub-shock appears only for the shock velocity greater than the maximum characteristic velocity, there must exist some special property of differential system of ET theories, which is still obscure. + +If we multiply the system (3) by the left eigenvector $\mathbf{l}$ of $\mathbf{A}$ corresponding to a given eigenvalue $\lambda$, we obtain + +$$ \mathbf{l} \cdot \frac{d\mathbf{U}}{dz} = \frac{\mathbf{l} \cdot \mathbf{f}}{\lambda - s}. $$ + +To make the solution regular, when the eigenvalue $\lambda$ approaches to $s$, $\mathbf{l} \cdot \mathbf{f}$ also must tend to 0. This means that the differential system of ET theories needs to satisfy some special condition between productions and the main part of the operator and this condition may be more restrictive than the K-condition. The identification of this condition is still an open problem and we will try to give an answer in the future. + +## ACKNOWLEDGMENTS + +This work was partially supported by JSPS KAKENHI Grant Number JP16K17555 (S. T.) and by National Group of Mathematical Physics GNFM-INdAM (T. R.). + +[1] I. Müller and T. Ruggeri, *Rational Extended Thermodynamics*, 2nd edn. (Springer, New York, 1998). + +[2] T. Ruggeri and M. Sugiyama, *Rational Extended Thermodynamics beyond the Monatomic Gas*. (Springer, Cham, Heidelberg, New York, Dordrecht, London, 2015). + +[3] H. Grad, Comm. Pure Appl. Math. **2**, 331 (1949). + +[4] H. Grad, Comm. Pure Appl. Math. **5**, 257 (1952). + +[5] T. Ruggeri, Phys. Rev. E **47**, 4135 (1993). + +[6] W. Weiss, Phys. Rev. E **52**, R5760 (1995). + +[7] G. 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Fluids **26**, 016103 (2014). + +[16] S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Int. J. Non-Linear Mech. **79**, 66 (2016). + +[17] S. Kosuge, K. Aoki and T. Goto, AIP Conference Proceedings **1786**, 180004 (2016). + +[18] M. Bisi, G. Martalò and G. Spiga, Acta Appl. Math. **132**(1), 95-105 (2014). + +[19] F. Conforto, A. Mentrelli and T. Ruggeri, Ricerche di Matematica, **66** (1) 221231 (2017). + +[20] G. Boillat, Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systémes hyper- \ No newline at end of file diff --git a/samples/texts/4730718/page_5.md b/samples/texts/4730718/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..5750d2a5b706c6457a99dd72c8880ea66178ab86 --- /dev/null +++ b/samples/texts/4730718/page_5.md @@ -0,0 +1,47 @@ +boliques, C. R. Acad. Sci. Paris A 278, 909 (1974). + +[21] T. Ruggeri, A. Strumia, Ann. Inst. H. Poincaré, Section A 34, 65 (1981). + +[22] G. Boillat and T. Ruggeri, Arch.Rat. Mech. Anal. **137**, 305 (1997). + +[23] T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Continuum Mech. Thermodyn. **25**, 727 (2013). + +[24] T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Phys. Lett. A **377**, 2136 (2013). + +[25] T. Arima, T. Ruggeri, M. Sugiyama and S. Taniguchi, Annals Phys. **372**, 83 (2016). + +[26] T.-P. Liu, Commun. Pure Appl. Math. **30**, 767 (1977); Commun. Math. Phys. **55**, 163 (1977). + +[27] F. Brini, T. Ruggeri, in *Proceedings of the 10th International Conference on Hyperbolic Problems (HYP2004)*, Osaka, 13-17 Sept 2004, vol. I, p. 319 Yokohama Publisher Inc., Yokohama, (2006). + +[28] F. Brini and T. Ruggeri, Suppl. Rend. Circ. Mat. Palermo II **78**, 31 (2006). + +[29] A. Mentrelli and T. Ruggeri, Suppl. Rend. Circ. Mat. Palermo II **78**, 201 (2006). + +[30] T.-P. Liu, in *Recent Mathematical Methods in Nonlinear Wave Propagation*, ed. by T. Ruggeri. Lecture Notes in Mathematics, vol. 1640, pp. 103-136 Springer, Berlin, (1996). + +[31] E. Toro, *Riemann Solvers and Numerical Methods for Fluid Dynamics*. Springer, Berlin, (2009). + +[32] F. Brini and T. Ruggeri, in *Proceedings XII Int. Conference on Waves and Stability in Continuous Media*. Monaco, R. et al. (eds.) World Scientific, Singapore, pp. 102-108 (2004). + +[33] S. F. Liotta, V. Romano and G. Russo, SIAM J. Numer. Anal. **38**, 1337 (2000). + +[34] We adopt different definition of the sign of the entropy from the one adopted in [29]. + +[35] Y. Shizuta, S. Kawashima, Hokkaido Math. J. **14**, 249-275 (1985). + +[36] S. Kawashima, Proc. Roy. Soc. Edimburgh **106A**, 169 (1987). + +[37] B. Hanouzet, R. Natalini, Arch. Rat. Mech. Anal. **169**, 89-117 (2003). + +[38] W.