diff --git a/samples/texts/102047/page_11.md b/samples/texts/102047/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..5644dccadd6c80bc010b8adfce11d1a21d3fe60b --- /dev/null +++ b/samples/texts/102047/page_11.md @@ -0,0 +1,26 @@ +[DM2] W. Dahmen and C. A. Micchelli, *On the local linear independence of translates of a box spline*, Studia Math., **82** (1985), 243–263. + +[DM3] —, *Multivariate E-splines*, Advances in Math., **76** (1989), 33–93. + +[DR1] N. Dyn and A. Ron, *Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems*, Trans. Amer. Math. Soc., **319** (1990), 381–404. + +[DR2] —, *On multivariate polynomial interpolation*, in *Algorithms for Approximation II*, J. C. Mason, M. G. Cox (eds.), Chapman and Hall, London 1990, 177–184. + +[G] J. A. Gregory, *Interpolation to boundary data on the simplex*, CAGD, 2 (1985), 43–52. + +[J] R. Q. Jia, *A dual basis for the integer translates of an exponential box spline*, preprint 1988. + +[R] A. Ron, *Relations between the support of a compactly supported function and the exponential-polynomials spanned by its integer translates*, Constructive Approx., **5** (1989), 297–308. + +Received April 17, 1989. The first author was supported by the National Science Foundation under Grant No. DMS-8701275 and by the United States Army under Contract No. DAAL03-87-K-0030. + +UNIVERSITY OF WISCONSIN +MADISON, WI 53706 + +TEL-AVIV UNIVERSITY +TEL-AVIV, ISRAEL + +AND + +UNIVERSITY OF WISCONSIN +MADISON, WI 53706 \ No newline at end of file diff --git a/samples/texts/102047/page_13.md b/samples/texts/102047/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/samples/texts/102047/page_14.md b/samples/texts/102047/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..eafdb06b59ee17202cd01b8e22c41bea0369e5e6 --- /dev/null +++ b/samples/texts/102047/page_14.md @@ -0,0 +1,39 @@ +We assume that $X$ spans all of $\mathbb{R}^s$. Then the only point common to all $h \in \mathbb{H}(X)$ is 0, and consequently, the variety of $I^X$ (i.e., the set of common zeros of all the polynomials in $I^X$) consists of 0 alone: + +$$ \mathcal{V}_{I^X} = \{\emptyset\}. $$ + +This implies that the codimension of $I^X$ in the space $\Pi$ of all polynomials in $s$ variables (i.e., the dimension of the quotient space $\Pi/I^X$) is finite, and that its kernel $I^X \perp$ is a finite-dimensional polynomial space, whose dimension equals the codimension of $I^X$. Moreover, $I^X \perp$ is *stratified*, i.e., spanned by its homogeneous elements, since $I^X$ is generated by homogeneous polynomials. [BR2] is a ready reference for these known facts. + +Here, to recall the definition, the kernel $I \perp$ of an ideal $I$ is the set + +$$ (2.2) \qquad \{f \in \mathcal{P}'(\mathbb{R}^s) : p(D)f = 0, \forall p \in I\} $$ + +of all distributions annihilated by the set of differential operators induced by $I$. In particular, since $I^X$ is generated by + +$$ p_h = \langle h^\perp, \cdot \rangle^{(X\setminus h)}, \quad h \in \mathbb{H}(X), $$ + +$I^X \perp$ consists of the solutions $f$ of the system of linear differential equations + +$$ (2.3) \qquad (D_{h^\perp})^{*(X\setminus h)} f = 0, \quad \forall h \in \mathbb{H}(X). $$ + +This section is devoted to a proof of the fact that $I^X \perp$ equals the polynomial space + +$$ (2.4) \qquad \mathcal{P}(X) := \operatorname{span}\{p_V : V \subset X, \operatorname{span}(X\setminus V) = \mathbb{R}^s\}, $$ + +with + +$$ p_V := \prod_{v \in V} \langle v, \cdot \rangle. $$ + +In the proof, the multiset + +$$ (2.5) \qquad \mathbb{B}(X) := \{B \subset X : B \text{ invertible}\} $$ + +of all bases contained in $X$ plays an important role. We use the abbreviation + +$$ (2.6) \qquad b(X) := \# \mathbb{B}(X). $$ + +(2.7) **THEOREM.** The kernel $I^X \perp$ of $I^X$ coincides with $\mathcal{P}(X)$, and + +$$ (2.8) \qquad \dim I^X \perp = b(X). $$ + +The theorem follows from the following three lemmata. \ No newline at end of file diff --git a/samples/texts/102047/page_16.md b/samples/texts/102047/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..cc0a802ec273cb13a626ba9aa41345b6609c43d3 --- /dev/null +++ b/samples/texts/102047/page_16.md @@ -0,0 +1,47 @@ +with $\operatorname{span} X' = \mathbb{R}^s$, and consider the map + +$$T: I^{X'} \to \prod_{h \in \mathbb{H}(X')} P_h : q \mapsto (p_h(D)q)_{h \in \mathbb{H}(X')},$$ + +where $p_h := p_{h, X'} = \langle h^\perp, \cdot \rangle^{(X'\setminus h)}$ are the generators of $I^{X'}$ and +$P_h := p_h(D)(I^{X'}\perp)$. Then + +$$\dim I^{X'} \leq \dim I^{X'} \perp + \sum_{h \in \mathbb{H}(X')} \dim P_h,$$ + +since + +$$\ker T = (I^{X'} \perp) \cap (I^{X} \perp) \subseteq I^{X'} \perp.$$ + +Consequently, by induction, + +$$\dim(I^{X'} \perp) \leq b(X') + \sum_{h \in \mathbb{H}(X')} \dim P_h.$$ + +This proves that + +$$\dim I^{X'} \perp \leq b(X),$$ + +provided we can prove that + +$$\sum_{h \in \mathbb{H}(X')} \dim P_h \leq \#\{B \in \mathcal{B}(X) : x \in B\}.$$ + +In particular, it is sufficient to prove that, for all $h \in \mathbb{H}(X')$, + +$$ (2.12) \qquad P_h \subset I^{X_h} \perp $$ + +with + +$$X_h := (X \cap h) \cup x.$$ + +For, (2.12) implies that + +$$\dim P_h \leq \dim I^{X_h} \perp \leq b(X_h),$$ + +(the last inequality by induction), while + +$$\sum_{h \in \mathbb{H}(X')} b(X_h) = \#\{B \in \mathcal{B}(X) : x \in B\}.$$ + +The claim (2.12) is trivial in case $x \in h$, since then $X'\setminus h = X\setminus h$, +and therefore $p_{h,X'} = p_{h,X}$ and so $P_h = \{0\}$ in that case. + +We now prove (2.12) for the contrary case, i.e., the case when $x \notin h$. We have to show that for every $k \in \mathbb{H}(X_h)$ + +$$ (2.13) \qquad p_{k, X_h}(D)P_h = \{0\}, $$ \ No newline at end of file diff --git a/samples/texts/102047/page_17.md b/samples/texts/102047/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..5679020e468d51128af3a8e8c1de98a819ed4087 --- /dev/null +++ b/samples/texts/102047/page_17.md @@ -0,0 +1,35 @@ +with + +$$p_{k, X_h} = (k^{\perp}, \cdot)'^{(X_h \setminus k)}.$$ + +If $x \notin k$, then $k = h$ and (since $X_h \setminus h = \{x\})$ $p_{k, X_h}$ is a linear polynomial, and there is nothing to prove since then $p_{k, X_h} p_{h, X'} = p_{h, X'}$, while $p_{h, X'}$ annihilates $I^X \perp$ by definition of $I^X \perp$. For the contrary case that $x \in k$, we need to prove that + +$$ (2.14) \qquad D_{k^\perp}^m D_{h^\perp}^{A-m} (I^X \perp) = 0, $$ + +with + +$$ m := \#(X_h \setminus k), \quad A - m := \#(X' \setminus h). $$ + +For this, it is sufficient to show that + +$$ (2.15) \qquad K^m H^{A-m} \in \text{ideal}\{L^{a(l)}\}_l, $$ + +with $l$ running over all hyperplanes spanned by elements of $X$ and containing $k \cap h$, and with + +$$ K := \langle k^{\perp}, \cdot \rangle, \quad H := \langle h^{\perp}, \cdot \rangle, \quad L := \langle l^{\perp}, \cdot \rangle, \\ a(l) := \#(X \setminus l). $$ + +For, (2.14) follows from (2.15) since each generator $L^{a(l)}$ of the ideal in (2.15) appears also in the defining set (2.1) of generators of $I^X$, hence annihilates $I^X \perp$. + +We prove (2.15) by writing each polynomial $L$ as a linear combination of the linear polynomials $K$ and $H$, thereby obtaining the general homogeneous element of degree $A$ of the ideal in (2.15) in the form + +$$ \sum_l (\alpha K + \beta H)^{a(l)} r_l, $$ + +with $r_l$ a homogeneous polynomial of degree + +$$ \mu(l) := A - a(l), $$ + +all $l$. We then show that the resulting linear system (for the coefficients of the various $r_l$) has a solution by showing that its coefficient matrix is the transpose of the matrix which occurs in (univariate polynomial) Hermite interpolation. + +Here are the details. + +Since each $l$ contains $k \cap h$, $l^\perp$ can be written uniquely as a linear combination of $k^\perp$ and $h^\perp$. We find it convenient in the sequel to have the weights in this linear combination sum to 1, i.e., to have $l^\perp$ in the affine hull of $k^\perp$ and $h^\perp$. This we can achieve by first \ No newline at end of file diff --git a/samples/texts/102047/page_18.md b/samples/texts/102047/page_18.md new file mode 100644 index 0000000000000000000000000000000000000000..44ef5b79ce3e4e0a0e3119ff4f25c1e3935c682f --- /dev/null +++ b/samples/texts/102047/page_18.md @@ -0,0 +1,42 @@ +choosing the (signed) magnitudes of the nonzero vectors $k^{\perp}$ and $h^{\perp}$ so that their difference is not perpendicular to any of the finitely many $l$. Then, for each $l$, we choose the nonzero vector $l^{\perp}$ to lie on the line through $k^{\perp}$ and $h^{\perp}$, i.e., so that + +$$l^{\perp} := (1 - \beta)k^{\perp} + \beta h^{\perp}$$ + +for some $\beta = \beta_l$. Then $L = (1 - \beta)K + \beta H$; hence + +$$L^a = \sum_j K^j H^{a-j} B_j^a(\beta),$$ + +with + +$$B_j^r(t) := \binom{r}{j} (1-t)^j t^{r-j}$$ + +the polynomials appearing in the Bernstein form. Since $DB_j^r = r(B_{j-1}^{r-1} - B_{j-1}^{r-1})$, we have more generally + +$$((a+i)!/a!)(H-K)^i L^a = \sum_j K^j H^{a+i-j} D^i B_j^{a+i}(\beta).$$ + +This means that we have available in our ideal an expression of the form + +$$\sum_l \sum_{i=0}^{\mu(l)} (H-K)^i L^{A-i} c(l, i) = \sum_j K^j H^{A-j} \sum_l \sum_{i=0}^{\mu(l)} D^l B_j^A(\beta) c(l, i)$$ + +to match the monomial $K^m H^{A-m}$. Such a match is possible provided the linear system of conditions imposed thereby on the coefficients $c(l, i)$ is solvable. + +We begin the proof that this linear system is indeed solvable by showing that its coefficient matrix is square. With the abbreviation $k' := h \cap k$, we compute + +$$m = \#(X_h \setminus k) = \#((X \cap h) \setminus k') = \#((X' \cap h) \setminus k'),$$ + +and therefore + +$$ (2.16) \qquad A = \#((X' \cap h) \setminus k') + \#(X' \setminus h) = \#(X' \setminus k'). $$ + +Also, + +$$ +\begin{aligned} +\mu(l) &= A - a(l) = \#(X' \setminus k') - \#(X \setminus l) = -1 + \#(X \setminus k') - \#(X \setminus l) \\ +&= \#((X \cap l) \setminus k') - 1. +\end{aligned} +$$ + +Therefore, the number of unknowns is + +$$\sum_l (\mu(l) + 1) = \sum_l \#((X \cap l) \setminus k') = \#(X \setminus k') = A + 1,$$ \ No newline at end of file diff --git a/samples/texts/102047/page_19.md b/samples/texts/102047/page_19.md new file mode 100644 index 0000000000000000000000000000000000000000..931fac67d456c52b1fe77062bf6f94f5c10a0af9 --- /dev/null +++ b/samples/texts/102047/page_19.md @@ -0,0 +1,32 @@ +i.e., equal to the number of equations. Here, the sums are over all $l \in H(X)$ which contain $k'$, which implies that $X \setminus k'$ is the disjoint union of the sets $(X \cap l) \setminus k'$ and so justifies the second equality. + +Now organize the unknowns by $l$ and, within $l$, by $i=0, \dots, \mu(l)$, and order the equations by $j=0, \dots, A$. Then the matrix consists, more precisely, of one block of columns for each $l$, with the $i$th column (in the block for $l$) containing the value at $\beta = \beta_i$ of the $i$th derivative of all the polynomials $B_j^A$, $j=0, \dots, A$, $i=0, \dots, \mu(l)$. Hence our matrix is the transpose of the matrix which occurs in the linear system for the determination of the Bernstein form of the polynomial in $\Pi_A$ which agrees with some function $(\mu(l)+1)$-fold at $\beta = \beta_i$, all $l$. Since such univariate Hermite interpolation is correct (since $\beta_i \neq \beta_{i'}$ for $l \neq l'$), the invertibility of our matrix follows. $\square$ + +We note that (2.7) Theorem now allows us to conclude that all inequalities appearing in its proof must be equalities. This implies, e.g., that + +$$I^{X'} \subset I^X \text{ whenever } X' \subset X,$$ + +and that $p_{h,X'}(D)(I^X \perp) = I^{h \cup X} \perp$. Another immediate consequence is the following + +(2.17) COROLLARY [DM3], [DR1]. + +$$\dim \mathcal{P}(X) = b(X).$$ + +Furthermore, + +$$ (2.18) \qquad d(X) := \min\{\#(X\setminus h) : h \in \mathbb{H}(X)\} $$ + +is the least degree of the generators of $I^X$; hence, since $I^X \perp = \mathcal{P}(X)$, +we have the following. + +(2.19) COROLLARY [DR1]. With $d(X)$ as in (2.18), + +$$\Pi_{ 0$ for all $x \in X\backslash h$; hence + +$$ c_h := \sum_{x \in X \backslash h} \langle h^{\perp}, x \rangle = \#X \backslash h. $$ + +Consequently, with $z$ chosen so as to satisfy $c_h - 1 < \langle h^\perp, z \rangle < c_h$, $\nu_X(z)$ must lie in the union of the hyperplanes + +$$ \langle h^{\perp}, \cdot \rangle = j, \quad j = 0, \dots, c_h - 1. $$ + +Moreover, $\#\nu_X = \dim \mathcal{P}(X)$, because of (2.17) Corollary and the following. + +(3.11) **RESULT [DM2].** If $X$ is unimodular, then + +$$ \#\nu_X = b(X). $$ + +This establishes the following theorem. + +(3.12) **THEOREM.** If $X$ is unimodular, then $(\exp_{\nu_X})_\downarrow = \mathcal{P}(X)$. In particular, $\nu_X$ is correct for $\mathcal{P}(X)$, and $\mathcal{P}(X)$ is of least degree among all polynomial spaces for which $\nu_X$ is correct. + +Note that only the inequality $\#\nu_X \ge b(X)$ was needed in the proof of (3.12) Theorem. As a matter of fact, the converse inequality is a consequence of the theorem. + +**4. The duality between $\mathcal{H}(X)$ and $\mathcal{P}(X)$.** The ideal $I_X$ and its kernel $\mathcal{P}(X)$ are intimately related to another ideal $I_X$ and its kernel $\mathcal{H}(X)$, which play a fundamental role in the theory of box splines. In the following, we review some of the basics about $I_X$ and $\mathcal{H}(X)$ and draw several connections between the two settings. \ No newline at end of file diff --git a/samples/texts/102047/page_6.md b/samples/texts/102047/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..21485bafdaa60fe8e797feba6fb5e60bbf1f76b4 --- /dev/null +++ b/samples/texts/102047/page_6.md @@ -0,0 +1,35 @@ +polynomials, we may assume that this $p$ is homogeneous. But then +$p^*(f) = p^*(f_\downarrow) \neq 0$, showing that the linear map $H \to P^*: f \mapsto (p \mapsto p^*(D)f(0))$ is 1-1; hence $P \to H^*: p \mapsto p^|_H$ is onto. Since $\dim H = \dim H_\downarrow$, and $\dim H_\downarrow = \dim P$ by assumption, the theorem follows. $\square$ + +We note that the converse of (4.3) Theorem does not hold, in gen- +eral. For, it is easy to make up a nonhomogeneous polynomial space +$H$ together with a homogeneous $P$ dual to it, for which the condi- +tions $\dim(\Pi_j \cap P) = \dim(\Pi_j \cap H_\downarrow)$, all $j$, fail to hold, while these +conditions are necessary for $P$ and $H_\downarrow$ to be dual, according to the +following proposition of use later. + +(4.4) PROPOSITION. If the homogeneous polynomial spaces Q and R are dual to each other, then + +$$ +(4.5) \quad \dim(\Pi_j \cap Q) = \dim(\Pi_j \cap R) +$$ + +for all $j$. + +*Proof*. Indeed, if (4.5) is violated for some (minimal) $j$ and, say, $\dim(\Pi_j \cap Q) > \dim(\Pi_j \cap R)$, then there exists a *homogeneous* polynomial $q \in Q$ of degree $j$ for which $q^*$ vanishes on all homogeneous polynomials in $R$ of degree $j$, and hence vanishes on all of $R$, in contradiction to the duality between $Q$ and $R$. $\square$ + +With this, the meaning of the following result is clear. + +(4.6) **RESULT** [DM3]¹, [DR1]. *The polynomial spaces *$\mathcal{P}(X)$* and *$\mathcal{H}(X)$* are dual to each other. + +In (3.12) Theorem, the space $\mathcal{P}(X)$ has been identified as the least +space for certain interpolation problems. In [BR2] the space $\mathcal{H}(X)$ +has been identified as the least space for other interpolation prob- +lems. We now make use of the duality between $\mathcal{P}(X)$ and $\mathcal{H}(X)$ +to connect $\mathcal{H}(X)$ with the interpolation problems associated with +$\mathcal{P}(X)$ and vice versa. As a preparation, we procure a class of spaces- +whose corresponding least space is $\mathcal{H}(X)$ in much the same way in +which we obtained suitable exponential spaces $\exp_{\nu_x}$ whose least is +$\mathcal{P}(X)$: We perturb the linear factors of the set of generators for the + +¹The authors in [DM3] attribute the result to Hakopian. \ No newline at end of file diff --git a/samples/texts/102047/page_9.md b/samples/texts/102047/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..447c34df550c33aa0c73a105979da6737bf51eab --- /dev/null +++ b/samples/texts/102047/page_9.md @@ -0,0 +1,39 @@ +the following + +(5.2) COROLLARY. Let $X$ be a unimodular set of vectors. Then there exists a function (actually many) $\psi = \psi_X$ which is supported in $\Omega_X$ and satisfies + +$$\Pi(\psi) = \mathcal{P}(X).$$ + +The above corollary provides no information about the smoothness +of the compactly supported $\psi$. Yet, it is known [BH2] that, at least +for the special case of the three-direction mesh with equal multiplic- +ities, no piecewise-$\mathcal{P}(X)$ function supported on $\Omega_X$ can match the +smoothness of the corresponding box spline $M_X$. + +The box spline $M_X$ is a smooth function supported on $\Omega_X$. Hence, +one may hope that there exist functions $\phi$ supported on $\Omega_X$ which +are less smooth than $M_X$, yet their corresponding $\Pi(\phi)$ is "better" +in the sense that it contains some of the polynomials of lower degrees +which were missing in $\Pi(M_X) = \mathcal{H}(X)$. [R; Cor. 4.2] gives a partial +negative answer to that hope by showing that for a unimodular $X$ and +a function $\phi$ supported in $\Omega_X$, if $\hat{\phi}(0) \neq 0$ and $\Pi_j \subset \Pi(\phi)$ for some +$j$, then $\Pi_j \subset \mathcal{H}(X) = \Pi(M_X)$. The following result improves that +corollary. + +(5.3) COROLLARY. Let $X$ be a unimodular set of directions, $M_X$ +the corresponding box spline. Let $\phi$ be a compactly supported function +satisfying + +$$\operatorname{supp} \phi \subset \operatorname{supp} M_X,$$ + +and + +$$\hat{\phi}(0) \neq 0.$$ + +Then, for each $j$, + +$$\dim(\Pi_j \cap \Pi(\phi)) \leq \dim(\Pi_j \cap \Pi(M_X)).$$ + +*Proof.* Let $\nu$ be one of the sets $\nu_X(z)$ associated with $X$. By (3.12) Theorem, $(\exp_\nu)_\downarrow = \mathcal{P}(X)$. Now we may apply (5.1) Result to conclude that $\nu$ is total for $\Pi(\phi)$, and hence $\Pi(\phi)$ can be extended to a space $Q$ for which $\nu$ is correct. Since $\mathcal{P}(X)$ is the least space of $\exp_\nu$, it satisfies the least degree property (3.4), thus we conclude that for every $j$ + +$$ (5.4) \quad \dim(\Pi_j \cap \Pi(\phi)) \le \dim(\Pi_j \cap Q) \le \dim(\Pi_j \cap \mathcal{P}(X)). $$ \ No newline at end of file diff --git a/samples/texts/1072043/page_17.md b/samples/texts/1072043/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..76a71fa5dbad769af0b94a0927a02bcddd1bb893 --- /dev/null +++ b/samples/texts/1072043/page_17.md @@ -0,0 +1,25 @@ +### 6.2.3 Power factor, apparent power, active power and reactive power + +The power factor, the apparent power S (VA), the active power P (W), and the reactive power Q (Var) are related through the equations + +$$PF = \cos \phi = \frac{P(W)}{S(VA)} \qquad (21)$$ + +$$S = VI' \qquad (22)$$ + +$$P = S \cos \phi \qquad (23)$$ + +$$Q = S \sin \phi \qquad (24)$$ + +### 6.2.4 Harmonic calculation + +Total harmonic distortion of voltage ($THD_v$) and current ($THD_i$) can be calculated by the Equations 25 and 26, respectively. + +$$\%THD_{i} = \frac{\sqrt{\sum_{h=2}^{\infty} I_{h(rms)}^{2}}}{I_{1(rms)}} \times 100\% \qquad (25)$$ + +$$\%THD_{v} = \frac{\sqrt{\sum_{h=2}^{\infty} V_{h(rms)}^{2}}}{V_{1(rms)}} \times 100\% \qquad (26)$$ + +Where $V_h (\text{rms})$ is RMS value of h th voltage harmonic, $I_h (\text{rms})$ RMS value of h th current harmonic, $V_1 (\text{rms})$ RMS value of fundamental voltage and $I_1 (\text{rms})$ RMS value of fundamental current + +
ParameterSteady state FVHC conditionTransient step down condition
ExperimentalModeling% ErrorExperimentalModeling% Error
Vrms (V)218.31218.040.12217.64218.20-0.26
Irms (A)23.1023.21-0.484.474.450.45
Frequency (Hz)50500.0050.0050.000.00
Power Factor0.990.990.000.990.990.00
THDv (%)1.151.2-4.351.181.24-5.08
THDi (%)3.253.124.003.533.68-4.25
S (VA)5044.385060.7-0.32972.85970.990.19
P (W)4993.945010.1-0.32963.12961.280.19
Q (Var)711.59713.85-0.32137.24136.970.19
V p.u.0.990.990.000.980.99-1.02
+ +Table 4. Comparison of measured and modeled electrical parameters of the FVHC condition and the transient step down condition \ No newline at end of file diff --git a/samples/texts/1072043/page_21.md b/samples/texts/1072043/page_21.md new file mode 100644 index 0000000000000000000000000000000000000000..c84402e7e249e62b65db1091aa10cb014dbdb1f7 --- /dev/null +++ b/samples/texts/1072043/page_21.md @@ -0,0 +1,29 @@ +[6] Muh. Imran Hamid and Makbul Anwari, "Single phase Photovoltaic Inverter Operation Characteristic in Distributed Generation System", Distributed Generation, Intech book, 2010 + +[7] Vu Van T., Driesen J., Belmans R., Interconnection of distributed generators and their influences on power system. International Energy Journal, vol. 6, no. 1, part 3, pp. 127-140, 2005. + +[8] A. Moreno-Munoz, J.J.G. de-la-Rosa, M.A. Lopez-Rodriguez, J.M. Flores-Arias, F.J. Bellido-Outerino, M. Ruiz-de-Adana, "Improvement of power quality using distributed generation", Electrical Power and Energy Systems 32 (2010) 1069-1076 + +[9] P.R.Khatri, V.S.Jape, N.M.Lokhande, B.S.Motling, "Improving Power Quality by Distributed Generation" Power Engineering Conference, 2005. IPEC 2005. The 7th International. + +[10] Vu Van T., Driesen J., Belmans R., Power quality and voltage stability of distribution system with distributed energy resources. International Journal of Distributed Energy Resources, vol. 1, no. 3, pp. 227-240, 2005. + +[11] Woyte A., Vu Van T., Belmans R., Nijs J., Voltage fluctuations on distribution level introduced by photovoltaic systems. IEEE Transactions on Energy Conversion, vol. 21, no. 1, pp. 202-209, 2006. + +[12] Thongpron, K.Kirtara, Effects of low radiation on the power quality of a distributed PV-grid connected system, Solar Energy Materials and Solar Cells Solar Energy Materials and Solar Cells, Vol. 90, No. 15. (22 September 2006), pp. 2501-2508. + +[13] S.K. Khadem, M.Basu and M.F.Conlon, "Power quality in Grid connected Renewable Energy Systems: Role of Custom Power Devices", International conference on Renewable Energies and Power quality (ICREPQ'10), Granada, Spain, 23rd to 25th March, 2010 + +[14] Mohamed A. Eltawil Zhengming Zhao, "Grid-connected photovoltaic power systems: Technical and potential problems – A review" Renewable and Sustainable Energy Reviews Volume 14, Issue 1, January 2010, Pages 112-129 + +[15] Soeren Baekhoej Kjaer, et al., "A Review of Single-Phase Grid-Connected inverters for Photovoltaic Modules, EEE Transactions and industry applications, Transactions on Industry Applications, Vol. 41, No. 5, September 2005 + +[16] F. Blaabjerg, Z. Chen and S. B. Kjaer, "Power Electronics as Efficient Interface in Dispersed Power Generation Systems." IEEE Trans. on Power Electronics 2004; vol.19 no. 5. Pp. 1184-1194. 2000. + +[17] Chi, Kong Tse, "Complex behavior of switching power converters", Boca Raton : CRC Press, c2004 + +[18] Giraud, F., Steady-state performance of a grid-connected rooftop hybrid wind-photovoltaic power system with battery storage, IEEE Transactions on Energy Conversion, 2001. + +[19] G.Saccomando, J.Svensson, "Transient Operation of Grid-connected Voltage Source Converter Under Unbalanced Voltage Conditions", IEEE Industry Applications Conference, 2001. + +[20] Li Wang and Ying-Hao Lin, "Small-Signal Stability and Transient Analysis of an Autonomous PV System", Transmission and Distribution Conference and Exposition, 2008. \ No newline at end of file diff --git a/samples/texts/2395852/page_3.md b/samples/texts/2395852/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..9c4d93fd82cfb993f8f3d85c8f2d0fc7e2246230 --- /dev/null +++ b/samples/texts/2395852/page_3.md @@ -0,0 +1,7 @@ +in order to check whether they apply to $\mathcal{ALC}_{RA\ominus}$. The discussion provides some key-insights into the high expressiveness of composition-based role axioms. + +**$\mathcal{ALC}_{R+}$**: The description logic $\mathcal{ALC}_{R+}$ augments $\mathcal{ALC}$ (see [17]) with transitively closed roles. A role $R$ may be declared as transitively closed which enforces (for every model $\mathcal{I}$) $R^\mathcal{I} = (R^\mathcal{I})^+$. $\mathcal{ALC}_{R+}$ is obviously a proper sub-fragment of $\mathcal{ALC}_{RA\ominus}$, since a role $R$ can be declared as transitively closed with the role axiom $R \circ R \sqsubseteq R$, which enforces $(R^\mathcal{I})^+ \subseteq R^\mathcal{I}$ and therefore $R^\mathcal{I} = (R^\mathcal{I})^+$. The concept satisfiability problem of $\mathcal{ALC}_{R+}$ is decidable and PSPACE-complete. $\mathcal{ALC}_{R+}$ is basically just a syntactic variant of the multi-modal logic $K4_n$, with $n$ transitive accessibility relations; plain $\mathcal{ALC}$ corresponds to $K_n$ (see [18]). The $n$ accessibility relations correspond to $n$ different roles. The only difference between $\mathcal{ALC}_{R+}$ and $K4_n$ is that the latter requires that all $n$ accessibility relations are transitively closed, whereas the transitive closure of a role is optionally in $\mathcal{ALC}_{R+}$. + +**$\mathcal{ALC}_+$ and $\mathcal{ALC}_\oplus$:** As Sattler points out, $\mathcal{ALC}_{R+}$ is not capable to distinguish “direct” and “indirect” successors of a transitively closed role $R$. Baader had already introduced the language $\mathcal{ALC}_+$ (see [1]) which provides a transitive closure operator (in fact, $\mathcal{ALC}_+$ is more or less a notational variant of the Propositional Dynamic Logic, PDL, see [18]). Both a “generating” role $R$ and its transitive closure $+(R)$ can be distinguished and used separately within concepts. $(+(R))^\mathcal{I} = (R^\mathcal{I})^+$ is enforced. $\mathcal{ALC}_+$ is no longer a subset of FOPL, since the transitive closure of a role cannot be expressed in FOPL (this violates the compactness of FOPL). However, it can be expressed in FOPL that a role $R$ is transitively closed: $\forall x, y, z : R(x, y) \wedge R(y, z) \Rightarrow R(x, z)$. There is no way to simulate the expressiveness of $\mathcal{ALC}_+$ in $\mathcal{ALC}_{RA\ominus}$, since the latter is still a subset of FOPL, but the former is not. The concept satisfiability problem of $\mathcal{ALC}_+$ is decidable and EXPTIME-complete. In the search for a computationally less expensive logic, Sattler introduced the language $\mathcal{ALC}_\oplus$ (see [17]), which replaces the transitive closure operator “+” with the so-called “transitive orbit” operator “⊕”. Like the “+”-operator, the transitive orbit operator can be applied to roles. Applied to a role $R$, the role $⊕(R)$ is interpreted as some relation being a superset of the transitive closure of the generating role $R$, but not necessarily the smallest one: only $(R^\mathcal{I})^+ \subseteq (⊕(R))^\mathcal{I}$ is granted. The concept satisfiability problem of $\mathcal{ALC}_\oplus$ is decidable but unfortunately, as Sattler has shown, EXPTIME-complete (as for $\mathcal{ALC}_+$, too). + +We show that $\mathcal{ALC}_\oplus$ is subsumed by $\mathcal{ALC}_{RA\ominus}$ by reducing the concept satisfiability problem of $\mathcal{ALC}_\oplus$ to the concept satisfiability problem of $\mathcal{ALC}_{RA\ominus}$: given an $\mathcal{ALC}_\oplus$ concept C, we construct a concept C' and a role box $\mathfrak{R}'$ such that \ No newline at end of file diff --git a/samples/texts/2395852/page_5.md b/samples/texts/2395852/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..98d25b8add36ea9e753e9c0c49204a17aacf40fb --- /dev/null +++ b/samples/texts/2395852/page_5.md @@ -0,0 +1,11 @@ +The so-called *Guarded Fragment* (*GF*) as introduced by Andréka, van Benthem and Németi is another fragment of FOPL that is decidable (it even has the finite tree model property). We will not formally discuss it here (see [9, 8]). However, its prominent feature is that the number of variables is *not* bounded, as long as certain syntactic restrictions on the use of the quantifiers are obeyed. Grädel suggested to use the guarded fragment as the basis for a new family of *n*-ary DLs (see [8]). Since *FO*³ is undecidable, but *GF*³ (the guarded fragment with three variables) is decidable, decidability for *ALCRA*ø would follow if *ALCRA*ø was expressible in *GF*³. A few informal words regarding the guarded fragment seem to be appropriate: when translating propositional modal logics (for example, *ALC* resp. *Kn* ) into FOPL, one observes that the quantifiers are always used in a certain *guarded way*. The quantifiers appear only in “patterns” of the form ∀x, y : *R(x,y)* ⇒ *C(y)* and ∃x, y : *R(x,y)* ∧ *C(y)*. Here, the atom *R(x,y)* is used as a *guard*. This observation was generalized into the guarded fragment and observed to be responsible for the nice computational properties resp. decidability of many modal logics (and the guarded fragment as well). More specifically, the guard must always be an atom (complex formulas may not be guards) and must contain all variables that appear in the subsequent of the formula “behind” the guard. The formulas ∀x, y : *R(x,y)* ⇒ *C(y)* and ∃x, y : *R(x,y)* ∧ *C(y)* are therefore in *GF*², where *GF*² is the guarded fragment with two variables: *GF*² = *F0*²∩*GF*. If all non-unary atoms are only used as guards, the *GF* formula is said to be *monadic*. This is obviously the case for *ALC*, since the binary relations occur solely as guards. Obviously, the transitivity axiom ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* is *not* in the *GF*. The loosely guarded fragment (*LGF*) is a generalization of the guarded fragment by additionally allowing not only atoms (like *R(x,y*) being guards, but also conjunctions of atoms. However, the transitivity axiom is not even in the *LGF*, since it is additionally required that there must be a conjunct using *x* and *z* in *one guard atom*; e.g. ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ∧ *S(x,z)* ⇒ *R(x,z)* is in the *LGF*, but ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* is not. Grädel has even shown that it is impossible to express that a relation is transitively closed within the guarded or the loosely guarded fragment. The transitivity axiom ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* cannot be expressed by any means (see [9]). But this means that *ALCRA*ø is not in the *LGF*. + +Therefore, the (loosely) guarded fragment has been extended by transitivity +on an “extra-logical” level (since transitivity is not expressible within the logic +itself), and the following results have been obtained:³ + +• $GF^3$ with transitive relations is undecidable (see [9]). + +• $\mathit{LGF}_-$ with one transitive relation is undecidable (see [7]). + +³A minus suffix indicates that the logic does not provide equality. \ No newline at end of file diff --git a/samples/texts/2395852/page_6.md b/samples/texts/2395852/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..3efe5fe4565aea6a1bff4ecc718bd16edbbbac80 --- /dev/null +++ b/samples/texts/2395852/page_6.md @@ -0,0 +1,13 @@ +Figure 3: RCC8 Qualitative Spatial Relationships: *EQ* = Equal, *DC* = Disconnected, *EC* = Externally Connected, *PO* = Partial Overlap, *TPP* = Tangential Proper Part, *NTPP* = Non-Tangential Proper Part. Read the relations as *TPP*(A, B), *NTPP*(A, B) etc. *TPP* and *NTPP* have corresponding inverse relationships: *TPPI* and *NTPPI*, e.g. *TPPI*(B, A), *NTPPI*(B, A). + +• Even $GF^2$ with transitive relations is undecidable (see [7]). + +• *Monadic* $GF^2$ with binary transitive, symmetric and/or reflexive relations is *decidable* (see [7]). + +None of these results is applicable in the case of $\mathcal{ALC}_{RA\ominus}$. The most important result concerning $\mathcal{ALC}_{RA\ominus}$ is the last one, since $\mathcal{ALC}$ is in monadic $GF^2$, and the role box allows one to express, for example, transitivity. However, the role boxes of $\mathcal{ALC}_{RA\ominus}$ can express a lot more than transitivity. Therefore, this result implies the decidability of, e.g. $\mathcal{ALC}_{R+}$, but not of $\mathcal{ALC}_{RA\ominus}$. In fact, a much more general result has been shown by Ganzinger et al. (see [7]), but it does not apply to axioms of the form $\forall x, y, z : R(x, y) \land S(y, z) \Rightarrow T(x, z)$. + +# 4 Spatial Reasoning With $\mathcal{ALC}_{RA\ominus}$ + +A widely accepted approach in the field of spatial reasoning for describing spatial relationships between two-dimensional objects in the plane is to describe their spatial interrelationship qualitatively instead of describing their metrical and/or geometrical attributes. Examples for qualitative spatial calculi fitting into this category are the well-known RCC8 calculus (see [16]) and the so-called Egenhofer-relations (see [6]). In the case of RCC8, we can distinguish 8 disjoint – pairwise exclusive – base relations that describe purely topological aspects of the scene, *exhaustively* covering the space of all possibilities (see Figure 3). Informally speaking this means that between every two objects in the plane *exactly one* of the RCC8 relations holds. + +Given a set of base relations, e.g. the RCC8 relations, the most important inference problem is the following: given three regions *a*, *b* and *c* in the plane, and the relations *R*(*a*, *b*), *S*(*b*, *c*) between them, what can be deduced about the possible relationships between *a* and *c*? This basic inference task is usually given by \ No newline at end of file diff --git a/samples/texts/2395852/page_8.md b/samples/texts/2395852/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..8ce338b2fcc2fe0a6629e0590276720364f22a27 --- /dev/null +++ b/samples/texts/2395852/page_8.md @@ -0,0 +1,7 @@ +Figure 5: Illustration of a model of *special\_figure* + +Then, the question is: does *figure\_touching\_a\_figure* subsume *special\_figure*, or equivalently, is *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ¬(*figure* ⊓ ∃*EC*.*figure*) unsatisfiable w.r.t. a role box $\mathfrak{R}$ corresponding to the RCC8 composition table? + +After pushing the negation sign inwards and removing the obviously contradictory disjunct from the resulting disjunction, the concept *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ∀*EC*.¬*figure* must be unsatisfiable then. Please consider Figure 5 which illustrates a “model” of *special\_figure*, with $a \in special\_figure^I$, $b \in (figure \sqcap \exists EC circle)^I$, and $c \in circle^I$, with $\langle a, b \rangle \in TPPI^I$, $\langle b, c \rangle \in EC^I$; please note that $TPPI \circ EC \subseteq EC \sqcup PO \sqcup TPPI \sqcup NTPPI \in \mathfrak{R}$. Due to the definition of *special\_figure*, it can be seen that in every model $\langle a, c \rangle \in EC^I$ must hold. But then, due to ∀*EC*.¬*figure*, it is obviously the case that *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ∀*EC*.¬*figure* has no models and is therefore unsatisfiable. This shows that *special\_figure* is indeed subsumed by *figure\_touching\_a\_figure*. + +Please note that there are also other description logics suitable for spatial reasoning tasks, namely the language $\mathcal{ALCRP}(D)$ (see [11]). However, unrestricted $\mathcal{ALCRP}(D)$ is undecidable (see [15]), and its decidable fragment suffers from very strong syntax-restrictions, dramatically pruning the space of allowed concept expressions. In fact, the finite model property is ensured in re- \ No newline at end of file diff --git a/samples/texts/2448265/page_1.md b/samples/texts/2448265/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..0274ffae4e7b071d95a99a96b890aa1744f9525c --- /dev/null +++ b/samples/texts/2448265/page_1.md @@ -0,0 +1,17 @@ +# Ambiguity in the m-bonacci numeration system + +Petra Kocábová, Zuzana Masáková, Edita Pelantová + +► To cite this version: + +Petra Kocábová, Zuzana Masáková, Edita Pelantová. Ambiguity in the m-bonacci numeration system. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2007, 9 (2), pp.109-123. hal-00966537 + +HAL Id: hal-00966537 + +https://hal.inria.fr/hal-00966537 + +Submitted on 26 Mar 2014 + +**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. + +L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. \ No newline at end of file diff --git a/samples/texts/2448265/page_10.md b/samples/texts/2448265/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..22ab8fce0337df62577c98307d457736acf91305 --- /dev/null +++ b/samples/texts/2448265/page_10.md @@ -0,0 +1,22 @@ +# Ambiguity in the *m*-bonacci numeration system + +Petra Kocábová and Zuzana Masáková and Edita Pelantová + +*Department of Mathematics FNSPE, Czech Technical University* +*Trojanova 13, 120 00 Praha 2, Czech Republic* + +*received 27 Dec 2004, revised 23 Apr 2005, accepted 19 May 2005.* + +We study the properties of the function $R^{(m)}(n)$ defined as the number of representations of an integer $n$ as a sum of distinct $m$-Bonacci numbers $F_k^{(m)}$, given by $F_i^{(m)} = 2^{i-1}$, for $i \in \{1, 2, \dots, m\}$, $F_{k+m}^{(m)} = F_{k+m-1}^{(m)} + F_{k+m-2}^{(m)} + \dots + F_k^{(m)}$, for $k \ge 1$. We give a matrix formula for calculating $R^{(m)}(n)$ from the greedy expansion of $n$. We determine the maximum of $R^{(m)}(n)$ for $n$ with greedy expansion of fixed length $k$, i.e. for $F_k^{(m)} \le n < F_{k+1}^{(m)}$. Unlike the Fibonacci case $m=2$, the values of the maxima are not related to the sequence $(F_k^{(m)})_{k \ge 1}$. We describe the palindromic structure of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$, which is richer than in the case of Fibonacci numeration system. + +**Keywords:** numeration system, generalized Fibonacci numbers, greedy expansion, palindromes + +## 1 Introduction + +Any strictly increasing sequence $(G_k)_{k \in \mathbb{N}}$, with $G_k \in \mathbb{N}$, $G_1 = 1$, defines a system of numeration where every positive integer can be written as a linear combination $\sum a_k G_k$, where $a_k \in \mathbb{N}_0$, see for instance [8]. Some sequences $(G_k)_{k \in \mathbb{N}}$ have even nicer property: Every positive integer can be expressed as a sum of distinct elements of the sequence $(G_k)_{k \in \mathbb{N}}$. The necessary and sufficient condition so that it is possible is that the sequence satisfies $G_1 = 1$ and $G_n - 1 \le \sum_{i=1}^{n-1} G_i$ for all $n \in \mathbb{N}$. Example of such a sequence is $(2^{k-1})_{k \ge 1}$ or $(F_k)_{k \ge 1}$, the sequence of Fibonacci numbers. The expression of $n$ in the form + +$$n = G_{i_s} + G_{i_{s-1}} + \cdots + G_{i_1}, \quad \text{where } i_s > i_{s-1} > \cdots > i_1 \ge 1,$$ + +is called a representation of $n$ in the numeration system $(G_k)_{k \in \mathbb{N}}$. This representation can be written using a sequence of coefficients $(a_k)_{k \in \mathbb{N}} \in \{0, 1\}^\mathbb{N}$ as $n = \sum_{i=1}^\infty a_i G_i$, where only a finite number of elements of the sequence $(a_k)_{k \in \mathbb{N}}$ are non-zero. The maximal index $k$ such that $a_k \neq 0$ are called the length of the representation. The representation can be coded by the word $a_k a_{k-1} \cdots a_1$ in the alphabet $\{0, 1\}$. In the writing of number representations we adopt the usual convention that the concatenation of $l$ copies of a finite word is written $w^l$, for $l = 0, 1, 2, \dots$. For example, the representation of the number $n = G_{k+2} + G_k$ is coded by the word $1010^{k-1}$. + +If $(G_k)_{k \in \mathbb{N}} = (2^{k-1})_{k \ge 1}$, then the representation of every integer $n$ in the system $(G_k)_{k \in \mathbb{N}}$ is unique, and the word $a_k a_{k-1} \cdots a_1$ is the binary expansion of $n$. If we choose for $(G_k)_{k \in \mathbb{N}}$ the \ No newline at end of file diff --git a/samples/texts/2448265/page_11.md b/samples/texts/2448265/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..27db04e84e875b05e2bc6ba816472b435775f3b3 --- /dev/null +++ b/samples/texts/2448265/page_11.md @@ -0,0 +1,28 @@ +Fibonacci sequence $G_k = F_k$, given by the recurrence $F_{k+2} = F_{k+1} + F_k$, $F_0 = F_1 = 1$, then most integers have several representations. The number of distinct representations, denoted by $R(n)$, is a function studied by many authors [1, 2, 5, 11]. + +On the set of representations of a given integer $n$ in the system $(G_k)_{k \in \mathbb{N}}$ one can introduce the lexicographic order in the following way: We say that the representation $v_l v_{l-1} \cdots v_1$ of the number $n$ is greater than the representation $u_k u_{k-1} \cdots u_1$ of $n$ if $l > k$ or $l = k$ and the first non-zero element in the sequence $v_k - u_k, v_{k-1} - u_{k-1}, \cdots, v_1 - u_1$ is positive. This order is sometimes called the radix order. The lexicographically greatest representation of a given number $n$ is called the greedy expansion of $n$. + +In this paper we study the measure of ambiguity of the representation of integers in the generalized Fibonacci numeration systems, the so-called *m*-Bonacci systems defined for $m \ge 2$ by recurrence + +$$ +\begin{aligned} +& F_1^{(m)} = 1, \quad F_2^{(m)} = 2, \quad \dots, \quad F_m^{(m)} = 2^{m-1}, \\ +& F_{k+m}^{(m)} = F_{k+m-1}^{(m)} + F_{k+m-2}^{(m)} + \dots + F_k^{(m)}, \quad \text{for } k \ge 1. +\end{aligned} +\tag{1} $$ + +The 2-Bonacci sequence is thus the ordinary Fibonacci sequence; 3-Bonacci sequence is usually called the Tribonacci sequence. Combinatorial properties of the *m*-Bonacci numeration system have been discussed in [7], in order to study the Garsia entropy connected with Pisot numbers $\beta$ fulfilling $\beta^m = \beta^{m-1} + \cdots + \beta + 1$. + +The *m*-Bonacci numeration systems are studied in [6] from the point of view of automata theory. It is proven that addition of integers written in the *m*-Bonacci numeration system can be performed by means of a finite state automaton, whereas it is impossible to convert an *m*-Bonacci representation of an integer into its standard binary expansion by a finite state automaton. + +It has been shown already in [10] that every non-negative integer $n$ can be represented as a sum of distinct elements of the *m*-Bonacci sequence. Such representation of $n$ may not be unique. We denote by $R^{(m)}(n)$ the number of different representations of $n$. The recurrence relation for *m*-Bonacci numbers ensures that starting from an arbitrary representation of $n$ we can get any other representation of $n$ by interchanging $10^m \leftrightarrow 01^m$ or vice versa in the word coding the representation of $n$. + +Obviously, the lexicographically greatest (greedy) representation of $n$ does not contain the block $1^m$. Let us denote the greedy expansion of $n$ in the numeration system $(F_k^{(m)})_{k \ge 1}$ by $\langle n \rangle_m$. It can be written in the form + +$$ \langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} 10^{r_{s-2}} \cdots 10^{r_2} 10^r, \quad \text{where } r_i \in \mathbb{N}_0, $$ + +and for every $i$ such that $m-1 \le i \le s$ we have $r_{i-m+2} + r_{i-m+3} + \cdots + r_i \ge 1$. The length of the greedy expansion $\langle n \rangle_m$ is $s+r_s+r_{s-1}+\cdots+r_1$. If the lengths of $\langle n \rangle_m$ is $k$, then every other representation of $n$ has the length either $k$ or $k-1$. Representations of length $k$ are called ‘long’ representations and their number is denoted by $R_1^{(m)}(n)$; the other representations are called ‘short’ and their number is denoted by $R_0^{(m)}(n)$. Obviously, we have + +$$ R^{(m)}(n) = R_0^{(m)}(n) + R_1^{(m)}(n). $$ + +The aim of the paper is to study the properties of the function $R^{(m)}(n)$. First we show that the Berstel matrix formula [1] for calculation of the value $R^{(2)}(n)$ from the greedy expansion $\langle n \rangle_m$ can \ No newline at end of file diff --git a/samples/texts/2448265/page_12.md b/samples/texts/2448265/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..d214505c92d9d2a0ecdbf400be871d5c93bebddf --- /dev/null +++ b/samples/texts/2448265/page_12.md @@ -0,0 +1,21 @@ +be generalized for $m \ge 3$. In the next section we focus on the study of the segment of the sequence $R^{(m)}(n)$ for $F_k^{(m)} \le n < F_{k+1}^{(m)}$, i.e. for such integers $n$ whose greedy expansion has constant length $k$. For the Fibonacci numeration system it is known [2, 5] that among numbers with a fixed length $k$ of the greedy expansion only $n = F_{k+1}^{(2)} - 1$ satisfies $R^{(2)}(n) = 1$, and moreover, the segments of the sequence $R^{(2)}(n)$ between two unit values are palindromes. For the $m$-Bonacci numeration system with $m \ge 3$ we show that the number of integers $n$ in the segment $[F_k^{(m)}, F_{k+1}^{(m)})$ with a unique representation $R^{(m)}(n) = 1$ is equal to the $(m-1)$-Bonacci number $F_k^{(m-1)}$. Thus the number of 1's in the corresponding segment of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ increases, however, we show that the palindromic structure of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ remains preserved. + +In the rest of the paper we determine the maximum of the function $(R^{(m)}(n))_{n \in \mathbb{N}}$ in the mentioned segment. For $m=2$, i.e. the Fibonacci numeration system, the maxima have been determined in [11], + +$$ \max\{R^{(2)}(n) \mid F_k^{(2)} \le n < F_{k+1}^{(2)}\} = \begin{cases} F_{\frac{k+1}{2}}^{(2)} & \text{for } k \text{ odd,} \\ 2F_{\frac{k-2}{2}}^{(2)} & \text{for } k \text{ even.} \end{cases} $$ + +We shall thus concentrate on determining the values of the maxima for $m \ge 3$. Unlike the Fibonacci case, the values of the maxima are not related to the sequence $(F_k^{(m)})_{k \ge 1}$. + +## 2 The number of representations of $n$ in the $m$-Bonacci system + +The number of representations of a given integer $n$ is related to the possible interchanges $10^m \leftrightarrow 01^m$ in the greedy expansion of $n$. For example, if $\langle n \rangle_m$ is of length $k \le m$, then no interchange is possible and we have $R^{(m)}(n) = 1$. If the length of $\langle n \rangle_m$ is $m+1$, then only $\langle n \rangle_m = 10^m$ admits such an interchange. It follows that + +$$ R^{(m)}(n) = 1, \quad \text{for } 1 \le n \le F_{m+2}^{(m)} - 1, \quad n \ne F_{m+1}^{(m)}, \\ R^{(m)}(F_{m+1}^{(m)}) = 2. \tag{2} $$ + +The aim of this section is to derive a compact formula for calculating the values of the function $R^{(m)}(n)$. Both the formula and its proof are slight generalizations of the result of [1, 5] for the case $m=2$. Consider therefore $m \ge 3$. + +First we state several simple observations, which transpose the calculation of the value $R_0^{(m)}(n)$ and $R_1^{(m)}(n)$ for an integer $n$ with $s$ 1's in its greedy expansion to calculation of $R_0^{(m)}(n)$ and $R_1^{(m)}(n)$ for some $\tilde{n}$ whose greedy expansion has strictly smaller number $\tilde{s} < s$ of 1's. In the following, we shall identify the writing $R^{(m)}(n)$ with $R^{(m)}(\tilde{n})$, where $\tilde{n}$ is the word in the alphabet $\{0,1\}$ coding the greedy expansion of $n$, i.e. a word starting with 1. + +**Fact 2.1** If $0 \le l \le m-2$, then $R_0^{(m)}(10^l w) = 0$, therefore $R_1^{(m)}(10^l w) = R^{(m)}(\tilde{w})$. In matrix form, + +$$ \begin{pmatrix} R_0^{(m)}(10^l w) \\ R_1^{(m)}(10^l w) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(\tilde{w}) \\ R_1^{(m)}(\tilde{w}) \end{pmatrix}. $$ \ No newline at end of file diff --git a/samples/texts/2448265/page_13.md b/samples/texts/2448265/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..c657d326f6aa5aa516b5f181a0b44c6b4964f166 --- /dev/null +++ b/samples/texts/2448265/page_13.md @@ -0,0 +1,59 @@ +**Remark 2.2** + +(i) Note that $R_0^{(m)}(10^l w) = 0$ does not imply $l \le m-2$. It is not difficult to see that $R_0^{(m)}(w) = 0$ +implies that the word $w$ is of the form $w = (10^{m-1})^s \tilde{w}$, where $s \ge 0$ and $\tilde{w}$ is either the +empty word, or a word of the form $\tilde{w} = 10^l \tilde{w}$ for $l \le m-2$. Therefore the smallest $n_1$ such +that $\langle n_1 \rangle_m = k$ and for which $R_0^{(m)}(n_1) = 0$ is the number with greedy expansion + +$$ +\langle n_1 \rangle_m = \begin{cases} (10^{m-1})^s, & \text{if } s := \frac{k}{m} \in \mathbb{N}, \\ (10^{m-1})^s 10^{k-ms-1}, & \text{if } s := \lfloor \frac{k}{m} \rfloor \neq \frac{k}{m}. \end{cases} \tag{3} +$$ + +At the same time, for every $n$ such that $n_1 \le n < F_{k+1}^{(m)}$ we have $R_0^{(m)}(n) = 0$. + +(ii) In the word $\langle n_1 \rangle_m$ of the form (3) one cannot perform any interchange $10^m \leftrightarrow 01^m$, and therefore $R^{(m)}(n_1) = 1$. We have thus found the smallest number $n$ such that $F_k^{(m)} < n < F_{k+1}^{(m)}$ and $R^{(m)}(n) = 1$. Note that in the Fibonacci numeration system $R_0^{(2)}(n) = 0$ already implies $R^{(2)}(n) = 1$. For $m \ge 3$ this is not valid. As an example, consider $\langle n \rangle_m = 110^{k-2}$ for $k \ge m+2$. Such $n$ satisfies $R_0^{(m)}(n) = 0$ and $R^{(m)}(n) \ge 2$. + +(iii) Let us express explicitly the value of $n_1$. Every 1 in the word $\langle n_1 \rangle_m$ at the position $i > m$ represents the number + +$$ +F_i^{(m)} = F_{i-1}^{(m)} + F_{i-2}^{(m)} + \dots + F_{i-m}^{(m)}. \\ +\text{The 1 at a position } i \le m \text{ represents} +$$ + +$$ +F_i^{(m)} = 2^{i-1} = 1 + F_{i-1}^{(m)} + F_{i-2}^{(m)} + \cdots + F_1^{(m)}. +$$ + +The number $n_1$ with greedy expansion of the form (3) is therefore equal to + +$$ +n_1 = 1 + \sum_{i=1}^{k-1} F_i^{(m)} . +$$ + +**Fact 2.3** $R_0^{(m)}(10^{m-1}w) = R_0^{(m)}(w)$ and $R_1^{(m)}(10^{m-1}w) = R^{(m)}(w)$. In a matrix form, + +$$ +\begin{pmatrix} R_0^{(m)}(10^{m-1}w) \\ R_1^{(m)}(10^{m-1}w) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(w) \\ R_1^{(m)}(w) \end{pmatrix}. +$$ + +**Fact 2.4** If $l \ge m$, then we have $R_0^{(m)}(10^l w) = R^{(m)}(10^{l-m} w)$ and $R_1^{(m)}(10^l w) = R_1^{(m)}(10^{l-m} w)$. +In a matrix form, + +$$ +\begin{pmatrix} R_0^{(m)}(10^l w) \\ R_1^{(m)}(10^l w) \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(10^{l-m} w) \\ R_1^{(m)}(10^{l-m} w) \end{pmatrix}. +$$ + +**Lemma 2.5** Let $\langle n \rangle_m = 10^l w$, where $w = (\tilde{n})_m$ for some integer $\tilde{n}$. Then + +$$ +\left(\begin{array}{c} +R_{0}^{(m)}(10^{l} w) \\ +R_{1}^{(m)}(10^{l} w) +\end{array}\right)=\left(\begin{array}{cc} +\left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\ +1 & 1 +\end{array}\right)\left(\begin{array}{c} +R_{0}^{(m)}(w) \\ +R_{1}^{(m)}(w) +\end{array}\right). +$$ \ No newline at end of file diff --git a/samples/texts/2448265/page_14.md b/samples/texts/2448265/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..ae4c56b5ab369a709f5bca468520cce1a5cd8a9c --- /dev/null +++ b/samples/texts/2448265/page_14.md @@ -0,0 +1,33 @@ +**Proof:** Let us write $l = am+b$, where $b \in \{0, 1, \dots, m-1\}$. If $b \le m-2$, then for the calculation of the values $R_0^{(m)}(10^l w)$, $R_1^{(m)}(10^l w)$ one uses $a$ times Fact 2.4 and then Fact 2.1. Since + +$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^a \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} a & a \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} \left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\ 1 & 1 \end{pmatrix}, $$ + +the statement is proved. + +If $b=m-1$, we use $a$ times Fact 2.4 and then Fact 2.3. The matrix identity + +$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^a \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} a+1 & a \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} \left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\ 1 & 1 \end{pmatrix}, $$ + +completes the proof. $\square$ + +In order to derive the formula for calculation of $R^{(m)}(n)$, we need to derive the values $R_0^{(m)}(n)$, $R_1^{(m)}(n)$ for integers $n$ with only one 1 in their greedy expansion. It is easy to see that $R_1^{(m)}(10^l) = 1$ and $R_0^{(m)}(10^l) = [\frac{l}{m}]$, which can be written by + +$$ \begin{pmatrix} R_0^{(m)}(10^l) \\ R_1^{(m)}(10^l) \end{pmatrix} = \begin{pmatrix} \left[ \frac{l+1}{m} \right] & \left[ \frac{l}{m} \right] \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$ + +Since $R^{(m)}(n) = R_0^{(m)}(n) + R_1^{(m)}(n)$, we can formulate the result. For that we introduce the following notation, + +$$ M(l) = M_m(l) := \begin{pmatrix} \left\lfloor \frac{l+1}{m} \right\rfloor & \left\lfloor \frac{l}{m} \right\rfloor \\ 1 & 1 \end{pmatrix}. \qquad (4) $$ + +**Theorem 2.6** Let $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ be the greedy expansion of the integer $n$ in the *m*-Bonacci numeration system. Then + +$$ R^{(m)}(n) = (1\ 1) M(r_s) M(r_{s-1}) \cdots M(r_1) \binom{0}{1}. \qquad (5) $$ + +### 3 Integers with a unique representation in the *m*-Bonacci numeration system + +In order that an integer $n$ has only one representation in the *m*-Bonacci numeration system, the lexicographically greatest and the lexicographically smallest representation must coincide. Consider $n$ in the interval $[F_k^{(m)}, F_{k+1}^{(m)})$. The word coding the greedy expansion of $n$ has the form + +$$ u_k u_{k-1} \cdots u_1, \quad \text{where } u_1, \ldots, u_{k-1} \in \{0, 1\} \text{ and } u_k = 1. \qquad (6) $$ + +If $k < m$, then an arbitrary word of the above form is a greedy expansion of some integer $n$. At the same time it is obvious that in such a word one cannot perform any interchange $10^m \leftrightarrow 01^m$ and therefore this integer $n$ has only one *m*-Bonacci representation. Let + +$$ U_k^{(m)} = \#\{n \mid F_k^{(m)} \le n < F_{k+1}^{(m)}, R^{(m)}(n) = 1\}. $$ \ No newline at end of file diff --git a/samples/texts/2448265/page_15.md b/samples/texts/2448265/page_15.md new file mode 100644 index 0000000000000000000000000000000000000000..e028aed474429bee832fac0d239d61600fc1d2e8 --- /dev/null +++ b/samples/texts/2448265/page_15.md @@ -0,0 +1,40 @@ +We have derived that + +$$U_k^{(m)} = 2^{k-1}, \quad \text{for } k = 1, 2, \dots, m-1. \tag{7}$$ + +Consider now $k \ge m$. A word of length $k$ satisfying (6) is a greedy expansion of some integer $n$, if and only if it does not contain the string $1^m$. In order that no interchange $10^m \leftrightarrow 01^m$ is possible in this word, so that $R^{(m)}(n) = 1$, the word cannot contain the string $0^m$. Therefore $U_k^{(m)}$ is equal to the number of words $u_k u_{k-1} \cdots u_1$ such that + +$$\begin{align} +& u_1, \dots, u_{k-1} \in \{0,1\}, \quad u_k = 1, \quad \text{and} \nonumber \\ +& u_k u_{k-1} \cdots u_1 \text{ does not contain the strings } 0^m, 1^m. \tag{8} +\end{align}$$ + +In order to determine $U_k^{(m)}$, we divide words satisfying (8) into $2(m-1)$ disjoint groups according to their suffix + +$$v \in S := \{10, 10^2, 10^3, \ldots, 10^{m-1}, 01, 01^2, 01^3, \ldots, 01^{m-1}\}.$$ + +The number of words satisfying (8) with suffix $v$ will be denoted by $A_k^v$. Obviously, + +$$U_k^{(m)} = \sum_{v \in S} A_k^v.$$ + +Since every word $w$ of length $k$ satisfying (8) is of the form $w = \tilde{w}0$ or $w = \tilde{w}1$, where $\tilde{w}$ is a word of length $k-1$ satisfying (8), we obtain recurrence relations + +$$A_k^{10} = A_{k-1}^{01} + A_{k-1}^{012} + \cdots + A_{k-1}^{01^{m-1}}, \tag{9}$$ + +$$A_k^{10l} = A_{k-1}^{10l-1}, \quad \text{for } l = 2, 3, \ldots, m-1, \tag{10}$$ + +$$A_k^{01} = A_{k-1}^{10} + A_{k-1}^{102} + \cdots + A_{k-1}^{10^{m-1}}, \tag{11}$$ + +$$A_k^{01l} = A_{k-1}^{01l-1}, \quad \text{for } l = 2, 3, \ldots, m-1. \tag{12}$$ + +Equations (9) and (11) imply that $U_k^{(m)} = A_{k+1}^{10} + A_{k+1}^{01}$. From (10) we obtain $A_k^{10l} = A_{k-l+1}^{10}$ for $l=2,3,\dots,m-1$. Similarly, from (12) we obtain $A_k^{01l} = A_{k-l+1}^{01}$ for $l=2,3,\dots,m-1$. Substituting this into (9) and (11) and taking sum, we obtain + +$$U_k^{(m)} = A_{k+1}^{10} + A_{k+1}^{01} = (A_k^{10} + A_k^{01}) + (A_{k-1}^{10} + A_{k-1}^{01}) + \cdots + (A_{k-m+2}^{10} + A_{k-m+2}^{01}).$$ + +The sequence $(A_{k+1}^{10} + A_{k+1}^{01})_{k \in \mathbb{N}} = (U_k^{(m)})_{k \in \mathbb{N}}$ thus satisfies the same recurrence relation as the $(m-1)$-Bonacci sequence $F_k^{(m-1)}$. It has even the same initial conditions (cf. (1) and (7)). We have thus derived the following statement. + +**Proposition 3.1** For $m \ge 3$ the number of integers $n$ with greedy expansion of length $k$ having unique representation in the $m$-Bonacci numeration system is equal to the $k$-th element of the $(m-1)$-Bonacci system. Formally, + +$$\#\{n \mid F_k^{(m)} \le n < F_{k+1}^{(m)} \text{ and } R^{(m)}(n) = 1\} = F_k^{(m-1)}.$$ + +A general theory for counting the number of words with forbidden strings is developed in [9]. \ No newline at end of file diff --git a/samples/texts/2448265/page_16.md b/samples/texts/2448265/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..2c6ca63a073f779fbbeb6225ffd18834dd5691ec --- /dev/null +++ b/samples/texts/2448265/page_16.md @@ -0,0 +1,40 @@ +# 4 Palindromic structure of $R^{(m)}(n)$ + +Let us recall that transition between different representations of the same integer $n$ is allowed by the interchange $10^m \leftrightarrow 01^m$. Note that the block $10^m$ is the complement of the block $01^m$, in the sense that every 1 is substituted by 0 and every 0 is substituted by 1. Taking complement of the word $\langle n \rangle_m = 1u_{k-1} \cdots u_1$, we obtain the word $(1-u_{k-1})(1-u_{k-2})\cdots(1-u_1)$, which is an $m$-Bonacci representation of an integer, which we denote by $\bar{n}$. It is obvious that + +$$R^{(m)}(n) = R^{(m)}(\bar{n}).$$ + +Since + +$$n + \bar{n} = \sum_{i=1}^{k} F_{i}^{(m)} \qquad (13)$$ + +the center of the symmetry of the function $R^{(m)}(n)$ is in the value $c = \frac{1}{2} \sum_{i=1}^{k} F_i^{(m)}$. Thus the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ contains a palindrome, which ends with the value $R^{(m)}(F_{k+1}^{(m)} - 1)$ and starts with the value $R^{(m)}(\sum_{i=1}^{k} F_i^{(m)} - F_{k+1}^{(m)} + 1)$. Note that the center of the symmetry $c$ satisfies $F_k^{(m)} < c < F_{k+1}^{(m)}$ for $k \ge m + 2$. According to (2), the values $R^{(m)}(1), \dots, R^{(m)}(F_{m+2}^{(m)} - 1)$ are all equal to 1 except $R^{(m)}(F_{m+1}^{(m)}) = 2$, thus only $k \ge m + 2$ is interesting. + +**Remark 4.1** For $m=2$, i.e. for the Fibonacci sequence, we have $\sum_{i=1}^k F_i^{(2)} = F_{k+2}^{(2)} - 2$. Thus the beginning of the palindrome is at $F_k^{(2)} - 1$ and the end at $F_{k+1}^{(2)} - 1$. + +**Remark 4.2** For $m \ge 3$, we have for the starting index of the palindrome + +$$\sum_{i=1}^{k} F_{i}^{(m)} - F_{k+1}^{(m)} + 1 < F_{k-1}^{(m)}.$$ + +Therefore having calculated the values of the function $R^{(m)}(n)$ for $n \le F_k^{(m)} - 1$, most of the values $R^{(m)}(n)$ for $F_k^{(m)} \le n < F_{k+1}^{(m)}$ can be obtained from the palindromic structure. + +Let us determine the smallest number $n_0 \in [F_k^{(m)}, F_{k+1}^{(m)})$, whose complement $\bar{n}_0$ lies in the range $[1, F_k^{(m)})$, where we assume having the knowledge of the values of $R^{(m)}$. Obviously $\bar{n}_0 = F_k^{(m)} - 1$ and from (13) we have $n_0 = \sum_{i=1}^{k-1} F_i^{(m)} + 1$. For $k \ge m+2$ we have $n_0 > F_k$. Note that $n_0$ is the same as the number $n_1$ from Remark 2.2. Thus the values $R^{(m)}(n)$ for $\bar{n}_0+1 = F_k^{(m)} \le n \le n_0-1$ are not equal to 1. The sequence $R^{(m)}(\bar{n}_0+1), R^{(m)}(\bar{n}_0+2), \dots, R^{(m)}(n_0-1)$ is a palindrome which does not contain the number 1. + +**Example 4.3** For the Tribonacci numeration system, i.e. for $m=3$, the values of the function $R_3^{(m)}$ between $F_7^{(3)} = 44$ and $F_8^{(3)} - 1 = 80$ are the following. + +$$ +\begin{array}{ccc} +&R^{(3)}(44) & n_0 = 52 \\ +&\downarrow & \\ +&322222231111122111112111222211121111 \\ +&\uparrow & \\ +\text{center of the palindrome} +\end{array} +\qquad +\begin{array}{ccc} +&R^{(3)}(80) \\ +&\downarrow \\ +& \\ +\downarrow +\end{array} +$$ \ No newline at end of file diff --git a/samples/texts/2448265/page_17.md b/samples/texts/2448265/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..41295cb5c98b037d795eb22ad1d79bba40378d27 --- /dev/null +++ b/samples/texts/2448265/page_17.md @@ -0,0 +1,37 @@ +Note that the value 1 appears in the line 21 times, where $21 = F_7^{(2)}$ as corresponds to Proposition 3.1. The line does not show the entire palindrome; the missing values are $R^{(3)}(15), \dots, R^{(3)}(43)$. + +We end this section with a theorem whose proof for $m = 2$ can be found in [2, 5]. The proof for $m \ge 3$ follows by induction on the length of the greedy expansion of $n$ from Remark 4.2. + +**Theorem 4.4** The segment of the sequence $R^{(m)}(n)$ between two consecutive 1's forms a palindrome, i.e. if $R^{(m)}(p) = R^{(m)}(q) = 1$ and $R^{(m)}(n) > 1$ for all $n, p < n < q$, then the sequence $R^{(m)}(p), R^{(m)}(p+1), \dots, R^{(m)}(q-1), R^{(m)}(q)$ is invariant under mirror image. + +## 5 Maxima of the function $R^{(m)}(n)$ + +The aim of this section is to determine the maximal value of the function $R^{(m)}$ on integers with a fixed length of the greedy expansion. Denote + +$$ \text{Max}(k) := \max\{R^{(m)}(n) \mid F_k^{(m)} \le n < F_{k+1}^{(m)}\}. $$ + +The values $\text{Max}(k)$ for small $k$ can be determined easily. We will use them as the initial step for the proof of the main theorem, which will be done by induction. + +* If $k \le m$, then in the expansion of the length $k$ one cannot perform any interchange $10^m \leftrightarrow 01^m$. Thus + +$$ \text{Max}(k) = 1, \quad \text{for } 1 \le k \le m. $$ + +* For $m < k \le 2m$ one can perform at most one interchange $10^m \leftrightarrow 01^m$ and therefore + +$$ \text{Max}(k) = 2, \quad \text{for } m + 1 \le k \le 2m. $$ + +* For $k = 2m+1$ one can perform in the strings $10^{m-1}10^m$ and $10^{2m}$ two interchanges in a given order. Therefore + +$$ \text{Max}(2m+1) = 3. $$ + +* For $2m+1 < k \le 3m$ one can perform on suitable chosen words two independent interchanges. Therefore + +$$ \text{Max}(k) = 4, \quad \text{for } 2m+2 \le k \le 3m. $$ + +* For $k = 3m+1$ we can see by similar arguments that + +$$ \text{Max}(3m+1) = 5. $$ + +In order to obtain a lower bound on $\text{Max}(k)$, we determine the value $R^{(m)}(n)$ on the integers represented by the following words: for $k = a(m+1), a(m+1)+1, \dots, a(m+1)+m-2$ consider $n$ with the greedy expansion of the form + +$$ \langle n \rangle_m = (10^m)^a, \ 1(10^m)^a, \ 10(10^m)^a, \ \dots, 10^{m-2}(10^m)^a. $$ \ No newline at end of file diff --git a/samples/texts/2448265/page_4.md b/samples/texts/2448265/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..c4353f7aebccaf9c3a15e121d6d5c11ba72c4ae3 --- /dev/null +++ b/samples/texts/2448265/page_4.md @@ -0,0 +1,29 @@ +**Proof:** The string $(10^m)^2 10^{2m-3}$ has the same length as $10^{m-1} 10^{3m-1}$ and the corresponding matrix, $M^2(m)M(2m-3) = \binom{4}{4}$, majores the matrix $M(m-1)M(3m-1) = \binom{3}{4}$ corresponding to the string $10^{m-1} 10^{3m-1}$. $\square$ + +**Claim 5.6** *The string $10^{m-1}10^{m-1}$ is forbidden for maximality.* + +**Proof:** The string $10^{2m-1}$ has the same length as $10^{m-1}10^{m-1}$ and the corresponding matrix $M(2m-1) = \binom{2}{1}$ majores the matrix $M^2(m-1) = \binom{1}{2}$ corresponding to the string $10^{m-1}10^{m-1}$. $\square$ + +**Claim 5.7** *The string $10^{m-1}10^{2m-1}10^{m-1}$ is forbidden for maximality.* + +**Proof:** The string $(10^m)^2 10^{2m-3}$ has the same length and the corresponding matrix $M^2(m)M(2m-3) = \binom{4}{4}$ majores the matrix $M(m-1)M(2m-1)M(m-1) = \binom{3}{5}$ corresponding to the string $10^{m-1}10^{2m-1}10^{m-1}$. $\square$ + +**Claim 5.8** *The string $10^{m-1}10^{2m-1}10^{2m-1}$ is forbidden for maximality for $m \ge 4$.* + +**Proof:** The string $10^{m-1}10^{2m-1}10^{2m-1}$ has the length $5m$ and the corresponding matrix is $M(m-1)M(2m-1)M(m-1) = \binom{5}{8}$. Such matrix is majored by the matrix $M^3(m)M(2m-4) = \binom{8}{8}$ corresponding to the string $(10^m)^3 10^{2m-4}$, which has the same length $5m$. $\square$ + +**Claim 5.9** *The string $10^{m-1}10^{2m-1}10^{3m-1}$ is forbidden for maximality.* + +**Proof:** The string $10^{m-1}10^{2m-1}10^{3m-1}$ with the corresponding matrix is $M(m-1)M(2m-1)M(3m-1) = \binom{7}{11}\binom{5}{8}$. has the length as the string $10^{2m} 10^{2m} 10^{2m-3}$ whose matrix is $M^2(2m)M(2m-3) = \binom{12}{6}\binom{12}{6}$. $\square$ + +**Remark 5.10** In searching the maximal values $\text{Max}(k)$ for $k \ge m+1$ one can restrict the consideration to integers $n$ such that in their greedy expansion $\langle n \rangle_m = 10^{r_s} \cdots 10^{r_1}$ the coefficient $r_1$ satisfies $r_1 = m$ or $r_1 = 2m$. For, $r_1 \le m-1$ implies that $R^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_3} 10^{r_2+r_1+1}) \ge R^{(m)}(n)$, as follows from the matrix formula. Similarly, $r_1 = am+b$ with $a \ge 1$, $b \in \{1, \dots, m-1\}$, implies that $R^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_2+b} 10^{am}) \ge R^{(m)}(n)$. Claim 5.4 moreover implies that $r_1 \in \{m, 2m\}$. + +**Theorem 5.11** Let $m, a$ be integers $m \ge 3$, $a \ge 1$. Then + +$$ +\begin{align*} +\operatorname{Max}(a(m+1)+b) &= 2^a && \text{for } b \in \{0, 1, \dots, m-2\}, \\ +\operatorname{Max}(a(m+1)+b-1) &= 2^a + 2^{a-2} && \text{if } a \ge 2, \\ +\operatorname{Max}(a(m+1)+b) &= 2^a + 2^{a-1}. && +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/2448265/page_5.md b/samples/texts/2448265/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..2eb34384feb0f77d1490d7a205bb5125411a819f --- /dev/null +++ b/samples/texts/2448265/page_5.md @@ -0,0 +1,33 @@ +**Proof:** The proof is done by induction on the length $k = a(m+1)+b$ of the greedy expansion. The veracity of the statement for the initial values has been established at the beginning of this section. We have also proved that the maxima are greater or equal to the mentioned values. It is therefore sufficient to show that these values are also upper bounds on the maxima. + +Assume that $n$ is the argument of the maximum $\text{Max}(k)$. We show that the structure of strings of 0's in the greedy expansion $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ is only of certain form. First suppose that $\text{R}_0^{(m)}(n) = 0$. Then + +$$\text{Max}(k) = \text{R}^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}) = \text{R}^{(m)}(10^{r_{s-1}} \cdots 10^{r_1}) \le \text{Max}(k - r_s - 1),$$ + +and the statement follows from the induction hypothesis. It is therefore sufficient to consider $n$ such that $\text{R}_0^{(m)}(n) \ge 1$. According to Remark 2.2, the greedy expansion $\langle n \rangle_m$ of $n$ is lexicographically smaller than $\langle n_1 \rangle_m$. Together with Claim 5.6 it implies that + +$$\langle n \rangle_m = (10^{m-1})^x 10^y w, \quad \text{where } x \in \{0, 1\}, y \ge m, \tag{16}$$ + +and $w$ is the empty word or the greedy expansion of an integer. + +We show that the coefficients $r_i$ (and in particular the exponent $y$) can take only certain values. If there exists an index $i$ such that $0 \le r_i \le m-2$, then (16) implies $i < s$. Since $M(r_i) = \binom{0}{1} \binom{0}{1}$, we have $M(r_{i+1})M(r_i) = \left(\binom{r_{i+1}}{1} \binom{r_{i+1}}{1}\right)$. This implies + +$$\text{Max}(k) = R^{(m)}(10^{r_s} \cdots 10^{r_1}) \le R^{(m)}(10^{r_s} \cdots 10^{r_{i+1}} 10^{r_{i-1}} \cdots 10^{r_1}) \le \text{Max}(k - r_i - 1),$$ + +and the statement follows from the induction hypothesis. + +Similarly, if there exists $i$ such that $m+1 \le r_i \le 2m-2$, then $M(r_i) = M(m)$, and therefore $\text{Max}(k) \le \text{Max}(k-r_i+m)$, and again the statement follows from the induction hypothesis. Therefore using Claim 5.4 and Remark 5.10 we can restrict our consideration to coefficients $r_s, r_{s-1}, \dots, r_2 \in \{m-1, m, 2m-1, 2m, 3m-1\}$ and $r_1 \in \{m, 2m\}$. + +It follows that $y$ in (16) takes only values $y \in \{m, 2m-1, 2m, 3m-1\}$. We shall now discuss the possibilities according to the values of $x$ and $y$. + +$x=1$: Let us discuss the case $x=1$. The condition $y \ge m$ and Claim 5.5 say that $y \in \{m, 2m, 2m-1\}$. + +• Let $\langle n \rangle_m = 10^{m-1} 10^m w$, where the length of the word $w$ is $k-(2m+1)$. Since + +$$(1\ 1)M(r_s)M(r_{s-1}) = (1\ 1)\binom{1\ 0}{1\ 1}\binom{1\ 1}{1\ 1} = 3(1\ 1),$$ + +we have $\text{Max}(k) = 3\text{R}^{(m)}(w) \le 3\text{Max}(k-2m-1)$. We express $k = a(m+1)+b$, where $b \in \{0, 1, \dots, m\}$. Therefore + +$$\text{Max}(a(m+1)+b) \le 3\text{Max}((a-2)(m+1)+b+1).