-A. Yong, Arch. Rat. Mech. Anal. **172** (2), 247 (2004). + +[39] T. Ruggeri and D. Serre, Quarterly of Applied Math. **62** (1), 163-179 (2004). + +[40] S. Bianchini, B. Hanouzet and R. Natalini, IAC Report **79** (2005). + +[41] T. Ruggeri, Il Nuovo Cimento B **119** (7-9), 809-821 (2004). + +[42] P. D. Lax, Comm. Pure Appl. Math. **10**, 537 (1957). + +[43] C. Dafermos, *Conservation Laws in Continuum Physics*, 2nd ed. (Springer Verlag, Berlin, 2005). \ No newline at end of file diff --git a/samples/texts/4730718/page_9.md b/samples/texts/4730718/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..929bfdc962cebd20e59377369a0724fdf67184ca --- /dev/null +++ b/samples/texts/4730718/page_9.md @@ -0,0 +1,61 @@ +FIG. 2. Profiles of the dimensionless mass density $\hat{\rho} \equiv \rho/\rho_0$ with $\rho_0$ being the mass density in the unperturbed state. Here $\hat{z}$ is the dimensionless position defined by $\hat{z} \equiv z/(c_0\tau_{\Pi})$ and $D = 7$. $M_0 = 1.5$ (left) and $M_0 = 2$ (right). + +where $c_0$ is the sound velocity in the unperturbed state. +As is well known, except for contact shocks, the solution +of the RH equations for Euler fluids is: + +$$ +\begin{aligned} +V_1 &= V_0 - \frac{2}{\gamma+1} V_0 \frac{M_0^2 - 1}{M_0^2}, & V &= \frac{1}{\rho}, \\ +v_1 &= v_0 + \frac{2c_0}{\gamma+1} \frac{M_0^2 - 1}{M_0}, && (18) \\ +T_1 &= T_0 + 2T_0 \frac{(M_0^2 - 1)(\gamma M_0^2 + 1)(\gamma - 1)}{M_0^2(1+\gamma)^2}. +\end{aligned} + $$ + +It is also well known that we should take $M_0 > 1$ for +obtaining the solution of a stable shock wave. + +Let us consider, without loss of generality, $v_0 = 0$ due +to the Galilean invariance and let us define the dimen- +sionless characteristic velocities as $\hat{\lambda} \equiv \lambda/c_0$. By consid- +ering only the two waves propagating in the positive $x$ +directions and taking (16), (17) and (18) into account, +we obtain the dimensionless characteristic velocities in +the unperturbed constant state $\mathbf{U}_0$ and in the perturbed +constant state $\mathbf{U}_1$: + +$$ +\begin{align*} +\hat{\lambda}_0^{(1)} &= \Delta^{(1)}, & \hat{\lambda}_0^{(2)} &= \Delta^{(2)} \\ +\hat{\lambda}_1^{(1)} &= \frac{v_1}{c_0} + \frac{c_1}{c_0} \Delta^{(1)}, & \hat{\lambda}_1^{(2)} &= \frac{v_1}{c_0} + \frac{c_1}{c_0} \Delta^{(2)}. +\end{align*} + $$ + +The former two are constant, while the latter two depend +on $M_0$. In the present case, the necessary condition for +existence of sub-shock (14) expressed by the dimension- +less variables reads: + +$$ \hat{\lambda}_0 < M_0 < \hat{\lambda}_1(M_0) < \hat{\lambda}_0^{\max}. \quad (19) $$ + +Figure 1 shows the dependence of the dimensionless char- +acteristic velocities in the perturbed state $\hat{\lambda}_1^{(1)}$ and $\hat{\lambda}_1^{(2)}$ on + +the Mach number $M_0$ in the cases of $D=3$ (monatomic +gas) and of $D=7$. It was proven that, in the limit of +$D \to 3$, the solutions for rarefied polyatomic gases con- +verge to the ones for rarefied monatomic gases when we +impose an appropriate initial condition, which is compat- +ible with monatomic gases [24, 25]. + +For $D = 7$, we have $\hat{\lambda}_0^{(1)} \approx 0.773809$ and $\hat{\lambda}_0^{(2)} = \hat{\lambda}_0^{\max} \approx 1.74093$. In contrast to the case $D = 3$, for $D = 7$, as we increase the Mach number from unity, the first characteristic velocity $\hat{\lambda}_1^{(1)}$ evaluated in the perturbed state $\mathbf{U}_1$ meets the shock velocity at $M_0 \approx 1.31579$ before the fastest characteristic velocity in the unperturbed state. Therefore (19) is satisfied for $1.31579 < M_0 < 1.74093$ and, in principle, the sub-shock formation with smaller shock velocity than the maximum characteristic velocity may exist in this range. However, as we will see in the next section, $M_0 = 1.31579$ is a regular singular point and no sub-shock arises until we reach $M_0 > 1.74093$, i.,e., until the shock velocity becomes larger than the maximum characteristic velocity evaluated in equilibrium state in front of the shock! + +A. Numerical analysis + +The shock structure was studied in [11] for a non-polytropic rarefied gas by solving the ODE system (3) numerically for Mach numbers less than 1.47 and the agreement between theoretical predictions and the experimental results is excellent with respect to previous theories. + +In order to obtain the shock-structure solution also for +large Mach number, in the present analysis, instead of +solving the ODE system (3), we use a different proce- +dure solving ad hoc Riemann problem for the PDE sys- +tem (15) according with the conjecture about the large- \ No newline at end of file diff --git a/samples/texts/4943237/page_10.md b/samples/texts/4943237/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..dca290ca722d716e588057370338d612549addc2 --- /dev/null +++ b/samples/texts/4943237/page_10.md @@ -0,0 +1,10 @@ +Figure 2: state of SQ, SN and the associated run, at an iteration of the algorithm + +$m$. Moreover, since the different branches of the tree are independent, we only have to keep one of them in memory at a time. Note that the construction of the automaton is interleaved with the emptiness test, so we also never keep the whole automaton in memory, but only the states which are relevant for the current branch. + +In order to remember the backtracking information for the depth first traversal, we use two data structures: **SQ** is a stack storing, for every predecessor of the current node, the transition which led to that node, and thus it contains the required node labels for the nodes of the current branch and their siblings. **SN** is another stack recording the current path by storing, for every level of the tree, the number of the node on the current path. If we refer to the elements in **SN** by **SN**(1)(the bottom element), ..., **SN**(d)(the top element), the next node to be checked is **SN**(1)·**SN**(2)···**SN**(d) + 1 (d is the depth of **SN**). Thus, **SQ** ∈ (Q^k)*, +because every transition is a k-ary tuple, and **SN** ∈ {1, ..., k}*. + +Figure 2 shows the values stored in each of the stacks SQ and SN at the beginning of an iteration, and their relation with the traversal of the run. The circled nodes represent the path followed to reach the node about to be checked. The values of the elements of the stack are shown next to the depth in the run to which they correspond. For this reason, the stacks appear backwards, with their bottom element at the top of the figure, and *vice versa*. + +After starting the algorithm, we first guess an initial transition. If we can find one, we push the labels of the nodes 1, ..., k onto SQ and the number 0 onto SN. Then we enter the while loop. As long as the stacks are not empty, we take the top elements of both stacks. If $n > k$ in line 11, this indicates that we have checked all nodes on this level, and we backtrack without pushing anything on the stacks, which means that we will continue at the next upper level in the next loop. Otherwise, we store the information that we have to check our next sibling \ No newline at end of file diff --git a/samples/texts/4943237/page_11.md b/samples/texts/4943237/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..74569583304ae409ee03d859d3551bbd4501b6f5 --- /dev/null +++ b/samples/texts/4943237/page_11.md @@ -0,0 +1,17 @@ +by pushing the same tuple of states onto SQ and the incremented number *n* onto +SN. If the current node belongs to $Q_0$ (line 14), we backtrack, which means that +we will continue with the next sibling. Otherwise, we try to guess a transition +from this node, and if we can find one, we push the required node labels for the +children of the current node onto SQ and the value 0 onto SN (line 20), which +means that we will descend to the first child of the current node in the next +loop. + +**Theorem 8** The emptiness problem of the language accepted by an *m*-segmentable Büchi automaton $\mathcal{A} = (Q, \Delta, I, F)$ over *k*-ary trees can be decided by a non-deterministic algorithm using space $O(\log(\#Q) \cdot m \cdot k)$. + +**Proof.** In order to show soundness, we will prove the claim “if the algorithm processes a node *n* or backtracks without descending into *n*, then there is a run *r* in which *n* is labelled with the same state as in the algorithm” by induction over the iterations of the while loop. Initially, if the algorithm does not answer “empty”, there is a transition (*q*₀, *q*₁, ..., *q*ₖ) from an initial state, which can serve as root of the run *r*, and for which the states 1, ..., *k* of *r* can be labelled with *q*₁, ..., *q*ₖ. + +If the algorithm has reached a node $n = n_0 \cdot n_1 \cdot \dots \cdot n_\ell$ without failing, it follows by induction hypothesis that each of the previously visited nodes corresponds to a node in $r$. Now there are two possibilities: firstly, if $r(n) \in Q_0$, then, since $\mathcal{A}$ is $Q_0$-looping, there exists a $k$-ary subtree rooted at $n$ all of whose states are accepting. Otherwise, since the algorithm does not answer “empty”, there is a transition $(r(n), q'_1, \dots, q'_k)$, and we can use the same transition in the construction of a run. + +In order to show completeness, we will prove the claim "if there exists a run, the algorithm can reach or skip every node in {1, ..., k}* without failing" by induction over the structure of the run *r*. Since there is a run, we can guess an initial transition, and the nodes of the first level have the same labels in the algorithm as in *r*. If we have reached a node *n* which corresponds to the node *n* in *r* with *r(n) = q*, there are again two possibilities: if *q* ∈ $Q_0$, the algorithm will backtrack and skip over all successor nodes of *n*. Otherwise, since *r* is a run, there exists a transition (*q*, $q'_1$, ..., $q'_k$), which