$$ + +We distinguish: \ No newline at end of file diff --git a/samples/texts/2448265/page_6.md b/samples/texts/2448265/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..22d75159babde8aefc822e95028ef3818f7fe4e7 --- /dev/null +++ b/samples/texts/2448265/page_6.md @@ -0,0 +1,47 @@ +- If $b \in \{0, 1, \dots, m-2\}$, then using the induction hypothesis + +$$ \text{Max}(a(m+1)+b) \leq 3(2^{a-2} + 2^{a-4}) < 2^a, $$ + +as required. + +- If $b=m-1$, then similarly $\text{Max}(a(m+1)+m-1) \leq 3(2^{a-2} + 2^{a-3}) < 2^a + 2^{a-2}$. + +- If $b=m$, then $\text{Max}(a(m+1)+m) \leq 3 \cdot 2^{a-1} = 2^a + 2^{a-1}$. + +* Let $\langle n \rangle_m = 10^{m-1}10^{2m}w$, where the length of the word $w$ is $k - (3m+1)$. Since + +$$ (1 \ 1)M(r_s)M(r_{s-1}) = (1 \ 1)\binom{1 \ 0}{1 \ 1}\binom{2 \ 2}{1 \ 1} = 5(1 \ 1), $$ + +we have $\text{Max}(k) = 5R^{(m)}(w) \leq 5\text{Max}(k - 3m - 1)$. We again express $k = a(m+1)+b$, where $b \in \{0, 1, \dots, m\}$. We have therefore + +$$ \text{Max}(a(m+1)+b) \leq 5\text{Max}((a-3)(m+1)+b+2). $$ + +We distinguish: + +- If $b \in \{0, 1, \dots, m-2\}$, then using the induction hypothesis + +$$ \text{Max}(a(m+1)+b) \leq 5(2^{a-3} + 2^{a-4}) < 2^a, $$ + +as required. + +- If $b=m-1$, then $\text{Max}(a(m+1)+m-1) \leq 5(2^{a-2}) = 2^a + 2^{a-2}$. + +- If $b=m$, then $\text{Max}(a(m+1)+m) \leq 5 \cdot 2^{a-2} < 2^a + 2^{a-1}$. + +* Let now $\langle n \rangle_m = 10^{m-1}10^{2m-1}w$. Claims 5.7, 5.8 and 5.9 imply that the word $w$ is of the form $w = 10^m \tilde{w}$ or $w = 10^{2m} \tilde{w}$. Thus we distinguish: + +- Let $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^m \tilde{w}$. Since $(1\ 1)M(m-1)M(2m-1)M(m) = 8(1\ 1)$ and the length of the word $\tilde{w}$ is $k - 4m - 1$, we have + +$$ \text{Max}(k) \leq 8\text{Max}(k - 4m - 1) \leq 2^3\text{Max}(k - 3(m + 1)), $$ + +which implies the desired result. + +- Let $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^m \tilde{w}$. Since $(1\ 1)M(m-1)M(2m-1)M(2m) = 13(1\ 1)$ and the length of the word $\tilde{w}$ is $k - 5m - 1$, we have + +$$ \text{Max}(k) \leq 13\text{Max}(k - 5m - 1) \leq 2^4\text{Max}(k - 4(m + 1)), $$ + +which implies the desired result. + +Note that Claim 5.8 is valid only for $m \geq 4$. + +$x=0$: Let us study the case $x=0$. We have to consider $y \in \{m, 2m-1, 2m, 3m-1\}$. \ No newline at end of file diff --git a/samples/texts/2448265/page_7.md b/samples/texts/2448265/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..9283ce4de69a578448fe5370f34aa07b7119e061 --- /dev/null +++ b/samples/texts/2448265/page_7.md @@ -0,0 +1,23 @@ +* Let $\langle n \rangle_m = 10^m w$. Since $(1\ 1)M(m) = 2(1\ 1)$, we have $\text{Max}(k) = 2\text{Max}(k - (m+1))$, what was to be proved. + +* Let $\langle n \rangle_m = 10^y w$, where $y \in \{2m-1, 2m, 3m-1\}$. In this case the complement of $n$ has the greedy expansion $\langle \bar{n} \rangle_m = 10^{m-1}\tilde{w}$ or $10^m\tilde{w}$, where $\tilde{w}$ is a greedy expansion of an integer. Such cases were already discussed before. Since $R^{(m)}(n) = R^{(m)}(\bar{n})$, the case is solved. + +This completes the proof for $m \ge 4$. Recall that the assumption $m \ge 4$ was used at one point of the discussion. For $m=3$ we have to consider $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^{2m-1}\tilde{w} = 10^210^510^5\tilde{w}$. The discussion splits into cases according to the prefix of $\tilde{w}$. Necessarily, $\tilde{w} = (10^5)^k 10^{r_i} \cdots 10^{r_1}$ for some $k \ge 0$, $r_i \ne 5$. Since it has been shown that $r_1 \in \{3, 6\}$, we must have $i \ge 1$. + +If $r_i = 3$, then $\langle n \rangle_3$ contains the string $10^510^510^3$. The corresponding matrix is $M(5)M(5)M(3) = {8 \choose 5}$ which is majored by ${8 \choose 8} = M(3)M(3)M(3)$ corresponding to the string $10^310^310^310^3$ of the same length as $10^510^510^3$. Thus $10^510^510^3$ is forbidden for maximality. + +If $r_i = 6$, then $\langle n \rangle_3$ contains the string $10^510^510^6$. The corresponding matrix is $M(5)M(5)M(6) = {13 \choose 8}$ which is majored by ${16 \choose 8} = M(6)M(3)M(3)$ corresponding to the string $10^610^310^310^3$ of the same length as $10^510^510^6$. Thus $10^510^510^6$ is forbidden for maximality. + +If $r_i = 8$, then $\langle n \rangle_3$ contains the string $10^510^8$. The corresponding matrix is $M(5)M(8) = {7 \choose 4}$ which is majored by ${8 \choose 4} = M(6)M(3)M(3)$ corresponding to the string $10^610^310^3$ of the same length as $10^510^8$. Thus $10^510^8$ is forbidden for maximality. + +Last, if $r_i = 2$, then $\langle n \rangle_3$ contains the string $10^2(10^5)^j 10^2$ for some $j \ge 2$. The corresponding matrix is $M(2)M^2(5)M(2) = (F_{2j} F_{2j-1} \ F_{2j+1} F_{2j-2})$ which is majored by $(F_{2j+2} F_{2j+1} \ F_{2j+2} F_{2j-2}) = M^{j+1}(5)$ corresponding to the string $(10^5)^{j+1}$ of the same length as $10^2(10^5)^j 10^2$. Thus $10^2(10^5)^j 10^2$ is forbidden for maximality. $\square$ + +# 6 Comments and open problems + +Although the numeration systems related to *m*-Bonacci numbers have been extensively studied from many different points of view, there remains a number of problems to be explored, even in the most simple Fibonacci case *m* = 2. + +One of these problems is to find a closed formula for the sequence *A(n*) giving the least integer having *n* representations as sums of distinct Fibonacci numbers, which is a sort of inverse to the function *R(2)(n)*. Some results about *A(n*) are given in [3, 4]. However, according to our knowledge, analogous function for *m*-Bonacci numeration system has never been studied. + +Another interesting question related to $R^{(2)}$ is the function rk($n$) defined in [5], counting the number of occurrences of a value $n$ among numbers $R^{(2)}(F_k)$, $R^{(2)}(F_k + 1)$, ..., $R^{(2)}(F_{k+1} - 1)$ for sufficiently large $k$. The authors of [5] show that the function is well defined, give a recurrent formula using the Euler function and exact value for $n$ prime. The function rk($n$) illustrates the exceptionality of the Fibonacci case $m=2$, because similar function cannot be defined if $m \ge 3$. We have already seen that the number of occurrences of the value $R^{(m)}(n) = 1$ among $R^{(m)}(F_k)$, $R^{(m)}(F_k + 1)$, ..., $R^{(m)}(F_{k+1} - 1)$ increases with $k$ to infinity. Similarly, it can be shown for other values $R^{(m)}(n)$. + +The literature often concentrates on the study of ambiguity in generalized Fibonacci numeration systems, where one allows coefficients only in $\{0, 1\}$. When omitting the limitation on \ No newline at end of file diff --git a/samples/texts/2448265/page_8.md b/samples/texts/2448265/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..187fcc421485ce7d98ab5dc84707e58c447b2826 --- /dev/null +++ b/samples/texts/2448265/page_8.md @@ -0,0 +1,34 @@ +the coefficients, the problem becomes much more difficult. Even in case of the usual Fibonacci +system, no compact formula is known for the so-called Fibagonci sequence $(B(n))_{n \in \mathbb{N}}$ counting +the number of representations of $n$ as sum of (possibly repeating) Fibonacci numbers. + +One can also ask the question about numeration systems which allow coefficients ≥ 2 even in the greedy expansion of an integer. An example of these is the Ostrowski numeration system based on sequences defined by linear recurrences of second order with non-constant coefficients. Such numeration systems have been considered by Berstel [1] who shows that a formula similar to (5) is valid for counting the number of representations of $n$. Other properties of these numeration systems are to be explored. + +Acknowledgements + +The authors acknowledge partial support by Czech Science Foundation GA ČR 201/05/0169, and +by the grant LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic. + +References + +[1] J. Berstel, *An excercise on Fibonacci representations*, RAIRO Theor. Inf. Appl. **35** (2001) 491–498. + +[2] M. Bicknell-Johnson, D. C. Fielder, *The number of representations of N using distinct Fibonacci numbers, counted by recursive formulas*, Fibonacci Quart. **37** (1999) 47–60. + +[3] M. Bicknell-Johnson, *The least integer having p Fibonacci representations, p prime*, Fibonacci Quart. **40** (2002) 260–265. + +[4] M. Bicknell-Johnson, *The smallest positive integer having Fk representations as sums of distinct Fibonacci numbers*, in Applications of Fibonacci numbers **8** (1999) 47–52. + +[5] M. Edson, L. Zamboni, *On representations of positive integers in the Fibonacci base*, Theoret. Comput. Sci. **326** (2004), 241–260. + +[6] C. Frougny, *Fibonacci representations and finite automata*, IEEE Trans. Inform. Theory, **37** (1991), 393–399. + +[7] P.J. Grabner, P. Kirschenhofer, R.F. Tichy, *Combinatorial and arithmetical properties of linear numeration systems*, Combinatorica **22** (2002), 245–267. + +[8] P.J. Grabner, P. Liardet, R.F. Tichy, *Odometers and systems of numeration*, Acta Arith. **70** (1995), 103–123. + +[9] L.J. Guibas, A.M. Odlyzko, *String overlaps, pattern matching, and nontransitive games*, J. Combin. Theory Ser. A **30** (1981), 208–208. + +[10] W.H. Kautz, *Fibonacci codes for synchronization control*, IEEE Trans. Inform. Theory, **11** (1965), 284–292. + +[11] P. Kocábová, Z. Masáková, E. Pelantová, *Integers with a maximal number of Fibonacci representations*, RAIRO Theor. Inf. Appl. **39** (2005), 343–359. \ No newline at end of file diff --git a/samples/texts/2448265/page_9.md b/samples/texts/2448265/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/samples/texts/3332461/page_2.md b/samples/texts/3332461/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..376841612338d54b4a2c762bfc36428fd3ed7e1d --- /dev/null +++ b/samples/texts/3332461/page_2.md @@ -0,0 +1,14 @@ +The A 1g to B 1u and A 1g to B 2u transitions are symmetry forbidden and thus have a lower probability which is evident from the lowered intensity of their bands. The singlet A 1g to triplet B 1u transition is both symmetry forbidden and spin forbidden and therefore has the lowest intensity. This transition is forbidden by spin arguments ... + +## Raman scattering - Wikipedia + +Chapter 4 Symmetry and Group Theory 33 ... planes, C v. c. A screw has no symmetry operations other than the identity, for a C1 classification. d. The number 96 (with the correct type font) has a C2 axis perpendicular to the plane of the paper, making it C2h. e. Your choice-the list is too long to attempt to answer it here. + +## Selection rule - Wikipedia + +For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. Lectures by Walter Lewin. They will make you ? Physics. Recommended for you + +## Symmetry And Spectroscopy K V + +Chapter 7 - Symmetry and Spectroscopy - Molecular Vibrations - p. 1 - +7. Symmetry and Spectroscopy - Molecular Vibrations 7.1 Bases for \ No newline at end of file diff --git a/samples/texts/3332461/page_3.md b/samples/texts/3332461/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..40d54e03349593b0cd450490bfa09d581d5cb643 --- /dev/null +++ b/samples/texts/3332461/page_3.md @@ -0,0 +1,15 @@ +molecular vibrations We investigate a molecule consisting of N atoms, +which has 3N degrees of freedom. Taking ... Symmetry of wavefunction +is equal to symmetry of Q k, i.e. + +**Symmetry: IR and Raman Spectroscopy** + +102 CHAPTER4. GROUPTHEORY In group theory, the elements considered are symmetry operations. For a given molecular system described by the Hamiltonian $H^$, there is a set of symmetry operations $O^i$ which commute with $H$: $O^i H^=O$. + +**Symmetry And Spectroscopy Of Molecules - K Veera Reddy ...** + +"The authors use an informal but highly effective writing style to present a uniform and consistent treatment of the subject matter." – Journal of Chemical Education. The primary focus of this text is to introduce students to vibrational and electronic spectroscopy, presenting applications of ... + +**Symmetry and Spectroscopy of Molecules: K. Veera Reddy ...** + +Raman scattering or the Raman effect / ? r ?? m ?n / is the inelastic scattering of photons by matter, meaning that there is an exchange of energy and a change in the light's direction. Typically this involves vibrational energy being gained by a molecule as incident photons from \ No newline at end of file diff --git a/samples/texts/3332461/page_4.md b/samples/texts/3332461/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..08fad325ca1049eb6883384b75aaffe8b7848831 --- /dev/null +++ b/samples/texts/3332461/page_4.md @@ -0,0 +1,15 @@ +a visible laser are shifted to lower energy. + +**K Veera Reddy - AbeBooks** + +where k is the bond force constant and m is the reduced mass for two nuclei of masses m1 and m2. $= 1 + 1$ m m1 m2 1 This yields the quantized vibrational level scheme shown in Figure 5.1 A. Because transitions between the v = 0 and v = 1 levels dominate in infrared or Raman spectroscopy, the harmonic + +**Group Theory in Spectroscopy - Elsevier** + +In vibrational spectroscopy, transitions are observed between different vibrational states. In a fundamental vibration, the molecule is excited from its ground state (v = 0) to the first excited state (v = 1). The symmetry of the ground-state wave function is the same as that of the molecule. + +**Infrared: Theory - Chemistry LibreTexts** + +How symmetric and asymmetric stretching of two identical groups can lead to two distinct signals in IR spectroscopy. Created by Jay. Watch the next lesson: h... + +**Electronic Spectroscopy: Interpretation - Chemistry LibreTexts** \ No newline at end of file diff --git a/samples/texts/3332461/page_8.md b/samples/texts/3332461/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..f3afde99e2b617d6cebfc05624d6944ee5c4943b --- /dev/null +++ b/samples/texts/3332461/page_8.md @@ -0,0 +1,5 @@ +R, the symmetry-defined types (irreducible representations, or irreps, +in the language of group theory) ? i, and, lastly, the order of the +group h (Table 7.1). + +Copyright code : [8722e4f451d30ae594c19503b1e9716d](https://www.citationstyle.org/8722e4f451d30ae594c19503b1e9716d) \ No newline at end of file diff --git a/samples/texts/3392042/page_1.md b/samples/texts/3392042/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..4aef27904b3c1f11952d027ce4376c50b419ffaa --- /dev/null +++ b/samples/texts/3392042/page_1.md @@ -0,0 +1,33 @@ +HADWIGER'S THEOREM FOR DEFINABLE FUNCTIONS + +Y. BARYSHNIKOV, R. GHRIST, AND M. WRIGHT + +**ABSTRACT.** Hadwiger's Theorem states that $\mathbb{E}_n$-invariant convex-continuous valuations of definable sets in $\mathbb{R}^n$ are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable $\mathbb{R}$-valued functions on $\mathbb{R}^n$. This generalizes intrinsic volumes to (dual pairs of) non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems. + +# 1. INTRODUCTION + +Let $\mathbb{R}^n$ denote Euclidean $n$-dimensional space. A *valuation* on a collection $\mathcal{S}$ of subsets of $\mathbb{R}^n$ is an +additive function $\nu : \mathcal{S} \to \mathbb{R}$: + +$$ +(1) \quad \nu(A) + \nu(B) = \nu(A \cap B) + \nu(A \cup B) \quad \text{whenever } A, B, A \cap B, A \cup B \in \mathcal{S}. +$$ + +Valuation v is $\mathbb{E}_n$-invariant if $v(\varphi A) = v(A)$ for all $A \in S$ and $\varphi \in \mathbb{E}_n$, the group of Euclidean (or rigid) motions in $\mathbb{R}^n$. A classical theorem of Hadwiger [16] states that the $\mathbb{E}_n$-invariant continuous valuations on compact convex sets $S$ in $\mathbb{R}^n$ (here a valuation is *continuous* with respect to convergence of sets in the Hausdorff metric) form a finite-dimensional $\mathbb{R}$-vector space generated by intrinsic volumes $\mu_k$, $k = 0, \dots, n$. + +**Theorem 1** (Hadwiger). Any $\mathbb{E}_n$-invariant continuous valuation $v$ on compact convex subsets of $\mathbb{R}^n$ is a linear combination of the intrinsic volumes: + +$$ +(2) \qquad \nu = \sum_{k=0}^{n} c_k \mu_k, +$$ + +for some constants $c_k \in \mathbb{R}$. If $\nu$ is homogeneous of degree $k$, then $\nu = c_k \mu_k$. + +The intrinsic volumes¹ $\mu_k$ are characterized uniquely by (1) $\mathbb{E}_n$ invariance, (2) normalization with respect to a closed unit ball, and (3) homogeneity: $\mu_k(\lambda \cdot A) = \lambda^k(A)$ for all $A \in \mathcal{S}$ and $\lambda \in \mathbb{R}^+$. These measures generalize Euclidean $n$-dimensional volume ($\mu_n$) and Euler characteristic ($\mu_0$). + +This paper extends Hadwiger's Theorem to similar valuations on functions instead of sets. Section 2 gives background on the definable (o-minimal) setting that lifts Hadwiger's Theorem to tame, non-convex sets and then to constructible functions; there, we also review the convex-geometric, integral-geometric, and sheaf-theoretic approaches to Hadwiger's Theorem. In Section 4, we consider definable functions Def($\mathbb{R}^n$) as $\mathbb{R}$-valued functions with tame graphs, and correspondingly define dual pairs of (typically) non-linear "integral" operators $\int \cdot |\mathrm{d}\mu_k|$ and $\int \cdot [\mathrm{d}\mu_k]$ mapping Def($\mathbb{R}^n$) $\to \mathbb{R}$ as generalizations of intrinsic volumes, so that $\int 1_A [\mathrm{d}\mu_k] = \mu_k(A) = \int 1_A [\mathrm{d}\mu_k]$ for + +Key words and phrases. valuations, Hadwiger measure, intrinsic volumes, Euler characteristic. +This work supported by DARPA # HR0011-07-1-0002 and by ONR N000140810668. + +¹Intrinsic volumes are also known in the literature as *Hadwiger measures*, *quermassintegrale*, *Lipschitz-Killing curvatures*, *Minkowski functionals*, and, likely, more. \ No newline at end of file diff --git a/samples/texts/3392042/page_10.md b/samples/texts/3392042/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..1e0101edda696a088db1969f31d38602287d7281 --- /dev/null +++ b/samples/texts/3392042/page_10.md @@ -0,0 +1,21 @@ +FIGURE 1. Conormal cycles of the point $p$, the open interval $(a, b)$, and the closed interval $[c, d]$ illustrate the additivity of the conormal cycle. + +**Continuity:** The flat norm on conormal cycles yields a topology on definable subsets on which the intrinsic volumes are continuous. For definable subsets A and B, define the *flat metric* by + +$$ (13) \qquad d_b(A, B) = |(\mathbf{C}^A - \mathbf{C}^B) \cap B_1^*\mathbb{R}^n|_b, $$ + +thereby inducing the *flat topology*. (That this is a metric follows from $\mathbf{C}^-$ being an injection on definable subsets.) For any $T \in \Omega_n$ and $\omega \in \Omega_c^n$, both supported on $B_1^*\mathbb{R}^n$: + +$$ (14) \qquad |T(\omega)| \le |T|_b \cdot \max \left\{ \sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega| \right\}. $$ + +Since the intrinsic volumes can be represented by integration of bounded forms over the intersection of the conormal cycle with the unit ball bundle, the intrinsic volumes are continuous with respect to the flat topology. We remark also that for the *convex* constructible sets, the flat topology is equivalent to the one given by the Hausdorff metric. + +### 3. INTRINSIC VOLUMES FOR CONSTRUCTIBLE FUNCTIONS + +It is possible to extend the intrinsic volumes beyond definable sets. The *constructible functions*, CF, are functions $h: \mathbb{R}^n \to \mathbb{R}$ with discrete image and definable level sets. By abuse of terminology, CF will always refer to compactly supported definable functions with *finite* image in $\mathbb{R}$. + +As the integral with respect to the Euler characteristics is well defined for constructible functions, one can extend the intrinsic volumes to constructible functions using the slicing definition above: + +$$ (15) \qquad \mu_k(h) = \int_{S_{n,n-k}} \int_{\mathbb{R}^n/L} \left( \int_{L+x} h d\chi \right) dx d\gamma(L). $$ + +In so doing, one obtains, e.g., the following generalization of the Poincaré theorem for Euler characteristic. \ No newline at end of file diff --git a/samples/texts/3392042/page_11.md b/samples/texts/3392042/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..07ae928324201bd3e559b383fad9fd7a75a5f635 --- /dev/null +++ b/samples/texts/3392042/page_11.md @@ -0,0 +1,29 @@ +We need the Verdier duality operator in CF, which is defined e.g. in [25]. Briefly, the dual of $h \in \mathbf{CF}$ is a function $\mathbf{D}h$ whose value at $x_0$ is given by + +$$ (16) \qquad (\mathbf{D}h)(x_0) = \lim_{\epsilon \to 0} \int_{\mathbb{R}^n} \mathbf{1}_{B(x_0, \epsilon)} h \, d\chi, $$ + +where the integral is with respect to Euler characteristic (see also [6]), and $B(x_0, \epsilon)$ is the n-dimensional ball of radius $\epsilon$ centered at $x_0$. In many cases, this duality swaps interiors and closures. For example, if A is a convex open set with closure $\bar{A}$, then $\mathbf{D1}_A = \mathbf{1}_{\bar{A}}$ and $\mathbf{D1}_{\bar{A}} = \mathbf{1}_A$. + +**Proposition 4.** For a constructible function $h$ on $\mathbb{R}^n$, $h \in \mathbf{CF}$, and $\mathbf{D}$ the Verdier duality operator in CF, + +$$ (17) \qquad \int_{\mathbb{R}^n} h \, d\mu_k = (-1)^{n-k} \int_{\mathbb{R}^n} \mathbf{D}h \, d\mu_k. $$ + +*Proof.* The result holds in the case $k=0$ (see [25]). From Equation (15), $\mu_k$ is defined by integration with respect to $d\chi$ along codimension-k planes, followed by the integration over the planes. By Sard's theorem, for (Lebesgue) almost all $L \in \mathcal{S}_{n,n-k}$ and $x \in \mathbb{R}^n/L$, the level sets of $h$ are transversal to $L+x$, whence, by Thom's second isotopy lemma, [28], + +$$ (18) \qquad \int_{L+x} h \, d\chi = (-1)^{n-k} \int_{L+x} \mathbf{D}h \, d\chi $$ + +for almost all L and x. Integration over $\mathcal{P}_{n,n-k}$ finishes the proof. $\square$ + +*Remark 5.* If definable sets $A, \bar{A} \subset \mathbb{R}^n$ satisfy $\mathbf{D1}_A = \mathbf{1}_{\bar{A}}$, then Proposition 4 implies + +$$ (19) \qquad \mu_k(A) = (-1)^{n-k} \mu_k(\bar{A}). $$ + +### 4. INTRINSIC VOLUMES FOR DEFINABLE FUNCTIONS + +The next logical step, lifting from constructible to definable functions, is the focus of this paper. Let $\textbf{Def}(\mathbb{R}^n)$ denote the *definable functions* on $\mathbb{R}^n$, that is, the set of functions $h : \mathbb{R}^n \to \mathbb{R}$ whose graphs are definable sets in $\mathbb{R}^n \times \mathbb{R}$ which coincide with $\mathbb{R}^n \times \{0\}$ outside of a ball (thus compactly supported and bounded). In [7], integration with respect to Euler characteristic $\mu_0$ was lifted to a dual pair of nonlinear “integrals” $\int \cdot |\mathrm{d}\chi|$ and $\int \cdot [\mathrm{d}\chi]$ via the following limiting process, now extended to $\mu_k$: + +**Definition 6.** For $h \in \textbf{Def}(\mathbb{R}^n)$, the *lower* and *upper* Hadwiger integrals of $h$ are, respectively, + +$$ (20) \qquad \begin{aligned} \int h \rfloor d\mu_k &= \lim_{m \to \infty} \frac{1}{m} \int |\mathrm{mh}| \, d\mu_k, \text{ and} \\ \int h \lceil d\mu_k \rceil &= \lim_{m \to \infty} \frac{1}{m} \int [\mathrm{mh}] \, d\mu_k. \end{aligned} $$ + +For $k=n$ these two definitions agree with each other and with the Lebesgue integral; for all $ks]}_-, C^{[g>s]}_-) \, ds. $$ + +The distinct topologies induced by the lower and upper flat metrics are the *lower* and *upper flat topologies* on definable functions. A valuation on definable functions is *lower-* or *upper-continuous* if it is continuous in the lower or upper flat topology, respectively. + +Note that the integrals in (28) and (29) are well-defined because they may be written with finite bounds, as it suffices to integrate between the minimum and maximum values of f and g. These metrics extend the flat metric on definable sets, for they reduce to (13) when f and g are characteristic functions. + +*Remark 9.* Definition 8 does result in *metrics*. If $\underline{d}_b(f,g) = 0$, then $|C^{[f \ge s]} - C^{[g \ge s]}|_b = 0$ only for $s$ in a set of Lebesgue measure zero. However, if the excursion sets of f and g agree almost everywhere, then all excursion sets of f and g agree, and thus $f = g$. For, if $\{s_i\}_i$ is a sequence of negative real numbers converging 0, and $\{f \ge s_i\} = \{g \ge s_i\}$ for all i, then: + +$$ \{f \ge 0\} = \bigcap_i \{f \ge s_i\} = \bigcap_i \{g \ge s_i\} = \{g \ge 0\}. $$ \ No newline at end of file diff --git a/samples/texts/3392042/page_14.md b/samples/texts/3392042/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..334f148220d077030582953af2d0c024d0639526 --- /dev/null +++ b/samples/texts/3392042/page_14.md @@ -0,0 +1,33 @@ +The result for $\bar{d}_b(f, g)$ follows similarly from the observation that $\{f > 0\} = \bigcup_{s>0}\{f > s\}$. + +*Remark 10.* That the lower and upper flat topologies are distinct can be seen by noting that for the identity function $f$ on the interval $[0, 1]$, the sequence of lower step functions $g_m = \frac{1}{m!} \lfloor mf \rfloor$ converges (as $m \to \infty$) to $f$ in the lower flat topology, but not in the upper flat topology. Dually, upper step functions converge in the upper flat topology, but not in the lower. + +**Lemma 11.** The lower and upper Hadwiger integrals are lower- and upper-continuous, respectively. + +*Proof.* Let $f, g \in \text{Def}(\mathbb{R}^n)$ be supported on $X \subset \mathbb{R}^n$. The following inequality for the lower integrals is via (14): + +$$ +\begin{aligned} +\left|\int f \rfloor d\mu_k\rfloor - \int g \rfloor d\mu_k\rfloor\right| &= \left|\int_{-\infty}^{\infty} (\mu_k\{f \ge s\} - \mu_k\{g \ge s\}) ds\right| \\ +&\le \int_{-\infty}^{\infty} \left|\left(C^{f \ge s}\right) - \left(C^{g \ge s}\right)\right| \cap B_1^*\mathbb{R}^n|_b \cdot \max \left\{\sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega|\right\} \\ +&= \underline{d}_b(f, g) \cdot \max \left\{\sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega|\right\} +\end{aligned} +$$ + +Since $\omega_k$ and $d\omega_k$ are bounded, we have continuity of the lower integrals in the lower flat topology. The proof for the upper integrals is analogous. $\square$ + +For *constructible* functions, the lower and upper flat topologies of the previous section are equivalent. Thus, we may refer to the *flat topology* on constructible functions without specifying *upper* or *lower*. A valuation on constructible functions is *conormal continuous* if it is continuous with respect to the flat topology. Conormal continuity is the same as “smooth” in the Alesker sense [3, 4], but distinct from continuity in the topology induced by the Hausdorff metric on definable sets. + +## 6. HADWIGER'S THEOREM FOR FUNCTIONS + +A dual pair of Hadwiger-type classifications for (lower-/upper-) continuous Euclidean-invariant valuations is the goal of this paper. + +**Lemma 12.** If $\nu: \text{CF}(\mathbb{R}^n) \to \mathbb{R}$ is a (conormal) continuous valuation on constructible functions, invariant with respect to the right action by Euclidean motions, then $\nu$ is of the form: + +$$ \nu(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) d\mu_k. $$ + +for some coefficient functions $c_k : \mathbb{R} \to \mathbb{R}$ with $c_k(0) = 0$. + +*Proof.* For the class of indicator functions for convex sets $\{h = r \cdot 1_A : r \in \mathbb{Z}\}$ and $A \subset \mathbb{R}^n$ definable, continuity of $\nu$ in the flat topology implies that $\nu$ is continuous in the Hausdorff topology. Since convex tame sets are dense (in Hausdorff metric) among convex sets in $\mathbb{R}^n$, Hadwiger's Theorem for sets implies that + +$$ (30) \qquad \nu(r \cdot 1_A) = \sum_{k=0}^{n} c_k(r) \mu_k(A), $$ \ No newline at end of file diff --git a/samples/texts/3392042/page_2.md b/samples/texts/3392042/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..c3e2ca76b3f4c3057095e1860ad795fb13659d05 --- /dev/null +++ b/samples/texts/3392042/page_2.md @@ -0,0 +1,39 @@ +where $c_k(r)$ are constants that depend only on $v$, not on $A$. Conormal continuity implies that the valuation $v(A)$ is the integral of the linear combination of the forms $\omega_k$ (defined in (12)), + +$$ (31) \qquad \sum_{k=0}^{n} c_k(r) \alpha_k $$ + +over $\mathbb{C}^n$. + +Now suppose $h = \sum_{i=1}^{m} r_i 1_{A_i}$ is a finite sum of indicator functions of disjoint definable subsets $A_1, \dots, A_m$ of $\mathbb{R}^n$ for some integer constants $r_1 < r_2 < \dots < r_m$. By equation (30) and additivity, + +$$ (32) \qquad v(h) = \sum_{k=0}^{n} \sum_{i=1}^{m} c_k(r_i) \mu_k(A_i). $$ + +We can rewrite equation (32) in terms of excursion sets of $h$. Let $B_i = \cup_{j \ge i} A_j$. That is, $B_i = \{h \ge r_i\}$ and $B_i = \{h > r_{i-1}\}$. Then the valuation $v(h)$ can be expressed as: + +$$ (33) \qquad v(h) = \sum_{k=0}^{n} \sum_{i=1}^{m} (c_k(r_i) - c_k(r_{i-1})) \mu_k(B_i), $$ + +where $c_k(r_0) = 0$. Thus, a valuation of a constructible function can be expressed as a sum of finite differences of valuations of its excursion sets. Equivalently, equation (33) can be written in terms of constructible Hadwiger integrals: + +$$ (34) \qquad v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \, d\mu_k. $$ + +Since we require that a valuation of the zero function is zero, it must be that $c_k(0) = 0$ for all $k$. $\square$ + +Note that Lemma 12 holds for functions of the form $h = \sum_{i=1}^{m} r_i 1_{A_i}$ where the $A_i$ are definable and the $r_i \in \mathbb{R}$ are not necessarily integers. + +In writing an arbitrary valuation on definable functions as a sum of Hadwiger integrals, the situation becomes complicated if the coefficient functions $c_k$ are decreasing on any interval. The following proposition illustrates the difficulty: + +**Proposition 13.** Let $c: \mathbb{R} \to \mathbb{R}$ be a continuous, strictly decreasing function. Then, + +$$ (35) \qquad \lim_{m \to \infty} \int_{\mathbb{R}^n} c\left(\frac{1}{m} \lceil mh \rceil\right) \, d\mu_k = \lim_{m \to \infty} \int_{\mathbb{R}^n} \frac{1}{m} \lfloor mc(h) \rfloor \, d\mu_k. $$ + +*Proof.* On the left side of equation (35), we integrate $c$ composed with upper step functions of $h$: + +$$ \int_{\mathbb{R}^n} c\left(\frac{1}{m}\lceil mh \rceil\right) d\mu_k = \sum_{i \in \mathbb{Z}} c\left(\frac{i}{m}\right) \cdot \mu_k\left\{\frac{i-1}{m} < h \le \frac{i}{m}\right\}. $$ + +On the right side of equation (35), we integrate lower step functions of the composition $c(h)$: + +$$ \int_{\mathbb{R}^n} \frac{1}{m} [\mathrm{mc}(h)] d\mu_k = \sum_{t \in \mathbb{Z}} \frac{t}{m} \cdot \mu_k\left\{\frac{t}{m} \le c(h) < \frac{t+1}{m}\right\}. $$ + +Since $c$ is strictly decreasing, $c^{-1}$ exists. There exists a discrete set + +$$ S = \left\{ c^{-1}\left(\frac{t}{m}\right) \mid t \in \mathbb{Z} \right\} \cap \{\text{neighborhood around range of } h\}. $$ \ No newline at end of file diff --git a/samples/texts/3392042/page_4.md b/samples/texts/3392042/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..7b9e5639d5fd1539de621bb3269c540776a7158d --- /dev/null +++ b/samples/texts/3392042/page_4.md @@ -0,0 +1,59 @@ +We can alternately express equation (37) as + +$$ +(38) \qquad v(h_m) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h_m) \lfloor d\mu_k \rfloor, +$$ + +where we choose lower rather than upper integrals since $v$ is continuous in the lower flat topology. +Continuity of $v$, and convergence of $h_m$ to $h$, in the lower flat topology imply that $v(h_m)$ converges +to $v(h)$ as $h \to \infty$. More specifically, + +$$ +(39) \qquad v(h) = \lim_{m \to \infty} v(h_m) = \sum_{k=0}^{n} \lim_{m \to \infty} \int_{\mathbb{R}^n} c_k(h_m) \lfloor d\mu_k \rfloor. +$$ + +By continuity of the lower Hadwiger integrals (Lemma 11) and the $c_k$, Equation (39) becomes + +$$ +(40) \qquad v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k \left( \lim_{m \to \infty} h_m \right) \lfloor d\mu_k \rfloor = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \lfloor d\mu_k \rfloor. +$$ + +The proof for the upper valuation is analogous. + +**Corollary 15.** Any $\mathbb{E}_n$-invariant valuation both upper- and lower-continuous is a weighted Lebesgue integral. + +*Proof.* Integration with respect to $\lfloor d\mu_k \rfloor$ and $\lceil d\mu_j \rceil$ are independent unless $k = j = n$. For any $v$ +both upper- and lower-continuous, we have + +$$ +v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \lfloor d\mu_k \rfloor = \sum_{k=0}^{n} \int_{\mathbb{R}^n} \bar{c}_k(h) \lceil d\mu_k \rceil +$$ + +for some functions $c_k$ and $\bar{c}_k$. + +Lower and upper Hadwiger integrals with respect to $\mu_k$ are unequal, except when $k=n$, implying +that $\underline{c}_k = \bar{c}_k = 0$ for $k=0, 1, \dots, n-1$, and $\overline{c}_n = \bar{c}_n$. Therefore, + +$$ +v(h) = \int_{\mathbb{R}^n} c(h) \, d\mu_n +$$ + +for some continuous function $c : \mathbb{R} \rightarrow \mathbb{R}$, and with $d\mu_n = [\mathrm{d}\mu_n] = [\mathrm{d}\mu_n]$ denoting Lebesgue measure. $\square$ + +7. SPECULATION + +The present constructions are potentially applicable to generalizations of current applications of +intrinsic volumes. One such recent application is to the dynamics of cellular structures, such as +crystals and foams in microstructure of materials. The cells in such structures often change shape +and size over time in order to minimize the total energy level in the system. Let $C = \bigcup_{i=0}^{n} C_i$ be a +closed $n$-dimensional cell, with $C_i$ denoting the union of all $i$-dimensional features of the cell: *i.e.*, +$C_0$ is the set of vertices, $C_1$ the set of edges, etc. MacPherson and Srolovitz found that when the +cell structure changes by a process of *mean curvature flow*, the volume of the cell changes according +to + +$$ +(41) \qquad \frac{\mathrm{d}\mu_n}{\mathrm{d}t}(C) = -2\pi M\gamma \left( \mu_{n-2}(C_n) - \frac{1}{6}\mu_{n-2}(C_{n-2}) \right) +$$ + +where M and γ are constants determined by the material properties of the cell structure [22]. +Replacing the intrinsic volumes of cells with Hadwiger integrals may (1) lead to interesting dynamical systems on the (singular) foliations (by the level sets of a piece-wise smooth function, and \ No newline at end of file diff --git a/samples/texts/3392042/page_5.md b/samples/texts/3392042/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..4b3a6a8b26ce773e67b28e11a960e0bf502af92d --- /dev/null +++ b/samples/texts/3392042/page_5.md @@ -0,0 +1,63 @@ +(2) allow for description of evolution of real-valued physical fields (temperature, density, etc.) of cells. + +A more widely-known application of the intrinsic volumes is in the formulas for expected Euler characteristic of excursion sets in Gaussian random fields [1, 2]. These formulae and the associated Gaussian kinematic formula [2] rely crucially on the intrinsic volumes of excursion sets. It is already recognized in recent work [8] that the definable Euler measure $[d\chi] = [d\mu_0]$ is relevant to Gaussian random fields: we strongly suspect that the other definable Hadwiger measures $[d\mu_k]$ and $[d\mu_k]$ of this paper are immediately applicable to Gaussian random fields. + +## REFERENCES + +[1] R. Adler, *The Geometry of Random Fields*, Wiley, 1981; reprinted by SIAM, 2009. + +[2] R. Adler and J. Taylor, "Topological Complexity of Random Functions", Springer Lecture Notes in Mathematics, Vol. 2019, Springer, 2011. + +[3] S. Alesker, "Theory of valuations on manifolds: a survey," *Geometric and Functional Analysis*, **17**(4), 2007, 1321–1341. + +[4] S. Alesker, "Valuations on manifolds and integral geometry," *Geometric and Functional Analysis*, **20**(5), 2010, 1073–1143. + +[5] A. Bernig, "Algebraic Integral Geometry," *Global Differential Geometry*, edited by C Bär, J. Lohkamp, and M. Schwarz, Springer, 2012. + +[6] Y. Baryshnikov and R. Ghrist, "Target enumeration via Euler characteristic integration," *SIAM J. Appl. Math.*, **70**(3), 2009, 825–844. + +[7] Y. Baryshnikov and R. Ghrist, "Definable Euler integration," *Proc. Nat. Acad. Sci.*, **107**(21), May 25, 9525-9530, 2010. + +[8] O. Bobrowski and M. Strom Borman, "Euler Integration of Gaussian Random Fields and Persistent Homology," 2011, arXiv:1003.5175. + +[9] J. Cheeger, W. Müller, and R. Schrader, "On the curvature of piecewise flat spaces," *Comm. Math. Phys.* **92**(3), 1984, 405–454. + +[10] M. Coste, An Introduction to o-minimal Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000, http://www.ihp-raag.org/publications.php. + +[11] H. Federer, *Geometric Measure Theory*, Springer 1969. + +[12] J. Fu, "Curvature measures of subanalytic sets", Amer. J. Math., **116**, (1994), 819-890. + +[13] J. Fu, "Notes on Integral Geometry," 2011, http://www.math.uga.edu/~fu/notes.pdf. + +[14] R. Ghrist and M. Robinson, "Euler-Bessel and Euler-Fourier transforms," *Inv. Prob.*, to appear. + +[15] Guesin-Zade, "Integration with respect to the Euler characteristic and its applications," Russ. Math. Surv., **65**:3, 2010, 399–432. + +[16] H. Hadwiger, "Integralsätze im Konvexring," Abh. Math. Sem. Hamburg, **20**, 1956, 136–154. + +[17] D. A. Klain and G.