the algorithm can guess, and therefore it will not fail. + +Regarding memory consumption, observe that the SQ stack contains, for every level, *k* states, each of which can be represented using space logarithmic in the number of states, e.g. by using binary coding. Since $\mathcal{A}$ is *m*-segmentable, there can be at most *m* tuples before the current state $q_n$ belongs to $Q_0$, thus the size of SQ is bounded by $\log(\#Q) \cdot m \cdot k$. SN stores at most *m* numbers between 0 and *k*, so the algorithm uses space logarithmic in the size of $\mathcal{A}$. ■ \ No newline at end of file diff --git a/samples/texts/4943237/page_12.md b/samples/texts/4943237/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..70932f421289be8189efd816ef9d8b51250810f4 --- /dev/null +++ b/samples/texts/4943237/page_12.md @@ -0,0 +1,17 @@ +The condition that an automaton $\mathcal{A}$ is $m$-segmentable is rather strong since it requires the transition relation to reduce the class with every possible transition, and thus e.g. the automaton $\mathcal{A}_{C,T}$ in Definition 4 cannot easily be proved to be segmentable even if the TBox is empty. The reason for this is that $\mathcal{A}_{C,T}$ does not require the Hintikka sets of the successor states to use only a lower quantification depth. However, in order to test emptiness, we only need the *existence* of such a transition. This is the idea behind the generalisation in the following definition. + +**Definition 9 (Weakly-m-segmentable, reduced.)** A Büchi automaton $\mathcal{A} = (Q, \Delta, I, F)$ is called weakly-$m$-segmentable if there exists a partition $Q_0, Q_1, \dots, Q_m$ of $Q$ such that $\mathcal{A}$ is $Q_0$-looping and for every $q \in Q$ there exists a function $f_q: Q \to Q$ which satisfies the following conditions: + +1. if $(q, q_1, \dots, q_k) \in \Delta$, then $(q, f_q(q_1), \dots, f_q(q_k)) \in \Delta$, and if $q \in Q_n$, then $f_q(q_i) \in Q_{*T*, because in the definition +of the interpretation for the role names, we consider only the successors that will +satisfy the existential restrictions of a node, and pay no attention to any other +possible successors. The universal restrictions are then immediately satisfied by +the definition of C, *T*-compatible. + +The “only-if” direction does not require significant modifications, because if +there is a model for a concept *C* and a TBox *T*, the node labels of the Hintikka +tree constructed as in [LS00] can easily be extended to reflect the constraints +imposed by the TBox. +■ + +With this result, we can use automata operating on Hintikka trees to test +for the existence of models. As mentioned before, we can omit the labelling of +the tree, since we are only interested in the existence of a model and all relevant +information to answer this question is kept in the transition relation. + +**Definition 4 (Automaton AC,T.)** For a concept *C* and a TBox *T*, let *k* be the number of existential formulas in sub(*C*, *T*). Then the looping automaton *A**C*,T* = (*Q*, *Δ*, *I*) is defined as follows: *Q* = {*S* ⊆ sub(*C*, *T*) | *S* is a *T*-expanded Hintikka set}; *Δ* = {*(*S*, *S*1, ..., *S**k*) | (*S*, *S*1, ..., *S**k*) is *C*, *T*-compatible}; *I* = {*S* ∈ *Q* | *C* ∈ *S*}. + +Using $\mathcal{A}_{C,T}$, we can reduce the satisfiability problem for $\mathcal{ALC}$ to the (non-) emptiness problem of $\mathcal{L}(\mathcal{A}_{C,T})$. Since these results are well known, they will not be formally proved. + +**Theorem 5** The language accepted by the automaton $\mathcal{A}_{C,\mathcal{T}}$ is empty iff $C$ is unsatisfiable w.r.t. $\mathcal{T}$. + +**Corollary 6** Satisfiability of $\mathcal{ALC}$ concepts w.r.t. general TBoxes is decidable in EXPTIME. + +For general TBoxes, this complexity bound is tight [Spa93], but in the special case of an empty or acyclic TBoxes, it can be improved to PSPACE. Usually, this is proved using a tableau algorithm, but in the next section we will show how the special properties of acyclic TBoxes can be used to perform the emptiness test of the automaton with logarithmic space. + +# 3 Segmentable automata + +In this section we will show how the space efficiency for the construction and +the emptiness test of the automaton can be improved under specific conditions. +The idea is to define a hierarchy of states and ensure that the level of the state +decreases with every transition. In Section 4 we will then show how the role +depth of concepts can be used to define this hierarchy. \ No newline at end of file diff --git a/samples/texts/4943237/page_9.md b/samples/texts/4943237/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..11e14e354281c0d8e956d5317ba1aa0c3291035f --- /dev/null +++ b/samples/texts/4943237/page_9.md @@ -0,0 +1,59 @@ +1: guess an initial state $q \in I$ + +2: **if** there is a transition from $q$ **then** + +3: guess a transition $(q, q_1, \dots, q_k) \in \Delta$ + +4: **else** + +5: **return** "empty" + +6: **end if** + +7: push ($\mathbf{SQ}$, $(q_1, \dots, q_k)$); push ($\mathbf{SN}$, 0) + +8: **while** $\mathbf{SN}$ is not empty **do** + +9: $(q_1, \dots, q_k) := \text{pop}(\mathbf{SQ})$ + +10: $n := \text{pop}(\mathbf{SN}) + 1$ + +11: **if** $n \le k$ **then** + +12: push($\mathbf{SQ}$, $(q_1, \dots, q_k)$) + +13: push($\mathbf{SN}$, $n$) + +14: **if** $q_n \notin Q_0$ **then** + +15: **if** there is a transition from $q_n$ **then** + +16: guess a transition $(q_n, q_1', \dots, q_k')$ + +17: **else** + +18: **return** "empty" + +19: **end if** + +20: push($\mathbf{SQ}$, $(q_1', \dots, q_k'$)) + +21: push($\mathbf{SN}$, 0) + +22: **end if** + +23: **end if** + +24: **end while** + +25: **return** "not empty" + +Figure 1: Emptiness test for segmentable automata + +**Definition 7 (Q₀-looping, m-segmentable.)** Let $\mathcal{A} = (Q, \Delta, I, F)$ be a Büchi automaton over $k$-ary trees and $Q_0 \subseteq F$. We call $\mathcal{A} \textit{ Q}_0\textit{-looping}$ if for every $q \in Q_0$ there exists a set of states $\{q_1, \dots, q_k\} \subseteq Q_0$ such that $(q, q_1, \dots, q_k) \in \Delta$. + +An automaton $\mathcal{A} = (Q, \Delta, I, F)$ is called *m-segmentable* if there exists a partition $Q_0, Q_1, \dots, Q_m$ of $Q$ such that $\mathcal{A}$ is $Q_0$-looping and, for every $(q, q_1, \dots, q_k) \in \Delta$, it holds that if $q \in Q_n$, then $q_i \in Q_{ 0$ and four particles (N=4). Open(filled) circles denote unoccupied(occupied) sites. These microstates minimize the system energy forming N+1 degenerate ground states. The system can be split in two partitions A and B in order to calculate its quantum correlations (entanglement). b) First nearest neighbor hopping allows transition between the ground states lifting their degeneracy, unlike second nearest neighbor hopping which preserves it. c) The microstates for attractive interactions $U < 0$ and $f = 1/2$. The stacking of all the particles minimizes their energy giving N+1 ground states. + +sites can never be simultaneously occupied like those in Eq. (3). + +A few notes about the properties of the system that we have considered. If we consider our system as a quantum fluid, then the encircled areas in Fig. 1a would be incompressible inside each partition, since bringing two particles on neighboring sites requires overcoming the energy gap U. The half-filled system can be mapped onto a XXZ chain of spins S=1/2 [43]. By using this analogy we can see that the ground state of our model contains hidden anti-ferromagnetic order. This can be understood easily by labeling occupied(unoccupied) sites as spin up(down). Then the spin alternates between up and down, as in a chain containing anti-ferromagnetic order. This is essentially a consequence of the microscopic rule that there must be at least one unoccupied site between all the particles. The idea can be applied to any filling, since we can condense successive unoccupied sites into one. + +## 3 Entanglement + +The entanglement in the strong interaction or the CDW limit, is governed by the different ways the particles organize to form the ground states. In essence, the quantum superposition of the enumerative combinatorics of the point-particles, described by Eq. (3), is creating quantum mechanical correlations in the system. Therefore we can estimate the entanglement, without diagonalizing the + +full Hamiltonian of the system, but by examining the microstructures inside the Fock states that contribute in the ground state wavefunction. The advantage of this approach compared to other methods like numerical diagonalization of the full Hamiltonian, is that we can study a relatively larger number of particles. + +A well established approach to quantify the quantum correlations is to split a system in two partitions, say A and B forming this way a composite system. Then the entanglement between these partitions can be estimated via the reduced density matrix of partition A, $\rho_A \equiv tr_B|\Psi\rangle\langle\Psi|$ after tracing out the rest of the system, that is partition B. The elements of the reduced density matrix can be calculated via $\rho_A^{ij} = \sum_{k \in B} \Psi_{ik}^* \Psi_{jk}$, where $\Psi_{ik} = \frac{1}{\sqrt{D}}$ is the amplitude for each partitioned ground state $|ik\rangle$, where i(k) is the corresponding microstate in A(B). Moreover the von Neumann entanglement entropy can be calculated + +$$S_A \equiv -tr(\rho_A \ln(\rho_A)). \qquad (8)$$ + +The scaling of the entanglement entropy provides information about the strength of entanglement in 1D quantum systems. For critical 1D phases it has been shown that the entropy diverges logarithmically with increasing the partition size, while it saturates/converges for non-critical phases [5]. The logarithmic divergence implies stronger entanglement than the converging case. + +In the case of half-filling, the density matrix obtains a simple form that allows the calculation of the entanglement entropy analytically as follows. We start by noticing that each subsystem A or B is half the size of the full composite system, with N/2 particles distributed in N sites. In order to calculate the element $\rho_A^{11}$ of the density matrix, we can fix the microstate inside A at $|1\rangle$ and then count the different microstates in B. These are N/2 + 1, as if B was an isolated system and we wanted to obtain the number of its ground states. Then we multiply by the square modulus of the normalization factor $\frac{1}{\sqrt{D}} = \frac{1}{\sqrt{N+1}}$ and obtain $\rho_A^{11} = \frac{N+2}{2(N+1)}$. All the other elements of the density matrix $\rho_A^{ij}$ for i,j $\neq$ 1 are equal to $\frac{1}{N+1}$, since the rest of the microstates in A appear only once in the ground state of the composite system. For convenience we define $x_1 = \frac{N+2}{2(N+1)}$, $x_2 = \frac{1}{N+1}$. Then the only two non-zero eigenvalues of the reduced matrix $\rho_A$ are + +$$\rho_{A_{1,2}} = \frac{1}{2} \left( x_1 + ax_2 \pm \sqrt{(x_1 - ax_2)^2 + 4ax_2^2} \right), \quad (9)$$ + +where $a = N/2$. The entropy of the subsystem A in terms of the eigenvalues of the density matrix becomes + +$$S_A = -\rho_{A_1} \ln \rho_{A_1} - \rho_{A_2} \ln \rho_{A_2}. \qquad (10)$$ + +At the thermodynamic limit $N \to \infty$ the above eigenvalues become $\rho_{A_{1,2}} = 1/2$, resulting in the entropy + +$$S_A = \ln 2. \qquad (11)$$ + +This is the entanglement entropy value for a maximally entangled Bell state of two spins in the singlet state. \ No newline at end of file diff --git a/samples/texts/6720294/page_5.md b/samples/texts/6720294/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..83ac7fb3958eda6fd3520413dccc5c522e94e49a --- /dev/null +++ b/samples/texts/6720294/page_5.md @@ -0,0 +1,19 @@ +The convergence at the thermodynamic limit implies semi-local correlations, that result in weak entanglement as in the non-critical phases of XY spin chains [5]. This result agrees with the numerical calculation of $S_A$ versus N using the ground state Eq. (3), shown in Fig. 3(a), where the dashed line is the analytical result Eq. (6). Also, we have plotted $S_A$ using the amplitudes of Eq. (7) for the lowest energy state with $j = D$, shown as empty circles, which gives the same result as the $j = 1$ case. At the thermodynamic limit both cases, converge asymptotically to the value $S_A = \ln2$. + +Note that the steps above are valid for even number of particles N, half of which go at each partition. Following a similar method we can derive that $S_A = \ln2$ for odd N also. + +The origins of the weak entanglement can be understood as follows. We could imagine the partitions A and B, as two isolated many-body systems. Each one is half the size of the full composite system, with $N/2$ particles distributed among N sites. As for the full system, the ground states of each partition are determined by the different microstates that minimize the energy, which requires that there must be at least one empty site between two particles. These states form the Hilbert space $\mathcal{H}_G^{A(B)}$ of each isolated partition. The short range interaction plays an important role at the boundary between A and B, when they are put together to form the full system. As an example consider the system shown in Fig 1(a). Each partition contains two particles and four sites. When we form the ground state of the full system, the combinations of states $|1001\rangle|1010\rangle$, $|1001\rangle|1001\rangle$, $|0101\rangle|1010\rangle$ and $|0101\rangle|1001\rangle$ have to be excluded. In other words, the ground state Hilbert space of the full system $\mathcal{H}_G$ is not simply the tensor product of the individual spaces of A and B, that is, $\mathcal{H}_G \neq \mathcal{H}_G^A \otimes \mathcal{H}_G^B$. The two partitions become weakly entangled, due to the local particle interaction at their boundary. + +In Fig. 3(a) we show $S_A$ for lower fillings ($f=1/3, 1/4$) calculated numerically using the ground state Eq. (3). For all these cases $S_A$ diverges logarithmically, as in the critical phases of XY spin chains [5, 11], implying stronger entanglement than $f=1/2$. The same logarithmic behavior is observed if we use Eq. (7) to calculate $S_A$ (for $j=1$ and $j=D$), represented by empty triangles in Fig. 3(a). This is an indication of the larger complexity of the particle configurations due to the larger spatial freedom compared to the half-filling. We remark that for $f<1/2$, the particle number at each partition is not conserved. Therefore we cannot form the ground state Hilbert space of the full system $\mathcal{H}_G$ as a tensor product of the individual spaces of two isolated partitions A and B, even if we remove the interaction at the boundary between them, unlike the half-filled case. This means that we cannot completely remove the entanglement with local operations, which is another indication of the strong entanglement, that induces long-range spatial correlations. + +The different entanglements are related to the spatial freedom of the particles at the respective fillings. In a low filled system we can freely add an additional particle with- + +out requiring more energy, as long as there is at least one unoccupied site between it and the rest of the particles. This way we can fill the empty space of the system with $\text{Int}[(M+1)/2 - N]$ particles (for even M) that will go at the ground state without affecting its energy, or the energy gap from the excited states. In this sense, the low fillings in our model correspond to a transition towards a superfluid phase[44], i.e. the system at low fillings lies in a critical regime. This could explain the logarithmic divergence of the entanglement entropy with the partition size, which occurs in general at the critical regime of 1D many-body systems. In the half-filled case on the other hand the system lies in a Mott insulating phase, since we cannot add an additional particle without exciting the system. Therefore the different scaling behavior of the entanglement entropy for the half-filled and low filled cases, is related to the different phases the system lies at these fillings. + +Note that a logarithmic divergence of the entropy is also expected when $U=0$, i.e., by removing the spatial constraint of an empty site between all the particles, leaving only the hardcore boson constraint of one particle per site. In this case all possible spatial configurations of the particles are allowed in Eq. (3), which will contain the full entanglement of the whole system. Our approach allows additional control of the entanglement, due to splitting of the system in isolated Hilbert spaces that contain different entanglement strengths, according to the microstructure inside the corresponding Fock states. + +Another way to detect the entanglement is by measuring the purity of the quantum state with density matrix $\rho$ which can be quantified by $\text{Tr}(\rho^2)[40]$. When $\text{Tr}(\rho^2) < 1$ the state is mixed which means that it is not quantum mechanically fully consistent and contains statistical fluctuations. When this happens for the reduced density matrix of a partition of a quantum system, entanglement with the rest of the system is implied. We have found $\text{Tr}(\rho_A^2) < 1$ for all the fillings studied ($f = 1/2, 1/3, 1/4$), which is an additional indication of the entanglement present in our system. + +Our results bear resemblance to topological ordered phases of matter, which is usually identified through its highly degenerate ground state and strong long-range entanglement properties. However we are not able to obtain both these properties simultaneously. For example the half-filled system has a highly degenerate ground state, that is not lifted by a second nearest neighbor hopping, but lacks the strong entanglement. On the other hand at lower filling the degeneracy is lifted for nearest and second nearest hopping, but the entanglement of these ground states, becomes stronger, as indicated by the logarithmic divergence of the entanglement entropy. + +In addition we have found that most of the eigenvalues of the density matrix are doubly degenerate as in the Haldane phase of spin chains where string order and entanglement are present[11, 12, 21]. \ No newline at end of file diff --git a/samples/texts/6720294/page_7.md b/samples/texts/6720294/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..6a3336b3e7ed1bdf350a5f81844cefd03c8b0eab --- /dev/null +++ b/samples/texts/6720294/page_7.md @@ -0,0 +1,107 @@ +ther analysis is needed, through the calculation of topo- +logical non-local measures like the string-order parameter +for example[11, 12, 21]. + +5 Attractive interactions + +We briefly analyze the case of negative U, that is, for at- +tractive interactions between the particles. In this case the +ground state is obtained by stacking all the particles to- +gether at neighboring sites, which minimizes their energy +at $E = -U(N-1)$. These states are separated by a large +gap U from the first excited states. An example of the mi- +crostates can be seen in Fig. 1(c). The degeneracy of these +ground states is preserved for both first and nearest neigh- +bor hopping as can be seen in Fig. 2, unlike the repulsive +interaction case. For the half-filled case the entanglement +entropy in Fig. 3 follows the limit where the number of +the different particle configurations in subsystems A and +B is equal to the corresponding number of microstates. At +this limit the reduced density matrix $\rho_A$ is diagonal, re- +sulting in maximum entropy $S_A = \ln(N+1)$ which is the +corresponding dashed curve in Fig. 3(a). For lower fillings +the entropy is reduced but still diverges logarithmically. + +6 Summary and Conclusions + +To summarize, we have investigated the self-organization +of strongly interacting spinless particles confined in one- +dimension. The spatial freedom of the particles allows +them to organize in various spatial configurations (Fock +states), corresponding to different energy bands, that are +separated by large gaps. These states can be realized in +spinless Hubbard chains of hardcore bosons, which have +gapped energy spectrum with a highly degenerate ground +state. + +By considering a superposition of the many-body Fock +states for the lowest energy of the system (ground state), +we have calculated the resulting quantum correlations. We +have done that by splitting the system in two partitions +and then calculating the scaling of the entanglement en- +ropy of each partition with the system size. We have +shown that the strength of the entanglement depends on +the spatial freedom of the particles, which is determined +by the system filling. For a half-filled system, the entan- +glement resembles that of two spins in a singlet maximally +entangled Bell state. At low fillings the entanglement be- +comes stronger as indicated by the logarithmic scaling of +the entanglement entropy of each partition. In addition we +found that confining the system by using open boundary +conditions, leads to edge modes at the ends of the system +at low fillings. These edge modes can be correlated with +each other, due to the strong long-range entanglement ap- +pearing at the respective fillings. + +Our results show that 1D Hubbard models of spin- +less hardcore bosons, at the strong interaction limit or +the charge density wave limit can have spectrally isolated +ground states with entanglement determined by their fill- +ing, despite the strong localization of the particles. These + +orders are created due to the spatial freedom of the par- +ticles which result in formation of many-body states with +rich micro-structures. Essentially this mechanism could be +used as a way to tune the entanglement in this type of +systems, by controlling the empty space in the system, +without the application of external fields. + +A natural extension of the current analysis would be to investigate the many-body orders and the entanglement in the different excited states that are separated by energy gaps, which contain many-body states with richer structures. In addition it would be interesting to examine the relevant many-body orders in systems of higher dimension. + +As a final note, we would like to remark that our results contribute to the idea, that entangled states with non-trivial features, can be created via the collective behaviors in many-body systems with relatively simple microscopic rules. Such well known examples are for instance, the Haldane phase of integer spin chains[8,9], the Majorana modes in the Kitaev chain[10] and the topological order in the toric code[19]. Apart from its fundamental significance, this approach might be useful in the on-going research on the experimental realization of different topological phases in cold-atomic and photonic systems. + +We would like to thank Ara Go, Guang-Yu Guo, Alexander Schnell, Vladislav Popkov and Daw-Wei Wang for useful comments. We acknowledge resources and financial support provided by the National Taiwan University, the Ministry of Science and Technology, the National Center for Theoretical Sciences of R.O.C. Taiwan and the Center for Theoretical Physics of Complex Systems in Daejeon Korea under the project IBS-R024-D1. + +References + +1. D. C. Tsui, H. L. Stormer, A. C. Gossard, Phys. Rev. Lett. **1982**, *48*, 1559. + +2. R. B. Laughlin, Phys. Rev. Lett. **1983**, *50*, 1395. + +3. J. Bardeen, L. N. Cooper, J. R. 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