-C. Rota, *Introduction to Geometric Probability*, Cambridge, 1997. + +[18] M. Kashiwara, "Index theorem for constructible sheaves," *Astrisque*, **130**, 1985, 193–209. + +[19] D. A. Klain, "A Short Proof of Hadwiger's Characterization Theorem," Mathematika, **42**, 1995, 329–339. + +[20] M. Kashiwara and P. Schapira, *Sheaves on Manifolds*, Springer, 1990. + +[21] M. Ludwig, "Valuations on function spaces," Adv. Geom., **11**, (2011), 745–756. + +[22] R. D. MacPherson and D. J. Srolovitz, "The von Neumann relation generalized to coarsening of three-dimensional microstructures," Nature, **446**, 2007, 1053–1055. + +[23] L. I. Nicolaescu, "Conormal Cycles of Tame Sets," preprint, 2010, http://www.nd.edu/~lnicolae/conormal.pdf. + +[24] L. I. Nicolaescu, "On the Normal Cycles of Subanalytic Sets," Ann. Glob. Anal. Geom., **39**, 2011, 427–454. + +[25] P. Schapira, "Operations on constructible functions," J. Pure Appl. Algebra, **72**, 1991, 83–93. + +[26] J. Schürmann, *Topology of Singular Spaces and Constructible Sheaves*, Birkhäuser, 2003. + +[27] S. H. Schanuel, "What is the Length of a Potato?" in Lecture Notes in Mathematics, Springer, 1986, 118–126. + +[28] M. Shiota, *Geometry of subanalytic and semialgebraic sets*, Birkhäuser, 1997. + +[29] L. Van den Dries, *Tame Topology and O-Minimal Structures*, Cambridge University Press, 1998. \ No newline at end of file diff --git a/samples/texts/3392042/page_6.md b/samples/texts/3392042/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..27ddb22ae8eb6a2b07fd81ccf69d6c1e3cfd8001 --- /dev/null +++ b/samples/texts/3392042/page_6.md @@ -0,0 +1,11 @@ +DEPARTMENTS OF MATHEMATICS AND ELECTRICAL AND COMPUTING ENGINEERING, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA IL, USA + +*E-mail address:* ymb@uiuc.edu + +DEPARTMENTS OF MATHEMATICS AND ELECTRICAL/SYSTEMS ENGINEERING, UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA PA, USA + +*E-mail address:* ghrist@math.upenn.edu + +DEPARTMENT OF MATHEMATICS, HUNTINGTON UNIVERSITY, HUNTINGTON IN, USA + +*E-mail address:* mwright@huntington.edu \ No newline at end of file diff --git a/samples/texts/3392042/page_7.md b/samples/texts/3392042/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..6ceed6937665683111df10c571f0be56d7c26405 --- /dev/null +++ b/samples/texts/3392042/page_7.md @@ -0,0 +1,27 @@ +all A definable. These integrals are $\mathbb{E}_n$-invariant and satisfy generalized homogeneity and additivity conditions reminiscent of intrinsic volumes; they are furthermore compact-continuous with respect to a dual pair of topologies on functions. This culminates in Section 6 in a generalization of Hadwiger's Theorem for functions: + +**Theorem 2 (Main).** Any $\mathbb{E}_n$-invariant definably lower- (resp. upper-) continuous valuation $v : \text{Def}(\mathbb{R}^n) \to \mathbb{R}$ is of the form: + +$$ (3) \qquad v(h) = \sum_{k=0}^{n} \left( \int_{\mathbb{R}^n} c_k \circ h \rfloor d\mu_k \right) $$ + + resp., integrals with respect to $d\mu_k$) for some $c_k \in C(\mathbb{R})$ continuous and monotone, satisfying $c_k(0) = 0$. + +The $k=0$ intrinsic measure $d\mu_0$ is a recent generalization of $d\chi$ of Euler characteristic [7] shown to have applications to signal processing [14] and Gaussian random fields [8]. The $k=n$ intrinsic measure $d\mu_n$ is Lebesgue volume. The measures come in dual pairs $d\mu_k$ and $d\mu_k$ as a manifestation of (Verdier-Poincaré) duality. Our results yield the following: + +**Corollary 3.** Any $\mathbb{E}_n$-invariant valuation, both upper- and lower-continuous, is a weighted Lebesgue integral. + +We conclude the Introduction remarking that over the past few years several very interesting papers by M. Ludwig, A. Tsang and others appeared, that deal with valuations (often tensor- or set-valued) on various functional spaces (such as $L_p$ and Sobolev spaces): for a recent report, see [21]. Our approach deviates from this circle of results primarily in the choice of the functional space: definable functions form a distinctly different domain for the valuation. The quite fine topologies we impose on the definable functions yield a rich supply of the valuations continuous in these topologies. + +## 2. BACKGROUND + +2.1. **Euler characteristic.** Intrinsic volumes are built upon the Euler characteristic. Among the many possible approaches to this topological invariant — combinatorial [17], cohomological [15], sheaf-theoretic [25, 26], we use the language of o-minimal geometry [29]. An *o-minimal structure* is a sequence $\O = (O_n)_n$ of Boolean algebras of subsets of $\mathbb{R}^n$ which satisfy a few basic axioms (closure under cross products and projections; algebraic basis; and $O_1$ consists of finite unions of points and open intervals). Examples of o-minimal structures include the semialgebraic sets, globally subanalytic sets, and (by slight abuse of terminology) semilinear sets; more exotic structures with exponentials also occur [29]. The details of o-minimal geometry can be ignored in this paper, with the following exceptions: + +(1) Elements of $\O$ are called *tame* or, more properly, *definable* sets. + +(2) A mapping between definable sets is definable if and only if its graph is a definable set. + +(3) The basic equivalence relation on definable sets is definable bijection; these are not necessarily continuous. + +(4) The Triangulation Theorem [29, Thm 8.1.7, p. 122]: any definable set $Y$ is definably equivalent to a finite disjoint union of open simplices $\{\sigma\}$ of different dimensions. + +(5) The Hardt Theorem [29]: for $f: X \to Y$ definable, $Y$ has a triangulation into open simplices $\{\sigma\}$ such that $f^{-1}(\sigma)$ is homeomorphic to $U_\sigma \times \sigma$ for $U_\sigma$ definable, and, on this inverse image, $f$ acts as projection. \ No newline at end of file diff --git a/samples/texts/3392042/page_8.md b/samples/texts/3392042/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..023192f14ec84af59106a24b2cb78b66d97db35f --- /dev/null +++ b/samples/texts/3392042/page_8.md @@ -0,0 +1,45 @@ +For more information, the reader is encouraged to consult [29, 10, 24]. + +The (o-minimal) Euler characteristic is the valuation $\chi$ that evaluates to $(-1)^k$ on an open k-simplex. +It is well-defined and invariant under definable bijection [29, Sec. 4.2], and, among definable sets +of fixed dimension (the dimension of the largest cell in a triangulation), is a complete invariant +of definable sets up to definable bijection. Note that the o-minimal Euler characteristic coincides +with the homological Euler characteristic (alternating sum of ranks of homology groups) on com- +pact definable sets. For locally compact definable sets, it has a cohomological definition (alternat- +ing sum of ranks of compactly-supported sheaf cohomology), yielding invariance with respect to +proper homotopy. + +**2.2. Intrinsic volumes.** Intrinsic volumes have a rich history (see, e.g., [5, 9, 17, 27]) and as many formulations as names, including the following: + +**Slices:** One way to define the intrinsic volume $\mu_k(A)$ of a definable set $A$ is in terms of the Euler characteristic of all slices of $A$ along affine codimension-k planes: + +$$ +(4) \qquad \mu_k(A) = \int_{\mathcal{P}_{n,n-k}} \chi(A \cap P) \, d\lambda(P), +$$ + +where $\lambda$ is the following measure on $\mathcal{P}_{n,n-k}$, the space of affine $(n-k)$-planes in $\mathbb{R}^n$. Each affine subspace $P \in \mathcal{P}_{n,n-k}$ is a translation of some linear subspace $L \in \mathcal{G}_{n,n-k}$, the Grassmannian of $(n-k)$-dimensional subspaces of $\mathbb{R}^n$. That is, $P$ is uniquely determined by $L$ and a vector $\mathbf{x} \in L^\perp$, such that $P = L + \mathbf{x}$. Thus, we can integrate over $\mathcal{P}_{n,n-k}$ by first integrating over $L^\perp$ and then over $\mathcal{G}_{n,n-k}$. Equation (4) is equivalent to + +$$ +(5) \qquad \mu_k(A) = \int_{\mathcal{G}_{n,n-k}} \left( \int_{\mathbb{R}^n/L} \chi(A \cap (L+x)) \, dx \right) d\gamma(L), +$$ + +where $L \in \mathcal{G}_{n,n-k}$, the factorspace $\mathbb{R}^n/L$ is given the natural Lebesgue measure, and $\gamma$ is the Haar (i.e. SO(n)-invariant) measure on the Grassmannian, scaled appropriately. + +**Projections:** Dual to the above definition, one can express $\mu_k$ in terms of projections onto k-dimensional linear subspaces: for any definable $A \subset \mathbb{R}^n$ and $0 \le k \le n$, + +$$ +(6) \qquad \mu_k(A) = \int_{\mathcal{G}_{n,k}} \left( \int_L \chi(\pi_L^{-1}(x)) \right) dx d\gamma(L) +$$ + +where $L \in \mathcal{G}_{n,k}$ and $\pi_L^{-1}(x)$ is the fiber over $x \in L$ of the orthogonal projection map $\pi : A \to L$. For +A convex, Equation (6) reduces to + +$$ +\mu_k(A) = \int_{\mathcal{G}_{n,k}} \mu_k(A|L) \, d\gamma(L) +$$ + +where the integrand is the k-dimensional (Lebesgue) volume of the projection of A onto a k-dimensional subspace L of $\mathbb{R}^n$. + +**2.3. Normal, conormal, and characteristic cycles.** Perspectives from geometric measure theory and sheaf theory are also relevant to the definition of intrinsic volumes. In this section, we restrict to the o-minimal structure of globally subanalytic sets and use analytic tools based on geometric measure theory, following Alesker [3, 4], Fu [12], Nicolaescu [23, 24] and many others. + +Let $\Omega_c^k(\mathbb{R}^n)$ be the space of differential k-forms on $\mathbb{R}^n$ with compact support. Let $\Omega_k(\mathbb{R}^n)$ be the space of k-currents — the topological dual of $\Omega_c^k(\mathbb{R}^n)$. Given any k-current $T \in \Omega_k(\mathbb{R}^n)$, the \ No newline at end of file diff --git a/samples/texts/3392042/page_9.md b/samples/texts/3392042/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..85cbd5df919831c956b3026663ec58bef204bc71 --- /dev/null +++ b/samples/texts/3392042/page_9.md @@ -0,0 +1,33 @@ +boundary of $T$ is $\partial T \in \Omega_{k-1}(\mathbb{R}^n)$ defined as the adjoint to the exterior derivative $\partial$. A cycle is a current with null boundary. + +It is customary to use the *flat topology* on currents [11]. The mass of a k-current $T$ is + +$$ (7) \qquad \mathcal{M}(T) = \sup \left\{ T(\omega) : \omega \in \Omega_c^k(\mathbb{R}^n) \text{ and } \sup_{x \in \mathbb{R}^n} |\omega(x)| \le 1 \right\} $$ + +($|\omega|$ is the usual norm), which generalizes the volume of a submanifold. The *flat norm* of a k-current $T$ is + +$$ (8) \qquad |T|_b = \inf\{\mathcal{M}(R) + \mathcal{M}(S) : T = R + \partial S, R \in \mathcal{R}_k(\mathbb{R}^n), S \in \mathcal{R}_{k+1}(\mathbb{R}^n)\}, $$ + +where $\mathcal{R}_k$ is the space of rectifiable k-currents. The flat norm quantifies the minimal-mass decomposition of a k-current $T$ into a k-current $R$ and the boundary of a $(k+1)$-current $S$. + +**Normal cycle:** The normal cycle of a compact definable set $A$ is a definable $(n-1)$-current $N^A$ on the unit sphere cotangent bundle $U^*\mathbb{R}^n \cong S^{n-1} \times \mathbb{R}^n$ that is Legendrian with respect to the canonical 1-form $\alpha$ on $T^*\mathbb{R}^n$. The normal cycle generalizes the unit normal bundle of an embedded submanifold to compact definable sets. The normal cycle is additive: for $A$ and $B$ compact and definable, + +$$ (9) \qquad N^{A\cup B} + N^{A\cap B} = N^A + N^B. $$ + +The intrinsic volume $\mu_k$ is representable as integration of a particular (non-unique) form $\alpha_k \in \Omega^{n-1}U^*\mathbb{R}^n$ against the normal cycle: + +$$ (10) \qquad \mu_k(A) = \int_{N^A} \alpha_k. $$ + +Fu [12] gives a formula for the normal cycle in terms of stratified Morse theory; Nicolaescu [24] gives a nice description of the normal cycle from Morse theory. + +**Conormal cycle:** The conormal cycle (also known as the characteristic cycle [17, 20, 26]) of a compact definable set $A$ is a Lagrangian $n$-current $C^A$ on $T^*\mathbb{R}^n$ that generalizes the cone of the unit normal bundle. Indeed, the conormal cycle is the cone over the normal cycle. An intrinsic description for $A$ a submanifold-with-corners is that the conormal cycle is the union of duals to tangent cones at points of $A$. The conormal cycle is additive: for $A$ and $B$ definable, + +$$ (11) \qquad C^{A\cup B} + C^{A\cap B} = C^A + C^B. $$ + +The intrinsic volume $\mu_k$ is representable as integration of a certain (non-unique) form $\omega_k \in \Omega^n T^*\mathbb{R}^n$ (supported by a bounded neighborhood of the zero section of the cotangent bundle) against the conormal cycle: + +$$ (12) \qquad \mu_k(A) = \int_{C^A} \omega_k. $$ + +As the conormal cycles are cones, one can always rescale the forms $\omega_k$ so that they are supported in a given neighborhood of the zero section of the cotangent bundle. We fix the neighborhood once and for all, and will assume henceforth that all $\omega_k$ are supported in the unit ball bundle $B_1^*(\mathbb{R}^n) := \{|P| \le 1\} \subset T^*\mathbb{R}^n$. + +The microlocal index theorem [20, 26] gives an interpretation of the conormal cycle in terms of stratified Morse theory. \ No newline at end of file diff --git a/samples/texts/3845339/page_5.md b/samples/texts/3845339/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..af51f733a076beed96bff19b76e3158930b8e117 --- /dev/null +++ b/samples/texts/3845339/page_5.md @@ -0,0 +1,103 @@ +This is to say that if (V,W) is a real solution of matrix equation (18), then $(-Q_n^{-1}VQ_p, -Q_r^{-1}WQ_p)$ is also a real solution of matrix equation (18). Similarly, we can prove that $(-S_n^{-1}VS_p, -S_r^{-1}WS_p)$ is also a real solution of quaternion matrix equation (18). In this case, the conclusion can be obtained along the line of the proof of Theorem 4.2 in [13]. $\square$ + +**Theorem 8.** Given the quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times r}$, let + +$$ +\begin{align} +f_{(I,A_\sigma)}(s) &= \det(I_{4n} - sA_\sigma) = \sum_{k=0}^{2n} a_{2k}s^{2k}, \tag{29} \\ +p_{A_\sigma}(s) &= \sum_{k=0}^{2n} a_{2k}s^k. +\end{align} +$$ + +Then the matrices $X \in Q^{n \times p}$, $Y \in Q^{r \times p}$ are given by + +$$ +\begin{align} +X &= \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k} (A\hat{A})^{s-k} CZ(\hat{B}B)^s \nonumber \\ + &\quad + \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k} (A\hat{A})^{s-k} A\hat{C}\hat{Z}B(\hat{B}B)^s, \tag{30} \\ +Y &= Z p_{A_\sigma}(\hat{B}B), +\end{align} +$$ + +in which Z is an arbitrary quaternion matrix. + +*Proof.* If Yakubovich quaternion *j*-conjugate matrix equation (17) has solution $(X, Y)$, then real representation matrix equation (18) has solution $(V, W) = (X_σ, Y_σ)$ with the free parameter $Z_σ$. By Theorems 2 and 7, we have + +$$ +\begin{align*} +X_{\sigma} &= \sum_{k=0}^{2n-1} \sum_{j=0}^{k} \alpha_j A_{\sigma}^{j-k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^j \\ +&= \sum_{k=0}^{2n-1} \sum_{j=2k}^{4n-1} \alpha_{2k} A_{\sigma}^{j-2k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^j \\ +&= \sum_{k=0}^{2n-1} \alpha_{2k} \left[ \sum_{s=k}^{2n-1} A_{\sigma}^{2s-2k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^{2s} \right. \\ +&\qquad \left. + \sum_{s=k}^{2n-1} A_{\sigma}^{2s-2k+1} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^{2s+1} \right] \\ +&= \sum_{k=0}^{2n-1} \alpha_{2k} \\ +&\quad \times \left[ \sum_{s=k}^{2n-1} \left( (A\hat{A})^{s-k} \right)_{\sigma} P_n C_{\sigma} P_r Z_{\sigma} \left( (\hat{B}B)^s \right)_{\sigma} P_p \right. \\ +&\qquad \left. + \sum_{s=k}^{2n-1} \left( (A\hat{A})^{s-k} \right)_{\sigma} P_n A_{\sigma} C_{\sigma} P_r Z_{\sigma} B_{\sigma} \left( (\hat{B}B)^s \right)_{\sigma} P_p \right] +\end{align*} +$$ + +$$ += \sum_{k=0}^{2n-1} \alpha_{2k} CZ(\hat{B}\hat{B})^s +$$ + +$$ ++ \sum_{s=k}^{2n-1} ((A\hat{A})^{s-k}) A\hat{C}\hat{Z}\hat{B}(B\hat{B})^s +$$ + +(31) + +In addition, by Proposition 5, $f_{(I,A_σ)}(s)$ is a real polynomial and $f_{(I,A_σ)}(B_σ) = (p_{A_σ}(B\hat{B}))_σP_p$. So according to Proposition 3, we obtain + +$$ +Y_σ = Z_σ f_{(I,A_σ)}(B_σ) = Z_σ (p_{A_σ}(B\hat{B}))_σ P_p = (Z_p p_{A_σ}(B\hat{B}))_σ. +$$ + +Thus, the conclusion above has been proved. $\square$ + +In the following, we provide an equivalent statement of Theorem 8. + +**Theorem 9.** Given quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times r}$, let + +$$ +\begin{align} +f_{(I,A_\sigma)}(s) &= \det(I_{4n} - sA_\sigma) = \sum_{k=0}^{2n} a_{2k}s^{2k}, \tag{33} \\ +p_{A_\sigma}(s) &= \sum_{k=0}^{2n} a_{2k}s^k. +\end{align} +$$ + +Then the matrices $X \in Q^{n \times p}$, $Y \in Q^{r \times p}$ given by (30) have the following equivalent form: + +$$ +\begin{align} +X &= Q_c(A\hat{A}, C, 2n) S_r(I, A_\sigma) Q_o(B\hat{B}, Z, 2n) \nonumber \\ + &\quad + Q_c(A\hat{A}, A\hat{C}, 2n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, \hat{Z}\hat{B}, 2n), \tag{34} \\ +Y &= Z p_{A_\sigma}(\hat{B}\hat{B}), +\end{align} +$$ + +in which Z is an arbitrary quaternion matrix. + +*Proof.* By the direct computation, we have + +$$ +\begin{align} +& \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k}(A\hat{A})^{s-k} CZ(\hat{B}\hat{B})^s \\ +&= Q_c(A\hat{A}, C, n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, Z, 2n), +\end{align} +$$ + +$$ ++ \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k}(A\hat{A})^{s-k} A\hat{C}\hat{Z}\hat{B}(B\hat{B})^s +$$ + +$$ += Q_c(A\hat{A}, A\hat{C}, 2n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, \hat{Z}\hat{B}, 2n). +$$ + +Thus, the first conclusion has been proved. With this the second conclusion is obviously true. $\square$ + +Finally, we consider the solution to the so-called Kalman-Yakubovich *j*-conjugate quaternion matrix equation + +$$ X - A\tilde{x}_B = C. +$$ + +Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36). \ No newline at end of file diff --git a/samples/texts/3953766/page_1.md b/samples/texts/3953766/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..c7b9f97c643d0a811fb05c8650575709953dbd91 --- /dev/null +++ b/samples/texts/3953766/page_1.md @@ -0,0 +1,27 @@ +Liquid crystal elastomers wrinkling + +Alain Goriely* + +L. Angela Mihai† + +* Mathematical Institute, University of Oxford, Oxford, UK + +† School of Mathematics, Cardiff University, Cardiff, UK + +May 14, 2021 + +Abstract + +When a liquid crystal elastomer layer is bonded to an elastic layer, it creates a bilayer with interesting properties that can be activated by applying traction at the boundaries or by optothermal stimulation. Here, we examine wrinkling responses in three-dimensional nonlinear systems containing a monodomain liquid crystal elastomer layer and a homogeneous isotropic incompressible hyperelastic layer, such that one layer is thin compared to the other. The wrinkling is caused by a combination of mechanical forces and external stimuli. To illustrate the general theory, which is valid for a range of bilayer systems and deformations, we assume that the nematic director is uniformly aligned parallel to the interface between the two layers, and that biaxial forces act either parallel or perpendicular to the director. We then perform a linear stability analysis and determine the critical wave number and stretch ratio for the onset of wrinkling. In addition, we demonstrate that a plate model for the thin layer is also applicable when this is much stiffer than the substrate. + +**Key words:** liquid crystal elastomers, finite deformation, instabilities, wrinkling, bilayer system, thin film. + +# 1 Introduction + +Liquid crystal elastomers (LCEs) are complex materials that combine the elasticity of polymeric solids with the self-organisation of liquid crystalline structures [31, 39]. Due to their molecular architecture, consisting of cross-linked networks of polymeric chains containing liquid crystal mesogens, these materials are capable of interesting mechanical responses, including relatively large nonlinear deformations which may arise spontaneously and reversibly under certain external stimuli (e.g., heat, solvents, electric or magnetic field). These qualities suggest many avenues for technological applications, such as soft actuators and soft tissue engineering, but more research efforts are needed before they can be exploited on an industrial scale [32, 49, 54, 60, 67, 81, 84, 86, 92–94]. + +In particular, the occurrence of material microstructures and the formation of wrinkles in response to external forces or optothermal stimulation are important phenomena that require further investigation. A survey on surface wrinkling in various materials is provided in [58]. The utility of surface instabilities in measuring material properties of thin films is reviewed in [23]. Within the framework of elasticity, relevant experimental and theoretical studies of wrinkling responses in LCEs are as follows: + +(I) *Wrinkling and shear stripes formation in liquid crystal elastomers:* + +(I.1) *Experimental results.* In [40, 55, 76, 101], stretched LCE samples were reported to display director re-orientation and microstructural shear stripes without wrinkling near the clamped ends, unlike in purely elastic samples. In [1, 2], spontaneous formation and reorientation of surface wrinkles under temperature changes were observed for stiff isotropic elastic plates bonded to a monodomain nematic elastomer foundation. In this case, wrinkles formed either parallel or perpendicular to the nematic director, depending on the temperature at \ No newline at end of file diff --git a/samples/texts/3953766/page_10.md b/samples/texts/3953766/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..fa30ad36a1a549b03c68af249da9bcb732261468 --- /dev/null +++ b/samples/texts/3953766/page_10.md @@ -0,0 +1,31 @@ +case with infinitely deep substrate. Typically, in wrinkling problems where the substrate is about 10 times thicker than the film, depth effects are negligible. Hence, our analysis applies to this situation. + +To illustrate the general theory, we specialised to a biaxial stretch resulting from a combination of elastic and natural deformations. In this case, we showed that, if the liquid crystal elastomer is at high temperature then cooled, it extends along the director and contracts in the perpendicular direction causing wrinkling parallel to the director, and if it is at low temperature then heated, it contracts along the director and extends in the perpendicular direction causing wrinkling perpendicular to the director. At constant temperature, we found that, if the compressive force acts along the director, it can first cause the director to rotate and align perpendicular to the compressive force, then wrinkling parallel to the rotated director can form. We further considered the case when the film deformation is approximated by a plate theory, and demonstrated the validity of this approximation when the film is much stiffer than the substrate. + +From classical wrinkling problems, the linear stability analysis provides an accurate picture of the material behaviour when the film is much stiffer than the substrate, i.e., when the parameter $\beta$ is large, as the instability is supercritical [3]. For smaller values of the stiffness ratio $\beta$, we expect that this instability will be replaced by a subcritical bifurcation and, possibly, a creasing instability. However, a full weakly nonlinear analysis needs to be carried out to study this regime. Such an analysis will be more meaningful if there is a suitable experimental situation of interest. + +# A Stress tensors in nematic elastomers + +For a given hyperelastic material, the Cauchy stress tensor (representing the internal force per unit of deformed area acting within the deformed solid) takes the form + +$$ \mathbf{T} = (\det \mathbf{A})^{-1} \frac{\partial W}{\partial \mathbf{A}} \mathbf{A}^T - p\mathbf{I} = \mu\mathbf{B} - p\mathbf{I}, \quad (\text{A.1}) $$ + +where $p$ denotes the Lagrange multiplier for the internal constraint $\det \mathbf{A} = 1$ [66, pp. 198-201]. The Cauchy stress tensor $\mathbf{T}$ defined by (A.1) is symmetric and coaxial (i.e., it has the same eigenvectors) with the left Cauchy-Green tensor given by (6), and with the Almansi strain tensor given by (7). + +The corresponding first Piola-Kirchhoff stress tensor (representing the internal force per unit of undeformed area acting within the deformed solid) is equal to + +$$ \mathbf{P} = \mathbf{T}\mathrm{Cof}(\mathbf{A}) = \frac{\partial W}{\partial \mathbf{A}} - p\mathbf{A}^{-T} = \mu\mathbf{A} - p\mathbf{A}^{-T}, \quad (\text{A.2}) $$ + +where $\mathrm{Cof}(\mathbf{A}) = (\det \mathbf{A}) \mathbf{A}^{-T}$ is the cofactor of $\mathbf{A}$. This stress tensor is not symmetric in general. + +The associated second Piola-Kirchhoff stress tensor is + +$$ \mathbf{S} = \mathbf{A}^{-1}\mathbf{P} = \mathbf{A}^{-1}\mathbf{T}\mathrm{Cof}(\mathbf{A}) = 2\frac{\partial W}{\partial \mathbf{C}} - p\mathbf{C}^{-1} = \mu\mathbf{I} - p\mathbf{C}^{-1}, \quad (\text{A.3}) $$ + +and this is symmetric and coaxial with the right Cauchy-Green tensor defined by (6) and with the Green-Lagrange strain tensor defined by (8). + +Next, we derive the stress tensors of a deformed LCE, with the strain-energy function described by (9), in terms of the stresses in the base polymeric network when the nematic director is ‘free’ to rotate relative to the elastic matrix (see also [63]). In this case, **F** and **n** are independent variables, and the Cauchy stress tensor for the LCE is calculated as follows, + +$$ \begin{align*} \mathbf{T}^{(nc)} &= J^{-1} \frac{\partial W^{(nc)}}{\partial \mathbf{F}} \mathbf{F}^T - p^{(nc)} \mathbf{I} \\ &= J^{-1} \mathbf{G}^{-T} \frac{\partial W}{\partial \mathbf{A}} \mathbf{A}^T \mathbf{G}^T - p^{(nc)} \mathbf{I} \\ &= J^{-1} \mathbf{G}^{-1} \mathbf{T} \mathbf{G}, \end{align*} \quad (\text{A.4}) $$ + +where $\mathbf{T}$ is the elastic Cauchy stress defined by (A.1), $J = \det \mathbf{F}$, and the scalar $p^{(nc)}$ (the hydrostatic pressure) represents the Lagrange multiplier for the internal constraint $J = 1$. \ No newline at end of file diff --git a/samples/texts/3953766/page_11.md b/samples/texts/3953766/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..92d8774241767fe4a239f383f261f1887dee739f --- /dev/null +++ b/samples/texts/3953766/page_11.md @@ -0,0 +1,41 @@ +Because the Cauchy stress tensor $\mathbf{T}^{(nc)}$ given by (A.4) is not symmetric in general [5, 72, 98], the following additional condition must hold [5, 98], + +$$ \frac{\partial W^{(nc)}}{\partial \mathbf{n}} = \mathbf{0}. \qquad (A.5) $$ + +By the principle of material objectivity stating that constitutive equations must be invariant under changes of frame of reference, this is equivalent to (see [5] for details), + +$$ \frac{1}{2} (\mathbf{T}^{(nc)} - \mathbf{T}^{(nc)T}) \mathbf{n} = \mathbf{0}, \qquad (A.6) $$ + +where $(\mathbf{T}^{(nc)} - \mathbf{T}^{(nc)T})/2$ represents the skew-symmetric part of the Cauchy stress tensor. + +The corresponding first Piola-Kirchhoff stress tensor for the LCE is equal to + +$$ \mathbf{P}^{(nc)} = \mathbf{T}^{(nc)} \mathrm{Cof}(\mathbf{F}) = \mathbf{G}^{-1} \mathbf{T} \mathbf{A}^{-T} = \mathbf{G}^{-1} \mathbf{P}, \qquad (A.7) $$ + +where **P** is the elastic first Piola-Kirchhoff stress given by (A.2). + +The associated second Piola-Kirchhoff stress tensor is + +$$ \mathbf{S}^{(nc)} = \mathbf{F}^{-1}\mathbf{P}^{(nc)} = \mathbf{A}^{-1}\mathbf{G}^{-2}\mathbf{P} = \mathbf{A}^{-1}\mathbf{G}^{-2}\mathbf{A}\mathbf{S}, \qquad (A.8) $$ + +where **S** is the elastic second Piola-Kirchhoff stress tensor given by (A.3). + +The strain-energy density (5) takes the equivalent form + +$$ W(\alpha_1, \alpha_2, \alpha_3) = \frac{\mu}{2} (\alpha_1^2 + \alpha_2^2 + \alpha_3^2 - 3), \qquad (A.9) $$ + +where $\{\alpha_i^2\}_{i=1,2,3}$ are the eigenvalues of the Cauchy-Green tensor given by (6). Then, for LCEs, where the nematic director can rotate freely relative to the elastic matrix, the strain-energy function given by (9) takes the equivalent form + +$$ W^{(nc)}(\lambda_1, \lambda_2, \lambda_3, \mathbf{n}) = \frac{\mu}{2} \left\{ a^{1/3} \left[ \lambda_1^2 + \lambda_2^2 + \lambda_3^2 - (1-a^{-1}) \mathbf{n} \cdot \left( \sum_{i=1}^3 \lambda_i^2 \mathbf{e}_i \otimes \mathbf{e}_i \right) \mathbf{n} \right] - 3 \right\}, \quad (A.10) $$ + +where $\{\lambda_i^2\}_{i=1,2,3}$ and $\{\mathbf{e}_i\}_{i=1,2,3}$ denote the eigenvalues and eigenvectors, respectively, of the tensor + +$$ \mathbf{F}\mathbf{F}^T = \sum_{i=1}^{3} \lambda_i^2 \mathbf{e}_i \otimes \mathbf{e}_i. \qquad (A.11) $$ + +## B The 3D equilibrium equations + +In this appendix, we show that the elastic energy of an ideal LCE is equal to the elastic energy of its polymeric network (see also [63]). We consider a solid LCE body characterised by the strain-energy function defined by (9), and occupying a compact domain $\bar{\Omega} \subset \mathbb{R}^3$, such that the interior of the body is an open, bounded, connected set $\Omega \subset \mathbb{R}^3$, and its boundary $\partial\Omega = \bar{\Omega} \setminus \Omega$ is Lipschitz continuous (in particular, we assume that a unit normal vector **N** exists almost everywhere on $\partial\Omega$). The elastic energy stored by the body is equal to [66, p. 205] + +$$ E = \int_{\Omega} [W^{(nc)}(\mathbf{F}, \mathbf{n}) - p^{(nc)}(\det \mathbf{F} - \det \mathbf{G})] dV, \qquad (B.1) $$ + +where $p^{(nc)}(\det \mathbf{F} - \det \mathbf{G})$ enforces the condition that $\det \mathbf{F} = \det \mathbf{G}$, with $\det \mathbf{G}$ representing the local change of volume due to the 'spontaneous' deformation. \ No newline at end of file diff --git a/samples/texts/3953766/page_12.md b/samples/texts/3953766/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..f4898c54a2fd46fb2356d4d9a46c7369e63fa311 --- /dev/null +++ b/samples/texts/3953766/page_12.md @@ -0,0 +1,13 @@ +which the bilayer was prepared. It was further suggested that controlled surface wrinkling patterns could be employed to characterise the mechanical properties of thin polymer films deposited on the LCE. Surface wrinkles in thin liquid crystal polymer films on a glass substrate were reported in [50]. In [85], snake-mimicking soft actuators composed of a bilayered liquid crystal elastomer ribbon were described. + +**(I.2) Models based on the linear elasticity theory.** In [22, 69], the formation of shear stripes microstructure and fine-scale wrinkles in stretched LCEs were studied using a Koiter-type theory. Namely, for physically relevant parameters, it was shown that the microstructure was finer than wrinkles, and that these instabilities occurred for distinct mesoscale stretches. For LCE plates with or without a compliant substrate, in [53], such instabilities were considered within the theoretical framework of Föppl-von Kármán. In [72], small-amplitude wrinkles of a compressed isotropic elastic film on a soft nematic elastomer substrate were treated. The formation and evolution of wrinkling in a solid liquid crystalline film attached to a compliant elastic substrate was explored numerically in [44, 95–97, 100]. Finite element simulations of shear striping were presented in [99]. + +## (II) Wrinkling of isotropic elastic materials: + +**(II.1) Models based on the linear elasticity theory.** Wrinkling of a stretched isotropic elastic membrane, with zero bending stiffness, was examined in [68, 73, 74]. A stretched Koiter-type plate was considered in [77]. A stretched neo-Hookean sheet was investigated in [65]. In [6–8, 16, 24, 47], a Föppl-von Kármán plate bonded to a compliant infinitely deep linearly elastic foundation under equi-biaxial compression was analysed. Wrinkling of twisted ribbons was addressed in [51], inspired by [9]. Large-amplitude wrinkling of a thin linearly elastic film attached to a pre-stretched substrate modelled by an Ogden strain-energy function was treated in [75]. Multi-layer systems were studied in [56]. + +**(II.2) Models based on finite strain elasticity, with either compression or pre-stretch causing wrinkling.** The first linear stability analysis of a nonlinear elastic half-space under compression in a direction parallel to its surface was developed in [13]. In [17], the first nonlinear analysis of a bilayer system formed from a stiff nonlinear hyperelastic film bonded to a nonlinear hypelastic substrate and subjected to uniaxial compression parallel to the interface was provided. A two-layer system comprising a neo-Hookean film bonded to an infinitely deep neo-Hookean substrate, with the film/substrate stiffness ratio ranging from the limit in which the film and substrate have the same elastic modulus to a very stiff film on a compliant substrate, subjected to a combination of compression and pre-stretched, was examined in [21]. The influence of the relative stiffness in a two-layer system of neo-Hookean materials was explored in [48]. In [43], a uniaxially compressed neo-Hookean thin film bonded to a neo-Hookean substrate of comparable stiffness was studied. In [18, 42], the exact nonlinear elasticity theory combined with an asymptotic perturbation procedure was employed to derive the critical stretch value at which period-doubling secondary bifurcation under uniaxial compression occurred. + +## (III) Wrinkling in growing biological tissues: + +**(III.1) Models based on plane strain finite elasticity, where the physical system is reduced to a two-dimensional problem, with either compression or growth causing wrinkling.** A semi-analytical approach for the study of the nonlinear buckling behaviour of a growing soft layer was developed in [27]. In [15], a model for a growing thin neo-Hookean layer bonded to an infinitely deep neo-Hookean substrate was proposed, where the stiffness ratio between the layer and substrate was relatively low while the thin layer was stiffer. A system formed from a neo-Hookean layer attached to a stiffer or softer neo-Hookean substrate undergoing differential growth was treated in [10]. Bilayer film-substrate systems of neo-Hookean materials with different stiffness ratios, subjected to combined compression and growth conditions were studied systematically in [46]. In [57], multi-layer systems were considered. A \ No newline at end of file diff --git a/samples/texts/3953766/page_17.md b/samples/texts/3953766/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..6b21f2c99121a157fc370da968f6ff011858e7dc --- /dev/null +++ b/samples/texts/3953766/page_17.md @@ -0,0 +1,33 @@ +[28] Cirak F, Long Q, Bhattacharya K, Warner M. 2014. Computational analysis of liquid crystalline elastomer membranes: Changing Gaussian curvature without stretch energy, International Journal of Solids and Structures 51(1), 144-153 (doi: 10.1016/j.ijsolstr.2013.09.019). + +[29] Clarke SM, Hotta A, Tajbakhsh AR, Terentjev EM. 2001. Effect of crosslinker geometry on equilibrium thermal and mechanical properties of nematic elastomers, Physical Reviews E 64, 061702 (doi: 10.1103/PhysRevE.64.061702). + +[30] Conti S, DeSimone A, Dolzmann G. 2002. Soft elastic response of stretched sheets of nematic elastomers: a numerical study, Journal of the Mechanics and Physics of Solids 50, 1431-1451. + +[31] de Gennes PG. 1975. Physique moléculaire - réflexions sur un type de polymères nématiques, Comptes rendus de l'Académie des Sciences B 281, 101-103. + +[32] de Haan LT, Schenning AP, Broer DJ. 2014. Programmed morphing of liquid crystal networks, Polymer 55(23), 5885-5896 (doi: 10.1016/j.polymer.2014.08.023). + +[33] DeSimone A. 1999. Energetics of fine domain structures, Ferroelectrics 222(1), 275-284 (doi: 10.1080/00150199908014827). + +[34] DeSimone A, Dolzmann G. 2000. Material instabilities in nematic elastomers, Physica D 136(1-2), 175-191 (doi: S0167-2789(99)00153-0). + +[35] DeSimone A, Dolzmann G. 2002. Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Archive of Rational Mechanics and Analysis 161, 181-204 (doi: 10.1007/s002050100174). + +[36] DeSimone A, Teresi L. 2009. Elastic energies for nematic elastomers, The European Physical Journal E 29, 191-204 (doi: 10.1140/epje/i2009-10467-9). + +[37] Diaz-Calleja R, Riande E. 2012. Biaxially stretched nematic liquid crystalline elastomers, The European Physical Journal E 35, 2-12. + +[38] Finkelmann H, Greve A, Warner M. 2001. The elastic anisotropy of nematic elastomers, The European Physical Journal E 5, 281-293 (doi: 10.1007/s101890170060). + +[39] Finkelmann H, Kock HJ, Rehage G. 1981. Investigations on liquid crystalline polysiloxanes 3, Liquid crystalline elastomers - a new type of liquid crystalline material, Die Makromolekulare Chemie, Rapid Communications 2, 317-322 (doi: 10.1002/marc.1981.030020413). + +[40] Finkelmann H, Kundler I, Terentjev EM, Warner M. 1997. Critical stripe-domain instability of nematic elastomers, Journal de Physique II 7, 1059-1069 (doi: 10.1051/jp2:1997171). + +[41] Fried E, Sellers S. 2006. Soft elasticity is not necessary for striping in nematic elastomers, Journal of Applied Physics 100, 043521. + +[42] Fu YB, Cai ZX. 2015. An asymptotic analysis of the period-doubling secondary bifurcation in a film/substrate bilayer, SIAM Journal on Applied Mathematics 75(6), 2381-2395 (doi: 10.1137/15M1027103). + +[43] Fu YB, Ciarletta P. 2015. Buckling of a coated elastic half-space when the coating and substrate have similar material properties, Proceedings of the Royal Society A 471, 20140979 (doi: 10.1098/rspa.2014.0979). + +[44] Fu C, Xu F, Huo Y. 2018. Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates, International Journal of Solids and Structures 132-133, 264277 (doi: 10.1016/j.ijsolstr.2017.10.018). \ No newline at end of file diff --git a/samples/texts/3953766/page_3.md b/samples/texts/3953766/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..728aa18260fc18c50718fe88e57501070e61c57a --- /dev/null +++ b/samples/texts/3953766/page_3.md @@ -0,0 +1,33 @@ +The biaxial deformation described by (51) was previously addressed in [62, 64] (see also [30, 35, 36]) where shear-striping pattern formation and the accompanying “soft elasticity” phenomenon [82] were analysed in detail. We will rely those results for our subsequent analysis. + +For a systematic treatment of the effect of biaxial forces acting either parallel or perpendicular to the nematic director, while the director remains parallel to the contact surface between the film and the substrate throughout deformation, we further require that + +$$ \lambda_2 > \max\{\lambda_1, \lambda_3\}, \quad (53) $$ + +which, together with (51), implies + +$$ 0 < \lambda < \min\{a^{1/12}, a^{1/3}\}. \quad (54) $$ + +This condition prevents the director from rotating out-of-plane, and also assumes that the largest principal stretch is in the second direction. The former is a reasonable physical assumption, while the latter is imposed without restricting the generality of the problem if we consider the reference director to be uniformly aligned either parallel or perpendicular to the second direction. The orthogonal situation, with $\lambda_1 > \lambda_2$, then leads to analogous results when the reference director aligns either parallel or perpendicular to the first direction. + +We are interested in the solutions $\lambda = \lambda_{cr}$ of equation + +$$ \det \mathbf{M} = 0 \quad (55) $$ + +that satisfy (54). + +The associated critical second Piola-Kirchhoff stress tensors (see Appendix B) then take the form + +$$ \mathbf{S}_{\text{cr}} = \mu_k \operatorname{diag} \left( 1 - \frac{1}{a^{1/3} \lambda_{\text{cr}}^2}, 1 - \frac{\lambda_{\text{cr}}^2}{a^{2/3}}, 0 \right) \quad \text{for the elastic layer,} \quad (56) $$ + +$$ \mathbf{S}_{\text{cr}}^{(\text{nc})} = \mu_k \operatorname{diag} \left( 1 - \frac{g_1^2}{\lambda_{\text{cr}}^2}, 1 - \frac{g_2^2 \lambda_{\text{cr}}^2}{a^{1/3}}, 0 \right) \quad \text{for the LCE layer.} \quad (57) $$ + +The general formulae for the above stress tensors are given by (A.3) and (A.8), respectively. + +## 5.1 Hyperelastic film on LCE substrate + +When the bilayer system is formed by a hyperelastic film and a liquid crystal elastomer substrate, the entries of the 6 × 6 matrix M in (50) are given in Table 2 where we take $h = 1$ without loss of generality. We are interested in solving equation (55). Fixing $a$, for each finite value of $β$, the solutions of $\det M = 0$, viewed as an equation in $\lambda$, provide solution curves $\lambda = \lambda(q)$. In compression, the critical stretch ratio is the maximal value of $\lambda$ of this function. We refer to this extremal value and the value $q$ where it is achieved as the pair $(\lambda_{cr}, q_{cr}) = (\lambda_{cr}(\beta), q_{cr}(\beta))$. We have made the dependence on $\beta$ explicit to indicate that we are primarily interested in the change of these critical values with the stiffness ratio $\beta$. Equivalently, the pair $(\lambda_{cr}(\beta), q_{cr}(\beta))$ is obtained as a solution of the system + +$$ \mathcal{E}(\lambda, q; \beta, a) \equiv \{\det \mathbf{M} = 0, \partial_q (\det \mathbf{M}) = 0\}. \quad (58) $$ + +To find these values asymptotically, we first consider the case of very large $\beta$. In the limit $\zeta = 1/\beta \to 0$, the critical wave number is $q_{cr} = 0$ and $\lambda_{cr} = \lambda_0$ is obtained as a solution of $\det M = 0$. Then, we substitute $\lambda_{cr} = \lambda_0 + \lambda_1\zeta^{\nu_1}$, $q_{cr} = q_1\zeta^{\nu_2}$ and expand $\mathcal{E}$ in powers of $\zeta$ (assuming $\nu_1$ and $\nu_2$ positive) to find the correct balance of exponents (for our problem, we found the scaling $\nu_1 = -2/3$ and $\nu_2 = -1/3$). Once these are obtained, a regular asymptotic expansion in powers of $\zeta$ is substituted in $\mathcal{E}$, and solved to arbitrary order. However, this solution is only valid locally near $\zeta = 0$. To obtain the entire curve numerically for a given value of $a$, we fix $\beta$ and solve $\mathcal{E}$ for $\lambda$ and $q$ simultaneously, by Newton's method, to derive their critical values. Explicitly, we start with a large value of $\beta \approx 100$ for which the asymptotic solution provides an initial guess that is very close to the numerical root. Once the numerical solution with this value of $\beta$ is known, we use it to solve for nearby values and, by regular continuation method, obtain the entire curve. \ No newline at end of file diff --git a/samples/texts/3953766/page_4.md b/samples/texts/3953766/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..498917ec5391f9d87a9700248c277ea292c077cb --- /dev/null +++ b/samples/texts/3953766/page_4.md @@ -0,0 +1,13 @@ +Table 2: The coefficients {$M_{ij}$}$_{i,j=1,...,6}$ of the homogeneous system of linear equations (50), for a hyperelastic film on a liquid crystal elastomer substrate, with {$\lambda_i$}$_{i=1,2,3}$ given by (51) and {$\alpha_i$}$_{i=1,2,3}$ given by (52). + +
Mijj = 1j = 2j = 3j = 4j = 5j = 6
i = 11111-1-1
i = 21a1/6λ-1-a1/6λ-1-a1/6λ/g12g2
i = 3(2μf/a1/3)μf(λ2 + 1/a1/3)(2μf/a1/3)μf(λ2 + 1/a1/3)(-2μsg12g22/a1/3)-μs(λ12 + g12g22/a1/3)
i = 4-μf(λ2 + 1/a1/3)(-2μfλ/a1/6)μf(λ2 + 1/a1/3)(2μfλ/a1/6)μs(λ22 + g22g12/a1/3)(2μsλg2/a1/6)
i = 5(2/a1/3eq)(λ2 + 1/a1/3)eqλλa1/6(2/a1/3e-q)(λ2 + 1/a1/3)e-qλλa1/600
i = 6-(λ2 + 1/a1/3)eq-2eqλλa1/6(λ2 + 1/a1/3)e-q(2e-qλλa1/6)0(2e-qλλa1/6) / a1/6
+ +**Case 1.** First, we assume that the director is initially aligned in the second direction, such that $g_1 = a^{-1/6}$ and $g_2 = a^{1/3}$. For all $a > 0$, in the asymptotic limit of large stiffness ratio $\beta = \mu_f/\mu_s$, we obtain the following critical stretch ratio $\lambda_{cr}$ and critical wave number $q_{cr}$, respectively: + +$$ \lambda_{cr} = a^{-1/6} - \frac{a^{1/18}}{2} \left(\frac{3}{\beta}\right)^{2/3} + O\left(\beta^{-4/3}\right), \quad (59) $$ + +$$ q_{cr} = a^{1/9} \left( \frac{3}{\beta} \right)^{1/3} + O\left( \beta^{-1} \right). \quad (60) $$ + +Figure 4: Case 1 (elastic film on LCE substrate, with director aligned in the second direction): The critical stretch ratio $0 < \lambda_{cr} < \min\{a^{1/12}, a^{1/3}\}$ and critical wave number $q_{cr}$ 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 $0 < q < \infty$, we can solve equation (55) numerically (for $q \to \infty$, see Appendix C). Figure 4 illustrates the critical stretch ratios $\lambda_{cr}$ and critical wave numbers $q_{cr}$ as functions of the relative stiffness ratio $\beta$: \ No newline at end of file diff --git a/samples/texts/3953766/page_5.md b/samples/texts/3953766/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..ab7a33c85951469bca4ee05bf3a318799e8472dd --- /dev/null +++ b/samples/texts/3953766/page_5.md @@ -0,0 +1,27 @@ +* When $a < 1$, wrinkling parallel to the director is generated at the critical stretch ratio $0 < \lambda_{cr} < a^{1/3}$; + +* When $a > 1$, wrinkles parallel to the director is produced at the critical stretch ratio $0 < \lambda_{cr} < a^{1/12}$. + +For example, when $a > 1$ and the LCE is at high temperature then cooled, it extends along the director and contracts in the perpendicular direction, causing wrinkling parallel to the director (see Figure 1). A similar behaviour produced experimentally is discussed in [1,2]. + +At constant temperature, microstructural shear striping patterns can also occur in the prolate LCE substrate if the director rotates [30, 34, 36, 37, 41, 62, 64, 71, 83, 98]. To verify whether shear striping is likely in our system, we assume that the LCE substrate is in its natural state before the film is attached, i.e., $\lambda = a^{-1/6}$ in the reference configuration. Starting from the reference state with $g_1 = a^{-1/6}$ and $g_2 = a^{1/3}$, analysis similar to that carried out in [62, 64] shows that this state is unstable and the director rotates, while shear stripes develop, as $\lambda$ increases within the interval $(a^{-1/6}, a^{1/12})$. Then, decreasing the value of $\lambda < a^{-1/6}$ will not produce rotation of the director in the substrate, i.e., no striping pattern will form. + +**Case 2.** Second, we assume that the director is initially aligned in the first direction, such that $g_1 = a^{1/3}$ and $g_2 = a^{-1/6}$. For large stiffness ratio $\beta = \mu_f/\mu_s$, the asymptotic results read: + +$$ \lambda_{cr} = a^{-1/6} - \frac{a^{-5/6}}{2} \left( \frac{a^{5/6} + a^{1/3}}{2} \right)^{2/3} \left( \frac{3}{\beta} \right)^{2/3} + O(\beta^{-4/3}), \quad (61) $$ + +$$ q_{cr} = a^{-2/9} \left( \frac{a^{1/2} + 1}{2} \right)^{1/3} \left( \frac{3}{\beta} \right)^{1/3} + O(\beta^{-1}). \quad (62) $$ + +For finite values of $0 < q < \infty$ (see Appendix C for $q \to \infty$), Figure 5 presents the critical stretch ratios $\lambda_{cr}$ and critical wave numbers $q_{cr}$ as functions of the stiffness ratio $\beta$: + +* When $a < 1$, wrinkling is produced at the critical stretch ratio $0 < \lambda_{cr} < a^{1/3}$. + +For example, when $a < 1$ and the LCE is at low temperature then heated, it contracts along the director and extends in the perpendicular direction, causing wrinkling perpendicular to the director (see Figure 2). A similar behaviour obtained experimentally is discussed in [1,2]. + +* When $a > 1$, wrinkling perpendicular to the director is induced at the critical stretch ratio $0 < \lambda_{cr} < a^{1/12}$. + +At constant temperature, we assume that the LCE substrate is in its natural state before the film is attached, i.e., $\lambda = a^{1/3}$ in the reference configuration. This state is unstable and the director rotates, while shear stripes form, as $\lambda$ decreases within the interval $(a^{1/12}, a^{1/3})$ [62]. For $\lambda < a^{1/12}$, (54) is satisfied and the problem reduces to that of Case 1. Visual inspection of the critical stretch ratios in Figures 4 (top-left) and 5 (top-left), and comparison of the asymptotic values given by (59) and (61) further suggest that the critical stretch ratios $\lambda_{cr}$ in Case 1 are smaller than those in Case 2. We interpret this as “delay” in wrinkling due to director rotation. + +## 5.2 LCE film on hyperelastic substrate + +For a bilayer system consisting of a liquid crystal elastomer film on a hyperelastic substrate, the entries of the 6 × 6 matrix M in (50) are given in Table 3, again with $h = 1$. \ No newline at end of file diff --git a/samples/texts/3953766/page_8.md b/samples/texts/3953766/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..ff37da0b8778d0cf14c24ba506bd031f3531162e --- /dev/null +++ b/samples/texts/3953766/page_8.md @@ -0,0 +1,15 @@ +Figure 7: Case 4 (LCE film, with director aligned in the first direction, on elastic substrate): The critical stretch ratio $0 < \lambda_{cr} < \min\{a^{1/12}, a^{1/3}\}$ and critical wave number $q_{cr}$ as functions of relative stiffness ratio $\beta = \mu_f/\mu_s$ when $g_1 = a^{1/3}$ and $g_2 = a^{-1/6}$. 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 7 shows the critical stretch ratios $\lambda_{cr}$ and critical wave numbers $q_{cr}$ as functions of the stiffness ratio $\beta$: + +* When $a < 1$, wrinkles perpendicular to the director appear at the critical stretch ratio $0 < \lambda_{cr} < a^{1/3}$; + +* When $a > 1$, wrinkles perpendicular to the director form at the critical stretch ratio $0 < \lambda_{cr} < a^{1/12}$. + +At constant temperature, the discussion is similar to that of Case 2. Then, visual inspection of the critical stretch ratios in Figures 6 (top-left) and 7 (top-left), and comparison of the asymptotic values given by (63) and (65) suggest that the critical stretch ratios $\lambda_{cr}$ in Case 3 are smaller than in Case 4, which can be interpreted as wrinkling being “delayed” while the director rotates. + +## 6 Approximate results based on plate models + +In [63], we derived a Föppl-von Kármán-type constitutive model for a liquid crystalline solid which is sufficiently thin so that it can be approximated by a plate equation. In our derivation, we relied on the following assumptions: (i) surface normals to the plane of the plate remain perpendicular to the plate after deformation; (ii) changes in the thickness of the plate during deformation are negligible; (iii) the stress field in the deformed plate is parallel to the mid-surface. For a wrinkling film, the plate equation takes the general form [24] + +$$ \frac{\mu_f}{3} h^3 \xi^{iv} - Sh\xi'' = f_s, \quad (67) $$ \ No newline at end of file diff --git a/samples/texts/4083386/page_1.md b/samples/texts/4083386/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..0a87034aa0b05b43316c1a22d37a2ab3876ade2d --- /dev/null +++ b/samples/texts/4083386/page_1.md @@ -0,0 +1,36 @@ +# Buried Targets +## Case 6b +### A wire at the ground interface vicinity + +Philippe. Pouliguen¹, Yannick Béniguel² + +1 DGA / DS / MRIS, Paris +2 IEEA, Courbevoie + +**1. A wire at the ground interface vicinity** + +The calculation to be performed is a near field calculation. The diffracting structure is a wire which will be either located on the ground or slightly buried. + +- The wire length is 100 m. Its radius is 0.75 mm. + +- The antenna phase center is located 2 meters above the ground level. + +- The antenna pattern is Gaussian; its aperture (3 dB) half angles are 40° in the horizontal plane and 5° in the vertical plane. Its amplitude is one V/m at its maximum. + +- The angle $β$ between the horizontal axis and the antenna Line of Sight (the pointing direction) is 10°. + +- Ground : Flat with electrical characteristics ($epsilon = 15$ ; $sigma = 0.01$ s / m) + +- Polarizations : VV and HH ($\theta$ and $\phi$) + +**2.1 Case 6b.1 Moving antenna perpendicular to the wire** + +Figure 2 moving antenna / RCS of a wire on the ground + +- The calculation will be performed at 500MHz + +- The distance is varying from 30m to 1m with a step equal to 1m. + +a) The wire is laid on the ground surface. The distance between the wire center at any point along it, to the ground is consequently equal to its radius i.e. 0.75 mm + +b) The wire is slightly buried. The distance between the wire center at any point along it, to the ground surface is 1.5 cm. \ No newline at end of file diff --git a/samples/texts/4943237/page_1.md b/samples/texts/4943237/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..94e0ff1afc2af1c507847972ffb9af16c0e6fa1c --- /dev/null +++ b/samples/texts/4943237/page_1.md @@ -0,0 +1,19 @@ +PSPACE Automata for Description Logics + +Jan Hladík* + +Rafael Peñaloza† + +Abstract + +Tree automata are often used for satisfiability testing in the area of description logics, which usually yields EXPTIME complexity results. We examine conditions under which this result can be improved, and we define two classes of automata, called segmentable and weakly-segmentable, for which emptiness can be decided using space logarithmic in the size of the automaton (and thus polynomial in the size of the input). The usefulness of segmentable automata is demonstrated by reproving the known PSPACE result for satisfiability of $\mathcal{ALC}$ concepts with respect to acyclic TBoxes. + +# 1 Introduction + +Tableau- and automata-based algorithms are two mechanisms which are widely used for testing satisfiability in description logics (DLs). Aside from considerations regarding implementation (most of the efficient implementations are tableau-based) and elegance (tableaus for expressive logics require a blocking condition to ensure termination), there is also a difference between the complexity results which can be obtained “naturally”, i.e. without using techniques with the sole purpose of remaining in a specific complexity class. With tableaus, the natural complexity class is usually NEXPTIME, e.g. for $\mathcal{SHIQ}$ [HST00, BS01] or $\mathcal{SHIQ}(D)$ [Lut04], although this result is not optimal in the former case. With automata, one usually obtains an EXPTIME result, e.g. for $\mathcal{ALC}$ with general TBoxes [Sch94]. + +Previously, we examined which properties of a NEXPTIME tableau algorithm make sure that the logic is decidable in EXPTIME, and we defined a class of tableau algorithms for which an EXPTIME automata algorithm can be automatically derived [BHLW03]. For EXPTIME automata, a frequently used + +*Automata Theory, TU Dresden (jan.hladik@tu-dresden.de) + +†Intelligent Systems, Uni Leipzig (rpenalozan@yahoo.com) \ No newline at end of file diff --git a/samples/texts/4962236/page_11.md b/samples/texts/4962236/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..0c6177594793b800b358bb48b299a4d4b34c251a --- /dev/null +++ b/samples/texts/4962236/page_11.md @@ -0,0 +1,48 @@ +**DEFINITION 2.2 ([1, 12]).** If the matrix $A(t)$ is T-periodic, then the linear system + +$$y' = A(t)y$$ + +is said to be *noncritical* with respect to $T$ if it has no periodic solution of period $T$ except the trivial solution $y=0$. + +Throughout this paper it is assumed that system (1.4) is noncritical. Let $\Phi(t)$ be the fundamental solution matrix of (1.4) with $\Phi(0) = I$, where $I$ is the $n \times n$ identity matrix. Then we have the following results [1, 12]. + +(i) $\det \Phi(t) \neq 0$; + +(ii) there exists a constant matrix $B$ such that $\Phi(t+T) = \Phi(t)e^{BT}$, by Floquet theory; + +(iii) system (1.4) is noncritical if and only if $\det(I - \Phi(T)) \neq 0$. + +**LEMMA 2.2.** Let $\Phi(t)$ be the fundamental solution matrix of (1.4) with $\Phi(0) = I$. Then + +$$ (2.1) \qquad \Phi(t)[(\Phi(t) - \Phi(t+T))^{-1} - (\Phi(t-T) - \Phi(t))^{-1}] = I. $$ + +*Proof.* We have + +$$ +\begin{aligned} +\Phi(t)[(\Phi(t) - \Phi(t+T))^{-1} - (\Phi(t-T) - \Phi(t))^{-1}] \\ +&= [(\Phi(t) - \Phi(t+T))\Phi^{-1}(t)]^{-1} - [(\Phi(t-T) - \Phi(t))\Phi^{-1}(t)]^{-1} \\ +&= (I - \Phi(t+T)\Phi^{-1}(t))^{-1} - (\Phi(t-T)\Phi^{-1}(t) - I)^{-1} \\ +&= (I - \Phi(t)e^{BT}\Phi^{-1}(t))^{-1} - (\Phi(t)e^{-BT}\Phi^{-1}(t) - I)^{-1} \\ +&= \Phi(t)e^{-BT}\Phi^{-1}(t)(\Phi(t)e^{BT}\Phi^{-1}(t) - I)^{-1} - (\Phi(t)e^{BT}\Phi^{-1}(t) - I)^{-1} \\ +&= (\Phi(t)e^{-BT}\Phi^{-1}(t) - I)(\Phi(t)e^{BT}\Phi^{-1}(t) - I)^{-1} = I. \blacksquare +\end{aligned} +$$ + +In order to obtain the existence of periodic solutions of system (1.3), we make the following preparations. + +Define + +$$ \mathrm{PC}(\mathbb{R}, \mathbb{R}^n) = \{ y : \mathbb{R} \to \mathbb{R}^n \mid y_i|_{(t_j, t_{j+1})} \in C(t_j, t_{j+1}), \exists y(t_j^-) = y(t_j), y(t_j^+), \\ j \in \mathbb{Z}, i = 1, \dots, n \} $$ + +and set + +$$ X = \{ y \in PC(\mathbb{R}, \mathbb{R}^n) : y(t+T) = y(t) \text{ for all } t \} $$ + +with the norm defined by $\|y\| = \sup_{t \in \mathbb{R}} |y(t)|_0 = \sup_{t \in [0,T]} |y(t)|_0$, where $|y(t)|_0 = \sum_{i=1}^n |y_i(t)|$. Then $X$ is a Banach space. + +The following lemma is fundamental to our discussion: + +**LEMMA 2.3.** A function $y$ is a $T$-periodic solution of (1.3) if and only if $y$ is a $T$-periodic solution of the integral equation + +$$ (2.2) \qquad y(t) = \int_t^{t+T} G(t,r)g(r,y_r) dr + \sum_{j: t_j \in [t,t+T)} G(t,t_j)I_j(y(t_j)), $$ \ No newline at end of file diff --git a/samples/texts/4962236/page_12.md b/samples/texts/4962236/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..ab2af1d5619f85fe2e4530506c10557ff9247dae --- /dev/null +++ b/samples/texts/4962236/page_12.md @@ -0,0 +1,56 @@ +where + +$$ +(2.3) \quad G(t, u) = \Phi(t)(\Phi(u-T) - \Phi(u))^{-1} := (G_{ik})_{n \times n}, \quad u \in [t, t+T). +$$ + +Proof. If y is a T-periodic solution of (1.3), then for any $t \in \mathbb{R}$, there exists $j \in \mathbb{Z}$ such that $t_j$ is the first impulsive point after $t$. Let $\Phi(t)$ be a fundamental solution of system (1.4). Since $\Phi(t)\Phi^{-1}(t) = I$, it follows that + +$$ +\begin{align*} +0 &= \frac{d}{dt}(\Phi(t)\Phi^{-1}(t)) = \frac{d}{dt}(\Phi(t))\Phi^{-1}(t) + \Phi(t)\frac{d}{dt}(\Phi^{-1}(t)) \\ + &= (A(t)\Phi(t))\Phi^{-1}(t) + \Phi(t)\frac{d}{dt}(\Phi^{-1}(t)) = A(t) + \Phi(t)\frac{d}{dt}(\Phi^{-1}(t)). +\end{align*} +$$ + +This implies that + +$$ +(2.4) \qquad \frac{d}{dt}(\Phi^{-1}(t)) = -\Phi^{-1}(t)A(t). +$$ + +By (2.4), we have + +$$ +\begin{equation} +\begin{aligned} +(2.5) \quad \frac{d}{dt}(\Phi^{-1}(t)y(t)) &= \frac{d}{dt}(\Phi^{-1}(t))y(t) + \Phi^{-1}(t) \frac{d}{dt}(y(t)) \\ +&= -\Phi^{-1}(t)A(t)y(t) + \Phi^{-1}(t)[A(t)y(t) + g(t, y_t)] \\ +&= \Phi^{-1}(t)g(t, y_t). +\end{aligned} +\end{equation} +$$ + +An integration of (2.5) from $t$ to $s$ for $s \in [t, t_j]$, $j \in \mathbb{Z}$, yields + +$$ +y(s) = \Phi(s) \int_{t}^{s} [\Phi^{-1}(r)g(r, y_r)] dr + \Phi(s)\Phi^{-1}(t)y(t), +$$ + +so + +$$ +y(t_j) = \Phi(t_j) \int_t^{t_j} [\Phi^{-1}(r)g(r, y_r)] dr + \Phi(t_j)\Phi^{-1}(t)y(t), \quad j \in \mathbb{Z}. +$$ + +Again, integrating (2.5) over $(t_j, t_{j+1}]$, $j \in \mathbb{Z}$, we get + +$$ +\begin{align*} +y(s) &= \Phi(s)\Phi^{-1}(t_j)y(t_j^+) + \int_{t_j}^{s} \Phi(s)\Phi^{-1}(r)g(r, y_r) dr \\ +&= \Phi(s)\Phi^{-1}(t_j)[y(t_j^-) + I_j(y(t_j))] + \int_{t_j}^{s} \Phi(s)\Phi^{-1}(r)g(r, y_r) dr \\ +&= \Phi(s)\Phi^{-1}(t)y(t) + \Phi(s) \left[ \int_{t}^{t_j} \Phi^{-1}(r)g(r, y_r) dr + \int_{t_j}^{s} \Phi^{-1}(r)g(r, y_r) dr \right] \\ +&\quad + \Phi(s)\Phi^{-1}(t_j)I_j(y(t_j)) \\ +&= \Phi(s)\Phi^{-1}(t)y(t) + \Phi(s) \int_{t}^{s} \Phi^{-1}(r)g(r, y_r) dr + \Phi(s)\Phi^{-1}(t_j)I_j(y(t_j)). +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/4962236/page_13.md b/samples/texts/4962236/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..66006a043699d6baa253b87d913371ed2ba6682d --- /dev/null +++ b/samples/texts/4962236/page_13.md @@ -0,0 +1,41 @@ +Repeating the above process for $s \in [t, t+T]$, we obtain + +$$ +\begin{aligned} +y(s) = \Phi(s)\Phi^{-1}(t)y(t) &+ \Phi(s) \int_{t}^{s} \Phi^{-1}(r)g(r, y_r) dr \\ +& + \sum_{j: t_j \in [t,s)} \Phi(s)\Phi^{-1}(t_j)I_j(y(t_j)). +\end{aligned} + $$ + +Let $s = t + T$ in the above equality. Then + +$$ +\begin{aligned} +y(t+T) = \Phi(t+T)\Phi^{-1}(t)y(t) &+ \Phi(t+T) \int_{t}^{t+T} \Phi^{-1}(r)g(r, y_r) dr \\ +& + \sum_{j: t_j \in [t,t+T)} \Phi(t+T)\Phi^{-1}(t_j)I_j(y(t_j)). +\end{aligned} + $$ + +It follows from $y(t+T) = y(t)$ that + +$$ +\begin{aligned} +& (I - \Phi(t+T)\Phi^{-1}(t))y(t) \\ +& \quad = \Phi(t+T) \int_{t}^{t+T} \Phi^{-1}(r)g(r, y_r) dr + \Phi(t+T) \sum_{j: t_j \in [t,t+T)} \Phi^{-1}(t_j)I_j(y(t_j)). +\end{aligned} + $$ + +So we get + +$$ +\begin{aligned} +y(t) &= (I - \Phi(t+T)\Phi^{-1}(t))^{-1}\Phi(t+T) \left[ \int_{t}^{t+T} \Phi^{-1}(r)g(r, y_r) dr \right. \\ +&\qquad \left. + \sum_{j: t_j \in [t,t+T)} \Phi^{-1}(t_j)I_j(y(t_j)) \right] \\ +&= (\Phi^{-1}(t+T) - \Phi^{-1}(t))^{-1} \left[ \int_{t}^{t+T} \Phi^{-1}(r)g(r, y_r) dr \right. \\ +&\qquad \left. + \sum_{j: t_j \in [t,t+T)} \Phi^{-1}(t_j)I_j(y(t_j)) \right] \\ +&= \Phi(t)(e^{-BT} - I)^{-1} \left[ \int_{t}^{t+T} \Phi^{-1}(r)g(r, y(r), y(r-\tau(r))) dr \right. \\ +&\qquad \left. + \sum_{j: t_j \in [t,t+T)} \Phi^{-1}(t_j)I_j(y(t_j)) \right] \\ +&= \int_{t}^{t+T} \Phi(t)(e^{-BT} - I)^{-1}\Phi^{-1}(r)g(r, y_r) dr \\ +&\quad + \sum_{j: t_j \in [t,t+T)} \Phi(t)(e^{-BT} - I)^{-1}\Phi^{-1}(t_j)I_j(y(t_j)) +\end{aligned} + $$ \ No newline at end of file diff --git a/samples/texts/4962236/page_14.md b/samples/texts/4962236/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..75a69cdeafd5b3f73be903466dd1cbd018709276 --- /dev/null +++ b/samples/texts/4962236/page_14.md @@ -0,0 +1,33 @@ +$$ = \int_{t}^{t+T} \Phi(t)(\Phi(r-T) - \Phi(r))^{-1}g(r, y_r) dr + \sum_{j: t_j \in [t, t+T)} \Phi(t)(\Phi(t_j - T) - \Phi(t_j))^{-1}I_j(y(t_j)). $$ + +Consequently, let $y$ be a $T$-periodic solution of (2.2). If $t \neq t_i$, $i \in \mathbb{Z}$, then by (2.2), we have + +$$ y(t) = \Phi(t) \left[ \int_{t}^{t+T} (\Phi(r-T) - \Phi(r))^{-1} g(r, y_r) dr + \sum_{j: t_j \in [t, t+T)} (\Phi(t_j - T) - \Phi(t_j))^{-1} I_j(y(t_j)) \right]. $$ + +Therefore, by (2.1), we obtain + +$$ +\begin{aligned} +y'(t) &= \Phi'(t) \left[ \int_{t}^{t+T} (\Phi(r-T) - \Phi(r))^{-1} g(r, y_r) dr \right. \\ +&\quad + \sum_{j: t_j \in [t, t+T)} (\Phi(t_j - T) - \Phi(t_j))^{-1} I_j(y(t_j)) \\ +&\quad \left. + \Phi(t) [( \Phi(t) - \Phi(t+T) )^{-1} g(t+T, y_{t+T}) \right. \\ +&\quad \left. - ( \Phi(t-T) - \Phi(t) )^{-1} g(t, y_t) \right] \\ +&= A(t)\Phi(t) \left[ \int_{t}^{t+T} (\Phi(r-T) - \Phi(r))^{-1} g(r, y_r) dr \right. \\ +&\quad + \sum_{j: t_j \in [t, t+T)} (\Phi(t_j - T) - \Phi(t_j))^{-1} I_j(y(t_j)) \\ +&\quad + \Phi(t) [(\Phi(t) - \Phi(t+T))^{-1} - (\Phi(t-T) - \Phi(t))^{-1}] g(t, y_t) \\ +&= A(t)y(t) + g(t, y_t). +\end{aligned} +$$ + +If $t = t_i$, $i \in \mathbb{Z}$, then by (2.1) and (2.2), we get + +$$ +\begin{aligned} +y(t_i^+) - y(t_i^-) &= \sum_{j: t_j \in [t_i^+, t_i^+ + T)} \Phi(t_i)(\Phi(t_j - T) - \Phi(t_j))^{-1} I_j(y(t_j)) \\ +&\quad - \sum_{j: t_j \in [t_i^-, t_i^- + T)} \Phi(t_i)(\Phi(t_j - T) - \Phi(t_j))^{-1} I_j(y(t_j)) \\ +&= \Phi(t_i)[(\Phi(t_i) - \Phi(t_i + T))^{-1} - (\Phi(t_i - T) - \Phi(t_i))^{-1}] I_j(y(t_j)) \\ +&= I_j(y(t_j)). +\end{aligned} +$$ + +So $y$ is also a $T$-periodic solution of (1.3). $\blacksquare$ \ No newline at end of file diff --git a/samples/texts/4962236/page_6.md b/samples/texts/4962236/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..e89f0dac95d784524a9834d597fa18aac7b1c83c --- /dev/null +++ b/samples/texts/4962236/page_6.md @@ -0,0 +1,45 @@ +which implies that + +$$ +\begin{align*} +\alpha(Hy) &= \inf_{t \in [0,T]} |(Hy)(t)|_0 \ge A \left( \int_0^T |g(s, y_s)|_0 ds + \sum_{j=1}^p |I_j(y(t_j))|_0 \right) \\ +&> A \frac{b}{B} = \delta b = a, +\end{align*} +$$ + +which means that condition (iii) of Lemma 2.1 holds. + +Therefore, by Lemma 2.1, the operator *H* has at least three fixed points +$x_1, x_2, x_3 \in \bar{K}_c$ such that + +$x_1 \in K_d, \quad x_2 \in \{x \in K(\alpha, a, c), \alpha(x) > a\}, \quad x_3 \in \bar{K}_c \setminus \alpha(K(\alpha, a, c) \cup \bar{K}_d).$ + +The proof of Theorem 3.1 is complete. ■ + +**REMARK 3.1.** From the proof of Theorem 3.1, one can easily see that if condition (i) is replaced by + +(i') if $g^0 = I^0 = 0$, and $g^\infty = I^\infty = 0$, + +then the conclusion of Theorem 3.1 remains valid. + +**Acknowledgements.** This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183 and the Natural Sciences Foundation of Yunnan Province. + +References + +[1] M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl. 331 (2007), 1175–1186. + +[2] D. Jiang, J. Wei and B. Zhang, *Positive periodic solutions of functional differential equations and population models*, Electron. J. Differential Equations 2002, no. 71, 13 pp. + +[3] R. W. Leggett and L. R. Williams, *Multiple positive fixed points of nonlinear operators on ordered Banach spaces*, Indiana Univ. Math. J. 28 (1979), 673–688. + +[4] X. Li, X. Lin, D. Jiang and X. Zhang, *Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects*, Nonlinear Anal. 62 (2005), 683–701. + +[5] X. Li, X. Zhang and D. Jiang, *A new existence theory for positive periodic solutions to functional differential equations with impulse effects*, Comput. Math. Appl. 51 (2006), 1761–1772. + +[6] Y. K. Li, *Positive periodic solutions of nonlinear differential systems with impulses*, Nonlinear Anal. 68 (2008), 2389–2405. + +[7] Y. K. Li and Z. W. Xing, *Existence and global exponential stability of periodic solution of CNNs with impulses*, Chaos Solitons Fractals 33 (2007), 1686–1693. + +[8] Y. K. Li and L. Zhu, *Positive periodic solutions of nonlinear functional differential equations*, Appl. Math. Comput. 156 (2004), 329–339. + +[9] P. De Nápoli and M. C. Mariani, *Three solutions for quasilinear equations in $\mathbb{R}^n$ near resonance*, Electron. J. Differential Equations Conf. 6 (2001), 131–140. \ No newline at end of file diff --git a/samples/texts/4962236/page_7.md b/samples/texts/4962236/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..18f3160cdefb067fb6549ce74753b62958fbfb0a --- /dev/null +++ b/samples/texts/4962236/page_7.md @@ -0,0 +1,23 @@ +[10] J. J. Nieto, *Basic theory for nonresonance impulsive periodic problems of first order*, J. Math. Anal. Appl. 205 (1997), 423-433. + +[11] H. Sun and Y. Wang, *Existence of positive solutions for $p$-Laplacian three-point boundary-value problems on time scales*, Electron. J. Differential Equations 2008, no. 92, 14 pp. + +[12] W. Walter, *Ordinary Differential Equations*, Springer, New York, 1991. + +[13] N. Zhang, B. Dai and X. Qian, *Periodic solutions for a class of higher-dimension functional differential equations with impulses*, Nonlinear Anal. 68 (2008), 629-638. + +[14] X. Zhang, J. Yan and A. Zhao, *Existence of positive periodic solutions for an impulsive differential equation*, ibid. 68 (2008), 3209-3216. + +[15] J. W. Zhou and Y. K. Li, *Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects*, ibid. 71 (2009), 2856-2865. + +Yongkun Li (corresponding author), Changzhao Li, Juan Zhang + +Department of Mathematics +Yunnan University +Kunming, Yunnan 650091, People's Republic of China +E-mail: yklie@ynu.edu.cn + +Received 6.4.2009 +and in final form 2.9.2009 + +(2001) \ No newline at end of file diff --git a/samples/texts/4962236/page_8.md b/samples/texts/4962236/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/samples/texts/4962236/page_9.md b/samples/texts/4962236/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..65acdd6a47063a251e7d9648d9a1784742dfd955 --- /dev/null +++ b/samples/texts/4962236/page_9.md @@ -0,0 +1,40 @@ +of functional differential equations + +$$ +(1.2) \qquad \dot{x}(t) = A(t)x(t) + f(t, x_t), +$$ + +where $A(t) = \operatorname{diag}[a_1(t), \dots, a_n(t)]$, $a_j \in C(\mathbb{R}, \mathbb{R})$ is $\omega$-periodic, $f$ is a function defined on $\mathbb{R} \times BC(\mathbb{R}, \mathbb{R}^n)$, and $f(t, x_t)$ is $\omega$-periodic whenever $x$ is $\omega$-periodic, where $BC(\mathbb{R}, \mathbb{R}^n)$ denotes the Banach space of bounded continuous functions $\phi: \mathbb{R} \to \mathbb{R}^n$ with the norm $\|\phi\| = \sup_{\theta \in \mathbb{R}} \sum_{j=1}^n |\phi_j(\theta)|$ where $\phi = (\phi_1, \dots, \phi_n)^T$, and $\omega > 0$ is a constant. If $x \in BC(\mathbb{R}, \mathbb{R}^n)$, then $x_t \in BC(\mathbb{R}, \mathbb{R}^n)$ for $t \in \mathbb{R}$ is defined by $x_t(\theta) = x(t + \theta)$ for $\theta \in \mathbb{R}$. + +However, to the best of our knowledge, there are few papers published on the multiple existence of positive periodic solutions for higher-dimensional functional differential equations with impulses; moreover, the existing results on the existence of periodic solutions for system (1.2) with or without impulses all assume that the coefficient matrix $A(t)$ is diagonal. Motivated by the above, in this paper, we are concerned with the following system: + +$$ +\begin{equation} +\begin{aligned} +& (1.3) \quad \left\{ + \begin{array}{@{}l@{\quad}l@{}} + y'(t) &= A(t)y(t) + g(t, y_t), && t \neq t_j, j \in \mathbb{Z}, \\ + y(t_j^+) &= y(t_j^-) + I_j(y(t_j)), & + \end{array} + \right. +\\[1em] +& \text{where } A(t) = (a_{ij}(t))_{n \times n} \text{ is a nonsingular matrix with continuous real-valued} +\end{aligned} +\end{equation} +$$ + +functions as entries and $A(t+T) = A(t)$; $g = (g_1, \ldots, g_n)^T$ is a functional $\mathbb{R} \times BC(\mathbb{R}, \mathbb{R}^n) \rightarrow \mathbb{R}^n$, satisfying $g(t+T, y_{t+T}) = g(t, y_t)$ for all $t \in \mathbb{R}$, $y_t \in BC(\mathbb{R}, \mathbb{R}^n)$, and if $y \in BC(\mathbb{R}, \mathbb{R}^n)$ then for any $t \in \mathbb{R}$, $y_t \in BC(\mathbb{R}, \mathbb{R}^n)$ is defined by $y_t(s) = y(t+s)$ for $s \in \mathbb{R}$; $y(t_j^+), y(t_j^-) = y(t_j)$ represent the right and the left limit of $y(t)$ at $t_j$, $j \in \mathbb{Z}$, respectively; and $I_j = (I_j^1, I_j^2, \ldots, I_j^n)^T \in C(\mathbb{R}^n, \mathbb{R}^n)$, $j \in \mathbb{Z}$. We assume that there exists an integer $p > 0$ such that $t_{j+p} = t_j + T$, $I_{j+p} = I_j$, $j \in \mathbb{Z}$, where $0 < t_1 < \cdots < t_p < T$. + +The Leggett–Williams multiple fixed point theorem [3] has proved to be a successful technique for dealing with the existence of three positive solu- tions of two, three or multi-point boundary value problems for differential equations (see [8, 9, 11]). Our main aim is (by using the Leggett–Williams theorem) to study the existence of at least three positive periodic solutions of (1.3). In the analysis we use the fundamental solution matrix of + +$$ +(1.4) \qquad y' = A(t)y +$$ + +and convert system (1.3) into an integral equation. Then we employ the +Leggett–Williams theorem to show the existence of triple positive periodic +solutions of (1.3). Our methods are different from those used in [2, 5, 8, 9]. + +In this paper, for each $x = (x_1, \dots, x_n)^T \in \mathbb{R}^n$, the norm of $x$ is defined as $|x|_0 = \sum_{i=1}^n |x_i|$. For matrices $A, B$, the notation $A > B$ ($A \le B$) means that each pair of corresponding entries of $A$ and $B$ satisfies the inequality ">$" ("$\le$"). In particular, $A$ is called a *positive matrix* if $A > 0$. + +In what follows, we assume that \ No newline at end of file diff --git a/samples/texts/5161315/page_1.md b/samples/texts/5161315/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..fb3652acf9debcad25c96d27ccdaec0e73fd21c7 --- /dev/null +++ b/samples/texts/5161315/page_1.md @@ -0,0 +1,27 @@ +UNIQUENESS OF SOLUTIONS TO A GAS-DISK INTERACTION SYSTEM + +ANTON IATCENKO AND WEIRAN SUN + +**ABSTRACT.** In this paper we give an elementary proof of uniqueness of solutions to a gas-disk interaction system with diffusive boundary condition. Existence of near-equilibrium solutions for this type of systems with various boundary conditions has been extensively studied in [1–8, 10]. However, the uniqueness has been an open problem, even for solutions near equilibrium. Our work gives the first rigorous proof of the uniqueness among solutions that are only required to be locally Lipschitz; in particular, it holds for solutions far from equilibrium states. + +**Keywords:** kinetic equations, integro-differential equations, uniqueness, gas-body interaction, friction. + +# 1. INTRODUCTION + +The main goal of this paper is to show uniqueness of solutions to a gas-disk interaction system. This system describes the motion of a disk immersed in a collisionless gas. Among many ways to model the friction between the gas and the disk, the simplest one is to assume that the friction is proportional to the velocity of the disk. In this scenario the velocity of the disk can be found by solving a linear ODE. Here we consider a more realistic model as in [1–8, 10–13], where the evolution of the gas and the disk satisfies a coupled system of integro-differential equations. The coupling is through collisions of gas particles with the disk: these collisions produce a drag force on the disk through momentum exchange and provide a boundary condition for the gas. + +In this paper we make a simplifying assumption that the disk is infinite. Together with assumed symmetry this lets us reduce the whole system to one dimension, thus making the disk a single point moving along the horizontal axis. To specify the model we let $f(x, v, t)$ be the density function of the gas that evolves according to the free transport equation away from the disk: + +$$ \partial_t f + v \partial_x f = 0, \quad f(x, v, 0) = \phi_0(v), \tag{1.1} $$ + +where $(x,v,t) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}^+$ are position, velocity, and time respectively. Denote the position of the disk at time $t$ by $\eta(t)$ and its velocity by $p(t)$. The interaction of the gas with the disk is described by a diffusive boundary condition: + +$$ f_R^+(\eta(t), v, t) = 2e^{-(v-p(t))^2} \int_{-\infty}^{p(t)} (p(t) - w) f_R^-(\eta(t), w, t) dw, \quad v > p(t), \tag{1.2} $$ + +$$ f_L^+(\eta(t), v, t) = 2e^{-(p(t)-v)^2} \int_{p(t)}^{\infty} (w-p(t)) f_L^-(\eta(t), w, t) dw, \quad v < p(t), \tag{1.3} $$ + +where the sub-indices *R* and *L* denote the right and left sides of the disk. Throughout the paper superscripts + and – on the density functions denote the postcollisional and precollisional distributions respectively, understood as one-sided limits: + +$$ f^{\pm}(\eta(t), v, t) = \lim_{\epsilon \to 0^+} f^{\pm}(\eta(t) \pm \epsilon v, v, t \pm \epsilon). \tag{1.4} $$ + +2010 Mathematics Subject Classification. 35A02, 35Q70, 35Q82. \ No newline at end of file diff --git a/samples/texts/5161315/page_10.md b/samples/texts/5161315/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..4f071206e6fc4133134619747c2ccbb8b7c7d8d7 --- /dev/null +++ b/samples/texts/5161315/page_10.md @@ -0,0 +1,42 @@ +for $n=1$ it becomes + +$$|f_2^{(p)}(s,t) - f_2^{(q)}(s,t)| \le \|p-q\| (2Q_5(\alpha_1(s) + \alpha_0(s)) + Q_8) \le 3\|p-q\|Q_5(\alpha_1(s) + \alpha_0(s)),$$ + +while for $n \ge 2$ we get + +$$|f_{n+1}^{(p)}(s,t) - f_{n+1}^{(q)}(s,t)| \le \|p-q\| (2 \cdot 3^{n-1} Q_5 (\alpha_n(s) + \alpha_{n-1}(s))) \le \|p-q\| 3^n Q_5 (\alpha_n(s) + \alpha_{n-1}(s)).$$ + +Choosing $Q_6 = \max\{Q_4, Q_5\}$ unifies all cases. Estimate (4.2) follows by summing the bounds in (4.1): + +$$ +\begin{align*} +|f_{\text{rec}}^{(p)}(t) - f_{\text{rec}}^{(q)}(t)| & \le \sum_{n=1}^{\infty} |f_n^{(p)}(t) - f_n^{(q)}(t)| \\ +& \le \left( Q_6 \sum_{n=0}^{\infty} 3^n (\alpha_n(s) + \alpha_{n-1}(s)) \right) \|p-q\| \\ +& = 4Q_6 e^{6M^2 s} \|p-q\|. \quad \square +\end{align*} +$$ + +The Lipschitz property of the drag force is an immediate consequence of Proposition 4.1: + +**Proposition 4.2.** Given two disk velocity profiles $p$ and $q$, let $G_p$ and $G_q$ be the corresponding drag forces defined in (2.3). Then for any $T > 0$ there exists a constant $L_T$ such that + +$$\|G_p - G_q\| \le L_T \|p - q\|.$$ + +*Proof.* Recall that we decompose $G_p = G_{p,0} + G_{p,rec}$ and $G_q = G_{q,0} + G_{q,rec}$. The Lipschitz bounds for $|G_{p,0} - G_{q,0}|$ and $|F(\eta_p(t), t) - F(\eta_q(t), t)|$ can be derived by direct estimates, so we focus on the re-collision part. We only give a sketch of the proof since it is very similar (and at times easier) to the one for (4.1). To simplify the notation we let + +$$A_p(s,t) = (p(t) - v_p(s,t))^2 + \frac{\sqrt{\pi}}{2}(p(t) - v_p(s,t)).$$ + +Then the difference becomes + +$$ +\begin{align*} +G_{p,\mathrm{rec}}(t) - G_{q,\mathrm{rec}}(t) &= \int_0^t \frac{\partial v_p(s,t)}{\partial s} A_p(s,t) f_{\mathrm{rec}}^{(p)}(s,t) \mathrm{d}s - \int_0^t \frac{\partial v_q(s,t)}{\partial s} A_q(s,t) f_{\mathrm{rec}}^{(q)}(s,t) \mathrm{d}s \\ +&= \int_0^t \left( \frac{\partial v_p(s,t)}{\partial s} - \frac{\partial v_q(s,t)}{\partial s} \right) A_p(s,t) f_{\mathrm{rec}}^{(p)}(s,t) \mathrm{d}s \\ +&\quad + \int_0^t \frac{\partial v_q(s,t)}{\partial s} (A_p(s,t) - A_q(s,t)) f_{\mathrm{rec}}^{(p)}(s,t) \mathrm{d}s \\ +&\quad + \int_0^t \frac{\partial v_q(s,t)}{\partial s} A_q(s,t) (f_{\mathrm{rec}}^{(p)}(s,t) - f_{\mathrm{rec}}^{(q)}(s,t)) \mathrm{d}s =: K_1 + K_2 + K_3. +\end{align*} +$$ + +A Lipschitz bound for $K_1$ follows from the same calculation as (4.3) in the proof of proposition 4.1. A Lipschitz bound for $K_2$ follows from the definition of $v_p$, and a Lipschitz bound for $K_3$ follows from (4.2) together with the bounds for $\partial_s v_q$ in Lemma 2.3. The combination of the three bounds for $K_1, K_2, K_3$ gives the Lipschitz bound for $G_{\mathrm{rec}}$. $\square$ + +*Remark 4.1.* Strictly speaking, the drag force in Proposition 4.2 is only the contribution from the right-side of the disk. However, as mentioned earlier, a similar Lipschitz property holds for the full drag force defined in (1.7), since the estimates for the left side follow from a similar argument. The main modification needed \ No newline at end of file diff --git a/samples/texts/5161315/page_11.md b/samples/texts/5161315/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..b117f845980c33e8e0c9f3ffd633d03f68bfa61b --- /dev/null +++ b/samples/texts/5161315/page_11.md @@ -0,0 +1,47 @@ +for the left side is to change the definition of the modified average velocity $\bar{v}(s,t)$ into + +$$ \bar{v}(s,t) = \max_{\tau \in [s,t]} \langle p \rangle_{s,t}. $$ + +Since replacing minimum with maximum does not affect the properties of the modified average velocity in Lemma 2.3, the rest of the estimates remain the same. + +We now have all the ingredients to prove the main result of this paper: + +*Proof of Theorem 1.1.* For any $T > 0$, by Proposition 4.2 and the assumption that the external force $F(\cdot, t)$ is Lipschitz, we have + +$$ \|p-q\|_{L^{\infty}(0,T)} \le (L_T + \text{Lip}(F)) \int_0^T \|p-q\|_{L^{\infty}(0,t)} dt, $$ + +which gives $p=q$ on $[0, T]$ by Gronwall's inequality. Meanwhile, for a given $p$, the density function $f$ on the disk can be written explicitly using the decomposition established in Section 2.2.1: + +$$ f_R(\eta(t), v, t) = \sum_{n=0}^{\infty} f_{R,n}(\eta(t), v, t), \quad f_L(\eta(t), v, t) = \sum_{n=0}^{\infty} f_{L,n}(\eta(t), v, t), $$ + +together with the initial condition $f(x,v,0) = \phi_0(v)$. Therefore, the boundary conditions in (1.2)-(1.3) are uniquely defined, which combined with the free transport equation (1.1) gives a unique solution for $f$. We thus obtain a unique solution to the full gas-disk system. □ + +**Acknowledgements:** The authors want to thank Ralf Wittenberg for fruitful discussions on this problem and pointing out a mistake in an earlier version. The research of W.S. is supported by NSERC Discovery Individual Grant R611626. + +REFERENCES + +[1] K. Aoki, G. Cavallaro, C. Marchioro, and M. Pulvirenti, *On the motion of a body in thermal equilibrium immersed in a perfect gas*, ESAIM:M2AN **42** (2008), 263–275. + +[2] S. Caprino, G. Cavallaro, and C. Marchioro, *On a microscopic model of viscous friction*, Math. Models Methods Appl. Sci. **17** (2007), no. 9, 1369–1403. + +[3] S. Caprino, C. Marchioro, and M. Pulvirenti, *Approach to equilibrium in a microscopic model of friction*, Comm. Math. Phys. **264** (2006), 167–189. + +[4] G. Cavallaro, *On the motion of a convex body interacting with a perfect gas in the mean-field approximation*, Rendiconti di Matematica, Serie VII **27** (2007), 123–145. + +[5] X. Chen and W. Strauss, *Approach to equilibrium of a body colliding specularly and diffusely with a sea of particles*, Arch. Rational Mech. Anal. **211** (2014), 879–910. + +[6] X. Chen and W. Strauss, *Velocity reversal criterion of a body immersed in a sea of particles*, Comm. Math. Phys. **338** (2015), 139–168. + +[7] K. Koike, *Motion of a rigid body in a special Lorentz gas: loss of memory effect*, J. Stat. Phys. **172** (2018), 795–823. + +[8] K. Koike, *Wall effect on the motion of a rigid body immersed in a free molecular flow*, Kinet. Relat. Mod. **11** (2018), no. 3, 441–467. + +[9] Giovanni Leoni, *A first course in Sobolev spaces*, Graduate Studies in Mathematics, vol. 105, AMS, 2009. + +[10] F. Sisti and C. Ricciuti, *Effects of concavity on the motion of a body immersed in a Vlasov gas*, SIAM J. Math. Anal. **46** (2014), no. 6, 3579–3611. + +[11] T. Tsuji and K. Aoki, *Decay of a linear pendulum in a free-molecular gas and in a special Lorentz gas*, J. Stat. Phys. **146** (2012), no. 3, 620–645. + +[12] T. Tsuji and K. Aoki, *Moving boundary problems for a rarefied gas: spatially one-dimensional case*, J. Comp. Phys. **250** (2013), 574–600. + +[13] T. Tsuji and K. Aoki, *Decay of a linear pendulum in a collisional gas: spatially one-dimensional case*, Phys. Rev. E **89** (2014), 052129. \ No newline at end of file diff --git a/samples/texts/5161315/page_12.md b/samples/texts/5161315/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..d4c842420ff38dcc9e8756441b79b2245ea7e488 --- /dev/null +++ b/samples/texts/5161315/page_12.md @@ -0,0 +1,25 @@ +The diffusive boundary conditions (1.2)-(1.3) essentially state that shape of the outgoing distribution is always Gaussian, with coefficients chosen to ensure the conservation of mass. Therefore, our model considers the case where collisions are instantaneous and the disk does not capture any finite mass of the gas via the collision process. + +We assume that the disk is acted on by an external force $F(x, t)$ and the drag force $G_p(t)$ generated through collisions with the gas particles (we have associated the drag force with the sub-index $p$ to emphasize its dependence on the disk velocity $p$). Then the motion of the disk is described by + +$$ \dot{p} = F(\eta(t), t) - G_p(t), \quad p(0) = p_0, \tag{1.5} $$ + +$$ \dot{\eta} = p(t), \qquad \eta(0) = 0. \tag{1.6} $$ + +We will write the total drag force as a combination of the drag forces due to particles colliding with the disk from the right and left: + +$$ G_p(t) = G_{p,R}(t) - G_{p,L}(t). \tag{1.7} $$ + +The signs are chosen to make both components of the drag positive. Physically speaking, $G_{p,L}$ accelerates the disk and $G_{p,R}$ decelerates it. Their exact expressions are derived from Newton's Second Law (see [5] for details): + +$$ G_{p,R}(t) := \int_{-\infty}^{p(t)} (p(t) - v)^2 f_R^-(t, \eta(t), v) \, dv + \int_{p(t)}^{\infty} (v - p(t))^2 f_R^+(t, \eta(t), v) \, dv, \tag{1.8} $$ + +$$ G_{p,L}(t) := \int_{-\infty}^{p(t)} (p(t) - v)^2 f_L^+(t, \eta(t), v) \, dv + \int_{p(t)}^{\infty} (v - p(t))^2 f_L^-(t, \eta(t), v) \, dv. \tag{1.9} $$ + +The evolution of the complete gas-disk system is governed by equations (1.1)-(1.9). We comment that the derivation of (1.8)-(1.9) relies on the Reynolds transport theorem, which assumes that the exchange of momentum between the gas and the disk can only happen through the fluxes of the gas moving into and out of the disk. Hence any particle that stays on the disk does not contribute to the momentum exchange or the drag force. We also note that to have an interaction with the disk the particle to the right (left) of it must be moving slower (faster) than the disk. + +Gas-body coupled systems have been extensively studied both numerically and analytically with pure diffusive, specular, and more generally, the Maxwell boundary conditions ([1–8, 10–13]). We refer the reader to a recent paper [7] for a comprehensive list of references. Among the central questions for these systems are their well-posedness and long-time behaviour. Regarding the long-time asymptotics, it is now fairly well-understood that due to the effect of re-collisions, the relaxation of the disks velocity toward its equilibrium state may not be exponential as in the simplified model where re-collisions are ignored. In fact, one may obtain algebraic decay rates [1–8, 10–13]. Moreover, depending on the shape of the body, such rates may or may not depend on the spatial dimension [4, 10]. + +The well-posedness issue, however, is less understood. To the best of our knowledge, existence of solutions has only been investigated for data near equilibrium states [1–8, 10] and uniqueness has been an open question even for these solutions. It is our goal in this paper to give a uniqueness proof for solutions to (1.1)-(1.9), where the disk velocity $p$ only needs to be in the natural space of locally Lipschitz functions. This includes solution spaces considered in [1, 5], as well as more general cases with solutions far from an equilibrium. The main result of this paper is + +**Theorem 1.1.** Suppose the initial density $\phi_0 \in L^1 \cap L^\infty(\mathbb{R})$ and the external force $F(x,t)$ is Lipschitz in $x$ with its Lipschitz coefficient independent of $t$. Then for any $p_0 \in \mathbb{R}$ there exists at most one solution $(\eta, p, f)$ to the system (1.1)-(1.9) such that $p$ is locally Lipschitz. \ No newline at end of file diff --git a/samples/texts/5161315/page_13.md b/samples/texts/5161315/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..e7948e7ee56edaf41993db1125bf47c384b6345d --- /dev/null +++ b/samples/texts/5161315/page_13.md @@ -0,0 +1,7 @@ +DEPARTMENT OF MATHEMATICS, SIMON FRASER UNIVERSITY, 8888 UNIVERSITY DR., BURNABY, BC V5A 1S6, CANADA + +*E-mail address: anton_iatcenko@sfu.ca* + +DEPARTMENT OF MATHEMATICS, SIMON FRASER UNIVERSITY, 8888 UNIVERSITY DR., BURNABY, BC V5A 1S6, CANADA + +*E-mail address: weiran.sun@sfu.ca* \ No newline at end of file diff --git a/samples/texts/5161315/page_14.md b/samples/texts/5161315/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..b69f99e1ee2e57ff54c4aaca7ebfbe1c9ef41600 --- /dev/null +++ b/samples/texts/5161315/page_14.md @@ -0,0 +1,33 @@ +Our main step in proving the main theorem is to show that the drag force due to recollisions, denoted by $G_{\text{rec}}$, is Lipschitz in the velocity $p$ (Proposition 4.2). The main difficulty for such estimate is the dependence the distribution of the recolliding particles on the entire history of the disk motion. We address this issue by taking advantage of the inherently recursive nature of the problem: the distribution of the particles colliding with the disk for the $n^{\text{th}}$ time at time $t$ is determined by the distribution of the particles colliding with the disk for the $(n-1)^{\text{th}}$ time at some earlier time $s$. Instead of trying to compute or estimate such $s$ for a given velocity $v$, we use a change of variables $v = v(s,t)$. This allows us to compare the particles that have collided with the disk at the same time in the past instead of comparing particles that have the same velocity at the current time. + +Three remarks are in order: first, we have assumed that the initial state of the gas is spatially homogeneous. This assumption can be dropped at the cost of adding more technicalities. Second, due to the essential step of change of variables, so far our technique is only applicable to the case with diffusive boundary conditions. Hence for systems with specular or Maxwell boundary conditions uniqueness is still an open question. Third, this paper only deals with the one-dimensional case with a collisionless gas, but we expect a similar strategy to be applicable in higher dimensions and for systems with simple collisions such as the special Lorenz gas in [11]. This will be subject to future investigation. + +The rest of the paper is laid out as follows: in Section 2 we state our assumptions, introduce partition of the density function and the change of variables, and reformulate the density function and the drag force into recursive forms. Section 2 contains the essential ideas and constructions that will be used in various estimates and the uniqueness proof in the later part. In Section 3 we obtain preliminary bounds on the density function using the recursive form. Finally, in Section 4 we establish the Lipschitz property of the drag force and give a proof of the uniqueness theorem. + +## 2. ASSUMPTIONS AND REFORMULATIONS + +In this section we state all the assumptions used to prove the uniqueness of the solution. We also introduce several reformulations of the density function $f$ as well as the drag term $G$. Most of the discussion here is built upon the understanding of the physics underlying the interactions of the gas particles with the disk. + +Throughout this paper we let $T$ be a fixed arbitrary time. + +### 2.1. Main Assumptions. +The assumptions on the system are + +(A0) Particles cannot penetrate the disk. + +(A1) *Assumptions on the gas:* the initial distribution $\phi_0 = \phi_0(v)$ satisfies + +(a) $\phi_0 \in L^\infty(\mathbb{R})$; + +(b) The zeroth, first and second moments of $\phi_0$ are finite: + +$$ \int_{\mathbb{R}} (1+v^2)\phi_0(v) \, dv < \infty. \quad (2.1) $$ + +(A2) *Assumption on the disk:* velocity of the disk is locally Lipschitz with + +$$ \|p\|_{L^\infty(0,T)} + \|\dot{p}\|_{L^\infty(0,T)} \leq M, \quad (2.2) $$ + +where $M$ may depend on $T$. + +### 2.2. Reformulation of the Model. +For the rest of the paper we will only consider the gas to be of right of the disk since the analysis for the gas to the left of the disk is analogous. This lets us drop the sub-indices $R$ and $L$ in (1.2)-(1.3) and (1.9)-(1.8). \ No newline at end of file diff --git a/samples/texts/5161315/page_15.md b/samples/texts/5161315/page_15.md new file mode 100644 index 0000000000000000000000000000000000000000..a2062f26fa40a21bca0199c0b8fdbb82c88827f1 --- /dev/null +++ b/samples/texts/5161315/page_15.md @@ -0,0 +1,60 @@ +We begin by simplifying the expression for the drag forces. The expression for the outgoing density in (1.2) allows us to write + +$$ +\begin{align*} +\int_{p(t)}^{\infty} (v-p(t))^2 f^{+}(\eta(t), v, t) \, dv &= 2 \left( \int_{-\infty}^{p(t)} (p(t)-v)f^{-}(\eta(t), v, t) \, dv \right) \int_{p(t)}^{\infty} (v-p(t))^2 e^{-(v-p(t))^2} \, dv \\ +&= \frac{\sqrt{\pi}}{2} \int_{-\infty}^{p(t)} (p(t)-v)f^{-}(\eta(t), v, t) \, dv, +\end{align*} +$$ + +so the expression for the drag force can be written as + +$$ +G_p(t) = \int_{-\infty}^{p(t)} \left( (p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v) \right) f^{-}(\eta(t), v, t) dv. \quad (2.3) +$$ + +2.2.1. *Partition of the density function.* To make the drag term more amiable to analysis, we introduce the idea of recursive scattering; for $x \neq \eta(t)$ let $f_n(x, v, t)$ be the density functions of the particles that have collided with the disk exactly $n$ times in the past. Away from the disk they satisfy the same free transport equation as $f$. For $x = \eta(t)$ we define $f_n^+(x, v, t)$ in terms of the one-sided limits similar to those in (1.4): + +$$ +f_n^{\pm}(\eta(t), v, t) = \lim_{\epsilon \to 0^+} f_n^{\pm}(\eta(t) \pm \epsilon v, v, t \pm \epsilon). \quad (2.4) +$$ + +The boundary conditions on $f_n$'s are similar to those for the full density function $f$, with the exception that +the collision with the disk now increments the sub-index. In particular, for $v > p(t)$ and $n \ge 0$ we write + +$$ +f_{n+1}^{+}(\eta(t), w, t) = 2e^{-(w-p(t))^2} \int_{-\infty}^{p(t)} (p(t)-v)f_{n}^{-}(\eta(t), v, t) dv. \quad (2.5) +$$ + +We also define $f_{\text{rec}}$ to be the density function of the particles that have collided with the disk in the past: + +$$ +f_{\text{rec}}(x, v, t) = \sum_{n \ge 1} f_n(x, v, t). \tag{2.6} +$$ + +Thus the full density function is decomposed as + +$$ +f(x, v, t) = \phi_0(v) + f_{\text{rec}}(x, v, t). +$$ + +Similarly, we define $G_{p,\text{rec}}$ to be the drag forces due to particles that have collided with the disk in the past: + +$$ +G_{p,\text{rec}}(t) = \int_{-\infty}^{p(t)} \left( (p(t)-v)^2 + \frac{\sqrt{\pi}}{2}(p(t)-v) \right) f_{\text{rec}}(\eta(t), v, t) \, dv. \quad (2.7) +$$ + +2.2.2. *Average Velocity.* We now address the possibility for the particles to collide with the disk multiple times. Throughout the paper we adopt the following notation for the average velocity of the disk on the time interval $[s, t]$: + +$$ +\langle p \rangle_{s,t} = \frac{1}{t-s} \int_s^t p(\tau) \, d\tau. +$$ + +It will play a significant role in the precollision conditions and the change of variables. We summarize a few +useful properties of the average velocity in the following lemma: + +**Lemma 2.1.** Suppose $t \in (0, T)$ and $s \in (0, t)$. Let $v(\cdot, \cdot)$ be the function defined as + +$$ +v(s,t) = \langle p \rangle_{s,t}. \tag{2.8} +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_16.md b/samples/texts/5161315/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..e8d452addf48187163f09ead24e60e9c598a8618 --- /dev/null +++ b/samples/texts/5161315/page_16.md @@ -0,0 +1,60 @@ +Let $M$ be the Lipschitz constant in (2.2). Then + +(a) $v(s,t)$ satisfies the bound $|v(s,t)| \le M;$ + +(b) the derivatives of $v(s,t)$ are + +$$ +\begin{equation} +\begin{aligned} +\frac{\partial v}{\partial s} &= \frac{v - p(s)}{t - s}, & \quad \frac{\partial v}{\partial t} &= \frac{p(t) - v}{t - s}; +\end{aligned} +\tag{2.9} +\end{equation} +$$ + +(c) the derivatives of $v$ satisfy the following estimates: + +$$ +\left|\frac{\partial v}{\partial s}\right| \le \frac{1}{2}M, \qquad \left|\frac{\partial v}{\partial t}\right| \le \frac{1}{2}M. \tag{2.10} +$$ + +*Proof*. Part (b) follows from direct computations. Parts (a) and (c) follow from Assumption (A2): + +$$ +|v(s,t)| = \left| \frac{1}{t-s} \int_s^t p(\tau) \, d\tau \right| \le \frac{1}{t-s} \int_s^t M \, d\tau = M, +$$ + +$$ +\left|\frac{\partial v}{\partial t}\right| = \frac{|p(t) - v(s,t)|}{t-s} \le \frac{1}{(t-s)^2} \int_s^t |p(t) - p(\tau)| \, d\tau \le \frac{\|\dot{p}\|_{L^\infty(0,t)}}{(t-s)^2} \int_s^t (t-\tau) \, d\tau = \frac{1}{2}M. +$$ + +The estimate for $\partial_s v$ is proved via a similar calculation. + +2.2.3. *Precollisional Velocities and Precollision Times.* In this section we prepare for the key step of change of variables. To illustrate the idea of change of variables, we consider for a moment a simplified case where $\dot{p}(t) > 0$ for all $t$. Then for each $t \in (0,T)$ the average velocity $\langle p \rangle_{s,t}$ is strictly increasing in $s$, and thus is a bijection between $[0,t]$ and $[\langle p \rangle_{0,t}, \langle p \rangle_{t,t}] = [\eta(t)/t, p(t)]$. This allows us to use the change of variables $v = v(s,t) = \langle p \rangle_{s,t}$ in (2.5) to obtain the following expression for $n \ge 1$ and $w > p(t)$: + +$$ +f_{n+1}^{+}(\eta(t), w, t) = 2e^{-(w-p(t))^2} \int_{\eta(t)/t}^{p(t)} (p(t)-v)f_{n}^{-}(\eta(t), v, t) dv \quad (2.11) +$$ + +$$ += 2e^{-(w-p(t))^2} \int_0^t \frac{\partial v}{\partial s}(p(t)-v(s,t)) f_n^-(\eta(t), v(s,t), t) ds. \quad (2.12) +$$ + +One immediate advantage of expression (2.12) is that it allows us to obtain an explicit recursive relationship between the sequence of outgoing densities {$f_n^+$}. Indeed, since the distribution density does not change between collisions, we have + +$$ +f_n^-(\eta(t), v(s,t), t) = f_n^+(\eta(s), v(s,t), s). +$$ + +This in turn implies + +$$ +f_{n+1}^{+}(\eta(t), w, t) = 2e^{-(w-p(s))^2} \int_0^t \frac{\partial v(s,t)}{\partial \tau} (p(t) - v(s,t)) f_n^+(\eta(s), v(s,t), s) ds. +$$ + +In Sections 3 and 4 we show the full usage of a similar recursive relation for obtaining the estimates for the density function and the drag term. + +Without the monotonicity assumption a proper change of variables requires much more work. The main difficulty is the non-injectivity of the mapping $v(\cdot, t)$ defined in (2.8). To handle it, we start by identifying that, among all the particles that are to collide with the disk at time $t$, which ones have had a collision in the past. Velocities of such particles will henceforth be called *precollisional*, to signify that the corresponding particles have previously collided with the disk. They must satisfy the following condition: + +There exists time $s \in [0, t)$ such that the particle and the disk have travelled the same distance over $[s, t]$ and $v < p(t)$. \ No newline at end of file diff --git a/samples/texts/5161315/page_17.md b/samples/texts/5161315/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..d7d3317e8b172d7edf95ef8ea961e29ed5ca5647 --- /dev/null +++ b/samples/texts/5161315/page_17.md @@ -0,0 +1,41 @@ +Since the velocity of the particle does not change between consecutive collisions, the above condition can be written as + +$$ (t-s)v = \int_s^t p(\tau) \, d\tau \quad \text{or} \quad v = \langle p \rangle_{s,t}. \qquad (2.13) $$ + +Introduce the notation + +$$ \underline{\kappa}(t) = \min_{s \in [0,t]} \langle p \rangle_{s,t}. $$ + +Then the precollisional velocities can be characterized as + +**Proposition 2.1.** Suppose a particle with velocity $v$ is colliding with the disk at time $t$ and $v \neq \underline{\kappa}(t)$. Then it has collided with the disk in the past if and only if + +$$ \underline{\kappa}(t) < v < p(t). \qquad (2.14) $$ + +*Proof.* Let $\underline{\kappa}(t) < v < p(t)$. Since $\langle p \rangle_{s,t}$ is a continuous function of $s$ for any $t$, it must obtain its minimum $\underline{\kappa}(t)$ at some $s^* \in [0, t]$. Assume $s^* < t$ and suppose for contradiction that the particle with velocity $v$ has not collided with the disk in the past. Let $\omega(s)$ be the position of the particle. Then + +$$ \omega(s) = \eta(t) - (t-s)v. $$ + +Since the particle is colliding with the disk from the right and could not have penetrated the disk by assumption (**A0**), it must have been to the right of the disk for all $s \in [0, t)$, that is + +$$ \omega(s) - \eta(s) \ge 0 \quad \forall s \in [0, t). $$ + +However, this condition is violated at $s = s^*$ since + +$$ \begin{aligned} \omega(s^*) - \eta(s^*) &= [\eta(t) - (t-s^*)v] - [\eta(t) - (t-s^*)\underline{\kappa}(t)] \\ &= [\underline{\kappa}(t) - v](t-s^*) < 0, \end{aligned} \qquad (2.15) $$ + +which is a contradiction. If $s^* = t$, then $\underline{\kappa}(t) = \langle p \rangle_{t,t} = p(t)$. This again violates (2.14). + +The converse is an immediate consequence of (2.13). □ + +Denote the set of all possible precollisional velocities by $\mathcal{V}_t$ where + +$$ \mathcal{V}_t = (\underline{\kappa}(t), p(t)). \qquad (2.16) $$ + +We now identify the times of the precollisions. + +*Definition 2.2 (Precollision Time).* Suppose a particle with velocity $v \in \mathcal{V}_t$ is to collide with the disk at time $t$. Then the time $s_v$ is a corresponding precollision time if $(v, s_v)$ satisfies (2.13) and the particle has been ahead of the disk for all $\tau \in (s, t)$, that is, + +$$ \eta(t) - (t-\tau)v \ge \eta(\tau) \quad \forall \tau \in (s,t). \qquad (2.17) $$ + +Note that this condition is a mathematical formulation of Assumption **(A0)**. \ No newline at end of file diff --git a/samples/texts/5161315/page_18.md b/samples/texts/5161315/page_18.md new file mode 100644 index 0000000000000000000000000000000000000000..71324f3333ec13cf87c026001a805454762e20b3 --- /dev/null +++ b/samples/texts/5161315/page_18.md @@ -0,0 +1,34 @@ +Let $\mathcal{N}_t$ be the set of all possible precollision times. To construct a bijection between $\mathcal{V}_t$ and $\mathcal{N}_t$ we need +a more explicit characterization of the latter. To this end, we first rewrite (2.17) as + +$$v \leq \langle p \rangle_{\tau,t} \quad \forall \tau \in (s,t), \tag{2.18}$$ + +which in turn implies + +$$v \leq \min_{\tau \in [s,t]} \langle p \rangle_{\tau,t}. \tag{2.19}$$ + +Motivated by (2.19), we define the modified average velocity $\underline{v}(s,t)$: + +$$\underline{v}(s,t) := \min_{\tau \in [s,t]} \langle p \rangle_{\tau,t}. \tag{2.20}$$ + +From the combination of (2.19) with (2.13) we conclude that $s_v$ is a precollision time corresponding to $v$ if and only if $v = v(s,t) = \underline{v}(s,t)$. Consequently, we have + +$$\mathcal{N}_t = \{ s \in [0, t] \mid v(s, t) = \underline{v}(s, t) \}. \tag{2.21}$$ + +Note the function $\underline{v}(\cdot, t)$ is monotonically, but not necessarily strictly, increasing. It can be thought of as the +tightest monotonically increasing lower envelope for $v(s, t)$; the notation $\underline{v}$ had been chosen to reflect that. +We give an example of $\mathcal{N}_t$ and $\underline{v}$ in Figure 1 to help intuitive understanding of their properties. + +*Notation.* For a given $t$, we will use $\underline{v}(\mathcal{A}, t)$ to denote the image of the set $\mathcal{A}$ under the map $\underline{v}(\cdot, t)$ and $\underline{v}^{-1}(\mathcal{B}, t)$ to denote the pre-image of the set $\mathcal{B}$ under the map $\underline{v}(\cdot, t)$. Note that the inversion is only performed in the first variable with the second variable $t$ fixed. + +We now establish properties of $\mathcal{N}_t$ and $\underline{v}$. A large part of the analysis is essentially Riesz's rising sun lemma [9] with a sign change. + +**Lemma 2.2.** Fix $t \in [0, T]$ and let $\mathcal{N}_t$ be the set defined in (2.21). Then $\mathcal{N}_t^c = [0, t] \setminus \mathcal{N}_t$ is open in $[0, t]$. + +*Proof.* Note that since $\underline{v}(t, t) = v(t, t) = p(t)$, we have $t \in \mathcal{N}_t$. We write $\mathcal{N}_t^c$ as + +$$\mathcal{N}_t^c = \left\{ s \in [0, t] \mid v(s,t) > \underline{v}(s,t) \right\} = \left\{ s \in [0, t] \mid v(s,t) > v(\tau, t) \text{ for some } \tau \in (s,t) \right\} = \bigcup_{\tau \in [0,t)} \mathcal{O}_{\tau},$$ + +where $\mathcal{O}_{\tau} = [0, \tau) \cap v^{-1}((v(\tau, t), \infty), t)$. Since $v(\cdot, t)$ is continuous, the pre-image $\mathcal{O}_{\tau}$ is open in the subspace topology on $[0, t]$. Therefore $\mathcal{N}_t^c$ is an open subset of $[0, t]$. $\square$ + +FIGURE 1. An example of the average velocity $v(s,t)$ and the corresponding modified average velocity $\underline{v}(s,t)$, with $t$ fixed. Note how the difference between the two informs the separation of the time domain into $\mathcal{N}_t$ and $\mathcal{N}_t^c$. \ No newline at end of file diff --git a/samples/texts/5161315/page_19.md b/samples/texts/5161315/page_19.md new file mode 100644 index 0000000000000000000000000000000000000000..edafec8cc10ca1247dd9fc23d480f41912f5f8e9 --- /dev/null +++ b/samples/texts/5161315/page_19.md @@ -0,0 +1,51 @@ +**Lemma 2.3 (Properties of $\underline{v}$).** Let $\underline{v}(s,t)$ be the modified velocity defined in (2.20). Then + +(a) $\underline{v}(\cdot, t) : [0, t] \to \mathcal{V}_t$ is continuous. + +(b) Let $(a,b)$ be a maximal connected component of $\mathcal{N}_t^c$. Then for all $s \in (a,b)$ we have + +$$ \underline{v}(a,t) = \underline{v}(s,t) = \underline{v}(b,t). $$ + +Consequently, $\partial_s\underline{v}(s,t) = 0$ on $\mathcal{N}_t^c$. + +(c) Restricting $\underline{v}$ to $\mathcal{N}_t$ does not change its range: $\underline{v}([0,t], t) = \underline{v}(\mathcal{N}_t, t)$. + +(d) For any fixed $t \in [0, T]$, $\underline{v}(s,t)$ is Lipschitz in $s$ with $|\partial_s\underline{v}(s,t)| \le M/2$ for almost every $s \in [0, T]$. + +(e) For any fixed $s \in [0, t]$, $\underline{v}(s,t)$ is Lipschitz in $t$ with $|\partial_t\underline{v}(s,t)| \le M$ for almost every $t \in [0, T]$. + +(f) Let $\mathcal{L}(A)$ be the Lebesgue measure of $A$ and define + +$$ \mathcal{D}_t := \left\{ s \in [0, t] \mid \frac{\partial \underline{v}(s,t)}{\partial s} \text{ exists} \right\}. $$ + +Then $\mathcal{L}(\underline{v}(\mathcal{D}_t^c, t)) = \mathcal{L}(v(\mathcal{D}_t^c, t)) = 0$. + +(g) For all $s \in \mathcal{N}_t \cap \mathcal{D}_t$ we have $\partial_s\underline{v}(s,t) = \partial_s v(s,t)$. + +*Proof.* (a) Let $\epsilon > 0$ be given. By the uniform continuity of $v(\cdot, t)$ on $[0, t]$, there exists $\delta > 0$ such that + +$$ |\underline{v}(s,t) - v(s', t)| < \epsilon \quad \text{whenever} \quad |s - s'| < \delta. \tag{2.22} $$ + +Without loss of generality, suppose $0 \le s - s' \le \delta$. Then by definition of $\underline{v}(\cdot, t)$ we have + +$$ +\begin{align*} +0 \le \underline{v}(s,t) - \underline{v}(s',t) &= \underline{v}(s,t) - \min_{\tau \in [s',s]} \{ \min_{\tau \in [s,s]} v(\tau,t), \underline{v}(s,t) \} \\ +&= (\underline{v}(s,t) - \min_{\tau \in [s',s]} v(\tau,t)) \operatorname{sgn}(\underline{v}(s,t) - \min_{\tau \in [s,s]} v(\tau,t)) \\ +&\le |v(s,t) - \min_{\tau \in [s',s]} v(\tau,t)| < \epsilon, +\end{align*} +$$ + +where the last inequality follows from (2.22). + +(b) Since $\underline{v}(\cdot, t)$ is non-decreasing we have $\underline{v}(s,t) \le \underline{v}(b,t)$. Suppose for contradiction that $\underline{v}(s,t) < \underline{v}(b,t)$. Then there must exist $\tau \in [s,b)$ such that $\underline{v}(s,t) = v(\tau,t)$, which in turn implies that $\underline{v}(\tau,t) = v(\tau,t)$. But then $\tau \in \mathcal{N}_t$ by the definition of $\mathcal{N}_t$, which is a contradiction since $\tau \in (a,b) \subseteq \mathcal{N}_t^c$. Equality $\underline{v}(a,t) = \underline{v}(s,t)$ follows from continuity of $\underline{v}(\cdot, t)$. + +(c) Take $s \in \mathcal{N}_t^c$ and let $(a,b)$ be the largest connected subset of $\mathcal{N}_t^c$ containing it. By part (b), we have $\underline{v}(s,t) = \underline{v}(b,t)$. Thus $\underline{v}(s,t) \in \underline{v}(\mathcal{N}_t, t)$ since $b \in \mathcal{N}_t$. + +(d) For $s \in \mathcal{N}_t^c$ let $(a_s, b_s)$ be the largest connected component of $\mathcal{N}_t^c$ containing $s$. In other words, the lower bound $a_s$ is the largest time less than $s$ such that $\underline{v}(a_s, t) = v(a_s, t)$. Similarly, the upper bound $b_s$ is the smallest time greater than $s$ such that $\underline{v}(b_s, t) = v(b_s, t)$. For $s \in \mathcal{N}_t$ we simply let $a_s = b_s = s$. Then for both cases we have + +$$ v(a_s, t) = \underline{v}(a_s, t) = \underline{v}(s, t) = \underline{v}(b_s, t) = v(b_s, t). $$ + +Let $\tau, \tau' \in [0, t]$ and assume, without loss of generality, that $\tau \le \tau'$. If $(\tau, \tau') \subseteq \mathcal{N}_t^c$ then $v(\tau', t) - v(\tau, t) = 0$ by part (b). Otherwise, by Lemma 2.1 we have + +$$ |\underline{v}(\tau', t) - \underline{v}(\tau, t)| = |v(a_{\tau'}, t) - v(b_{\tau'}, t)| \le \frac{M}{2} (a_{\tau'} - b_{\tau'}) \le \frac{M}{2} (\tau' - \tau). $$ \ No newline at end of file diff --git a/samples/texts/5161315/page_2.md b/samples/texts/5161315/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..35abc561067b811490d7cece99f249c8acbdb93a --- /dev/null +++ b/samples/texts/5161315/page_2.md @@ -0,0 +1,43 @@ +*Remark 2.1.* We have not yet discussed the relationship between the velocity of the particle that had precollided with the disk at time $s$ and the velocity of the disk itself at time $s$; one would expect the particle to be moving faster in that case. Indeed, combining (2.17) with (2.13) yields + +$$ \eta(s) + (\tau - s)v \geq \eta(\tau) \quad \forall \tau \in (s, t), $$ + +which can in turn be rewritten as $v \geq \langle p \rangle_{s,\tau}$ for all $\tau \in (s,t)$. Letting $\tau \to s$ gives $v \geq \langle p \rangle_{s,s} = p(s)$, so the particle is, at least, can not be slower than the disk. The case $v = p(s)$ is the grazing precollision. Since + +$$ \frac{\partial v(s,t)}{\partial s} = \frac{v(s,t) - p(s)}{t-s}, $$ + +all velocities that have had a grazing precollision and their corresponding (non-unique!) precollision times are collected in $W_t^c$ and $\Phi_t^c$ respectively. Since $\mathcal{L}(v(\Phi_t^c, t)) = 0$, particles that have had a grazing precollision have no effect on the dynamics of the disk, and thus can be safely excluded. + +**2.2.4. Change of Variables.** We now make a change variables in (2.7): by (2.16) and Theorem 2.3, we have + +$$ +\begin{align} +G_{p,\text{rec}}(t) &= \int_{V_t} \left( (p(t)-v)^2 + \frac{\sqrt{\pi}}{2}(p(t)-v) \right) f_{\text{rec}}^{-}(\eta(t), v, t) \, dv \\ +&= \int_{W_t} \left( (p(t)-v)^2 + \frac{\sqrt{\pi}}{2}(p(t)-v) \right) f_{\text{rec}}^{-}(\eta(t), v, t) \, dv \\ +&= \int_{\Phi_t} \frac{\partial v(s,t)}{\partial s} \left( (p(t)-v(s,t))^2 + \frac{\sqrt{\pi}}{2}(p(t)-v(s,t)) \right) f_{\text{rec}}^{-}(\eta(t), v(s,t), t) \, ds. \tag{2.24} +\end{align} +$$ + +Furthermore, since $\partial_s v(s,t)$ vanishes on $\Phi_t^c$, we can write + +$$ +\begin{align} +G_{p,\text{rec}}(t) &= \int_{\Phi_t} \frac{\partial v(s,t)}{\partial s} \left( (p(t)-v(s,t))^2 + \frac{\sqrt{\pi}}{2}(p(t)-v(s,t)) \right) f_{\text{rec}}^{-}(\eta(t), v(s,t), t) \, ds \\ +&= \int_{\Phi_t} (\ldots) \, ds + \int_{\Phi_t^c} (\ldots) \, ds \\ +&= \int_0^t \frac{\partial v(s,t)}{\partial s} \left( (p(t)-v(s,t))^2 + \frac{\sqrt{\pi}}{2}(p(t)-v(s,t)) \right) f_{\text{rec}}(s,t) \, ds, \tag{2.25} +\end{align} +$$ + +where, with a slight abuse of notation, we have written + +$$ f_{\text{rec}}(s,t) = \sum_{n \ge 1} f_n(s,t) = \sum_{n \ge 1} f_n^-(\eta(t), v(s,t), t) = \sum_{n \ge 1} f_n^+(v(s,t), s). \quad (2.26) $$ + +The last equality in (2.26) holds because the distribution density does not change between collisions. Note that in (2.25), the modified velocity $v(s,t)$ needs to appear only in the derivative since whenever $\partial_s v(s,t) \neq 0$, we have $v(s,t) = v(s,t)$. + +Making the same change of variables in (2.5) for $n \ge 1$ and using the notation in (2.26) lead us to the key recurrence relation: + +$$ f_{n+1}(s,t) = 2e^{-(v(s,t)-p(s))^2} \int_0^s \frac{\partial v(\tau,s)}{\partial s} (p(s)-v(\tau,s)) f_n(\tau,s) d\tau. \quad (2.27) $$ + +For future convenience, we define the density flux of $f_n$ as + +$$ j_n(s) = \int_0^s \frac{\partial v(\tau, s)}{\partial s} (p(s) - v(\tau, s)) f_n(\tau, s) d\tau, \quad (2.28) $$ \ No newline at end of file diff --git a/samples/texts/5161315/page_20.md b/samples/texts/5161315/page_20.md new file mode 100644 index 0000000000000000000000000000000000000000..d733f956b7bc36e702599444ed74b889124a7763 --- /dev/null +++ b/samples/texts/5161315/page_20.md @@ -0,0 +1,41 @@ +(e) Let $t' > t$ and fix $s \in [0, t]$. Since $v(s,t)$ is continuous, $\underline{v}(s,t) = v(\tau,t)$ for some $\tau \in [s,t]$. We have + +$$ \underline{v}(s, t') \leq v(\tau, t') \leq v(\tau, t) + \frac{M}{2}|t' - t| \implies \underline{v}(s, t') - \underline{v}(s, t) \leq \frac{M}{2}|t' - t|. $$ + +On the other hand, for all $\tau \in [0, t]$ we have + +$$ v(\tau, t') \geq v(\tau, t) - M|t' - t| \implies \underline{v}(s, t') \geq \underline{v}(s, t) - M|t' - t|, $$ + +Thus the function $\underline{v}(s,t)$ is Lipschitz in $t$. Consequently, $\partial_t\underline{v}(s,t)$ exist for almost all $t$ and $|\partial_t\underline{v}(s,t)| \leq M$. + +(f) Since $\underline{v}(s,t)$ is Lipschitz in $s$, it is almost everywhere differentiable and thus $\mathcal{L}(\mathcal{D}_t^c) = 0$. Since $\underline{v}(\cdot, t)$ is absolutely continuous, it possesses the Lusin property: $\mathcal{L}(\underline{v}(\mathcal{D}_t^c, t)) = 0$. The same argument holds for $v(\cdot, t)$. + +(g) Let $s \in N_t \cap D_t$. Then $\partial_s\underline{v}(s,t)$ is given by the definition of the classical derivative. Therefore, + +$$ \begin{aligned} \frac{\partial \underline{v}(s,t)}{\partial s} &= \lim_{\tau \to s^+} \frac{\underline{v}(\tau,t) - \underline{v}(s,t)}{\tau - s} = \lim_{\tau \to s^+} \frac{\underline{v}(\tau,t) - v(s,t)}{\tau - s} \leq \lim_{\tau \to s^+} \frac{v(\tau,t) - v(s,t)}{\tau - s} = \frac{\partial v(s,t)}{\partial s}, \\ \frac{\partial v(s,t)}{\partial s} &= \lim_{\tau \to s^-} \frac{v(s,t) - v(\tau,t)}{s - \tau} = \lim_{\tau \to s^-} \frac{\underline{v}(s,t) - v(\tau,t)}{s - \tau} \leq \lim_{\tau \to s^-} \frac{\underline{v}(s,t) - \underline{v}(\tau,t)}{s - \tau} = \frac{\partial \underline{v}(s,t)}{\partial s}. \end{aligned} $$ + +It now follows that $\partial_s\underline{v}(s,t) = \partial_s v(s,t)$. $\square$ + +Since the measure of the set $D_t^c$, as well as its images under both $\underline{v}(\cdot, t)$ and $v(\cdot, t)$, is zero, we can safely ignore it from now on. + +From Lemma 2.3(c) it follows that the map $\underline{v}(\cdot, t): N_t \to V_t$ is a surjection. However, it is not necessarily an injection, so further restriction is required. To show that the restriction we are about to make does not affect the dynamics of the disk we will need the following lemma from [9] (page 77): + +**Lemma 2.4 ([9]).** Let $I \subseteq \mathbb{R}$ be an interval and let $u: I \to \mathbb{R}$. Assume that there exists a set $E \subseteq I$ (not necessarily measurable) and $M \ge 0$ such that $u$ is differentiable for all $x \in E$, with + +$$ |u'(x)| \le M \quad \text{for all } x \in E. $$ + +Then $\mathcal{L}_\circ(u(E)) \le M\mathcal{L}_\circ(E)$, where $\mathcal{L}_\circ$ denotes the outer Lebesgue measure. + +We are now ready to make the restriction and create a bijection. + +**Theorem 2.3.** For any fixed $t \in [0, T]$, let $\underline{v}(\cdot, t)$ be the function defined in (2.20). Let + +$$ \Phi_t := \left\{ s \in [0, t] \mid \frac{\partial \underline{v}(s,t)}{\partial s} > 0 \right\} \quad \text{and} \quad W_t := \underline{v}(\Phi_t, t). \tag{2.23} $$ + +Then $\underline{v}(s,t) = v(s,t)$ for all $s \in \Phi_t$, the mapping $\underline{v}(\cdot, t) : \Phi_t \to W_t$ is a bijection and is strictly increasing, and $W_t$ contains almost all postcollisional velocities, that is, $\mathcal{L}(V_t\setminus W_t) = 0$. + +*Proof.* From Lemma 2.3(b) we know that $\partial_s\underline{v}(s,t) = 0$ for all $s \in N_t^c$, so it must be the case that $\Phi_t \subseteq N_t$. Furthermore, since $\underline{v}(\cdot, t)$ is a monotonically increasing function on the interval $[0, t]$, its restriction to $\Phi_t$ is strictly increasing and thus is a bijection between its domain and range. Choosing $I = [0, t]$, $u = \underline{v}(\cdot, t)$, $E = \Phi_t^c$ and $M = 0$ in Lemma 2.4 gives + +$$ \mathcal{L}_{\circ}(W_t^c) = \mathcal{L}_{\circ}(\underline{v}(\Phi_t^c, t)) = 0. $$ + +Hence $\mathcal{L}_{\circ}(\nu_t W_t) = 0$, so $\nu_t W_t$ is measurable and almost all postcollisional velocities are included in $W_t$. $\square$ \ No newline at end of file diff --git a/samples/texts/5161315/page_3.md b/samples/texts/5161315/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..6fc4cb9ad27dbe286ff524dec8ed70b8b1bda086 --- /dev/null +++ b/samples/texts/5161315/page_3.md @@ -0,0 +1,59 @@ +which allows us to write + +$$ +f_{n+1}(s,t) = 2e^{-(v(s,t)-p(s))^2} j_n(s). \tag{2.29} +$$ + +The change of variables does not apply to the particles that have not collided with the disk previously; +since such particles maintain the initial density distribution, we have + +$$ +f_1(s,t) = 2e^{-(v(s,t)-p(s))^2} \int_{-\infty}^{\underline{v}(0,s)} (p(s)-v)\phi_0(v) \, dv, \quad (2.30) +$$ + +$$ +G_{p,0}(t) = \int_{-\infty}^{\underline{v}(0,t)} \left( (p(t)-v)^2 + \frac{\sqrt{\pi}}{2}(p(t)-v) \right) \phi_0(v) \, dv. \quad (2.31) +$$ + +Recalling the definition of $G_{p,\text{rec}}$ in (2.7), we have constructed a decomposition of the drag force: + +$$ +G_p(t) = G_{p,0}(t) + G_{p,\text{rec}}. +$$ + +*Remark 2.2.* Note that even the particles that had no precollisions obey Assumption **(A0)** or, equivalently, the mathematical formulation in (2.17). This is why the effective integration domain in $G_{p,0}$ and $f_1(s,t)$ is $(-\infty, \underline{v}(0,t))$ instead of $(-\infty, p(t))$: the latter would allow the particles originally in front of the disk to fall behind the disk. + +3. PRELIMINARY BOUNDS + +For future convenience we define + +$$ +\alpha_n(s) = \frac{(2M^2s)^n}{n!}, \quad \alpha_0(s) = 1, \quad \alpha_{-1}(s) = 0. \tag{3.1} +$$ + +In this section we use the recurrence relation (2.27) to derive essential bounds on $f_n$ and its derivatives; they +are summarized in two propositions. + +**Proposition 3.1.** Let $j_n$ and $f_n$ be the iterative sequences given by (2.28) and (2.27)-(2.30) respectively. Let $M$ be the Lipschitz bound in Assumption **(A2)**. Then there exists a constant $Q_1$ that does not depend on $n$ such that for any $n \ge 1$ we have + +$$ +0 \le f_n(s,t) \le Q_1 \alpha_{n-1}(s) \quad \text{and} \quad 0 \le j_n(s) \le \frac{1}{2} Q_1 \alpha_n(s). \tag{3.2} +$$ + +Moreover, for each $s \in [0, t]$, the function $f_n(s, \cdot) \in C^1([0, T])$ with the bound + +$$ +\left| \frac{\partial}{\partial t} f_n(s,t) \right| \leq 4M^2 Q_1 \alpha_{n-1}(s). \quad (3.3) +$$ + +As a consequence, the function $f_{\text{rec}}$ defined in (2.6) satisfies + +$$ +0 \le f_{\text{rec}}(s,t) \le Q_1 e^{2M^2 s} \quad \text{and} \quad \left| \frac{\partial}{\partial t} f_{\text{rec}}(s,t) \right| \le 4M^2 Q_1 e^{2M^2 s}. \quad (3.4) +$$ + +Proof. First we derive the bound (3.2). For *n* = 1 we use the definition of *f*₁ in (2.30) to write + +$$ +0 \le f_1(s,t) \le 2 \int_{-\infty}^{\underline{v}(0,s)} (p(s)-v) \phi_0(v) dv \le 2 \int_{-\infty}^{M} (M-v) \phi_0(v) dv =: Q_1. +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_4.md b/samples/texts/5161315/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..43849743aca6404180d20bfe9090bb183dfa913d --- /dev/null +++ b/samples/texts/5161315/page_4.md @@ -0,0 +1,47 @@ +For $n \ge 2$ we apply (2.27) together with the definition of $\alpha_n$ in (3.1): + +$$ +\begin{align*} +f_n(s,t) &= 2e^{-(v(s,t)-p(s))^2} \int_0^s \frac{\partial v(\tau,s)}{\partial \tau} (p(s)-v(\tau,s)) f_{n-1}(\tau,s) \, d\tau \\ +&\le 2M^2 \int_0^s Q_1 \alpha_{n-2}(\tau) \, d\tau = Q_1 \alpha_{n-1}(s). +\end{align*} +$$ + +Note that the above step also gives the bound of $j_n$. Indeed, by its definition, + +$$ +j_n(s) \le \int_0^s \left| \frac{\partial v(\tau, s)}{\partial \tau} \right| |p(s) - v(\tau, s)| f_n(\tau, s) d\tau \le M^2 \int_0^s Q_1 \alpha_{n-1}(\tau) d\tau = \frac{1}{2} Q_1 \alpha_n(s). +$$ + +The bound (3.4) follows directly from the definition of $f_n$. Indeed, + +$$ +\left|\frac{\partial}{\partial t} f_n(s,t)\right| = 2 \left|\frac{\partial}{\partial t} e^{-(v(s,t)-p(s))^2} j_{n-1}(s)\right| = 2 |v(s,t) - p(s)| \left|\frac{\partial v(s,t)}{\partial t}\right| f_n(s,t) \le 4M^2 Q_1 \alpha_{n-1}(s). +$$ + +The estimates for $f_{\text{rec}}$ and $\partial_t f_{\text{rec}}$ follow by summing the bounds for $f_n$ and $\partial_t f_n$. $\square$ + +**Proposition 3.2.** For all $t \in [0,T]$ the function $f_n(s,t)$ is Lipschitz in $s$. As a result, it is almost everywhere differentiable in $s \in [0,t]$. Moreover, there exists a positive constant $Q_3$ that does not depend on $n$ such that + +$$ +\left| \frac{\partial f_{n+1}(s,t)}{\partial s} \right| \leq 3^n Q_3 (\alpha_n(s) + \alpha_{n-1}(s)), \quad n \geq 0, \tag{3.5} +$$ + +As a consequence, the function $f_{\text{rec}}$ defined in (2.6) satisfies + +$$ +\left| \frac{\partial f_{\text{rec}}(s,t)}{\partial s} \right| \leq 4Q_3 e^{6M^2 s}. \quad (3.6) +$$ + +Proof. Fix $t \in [0,T]$. For $n=0$, the definition of $f_1(s,t)$ in (2.30) shows it is Lipschitz in $s$ since $v(s,t)$, +$p(s)$, and $\underline{v}(0,s)$ are all Lipschitz in $s$. This allows us to obtain the desired bound by a direct calculation: + +$$ +\begin{align*} +\left| \frac{\partial}{\partial s} f_1(s,t) \right| &= \left| \frac{\partial}{\partial s} \left( 2e^{-(v(s,t)-p(s))^2} \int_{-\infty}^{\underline{v}(0,s)} (p(s)-v) \phi_0(v) dv \right) \right| \\ +&\leq \left| 2(v(s,t)-p(s)) \left( \frac{\partial v(s,t)}{\partial s} - \dot{p}(s) \right) \right| f_1(s,t) \\ +&\quad + 2e^{-(v(s,t)-p(s))^2} \left| \frac{\partial \underline{v}(0,s)}{\partial s} (p(s)-\underline{v}(0,s)) \right| \phi_0(\underline{v}(0,s)) \\ +&\quad + 2e^{-(v(s,t)-p(s))^2} \int_{-\infty}^{\underline{v}(0,s)} |\dot{p}(s)| \phi_0(v) dv \\ +&\leq 8M^2 Q_1 + 4M^2 \| \phi_0 \|_\infty + 2M \| \phi_0 \|_1 :=: Q_2. +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_5.md b/samples/texts/5161315/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..3d6a35f8a52cca28825cff3b19c24a849d28fc5d --- /dev/null +++ b/samples/texts/5161315/page_5.md @@ -0,0 +1,43 @@ +We now proceed by induction. Assume that the conclusion holds for $f_n$. Without loss of generality, assume $0 \le s < s' \le t$. Then + +$$ +\begin{align*} +j_n(s) - j_n(s') &= \int_0^s \frac{\partial v(\tau, s)}{\partial \tau} (p(s) - v(\tau, s)) f_n(\tau, s) \, d\tau - \int_0^{s'} \frac{\partial v(\tau, s')}{\partial \tau} (p(s') - v(\tau, s')) f_n(\tau, s') \, d\tau \\ +&= \int_{s'}^s \frac{\partial v(\tau, s)}{\partial \tau} (p(s) - v(\tau, s)) f_n(\tau, s) \, d\tau \\ +&\quad + \int_0^{s'} \left[ \frac{\partial v(\tau, s)}{\partial \tau} - \frac{\partial v(\tau, s')}{\partial \tau} \right] (p(s) - v(\tau, s)) f_n(\tau, s) \, d\tau \\ +&\quad + \int_0^{s'} \frac{\partial v(\tau, s')}{\partial \tau} [(p(s) - v(\tau, s)) - (p(s') - v(\tau, s'))] f_n(\tau, s) \, d\tau \\ +&\quad + \int_0^{s'} \frac{\partial v(\tau, s')}{\partial \tau} (p(s') - v(\tau, s')) [f_n(\tau, s) - f_n(\tau, s')] \, d\tau =: I_1 + I_2 + I_3 + I_4. +\end{align*} +$$ + +By Lemma 2.3(d) and Proposition 3.1, we obtain estimates of $I_1$, $I_3$ and $I_4$ as follows: + +$$ +\begin{align*} +|I_1| &= \left| \int_{s'}^s \frac{\partial v(\tau, s)}{\partial \tau} (p(s) - v(\tau, s)) f_n(\tau, s) \, d\tau \right| \le |s - s'| M^2 Q_1 \alpha_{n-1}(s), \\ +|I_3| &= \left| \int_0^{s'} \frac{\partial v(\tau, s')}{\partial \tau} [(p(s) - v(\tau, s)) - (p(s') - v(\tau, s'))] f_n(\tau, s) \, d\tau \right| \\ +&\le |s - s'| M^2 \int_0^{s'} Q_1 \alpha_{n-1}(\tau) \, d\tau \le |s - s'| \frac{Q_1}{2} \alpha_n(s'), +\end{align*} +$$ + +and + +$$ +\begin{align*} +|I_4| &= \left| \int_0^{s'} \frac{\partial v(\tau, s')}{\partial \tau} (p(s') - v(\tau, s')) [f_n(\tau, s) - f_n(\tau, s')] \, d\tau \right| \\ +&\le M^2 \int_0^{s'} |f_n(\tau, s) - f_n(\tau, s')| \, d\tau \\ +&\le M^2 |s - s'| \int_0^{s'} 4M^2 Q_1 \alpha_{n-1}(\tau) \, d\tau \\ +&\le |s - s'| 2M^2 Q_1 \alpha_n(s'). +\end{align*} +$$ + +To bound $I_2$ we note that by Lemma 2.3(d) and the induction hypothesis on $f_n$, the integrands $\underline{v}(\cdot, s)$, $\underline{v}(\cdot, s')$ and $(p(s) - v(\cdot, s))f_n(\cdot, s)$ are all Lipschitz. Hence we can integrate by parts and obtain + +$$ +\begin{align*} +I_2 &= \int_0^{s'} \left[ \frac{\partial v(\tau, s)}{\partial \tau} - \frac{\partial v(\tau, s')}{\partial \tau} \right] (p(s) - v(\tau, s)) f_n(\tau, s) \, d\tau \\ +&= \left[ (\underline{v}(\tau, s) - \underline{v}(\tau, s')) (p(s) - v(\tau, s)) f_n(\tau, s) \right]_{\tau=0}^{\tau=s} \\ +&\quad + \int_0^{s'} [\underline{v}(\tau, s) - \underline{v}(\tau, s')] \frac{\partial v(\tau, s)}{\partial \tau} f_n(\tau, s) \, d\tau \\ +&\quad - \int_0^{s'} [\underline{v}(\tau, s) - \underline{v}(\tau, s')] (p(s) - v(\tau, s)) \frac{\partial f_n(\tau, s)}{\partial \tau} \, d\tau. +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_6.md b/samples/texts/5161315/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..ed790319af1172713392304d72b0904cdd4fcd0d --- /dev/null +++ b/samples/texts/5161315/page_6.md @@ -0,0 +1,53 @@ +This gives the bound + +$$ +\begin{align*} +|I_2| & \le 2M|\underline{v}(0,s) - \underline{v}(0,s')|Q_1\delta_{1n} + \frac{M^2}{2}|s-s'| \int_0^{s'} Q_1\alpha_{n-1}(\tau) \, d\tau + 2|s-s'|M^2 \int_0^{s'} \left|\frac{\partial f_n(\tau,s)}{\partial\tau}\right| \, d\tau \\ +& \le 2|s-s'|M^2 Q_1\delta_{1n} + \frac{Q_1}{4}|s-s'|\alpha_n(s') + 2|s-s'|M^2 \int_0^{s'} \left|\frac{\partial f_n(\tau,s)}{\partial\tau}\right| \, d\tau, +\end{align*} +$$ + +where $\delta_{1n}$ is the Kronecker delta: $\delta_{1n} = 1$ when $n=1$ and vanishes otherwise. Combining the estimates for $I_1-I_4$, we have + +$$ +\frac{|j_n(s) - j_n(s')|}{|s - s'|} \leq Q_1 \alpha_n(s) \left(\frac{3}{4} + 2M^2\right) + 2M^2 Q_1 \alpha_{n-1}(s') + M^2 Q_1 \delta_{1n} + 2M^2 \int_0^{s'} \left|\frac{\partial f_n(\tau, s)}{\partial \tau}\right| d\tau. +$$ + +The right-hand side of the inequality above is bounded uniformly in $s$ and $s'$ since $\partial_\tau f_n(\tau, s) \in L^\infty(0, s)$ by +the induction assumption. Therefore, $j_n(s)$ is Lipschitz, and thus differentiable almost everywhere with + +$$ +\begin{aligned} +\left|\frac{\partial j_n(s)}{\partial s}\right| &= \lim_{s'\to s} \frac{|j_n(s) - j_n(s')|}{|s - s'|} \\ +&\le Q_1 \alpha_n(s) \left(\frac{3}{4} + 2M^2\right) + M^2 Q_1 \alpha_{n-1}(s) + 2M^2 Q_1 \delta_{1n} + 2M^2 \int_0^s \left|\frac{\partial f_n(\tau, s)}{\partial \tau}\right| d\tau. +\end{aligned} +$$ + +To derive the Lipschitz bound for $f_{n+1}$ we separate the two cases where $n=1$ and $n \ge 2$. For $n=1$ we have + +$$ +\begin{align*} +\left|\frac{\partial j_1(s)}{\partial s}\right| &\le Q_1 \alpha_1(s) \left(\frac{3}{4} + 2M^2\right) + M^2 Q_1 + 2M^2 Q_1 + 2M^2 \int_0^s Q_2 \, d\tau \\ +&\le \alpha_1(s) \left(\frac{3}{4} Q_1 + 2M^2 Q_1 + Q_2\right) + 3M^2 Q_1. +\end{align*} +$$ + +Applying the bound above in the definition of $f_2$ gives + +$$ +\begin{align*} +\left|\frac{\partial f_2(s,t)}{\partial s}\right| &= \left|\frac{\partial}{\partial s}\left(2e^{-(v(s,t)-p(s))^2} j_1(s)\right)\right| \\ +&\le 4|v(s,t)-p(s)| \left|\frac{\partial v(s,t)}{\partial s} - \dot{p}(s)\right| e^{-(v(s,t)-p(s))^2} |j_1(s)| + 2e^{-(v(s,t)-p(s))^2} \left|\frac{\partial j_1(s)}{\partial s}\right| \\ +&\le 6M^2 Q_1 + \alpha_1(s) \left(8M^2 Q_1 + 2\left(\frac{3}{4}Q_1 + 2M^2 Q_1 + Q_2\right)\right). +\end{align*} +$$ + +For $n \ge 2$, by using the bounds for $f_{n+1}$ and $\partial_s j_n$ we have + +$$ +\begin{align*} +\left|\frac{\partial f_{n+1}(s,t)}{\partial s}\right| &= \left|2\left(\frac{\partial}{\partial s}e^{-(v(s,t)-p(s))^2}\right)j_n(s) + 2e^{-(v(s,t)-p(s))^2}\frac{\partial}{\partial s}j_n(s)\right| \\ +&\le 2|v(s,t)-p(s)|\left|\frac{\partial v(s,t)}{\partial s}-\dot{p}(s)\right|f_{n+1}(s,t) + 2e^{-(v(s,t)-p(s))^2}\left|\frac{\partial}{\partial s}j_n(s)\right| \\ +&\le \alpha_n(s)Q_1\left(16M^2+\frac{3}{2}\right) + 2M^2Q_1\alpha_{n-1}(s) + 4M^2 \int_0^s \left|\frac{\partial f_n(\tau,s)}{\partial\tau}\right| d\tau. +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_7.md b/samples/texts/5161315/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..fd8eea1a27e51215303d156a6742e611bc3d01cc --- /dev/null +++ b/samples/texts/5161315/page_7.md @@ -0,0 +1,46 @@ +Using the induction assumption on $f_n$, the last integral term is bounded as + +$$ +\begin{aligned} +4M^2 \int_0^s \left| \frac{\partial f_n(\tau, s)}{\partial \tau} \right| d\tau &\le 4M^2 Q_3 3^{n-1} \int_0^s (\alpha_{n-1}(\tau) + \alpha_{n-2}(\tau)) d\tau \\ +&= 2 \cdot 3^{n-1} Q_3 (\alpha_{n-1}(s) + \alpha_n(s)). +\end{aligned} +$$ + +Hence, if we choose + +$$Q_3 = Q_1 \left( 16M^2 + \frac{3}{2} + 2M^2 Q_1 \right) + 2Q_2,$$ + +then for $n \ge 2$, we have + +$$\left|\frac{\partial f_{n+1}(s,t)}{\partial s}\right| \le Q_3 (\alpha_{n-1}(s) + \alpha_n(s)) + 2 \cdot 3^{n-1} Q_3 (\alpha_{n-1}(s) + \alpha_n(s)) = 3^n Q_3 (\alpha_{n-1}(s) + \alpha_n(s)),$$ + +which finishes the induction proof. Since $Q_3 > Q_2$, the bound above holds for $n = 0$ as well. + +The Lipschitz estimate (3.6) follows by summing the bounds (3.5): + +$$\left|\frac{\partial f_{\text{rec}}(s,t)}{\partial s}\right| \le \sum_{n=0}^{\infty} \left|\frac{\partial f_{n+1}(s,t)}{\partial s}\right| \le \sum_{n=0}^{\infty} 3^n Q_3 (\alpha_{n-1}(s) + \alpha_n(s)) = 4Q_3 e^{6M^2s}. \quad \square$$ + +### 4. PROOF OF UNIQUENESS + +In this section we prove the uniqueness theorem. An essential preliminary result is a Lipschitz bound for the density functions corresponding to different disk dynamics. Recall that $\|\cdot\|$ denotes the $L^\infty$-norm unless otherwise specified. We begin by showing that modified average velocity satisfies a Lipschitz bound + +**Lemma 4.1.** Let $p$ and $q$ be two Lipschitz velocity profiles and $\underline{v}_p$ and $\underline{v}_q$ be their associated modified velocities. Then + +$$|\underline{v}_p(s,t) - \underline{v}_q(s,t)| \le \|p-q\| \quad \text{for all } s,t.$$ + +*Proof.* For a fixed $s \in [0, t]$ and $t \in [0, T]$ suppose + +$$\underline{v}_p(s,t) = \langle p \rangle_{\tau_1, t} \quad \text{and} \quad \underline{v}_q(s,t) = \langle q \rangle_{\tau_2, t}.$$ + +Without loss of generality, assume that $\underline{v}_p(s,t) \ge \underline{v}_q(s,t)$. Then + +$$|\underline{v}_p(s,t) - \underline{v}_q(s,t)| = |\langle p \rangle_{\tau_1, t} - \langle q \rangle_{\tau_2, t}| \le |\langle p \rangle_{\tau_2, t} - \langle q \rangle_{\tau_2, t}| \le \|p-q\|. \quad \square$$ + +**Proposition 4.1.** Let $\{p, \eta^{(p)}, \{f_n^{(p)}\}_{n=1}^\infty, f_{\text{rec}}^{(p)}\}$ and $\{q, \eta^{(q)}, \{f_n^{(q)}\}_{n=1}^\infty, f_{\text{rec}}^{(q)}\}$ be two systems of disk-gas dynamics satisfying Assumptions **A0**-(**A2**). Then there exist a positive constant $Q_6$ that does not depend on $n$ such that the gas densities $\{f_n^{(p)}\}_{n=1}^\infty$ and $\{f_n^{(q)}\}_{n=1}^\infty$ satisfy the bound + +$$\left|f_{n+1}^{(p)}(s,t) - f_{n+1}^{(q)}(s,t)\right| \le 3^n Q_6 (\alpha_n(s) + \alpha_{n-1}(s)) \|p-q\|, \quad n \ge 0. \quad (4.1)$$ + +Consequently, for all $s \in [0, t]$ and $t \in [0, T]$ we have + +$$\left|f_{\text{rec}}^{(p)}(s,t) - f_{\text{rec}}^{(q)}(s,t)\right| \le 4Q_6 e^{6M^2s} \|p-q\|. \quad (4.2)$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_8.md b/samples/texts/5161315/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..1484ea9ffd70e14b31dfb561514ed2ed5bef32ce --- /dev/null +++ b/samples/texts/5161315/page_8.md @@ -0,0 +1,52 @@ +*Proof.* We show the bounds in (4.1) by a similar induction proof as for Proposition 3.2. First, the difference in density fluxes of $\phi_0$ satisfies + +$$ +\begin{align*} +|j_0^{(p)}(t) - j_0^{(q)}(t)| &= \left| \int_{-\infty}^{\underline{v}_p(0,t)} (p(t)-v)\phi_0(v) \,dv - \int_{-\infty}^{\underline{v}_q(0,t)} (q(t)-v)\phi_0(v) \,dv \right| \\ +&= \left| \int_{\underline{v}_q(0,t)}^{\underline{v}_p(0,s)} (p(t)-v)\phi_0(v) \,dv - \int_{-\infty}^{\underline{v}_q(0,t)} [(p(t)-v)-(q(t)-v)]\phi_0(v) \,dv \right| \\ +&\leq \|\underline{v}_q - \underline{v}_p\| \| \phi_0 \|_\infty + \|p-q\| \| \phi \|_1 = (\|\phi_0\|_\infty + \|\phi_0\|_1) \|p-q\|. +\end{align*} +$$ + +Therefore, we have + +$$ +\begin{align*} +|f_1^{(p)}(s,t) - f_1^{(q)}(s,t)| &= |2e^{-(v_p(s,t)-p(s))^2} j_0^{(p)}(s) - 2e^{-(v_q(s,t)-q(s))^2} j_0^{(q)}(s)| \\ +&\le 2 |e^{-(v_p(s,t)-p(s))^2} - e^{-(v_p(s,t)-p(s))^2}| |j_0^{(p)}(s)| + 2 |j_0^{(p)}(s) - j_0^{(q)}(s)| \\ +&\le Q_1 \|p-q\| + 2 \|p-q\| (\| \phi_0 \|_\infty + \| \phi_0 \|_1) \\ +&= \|p-q\| (Q_1 + 2\|\phi_0\|_\infty + 2\|\phi_0\|_1). +\end{align*} +$$ + +Thus by choosing $Q_4 = Q_1 + 2\|\phi_0\|_\infty + 2\|\phi_0\|_1$ we complete the proof for $n=1$. For $n \ge 1$ we have + +$$ +\begin{align*} +f_{n+1}^{(p)}(s,t) - f_{n+1}^{(q)}(s,t) &= 2e^{-(v(s,t)-p(s))^2} j_n^{(p)}(s) - 2e^{-(v(s,t)-p(s))^2} j_n^{(q)}(s) \\ +&\le 2 \left| e^{-(v_p(s,t)-p(s))^2} - e^{-(v_p(s,t)-p(s))^2} \right| |j_n^{(p)}(s) + 2 |j_n^{(p)}(s) - j_n^{(q)}(s)| \\ +&\le \|p-q\| \alpha_n(s) + 2 |j_n^{(p)}(s) - j_n^{(q)}(s)|. +\end{align*} +$$ + +We re-formulate the difference in density fluxes as + +$$ +\begin{align*} +j_n^{(p)}(t) - j_n^{(q)}(t) &= \int_0^t \frac{\partial v_p(s,t)}{\partial s} (p(t) - v_p(s,t)) f_n^{(p)}(s,t) ds - \int_0^t \frac{\partial v_q(s,t)}{\partial s} (q(t) - v_q(s,t)) f_n^{(q)}(s,t) ds \\ +&= \int_0^t \left[ \frac{\partial v_p(s,t)}{\partial s} - \frac{\partial v_q(s,t)}{\partial s} \right] (p(t) - v_p(s,t)) f_n^{(p)}(s,t) ds \\ +&\quad - \int_0^t \frac{\partial v_q(s,t)}{\partial s} [(p(t) - v_p(s,t)) - (q(t) - v_q(s,t))] f_n^{(p)}(s,t) ds \\ +&\quad - \int_0^t \frac{\partial v_q(s,t)}{\partial s} (q(t) - v_q(s,t)) [f_n^{(p)}(s,t) - f_n^{(q)}(s,t)] ds =: J_1 + J_2 + J_3. +\end{align*} +$$ + +By integration by parts and Proposition 3.2, we write $J_1$ as + +$$ +\begin{align} +J_1 &= \int_0^t \frac{\partial}{\partial s} [\underline{v}_p(s, t) - \underline{v}_q(s, t)] (p(t) - v_p(s, t)) f_n^{(p)}(s, t) ds \tag{4.3} \\ +&= \left[ (\underline{v}_p(s, t) - \underline{v}_q(s, t))(p(t) - v_p(s, t)) f_n^{(p)}(s, t) \right]_{s=0}^{s=t} \nonumber \\ +&\quad + \int_0^t \left[ \underline{v}_p(s, t) - \underline{v}_q(s, t) \right] \frac{\partial v_p(s, t)}{\partial s} f_n^{(p)}(s, t) ds \nonumber \\ +&\quad - \int_0^t \left[ \underline{v}_p(s, t) - \underline{v}_q(s, t) \right] (p(t) - v_p(s, t)) \frac{\partial f_n^{(p)}(s, t)}{\partial s} ds =: J_1^1 + J_1^2 - J_1^3. \nonumber +\end{align} +$$ \ No newline at end of file diff --git a/samples/texts/5161315/page_9.md b/samples/texts/5161315/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..538ac8b12bd0265aaafdc96568fe11a5e43d476b --- /dev/null +++ b/samples/texts/5161315/page_9.md @@ -0,0 +1,58 @@ +The boundary terms $J_1^1$ are only nonzero for $n=1$, so we write + +$$ +\begin{align*} +|J_1^1| &= \left| \left[ (\underline{v}_p(s,t) - \underline{v}_q(s,t))(p(t) - v_p(s,t))f_n^{(p)}(s,t) \right]_{s=0}^{s=t} \right| \\ +&= \left| (\underline{v}_q(0,t) - \underline{v}_p(0,t))(p(t) - v_p(0,t))f_n^{(p)}(0,t) \right| \le \|p-q\| 2MQ_1\delta_{1n}. +\end{align*} +$$ + +By Lemma 2.3, the second term $J_1^2$ satisfies + +$$ +|J_1^2| = \left| \int_0^t [\underline{v}_p(s,t) - \underline{v}_q(s,t)] \frac{\partial v_p(s,t)}{\partial s} f_n^{(p)}(s,t) ds \right| \leq \|p-q\| \frac{M}{2} \int_0^t Q_1 \alpha_{n-1}(s) ds = \|p-q\| \frac{Q_1}{4M} \alpha_n(t). +$$ + +Similarly, the third term $J_1^3$ is bounded as + +$$ +\begin{align*} +|J_1^3| &= \left| \int_0^t [\underline{v}_p(s,t) - \underline{v}_q(s,t)] (p(t) - v_p(s,t)) \frac{\partial f_n^{(p)}(s,t)}{\partial s} ds \right| \\ +&\le 2M \|p-q\| \int_0^t 3^{n-1} Q_3 (\alpha_{n-1}(s) + \alpha_{n-2}(s)) ds \\ +&= \|p-q\| \frac{3^{n-1} Q_3}{M} (\alpha_n(t) + \alpha_{n-1}(t)). +\end{align*} +$$ + +Combining estimates for $J_1^1$, $J_1^2$ and $J_1^3$ we get + +$$ +|J_1| \leq \frac{\|p-q\|}{M} \left( \frac{Q_1}{4} \alpha_n(t) + 3^{n-1} Q_3 (\alpha_n(t) + \alpha_{n-1}(t)) + 2M^2 Q_1 \delta_{1n} \right). +$$ + +The second term $J_2$ is bounded as + +$$ +\begin{align*} +|J_2| &= \left| \int_0^t \frac{\partial v_q(s,t)}{\partial s} [(p(t)-v_p(t)) - (q(t)-v_q(s,t))] f_n^{(p)}(s,t) ds \right| \\ +&\le \|p-q\| M \int_0^t Q_1 \alpha_{n-1}(s) ds = \|p-q\| \frac{Q_1}{2M} \alpha_n(t). +\end{align*} +$$ + +Using the induction assumption, we derive the bound for $J_3$ as + +$$ +\begin{align*} +|J_3| &= \left| \int_0^t \frac{\partial v_q(s,t)}{\partial s} (q(t) - v_q(s,t)) [f_n^{(p)}(s,t) - f_n^{(q)}(s,t)] ds \right| \\ +&\leq \|p-q\| M^2 \int_0^t 3^{n-1} Q_5 (\alpha_{n-1}(s) + \alpha_{n-2}(s)) ds \\ +&= \|p-q\| \frac{3^{n-1} Q_5}{2} (\alpha_n(t) + \alpha_{n-1}(t)). +\end{align*} +$$ + +Let $Q_5 = \frac{3Q_1}{2M} + 1 + \frac{2Q_3}{M} + 4MQ_1$. Then combining the estimates on $J_1$, $J_2$ and $J_3$ gives + +$$ +\begin{align*} +\left| f_{n+1}^{(p)}(s,t) - f_{n+1}^{(q)}(s,t) \right| &\le \|p-q\|\alpha_n(s) + 2(J_1 + J_2 + J_3) \\ +&\le \|p-q\|(2 \cdot 3^{n-1}Q_5(\alpha_n(s) + \alpha_{n-1}(s)) + Q_5\delta_{1n}); +\end{align*} +$$ \ No newline at end of file diff --git a/samples/texts/5378539/page_1.md b/samples/texts/5378539/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..2ea247ede659e8b351de5aa701b0f03badfbbc75 --- /dev/null +++ b/samples/texts/5378539/page_1.md @@ -0,0 +1,25 @@ +# A NOTE ON THE THEORY OF PRIMES + +J. E. SCHNEIDER + +In this paper we find those commutative rings for which the theory of primes is subsumed under classical ideal theory, that is, for which every finite prime is an ideal. The characterization is given in terms of domains with this property and they are shown to form a class of domains from number theory. In addition we give two characterizations of the primes of a subring of a global field. The first establishes them as the nontrivial preprimes whose complements are multiplicatively closed and the second relates the space of all primes to that of the quotient field. + +The concept of a prime for commutative rings with identity was introduced by Harrison in 1966. + +In what follows all rings are commutative and have a unity and all primes are finite. $X(R)$ denotes the set of primes of a ring $R$ and $X'(R)$ denotes the set of valuation preprimes (preprimes $T$ such that for each finite $E \subset R$, $T \cap E = \emptyset \Rightarrow$ there is $P \in X(R)$ with $T \subset P$ and $P \cap E = \emptyset$). For a preprime $T$ of $R$ which is closed under subtraction, define the idealizer $A(T)$ of $T$ in $R$ by $A(T) = \{a \in R : aT \subset T\}$. $A(T)$ is a subring of $R$ in which $T$ is an ideal. + +1. Call a ring a *C-ring* if every finite prime of it is an ideal. It is easy to check that the class of *C-rings* is closed under taking subrings and homomorphic images. + +**THEOREM 1.** The following are equivalent for a ring $R$: + +(1) $R$ is a *C-ring*; + +(2) $X(R) = \{\text{maximal ideals of } R\}$; + +(3) $R/P$ is a *C-ring*, for each minimal prime ideal $P$ of $R$; + +(4) $X'(R) = \text{Spec}(R)$. + +*Proof*. That (4) $\Rightarrow$ (2) $\Rightarrow$ (1) $\Rightarrow$ (3) is clear. In any case, $\text{Spec}(R) \subset X'(R)$ [1, Lemma 2.6]. Let $P \in X'(R)$. $P$ contains a minimal prime ideal $Q$ of $R$ and $P/Q \in X'(R/Q)$. Then $P/Q$ is the intersection of the primes of $R/Q$ which contain it; so, if $R/Q$ is a *C-ring*, then $P/Q \in \text{Spec}(R/Q)$ and $P \in \text{Spec}(R)$. + +Because of condition (3), we turn to the classification of *C-domains*. If $S$ denotes the ring of rational integers or a ring of polynomials in one variable over a finite field, then one checks that the polynomial \ No newline at end of file diff --git a/samples/texts/5378539/page_2.md b/samples/texts/5378539/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..b3aae2f81f616ac84a9075b8408903b55d0234e7 --- /dev/null +++ b/samples/texts/5378539/page_2.md @@ -0,0 +1,11 @@ +ring $S[X]$ has primes which are not ideals. Thus, since a subring of +a C-ring is also a C-ring, the transcendence degree of a C-domain is +zero or one, the latter only if it has nonzero characteristic. + +Let $R$ denote a domain whose quotient field $F'$ is absolutely algebraic and has characteristic zero. Let $R'$ denote the integral closure of $R$ and $S'$ the integral closure in $F$ of its prime subring. Let $S = R \cap S'$. $S'$ (and hence $S$) is a $C$-domain and $P \to P \cap S'$ gives a bijection from $X(F)$ onto $X(S')$. In fact, the finite dimentional case follows from Proposition 3.4 of [1] and the general case follows im-mediately from it. Since for $P \in X(F)$, $P \cap R'$ is an ideal of $R'$ if and only if $R' \subset A(P)$, we have that $R'$ is a $C$-domain if and only if $R' = S'$. Now $S' \subset R' \subset F$ yields that $P \to P \cap S'$ gives a bijection from $X(R')$ onto $X(S')$ and $P \to P \cap R'$ gives a bijection from $X(F)$ onto $X(R')$. Since $S \subset S'$ are both $C$-domains, $P \cap S \in X(S)$ for any $P \in X(R')$. + +LEMMA 2. If $P \in X(F)$, then $P \cap R \in X(R)$. + +*Proof*. Note that it suffices to consider the case where $F'$ is finite dimensional over its prime subfield. Moreover, since $P \cap R' \in X(R')$ for any $P \in X(F)$, as already noted, it suffices to show that $P \cap R \in X(R)$ for $P \in X(R')$. We may assume that $R \neq R'$, whence $S \neq S'$. Let $c(S)$ (resp. $c(R)$) denote the conductor of $S$ in $S'$ (resp. $R$ in $R'$). Recall that $(0) \neq c(S) \subset c(R)$. Now let $P \in X(R')$ and just suppose that $P \cap R$ is not in $X(R)$. Then $P \cap R \subset T \in X(R)$, since $P \cap R$ is a preprime of $R$. But $T$ is a preprime of $R'$, so $T \subset Q \in X(R')$. Then $P \cap R \subset T = Q \cap R$. Since $P \cap R \neq T$, we have $P \neq Q$ and $P \cap S' \neq Q \cap S'$. Now $P \cap S = P \cap R \cap S \subset Q \cap R \cap S = Q \cap S$. But $P \cap S' \in X(S')$ yields $P \cap S = (P \cap S') \cap S \in X(S)$. Thus $P \cap S = Q \cap S$. Hence, by [2, p. 91], $c(S) \subset P \cap Q$. By the approximation theorem, there is $\alpha \in F$ whose (normalized exponential) value is $-1$ at $P$, $+1$ at $Q$, and nonnegative otherwise. Note that $\alpha$ is in $R'$. Let $\beta \neq b \in c(S) \subset P \cap Q$ and let $\gamma$ be its (positive) value at $P$. Then $ba'' \in Q \cap S'$. Also $ba'' \in c(S)R' \subset c(R)R' \subset R$. Thus $ba'' \in Q \cap S' \cap R = Q \cap S = P \cap S$. But this a contradiction, since $ba''$ was constructed to have value zero at $P$. + +Now assume that $R$ is a $C$-domain. We will show that then $R'$ is a $C$-domain, so $R' = S'$ and $R$ is absolutely integral. Let $P \in X(R')$. We must show that $A(P) = R'$. If $R \neq R'$ we are done, so we assume that $R \neq R'$. By the lemma, $P \cap R \in X(R)$. But then $A(P) \cap R = A(P \cap R) = R$, so $R \subset A(P)$. Since $A(P)$ is integrally closed in $R'$, it is integrally closed in $F$. Then $R \subset A(P)$ yields $R' \subset A(P)$, whence $A(P) = R'$. \ No newline at end of file diff --git a/samples/texts/5378539/page_3.md b/samples/texts/5378539/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..ab50edde49afa08d741e053a01d3064b956fa9d2 --- /dev/null +++ b/samples/texts/5378539/page_3.md @@ -0,0 +1,9 @@ +We have shown that a domain of characteristic zero is a *C*-domain if and only if it is absolutely integral. An absolutely integral domain of nonzero characteristic is always a *C*-domain (such a domain has a unique prime and a unique proper ideal, namely (0)). We are faced with determining which domains of nonzero characteristic and transcendence degree one are *C*-domains. This case is handled in a fashion similar to the above case with the usual care necessary to handle a finite set of primes (those arising from the "infinite" valuations of the quotient field). If *R* is such a domain, *F* its quotient field, and *R'* its integral closure, then $P \to P \cap R'$ gives a bijection from *X*(*F*) onto *X*(*R*) or onto *X*(*R')* $\cup$ {(0)}, depending on whether there is not or is a unique valuation ring of *F* which does not contain *R*. For $P \in X(F)$, $P \cap R \in X(R)$ unless there is a unique valuation ring of *F* which does not contain *R* and *P* is its maximal ideal, in which case $P \cap R = (0)$. Now if *R* is a *C*-domain, then so is *R'.* But then *R'* (or equivalently *R*) is contained in all but one valuation ring of *F*. We omit the details. This completes the classification of *C*-domains. + +**THEOREM 3.** *R* is a *C*-domain if and only if *R* is absolutely integral or *R* has nonzero characteristic, has transcendence degree one, and is contained in all but one valuation ring of its quotient field. + +**REMARKS.** The hypothesis on the transcendence degree of $R$ is superfluous in Theorem 3, but we admit it to emphasize that $C$-domains lie in the realm of number theory. Since the Krull dimension of a $C$-domain is zero or one, the same is true for $C$-rings. Using Proposition 2.11 of [1], it is easy to check that $X(R) = \text{Spec}(R)$ if and only if $R/\sqrt{0}$ is a generalized Boolean ring. Theorem 1 answers a question arising in the theory of valuations of commutative rings introduced by Manis [3], namely, every valuation of a commutative ring is trivial if and only if it is a $C$-ring. + +2. Call a domain a *global ring* if its quotient field is a global field, that is, a finitely generated field of adjusted transcendence degree one (equals transcendence degree plus one or zero depending on whether the characteristic is zero or not). We seek to characterize the primes of a global ring. Since the infinite primes are easily seen to arise in the same way and correspond exactly to the infinite primes of its quotient field [1, Propositions 3.5 and 3.6], we will consider only finite primes. + +Let $R$ denote any domain of adjusted transcendence degree one \ No newline at end of file diff --git a/samples/texts/5378539/page_4.md b/samples/texts/5378539/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..97525b906af1896ccef65b6e82799dae595fa695 --- /dev/null +++ b/samples/texts/5378539/page_4.md @@ -0,0 +1,13 @@ +and $F$ its quotient field. Let $T$ denote a nonzero preprime of $R$ whose complement in $R$ is multiplicatively closed. Then $T$ is closed under subtraction, $A = A(T)$ also has adjusted transcendence degree one, and the complement of $T$ in $A$ is multiplicatively closed. Thus $T$ is a maximal ideal of $A$ and a (finite) prime of $A$. We seek to show that $T$ is a prime of $R$. We may assume that $A \neq R$, i.e., that $T$ is not an ideal of $R$. Let $S$ and $B$ denote the integral closures in $F$ of $R$ and $A$ respectively. Then $B \subset S$, $R \cap B = A$, and $B \neq S$. + +Let $E(R)$ denote the set of valuation rings of $F$ which do not contain $R$. Since $B \subset S$ and $B \neq S$, $E(R)$ is not empty. It suffices to consider the case where $R$ is a global ring and $E(R)$ is finite. In fact, assume that $T$ is properly contained in a prime $P$ of $R$. Let $a \in P$, $a \notin T$, and $0 \neq b \in T$. Let $R_1$ denote the subring of $R$ generated by $a$ and $b$. Then $R_1$ is a global ring (otherwise $\alpha$ is a root of unity and $1 \in P$). $E(R_1)$ is the set of valuation rings of the quotient field of $R_1$ which exclude $\alpha$ or $b$, so $E(R_1)$ is finite. $T \cap R_1$ is a nonzero preprime of $R_1$ whose complement in $R_1$ is multiplicatively closed, and $T \cap R_1$ is properly contained in the preprime $P \cap R_1$. + +We omit the proof of the following lemma since it is a straightforward application of the approximation theorem and the fact that the conductor of a global ring in its integral closure is not zero. + +**LEMMA 4.** Let $L$ denote a global ring and $V_1, \dots, V_n$ a set of valuation rings of the quotient field of $L$ which do not contain $L$. Then $L \cap V_1 \cap \dots \cap V_n$ is an irredundant intersection. + +Let $M = M(T, R) = \{V \in E(R) : A \subset V\}$. Since $B \neq S$, $M$ is not empty. Let $M = \{V_1, \dots, V_n\}$. Then $B = S \cap V_1 \cap \dots \cap V_n$ and $A = R \cap V_1 \cap \dots \cap V_n$. Let $P_i$ denote the maximal ideal of $V_i \in M$ and let $N = \{P_1, \dots, P_n\}$. Recall that $N \subset X(F)$. + +**LEMMA 5.** *Let* $T = A \cap P$, *for* $P \in N$. + +*Proof.* Just suppose that $T \neq A \cap P$, for all $P \in N$. Let $Q \in X(F)$ with $T \subset Q$. Since $T = A \cap Q$ and $T$ is an ideal and a prime of $A$, $A \subset A(Q)$. $Q \in N$, so $A(Q) \subset E(R)$. Thus $R \subset A(Q)$ and $R \cap Q$ is a maximal ideal of $R$. Since $T$ is not an ideal of $R$, $R \cap Q \neq T = A \cap Q$. Let $b \in R \cap Q$ with $b \in A \cap Q$. Now, for $P_i \in N$, $A \cap P_i$ is a maximal ideal of $A$ and since $A \cap P_i \neq T = A \cap Q$, there is $a_i \in A \cap P_i$ with $a_i \in A \cap Q$. Let $\alpha = a_1a_2\cdots a_n$. Let $\omega$ (resp. $\nu_i$) denote the normalized exponential valuation of $F$ associated with $Q$ (resp. $P_i$). Then $\omega(b) > 0$, $\omega(a) = 0$, and $\nu_i(a) > 0$. We fix $m > 0$ so that $\nu_i(a^m b) > 0$, for $1 \le i \le n$. Then $a^m b \in R \cap V_1 \cap \cdots \cap V_n \cap Q = A \cap Q = T$. But \ No newline at end of file diff --git a/samples/texts/5378539/page_5.md b/samples/texts/5378539/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..a884b63e1eef86611bd14876e6ee03d66d67d8eb --- /dev/null +++ b/samples/texts/5378539/page_5.md @@ -0,0 +1,16 @@ +$a, b \in R$, so $a^m b \in T$ implies that $a \in T$ or $b \in T$. By this contradiction +$T = A \cap P$, for some $P \in N$. + +Now reindex *N* if necessary so that *T* = *A* ∩ *P*ᵢ for 1 ≤ *i* ≤ *s*, and *T* ≠ *A* ∩ *P*ᵢ for *s* < *i* ≤ *n*. Just suppose *s* < *n*. Let *R*₁ = *R* ∩ *V*₁ ∩ ... ∩ *V*ₙ. *T* is a preprime of *R*₁ with multiplicatively closed complement in *R*₁. By Lemma 4, *A* ≠ *R*₁ and *M*(*T*, *R*₁) = {*V*₀+₁, ..., *V*ₙ}. Applying the first part of this proof to *T* and *R*₁, we get *T* = *R*₁ ∩ *P*ᵢ, for some *s* < *i* ≤ *n*. But then *T* = *T* ∩ *R* = *R* ∩ *P*ᵢ, a contradiction. + +LEMMA 6. Let L denote a global ring, T a nonzero preprime of L with a multiplicatively closed complement in L, and A the idealizer of T in L. Assume that E(L) is finite and that M(T, L) = {V}. Then T ∈ X(L). + +Proof. Let $P$ denote the maximal ideal of $V$. Then Lemma 5 yields $T = A \cap P$. But $M(T,L) = \{V\}$ yields $A = L \cap V$. Thus $T = L \cap V \cap P = L \cap P$. But if $L$ has characteristic zero, Lemma 2 yields $T = L \cap P \in X(L)$. If $L$ has nonzero characteristic, we consider $E(L)$. At least $V \in E(L)$. If $E(L) = \{V\}$, then $A = L \cap V$ is a finite field and $T = (0)$. Hence $E(L)$ is not a singleton and $T = L \cap P \in X(L)$. + +We can now give the first characterization. + +PROPOSITION 7. Let $R$ be a domain of adjusted transcendence degree one. A necessary and sufficient condition that a nonzero preprime of $R$ be a prime is that its complement be multiplicatively closed. + +Proof. Necessity holds for arbitrary commutative rings. [1, Proposition 2.1]. To prove the sufficiency, we have already noted that we may assume that $R$ is a global ring, that $E(R)$ is finite, and that the idealizer $A$ of $T$ in $R$ is not $R$. In the notation established above, $M(T, R) = \{V_1, \dots, V_n\}$. By Lemma 6, we need only show that $n=1$. Just suppose $n \ge 2$. Let $L = R \cap V_3 \cap \cdots \cap V_n$. Then $A = L \cap V_1 \cap V_2$. Let $R_1 = L \cap V_1$, so $A = R_1 \cap V_2$. By Lemma 6 applied to $R_1$ and $T$, we have that $T \in X(R_1)$; and by Lemma 5, $T = A \cap P_1 \subset R_1 \cap P_1$, so $T = P_1 \cap R_1$. Then, since $A$ is the idealizer of $T$ in $R_1$, $A = V_1 \cap R_1 = R_1$, in contradiction to Lemma 4. + +For a commutative ring $R$, $X(R)$ is topologized by taking the sets $U(a) = \{P \in X(R) : a \in P\}$, for all $a \in R$, as subbasic open sets. Let $R$ denote a global ring and $F$ its quotient field. We fix an element $x \in R$ so that $x$ is not absolutely algebraic, if $\text{char}(R) \neq 0$, and $x = 1$, otherwise. Let $F_0$ (resp. $R_0$) denote the subfield (resp. subring) of $F$ generated by $x$. Let $V = U(1/x) \subset X(F)$. Let $S$ (resp. $S_0$) denote the \ No newline at end of file diff --git a/samples/texts/5378539/page_6.md b/samples/texts/5378539/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..72ef5ab4442b1d24bfb8592eb9733ef2fbfcd099 --- /dev/null +++ b/samples/texts/5378539/page_6.md @@ -0,0 +1,31 @@ +integral closure of $R$ (resp. $R_0$) in $F$. We fix $c \in R_0$ with $cS \subset R$ and +$c \neq 0$. Note that for $P \in V$, $x \in A(P)$; so $c \in R_0 \subset A(P)$. + +For commutative rings $K \subset L$, let $\pi(L/K)$ denote the map from $X(L)$ to the power set of $K$ given by $\pi(L/K)(P) = P \cap K$. We have noted that $\pi(F/S_0)$ restricted to $V$ is a bijection onto the set of maximal ideals of $S_0$, and $\pi(F_0/R_0)$ restricted to $\{P \in X(F_0) : 1/x \in P\}$ is a bijection onto $X(R_0)$[cf. 1, Proposition 3.4 ff.]. Thus $S_0 \subset S \subset F$ implies that $\pi(F/S)$ restricted to $V$ is injective and that for $P \in V$, $\pi(F/S)(P) \neq (0)$. Then by proposition 7, $\pi(F/S)(V) \subset X(S)$. Note that, for $P, Q \in X(F)$ with $P \cap R = Q \cap R$, either both $P$ and $Q$ are in $V$ or both are not. In fact, $P \in V$ if and only if $P \cap R_0 \neq (0)$. + +LEMMA 8. Let $P, Q \in V$. If $P \not= Q$ and $P \cap R = Q \cap R$, then $c \in P \cap Q$. + +*Proof*. Let $T = (P \cap S) + (Q \cap S) - (P \cap S)(Q \cap S)$. $T$ is closed under addition and multiplication, but $T$ properly contains $P \cap S \in X(S)$. Thus $T$ is not a preprime of $S$ and hence $1 \in T$. Then $c \in cT \subset P \cap Q$. + +Since $X(F)$ is cofinite space, $C = \{P \in X(F) : 1/x \in P \text{ or } c/x \in P\}$ is +finite. By Lemma 8, $\pi(F/R)$ restricted to the complement $C'$ of $C$ +is injective. $X(R)$ lies in the range of $\pi(F/R)$, and we may choose a +subset $D$ of $C$ on which $\pi(F/R)$ is injective and so that $\pi(F/R)(D) =$ +$\pi(F/R)(C)$. Then $D \cup C'$ is a cofinite set and an open set, and $\pi(F/R)$ +restricted to $D \cup C'$ is a bijection onto $X(R)$. Since $X(F)$ and $X(R)$ +are cofinite spaces, this map is a homeomorphism. We have proven. + +PROPOSITION 9. Let $R$ denote a global ring and $F$ its quotient field. Then $X(R)$ is homeomorphic to an open (cofinite) subset of $X(F)$ and the homeomorphism is induced by $\pi(F/R)$. + +REFERENCES + +1. D. K. Harrison, *Finite and infinite primes for rings and fields*, Memoirs Amer. Math. Soc. **68** (1966). + +2. W. Krull, *Idealtheorie*, Springer, Berlin, 1935. + +3. M. E. Manis, *Extension of valuation theory*, announced in Bull. Amer. Math. Soc. **73** (1967), p. 735, and to appear. + +Received December 1968. This research was completed while the author held an NDEA Fellowship at the University of Oregon. + +THE PENNSYLVANIA STATE UNIVERSITY +UNIVERSITY PARK, PENNSYLVANIA \ No newline at end of file diff --git a/samples/texts/6192125/page_4.md b/samples/texts/6192125/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..84b850d0faa9c8cac991ceb7741a60becb2a2cbc --- /dev/null +++ b/samples/texts/6192125/page_4.md @@ -0,0 +1,17 @@ +we compute the spin gap for $\omega$ around $\omega_c \approx 1.5$, as shown in Fig. 4(c). The spin gap is finite in the VBS regime, but it is extrapolated to zero in the AF regime $\omega > \omega_c$, indicating the emergence of gapless spin-wave excitations as Goldstone modes of spin SU(2) symmetry breaking in the AF phase. Taken together, these results convincingly show that the occurrence of phonon-induced AF long-range order for $\omega > \omega_c$, where $\omega_c$ depends on $\lambda$. + +Evidences of AF ordering at $\omega > \omega_c(\lambda)$ are also obtained for various other EPC dimensionless parameters $\lambda$, from weak to strong, as plotted in Fig. 1. As the critical frequency $\omega_c(\lambda)$ increases monotonically with increasing $\lambda$, for a fixed frequency it is expected that the AF phases should emerge in the regime of $\lambda < \lambda_c$ where $\lambda_c$ is the critical EPC constant. Indeed, for the fixed frequency $\omega = 1.0$, AF ordering is observed in the regime of $\lambda < \lambda_c \approx 0.18$ from the crossing of the AF susceptibility ratio for different $L$, as shown in Fig. 4(b). The $(\pi, \pi)$ AF ordering fully gaps out the Fermi surface such that the ground state is an AF insulator for $\lambda < \lambda_c$. As mentioned earlier, the optical SSH model at half-filling respects the O(4) symmetry, giving rise to the degeneracy between spin AF and pseudospin AF (namely CDW/SC). The degeneracy can be lifted and spin AF is more favored by turning on a weak repulsive Hubbard interaction, as shown in the QMC simulations of models with a weak Hubbard interaction (see the SM for details). + +It is worth to understand heuristically why AF ordering emerges for small $\lambda$. For sufficiently small $\lambda$, one can treat electron-phonon coupling term $g$ as a weak perturbation and the second-order process in $g$ would generate a spin exchange process when the spin polarizations in the NN sites are opposite, as shown in Fig. 4(d). If the two spins on NN sites are parallel (namely forming a triplet), the exchange process is not allowed. Since this second-order spin-exchange process can gain energy, the spin-exchange interaction is antiferromagnetic. This provides a phonon mechanism to drive AF ordering, which is qualitatively different from the usual AF exchange mechanism of strong Hubbard Coulomb interaction. + +Note that AF ordering was not observed in an earlier QMC study of the 2D optical SSH phonon model [85]. There the absence of AF ordering is possibly due to the fact that the QMC simulations were at finite temperature and spin-SU(2) rotational symmetry in 2D cannot be spontaneously broken at any finite temperature. In contrast, we performed zero-temperature QMC simulations which can directly access properties of the ground state of the 2D phonon model and observe a spontaneous spin-SU(2) symmetry breaking. + +We have shown evidences of a direct QPT between the AF and VBS phases. It is natural to ask if the direct QPT between AF and VBS phases here is first order or continuous. Since AF and VBS phases break totally different symmetries, the QPT between them is putatively first-order in the Landau paradigm although it would + +be intriguing to explore if a deconfined quantum critical point (DQCP) [100, 101] occurs in this case. The phenomena of DQCP have been extensively studied for QPTs between Neel AF and columnar VBS [102–112]. More recently, it has been argued from duality relations that, at such transition point, the SO(5) symmetry might emerge at low energy [113–118]. However, the VBS order in the optical SSH phonon model studied here is the staggered one, for which the VBS $Z_4$ vortex is featureless, namely not carrying a spinon [119, 120]. Consequently, a (possibly weak) first-order transition instead of DQCP [121] would be expected for the QPT between AF and staggered VBS phases in the phonon model under study. + +**Conclusions and discussions:** We have systematically explored the ground-state phase diagram of the 2D optical SSH model taking account of full quantum phonon dynamics by zero-temperature QMC simulations. Remarkably, from the state-of-the-art numerically-exact simulations, we have shown that the optical SSH phonons can induce a Neel AF order when the phonon frequency is larger than a critical value ($\omega > \omega_c$) or the EPC constant is smaller than a critical value ($\lambda < \lambda_c$). The critical frequency $\omega_c$ can be much smaller than the band width $W$ for weak or moderate EPC constant $\lambda$, which makes the phonon mechanism of AF ordering practically feasible in realistic quantum materials. For instance, for the optical SSH model on the square lattice, we obtained $\omega_c/W \sim 0.1$ when $\lambda \approx 0.15$. + +As mentioned above, the role of EPC in understanding the physics of strongly correlated materials, including cuprate and Fe-based high-temperature superconductors, has attracted increasing attentions. We believe that our finding of optical SSH phonon induced AF order may shed new light on understanding the cooperative effects of electronic correlations and EPC on the nature of AF Mott physics. Following this, a natural question to ask is whether such SSH phonon can have crucial effect on unconventional superconductivity arising from doping an AF insulating phase [18–23]. In a follow-up work [122], we shall present evidences that quantum SSH optical phonons can substantially enhance the d-wave pairing. We believe that these findings pave an important step to understanding the interplay of EPC and electronic correlations in strongly correlated materials including high-temperature superconductors. + +*Acknowledgement.*—We would like to thank Steve Kivelson, Dung-Hai Lee, and Yoni Schattner for helpful discussions. This work is supported in part by the NSFC under Grant No. 11825404 (XC and HY), the MOSTC under Grant Nos. 2016YFA0301001 and 2018YFA0305604 (HY), the CAS Strategic Priority Research Program under Grant No. XDB28000000 (HY), Beijing Municipal Science and Technology Commission under Grant No. Z181100004218001 (HY), and the Gordon and Betty Moore Foundation’s EPiQS under Grant No. GBMF4545 (ZXL). \ No newline at end of file diff --git a/samples/texts/6192125/page_5.md b/samples/texts/6192125/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..4383d57da8b4bbbf7629bdc7b01d6c707f0bc482 --- /dev/null +++ b/samples/texts/6192125/page_5.md @@ -0,0 +1,89 @@ +* yaohong@tsinghua.edu.cn + +[1] R. E. Peierls, *Quantum Theory of Solids* (Oxford University, New York/London, 1955). + +[2] G. Grüner, *Rev. Mod. Phys.* **60**, 1129 (1988). + +[3] W. P. Su, J. R. Schrieffer, and A. J. Heeger, *Phys. Rev. Lett.* **42**, 1698 (1979). + +[4] A. J. Heeger, S. A. Kivelson, J. R. 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We calculate the spatial quantum correlations +that are created when the system is in a superposition of +such many-body Fock states. By splitting the system in +two partitions we calculate analytically and numerically +the reduced density matrix and the bipartite entangle- +ment, which is a way to quantify the many-body correla- +tions [6, 7]. Using this method, we show that the strength +of entanglement varies according to the spatial freedom of +the particles, determined by the system filling. In addition +edge modes appear at the ends of the system, when open +boundary conditions are considered, that can be corre- +lated with each other due to the entanglement. In overall, +we demonstrate a simple mechanism to create entangled +states in the strong interaction or the charge density wave +(CDW) limit of many-body systems, with controllable en- +tanglement strength. + +**2 Superpositions of Fock states with spatial constraints** + +For our study, we consider collections of point-like(localized) +particles arranged in different spatial configurations. A few +examples of such states can be seen in Fig. 1(a). These +states are known also, as charge density wave (CDW) +states that can carry collectively charge current through +the system. The following spatial constraints in the self or- +ganization of the particles are assumed. Firstly, only one +particle is allowed per site in an 1D chain of empty sites, +as in Hubbard models of hard-core bosons. Following this +rule there are many different ways to arrange the particles, +according to the empty space in the system. If no interac- +tion between the particles is assumed, then all the possible +configurations allowed will occupy the same energy E=0. +In order to split energetically these configurations, we can +add an additional spatial constraint, a nearest neighbor +interaction between the particles. This can be expressed +by a Hubbard-like Hamiltonian term in a chain with M +sites, + +$$ +H_U = U \sum_{i=1}^{M-1} n_i n_{i+1} \quad (1) +$$ + +where $n_i = c_i^\dagger c_i$ is the number operator, with $c_i^\dagger$ being the creation and annihilation operators for spin-less particles at site i. The interaction strength U is measured in units of eV and can be either positive or negative for repulsive or attractive interaction respectively. The particles interact with strength U only when they are occupying neighboring sites in the chain. We consider that the Hubbard chain terminates at sites 1 and M with hard-wall/open boundary conditions. The ratio of the number of particles N over the number of sites in the Hubbard chain M, determines the filling $f = N/M$ of the system. As we shall show the spatial freedom of the particles, at different fillings, can lead to surprisingly complex behaviors, even for this simple short-range interacting model. We restrict our study on many-body wavefunctions that + +are symmetric under exchange of two particles. In addi- +tion we have assumed that the particles cannot occupy +the same quantum state, i.e., only one particle is allowed +per site (occupation number $n_i$ can be either 0 or 1). This +is the case of the hardcore bosons[36,37,38,39] that can +be realized in cold atom and helium-4 systems experimen- +tally [31, 40, 41, 42]. The hardcore bosons satisfy the com- +mutation relation [$c_i, c_j^\dagger] = (1 - 2n_i)\delta_{ij}$. In order to char- +acterize the many-body states in the text we use 1(0) to +denote occupied(unoccupied) sites of the Hubbard chain. +Schematically we represent it with filledGORILLA circles. + +In the rest of the paper we focus on the repulsive inter- +action case $U > 0$, although we present some brief analysis +of the attractive interaction case $U < 0$ in section 5. The +eigenstates for $U > 0$ and $f \le 1/2$ are determined by the +possible Fock microstates that correspond to the discrete +energies $E = 0, U, 2U, \dots, (N-2)U, (N-1)U$. Each energy +contains states with different microstructures. + +In overall we can see that the interaction between the +particles splits the Hilbert space of the non-interacting +system (U=0) that contains all possible Fock states, in +subspaces containing states according to the different par- +ticle configurations allowed at each energy. For example +the first excited states contain at least one pair of par- +ticles at adjacent sites, for instance the state $|10110100\rangle$. +This state has energy $E = U$. In this paper we focus on the +ground states with energy $E = 0$ where all particles are +separated by at least one empty site, for example like the +state $|10101010\rangle$ as shown in Fig. 1(a). These are many- +body Fock states where neighboring sites can never be si- +multaneously occupied. This is valid when $f \le 1/2$ which +concerns all the results we present in our paper. + +The degree of degeneracy of the Fock states depends on +the filling $f=N/M$, which determines the spatial freedom +of the interacting particles. The number of allowed particle +configurations/microstates can be expressed mathemati- +cally with factorials via combinatorics, in the binomial +form + +$$ +D(N, M) = \binom{M-N+1}{N}, \qquad (2) +$$ + +for the states that have at least one empty site between +all the particles. + +If there are *N* particles in a system of *M* = 2*N* sites, +for half filling *f* = 1/2, then from Eq. (2) the number +of degenerate ground states is *N*+1. This can be seen in +Fig. 1(a) where we show a schematic representation of the +different states for a half-filled system with *N* = 4. Be- +low half-filling (*f* < 1/2), the particles are spatially less +restricted and therefore the number of ground states in- +creases. As we shall show this degree of spatial freedom +affects the entanglement properties of the ground state. +We remark that such many-body states can be realized +experimentally in cold-atomic systems[32]. + +The simplest way to construct the respective many- +body wavefunction, is to assume a linear superposition of \ No newline at end of file diff --git a/samples/texts/6720294/page_3.md b/samples/texts/6720294/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..0435e8f25080cad8802fe50b5e50eb80e1bb0650 --- /dev/null +++ b/samples/texts/6720294/page_3.md @@ -0,0 +1,43 @@ +the degenerate many-body states, + +$$ +|\Psi_G\rangle = \frac{1}{\sqrt{D}} \sum_{i_1=1}^{M_1} \prod_{n=2}^{N} \sum_{i_n=i_{n-1}+1}^{M_n} [1 - \delta_{i_n, i_{n-1}+1}] |\{i_1 \dots i_n\}\rangle. \quad (3) +$$ + +We have $M_n = M - N + n$ while index $i_n$ denotes the occupied sites in the Hubbard chain. The list $\{i_1, \dots, i_n\}$ has length $N$ and determines a Fock state $|\{i_1 \dots i_n\}\rangle$. This type of superposition with equal amplitudes for each microstate is a reasonably good approximation when the microstates are degenerate or nearly degenerate. A similar wavefunction can be used for describing the states of ferromagnetic spin chains with periodic boundary conditions (PBC)[23]. In addition Eq. (3) is a physically acceptable wavefunction for the system described by Eq. (1) since it results in spatially mirror symmetric particle density, which can be defined as, + +$$ +\overline{n_i} = \langle \Psi_G | n_i | \Psi_G \rangle . \tag{4} +$$ + +The wavefunction Eq. (3) satisfies $\overline{n_i} = \overline{n_{M-i+1}}$. Eq. (3) can be thought as an explicit ansatz for the ground state wave function of Eq. (1). + +Other statistical mixtures giving different amplitudes are also possible instead of Eq. (3). For example, the amplitudes will be modified when considering a hopping term of the type + +$$ +H_t = t_1 \sum_{i=1}^{M-1} c_i^\dagger c_{i+1} + t_2 \sum_{i=1}^{M-2} c_i^\dagger c_{i+2} + h.c. \quad (5) +$$ + +in addition to the interaction term Eq. (1), where $t_1$ is a nearest-neighbor hopping and $t_2$ is a next nearest-neighbor hopping. Note that in general there will be strong mixture between the Fock states at E=0 and the excited states, when adding this type of hopping term. However we expect the states studied in our paper to be a good approximation for strong interaction strength $U \gg t$. + +The effect of a weak first or second nearest hopping on the ground states of Eq. (1) can be understood perturbatively. Consider the half filled case. First nearest neighbor hopping will allow transitions between the Fock states, creating a small dispersion that will lift their degeneracy. For example, acting with a hopping term $c_6^\dagger c_5$ on the ground state $|10101010\rangle$ results in the state $|10101001\rangle$, as can be seen in Fig. 1(b). These degenerate ground states can be thought as the different sites in a tight-binding chain. A hopping between the sites will create a dispersion in the energy spectrum of the chain, generating a band structure, which is equivalent to lifting the degeneracy of the ground states for $t_1 = 0$. On the other hand acting with a second nearest neighbor hopping $c_i^\dagger c_{i+2}$ on the ground states, for any $i$, will result only in excited states, which contain at least a pair of particles occupying neighboring sites, contributing energy U, the energy gap that separates them from the ground states. Therefore the degeneracy of the ground states will not be lifted when adding weak second nearest neighbor hopping. Also, the particle configurations for each ground state will be preserved. Therefore, if the second nearest neighbor $t_2$ is + +considered only, then the Fock states remain degenerate at energy $E=0$ for half filling. This mechanism is demonstrated schematically in Fig. 1(b). + +We have verified the above results by applying degenerate first order perturbation theory. We have found that the energy of the perturbed system with first nearest neighbor hopping will be, + +$$ +E_G(j) \sim E_G^0 + 2t_1 \cos \left( \frac{\pi j}{D+1} \right), \quad (6) +$$ + +where D is the degeneracy (Eq. (2)) and j is an integer taking values j=1,2..D, for each of the perturbed ground states. This is the energy dispersion of a tight-binding chain with D sites and hardwall boundary conditions. Moreover each of these perturbed ground states can be written as linear combination of the unperturbed ones. The amplitudes are given by the corresponding wavefunction for a state j in the tight-binding chain, + +$$ +|\Psi_G(j)\rangle = \sqrt{\frac{2}{D+1}} \sum_{x=1}^{D} \sin\left(\frac{\pi j x}{D+1}\right) |x\rangle, \quad (7) +$$ + +where $|x\rangle$ is the lattice site, representing each of the unperturbed states $\{|i_1...i_n\}\}$ in the linear combination Eq. (3), running over the degenerate space of dimension D. Both results above are well known solutions for a XXZ spin chain, to which our model can be mapped at half-filling[43]. + +In order to check the effect of the hopping term on the ground state degeneracy, we have calculated the energy spectrum of the Hamiltonian including both the interaction term Eq. (1) and the hopping term Eq. (5), using numerical diagonalization, for a system with N=6 and U = 2 and for different fillings. The eigenvalues can be seen in Fig. 2. As we have argued above, for half-filling ($f=1/2$) the degeneracy is preserved for $t_1 = 0$ eV, $t_2 = 0.1$ eV while it is lifted for $t_1 = 0.1$ eV, $t_2 = 0$ eV. Although the degeneracy is lifted also for lower fillings ($f=1/3$), the energy gap from the excited states does not close. Note that the hopping we considered causes mixing of the ground states with the excited states unless $U \gg t_1, t_2$. + +In our analysis we consider a superposition of Fock states described by Eq. (3), which are the ground states of Eq. (1). We assume that an infinitesimally small hopping ($|U| \gg t$), for example one that does not create clustering of the particles, will not affect the micro-structures of these states, but might lift their degeneracy, as we have shown above. As long as the gap from the excited states at energy $E=U$ does not close, we can consider these nearly degenerate ground states as an isolated Hilbert space and study its entanglement properties by using the wavefunction Eq. (3) or Eq. (7). Our approach can be thought as a good approximation to estimate the entanglement properties, when spatial restrictions are imposed in the self-organization of the particles via different interaction terms. We expect that for a sufficiently strong nearest neighbor interaction term, the major contribution in the ground state to come from Fock states where neighboring \ No newline at end of file diff --git a/samples/texts/6720294/page_6.md b/samples/texts/6720294/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..a833121d46d96821d2883bc691aaaf27ef8f7057 --- /dev/null +++ b/samples/texts/6720294/page_6.md @@ -0,0 +1,15 @@ +**Fig. 2.** The energy levels for repulsive ($U > 0$) and attractive ($U < 0$) interaction with first ($t_1$) or second ($t_2$) nearest neighbor hopping. In all cases the ground states are isolated from the excited states since they are separated by a large energy gap. For repulsive interaction the ground state degeneracy is preserved only for $t_1 = 0, t_2 = 0.1$. + +**Fig. 3.** a) The entanglement entropy for a partition A of the full system, at different fillings $f$ versus the number of particles N. For half-filling $f = \frac{1}{2}$ the entropy converges at $\ln 2$ as two spins in a singlet maximally entangled Bell state. For lower fillings the entropy diverges logarithmically as $S_A \sim \ln N$, implying stronger entanglement. The last two points in the curve for $U > 0$ for $f = 1/4$ are calculated via extrapolation. The filled(open) triangles and circles, are calculated using Eq. (3)(Eq. (7)) b) The occupation probability $\overline{n}_i$ (particle density) versus each site i in the Hubbard chain, for N=6,8,10 and different fillings. Edge modes can be seen for $f < 1/2$, with uniform bulk density $\overline{n}_i \approx f$ and edge density that is a fraction $\overline{n}_{edge} \approx \frac{f}{1-f}$. The edge modes are not affected by N. + +## 4 Edge effects + +In order to obtain additional features of the ground states we calculate the occupation probability (particle density) for each site of the Hubbard chain, using Eq. (4). + +In Fig. 3(b) we show $\overline{n}_i$ for different fillings and number of particles N=6,8,10. For the half-filled case we observe fluctuations of the probability, as expected [45] due to the edges acting as impurities causing Friedel oscillations, that are larger at the two ends of the Hubbard chain. For the lowest fillings ($f = 1/3, 1/4$) however, the fluctuations smoothen out in the bulk of the Hubbard chain resulting in uniform density $\overline{n} \approx f$. + +In order to understand this uniform bulk density we can avoid the edge effects by applying periodic boundary conditions(PBC) in the system. This can be done by changing the upper summation limit to M in Eq. (1) and considering the condition $M + 1 \to 1$. Since we are interested in the $E=0$ states, we must remove the states that contain particles at the two edge sites of the chain $|1010...01\rangle$ as these will lift the energy of the system by $E=U$. We can use Eq. (2) to calculate the number of these states, which is $D(N-2, M-4)$. Therefore the number of ground states with PBC is $D_{PBC}(N,M) = D(N,M) - D(N-2, M-4)$. The corresponding particle density can be calculated by noticing that for $f \le 1/2$ every occupied site has to lie between two empty sites in a state of the type $|...010...\rangle$. Therefore, the density on every site, at the thermodynamic limit $N \to \infty$, becomes $\overline{n}_{bulk} = \frac{D(N-1,M-3)}{D_{PBC}(N,M)} = f$, as shown in Fig. 3(b) for low fillings. However when edges are present edge modes are formed due to the inclusion of the states of the type $|1010...01\rangle$, that have a particle on both edge sites. In this case, the occupation probability at the edge can be obtained by the ratio between the states that have one occupied site at the corresponding edge (site 1 or M) $D(N-1, M-2)$, over the total number of microstates $D(N,M)$, giving the edge density $\overline{n}_{edge} = \frac{D(N-1, M-2)}{D(N,M)}$. Using Eq. (2) we find that $\overline{n}_{edge} = \frac{N}{1+M-N} = \frac{1}{\frac{1}{N}+\frac{1}{J}}$ which becomes a fraction $\overline{n}_{edge} = \frac{f}{1-f}$ in the thermodynamic limit $N \to \infty$. This result agrees well with the edge density shown in Fig. 3(b), even for the relatively small number of particles shown (N=6,8,10). We note that the dip of $\overline{n}_i$ near the edge sites is due to the repulsive interaction U which reduces the probability to find a particle on the neighboring site to the edge, if the edge site is already occupied. + +The edge modes could be considered as excitations of the particle density which remains uniform at the bulk of the Hubbard chain. Each edge is contained in one of the partitions A or B which are quantum mechanically correlated/entangled, as we have shown. Therefore, we can assume that the edge modes at the opposite ends of the system can be entangled with each other. + +Combining the edge modes with the results obtained in the previous section, namely the ground state degeneracy and the variable entanglement, it is tempting to assume that our system contains topological order. 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